%!PS-Adobe-3.0
%%Title: (Microsoft Word - Arcavi et al.)
%%Creator: (Microsoft Word: PSPrinter 8.3.1)
%%CreationDate: (2:24 PM Sunday, January 10, 1999)
%%For: (Douglas Ulmer)
%%Pages: 90
%%DocumentFonts: Palatino-Bold Palatino-Roman Palatino-Italic Times-Roman
%%DocumentNeededFonts: Palatino-Bold Palatino-Roman Palatino-Italic Times-Roman
%%DocumentSuppliedFonts:
%%DocumentData: Clean7Bit
%%PageOrder: Ascend
%%Orientation: Portrait
%%DocumentMedia: Default 612 792 0 () ()
%ADO_ImageableArea: 30 31 582 761
%%EndComments
%%BeginDefaults
%%ViewingOrientation: 1 0 0 1
%%EndDefaults
userdict begin/dscInfo 5 dict dup begin
/Title(Microsoft Word - Arcavi et al.)def
/Creator(Microsoft Word: PSPrinter 8.3.1)def
/CreationDate(2:24 PM Sunday, January 10, 1999)def
/For(Douglas Ulmer)def
/Pages 90 def
end def end
/md 258 dict def md begin/currentpacking where {pop /sc_oldpacking currentpacking def true setpacking}if
%%BeginFile: adobe_psp_basic
%%Copyright: Copyright 1990-1996 Adobe Systems Incorporated. All Rights Reserved.
/bd{bind def}bind def
/xdf{exch def}bd
/xs{exch store}bd
/ld{load def}bd
/Z{0 def}bd
/T/true
/F/false
/:L/lineto
/lw/setlinewidth
/:M/moveto
/rl/rlineto
/rm/rmoveto
/:C/curveto
/:T/translate
/:K/closepath
/:mf/makefont
/gS/gsave
/gR/grestore
/np/newpath
14{ld}repeat
/$m matrix def
/av 83 def
/por true def
/normland false def
/psb-nosave{}bd
/pse-nosave{}bd
/us Z
/psb{/us save store}bd
/pse{us restore}bd
/level2
/languagelevel where
{
pop languagelevel 2 ge
}{
false
}ifelse
def
/featurecleanup
{
stopped
cleartomark
countdictstack exch sub dup 0 gt
{
{end}repeat
}{
pop
}ifelse
}bd
/noload Z
/startnoload
{
{/noload save store}if
}bd
/endnoload
{
{noload restore}if
}bd
level2 startnoload
/setjob
{
statusdict/jobname 3 -1 roll put
}bd
/setcopies
{
userdict/#copies 3 -1 roll put
}bd
level2 endnoload level2 not startnoload
/setjob
{
1 dict begin/JobName xdf currentdict end setuserparams
}bd
/setcopies
{
1 dict begin/NumCopies xdf currentdict end setpagedevice
}bd
level2 not endnoload
/pm Z
/mT Z
/sD Z
/realshowpage Z
/initializepage
{
/pm save store mT concat
}bd
/endp
{
pm restore showpage
}def
/endp1
{
pm restore
}def
/endp2
{
showpage
}def
/$c/DeviceRGB def
/rectclip where
{
pop/rC/rectclip ld
}{
/rC
{
np 4 2 roll
:M
1 index 0 rl
0 exch rl
neg 0 rl
:K
clip np
}bd
}ifelse
/rectfill where
{
pop/rF/rectfill ld
}{
/rF
{
gS
np
4 2 roll
:M
1 index 0 rl
0 exch rl
neg 0 rl
fill
gR
}bd
}ifelse
/rectstroke where
{
pop/rS/rectstroke ld
}{
/rS
{
gS
np
4 2 roll
:M
1 index 0 rl
0 exch rl
neg 0 rl
:K
stroke
gR
}bd
}ifelse
%%EndFile
%%BeginFile: adobe_psp_colorspace_level1
%%Copyright: Copyright 1991-1996 Adobe Systems Incorporated. All Rights Reserved.
/G/setgray ld
/:F1/setgray ld
/:F/setrgbcolor ld
/:F4/setcmykcolor where
{
pop
/setcmykcolor ld
}{
{
3
{
dup
3 -1 roll add
dup 1 gt{pop 1}if
1 exch sub
4 1 roll
}repeat
pop
setrgbcolor
}bd
}ifelse
/:Fx
{
counttomark
{0{G}0{:F}{:F4}}
exch get
exec
pop
}bd
/:rg{/DeviceRGB :ss}bd
/:sc{$cs :ss}bd
/:dc{/$cs xdf}bd
/:sgl{}def
/:dr{}bd
/:nmCRD{pop}bd
/:fCRD{pop}bd
/:ckcs{}bd
/:ss{/$c xdf}bd
/$cs Z
%%EndFile
level2 startnoload
%%BeginFile: adobe_psp_patterns_level1
%%Copyright: Copyright 1991-1996 Adobe Systems Incorporated. All Rights Reserved.
/patfreq Z
/patangle Z
/bk Z
/fg Z
/docolorscreen Z
/graystring Z
/pattransf{}def
/initQDpatterns
{
/patfreq 9.375 store
/patangle
1 0 $m defaultmatrix dtransform
exch atan
por not
{90 add}if
normland{180 add}if
store
:a
}def
/docolorscreen
/setcolorscreen where
{
pop/currentcolorscreen where
{
pop/setcmykcolor where
{
pop true
}{
false
}ifelse
}{
false
}ifelse
}{
false
}ifelse
def
/setgraypattern
{
/graystring xs
patfreq
patangle
{
1 add
4 mul
cvi
graystring
exch get
exch
1 add 4 mul
cvi
7 sub
bitshift
1 and
}setscreen
64 div setgray
}bd
/:b
{
/pattransf load settransfer
pop pop pop
setgraypattern
}bd
docolorscreen startnoload
/screensave 5 array def
/:a{currentgray currentscreen currenttransfer screensave astore pop}bd
/:e{screensave aload pop settransfer setscreen setgray}bd
/:d
{
pop pop pop
/pattransf load settransfer
setgraypattern 8{pop}repeat
}bd
/:c
/:d ld
docolorscreen endnoload docolorscreen not startnoload
/screensave 20 array def
/:a{currentcmykcolor currentcolorscreen currentcolortransfer screensave astore pop}bd
/:e{screensave aload pop setcolortransfer setcolorscreen setcmykcolor}bd
/rstring Z
/grstring Z
/blstring Z
/convroll{64 div 4 -1 roll}bd
/setcolorpattern
{
/graystring xs
/blstring xs
/grstring xs
/rstring xs
patfreq
patangle
{
1 add 4 mul cvi rstring
exch get exch 1 add 4 mul
cvi 7 sub bitshift 1 and
}
patfreq
patangle
{
1 add 4 mul cvi grstring
exch get exch 1 add 4 mul
cvi 7 sub bitshift 1 and
}
patfreq
patangle
{
1 add 4 mul cvi blstring
exch get exch 1 add 4 mul
cvi 7 sub bitshift 1 and
}
patfreq
patangle
{
1 add 4 mul cvi graystring
exch get exch 1 add 4 mul
cvi 7 sub bitshift 1 and
}
setcolorscreen
convroll convroll convroll convroll
setcmykcolor
}bd
/:d
{
pop pop pop
/pattransf load settransfer
pop pop setcolorpattern
}bd
/:c
/:d ld
docolorscreen not endnoload
%%EndFile
level2 endnoload level2 not startnoload
%%BeginFile: adobe_psp_patterns_level2
%%Copyright: Copyright 1990-1996 Adobe Systems Incorporated. All Rights Reserved.
/pmtx Z
/BGnd Z
/FGnd Z
/PaintData Z
/PatternMtx Z
/PatHeight Z
/PatWidth Z
/$d Z
/savecolor 4 array def
/savecolorspace Z
/patDecode Z
/:a{
mark 0 0 0 currentcolor savecolor astore pop cleartomark
/savecolorspace currentcolorspace store
$c setcolorspace
/patDecode
[currentcolor counttomark
dup 2 add 1
roll
cleartomark[exch
{0 1}repeat]
def
}bd
/:e{
savecolorspace setcolorspace
mark savecolor aload pop setcolor cleartomark
}bd
/initQDpatterns
{
gS
initmatrix
mT dup 4 get exch 5 get :T
1 0 dtransform round exch round exch idtransform
dup mul exch dup mul exch add sqrt
0 1 dtransform round exch round exch idtransform
dup mul exch dup mul exch add sqrt
neg
scale
0
por not{90 add}if
normland{180 add}if
rotate
matrix currentmatrix
gR
/pmtx xs
:a
}bd
/:t
{
14 dict begin
/BGnd xdf
/FGnd xdf
/PaintData xdf
/PatternType 1 def
/PaintType 1 def
/BBox[0 0 1 1]def
/TilingType 1 def
/XStep 1 def
/YStep 1 def
/PatternMtx[24 0 0 24 0 0]def
/PaintProc
BGnd null ne
{
{
begin
BGnd aload pop :F
0 0 1 1 rF
FGnd aload pop :F
24 24 true PatternMtx PaintData imagemask
end
}
}{
{
begin
FGnd aload pop :F
24 24 true PatternMtx PaintData imagemask
end
}
}ifelse
def
currentdict
PatternMtx
end
$c setcolorspace
gS pmtx setmatrix makepattern gR
}bd
/:u
{
14 dict begin
/$d 8 dict def
/PatternType 1 def
/PaintType 1 def
/BBox[0 0 1 1]def
/TilingType 1 def
/XStep 1 def
/YStep 1 def
/PaintData xdf
/PatHeight xdf
/PatWidth xdf
/PatternMtx[PatWidth 0 0 PatHeight 0 0]def
$d begin
/ImageType 1 def
/MultipleDataSource false def
/Height PatHeight def
/Width PatWidth def
/Decode patDecode def
/ImageMatrix PatternMtx def
/DataSource PaintData def
/BitsPerComponent 8 def
end
/PaintProc
{
begin
$d image
end
}def
currentdict
PatternMtx
end
gS $c setcolorspace pmtx setmatrix makepattern gR
}bd
/bk[1 1 1]def
/fg[0 0 0]def
/:b{
:t
setpattern
pop pop
}bd
/:d{
:t
setpattern
10{pop}repeat
}bd
/:c{
:u
setpattern
10{pop}repeat
}bd
%%EndFile
level2 not endnoload
level2 startnoload
%%BeginFile: adobe_psp_level1_basicimages
%%Copyright: Copyright 1990-1996 Adobe Systems Incorporated. All Rights Reserved.
/$i false def
/flipinvert
statusdict begin
version cvr 47.0 lt
end
def
/iw Z
/ih Z
/im_save Z
/setupimageproc Z
/polarity Z
/smoothflag Z
/$z Z
/bpc Z
/smooth_moredata Z
/datatype Z
/:f
{
/im_save save store
/datatype xs
$i flipinvert
and
xor
/polarity xs
/smoothflag xs
concat
/$z exch string store
/bpc xs
/ih xs
/iw xs
/smoothflag
smoothflag
bpc 1 eq and
smoothflag and
userdict/sc_smooth known and
vmstatus pop exch pop iw 3 mul sub 1000 gt and
iw 4 mul 7 add 8 idiv 4 mul 65535 le and
store
smoothflag{
iw
ih
$z
iw 7 add 8 idiv 4 mul string
iw 4 mul 7 add 8 idiv 4 mul string
true
false
sc_initsmooth
/iw iw 4 mul store
/ih ih 4 mul store
}if
/setupimageproc datatype 0 eq datatype 1 eq or{
smoothflag{
{
[
/smooth_moredata cvx[
currentfile
$z
{readstring readhexstring}datatype get
/pop cvx
]cvx[
$z
]cvx/ifelse cvx
/sc_smooth cvx
/smooth_moredata/exch cvx/store cvx
]cvx bind
/smooth_moredata true store
dup exec pop dup exec pop
}
}{
{
[
currentfile
$z
{readstring readhexstring}datatype get
/pop cvx
]cvx bind
}
}ifelse
}{
(error, can't use level2 data acquisition procs for level1)print flush stop
}ifelse
store
}bd
/:j{im_save restore}bd
/:g
{
1 setgray
0 0 1 1 rF
0 setgray
iw ih polarity[iw 0 0 ih 0 0]setupimageproc
imagemask
}bd
/:h
{
:Fx
0 0 1 1 rF
:Fx
iw ih polarity[iw 0 0 ih 0 0]setupimageproc
imagemask
}bd
/:i
{
:Fx
iw ih polarity[iw 0 0 ih 0 0]setupimageproc
imagemask
}bd
%%EndFile
level2 endnoload level2 not startnoload
%%BeginFile: adobe_psp_level2_basicimage
%%Copyright: Copyright 1990-1996 Adobe Systems Incorporated. All Rights Reserved.
/$j 9 dict dup
begin
/ImageType 1 def
/MultipleDataSource false def
end
def
/im_save Z
/setupimageproc Z
/polarity Z
/smoothflag Z
/bpc Z
/ih Z
/iw Z
/datatype Z
/:f
{
/im_save save store
/datatype xs
datatype 0 lt datatype 8 gt or{
(error, datatype out of range)print flush stop
}if
/setupimageproc{
{
currentfile
}
{
currentfile 0(%ADOeod)/SubFileDecode filter/ASCIIHexDecode filter
}
{
currentfile/RunLengthDecode filter
}
{
currentfile/ASCII85Decode filter/RunLengthDecode filter
}
{
currentfile/ASCII85Decode filter
}
{
currentfile/DCTDecode filter
}
{
currentfile/ASCII85Decode filter/DCTDecode filter
}
{
currentfile 6 dict dup begin
/K 0 def
/EndOfLine true def
/EncodedByteAlign true def
/Rows Height def
/Columns Width def
/EndOfBlock true def
end/CCITTFaxDecode filter
}
{
currentfile/ASCII85Decode filter 6 dict dup begin
/K 0 def
/EndOfLine true def
/EncodedByteAlign true def
/Rows Height def
/Columns Width def
/EndOfBlock true def
end/CCITTFaxDecode filter
}
}datatype get store
{
[1 0]
}{
[0 1]
}ifelse
/polarity xs
/smoothflag xs
concat
pop
/bpc xs
/ih xs
/iw xs
$c setcolorspace
}bd
/:j{im_save restore}bd
/:g
{
1 G
0 0 1 1 rF
0 G
$j dup begin
/Width iw def
/Height ih def
/Decode polarity def
/ImageMatrix[iw 0 0 ih 0 0]def
/DataSource setupimageproc def
/BitsPerComponent 1 def
/Interpolate smoothflag def
end
imagemask
}bd
/:h
{
:Fx
0 0 1 1 rF
:Fx
$j dup begin
/Width iw def
/Height ih def
/Decode polarity def
/ImageMatrix[iw 0 0 ih 0 0]def
/DataSource setupimageproc def
/BitsPerComponent 1 def
/Interpolate smoothflag def
end
imagemask
}bd
/:i
{
:Fx
$j dup begin
/Width iw def
/Height ih def
/Decode polarity def
/ImageMatrix[iw 0 0 ih 0 0]def
/DataSource setupimageproc def
/BitsPerComponent 1 def
/Interpolate smoothflag def
end
imagemask
}bd
%%EndFile
level2 not endnoload
%%BeginFile: adobe_psp_smooth
%%Copyright: Copyright 1991-1996 Adobe Systems Incorporated. All Rights Reserved.
/junk Z
/$z Z
userdict/sc_smooth known not
save
systemdict/eexec known
systemdict/cexec known and{
countdictstack mark
false
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{eexec}stopped{
cleartomark
countdictstack exch sub dup 0 gt{{end}repeat}{pop}ifelse
false
}{
{cleartomark pop true}{cleartomark pop false}ifelse
}ifelse
}{false}ifelse
exch restore and
level2 not and
vmstatus exch sub exch pop 15000 gt and
{
currentfile eexec
}{
/junk save store
/$z 4795 string store
currentfile $z readhexstring pop pop
{
currentfile $z readline not
{
stop
}if
(%ADOendeexec)eq
{
exit
}if
}bind loop
junk restore
}ifelse
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cleartomark
%ADOendeexec
%%EndFile
%%BeginFile: adobe_psp_uniform_graphics
%%Copyright: Copyright 1990-1996 Adobe Systems Incorporated. All Rights Reserved.
/@a
{
np :M 0 rl :L 0 exch rl 0 rl :L fill
}bd
/@b
{
np :M 0 rl 0 exch rl :L 0 rl 0 exch rl fill
}bd
/@c
{
moveto lineto stroke
}bd
/arct where
{
pop
}{
/arct
{
arcto pop pop pop pop
}bd
}ifelse
/x1 Z
/x2 Z
/y1 Z
/y2 Z
/rad Z
/@q
{
/rad xs
/y2 xs
/x2 xs
/y1 xs
/x1 xs
np
x2 x1 add 2 div y1 :M
x2 y1 x2 y2 rad arct
x2 y2 x1 y2 rad arct
x1 y2 x1 y1 rad arct
x1 y1 x2 y1 rad arct
fill
}bd
/@s
{
/rad xs
/y2 xs
/x2 xs
/y1 xs
/x1 xs
np
x2 x1 add 2 div y1 :M
x2 y1 x2 y2 rad arct
x2 y2 x1 y2 rad arct
x1 y2 x1 y1 rad arct
x1 y1 x2 y1 rad arct
:K
stroke
}bd
/@i
{
np 0 360 arc fill
}bd
/@j
{
gS
np
:T
scale
0 0 .5 0 360 arc
fill
gR
}bd
/@e
{
np
0 360 arc
:K
stroke
}bd
/@f
{
np
$m currentmatrix
pop
:T
scale
0 0 .5 0 360 arc
:K
$m setmatrix
stroke
}bd
/@k
{
gS
np
:T
0 0 :M
0 0 5 2 roll
arc fill
gR
}bd
/@l
{
gS
np
:T
0 0 :M
scale
0 0 .5 5 -2 roll arc
fill
gR
}bd
/@m
{
np
arc
stroke
}bd
/@n
{
np
$m currentmatrix
pop
:T
scale
0 0 .5 5 -2 roll arc
$m setmatrix
stroke
}bd
%%EndFile
%%BeginFile: adobe_psp_customps
%%Copyright: Copyright 1990-1996 Adobe Systems Incorporated. All Rights Reserved.
/$t Z
/$p Z
/$s Z
/$o 1. def
/2state? false def
/ps Z
level2 startnoload
/pushcolor/currentrgbcolor ld
/popcolor/setrgbcolor ld
/setcmykcolor where
{
pop/currentcmykcolor where
{
pop/pushcolor/currentcmykcolor ld
/popcolor/setcmykcolor ld
}if
}if
level2 endnoload level2 not startnoload
/pushcolor
{
currentcolorspace $c eq
{
currentcolor currentcolorspace true
}{
currentcmykcolor false
}ifelse
}bd
/popcolor
{
{
setcolorspace setcolor
}{
setcmykcolor
}ifelse
}bd
level2 not endnoload
/pushstatic
{
ps
2state?
$o
$t
$p
$s
$cs
}bd
/popstatic
{
/$cs xs
/$s xs
/$p xs
/$t xs
/$o xs
/2state? xs
/ps xs
}bd
/pushgstate
{
save errordict/nocurrentpoint{pop 0 0}put
currentpoint
3 -1 roll restore
pushcolor
currentlinewidth
currentlinecap
currentlinejoin
currentdash exch aload length
np clippath pathbbox
$m currentmatrix aload pop
}bd
/popgstate
{
$m astore setmatrix
2 index sub exch
3 index sub exch
rC
array astore exch setdash
setlinejoin
setlinecap
lw
popcolor
np :M
}bd
/bu
{
pushgstate
gR
pushgstate
2state?
{
gR
pushgstate
}if
pushstatic
pm restore
mT concat
}bd
/bn
{
/pm save store
popstatic
popgstate
gS
popgstate
2state?
{
gS
popgstate
}if
}bd
/cpat{pop 64 div setgray 8{pop}repeat}bd
%%EndFile
%%BeginFile: adobe_psp_basic_text
%%Copyright: Copyright 1990-1996 Adobe Systems Incorporated. All Rights Reserved.
/S/show ld
/A{
0.0 exch ashow
}bd
/R{
0.0 exch 32 exch widthshow
}bd
/W{
0.0 3 1 roll widthshow
}bd
/J{
0.0 32 4 2 roll 0.0 exch awidthshow
}bd
/V{
0.0 4 1 roll 0.0 exch awidthshow
}bd
/fcflg true def
/fc{
fcflg{
vmstatus exch sub 50000 lt{
(%%[ Warning: Running out of memory ]%%\r)print flush/fcflg false store
}if pop
}if
}bd
/$f[1 0 0 -1 0 0]def
/:ff{$f :mf}bd
/MacEncoding StandardEncoding 256 array copy def
MacEncoding 39/quotesingle put
MacEncoding 96/grave put
/Adieresis/Aring/Ccedilla/Eacute/Ntilde/Odieresis/Udieresis/aacute
/agrave/acircumflex/adieresis/atilde/aring/ccedilla/eacute/egrave
/ecircumflex/edieresis/iacute/igrave/icircumflex/idieresis/ntilde/oacute
/ograve/ocircumflex/odieresis/otilde/uacute/ugrave/ucircumflex/udieresis
/dagger/degree/cent/sterling/section/bullet/paragraph/germandbls
/registered/copyright/trademark/acute/dieresis/notequal/AE/Oslash
/infinity/plusminus/lessequal/greaterequal/yen/mu/partialdiff/summation
/product/pi/integral/ordfeminine/ordmasculine/Omega/ae/oslash
/questiondown/exclamdown/logicalnot/radical/florin/approxequal/Delta/guillemotleft
/guillemotright/ellipsis/space/Agrave/Atilde/Otilde/OE/oe
/endash/emdash/quotedblleft/quotedblright/quoteleft/quoteright/divide/lozenge
/ydieresis/Ydieresis/fraction/currency/guilsinglleft/guilsinglright/fi/fl
/daggerdbl/periodcentered/quotesinglbase/quotedblbase/perthousand
/Acircumflex/Ecircumflex/Aacute/Edieresis/Egrave/Iacute/Icircumflex/Idieresis/Igrave
/Oacute/Ocircumflex/apple/Ograve/Uacute/Ucircumflex/Ugrave/dotlessi/circumflex/tilde
/macron/breve/dotaccent/ring/cedilla/hungarumlaut/ogonek/caron
MacEncoding 128 128 getinterval astore pop
level2 startnoload
/copyfontdict
{
findfont dup length dict
begin
{
1 index/FID ne{def}{pop pop}ifelse
}forall
}bd
level2 endnoload level2 not startnoload
/copyfontdict
{
findfont dup length dict
copy
begin
}bd
level2 not endnoload
md/fontname known not{
/fontname/customfont def
}if
/Encoding Z
/:mre
{
copyfontdict
/Encoding MacEncoding def
fontname currentdict
end
definefont :ff def
}bd
/:bsr
{
copyfontdict
/Encoding Encoding 256 array copy def
Encoding dup
}bd
/pd{put dup}bd
/:esr
{
pop pop
fontname currentdict
end
definefont :ff def
}bd
/scf
{
scalefont def
}bd
/scf-non
{
$m scale :mf setfont
}bd
/ps Z
/fz{/ps xs}bd
/sf/setfont ld
/cF/currentfont ld
/mbf
{
/makeblendedfont where
{
pop
makeblendedfont
/ABlend exch definefont
}{
pop
}ifelse
def
}def
%%EndFile
%%BeginFile: adobe_psp_derived_styles
%%Copyright: Copyright 1990-1996 Adobe Systems Incorporated. All Rights Reserved.
/wi
version(23.0)eq
{
{
gS 0 0 0 0 rC stringwidth gR
}bind
}{
/stringwidth load
}ifelse
def
/$o 1. def
/gl{$o G}bd
/ms{:M S}bd
/condensedmtx[.82 0 0 1 0 0]def
/:mc
{
condensedmtx :mf def
}bd
/extendedmtx[1.18 0 0 1 0 0]def
/:me
{
extendedmtx :mf def
}bd
/basefont Z
/basefonto Z
/dxa Z
/dxb Z
/dxc Z
/dxd Z
/dsdx2 Z
/bfproc Z
/:fbase
{
dup/FontType get 0 eq{
dup length dict begin
dup{1 index/FID ne 2 index/UniqueID ne and{def}{pop pop}ifelse}forall
/FDepVector exch/FDepVector get[exch/:fbase load forall]def
}/bfproc load ifelse
/customfont currentdict end definefont
}bd
/:mo
{
/bfproc{
dup dup length 2 add dict
begin
{
1 index/FID ne 2 index/UniqueID ne and{def}{pop pop}ifelse
}forall
/PaintType 2 def
/StrokeWidth .012 0 FontMatrix idtransform pop def
/customfont currentdict
end
definefont
8 dict begin
/basefonto xdf
/basefont xdf
/FontType 3 def
/FontMatrix[1 0 0 1 0 0]def
/FontBBox[0 0 1 1]def
/Encoding StandardEncoding def
/BuildChar
{
exch begin
basefont setfont
( )dup 0 4 -1 roll put
dup wi
setcharwidth
0 0 :M
gS
gl
dup show
gR
basefonto setfont
show
end
}def
}store :fbase
}bd
/:mso
{
/bfproc{
7 dict begin
/basefont xdf
/FontType 3 def
/FontMatrix[1 0 0 1 0 0]def
/FontBBox[0 0 1 1]def
/Encoding StandardEncoding def
/BuildChar
{
exch begin
sD begin
/dxa 1 ps div def
basefont setfont
( )dup 0 4 -1 roll put
dup wi
1 index 0 ne
{
exch dxa add exch
}if
setcharwidth
dup 0 0 ms
dup dxa 0 ms
dup dxa dxa ms
dup 0 dxa ms
gl
dxa 2. div dup ms
end
end
}def
}store :fbase
}bd
/:ms
{
/bfproc{
dup dup length 2 add dict
begin
{
1 index/FID ne 2 index/UniqueID ne and{def}{pop pop}ifelse
}forall
/PaintType 2 def
/StrokeWidth .012 0 FontMatrix idtransform pop def
/customfont currentdict
end
definefont
8 dict begin
/basefonto xdf
/basefont xdf
/FontType 3 def
/FontMatrix[1 0 0 1 0 0]def
/FontBBox[0 0 1 1]def
/Encoding StandardEncoding def
/BuildChar
{
exch begin
sD begin
/dxb .05 def
basefont setfont
( )dup 0 4 -1 roll put
dup wi
exch dup 0 ne
{
dxb add
}if
exch setcharwidth
dup dxb .01 add 0 ms
0 dxb :T
gS
gl
dup 0 0 ms
gR
basefonto setfont
0 0 ms
end
end
}def
}store :fbase
}bd
/:mss
{
/bfproc{
7 dict begin
/basefont xdf
/FontType 3 def
/FontMatrix[1 0 0 1 0 0]def
/FontBBox[0 0 1 1]def
/Encoding StandardEncoding def
/BuildChar
{
exch begin
sD begin
/dxc 1 ps div def
/dsdx2 .05 dxc 2 div add def
basefont setfont
( )dup 0 4 -1 roll put
dup wi
exch dup 0 ne
{
dsdx2 add
}if
exch setcharwidth
dup dsdx2 .01 add 0 ms
0 .05 dxc 2 div sub :T
dup 0 0 ms
dup dxc 0 ms
dup dxc dxc ms
dup 0 dxc ms
gl
dxc 2 div dup ms
end
end
}def
}store :fbase
}bd
/:msb
{
/bfproc{
7 dict begin
/basefont xdf
/FontType 3 def
/FontMatrix[1 0 0 1 0 0]def
/FontBBox[0 0 1 1]def
/Encoding StandardEncoding def
/BuildChar
{
exch begin
sD begin
/dxd .03 def
basefont setfont
( )dup 0 4 -1 roll put
dup wi
1 index 0 ne
{
exch dxd add exch
}if
setcharwidth
dup 0 0 ms
dup dxd 0 ms
dup dxd dxd ms
0 dxd ms
end
end
}def
}store :fbase
}bd
/italicmtx[1 0 -.212557 1 0 0]def
/:mi
{
italicmtx :mf def
}bd
/:v
{
[exch dup/FontMatrix get exch
dup/FontInfo known
{
/FontInfo get
dup/UnderlinePosition known
{
dup/UnderlinePosition get
2 index 0
3 1 roll
transform
exch pop
}{
.1
}ifelse
3 1 roll
dup/UnderlineThickness known
{
/UnderlineThickness get
exch 0 3 1 roll
transform
exch pop
abs
}{
pop pop .067
}ifelse
}{
pop pop .1 .067
}ifelse
]
}bd
/$t Z
/$p Z
/$s Z
/:p
{
aload pop
2 index mul/$t xs
1 index mul/$p xs
.012 mul/$s xs
}bd
/:m
{gS
0 $p rm
$t lw
0 rl stroke
gR
}bd
/:n
{
gS
0 $p rm
$t lw
0 rl
gS
gl
stroke
gR
strokepath
$s lw
/setstrokeadjust where{pop
currentstrokeadjust true setstrokeadjust stroke setstrokeadjust
}{
stroke
}ifelse
gR
}bd
/:o
{gS
0 $p rm
$t 2 div dup rm
$t lw
dup 0 rl
stroke
gR
:n
}bd
%%EndFile
%%BeginFile: adobe_psp_dashes
%%Copyright: Copyright 1990-1996 Adobe Systems Incorporated. All Rights Reserved.
/:q/setdash ld
/:r{
np
:M
:L
stroke
}bd
/nodash[]def
/qdenddash
{
nodash 0 setdash
}bd
%%EndFile
/currentpacking where {pop sc_oldpacking setpacking}if end
%%EndProlog
%%BeginSetup
md begin
countdictstack[{
%%BeginFeature: *ManualFeed False
level2 {1 dict dup /ManualFeed false put setpagedevice}{statusdict begin /manualfeed false store end} ifelse
%%EndFeature
}featurecleanup
countdictstack[{
%%BeginFeature: *InputSlot Upper
%%EndFeature
}featurecleanup
countdictstack[{
%%BeginFeature: *PageRegion LetterSmall
level2 {
2 dict dup /PageSize [612 792] put dup /ImagingBBox [30 31 582 761] put setpagedevice
}{
/lettersmall where {pop lettersmall} {letterR} ifelse
} ifelse
%%EndFeature
}featurecleanup
(Douglas Ulmer; document: Microsoft Word - Arcavi et al.)setjob
/mT[1 0 0 -1 30 761]def
initQDpatterns
/sD 16 dict def
300 level2{1 dict dup/WaitTimeout 4 -1 roll put setuserparams}{statusdict/waittimeout 3 -1 roll put}ifelse
%%IncludeFont: Palatino-Bold
%%IncludeFont: Palatino-Roman
%%IncludeFont: Palatino-Italic
%%IncludeFont: Times-Roman
/f0_1/Palatino-Bold
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-.085(Teaching Mathematical Problem Solving:)A
129 71 :M
-.206(An Analysis of an Emergent Classroom Community)A
105 89 :M
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1.006 .101(Abraham Arcavi, Cathy Kessel, Luciano Meira, & John P. Smith III)J
250 143 :M
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-.168(Introduction)A
156 197 :M
-.227(An overview of the problem solving course)A
243 215 :M
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2.196 .22(Abraham Arcavi)J
186 233 :M
1.624 .162(Weizmann Institute of Science, Israel)J
89 287 :M
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-.19(Presenting and doing mathematics: An introduction to heuristics)A
249 305 :M
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1.712 .171(Luciano Meira)J
166 323 :M
1.242 .124(Universidade Federal de Pernambuco, Brazil)J
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-.117(Making the case for heuristics:)A
146 395 :M
-.296(Authority and direction in the inscribed square)A
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1.271 .127(John P. Smith III)J
196 431 :M
1.557 .156(Michigan State University, U.S.A.)J
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-.082(Practicing mathematical communication:)A
169 503 :M
-.199(Using heuristics with the magic square)A
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.508 .051(Cathy Kessel)J
170 539 :M
1.181 .118(University of California at Berkeley, U.S.A.)J
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-.326(Concluding discussion)A
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1.155 .116(Published in Alan Schoenfeld, Ed Dubinsky, and James Kaput \(Eds.\),)J
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.134 .013( )J
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1.375 .137(Research in)J
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2.06 .206(Collegiate Mathematics Education, III. )J
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2.387 .239( Providence, RI: American Mathematical)J
60 665 :M
.144(Society.)A
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-.168(Introduction)A
347 50 :M
f4_8 sf
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f1_9 sf
-.086(Toward the end of the semester I assigned the following. . . . As usual, the class broke into groups)A
96 91 :M
-.131(to work on the problem. One group became the staunch defenders of one conjecture, while a second)A
96 109 :M
-.07(group lobbied for another. The two groups argued somewhat heatedly, with the rest of the class)A
96 127 :M
-.064(following the discussion. Finally, one group prevailed, on what struck me as solid mathematical)A
96 145 :M
-.189(grounds. As is my habit, I did not reveal this but made my usual comment: \322OK, you seem to have)A
96 163 :M
-.073(done as much with this as you can. Shall I try to pull things together?\323 One of the students replied,)A
96 181 :M
-.014(\322Don\325t bother. We got it.\323 The class agreed. \(Schoenfeld, 1994, pp. 63-64\))A
60 209 :M
f1_12 sf
.974 .097(Two main goals of Alan Schoenfeld\325s problem solving course are illustrated by)J
60 227 :M
1.218 .122(this anecdote: That the class function as a \322mathematical community\323 advancing)J
60 245 :M
1.076 .108(and defending conjectures and proofs on mathematical grounds, and that the)J
60 263 :M
1.349 .135(locus of authority be the \322mathematical community,\323)J
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.144 .014( )J
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1.364 .136(not the teacher \(Schoenfeld,)J
60 281 :M
1.131 .113(1994, p. 65\). Such incidents are not common in undergraduate mathematics)J
60 299 :M
1.039 .104(classes, whether they are composed of elite mathematics majors or students)J
60 317 :M
1.021 .102(struggling through their first calculus course. All too often students seem passive,)J
60 335 :M
.191(disengaged,)A
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.089 .009( )J
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.784 .078(and untroubled by contradictions in their work.)J
60 365 :M
1.039 .104(After twelve or more years of schooling, most undergraduates have well-)J
60 383 :M
.9 .09(developed expectations for mathematics classes based on their many experiences of)J
60 401 :M
.982 .098(listening, taking notes, and learning procedures to solve standard problems. In)J
60 419 :M
1.293 .129(order to establish his \322classroom community,\323 Schoenfeld must convey his)J
60 437 :M
1.149 .115(nonstandard expectations for behavior and at the same time convince his students)J
60 455 :M
.84 .084(that he knows what he\325s talking about, that his course is of value, and that)J
60 473 :M
1.013 .101(heuristics as well as formalism are essential parts of doing mathematics. In other)J
60 491 :M
.916 .092(words, Schoenfeld must renegotiate the \322didactic contract\323 \(Brousseau, 1986\) with)J
60 509 :M
1.01 .101(his students. This contract includes, among other things, teachers\325 and students\325)J
60 527 :M
.804 .08(understandings of what is to be expected in the classroom: \322What assistance can)J
60 545 :M
.803 .08(the students reasonably expect from the teacher; what assistance can the students)J
60 563 :M
.938 .094(seek from each other; what level of explanation is the teacher obliged to provide;)J
60 576 :M
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( )S
60 573.48 -.48 .48 204.48 573 .48 60 573 @a
60 585 :M
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.034 .003(This paper is the product of a long and enjoyable collaboration that began in 1990, in Berkeley, California, and)J
60 597 :M
-.042(continued over six years and four continents \(thanks to e-mail\). Each major section was individually developed and)A
60 606 :M
.043 .004(thus has a single author, though all of us critiqued each section. The )J
f2_9 sf
.012(Introduction)A
f1_9 sf
.028 .003( and the )J
f2_9 sf
.091 .009(Concluding Discussion)J
60 615 :M
f1_9 sf
-.051(reflect our shared views, and each of us had some part in writing them. However, Abraham Arcavi, Luciano Meira,)A
60 624 :M
-.044(and Jack Smith would like to thank Cathy Kessel who composed these sections with unusual editorial care and)A
60 633 :M
-.257(wisdom.)A
60 651 :M
-.04(The authors thank the editors, Ed Dubinsky and Jim Kaput; the reviewers, Barbara Pence, Beth Warren, and one)A
60 660 :M
-.083(anonymous reviewer; and members of the Functions Group, Ilana Horn, Andrew Iszak, Sue Magidson, and Natasha)A
60 669 :M
-.047(Speer, for their help in improving the successive versions of this article.)A
60 687 :M
-.001(We owe special thanks to Alan Schoenfeld. This article would not have been possible without his cooperation. It is)A
60 696 :M
-.061(not easy to be the subject of any analysis, let alone one so prolonged. Schoenfeld not only cooperated with us, but did)A
60 705 :M
-.042(so with grace, tolerance, and generosity.)A
endp
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.766 .077(what questions can the teacher reasonably ask; what form of response will be)J
60 69 :M
.532 .053(considered satisfactory\323 \(Clarke, 1995 April\).)J
60 99 :M
.883 .088(Schoenfeld\325s situation is an instance of a more general problem: If a course differs)J
60 117 :M
1.078 .108(radically from students\325 previous courses, how can its instructor convince students)J
60 135 :M
1.133 .113(that the course is worthwhile and convey expectations for classroom behavior?)J
60 153 :M
.908 .091(This problem is often encountered by those who teach first-year undergraduate)J
60 171 :M
.426 .043(courses. It is also faced by teachers of reformed calculus courses \(Cipra, 1995;)J
60 189 :M
1.038 .104(Culotta, 1992; University of Michigan, 1993\). Students are uncomfortable with)J
60 207 :M
1.116 .112(changed teaching and expectations. At some universities this discomfort has)J
60 225 :M
.538 .054(brought calculus reform to a halt \(Cipra, 1995, p. 19\).)J
60 255 :M
1.142 .114(Like Schoenfeld, some instructors have solved the problem of how to establish)J
60 273 :M
.959 .096(desired classroom norms. Others have noted it, but not yet developed a solution.)J
60 291 :M
.939 .094(Those new to teaching may not be aware the problem exists. We think that our)J
60 309 :M
.919 .092(description of Schoenfeld\325s solution will be of interest to people in each of these)J
60 327 :M
.599 .06(categories and present this account, not as a prescription to be followed, but as an)J
60 345 :M
1.127 .113(example that might illuminate aspects of the difficult task of mathematics)J
60 363 :M
.237(teaching.)A
60 393 :M
.687 .069(We also hope to suggest an analytic way in which to talk about teaching and)J
60 411 :M
1.196 .12(attempt to make salient considerations that are sometimes overlooked. Instead of)J
60 429 :M
.893 .089(describing selected incidents from the class or characterizing Schoenfeld\325s)J
60 447 :M
.792 .079(\322teaching style,\323 we focus on making sense of all the teaching actions at the)J
60 465 :M
.722 .072(beginning of the course, describing actions in detail as well as providing a)J
60 483 :M
.822 .082(rationale. Our account offers a method of description as well as the description)J
60 501 :M
.214(itself.)A
60 531 :M
1.028 .103(In the fall of 1990, Schoenfeld taught his undergraduate course Mathematical)J
60 549 :M
1.309 .131(Problem Solving in the mathematics department of the University of California at)J
60 567 :M
.967 .097(Berkeley. The course is designed to provide students with an introduction to what)J
60 585 :M
1.099 .11(it means to think mathematically. It is an elective course. The prerequisite is one)J
60 603 :M
1.071 .107(semester of calculus or consent of instructor.)J
60 633 :M
.777 .078(In order to build an empirical base for his own in-depth analysis of the course,)J
60 651 :M
.755 .076(Schoenfeld asked the first author \(who had already attended this class at Berkeley)J
60 669 :M
.879 .088(in 1987\) to videotape each of the 29 two-hour class meetings. The first author)J
60 687 :M
.862 .086(attended and videotaped each class session, and the other three authors also)J
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1.162 .116(attended numerous class sessions. All four authors were members of Schoenfeld\325s)J
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.974 .097(research group)J
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f1_12 sf
.842 .084( at the time, and they became interested in the same question: How)J
60 87 :M
1.192 .119(does Schoenfeld create a classroom community of problem solvers in which)J
60 105 :M
1.264 .126(undergraduates learn to think and do mathematics?)J
60 135 :M
.958 .096(This question shapes our analysis. We focus on the initial stages of the)J
60 153 :M
.846 .085(community, using the videotape records of the first two class sessions as the)J
60 171 :M
1.048 .105(principal data. Other data include the remaining 27 videotapes, interviews with)J
60 189 :M
.847 .085(students after the class ended, audiotaped discussions with Schoenfeld, and)J
60 207 :M
1.21 .121(Schoenfeld\325s writings. To understand the emergence of this community, we)J
60 225 :M
1.317 .132(examine how Schoenfeld understood and made the connections between his)J
60 243 :M
1.336 .134(mathematics classroom and the professional community of practicing)J
60 261 :M
1.013 .101(mathematicians; how he stated and enacted his expectations for student)J
60 279 :M
1.107 .111(participation; how he introduced students to desired forms of mathematical)J
60 297 :M
.856 .086(discourse and activity; and how he introduced heuristics in the context of specific)J
60 315 :M
.244(problems.)A
60 345 :M
.859 .086(At this point it is important to delimit the scope of this paper.)J
96 381 :M
.784 .078(\245 It is not our intention to describe, compare, or ignore the design and)J
105 399 :M
1.506 .151(implementation of successful mathematics classroom practices. We)J
105 417 :M
.701 .07(believe that there are many such practices, but we decided to concentrate)J
105 435 :M
.887 .089(on this one because we were fortunate enough to observe, analyze, and)J
105 453 :M
1.001 .1(discuss it in depth. We hope this example will encourage other researchers)J
105 471 :M
1.063 .106(to offer similarly detailed accounts of mathematics teaching and learning.)J
96 507 :M
.823 .082(\245 We do not analyze the success of the course. This has been documented:)J
105 525 :M
.885 .089(Students learn to use heuristics successfully \(Schoenfeld, 1982; 1985\) and)J
105 543 :M
1.072 .107(the class becomes a \322mathematical community\323 \(Schoenfeld 1989a; 1991;)J
105 561 :M
-.033(1992a; 1992b; 1994\).)A
96 597 :M
1.062 .106(\245 We do not provide an analysis of how the classroom evolved over the)J
105 615 :M
.838 .084(semester, how students participated, how they learned, how they changed)J
105 633 :M
.974 .097(from being rather silent to being very involved. We decided to focus on)J
105 651 :M
.976 .098(Schoenfeld\325s teaching at the very beginning for two main reasons: first, the)J
105 669 :M
1.235 .123(analyses suggest some very interesting and counter-intuitive findings)J
60 684 :M
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( )S
60 681.48 -.48 .48 204.48 681 .48 60 681 @a
60 693 :M
f4_8 sf
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-.022(Schoenfeld\325s research group, the Functions Group at the University of California at Berkeley, has been involved)A
60 705 :M
-.072(since 1985 in a series of research and development studies of mathematical teaching and learning.)A
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.782 .078(about the initial stages of such a class; and second, if continued through)J
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.875 .088(the remaining 54 hours of tape, the analyses would result in a long book)J
105 87 :M
1.003 .1(rather than a long article.)J
60 117 :M
.944 .094(As one might expect, data in the form of videotapes and interviews require)J
60 135 :M
.96 .096(methods of analysis different from those used to examine data that are more)J
60 153 :M
1.177 .118(uniform and manageable. Our analytic method, often known as microanalysis, has)J
60 171 :M
1.152 .115(roots in cognitive science and ethnography. Schoenfeld, Smith, and Arcavi \(1993\))J
60 189 :M
.705 .071(describe it as striving \322for explanations that are both locally and globally consistent,)J
60 207 :M
.818 .082(acounting for as much observed detail as possible and not contradicting any other)J
60 225 :M
.999 .1(related explanations.\323 \(For general discussion of this method, see Schoenfeld, 1988;)J
60 243 :M
1.016 .102(Schoenfeld, Smith, & Arcavi, 1993; Schoenfeld, 1992b. For discussion of related)J
60 261 :M
.948 .095(methods in the analysis of teaching, see Schoenfeld et al., 1992; Schoenfeld,)J
60 279 :M
1.07 .107(Minstrell, & van Zee, 1996 April.\))J
60 309 :M
1.011 .101(After our initial collaborative analysis of the first two videotapes, each author)J
60 327 :M
.896 .09(selected instructional segments from those tapes to analyze in detail. Each analysis)J
60 345 :M
.963 .096(was discussed with the other authors. The results are the four main sections of this)J
60 363 :M
1.335 .133(article. Though we share a common interest in mathematics education, our)J
60 381 :M
.901 .09(backgrounds are different enough for us to provide a multilayered view of the)J
60 399 :M
.868 .087(class. Arcavi has a Master\325s degree in mathematics and a Ph.D. in mathematics)J
60 417 :M
.826 .083(education. He has taught secondary school mathematics for 10 years, in teacher)J
60 435 :M
1.067 .107(education programs for 10 years, and has been involved in curriculum)J
60 453 :M
1.099 .11(development and research on mathematics teaching and learning for the past 15)J
60 471 :M
.533 .053(years. Meira has a Master\325s degree in cognitive psychology and a Ph.D. in)J
60 489 :M
1.041 .104(mathematics education. He has teaching experience at the elementary, secondary)J
60 507 :M
1.199 .12(and tertiary levels, and has been involved in research on mathematics teaching)J
60 525 :M
.937 .094(and learning for the past ten years. Smith taught upper elementary, middle, and)J
60 543 :M
.969 .097(high school mathematics for 6 years, with a B.A. in mathematics before obtaining a)J
60 561 :M
.689 .069(Ph.D. in educational psychology. His research centers on detailed analyses of)J
60 579 :M
1.122 .112(student understanding and learning of precollege mathematics.)J
f2_12 sf
.11 .011( )J
f1_12 sf
.827 .083(Kessel has a Ph.D.)J
60 597 :M
1.012 .101(in mathematics, taught lower and upper division undergraduate courses as a)J
60 615 :M
.542 .054(teaching assistant, lecturer, and assistant professor for 14 years, and works as a)J
60 633 :M
1.613 .161(researcher in mathematics education.)J
60 663 :M
1.792 .179(In the first of these sections, )J
f2_12 sf
2.344 .234(An Overview of the Problem Solving Course)J
f1_12 sf
2.692 .269(, Arcavi)J
60 681 :M
.71 .071(provides a general description of the goals, curriculum, and pedagogy of the)J
60 699 :M
.656 .066(course, as well as some background on the 1990 class. In the second section)J
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2.602 .26(Presenting and Doing Mathematics: An Introduction to Heuristics)J
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2.069 .207(, Meira analyzes)J
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.933 .093(how mathematical problem solving was first discussed and enacted. Central to this)J
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1.564 .156(analysis are the distinctions among )J
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1.601 .16( mathematics, )J
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2.377 .238( mathematics,)J
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1.333 .133(and )J
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1.616 .162(presenting how to do)J
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1.812 .181( mathematics, which Meira uses to examine the complex)J
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1.637 .164(relationships among professional mathematics, school mathematics, and)J
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2.343 .234(Schoenfeld\325s course. Smith\325s section )J
f2_12 sf
2.032 .203(Making the Case for Heuristics: Authority and)J
60 159 :M
1.773 .177(Direction in the Inscribed Square)J
f1_12 sf
1.517 .152(, focuses on the solution to the second problem of)J
60 177 :M
.948 .095(the course, analyzing how heuristics were introduced and how the students\325 work)J
60 195 :M
1.015 .101(with them was managed. His emphasis on the role of Schoenfeld\325s leadership and)J
60 213 :M
.971 .097(authority in the early days of the class, shows how complex and counterintuitive)J
60 231 :M
1.409 .141(the initiation of a classroom community can be. In )J
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2.786 .279(Practicing Mathematical)J
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2.563 .256(Communication: Using Heuristics with the Magic Square)J
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2.267 .227(, Kessel discusses the)J
60 267 :M
1.034 .103(third problem of the course, focusing on Schoenfeld\325s use of traditional and non-)J
60 285 :M
1.141 .114(traditional mathematical language and discourse. Smith\325s and Kessel\325s sections)J
60 303 :M
.756 .076(discuss different aspects of the complex interplay between teacher authority and)J
60 321 :M
1.607 .161(communal judgment, and between traditional and nontraditional elements of)J
60 339 :M
1.389 .139(Schoenfeld\325s pedagogy. In the )J
f2_12 sf
.341(Concluding)A
f1_12 sf
.163 .016( )J
f2_12 sf
.321(Discussion)A
f1_12 sf
1.296 .13( we summarize what we)J
60 357 :M
.905 .091(consider to be the most important issues arising from these analyses and offer)J
60 375 :M
2.373 .237(some implications.)J
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2.196 .22(Abraham Arcavi)J
60 125 :M
1.213 .121(This section describes Schoenfeld's problem solving course, providing the context)J
60 143 :M
.918 .092(and background for the analysis of the following sections. The description includes)J
60 161 :M
.725 .072(his professional background, his goals for the course, the basic characteristics of the)J
60 179 :M
1.156 .116(\322classroom culture\323 he wants to create in order to achieve them, the curriculum)J
60 197 :M
.464 .046(and pedagogy, and some details about the 1990 class. The description is based on)J
60 215 :M
.786 .079(our observations of the 1987 and 1990 classes, on personal dialogues with)J
60 233 :M
.73 .073(Schoenfeld, and his written accounts \(Schoenfeld, 1983; 1985; 1988; 1989a; 1991;)J
60 251 :M
(1994\).)S
229 281 :M
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(Background and goals)S
60 311 :M
f1_12 sf
1.117 .112(Schoenfeld's conceptualization, design and teaching of his course draw upon three)J
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.895 .09(distinct but related fields: mathematics, cognitive science, and college teaching. As)J
60 347 :M
1.31 .131(a member of the mathematical community, he has examined the nature of his)J
60 365 :M
1.061 .106(own mathematical activity and the practices of the discipline itself \(Schoenfeld,)J
60 383 :M
.515 .051(1983; 1985; 1991; 1994\). His participation in that professional practice is directly)J
60 401 :M
.796 .08(reflected in his goals for his course. As a cognitive scientist, he has conducted)J
60 419 :M
1.427 .143(extensive research on the nature of mathematical problem solving and thinking)J
60 437 :M
.96 .096(\(Schoenfeld, 1985; 1987; 1992; Schoenfeld, Smith, & Arcavi, 1993\). This research has)J
60 455 :M
1.067 .107(provided detailed models of successful \(and unsuccessful\) problem solving which)J
60 473 :M
.933 .093(have directly influenced his teaching practice. As a teacher of college mathematics,)J
60 491 :M
.964 .096(he designed the problem solving course, taught, evaluated, and revised it over a)J
60 509 :M
.559 .056(period of more than 15 years \(Schoenfeld, 1983; 1985; 1991; 1994\).)J
60 539 :M
.937 .094(Though most students who take the course are among the successes of the)J
60 557 :M
1.019 .102(mathematics education system, they begin the course with very different)J
60 575 :M
.906 .091(expectations and practices from those envisioned by Schoenfeld. In the area of)J
60 593 :M
1.077 .108(problem solving, students\325 past experiences in mathematics consist mostly of)J
60 611 :M
.811 .081(generating \322the answer\323 to problems by applying procedures for manipulating)J
60 629 :M
1.106 .111(numerical and symbolic expressions. They have learned to view the professor)J
60 647 :M
.803 .08(\(and/or the textbook\) as the sole authorities in the classroom and to defer to this)J
60 665 :M
1.089 .109(external judgment on most issues. They have developed an ability to \322master\323)J
60 683 :M
.707 .071(facts and procedures for exams which are the accepted evidence of their)J
60 701 :M
1.159 .116(mathematical competence. However, given problems out of context they may well)J
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.888 .089(not know when to apply those facts and procedures. Moreover, as Schoenfeld)J
60 69 :M
.888 .089(\(1994, p. 43\) notes, \322For most of them, doing mathematics has meant studying)J
60 87 :M
.815 .082(material and working tasks set by others, with little or no opportunity for)J
60 105 :M
2.025 .202(invention or sustained investigations.\323)J
60 135 :M
1.038 .104(A major goal of Schoenfeld\325s problem solving course is to provide his students)J
60 153 :M
.781 .078(with the opportunity to engage in doing mathematics by creating and supporting a)J
60 171 :M
1.248 .125(\322classroom culture\323 in which students )J
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.299(can)A
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1.081 .108( solve problems given out of context,)J
60 189 :M
1.106 .111(judge the validity of their solutions without appealing to an external authority,)J
60 207 :M
1.451 .145(and have the opportunity for invention and sustained investigations. These)J
60 225 :M
1.122 .112(aspects of the course are consistent with those of the mathematical community.)J
175 255 :M
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.302 .03(Characteristics of the \322classroom culture\323)J
60 285 :M
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1.22 .122(The task of creating and nurturing the \322mathematics classroom culture\323 in which)J
60 303 :M
1.166 .117(students will have the experience of doing mathematics, has the following)J
60 321 :M
.123(characteristics.)A
60 351 :M
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2.529 .253(Development of a mathematical point of view)J
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2.93 .293(. According to Schoenfeld, doing)J
60 369 :M
.855 .086(mathematics is more than acquiring the primary tools \(e.g., facts and procedures\))J
60 387 :M
1.268 .127(and deploying them thoughtfully in solving problems. It involves looking at the)J
60 405 :M
.886 .089(world with \322mathematical spectacles\323 in a wide variety of problem)J
60 423 :M
1.21 .121(situations\321using mathematics to symbolize, abstract, model, prove or disprove)J
60 441 :M
1.08 .108(conjectures; perceiving connections across problems and results; and creating)J
60 459 :M
.963 .096(knowledge that is new to oneself or to the community. The search for and)J
60 477 :M
.921 .092(discussion of solutions to problems is not the only focus of the activity. Problems)J
60 495 :M
.892 .089(should also serve as springboards for generalization \(or specialization\), to establish)J
60 513 :M
1.618 .162(connections between mathematical domains, to reveal mathematical structure,)J
60 531 :M
.758 .076(and to pose new problems.)J
60 561 :M
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2.033 .203(Emphasis on processes as well as on results)J
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2.859 .286(. While emphasizing and honoring)J
60 579 :M
.827 .083(results that are new to the class, Schoenfeld gives greater priority to the reasoning)J
60 597 :M
.934 .093(processes that generated the results. Results that students cannot explain,)J
60 615 :M
.737 .074(regardless of their correctness, power, or appeal, are not valued. Tricks, results)J
60 633 :M
.646 .065(proven elsewhere \(\322we proved in Math 127 that . . . \323\), and \322rabbits pulled out of)J
60 651 :M
.975 .097(hats,\323 are dismissed in favor of presentations of accessible and non-technical)J
60 669 :M
1.122 .112(mathematical arguments that presume only what is known by all members of the)J
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1.084 .108(community. Explanations of how ideas are generated are highly valued, even)J
60 69 :M
1.088 .109(when they do not produce solutions.)J
60 99 :M
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.466(Communication)A
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1.767 .177(. Students\325 mathematical activity takes place in an inherently)J
60 117 :M
.93 .093(social milieu, where they work as individuals, as members of small groups and as)J
60 135 :M
1.065 .107(participants in whole-class activities. The classroom setting encourages and)J
60 153 :M
1.54 .154(supports various levels of oral and written mathematical communication: from)J
60 171 :M
.985 .099(expressions of raw ideas, suggestions, intuitions, or insights to top-level)J
60 189 :M
1.129 .113(descriptions of mathematical arguments and also final, polished, and airtight)J
60 207 :M
1.263 .126(mathematical presentations. Students are encouraged to evaluate, question, and)J
60 225 :M
.873 .087(criticize each other\325s suggestions and work, in both small-group and whole-class)J
60 243 :M
1.277 .128(activities. Schoenfeld plays a strong role in shaping this communication, ensuring)J
60 261 :M
1.145 .115(that students criticize the mathematics rather than each other \(Schoenfeld, 1994\).)J
60 291 :M
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2.256 .226(Leadership and authority)J
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1.722 .172(. As the established leader of this community,)J
60 309 :M
.889 .089(Schoenfeld's task requires both a detailed overall design and a continual day-to-)J
60 327 :M
.698 .07(day shaping. He sets the top-level goals, selects the initial problems, directs)J
60 345 :M
1.086 .109(students\325 work on those problems, models desirable mathematical actions and)J
60 363 :M
.896 .09(dispositions, and consults with individuals and groups of students. He states his)J
60 381 :M
.742 .074(goals openly and explicitly relates them to specific teaching actions and decisions.)J
60 399 :M
1.002 .1(However, his intention is to gradually transfer the locus of authority and)J
60 417 :M
1.225 .123(community leadership from himself to the students as they become more)J
60 435 :M
.712 .071(comfortable with him and the class. He starts by deferring to students in)J
60 453 :M
.993 .099(evaluating the validity of proposed solutions. He asks them to question and)J
60 471 :M
.997 .1(challenge any hidden or explicit parts of mathematical arguments that are)J
60 489 :M
1.067 .107(unconvincing, unclear, or based on implicit knowledge not shared by the whole)J
60 507 :M
.745 .075(class. Schoenfeld leads the class towards assuming responsibility for safeguarding)J
60 525 :M
.368 .037(standards, for what can be accepted as \322basic\323 shared knowledge and for the)J
60 543 :M
1.27 .127(completeness, coherence, and conviction of mathematical arguments. He does so)J
60 561 :M
.704 .07(by asking very direct questions and by providing initial modeling of desirable)J
60 579 :M
.236(actions.)A
60 609 :M
.944 .094(Though he expects students to play a substantial role in setting the mathematical)J
60 627 :M
1.206 .121(agenda, in presenting their thinking, and in evaluating each others\325 arguments, he)J
60 645 :M
.902 .09(reserves the right to encourage the class in certain directions and not others. He)J
60 663 :M
.563 .056(describes this as follows:)J
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.697 .07(I know what fruitful directions are that students are likely to [engage in], or)J
96 69 :M
.596 .06(can be nudged into, and on the basis of my general sense of what\325s)J
96 87 :M
1.069 .107(mathematically valuable, I\325m going to try, without letting the students)J
96 105 :M
1.065 .107(know it, to nudge the conversation in the direction of things which I)J
96 123 :M
1.03 .103(consider important, giving enough latitude to go where they think it\325s right.)J
96 141 :M
.753 .075(It\325s clear that that works in the sense that, in a number of classes they\325ve)J
96 159 :M
.812 .081(discovered mathematics which I didn\325t know, it was good mathematics . . .)J
96 177 :M
1.01 .101(on the other hand, I am nudging away from things that are frivolous, not)J
96 195 :M
.26 .026(necessarily dead ends \(because dead ends can be profitable\), and I try to do)J
96 213 :M
.837 .084(that in a way which is not terribly overt, but someone who really)J
96 231 :M
.757 .076(understands the mathematics and the goals for my class can clearly pick that)J
96 249 :M
1.176 .118(up. \(From an audiotaped conversation with Schoenfeld about his course,)J
96 267 :M
(May, 1991\))S
60 297 :M
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2.988 .299(Reflective mathematical practice)J
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2.116 .212(. Learning to solve problems and think)J
60 315 :M
1.343 .134(mathematically requires continuous reflection on the nature of that activity.)J
60 333 :M
1.019 .102(Questions that Schoenfeld first asks of students almost routinely, are intended to)J
60 351 :M
.933 .093(play a central role in developing that reflective capability. For example, Schoenfeld)J
60 369 :M
1.574 .157(has shown that skillful mathematical problem solving includes the development)J
60 387 :M
.865 .086(of a critical attitude toward mathematical argument: \322Is this airtight?,\323 \322Does it)J
60 405 :M
1.115 .111(convince me, a friend, an enemy?\323 \(Mason, Burton, & Stacey, 1982; Schoenfeld,)J
60 423 :M
.953 .095(1994\), \322Am I done with this problem?\323 Other questions help to develop the)J
60 441 :M
1.113 .111(mathematical point of view: \322How could this have been done in another way?,\323)J
60 459 :M
.93 .093(\322How can this result be generalized?,\323 \322Is this result similar to another we have)J
60 477 :M
1.008 .101(seen?\323 and so on.)J
60 507 :M
.969 .097(Later in the course, Schoenfeld also devotes time to develop what he calls)J
60 525 :M
1.088 .109(\322executive control of students\325 solution attempts.\323)J
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.112 .011( )J
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.852 .085(Briefly stated, control is \322a)J
60 543 :M
.964 .096(category of behavior [which] deals with the way individuals use the information)J
60 561 :M
.779 .078(potentially at their disposal. It focuses on major decisions about what to do in a)J
60 579 :M
.914 .091(problem, decisions that in and of themselves may \324make or break\325 an attempt to)J
60 597 :M
1.07 .107(solve a problem. Behaviors of interest include making plans, selecting goals and)J
60 615 :M
1.103 .11(subgoals, monitoring and assessing solutions as they evolve, and revising or)J
60 633 :M
1.026 .103(abandoning plans when the assessments indicate that such actions should be)J
60 651 :M
.237(taken\323)A
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.11 .011( )J
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.931 .093(\(Schoenfeld, 1985, p. 27\). Schoenfeld nurtures this behavior by asking)J
60 669 :M
.9 .09(students the following questions while they work: \(1\) \322What are you doing?,\323 \(2\))J
60 687 :M
.639 .064(\322Why are you doing it?,\323 and \(3\) \322How does it help you?\323 \(Schoenfeld, 1985; 1988;)J
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.846 .085(1992a\). Schoenfeld suggests that these kinds of questions are slowly internalized)J
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.906 .091(and become an integral part of the students\325 doing mathematics.)J
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.337(Curriculum)A
60 129 :M
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1.029 .103(The curriculum of the course consists of a collection of carefully chosen problems,)J
60 147 :M
1.352 .135(drawn from a wide variety of mathematical domains\321including number theory,)J
60 165 :M
.829 .083(Euclidean constructions, cryptarithmetic, calculus, algebra, and probability.)J
60 183 :M
.758 .076(Problems are presented to students on sheets distributed in class. In general, each)J
60 201 :M
1.119 .112(problem sheet does not have more than one or two problems from the same)J
60 219 :M
2.763 .276(mathematical domain.)J
60 249 :M
.924 .092(This aspect of the curriculum addresses one of the main course goals: That)J
60 267 :M
1.002 .1(students learn how to solve problems out of context. In traditional courses,)J
60 285 :M
.631 .063(problems and exercises are often sequenced in such a way that students can easily)J
60 303 :M
1.183 .118(find solution techniques. Thus problems are perceived as mere opportunities to)J
60 321 :M
.972 .097(exercise a pre-established and known technique. Schoenfeld deliberately chooses)J
60 339 :M
1.303 .13(not to sequence problems from the same mathematical domain consecutively. On)J
60 357 :M
.923 .092(the contrary, whenever he feels a technique or a solution strategy is understood,)J
60 375 :M
1.065 .106(he changes the type of problem, even giving examples in which the thoughtless)J
60 393 :M
.919 .092(application of a recently \322mastered\323 technique can lead to error or nowhere. Thus)J
60 411 :M
1.131 .113(the sequencing of the problems is consonant with his intention to teach students)J
60 429 :M
.954 .095(to approach problems as professionals do, namely without having explicit cues)J
60 447 :M
.66 .066(about the techniques to be used. Because of that, to a casual observer, it may seem)J
60 465 :M
.909 .091(that the design of the course has discontinuities and lacks coherence: a result)J
60 483 :M
.943 .094(reached in one class session may not be recalled or invoked until three or four)J
60 501 :M
1.094 .109(sessions later when the result is relevant, useful, or connected with the issues)J
60 519 :M
.032(discussed.)A
60 549 :M
1.247 .125(Since Schoenfeld is not constrained to \322covering\323 a predetermined amount of)J
60 567 :M
.759 .076(content, he can afford to allocate time flexibly so that work and discussion can)J
60 585 :M
1.243 .124(yield the maximum mathematical profit. Problems which can be solved in)J
60 603 :M
.896 .09(minutes with traditional \322show and tell\323 teaching are worked on and discussed as)J
60 621 :M
1.096 .11(long as they have mathematical substance, fulfilling one of the main course goals:)J
60 639 :M
1.457 .146(That students have the opportunity for invention and sustained investigations.)J
60 669 :M
.863 .086(What are problems and how are they chosen? Are there any criteria for good)J
60 687 :M
1.261 .126(problems? Schoenfeld regards problems as demanding, non-routine and)J
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1.073 .107(interesting mathematical tasks, which students want and like to solve, and for)J
60 69 :M
.914 .091(which they lack readily accessible means to achieve a solution \(Schoenfeld, 1985;)J
60 87 :M
.925 .093(1989a\). Problems selected for the course must satisfy five main criteria \(Schoenfeld,)J
60 105 :M
(1994\).)S
96 141 :M
.755 .075(\245 Without being trivial, problems should be accessible to a wide range of)J
105 159 :M
.843 .084(students on the basis of their prior knowledge, and should not require a)J
105 177 :M
.903 .09(lot of machinery and/or vocabulary.)J
96 195 :M
.847 .085(\245 Problems must be solvable, or at least approachable, in more than one)J
105 213 :M
1.166 .117(way. Alternative solution paths can illustrate the richness of the)J
105 231 :M
1.225 .123(mathematics, and may reveal connections among different areas of)J
105 249 :M
.397(mathematics.)A
96 267 :M
1.328 .133(\245 Problems should illustrate important mathematical ideas, either in terms)J
105 285 :M
1.07 .107(of the content or the solution strategies.)J
96 303 :M
1.06 .106(\245 Problem solutions should be constructible without tricks.)J
96 321 :M
.922 .092(\245 Problems should serve as first steps towards mathematical explorations,)J
105 339 :M
.949 .095(they should be extensible and generalizable; namely, when solved, they)J
105 357 :M
.828 .083(can serve as springboards for further explorations and problem posing.)J
60 387 :M
1.346 .135(A main topic in Schoenfeld's curriculum is, as already implied, )J
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.308(heuristics)A
f1_12 sf
.485(\321\322rules)A
60 405 :M
.995 .1(of thumb for successful problem solving, general suggestions that help an)J
60 423 :M
.783 .078(individual to understand a problem better or to make progress towards its)J
60 441 :M
1.137 .114(solution\323 \(Schoenfeld, 1985, p. 23\). Commonly used heuristics include: exploiting)J
60 459 :M
.671 .067(analogies, examining special cases, arguing by contradiction, working backwards,)J
60 477 :M
1.113 .111(decomposing and recombining, exploiting related problems, generalizing and)J
60 495 :M
.814 .081(specializing, and relaxing conditions in the problem \(see P\227lya, 1973 for a more)J
60 513 :M
1.034 .103(complete list\). The rationale for teaching heuristics is clear: expert problem solvers)J
60 531 :M
.852 .085(develop and rely on these strategies to make progress on difficult problems. Thus)J
60 549 :M
.798 .08(if heuristics can be taught, they may help students become better problem solvers.)J
60 567 :M
1.041 .104(Indeed, this hypothesis \(among others\) led Schoenfeld to develop the course.)J
60 597 :M
1.154 .115(Researchers in mathematics education have not found it easy to teach heuristics in)J
60 615 :M
.927 .093(the classroom \(e.g., Lester, Garofalo, & Kroll, 1989\). Schoenfeld has himself)J
60 633 :M
.838 .084(experienced difficulty at the college level. In his early work teaching problem)J
60 651 :M
1.258 .126(solving, he identified three main complications in the task of teaching heuristics)J
60 669 :M
1.276 .128(\(Schoenfeld, 1985\).)J
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.668 .067(\245 )J
f2_12 sf
1.665 .167(The specificity problem. )J
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1.654 .165(If heuristics are presented in their most general)J
105 69 :M
.814 .081(\(and useful\) form, students will be unable to apply them; if they are given)J
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1.073 .107(in more context-specific forms, their number explodes and only some can)J
105 105 :M
.48 .048(be taught.)J
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.802 .08(\245 )J
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2.828 .283(The implementation problem)J
f1_12 sf
2.305 .231(. Applying heuristics requires many steps,)J
105 141 :M
.955 .095(and therefore creates many opportunities for students to make fatal errors.)J
96 159 :M
.671 .067(\245 )J
f2_12 sf
2.028 .203(The resource problem.)J
f1_12 sf
1.567 .157( To be successful, students must know both the)J
105 177 :M
1.116 .112(appropriate heuristics and the mathematics required to solve the problem.)J
60 207 :M
1.283 .128(In his continuing development and revision of the course, Schoenfeld has had to)J
60 225 :M
.734 .073(address each of these problems. A major goal of this paper is to understand his)J
60 243 :M
.123(approach.)A
275 273 :M
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-.397(Effect)A
60 303 :M
f1_12 sf
.757 .076(Schoenfeld has discussed the course many times \(see [Schoenfeld 1983; 1985; 1988;)J
60 321 :M
.849 .085(1989a; 1991; 1994] for the most substantive discussions\). Some accounts have been)J
60 339 :M
1.088 .109(descriptive introductions built around vignettes: rich snapshots of the class)J
60 357 :M
.775 .077(working on particular problems, such as the magic square \(Schoenfeld, 1989a; 1991\))J
60 375 :M
.644 .064(or Pythagorean triples \(Schoenfeld 1988; 1991; 1994\). These accounts suggest that)J
60 393 :M
1.106 .111(his students are engaged in more productive sorts of mathematical thinking and)J
60 411 :M
.791 .079(activity than are typical of most undergraduates.)J
60 441 :M
1.288 .129(Students work collaboratively in groups with or without Schoenfeld's)J
60 459 :M
1.353 .135(presence\321indicating engagement and commitment to the enterprise. They stop)J
60 477 :M
1.205 .12(looking to him to evaluate the validity of their arguments, turning instead to their)J
60 495 :M
.876 .088(peers. They produce results that are new to them, surprising and interesting to)J
60 513 :M
1.039 .104(Schoenfeld, and occasionally publishable \(Schoenfeld, 1989b\). And most)J
60 531 :M
.984 .098(important, they learn to use heuristics effectively over a range of problems,)J
60 549 :M
1.467 .147(considering, pursuing, and monitoring multiple approaches.)J
60 579 :M
1.167 .117(Schoenfeld examined students\325 problem solving performance before and after the)J
60 597 :M
.867 .087(course using measures which ranged from paper-and-pencil tests to analyses of)J
60 615 :M
.65 .065(problem-solving protocols \(see Chapters 7, 8, 9, and 10 of Schoenfeld, 1985\). His)J
60 633 :M
.768 .077(results showed that students who completed the course \(1\) used a variety of)J
60 651 :M
.858 .086(heuristics effectively to solve challenging problems; \(2\) had a better sense of how)J
60 669 :M
.687 .069(to proceed and were less likely to \322plunge in\323 with the first approach that came to)J
60 687 :M
.952 .095(mind; \(3\) saw through the surface features to the deeper mathematical structure of)J
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.928 .093(problems; and \(4\) used heuristics to solve problems unlike those they had worked)J
60 69 :M
1.337 .134(previously in the course.)J
60 99 :M
.714 .071(On the first day of class, Schoenfeld described one of the measures he used in his)J
60 117 :M
.15(analysis:)A
96 153 :M
.84 .084(I gave an in-class final, and there were three parts to the final exam. The)J
96 171 :M
.76 .076(first part was problems like the problems we solved in class. No surprise,)J
96 189 :M
.631 .063(you expect people to do well on those. The second part was problems that)J
96 207 :M
.726 .073(could be solved by the methods that we used in class\321but ones for which if)J
96 225 :M
.986 .099(you looked at them you couldn't recognize that they had obvious features)J
96 243 :M
.753 .075(similar to the ones that we'd studied in class. So yes, you had the tools and)J
96 261 :M
.619 .062(techniques, but you had to be pretty clever about recognizing that they were)J
96 279 :M
.63 .063(appropriate. And, the class did pretty well on those too. Part three of the)J
96 297 :M
1.139 .114(final exam . . . There's a collection of books called the )J
f2_12 sf
2.682 .268(Hungarian Problem)J
96 315 :M
.391(Books)A
f1_12 sf
1.523 .152( which have some of the nastiest mathematical problems known to)J
96 333 :M
.826 .083(man and woman. I went through those, and as soon as I found a problem I)J
96 351 :M
.95 .095(couldn't make any sense of, whatsoever, I put it on the final. \(I know that)J
96 369 :M
.687 .069(makes you feel good.\) [Laughter from class, Schoenfeld smiles.] The class did)J
96 387 :M
.764 .076(spectacularly well, and actually wound up solving some problems I didn't)J
f1_9 sf
.146 .015(. . . .)J
264 417 :M
f0_12 sf
-.335(Pedagogy)A
60 447 :M
f1_12 sf
.64 .064(In the versions of the course we observed \(1987 and 1990\), the class was organized)J
60 465 :M
1.212 .121(into six principal modes: lectures, reflective presentations, student presentations,)J
60 483 :M
1.019 .102(small-group work, whole-class discussions, and individual work. In the first two)J
60 501 :M
.817 .082(class sessions all six)J
f2_12 sf
.11 .011( )J
f1_12 sf
.96 .096(modes occurred, although not exactly in the same proportions)J
60 519 :M
.759 .076(as throughout the semester. On the first day of the course Schoenfeld described)J
60 537 :M
1.021 .102(these modes to the students:)J
96 573 :M
.71 .071(Most days I'm going to walk in . . . and hand out a bunch of problems. I've)J
96 591 :M
.407 .041(got enough here to probably keep us busy for two days or so. And what)J
96 609 :M
.887 .089(you're seeing here is unusual, because you won't be seated in rows watching)J
96 627 :M
.803 .08(me talk. Instead you're going to break into groups of three or four or five,)J
96 645 :M
1.127 .113(and work on problems together. As you're working on them, I'll circulate)J
96 663 :M
1.238 .124(through the room, occasionally make comments about the kinds of things)J
96 681 :M
.835 .083(you're doing, respond to questions from you. But, by and large, I'll just)J
96 699 :M
.849 .085(nudge you to keep working on the problems.)J
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.779 .078(Then at some point I'll call us to order as a group, and we'll start discussing)J
96 87 :M
1.095 .109(the things that you've done, and talk about the things that you've pushed)J
96 105 :M
1.102 .11(and why; what's been successful, what hasn't. I'll mention a variety of)J
96 123 :M
.924 .092(specific mathematical techniques as we go through the problems. Many of)J
96 141 :M
1.111 .111(the problems are chosen so that they illustrate useful techniques. So you'll)J
96 159 :M
.88 .088(work on one for a while; may or may not make some progress; and then)J
96 177 :M
.721 .072(we'll talk about it. And as we talk about it what I'll do is indicate some of)J
96 195 :M
.939 .094(the problem solving strategies that I know, and that are in the literature,)J
96 213 :M
.905 .09(that might help you make progress on this problem, and progress on other)J
96 231 :M
.244(problems.)A
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.258(Lectures)A
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1.097 .11(. In contrast with most college classrooms, the lecture mode occurred)J
60 279 :M
1.4 .14(relatively infrequently. When Schoenfeld lectured, the lecture segments were)J
60 297 :M
.867 .087(relatively short and oriented toward particular goals: to provide background on)J
60 315 :M
1.198 .12(mathematical resources needed to make progress)J
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.12 .012( )J
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1.223 .122(\(e.g., mathematical induction\), to)J
60 333 :M
.643 .064(introduce heuristics, and to describe his goals for the class. He did not generally)J
60 351 :M
1.052 .105(present his own solutions to problems, except on the occasion that an important)J
60 369 :M
.839 .084(solution was not developed by the class. Because his lecture segments were short,)J
60 387 :M
.909 .091(pointed, and related to activities in other modes, many of the traditional effects of)J
60 405 :M
1.189 .119(the lecture\321e.g., student passivity and disengagement\321were not evident. \(More)J
60 423 :M
1.057 .106(details are provided throughout the following sections by Meira, Smith and)J
60 441 :M
.164(Kessel.\))A
60 471 :M
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2.977 .298(Reflective presentations)J
f1_12 sf
2.19 .219(. In this mode, Schoenfeld presented mathematical)J
60 489 :M
1.087 .109(commentaries to the class, interpreting segments of activity just completed and)J
60 507 :M
1.079 .108(highlighting important aspects. We characterize them as \322reflective\323 because they)J
60 525 :M
1.128 .113(directed students\325 attention to mathematically significant features of either)J
60 543 :M
.842 .084(Schoenfeld\325s or his students\325 actions. They differed from lectures because they)J
60 561 :M
.776 .078(engaged students as participants. They were unlike whole-class discussions because)J
60 579 :M
.927 .093(Schoenfeld pursued specific goals and directly controlled the flow. Reflective)J
60 597 :M
1.075 .107(presentations took quite different forms: e.g. modeling a problem solution to)J
60 615 :M
1.032 .103(illustrate a particular heuristic, to demonstrate a specific mathematical point, or to)J
60 633 :M
1.51 .151(highlight executive control in problem solving; recounting, and highlighting)J
60 651 :M
1.112 .111(aspects of students\325 presentations of their solutions; conducting \322post-mortem\323)J
60 669 :M
.925 .092(reviews of complete problem solutions \(see Schoenfeld, 1983 for a specific)J
60 687 :M
.983 .098(example\). Like lectures, they all involved significant forms of teaching by telling;)J
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1.151 .115(i.e., substantive insertions of content into the classroom discourse \(Ball & Chazan,)J
60 69 :M
.773 .077(1994\), but occurred in the broader context of problem solving. They provided)J
60 87 :M
.912 .091(students with a clear view of the reflective mathematical practices of a skilled)J
60 105 :M
1.175 .118(mathematician, an opportunity that is absent from many college classrooms.)J
60 135 :M
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2.402 .24(Student presentations)J
f1_12 sf
1.583 .158(. At appropriate junctures, students were invited to present)J
60 153 :M
1.233 .123(their solutions to assigned problems. During these presentations, Schoenfeld)J
60 171 :M
1.242 .124(avoided giving immediate verbal feedback and non-verbal evaluations of student)J
60 189 :M
.972 .097(success, though students initially expected such judgments \(see Smith\325s analysis of)J
60 207 :M
.788 .079(the inscribed square problem\). With a blank \322poker-face\323 he usually addressed the)J
60 225 :M
.893 .089(class with one of the following questions: \322What do you guys think?,\323 \322Does the)J
60 243 :M
.968 .097(class buy this argument?,\323 or \322Are you convinced?\323 These questions were)J
60 261 :M
.911 .091(routinely posed after each presentation to signal that students should not wait for)J
60 279 :M
1.505 .15(an external authoritative judgment. Student presentations also provided)J
60 297 :M
1.451 .145(opportunities to work on issues of mathematical exposition and communication;)J
60 315 :M
.99 .099(such as top-level descriptions of an argument vs. more polished and detailed)J
60 333 :M
1.439 .144(versions, comparing formal/symbolic and informal presentations, contrasting)J
60 351 :M
1.864 .186(convincing arguments with \322hand-waving.\323)J
60 381 :M
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2.039 .204(Small-group work)J
f1_12 sf
1.122 .112(. About 30% of each class was devoted to work in small groups)J
60 399 :M
.692 .069(of two to four students. Its purpose was to provide a stable and continuous context)J
60 417 :M
.776 .078(for students to engage collectively in problem solving. In the best of cases, this)J
60 435 :M
1.062 .106(collaborative work generated negotiation among the members of the group about)J
60 453 :M
.766 .077(approaches to pursue, allowed each student to calibrate his/her own)J
60 471 :M
1.365 .137(understanding of the mathematics involved with the other group members, and)J
60 489 :M
1.019 .102(promoted the disposition to listen to and learn from peers. In this mode,)J
60 507 :M
1.017 .102(Schoenfeld played the role of \322traveling consultant\323 and critic.)J
60 537 :M
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2.872 .287(Whole-class discussions)J
f1_12 sf
1.947 .195(. After an individual presentation or small-group work,)J
60 555 :M
1.117 .112(Schoenfeld often engaged the class in collective discussion. Sometimes the class)J
60 573 :M
.77 .077(attempted to solve a problem as a whole group, and, as in the small-group work,)J
60 591 :M
1.429 .143(Schoenfeld usually avoided immediate evaluation of the usefulness of the)J
60 609 :M
.52 .052(approach suggested by students, even when the approach could lead to a dead end.)J
60 627 :M
.984 .098(This mode had several purposes: it allowed all students to listen to each other's)J
60 645 :M
1.083 .108(questions, comments, and solution attempts. As students started to feel more)J
60 663 :M
.839 .084(comfortable with the class, it slowly became a forum in which they could openly)J
60 681 :M
1.142 .114(voice misunderstandings and/or requests for mathematical resources invoked by)J
60 699 :M
.871 .087(some and lacked by others. There were occasions later in the course in which the)J
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.998 .1(whole-class discussion also dealt with issues of mathematical elegance and)J
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.172(aesthetics.)A
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2.217 .222(Individual work)J
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1.705 .171(. Students had many opportunities to work individually before,)J
60 117 :M
1.006 .101(within, and after some of the modes described above. However, individual work)J
60 135 :M
1.216 .122(was the main mode for homework assignments, and the two take-home exams,)J
60 153 :M
.955 .096(on which students worked for about two weeks with the promise of not consulting)J
60 171 :M
1.095 .11(each other. Individual work consisted not only in solving problems, it also)J
60 189 :M
1.287 .129(included, as mentioned above, preparation for communication of results, either to)J
60 207 :M
.888 .089(a small group, the whole class, or \(in the case of the written take-home exams\) the)J
60 225 :M
.222(teacher.)A
60 255 :M
.634 .063(Students received specific guidelines about exams and grading. On the first day of)J
60 273 :M
.996 .1(class Schoenfeld told the students:)J
96 309 :M
.687 .069([A] week or two into the class I'll give you the opportunity to write out a)J
96 327 :M
.587 .059(problem or two for me so that I can get a sense of the kind of writing you do,)J
96 345 :M
.757 .076(and give you some feedback on the kind of writing I expect. The first main)J
96 363 :M
.769 .077(thing we do is: about half-way through the course I'll give you a two-week)J
96 381 :M
.788 .079(take-home. It'll consist of about ten problems and they will occupy you for a)J
96 399 :M
1.084 .108(long time. But you'll make progress on them and you'll do reasonably well)J
96 417 :M
1.257 .126(on them. And then, the final. Again, the department formally requires me)J
96 435 :M
.853 .085(to give an in-class final, so I usually wind up giving a one-problem in-class)J
96 453 :M
1.056 .106(final to meet the rules and regulations. That's about ten percent of the final)J
96 471 :M
.978 .098(exam grade. The rest of it is another take-home that you'll have two weeks)J
96 489 :M
1.025 .103(to work on. There are some funny rules, which are that:)J
96 525 :M
1.113 .111(What counts is not simply the answer, what counts is doing mathematics.)J
96 543 :M
.824 .082(And that means, among other things, if you can find two different ways to)J
96 561 :M
.699 .07(solve a problem, you'll get twice as much credit for it. If you can extend the)J
96 579 :M
1.042 .104(problem and generalize it and make it your own, you'll get even more. The)J
96 597 :M
.968 .097(bottom line is, I'd like to have you doing some mathematics and I will do)J
96 615 :M
.981 .098(everything I can\321including using grading\321as a device for having you do)J
96 633 :M
.299(that.)A
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-.023(The 1990 class)A
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.052(Students)A
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1.008 .101(Mathematical Problem Solving, listed in the university catalog as Math 67, is not a)J
60 141 :M
.952 .095(required course for any major. The course prerequisite is one semester of calculus)J
60 159 :M
1.226 .123(or consent of instructor.)J
60 189 :M
.816 .082(The students in the first two classes had a wide range of mathematical)J
60 207 :M
.513 .051(backgrounds \(see Table 1\). For example, Jeff,)J
302 204 :M
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.999 .1( a history major, had taken one)J
60 225 :M
.781 .078(semester of calculus three years ago, and Diane, a genetics major, had taken the)J
60 243 :M
.803 .08(calculus sequence. In contrast, Mitch was a graduate student in computer science,)J
60 261 :M
.914 .091(and Jes\234s, a fourth year applied math major.)J
60 291 :M
.746 .075(In the first week, the \322traffic\323 in and out of the class was relatively heavy; students)J
60 309 :M
.934 .093(were shopping for classes and adjusting their schedules. The university catalog)J
60 327 :M
.775 .077(had also listed the course as beginning one hour later than it did, thus adding to)J
60 345 :M
.891 .089(the traffic. Thirteen students attended all or part of the first session. Three new)J
60 363 :M
1.065 .106(students entered in the second session.)J
60 393 :M
.889 .089(The eight students who completed the course were all enrolled for credit. Six were)J
60 411 :M
1.243 .124(majors \(or intended majors\) in mathematics or computer science. Only one of)J
60 429 :M
1.029 .103(these students \(Jeff\) had a major outside of science, mathematics, and engineering.)J
60 447 :M
1.032 .103(Only one was female. The group comprised four European Americans, two Asian)J
60 465 :M
1.082 .108(Americans, and two Hispanics.)J
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.124 .012(Brief overviews of the first two class sessions)J
60 513 :M
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1.533 .153(Session #1)J
f1_12 sf
1.239 .124(. For the first twenty minutes Schoenfeld introduced the course: its)J
60 531 :M
.779 .078(history, its basic mechanics, the grading system \(a complete transcript is given in)J
60 549 :M
.663 .066(Appendix A\). He then distributed the first set of problems and asked students to)J
60 567 :M
1.005 .101(start working on them in groups. For the next twenty minutes most groups)J
60 585 :M
.951 .095(worked mainly on the first two problems: finding the sum of the telescoping series)J
60 603 :M
.795 .08(and inscribing a square in an arbitrary triangle. Forty minutes into the class,)J
60 621 :M
.805 .08(Schoenfeld called the class back and, for about twenty-five minutes, he discussed)J
60 693 :M
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60 690.48 -.48 .48 204.48 690 .48 60 690 @a
60 702 :M
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64 705 :M
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-.109(All students are referred to by pseudonyms.)A
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.485 .049(Table 1)J
133 69 :M
.856 .086(Background of Students Participating in Sessions 1 and 2)J
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-.176(Student)A
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-.015(Origin of Interest; Entry)A
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514 82 1 1 rF
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(Austin)S
101 118 :M
.085 .008(Saw description in the catalog)J
101 136 :M
.129 .013(Looking for such a course \322for years\323)J
101 154 :M
.181 .018(Entered at start of Session 2)J
281 118 :M
-.103(Third-year computer science major)A
281 136 :M
-.124(Calculus sequence, discrete math)A
281 154 :M
-.148(Audited one class of Putnam)A
392 151 :M
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-.043( course)A
55 101 1 1 rF
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60 173 :M
-.127(Devon)A
101 173 :M
.085 .008(Saw description in the catalog)J
101 191 :M
.181 .018(Entered at start of Session 1)J
281 173 :M
-.088(Third-year student with interests in math and computer)A
281 191 :M
-.031(science; part-time: Math 67 only course)A
281 209 :M
-.046(Older student, recent transfer to UCB)A
55 156 1 1 rF
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97 156 179 1 rF
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514 156 1 1 rF
55 157 1 54 rF
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60 228 :M
-.057(Don)A
101 228 :M
.085 .008(Saw description in the catalog)J
101 246 :M
.181 .018(Entered at start of Session 1)J
281 228 :M
-.05(Math major)A
281 246 :M
-.061(Calculus sequence plus 4 upper division courses)A
55 211 1 1 rF
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96 211 1 1 rF
97 211 179 1 rF
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55 212 1 36 rF
96 212 1 36 rF
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514 212 1 36 rF
60 265 :M
-.098(Jeff)A
101 265 :M
.085 .008(Saw description in the catalog)J
101 283 :M
.181 .018(Entered at start of Session 1)J
281 265 :M
-.05(Fourth-year history major; goal: teach history and math)A
281 283 :M
-.047(One semester of calculus as first-year student)A
55 248 1 1 rF
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96 248 1 1 rF
97 248 179 1 rF
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96 249 1 36 rF
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60 302 :M
-.089(Jes\234s)A
101 302 :M
.085 .008(Saw description in the catalog)J
101 320 :M
.181 .018(Entered at start of Session 1)J
281 302 :M
-.109(Fourth-year applied math major)A
281 320 :M
-.067(13 math courses)A
55 285 1 1 rF
56 285 40 1 rF
96 285 1 1 rF
97 285 179 1 rF
276 285 1 1 rF
277 285 237 1 rF
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60 339 :M
.306(Sasha)A
101 339 :M
.085 .008(Saw description in the catalog)J
101 357 :M
.181 .018(Entered at start of Session 1)J
281 339 :M
-.125(First-year student, intending computer science major)A
281 357 :M
-.146(Calculus, discrete math, Putnam course)A
281 375 :M
-.107(Math camps & 3 high school competitions)A
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-.192(Stephen)A
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-.046(Looking for some \322easy units\323)A
101 412 :M
.181 .018(Entered at start of Session 1)J
281 394 :M
-.098(Physics major)A
55 377 1 1 rF
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.093(Diane)A
101 431 :M
.228 .023(Looking for a \322fun class\323)J
101 449 :M
-.05(Attended Sessions 1 and 2)A
281 431 :M
-.091(Genetics major, calculus sequence)A
281 449 :M
-.122(Liked problem solving)A
55 414 1 1 rF
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.098(Richard)A
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281 468 :M
-.194(Computer science major, part-time student)A
281 486 :M
-.028(Calculus sequence, 1.5 years before)A
281 504 :M
.037 .004(Worried about \322rusty background\323)J
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.177(Mitch)A
101 523 :M
-.065(Read about the course)A
101 541 :M
.008 .001(Entered at start of Session 1, auditing)J
281 523 :M
-.124(Graduate student in computer science)A
281 541 :M
-.041(Calculus sequence, discrete math, linear and abstract)A
281 559 :M
.09(algebra)A
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.167(Sharon)A
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-.156(Heard about Schoenfeld via family member)A
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-.078(Attended Session 1 only, auditing)A
281 578 :M
-.156(Varied academic background; intending math major;)A
281 596 :M
.061 .006(Calculus, logic, and statistics;)J
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1.013 .101(the telescoping series problem. This discussion is analyzed in detail in Meira's)J
60 643 :M
.273(section.)A
60 666 :M
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60 663.48 -.48 .48 204.48 663 .48 60 663 @a
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-.081(The William Lowell Putnam Mathematical Competition is administered annually in December by the Mathematical)A
60 687 :M
-.058(Association of America to students who have not yet received a college degree \(Reznick, 1994, p. 19\). Neither of the)A
60 696 :M
-.109(students who attended the Putnam course \(H90, Honors Undergraduate Seminar in Mathematical Problem Solving,)A
60 705 :M
-.069(also offered in the fall of 1990\) mentioned taking the Putnam exam though they were interviewed early in 1991.)A
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.993 .099(The students then went back to work in their small groups, and Schoenfeld)J
60 69 :M
.992 .099(moved around the class monitoring the work of each group. He discovered that)J
60 87 :M
.947 .095(some groups did not fully understand the written statement of the inscribed)J
60 105 :M
.951 .095(square problem, so he explained the distinction between showing that the required)J
60 123 :M
1.014 .101(square exists and giving a construction. Approximately one hour and twenty-five)J
60 141 :M
.896 .09(minutes into the session, Schoenfeld again called the class back to a whole-class)J
60 159 :M
.91 .091(discussion of this problem which was interrupted when the class period ended.)J
60 177 :M
1.098 .11(Students left class with instructions to think about the problem at home.)J
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1.269 .127(Session #2)J
f1_12 sf
1.058 .106(. Schoenfeld began by emphasizing the importance of working as a)J
60 225 :M
1.045 .105(community and presenting results to one\325s peers. Upon request, one student)J
60 243 :M
.896 .09(volunteered to present his constructive proof that a square can be inscribed in an)J
60 261 :M
.898 .09(arbitrary triangle. His argument was fundamentally sound, but directed almost)J
60 279 :M
.975 .097(entirely to Schoenfeld who was standing at the side of the room. Schoenfeld noted)J
60 297 :M
.992 .099(this deference and explained that the class must become the judge of mathematical)J
60 315 :M
.847 .085(validity of proposed solutions. The student then addressed the class more directly,)J
60 333 :M
1.122 .112(repeating his solution with slightly more detail. This discussion lasted twenty-five)J
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1.148 .115(minutes. Smith analyzes in detail the way in which Schoenfeld directed the)J
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1.318 .132(discussion on this problem.)J
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.971 .097(Schoenfeld then drew students\325 attention to the third problem of the set, placing)J
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.629 .063(the integers from 1 to 9 to make a 3 by 3 magic square. Another student)J
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1.108 .111(volunteered to present his solution, and its correctness was immediately apparent.)J
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1.062 .106(The next forty minutes were spent in discussion of different solution paths that)J
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1.225 .122(could produce the same result. Schoenfeld led this discussion, involving students)J
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.996 .1(in substantive ways and introducing many new heuristics. Kessel analyzes this)J
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1.227 .123(discussion in detail.)J
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1.095 .109(The final fifteen minutes of the session were devoted to quick solutions of the)J
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.832 .083(next problems in the set, which will not be discussed in this paper. In sum, the)J
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1.051 .105(next three sections of this paper cover most of the whole-class instructional)J
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.66 .066(episodes of the first two two-hour class periods.)J
endp
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-.19(Presenting and doing mathematics: An introduction to heuristics)A
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1.712 .171(Luciano Meira)J
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1.027 .103(Solving problems is a considerable part of what mathematicians do, and learning)J
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1.15 .115(to solve problems is part of learning to think mathematically. Shaping the culture)J
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1.051 .105(of the classroom so that his students learn to think mathematically is the heart of)J
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.963 .096(Schoenfeld\325s teaching enterprise. Therefore his central goal is to create a classroom)J
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1.122 .112(community which embodies selected values, beliefs, and activities of the)J
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2.307 .231(professional mathematical community.)J
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1.193 .119(But within this parallel, Schoenfeld has also acknowledged the individual and)J
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1.315 .132(collective differences between professional mathematicians and the students)J
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1.092 .109(taking his course.)J
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1.275 .128(The class itself is a mathematical community \(better, a micro-community in)J
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1.225 .123(which certain mathematical values are highly prized\) in which the students)J
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.86 .086(interact with each other in ways very much like the ways that)J
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1.125 .112(mathematicians interact\321but at a level appropriate to their knowledge and)J
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1.081 .108(abilities. At their own level the students )J
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1.396 .14( mathematicians, engaged in the)J
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1.438 .144(practice of mathematical sense-making. They )J
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.316(do)A
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1.321 .132( mathematics, with the)J
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1.336 .134(same sense of engagement and involvement. The difference is that)J
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.996 .1(boundaries of understanding that they challenge are the boundaries of their)J
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1.459 .146(own \(community\325s\) understanding, rather than those of the mathematical)J
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1.176 .118(community at large. \(Schoenfeld, 1988, author\325s emphasis\))J
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1.168 .117(His characterization of the relevant differences centers on issues of mathematical)J
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.942 .094(background. Students\325 views of problems and significant results reflect their own)J
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1.366 .137(understanding, which is substantially more limited than professional)J
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1.215 .122(mathematicians\325. But are the differences between these two communities only a)J
60 581 :M
.745 .074(matter of what constitutes shared knowledge and problems at the edge of)J
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1.091 .109(collective understanding? We think not, especially in the first sessions of the class)J
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1.08 .108(when much \322shaping\323 of the community is being done. We propose that there are)J
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1.065 .107(other important \(and sometimes subtle\) differences that follow directly from a)J
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1.077 .108(second and equally obvious difference, that the classroom community is)J
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.875 .088(deliberately shaped and \322engineered\323 by Schoenfeld, whereas the professional)J
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1.303 .13(mathematical community has no recognized single authority or leader.)J
endp
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.946 .095(To explore the subtleties of these differences, we introduce the distinction between)J
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1.219 .122(\322presenting\323 and \322doing\323 mathematics. \322Doing mathematics\323 means engaging in)J
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1.261 .126(reasoning that reflects the thinking of mathematicians: resourcefully tackling and)J
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.935 .094(making progress on hard mathematical problems. In order to bring students to the)J
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1.217 .122(point where they can approximate mathematical doing, some \322presenting\323 must)J
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.966 .097(take place, in both traditional and less traditional forms. As will become evident)J
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.765 .076(below, we use \322presenting\323 to characterize different acts of teaching, though all)J
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1.228 .123(forms involve the display of some mathematical concept or part of mathematical)J
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.656 .066(practice for students.)J
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.946 .095(The contrast between doing and presenting mathematics is enacted by Schoenfeld)J
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1.093 .109(during work on the first problem, summing the telescoping series. We analyze)J
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.974 .097(this contrast in three consecutive segments of class activity: \(1\) his mock lecture on)J
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1.093 .109(the standard solution to the problem, where Schoenfeld critiques a traditional)J
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1.163 .116(form of teacher presentation in college mathematics classrooms; \(2\) his)J
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.894 .089(presentation of a heuristic-based solution as an important part of the practice \(the)J
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1.025 .103(\322doing\323\) of professional mathematicians; and \(3\) his lecture on and subsequent)J
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1.246 .125(use of mathematical induction to prove that the solution found is general.)J
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.434 .043(Caricaturing mathematics teaching as presenting: The mock lecture)J
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.989 .099(Finding the sum of the telescoping series is a well-known problem, appearing in)J
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.839 .084(most first-year calculus courses. It asks for the sum of the following terms,)J
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1.077 .108(value of heuristics as tools to unpack results which are either unknown or recalled)J
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1.137 .114(but not understood.)J
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.978 .098(After some twenty minutes of group work, where students worked on the)J
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.878 .088(telescoping series and other problems in the set, Schoenfeld called the class)J
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.909 .091(together to present a \322lecture\323 on the textbook solution. This was not just any)J
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.633 .063(lecture, but a play that caricatured the \322typical\323 calculus professor presenting the)J
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.941 .094(standard solution to the problem. We quote the transcript at length so that the)J
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.788 .079(content and tone of the play are clear. Note that the goal of differentiating between)J
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.955 .095(presenting and doing mathematics was explicit from the start.)J
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.954 .095(Let me show you what you were shown, by metamorphosing into the)J
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1.057 .106(typical calculus professor for three minutes, and lecturing on the solution to)J
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.603 .06(that problem as it\325s typically presented in a calculus class and then talk about)J
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.683 .068(the way it gets )J
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.639 .064(this. [Schoenfeld walks toward the door, turns, and starts back toward the)J
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.632 .063(board as if he were another person.])J
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.826 .083(All right. Well, the problem I asked you to look at was find the sum:)J
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.659 .066([writes rapidly, banging the chalk, on the board and states the formula)J
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.739 .074(We take the overall purpose of the play to be straightforward. In acting out an)J
60 183 :M
.985 .098(objectionable teaching practice, Schoenfeld sets the stage for presenting himself)J
60 201 :M
1.006 .101(and his course as a new and more positive mathematical experience for students.)J
60 219 :M
1.252 .125(We identify the following negative elements in this caricature of \322traditional\323)J
60 237 :M
.237(teaching:)A
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.61 .061(\245 The caricatured professor stated the problem exactly as it is printed in the)J
105 291 :M
.948 .095(problem set and began his lecture without any preliminary discussion of)J
105 309 :M
.98 .098(the problem that might engage students in the task. In traditional)J
105 327 :M
1.397 .14(mathematics classrooms, the curriculum \(problems and solutions\) is seen)J
105 345 :M
1.021 .102(as uniquely defining the activity.)J
96 363 :M
.664 .066(\245 He spoke and wrote on the board very rapidly. Such high speed deliveries)J
105 381 :M
1.246 .125(have additional inhibitory effects on student contributions, over and)J
105 399 :M
.766 .077(above the standard expectations, among both faculty and students,)J
105 417 :M
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96 435 :M
.921 .092(\245 He assumed that the key algebraic reformulation was obvious and never)J
105 453 :M
.716 .072(addressed the source of the insight, suggesting only that students could)J
105 471 :M
.954 .095(undertake the remedial task of \322checking\323 its validity.)J
96 489 :M
.856 .086(\245 He maintained a haughty and arrogant attitude throughout, but especially)J
105 507 :M
1.169 .117(in his implication that his students had seen the solution in calculus but)J
105 525 :M
.924 .092(apparently forgotten it and in his emphasized difference between)J
105 543 :M
1.226 .123(professors who know and students who memorize.)J
96 561 :M
.689 .069(\245 He asked no real questions during the lecture. The two queries posed to)J
105 579 :M
.96 .096(the class were not serious invitations to discussion, since he did not wait)J
105 597 :M
1.013 .101(for a student response. These \322questions\323 were merely rhetorical)J
105 615 :M
1.688 .169(ornaments in the lecture.)J
96 633 :M
.675 .067(\245 It was evident from the videotape record that he wrote the key algebraic)J
105 651 :M
.964 .096(steps on panels of the moveable chalkboard but then quickly removed)J
105 669 :M
1.072 .107(them from the students\325 view behind the fresh panels he was sliding into)J
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.816 .082(place. On one occasion, he covered a long computation just as he began to)J
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2.328 .233(summarize it.)J
60 99 :M
.71 .071(The mock lecture was clearly a set-up, a worst-case scenario of traditional)J
60 117 :M
.883 .088(mathematics teaching practices where teachers \322tell\323 students the facts and)J
60 135 :M
1.33 .133(procedures they think are important and students memorize them. Schoenfeld\325s)J
60 153 :M
.955 .095(enactment of this caricature communicates at least two related messages to his)J
60 171 :M
.884 .088(class: \322I know about your experience with the mathematics teachers, particularly)J
60 189 :M
.792 .079(college professors\323 and \322I will make your experience with me different.\323 As is)J
60 207 :M
1.085 .108(evident from the start, the mock lecture serves as counterpoint between students\325)J
60 225 :M
.756 .076(impoverished past experience and the yet-to-be seen, but allegedly real)J
60 243 :M
1.133 .113(mathematical practice. What properties of real mathematical practice are)J
60 261 :M
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60 279 :M
.896 .09(say that mathematics \322gets done\323?)J
60 309 :M
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1.066 .107(Immediately after the mock lecture, Schoenfeld presented a heuristic-based)J
60 357 :M
.932 .093(solution that \322serves as a window into the practice of mathematicians.\323 He began)J
60 375 :M
.776 .078(by declaring that mathematicians do not solve problems by recalling algebraic)J
60 393 :M
.867 .087(identities, but by applying well-known problem solving strategies:)J
96 429 :M
.872 .087(Now, if I called up any member of the math department at four in the)J
96 447 :M
.729 .073(morning and said, \322Hey, your house is on fire but before you leave, what\325s)J
96 465 :M
.73 .073(the sum of this series?\323 they could tell me because it\325s part of the)J
96 483 :M
1.254 .125(mathematicians\325 collective unconscious. If I gave them a slightly more)J
96 501 :M
.809 .081(complicated problem, being mathematicians, they\325d probably stop to solve it)J
96 519 :M
.801 .08(before they ran out of their house anyway\321they\325re a little weird that)J
96 537 :M
.736 .074(way\321and what they\325d do is )J
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.194(not)A
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96 555 :M
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96 573 :M
.728 .073(known and quite comfortable strategy to all of them, which is [begins)J
96 591 :M
.753 .075(writing] long-winded, but it\325s worth writing down.)J
96 609 :M
.593 .059([Writes as he speaks; the italicized text below is what also is written.])J
96 627 :M
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2.307 .231(If you\325ve got a problem you need to make sense of and it has an\321)J
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1.011(jargon)A
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2.716 .272(coming up!\321)J
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1.937 .194(integer parameter n,)J
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1.206 .121( that is something that takes on values . . .)J
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.797 .08(\(whole number values, and I\325ll be explicit about what that is in a minute\) . . .)J
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1.017 .102(try values of n = 1, 2, 3, 4, 5, . . . and see if you can find a pattern. )J
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2.814 .281(That pattern)J
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.54 .054(to make it pretty\) [he draws a box around what he\325s just written] and)J
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.637 .064(you can find a pattern.)J
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.47 .047(Heuristic strategy)J
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.347(short.)A
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.8 .08(this problem. \322This [problem] asks for the sum of )J
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.791 .079( terms, and you can ask: What\325s)J
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.763 .076(the sum of the first one, the first two, the first three etc.\323 He wrote the first four)J
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1.035 .104(partial sums and retrieved their results from the class,)J
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.586 .059(and suggested that the next sum would be )J
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1.325 .133(. With the emergent pattern in hand,)J
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.992 .099(he queried the class about verification.)J
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.975 .097( be deceiving, I mean this is pretty compelling evidence. But)J
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1 .1(where a compelling pattern doesn\325t necessarily come true.)J
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.832 .083(confirm that the pattern holds. How do I do that?)J
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2.277 .228(Math induction.)J
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.928 .093(Two features of this segment are worth comment. Schoenfeld\325s query to the class,)J
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.757 .076(\322Am I done?\323 was his first real question, a serious request for students\325)J
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1.648 .165(contributions to the solution.)J
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.833 .083( He invited students to collaborate, modeled a)J
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.793 .079(not ends, and drew a sharp contrast with the haughty, one-sided nature of his)J
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1.132 .113(mock lecture. This question signaled that his own teaching, not the traditional)J
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1.017 .102(calculus instructor\325s, was beginning. But with this overture to students, he also)J
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.996 .1(explicitly indicated that he had sure command of the course content \(some)J
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.697 .07(patterns would deceive\) and a ready proof of the pattern of partial sums. In short,)J
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1.014 .101(he presented the use of heuristics, a key component of real mathematical activity,)J
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.798 .08(and began to engage the students as thinkers in the real practice of mathematics,)J
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.967 .097(without straying off the path to a complete solution.)J
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1.118 .112(Following Sasha\325s suggestion, Schoenfeld asked:)J
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.798 .08(How many of you guys feel comfortable with induction? [pause, students)J
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.765 .077(are not visible or audible on the videotape at this point] OK, let me ask it the)J
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.87 .087(other way: How many of you feel uncomfortable with induction? [pause])J
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.379 .038(OK, good. [erases blackboard panel, pushes it up] I\325m not going to spend too)J
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.875 .088(much time on it in the course but it will occasionally be a useful tool. So I\325ll)J
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-.045(, were not questions)A
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-.056(speed of their response\321not the content\321was an issue.)A
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.729 .073(give you an example of how it works here and if it turns out to be an issue,)J
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.851 .085(then come talk to me in my office hours and we\325ll worry about it then. OK?)J
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1.056 .106(increasing requests for students\325 contributions to the evolving proof by induction.)J
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1.041 .104(The general idea of mathematical induction was stated in the standard manner but)J
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.829 .083(also characterized as \322the mystical algebraic formulation.\323 \322If you'd like to show)J
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.679 .068(, then it must also be true for )J
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.89 .089(on any given step of the staircase, the second inductive hypothesis allowed)J
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1.112 .111(where )J
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.564 .056(. . . Now in and of itself, that statement may or may not do you any good. It)J
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.839 .084(simply says, if you\325ve managed to show that it\325s true for some value, then)J
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.682 .068(it\325s true for the next value as well. . . . The problem is getting on the staircase)J
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.491 .049(in the first place. . . . That\325s why you need the first part. . . . You show that if)J
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.733 .073(the statement is true for )J
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.573 .057( 1, I can get on the staircase. . . . So, that's the two)J
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.925 .092(parts to an inductive argument: first there's the place where you get on the)J
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.597 .06(staircase; second, you can climb one step at a time.)J
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1.076 .108(The sum of the telescoping series was then solved for the third time. Schoenfeld)J
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.919 .092( equation and simplifying the resulting algebra. He)J
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.95 .095(The same as [inaudible] but with one more term.)J
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.258 .026( = )J
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.238(k)A
f1_12 sf
.353 .035( + 1, \()J
f2_12 sf
.238(k)A
f1_12 sf
.468 .047( + 1\)\()J
f2_12 sf
.238(k)A
f1_12 sf
.578 .058( + 2\). OK.)J
132 231 :M
.801 .08(So what I want to show is: 1 over 1 times 2, plus\321and I'll write)J
132 249 :M
.857 .086(the next to last term\321to make life easy for myself. The last)J
132 267 :M
.699 .07(term is 1 over )J
f2_12 sf
.272(k)A
f1_12 sf
.597 .06( + 1 [times] )J
f2_12 sf
.272(k)A
f1_12 sf
.683 .068( + 2. The next to last term is going)J
132 285 :M
.58 .058(to be what? . . . What\325s the right-hand side going to be if )J
f2_12 sf
.259(n)A
f1_12 sf
.374 .037( is)J
132 303 :M
.922 .092(equal to )J
f2_12 sf
.312(k)A
f1_12 sf
.601 .06( + 1?)J
55 307 72 18 rC
67 321 :M
1.299 .13(A student:)J
gR
gS 0 0 552 730 rC
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f2_12 sf
.352(k)A
f1_12 sf
.573 .057( + 1 over . . .)J
60 351 :M
1.134 .113(The solution of the telescoping series came to an end with Schoenfeld\325s transition)J
60 369 :M
.825 .082(back to the more general application context for Try Specific Values:)J
96 405 :M
.955 .096(When you see an )J
f2_12 sf
.381(n)A
f1_12 sf
1.028 .103(, sometimes it\325ll be explicit as it is here, sometimes it\325ll)J
96 423 :M
.608 .061(be implicit, you just look at it and say to yourself, \322Hey, it\325s )J
f2_12 sf
.17(really)A
f1_12 sf
.733 .073( a problem)J
96 441 :M
.971 .097(that has different values for )J
f2_12 sf
.347(n)A
f1_12 sf
.75 .075( equals 1, 2, 3, 4, 5, even though there\325s no )J
f2_12 sf
.419 .042(n )J
f1_12 sf
.544(in)A
96 459 :M
1.037 .104(the problem formulation.\323 Then, if you need to make progress on it, it often)J
96 477 :M
.814 .081(helps to look for systematic patterns.)J
260 507 :M
f0_12 sf
-.068(Discussion)A
60 537 :M
f1_12 sf
1.138 .114(Our characterization of the telescoping series problem has interwoven two levels)J
60 555 :M
.99 .099(of analysis of Schoenfeld\325s teaching. First, we have emphasized the many)J
60 573 :M
1.079 .108(introductions he makes in this first session: to his own teaching as different from)J
60 591 :M
.958 .096(other, content-oriented, classes; to heuristic strategies as a crucial component of)J
60 609 :M
.639 .064(problem solving; and to his mastery and expertise both as a teacher and as a)J
60 627 :M
.993 .099(mathematician. But none of these features is surprising, since they are described in)J
60 645 :M
.672 .067(Schoenfeld\325s own written accounts of the course \(1983; 1985; 1991; 1994\).)J
60 675 :M
1.374 .137(We believe our contribution consists in providing another level of analysis. We)J
60 693 :M
.982 .098(have used the distinction between \322doing\323 and \322presenting\323 to capture the more)J
endp
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1.324 .132(fundamental connection between his teaching and the practice of mathematicians)J
60 69 :M
1.099 .11(that underlies those introductions. Figure 1 presents a model of Schoenfeld\325s)J
60 87 :M
.954 .095(teaching with the telescoping series as relationships between doing and presenting)J
60 105 :M
.397(mathematics.)A
60 135 490 295 rC
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:j
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.855 .085(Figure 1)J
142 462 :M
.834 .083(Schoenfeld\325s pedagogy as doing and presenting mathematics)J
60 492 :M
1.183 .118(At the top level, the model distinguishes the practice of mathematicians)J
60 510 :M
1.187 .119(\(professional \322doing\323\) from teaching. Despite Schoenfeld\325s intentions to construct)J
60 528 :M
1.078 .108(a classroom community to parallel professional mathematical practice, he was a)J
60 546 :M
.757 .076(\322messenger\323 who modeled professional practice for his students, probably because)J
60 564 :M
.956 .096(he thinks that at the beginning, modeling is a way to help that community to)J
60 582 :M
.267(emerge.)A
f2_12 sf
.121 .012( )J
f1_12 sf
1.008 .101(In contrast to that professional \322doing,\323 Schoenfeld\325s teaching in this)J
60 600 :M
1.534 .153(segment of the class involved two forms of \322presenting\323: )J
f2_12 sf
.322(reflective)A
f1_12 sf
.779 .078( and )J
f2_12 sf
.342(rhetorical)A
f1_12 sf
(.)S
60 618 :M
.848 .085(We characterize his mock lecture as a rhetorical presentation because it was a)J
60 636 :M
1.025 .102(skillful performance that he enacted alone to encourage students to follow his lead)J
60 654 :M
.988 .099(in developing contrasting forms of classroom dialogue and practice. His reflective)J
60 672 :M
.976 .098(presentations, as exemplified in his discussions of Try Specific Values and)J
60 690 :M
1.214 .121(mathematical induction, were more frequent and characteristic of his teaching)J
endp
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1.204 .12(throughout the course. They involved his modeling of significant parts of skilled)J
60 69 :M
1.235 .123(problem solving including both knowledge and decision-making and his)J
60 87 :M
1.257 .126(solicitation of student contributions within these goal-directed activities.)J
60 117 :M
1.271 .127(We consider them as \322reflective\323 for different reasons. Some presentations)J
60 135 :M
.896 .09(introduced new knowledge \(e.g., Try Specific Values\) in particular problem)J
60 153 :M
1.145 .115(contexts and highlighted for students the importance of thinking when and where)J
60 171 :M
.885 .088(that knowledge can be used again \(recall Schoenfeld\325s closing comments on Try)J
60 189 :M
1.495 .15(Specific Values\). The reflection here involved the relationship between)J
60 207 :M
.855 .086(knowledge and the \(problem\) contexts where that knowledge is applied. In other)J
60 225 :M
.757 .076(cases, he modeled specific actions of a skilled problem solver, e.g., control)J
60 243 :M
1.155 .115(questions such as \322Am I done?\323 Reflection there involved an awareness of the)J
60 261 :M
1.062 .106(relation between the state of a person\325s evolving solution and the problem context.)J
60 279 :M
.971 .097(All such presentations contained \(1\) an introduction to a necessary \322tool of the)J
60 297 :M
.72 .072(trade\323 in one context, and \(2\) pointers to how knowledge and skill relate to a wider)J
60 315 :M
.735 .073(range of contexts. The intent of these presentations was to help students carry away)J
60 333 :M
1.099 .11(from the course a different form of classroom mathematical practice.)J
60 363 :M
.902 .09(Because they were the places where Schoenfeld presents new knowledge and skill)J
60 381 :M
1.131 .113(to students, it is important to contrast reflective presentations with simpler forms)J
60 399 :M
1.294 .129(of \322transmission\323 teaching common to college and precollege instruction in)J
60 417 :M
1.301 .13(mathematics. Both involve teachers\325 display of mathematics knowledge and skill)J
60 435 :M
.789 .079(for students who are understood to lack that knowledge and need it to make)J
60 453 :M
1.214 .121(further progress. We see elements of transmission in the introduction and use of)J
60 471 :M
1.311 .131(Try Specific Values and in the review of mathematical induction. Moreover, the)J
60 489 :M
1.152 .115(context and content of Schoenfeld\325s reflective presentations differed from)J
60 507 :M
.661 .066(traditional \322teaching by telling\323 as well as \322socratic teaching\323 in significant ways.)J
60 525 :M
1.11 .111(In emphasizing process in problem solving, he shifted the focus from)J
60 543 :M
2.344 .234(mathematical )J
f2_12 sf
1.565 .156(content )J
f1_12 sf
1.332 .133(to issues of how)J
f2_12 sf
.196 .02( )J
f1_12 sf
1.688 .169(mathematics is )J
f2_12 sf
1.287 .129(done. )J
f1_12 sf
1.766 .177(His solicitation of)J
60 561 :M
.827 .083(student contributions was a step toward fulfilling the expectation, stated during)J
60 579 :M
.934 .093(the first twenty minutes of the course, that students would soon take over and use)J
60 597 :M
.967 .097(the tools he presented without his assistance. In our view, it is the complex)J
60 615 :M
1.426 .143(interweaving of transmission and student participation in Schoenfeld\325s reflective)J
60 633 :M
1.016 .102(teaching combined with his explicit statements and illustrations of the goals of his)J
60 651 :M
.863 .086(actions that makes his course a good example of teaching towards sense-making.)J
60 681 :M
.93 .093(Finally, it is worth emphasizing that our analysis has not centered on the practice)J
60 699 :M
1.258 .126(of the mathematical community \(professional \322doing\323\), but on \322presenting\323 in the)J
endp
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.841 .084(classroom context. Indeed schools are unique and specialized contexts for)J
60 69 :M
1.318 .132(mathematical thinking. We think the analysis shows that the differences between)J
60 87 :M
1.353 .135(his classroom community and that of professional mathematicians lies not only in)J
60 105 :M
.868 .087(shared knowledge and skills, as Schoenfeld has suggested, but also in the )J
f2_12 sf
.275(nature)A
60 123 :M
.853 .085(and goals )J
f1_12 sf
1.168 .117(of the practices in these two contexts. His presentations of mathematical)J
60 141 :M
.99 .099(content and heuristics had clear instructional goals and employed rather tightly)J
60 159 :M
1.172 .117(controlled mechanisms for student participation. His intent was that students)J
60 177 :M
.974 .097(construct appropriate models of and beliefs about mathematical activity in the)J
60 195 :M
1.157 .116(professional community. But this teaching practice did not make his classroom)J
60 213 :M
1.18 .118(part of the professional mathematical community. Rather it helped to close the)J
60 231 :M
.804 .08(gap between the two,)J
f0_12 sf
.096 .01( )J
f1_12 sf
.808 .081(creating in students the sense of belonging and contributing)J
60 249 :M
1.269 .127(to a more authentic mathematical practice.)J
endp
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-.117(Making the case for heuristics:)A
146 95 :M
-.296(Authority and direction in the inscribed square)A
243 125 :M
f1_12 sf
1.271 .127(John P. Smith III)J
60 155 :M
1.295 .129(To teach his students to solve challenging problems, Schoenfeld must himself)J
60 173 :M
1.018 .102(solve a difficult instructional problem: how to introduce P\227lya-type heuristics so)J
60 191 :M
.706 .071(that students quickly appreciate their power and slowly learn to apply them)J
60 209 :M
.619 .062(productively across a wide range of problems.)J
60 239 :M
.86 .086(As his own past teaching and research has shown, this problem does not submit to)J
60 257 :M
.719 .072(easy solutions \(Schoenfeld, 1985; 1992a\). Students can struggle to see how and)J
60 275 :M
.665 .066(where to apply particular heuristic strategies because of their general character as)J
60 293 :M
.789 .079(\322rules of thumb.\323 Schoenfeld could address this part of the problem by presenting)J
60 311 :M
.873 .087(more specific versions of each strategy with clearer conditions of application. But if)J
60 329 :M
.93 .093(he did, the list of useful heuristics would become too long and cumbersome to)J
60 347 :M
.749 .075(teach and learn \(the \322specificity\323 problem\). So the generality of strategies \(and their)J
60 365 :M
.957 .096(attendant vagueness\) must be retained. Given the generality of these strategies,)J
60 383 :M
1.035 .104(students must be thoughtful in selecting, applying, and evaluating them, but such)J
60 401 :M
1.075 .107(thoughtfulness is difficult to teach. How, for example, should students evaluate)J
60 419 :M
1.278 .128(their work so that they avoid committing too much time to unproductive)J
60 437 :M
.876 .088(approaches? Even if students select productive heuristics, applying those strategies)J
60 455 :M
1.177 .118(usually involves many steps, and mistakes at any one point can undermine the)J
60 473 :M
1.365 .137(entire effort \(the \322implementation\323 problem\). Finally, the skillful use of heuristics)J
60 491 :M
.855 .085(is not neutral with respect to content knowledge. If students do not know or)J
60 509 :M
1.03 .103(cannot recall the necessary mathematical concepts or procedures, even workable)J
60 527 :M
1.103 .11(solution plans can fail \(the \322resource\323 problem\).)J
60 557 :M
.936 .094(Many curricular approaches to problem solving fail to take these problems)J
60 575 :M
.871 .087(seriously. It is common, especially at the pre-college level, either to cast problem)J
60 593 :M
.973 .097(solving as recreation\321a separate activity from the \322real\323 task of learning)J
60 611 :M
.595 .059(procedures \(e.g., Cooney, 1985\)\321or to teach students to practice and master each)J
60 629 :M
.699 .07(strategy separately. These efforts, as Schoenfeld \(1992a\) has argued, fundamentally)J
60 647 :M
1.547 .155(miss the mark.)J
96 683 :M
.827 .083(Problem solving in the spirit of P\227lya is learning to grapple with new and)J
96 701 :M
1.413 .141(unfamiliar tasks when the relevant solution methods \(even if only partially)J
endp
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1.232 .123(mastered\) are not known. When students are drilled in solution procedures)J
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.171 .017(. . . )J
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.624 .062(, they are not developing the broad set of skills P\227lya and other)J
96 87 :M
1.576 .158(mathematicians who cherish mathematical thinking have in mind. \(p. 354\))J
60 117 :M
1.276 .128(One central element in teaching problem solving is identified here: Students must)J
60 135 :M
.649 .065(regularly work on real problems, not \322exercises\323 that are clearly tied to standard)J
60 153 :M
1.034 .103(procedures or methods. But even with such problems in hand, how then can you)J
60 171 :M
1.281 .128(teach problem solving, introduce and highlight heuristics as important \322content,\323)J
60 189 :M
1.089 .109(and avoid the pitfalls identified above?)J
60 219 :M
1.052 .105(In this section we analyze one important step Schoenfeld made toward solving)J
60 237 :M
1.034 .103(this problem, using the solution and discussion of the challenging inscribed square)J
60 255 :M
1.029 .103(problem as data. Our main claim is that Schoenfeld\325s approach involved leading)J
60 273 :M
1.091 .109(the class through the solution in a carefully planned and directive manner. In so)J
60 291 :M
.838 .084(doing, he acted in accord with traditional classroom norms \(e.g., the teacher is the)J
60 309 :M
1.404 .14(mathematical authority\) that he aimed to undermine and change. Though his)J
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1.119 .112(students eventually chose their own approaches to problems and evaluated their)J
60 345 :M
.858 .086(attempts and solutions \(and those of their peers\), they were shown their way)J
60 363 :M
.999 .1(through these issues on this particular problem. Schoenfeld\325s choice to play the)J
60 381 :M
.715 .072(strong leader and director indicates that his actions and local goals early in the)J
60 399 :M
1.312 .131(course did not map onto his long-term intentions and achievements in any)J
60 417 :M
.691 .069(simple way. Getting the problem solving class \322off the ground\323 was a quite)J
60 435 :M
1.015 .102(different task than teaching it in its mature, stable form\321the stage emphasized in)J
60 453 :M
.694 .069(his written accounts \(Schoenfeld, 1988; 1989; 1991; 1994\).)J
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.54 .054(The problem and two relevant heuristics)J
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.709 .071(The task of inscribing a square in an arbitrary triangle was second on the problem)J
60 531 :M
.688 .069(sheet after the telescoping series. Its wording and accompanying diagrams are)J
60 549 :M
.542 .054(reproduced below \(Figure 2\).)J
96 585 :M
.965 .097(You are given the triangle on the left in the figure below. A friend of mine)J
96 603 :M
.877 .088(claims that he can inscribe a square in the triangle\321that is, that he can find)J
96 621 :M
.896 .09(a construction that results in a square, all four of whose corners lie on the)J
96 639 :M
.959 .096(sides of the triangle. Is there such a construction\321or)J
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.126 .013( )J
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1.094 .109(might it be impossible?)J
96 657 :M
.797 .08(Do you know for certain that there's an inscribed square? Do you know for)J
96 675 :M
1.148 .115(certain there's a construction that will produce it?)J
endp
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.822 .082(Is there anything special about the triangle you were given? That is, suppose)J
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.937 .094(you did find a construction. Will it work for all triangles, or only some?)J
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.855 .085(Figure 2)J
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1.223 .122(Statement of Problem 2:)J
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.84 .084(Inscribing a Square in an Arbitrary Triangle)J
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.952 .095(The questions posed in the problem statement raise the issues of existence and)J
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.782 .078(construction. First, there is the problem of showing that a square can be inscribed)J
60 312 :M
.91 .091(in the given triangle. But an existence proof may not necessarily lead to the)J
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.801 .08(construction of the square, so the question of whether the inscribed square can be)J
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1.178 .118(constructed using Euclidean ruler and compass techniques remains. The)J
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.964 .096(existence/construction distinction influenced the class\325s work in two ways.)J
60 384 :M
.868 .087(Students struggled at first to understand the problem statement, and a major part)J
60 402 :M
1 .1(of their confusion was their difficulty in separating these two issues. Schoenfeld)J
60 420 :M
.898 .09(also used the distinction to structure the discussion. He drew on his knowledge)J
60 438 :M
1.173 .117(that some existence arguments generate constructions more easily than others to)J
60 456 :M
.67 .067(support the students\325 progress toward a solution \(see Schoenfeld, 1985, pp. 84\32091)J
60 474 :M
.881 .088(for his analysis of the problem\).)J
60 504 :M
.96 .096(In contrast to the preceding problem \(summing the telescoping series\) and the)J
60 522 :M
.516 .052(subsequent one \(the 3 by 3 magic square\), the class found both parts of the inscribed)J
60 540 :M
.829 .083(square challenging. Much of the students\325 work prior to specific suggestions from)J
60 558 :M
.973 .097(their teacher was devoted simply to understanding the problem. The difficulty)J
60 576 :M
.808 .081(they experienced in getting started provided Schoenfeld with an early context for)J
60 594 :M
1.254 .125(demonstrating the power of heuristics in solving problems.)J
60 624 :M
.883 .088(Two related heuristics, both attributed to P\227lya, were introduced in solving this)J
60 642 :M
2.167 .217(problem. The very general strategy, )J
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1.584 .158(Look for a related problem that is easier to)J
60 660 :M
2.381 .238(solve and try to exploit its solution to solve the original problem)J
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2.25 .225(, was the first.)J
60 678 :M
1.057 .106(Schoenfeld\325s presentation of it was cautionary, if not somewhat negative. A)J
60 696 :M
.794 .079(number of \322easier, related\323 problems were generated and found to be either hard)J
endp
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.96 .096(to solve or difficult to exploit to solve the original problem. These cautions set up)J
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2.589 .259(the second heuristic, )J
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2.014 .201(If there is a special condition in the problem, relax that)J
60 87 :M
3.193 .319(condition and look for the desired solution in the resulting family of solutions.)J
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.804 .08(Two \322special conditions\323 are embedded in this problem: \(1\) that the inscribed)J
60 123 :M
.668 .067(figure must be a square \(a rectangle is easier\) and \(2\) that it must have all four)J
60 141 :M
1.173 .117(vertices on the triangle \(three vertices are easier\). Relaxation of either condition)J
60 159 :M
.587 .059(can \(and did\) produce an existence proof.)J
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.905 .091(The classroom solution)J
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.341(Overview)A
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1.204 .12(. Schoenfeld\325s work with this problem can be divided into four phases,)J
60 237 :M
1.117 .112(spanning some 70 minutes of class time. Some students worked on the problem)J
60 255 :M
.847 .085(during the first 20 minutes of group work prior to the discussion of the telescoping)J
60 273 :M
1.255 .125(series solution. Immediately following that discussion, Schoenfeld directed the)J
60 291 :M
.751 .075(class, again in groups, toward the inscribed square problem with the strong \322hint\323)J
60 309 :M
.979 .098(to Solve An Easier Related Problem)J
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.119 .012( )J
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.888 .089(\(Phase 1\). When he called the groups back)J
60 327 :M
.71 .071(together to discuss their progress \(Phase 2\), three different related problems were)J
60 345 :M
.519 .052(suggested, two by students, one by Schoenfeld. They were considered and)J
60 363 :M
1.135 .114(ultimately rejected. The second heuristic, Relax a Condition, was then introduced)J
60 381 :M
.886 .089(and applied in Phase 3. The two conditions were identified, and Schoenfeld used)J
60 399 :M
.721 .072(each to produce an existence proof. At the end of the first session, he sent the class)J
60 417 :M
.962 .096(away with some general directions to investigate one of the existence proofs more)J
60 435 :M
.867 .087(closely. The second class session opened with Devon\325s constructive proof \(Phase 4\).)J
60 453 :M
1.087 .109(Schoenfeld used his solution as context to state some features of the mathematical)J
60 471 :M
1.615 .161(community he desired.)J
60 501 :M
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2.885 .289(Clarifying the problem and presenting Solve an Easier Related Problem \(Phase 1\))J
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60 519 :M
1.093 .109(When the discussion of the telescoping series problem ended, Schoenfeld directed)J
60 537 :M
.595 .059(the class to the inscribed square.)J
96 573 :M
.878 .088(This [telescoping series] is a fairly straightforward example. We\325ll encounter)J
96 591 :M
.78 .078(a lot more later in the class that are not so straightforward. What I want to)J
96 609 :M
.694 .069(do to nudge you in the direction of a solution to problem 2, but not get you)J
96 627 :M
.892 .089(far enough yet most likely, is mention a second strategy and have you think)J
96 645 :M
.957 .096(about problem 2 a little bit. Which is: [writes and speaks] )J
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.938 .094(If you can\325t solve)J
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3.003 .3(the given problem, try to solve an easier related problem and then exploit)J
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1.936 .194(your solution.)J
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1.369 .137( That\325s a statement that\325s almost verbatim what it comes out)J
96 699 :M
.817 .082(of\321from P\227lya. See if you can use that to solve problem 2.)J
endp
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1.097 .11(The arrangement of the class at this point\321still in small groups\321is reproduced)J
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.566 .057(below \(Figure 3\).)J
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.855 .085(Figure 3)J
191 289 :M
1.202 .12(Arrangement of the Class in Small Groups)J
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.831 .083(Session 1: Inscribed Square)J
60 337 :M
1.143 .114(The class did not immediately move to this task. Snippets of conversation from)J
60 355 :M
.805 .081(the four groups indicate that each returned to discuss the telescoping series)J
60 373 :M
1.032 .103(solution, solve the series problem that was given just after the telescoping series)J
60 391 :M
.291(problem,)A
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1.067 .107( or discuss the principle of mathematical induction. About halfway into)J
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.959 .096(these 20 minutes, one group after another turned to the inscribed square problem.)J
60 427 :M
1.098 .11(Schoenfeld worked his way around the room, discussing the problem statement)J
60 445 :M
.82 .082(with each of the four groups. Members of two groups questioned him about what)J
60 463 :M
.847 .085(exactly the problem was asking. His responses emphasized the distinction between)J
60 481 :M
.989 .099(existence and construction. These quick \322check-ins\323 not only assisted students in)J
60 499 :M
.981 .098(understanding the problem but allowed Schoenfeld to observe the work of each)J
60 517 :M
.733 .073(group and see which particular \322easier related\323 problems they generated.)J
60 547 :M
1.028 .103(In opening the whole group discussion, Schoenfeld declared his expectation that)J
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.82 .082(students would come to the board in the next class session and present their work)J
60 583 :M
1.026 .103(on problems. He then focused their attention on the inscribed square problem)J
60 601 :M
.953 .095(with a question, \322What does problem 2 tell you to do?\323 When there was no)J
60 619 :M
.911 .091(immediate response from the class, he explained the two parts of the problem and)J
60 637 :M
.985 .098(used the example of an angle bisector to distinguish existence from construction.)J
60 655 :M
.677 .068(The existence of the angle bisector was established by considering all rays interior)J
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.972 .097(to the angle whose endpoints are the vertex and using a continuity argument. He)J
60 69 :M
1.008 .101(then drew students\325 attention back to Solve an Easier Related Problem, reminded)J
60 87 :M
1.039 .104(them that they would be reading P\227lya\325s discussion of it in )J
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.943 .094(How to Solve It)J
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.608 .061( \(a)J
60 105 :M
.95 .095(supplemental reading for the course\). But his stance was cautionary, \322... remember)J
60 123 :M
.591 .059(I said lots of people thought P\227lya didn\325t quite work and this is an example of why.)J
60 141 :M
1.009 .101(We need to push him a little bit further.\323 Even as he promoted heuristics, he)J
60 159 :M
.95 .095(hinted at difficulties in using them, in this case the specificity problem.)J
60 189 :M
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2.343 .234(Evaluating Solve an Easier Related Problem \(phase 2\))J
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2.44 .244(. When Schoenfeld asked a)J
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.847 .085(more specific question, \322What easier related problems did people try? I\325m)J
60 225 :M
.878 .088(curious,\323 three students responded. For each suggestion, Schoenfeld gave a quick)J
60 243 :M
.78 .078(verbal restatement and wrote the suggested approach on the left-hand blackboard)J
60 261 :M
1.142 .114(\(leaving the center board empty and available\). We reproduce the major elements)J
60 279 :M
.81 .081(of this dialogue below because it is crucial to understanding and interpreting his)J
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.123(approach.)A
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.765 .077(Relax the constraint on the square and try a rectangle.)J
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.74 .074(OK, let me get to that in a minute [chuckles]. OK, um. So the)J
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.673 .067(problem says, stick a square inside the triangle and each corner)J
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.824 .082(is on the triangle, one easier related problem is:)J
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.801 .08(Don\325t go for the)J
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.46 .046(square, go for a rectangle. [Writes \3221. Try a rectangle instead\323)J
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.962 .096(under the list heading \322Related Problems.\323])J
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.692 .069(What else did people try? I saw people doing different things,)J
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.627 .063(so I know that you tried.)J
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.605 .061(I tried to disprove it for an arbitrary triangle.)J
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.796 .08(OK. Try to find a counterexample. So, this is if you don't)J
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.581 .058(believe it\325s true, that\325s not part of the strategy but, [Writes as he)J
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.683 .068(speaks, \322Look for a specific counterexample,\323 well below)J
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.714 .071(suggestion #1] And that\325s a generally useful thing to do. [Some)J
132 567 :M
.943 .094(of AS\325s short commentary on the benefits of looking for specific)J
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1.082 .108(examples and counterexamples is omitted here.] Other things)J
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.708 .071(that people tried? Yup? [responding to student])J
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.835 .083(A circle. [This student was not visible on the videotape.])J
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.697 .07(OK. The problem was, stick in a square inside a triangle. An)J
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.494 .049(easier related problem might be, [writes as he speaks, \322Try a)J
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.723 .072(circle,\323 just below suggestion #1]. Other things that people)J
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.795 .08(tried? Yeah? [responding to Sasha])J
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.752 .075([AS appeals to the student who made the suggestion and)J
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.649 .065(clarifies that suggestion #2 was to inscribe a circle in a triangle.)J
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.87 .087(states it is not useful for solving the problem.])J
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1.213 .121(Other things people might have tried?)J
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.852 .085(I can mention at least one more that I thought I saw people)J
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.887 .089(doing, and that I've certainly seen before. Instead of making an)J
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.729 .073(arbitrary triangle, make a special kind of triangle, try either the)J
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.592 .059(isosceles or equilateral triangles. [He writes, \3223. Instead of an)J
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.7 .07(arbitrary triangle, try special triangles\321isosceles, equilateral.\323])J
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.835 .083(Let me leave number 1 alone for a short while. I'll get back to)J
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.658 .066(that and a couple of others, and talk about the general process)J
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.719 .072(and illustrate it with numbers 2 and 3. \324Cause this is a general)J
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.788 .079(discussion of, what happens when you try to use the)J
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.135(suggestions.)A
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.725 .073(The final written list of suggested approaches to problem 2 is reproduced in Figure)J
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.336 .034(4 below.)J
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.468 .047(1. Try a rectangle instead.)J
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.52 .052(2. Try a circle in a triangle.)J
96 481 :M
.672 .067(3. Instead of an arbitrary triangle, try special triangles\321isosceles, equilateral.)J
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.877 .088(Look for a specific counterexample.)J
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.855 .085(Figure 4)J
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1.286 .129( Schoenfeld\325s Restatements of Students\325)J
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.773 .077(Easier Related Problems for the Inscribed Square)J
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.903 .09(Two important teaching decisions are notable in this exchange; both are related to)J
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.932 .093(the task of structuring the discussion of Solve an Easier Related Problem and the)J
60 664 :M
.886 .089(problems it generated. First, Schoenfeld did not list the suggestions in the order in)J
60 682 :M
.913 .091(which they were given. The suggestion to look for a counterexample was listed)J
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.694 .069(below the others and separated from them by a squiggly line. The message seemed)J
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.607 .061(to be: this is a different sort of suggestion, and we should treat it differently.)J
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.915 .091(Second, he chose not to consider the suggestions in the order in which they were)J
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1.001 .1(given. At the end of the interchange, he declared his intention to \322discuss\323)J
60 105 :M
.577 .058(suggestions 2 and 3 before suggestion 1, though the latter was certainly a)J
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.774 .077(straightforward example of an \322easier related\323 problem. These choices suggest that)J
60 141 :M
1.005 .101(Schoenfeld had lessons he wanted to draw from this problem, that some student)J
60 159 :M
.848 .085(suggestions fit more easily with his plan than others, and that to draw out these)J
60 177 :M
.812 .081(lessons he needed to consider the students\325 suggestions in a particular order.)J
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.814 .081(To frame the discussion of the \322general process\323 of using related problems \(and)J
60 225 :M
1.026 .103(the attendant pitfalls\) he drew an application scheme for the second heuristic)J
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.638(on)A
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.685 .068(the board and identified a question relevant to each step. For the first step, if you)J
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.764 .076(can identify what seems to be an easier related problem, can you solve it? For the)J
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1.013 .101(second and perhaps more important step, does that solution help to solve the)J
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.892 .089(original problem? This scheme is reproduced in Figure 5 as he drew it on the)J
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-.051(board.)A
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1 G
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2.18 .218(you're here)J
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.974 .097(easier related)J
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.34(problem)A
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2.123 .212(solution to)J
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1.852 .185(original problem)J
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.855 .085(Figure 5)J
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1.891 .189(Schoenfeld\325s Application Scheme)J
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1.102 .11(for Solve an Easier Related Problem)J
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.829 .083(With this general frame before the class, he sketched a circle inscribed in a triangle,)J
60 529 :M
.881 .088(stated that the construction could be done \(referring again to work in high school)J
60 547 :M
.738 .074(geometry\), but declared that it could not be used to solve the original problem.)J
96 583 :M
.616 .062(So for that particular problem, this part is easy [tracing the arrow between)J
96 601 :M
.675 .067(the \322you\325re here\323 and the \322easier related problem\323 box] at least at the level)J
96 619 :M
.94 .094(of yes, you can take that step. . . . but )J
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1.587 .159(I\325ve never found anyone)J
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1.356 .136( who was)J
96 637 :M
.727 .073(actually able to take that particular thing, go from having a circle inscribed)J
96 655 :M
.609 .061(in a triangle and be able to use that to inscribe a square in a triangle. So the)J
96 673 :M
.766 .077(problem is that you can spend a fair amount of effort getting here [pointing)J
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1.476 .148(to the \322easier related problem\323 box] and then )J
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1.089 .109(I know of no way to get there)J
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.845 .084([pointing to the \322solution to original problem\323 box]. So that\325s an example of)J
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.564 .056(a stepping stone that only doesn\325t do you too much good because it only gets)J
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.455 .046(you halfway there. [emphasis added])J
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.862 .086(His treatment of the two special triangles which followed was similarly brief.)J
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.804 .08(Instead of drawing either an isosceles or equilateral triangle on the board, he)J
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.58 .058(simply stated that both possibilities fail by both criteria, easier and related.)J
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.798 .08(It looks like it should be easier to inscribe a, square in something nice and)J
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.832 .083(regular like an isosceles triangle instead of a random triangle, or maybe)J
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1.994 .199(even equilateral, but it turns out )J
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1.971 .197(I don't know of anyone who has actually)J
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2.454 .245(managed to do that in an easy way, and I don't know of anyone who\325s been)J
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2.602 .26(able to show how you can go from a solution of that to the general solution)J
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.035 .003([emphasis added])J
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1.184 .118(From these illustrations of the problematic nature of Solve an Easier Related)J
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1.111 .111(Problem, Schoenfeld stated his general point, \322When you\325re working on a)J
60 345 :M
1.025 .102(complicated problem that involves using a stepping stone, you want to think both)J
60 363 :M
.772 .077(about getting to the stepping stone and whether or not you can get from there on.\323)J
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.882 .088(The stage was now set for reformulating this heuristic in a more specific and)J
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2.246 .225(deterministic form.)J
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1.819 .182(Presenting Relax a Condition \(phase 3\))J
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1.709 .171(. Declaring his intention to help students be)J
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.96 .096(more specific about how they might generate easier related problems, Schoenfeld)J
60 465 :M
.827 .083(introduced the third heuristic of the day, \322a more elaborate version of P\227lya\325s)J
60 483 :M
.612 .061(strategy,\323 alternately speaking and writing on the board.)J
96 519 :M
.705 .071(Suppose the problem asks for something, that\325s what I mean by a specific)J
96 537 :M
.814 .081(condition in the problem you want, P\227lya says relax the condition, ask for)J
96 555 :M
.921 .092(something, ask for less. Since you\325re less demanding, there ought to be more)J
96 573 :M
.861 .086(solutions. There could be a whole family of them. So if you get a whole)J
96 591 :M
.822 .082(family of them, maybe you\325ll find the one you want in that family. [His)J
96 609 :M
.883 .088(written statement reads, \322If there is a special condition in the problem you)J
96 627 :M
1.011 .101(want, relax the condition\321ask for less. Since you\325re less demanding, there)J
96 645 :M
.998 .1(should be a whole )J
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1.079 .108( of solutions. Look for the one you want among)J
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.986 .099(them.\323] [his emphasis])J
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1.02 .102(Turning then to the issue of specific conditions, he asked, \322What does the problem)J
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.956 .096(ask you for?\323 A combination of student suggestions and Schoenfeld\325s)J
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.933 .093(interpretation produced two conditions which were also written on the board: \(1\))J
60 105 :M
.672 .067(the desired figure was a square and \(2\) all four of its vertices must lie on the)J
60 123 :M
.08(square.)A
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.703 .07(He chose to tackle the first condition, asking the class what was easier than a)J
60 171 :M
.797 .08(square. A student \(invisible to the camera\) responded quickly, \322A rectangle.\323)J
60 189 :M
.941 .094(When Schoenfeld asked how a rectangle could be inscribed in the triangle he had)J
60 207 :M
.99 .099(drawn on the board, Devon began to outline a construction using three)J
60 225 :M
.959 .096(perpendicular segments. Schoenfeld accepted and completed his procedure,)J
60 243 :M
.709 .071(quickly producing three different rectangles, including a \322short and fat\323 and a \322tall)J
60 261 :M
.709 .071(and skinny\323 example, as reproduced below in Figure 6.)J
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.855 .085(Figure 6)J
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1.132 .113(Three Inscribed Rectangles)J
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1.319 .132(From these three examples, Schoenfeld completed the continuity argument for the)J
60 463 :M
.769 .077(existence of the inscribed square: if the short, fat rectangle were transformed)J
60 481 :M
1.039 .104(continuously into the tall, skinny one, that process must generate a rectangle with)J
60 499 :M
.569 .057(equal base and height \(i.e., a square\) somewhere along the way. He added that this)J
60 517 :M
.817 .082(was \322actually the same continuity argument that I used before for the angle)J
60 535 :M
.892 .089(bisector.\323 But with this existence proof in hand, he then denied the possibility of)J
60 553 :M
1.037 .104(elaborating it into a constructive proof.)J
96 589 :M
1.779 .178(Now that\325s a nice existence proof. )J
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1.418 .142(I don't know how to turn that into a)J
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2.194 .219(constructive proof.)J
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1.412 .141( So that\325s actually argument number one, that\325s part of)J
96 625 :M
1.302 .13(the problem but not the whole problem and to this day, )J
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1.363 .136(I don't know how)J
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.634 .063(to take that nice, little existence proof and say, \322Yeah, you can use that and,)J
96 661 :M
.628 .063(out of that here is a sequence of things you can do with straightedge and)J
96 679 :M
.275 .027(compass.\323 [emphasis added])J
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.995 .1(He then turned to the second condition and asked for volunteers to generate)J
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.896 .09(squares with three vertices on the triangle that he\325d drawn on the board. Three)J
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.892 .089(students, Sasha, Devon, and Stephen, came to the board, each producing an)J
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.816 .082(\322inscribed\323 square of different size and orientation \(Figure 7\).)J
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.855 .085(Figure 7)J
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.9 .09(Three Students\325 Partially Inscribed Squares)J
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.877 .088(Stephen\325s construction assumed the top angle of the triangle was a right angle.)J
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.827 .083(After noting this flaw, Schoenfeld erased that square, leaving two. He asked once)J
60 339 :M
1.201 .12(more for other examples and, hearing no volunteers, noted his surprise that no)J
60 357 :M
.831 .083(one drew a square in the opposite corner to Devon\325s, because that \322normally)J
60 375 :M
.909 .091(happens.\323 He then drew increasingly larger squares with the same orientation as)J
60 393 :M
.783 .078(Devon\325s \(see Figure 8\) and asked, \322What you can tell me about that family, what)J
60 411 :M
1.118 .112(happens to the fourth corners?\323 Sharon responded, \322They're given any range of)J
60 429 :M
1.01 .101(sizes and then when you finally meet a distance where the, where the fourth one,)J
60 447 :M
1.042 .104(where the fourth vertice meets the triangle, you have four equal sides.\323)J
60 477 :M
.824 .082(Schoenfeld drew a squiggly locus connecting the fourth vertices of the squares)J
60 495 :M
.945 .094(\(Figure 8\) and restated Sharon\325s response in terms of the intersection of the locus)J
60 513 :M
1.126 .113(and the triangle, thus completing the second existence argument.)J
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.855 .085(Figure 8)J
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.86 .086(The Squiggly Locus Connecting the Fourth Vertices of \322Inscribed\323 Squares)J
endp
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.748 .075(With the end of class approaching, he directed the class to \322play with this example)J
60 69 :M
.849 .085(for Wednesday and see what you discover.\323 This \322assignment\323 seemed a clear)J
60 87 :M
1.021 .102(indication that the second existence argument was more likely to generate the)J
60 105 :M
1.439 .144(missing construction than the first.)J
60 135 :M
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1.25 .125(. At the start of the next class two days later)J
60 153 :M
.802 .08(Schoenfeld pointed the class back to the problem, \322We left with problem 2 partly)J
60 171 :M
.762 .076(solved and partly up in the air. Does anyone have anything to say about problem)J
60 189 :M
1.223 .122(2?\323 When Devon volunteered and came to the board to show his solution,)J
60 207 :M
.846 .085(Schoenfeld turned to the class and described some of his longer-term goals for the)J
60 225 :M
(class.)S
96 261 :M
.653 .065(One of things that I want to do during the course of the semester is get us)J
96 279 :M
1.279 .128(talking like a mathematical community and ultimately using the standards)J
96 297 :M
1.608 .161(of the mathematical community which means not like mumbling on the)J
96 315 :M
.638 .064(board, but instead being fairly clear, lucid, really making arguments clear so)J
96 333 :M
.687 .069(that all of us can understand precisely what\325s going on. So I\325m going to push)J
96 351 :M
1.122 .112(for those kind of standards in explanation, which means not just beautiful)J
96 369 :M
1.22 .122(finished products,)J
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.096 .01( )J
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.747 .075(but also explanations of how and why)J
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.096 .01( )J
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.904 .09(it\325s reasonable that)J
96 387 :M
.577 .058(you did what you did and things like that. OK?)J
60 417 :M
.925 .093(Figure 9 gives the location of the participants in the classroom at that point in)J
60 435 :M
1.454 .145(Session 2.)J
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.936 .094(Arrangement of the Class: Start of Session 2)J
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1.041 .104(Devon\325s Construction of the Inscribed Square)J
endp
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.753 .075(Devon\325s argument was based on his insight that the inscribed square could be)J
60 69 :M
.707 .071(produced by simply scaling another square up or down and therefore that the)J
60 87 :M
.815 .081(problem could be solved using similarity. He first explained how to construct a)J
60 105 :M
.938 .094(square with three vertices on the square and the fourth lying outside the triangle)J
60 123 :M
.679 .068(\(Figure 10, frame 1\).)J
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.807 .081(Figure 10)J
111 269 :M
1.059 .106(Devon\325s Drawings Illustrating the Inscribed Square Construction)J
60 299 :M
.945 .094(Then he drew the line from the far vertex of the triangle \(point E, in frame 2\))J
60 317 :M
1.05 .105(through the outside vertex of the square. \(This line is the accurate representation)J
60 335 :M
1.041 .104(of Schoenfeld's \322squiggly locus.\323\) The point of intersection of that line with the)J
60 353 :M
.821 .082(triangle \(point A' in frame 2\) is one vertex of the inscribed square, and the rest of)J
60 371 :M
.945 .094(the square A'B'D'C' can be constructed from that point by dropping perpendiculars)J
60 389 :M
.972 .097(to the other two sides of the triangle. Devon went to show how, via similar)J
60 407 :M
.683 .068(triangles \(e.g., \306 ABE ~ \306\312A'B'E'\), the fact that ABDC was a square guaranteed that)J
60 425 :M
1.986 .199(A'B'D'C' was also.)J
60 461 :M
1.171 .117(Throughout his presentation, Devon faced and addressed himself primarily to)J
60 479 :M
1.07 .107(Schoenfeld, not the class. Schoenfeld, still seated in the front corner of the room,)J
60 497 :M
1.017 .102(called students\325 attention to this phenomenon and suggested different standards)J
60 515 :M
1.106 .111(for their presentations in class.)J
96 551 :M
1.191 .119(A comment and a question: The comment is that what just happened in)J
96 569 :M
.626 .063(terms of [Devon\325s] behavior is exactly what happens on the second day of)J
96 587 :M
.753 .075(class every time I teach this course, which is that: I ask someone to do)J
96 605 :M
.828 .083(something at the board and he spends 90 percent of his time looking at me)J
96 623 :M
.639 .064(for approval. I\325m actually pretty good at playing poker and not revealing)J
96 641 :M
.748 .075(whether something is correct or not. I\325m going to do that a lot, because)J
96 659 :M
.876 .088(ultimately I don\325t )J
f2_12 sf
.276(want)A
f1_12 sf
.76 .076( to be the judge of what\325s right or wrong. The judge)J
96 677 :M
1.019 .102(of what\325s right or wrong in some sense is the mathematics and in another)J
96 695 :M
.724 .072(sense, it\325s the class. And what I want this to be is a community that develops)J
endp
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.853 .085(its own standards about mathematical correctness and it argues about when)J
96 69 :M
.944 .094(it buys an argument or not. So that\325s my comment. The question then is, did)J
96 87 :M
.728 .073(you guys buy what you\325ve seen? Is that sufficiently compelling that you all)J
96 105 :M
.93 .093(believe the construction that [Devon] suggested? [pause during which Mitch)J
96 123 :M
.446 .045(says, \322I\325m not really convinced it\325s a proof.\323] I saw three heads nodding [Jeff\325s)J
96 141 :M
.367 .037(was one], I saw a bunch that didn\325t react.)J
60 171 :M
.805 .08(Perhaps in response to Mitch\325s skepticism, Devon said that he \322could argue for it.\323)J
60 189 :M
.913 .091(His restatement added more detail, e.g., \306 ABE ~ \306\312A'B'E' and \306 ACE ~ \306\312A'C'E')J
60 207 :M
1.369 .137(guaranteed that A'C' = A'B', but included no major changes. When he finished,)J
60 225 :M
.927 .093(Schoenfeld again asked the class if they were convinced, and this time no one)J
60 243 :M
.838 .084(spoke up. Devon went back to his seat, and Schoenfeld returned to the center)J
60 261 :M
.912 .091(board and explained how he would generally evaluate students\325 presentations. His)J
60 279 :M
1.184 .118(evaluation of Devon\325s argument was different and explicitly positive, though with)J
60 297 :M
.232(qualifications.)A
96 333 :M
.804 .08(One of the things that I\325m going to do throughout the period of the course is)J
96 351 :M
.811 .081(ask nasty questions. Some of the times when I ask a nasty question that will)J
96 369 :M
.718 .072(indeed be true that it turns out that will be the case and that will knock your)J
96 387 :M
1.039 .104(argument apart. Some of the times it turns out that your argument\325s right)J
96 405 :M
.915 .092(and I\325m just being nasty [murmur of amusement from class]. That\325s because)J
96 423 :M
.687 .069(again the idea is what we\325re trying to do is make sure that the arguments are)J
96 441 :M
.563 .056(right. I buy this argument, and it needs a little bit of cleaning up maybe to be)J
96 459 :M
1.093 .109(comprehensible to anyone who hasn\325t had [Devon] explaining it to them.)J
96 477 :M
.649 .065(But the structure of it, I think, is pretty nice and straightforward. It\325s still a)J
96 495 :M
.696 .07(little bit, looks like a rabbit pulled out of a hat, in that you have a nice)J
96 513 :M
1.131 .113(explanation for something that\325s sort of presented there full-blown.)J
60 543 :M
1.187 .119(Then to connect Devon\325s proof to the previous day\325s line of development\321and in)J
60 561 :M
.97 .097(particular to Relax a Condition\321he showed how the proof could be interpreted in)J
60 579 :M
.829 .083(terms of relaxing conditions. Next came the standard \322final\323 query, \322Are we)J
60 597 :M
.911 .091(done?\323 A student \(possibly one of the two students who entered the room during)J
60 615 :M
1.291 .129(Devon\325s presentation\) responded, \322Think so.\323 Schoenfeld replied,)J
96 651 :M
.963 .096(You\325ll learn within two weeks that\325s almost always a rhetorical question.)J
96 669 :M
.774 .077(The answer is \322No\323 because there\325s still more we can do with that. Let me)J
96 687 :M
.608 .061([erases board], let me return to a point where we left off on Monday and that)J
endp
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.957 .096(actually will wind up with the same construction but might give a different)J
96 69 :M
.605 .06(idea of how it actually works.)J
60 99 :M
.724 .072(He reconstructed a series of partially inscribed squares in a triangle, highlighted the)J
60 117 :M
.851 .085(fourth vertices, and asked what might be true about this locus. A student \(not)J
60 135 :M
1.05 .105(visible on the videotape\) responded that it might be linear. Schoenfeld then left it)J
60 153 :M
.696 .07(for the class to verify that the locus was indeed a straight line.)J
260 197 :M
f0_12 sf
-.068(Discussion)A
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1.024 .102(Of the many instructional issues Schoenfeld addressed early in the course, none)J
60 245 :M
1.068 .107(was more important than the task of introducing heuristics so that students)J
60 263 :M
.903 .09(quickly appreciate their importance and gradually begin to use them intelligently.)J
60 281 :M
1.181 .118(What does Schoenfeld\325s management of the solution of the inscribed square)J
60 299 :M
1.023 .102(reveal about his approach to this teaching problem?)J
60 329 :M
.997 .1(Before turning to the solution itself, it is important to consider the place of the)J
60 347 :M
1.013 .101(inscribed square in the course as a whole. Two major components of Schoenfeld\325s)J
60 365 :M
1.157 .116(curriculum are his problems and the heuristics that he introduces with them. If)J
60 383 :M
.916 .092(students are ever to see heuristic strategies as problem solving tools worth)J
60 401 :M
.869 .087(learning, they must face problems that do not easily submit to techniques they)J
60 419 :M
.775 .077(already know. In contrast to the two other problems discussed in detail in the first)J
60 437 :M
.797 .08(week, most students found it difficult to make progress on the inscribed square.)J
60 455 :M
.952 .095(Because that problem stumped most of the class, Relax a Condition could then)J
60 473 :M
.912 .091(demonstrate the power of heuristics. Likewise, Try an Easier Related Problem)J
60 491 :M
.985 .098(illustrated that one needs experience, skill and even patience in order to use)J
60 509 :M
.946 .095(heuristics\321that they can\325t be applied in rote fashion. The discussion of the pitfalls)J
60 527 :M
1.018 .102(of easier related problems helped to clarify their non-deterministic, \322rule of)J
60 545 :M
.946 .095(thumb\323 character and show the importance of how you apply them. So)J
60 563 :M
1.282 .128(Schoenfeld\325s solution of the instructional problem required demonstrably hard)J
60 581 :M
1.269 .127(problems and quickly useful, if non-deterministic, heuristics.)J
60 611 :M
.745 .075(But difficult problems and potentially useful strategies are only part of the story.)J
60 629 :M
1.1 .11(Schoenfeld\325s extensive experience with the inscribed square problem provided)J
60 647 :M
.681 .068(well-grounded expectations about what students\325 likely responses would be \(e.g.,)J
60 665 :M
.933 .093(which easier related problems they would generate\). These expectations)J
60 683 :M
1.188 .119(complemented his knowledge of which solution)J
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.965 .097(paths would be more accessible to)J
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.945 .095(students than others \(e.g., which existence proofs led toward constructions\). This)J
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.813 .081(knowledge made it easier to recognize and interpret his students\325 suggestions and)J
60 69 :M
.863 .086(guide their efforts toward a successful conclusion.)J
60 99 :M
1.126 .113(These three top-level features \(problems, heuristics, and prior experience with the)J
60 117 :M
.902 .09(problems\) are all consistent with Schoenfeld\325s stated goals for the class. Solving)J
60 135 :M
.726 .073(problems \(as opposed to exercises\) supports his claim that the activity in and)J
60 153 :M
.965 .097(around the class reflects important aspects of professional mathematical practice.)J
60 171 :M
.88 .088(Learning to judge when, if, and how to apply particular heuristics is an important)J
60 189 :M
.806 .081(part of that practice. And the fact that extensive teaching experience with particular)J
60 207 :M
.914 .091(problems was central to using those problems productively reflects the complex)J
60 225 :M
1.292 .129(relationship between problems and heuristics. But his management of the)J
60 243 :M
.975 .097(solution itself, particularly his appropriation \(or not\) of students\325 suggestions,)J
60 261 :M
.792 .079(bears a more complex relationship to his goals for the class.)J
60 291 :M
.858 .086(It is important to recognize that the students played an active and substantive role)J
60 309 :M
.819 .082(in solving the inscribed square. Nearly half of the class contributed some piece of)J
60 327 :M
1.239 .124(the evolving solution, and Devon\325s work, especially his construction, was more)J
60 345 :M
1.152 .115(than simply \322a contribution.\323 Schoenfeld deliberately solicited their participation,)J
60 363 :M
.785 .078(but he also carefully organized and controlled it. Student input was solicited at)J
60 381 :M
.773 .077(certain points in the solution \(e.g., when a range of possible approaches was)J
60 399 :M
.984 .098(needed\) and not others, and their suggestions were assimilated into his)J
60 417 :M
1.115 .112(instructional plan. His role as instructional leader and, at crucial junctures,)J
60 435 :M
.891 .089(mathematical authority was central to the pace and process of the solution. He)J
60 453 :M
1.113 .111(orchestrated student participation within a relatively traditional model of roles for)J
60 471 :M
.707 .071(teachers and students, where teachers decide what choices to offer to students and)J
60 489 :M
.822 .082(when it is best to do so. These traditional elements of teaching appear\321at first)J
60 507 :M
1.072 .107(blush\321to run counter to his stated long-term goal of creating a mathematical)J
60 525 :M
1.292 .129(community where authority rests with the mathematics and community as a)J
60 543 :M
.445(whole.)A
60 573 :M
1.173 .117(Before attempting to resolve this apparent contradiction, we review the evidence)J
60 591 :M
1.147 .115(that undergirds these interpretative claims. First, what indicates that Schoenfeld)J
60 609 :M
.873 .087(came in with a pre-existing plan for solving the inscribed square? Though we did)J
60 627 :M
.958 .096(not question him at the time about his plans, he has written about his purposes for)J
60 645 :M
.994 .099(using the problem \(to demonstrate the difficulties of applying heuristics\), the)J
60 663 :M
.347 .035(easier related problems he expects to see \(try a rectangle, try a circle, try a special)J
60 681 :M
.849 .085(triangle\), the two existence proofs, and the difficulty of obtaining a construction)J
60 699 :M
1.044 .104(from one of them \(Schoenfeld, 1985, pp. 84\32091\). Most of the major elements of the)J
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1.007 .101(solution that he orchestrated with the class are present in this written account.)J
60 69 :M
.915 .091(Given the strong similarities between the two, it is difficult to doubt the existence)J
60 87 :M
.894 .089(of a detailed instructional plan. Schoenfeld chose the inscribed square as the)J
60 105 :M
1.003 .1(second problem of the course for particular reasons, and the main elements of the)J
60 123 :M
.907 .091(discussion were in place for him)J
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.118 .012( )J
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.946 .095(before the problem sheet was given to students.)J
60 153 :M
1.132 .113(This plan then became the framework for guiding the class through the solution)J
60 171 :M
.952 .095(and underscoring major points along the way. To achieve these goals in a)J
60 189 :M
.792 .079(reasonable time frame, he directed the class down certain pathways \(and not)J
60 207 :M
1.08 .108(others\) and used his own mathematical experience and authority to justify these)J
60 225 :M
1.207 .121(choices. As warrant for this interpretation, we summarize in chronological order)J
60 243 :M
.995 .099(the instances where Schoenfeld used his personal authority to direct the course of)J
60 261 :M
2.305 .23(the solution.)J
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.821 .082(\245 He wrote Devon\325s suggestion to seek a counterexample below the other)J
105 315 :M
.845 .085(suggestions and did not seriously consider it in the subsequent discussion.)J
96 333 :M
.759 .076(\245 He delayed dealing with Mitch\325s suggestion to relax the condition on the)J
105 351 :M
.534 .053(square and try a rectangle \(suggestion 1\) until he discussed suggestions 2)J
105 369 :M
.459 .046(and 3.)J
96 387 :M
.538 .054(\245 He asserted that inscribing a circle in the triangle could not be adapted to)J
105 405 :M
.735 .073(inscribing a square in a triangle.)J
96 423 :M
.777 .078(\245 He asserted that the solutions for isosceles and equilateral triangles were)J
105 441 :M
.623 .062(neither easy to produce nor to adapt to the arbitrary triangle.)J
96 459 :M
.509 .051(\245 He asserted that the first existence proof could not be adapted into a)J
105 477 :M
1.743 .174(constructive proof.)J
60 507 :M
1.102 .11(What can we learn about his teaching from these choices? First, given the match)J
60 525 :M
1.065 .106(with his plan for the solution, each move is an example of how teachers)J
60 543 :M
.913 .091(appropriate students\325 ideas and suggestions to their own plans \(Newman, Griffin,)J
60 561 :M
.615 .061(& Cole, 1989\). Appropriation is a tool for balancing the dual goals of engaging)J
60 579 :M
.973 .097(students\325 interest and participation and sustaining progress toward important)J
60 597 :M
1.185 .118(instructional goals. Second, while these instructional decisions were all explicit in)J
60 615 :M
.901 .09(the data, they were not all identical in character. The first two were management)J
60 633 :M
.638 .064(choices; they were decisions about what to take up for discussion and what to set)J
60 651 :M
1.097 .11(aside, more than direct evaluations of what could be done mathematically. The)J
60 669 :M
.965 .097(first decision was sensible since the search for counterexamples would have been)J
60 687 :M
1.168 .117(very difficult to assimilate into his plan. So Schoenfeld honored the suggestion in)J
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.712 .071(general terms and set it aside. The second decision was equally sensible since)J
60 69 :M
1.062 .106(developing this suggestion first would have removed the possibility of teaching)J
60 87 :M
.828 .083(the lesson about the pitfalls of easier related problems. In that sense, Mitch\325s)J
60 105 :M
.661 .066(suggestion to relax a condition on the square was too good.)J
60 135 :M
1.243 .124(The last four, however, all involved Schoenfeld\325s explicit judgment of what is)J
60 153 :M
.841 .084(possible and productive mathematically and were justified by appeal to his own)J
60 171 :M
1.283 .128(mathematical experience. Essentially they communicated the message, \322Trust me.)J
60 189 :M
.775 .077(I have explored this problem extensively, and I know its \324ins\325 and \324outs\325.\323 Like the)J
60 207 :M
1.104 .11(management of Easier Related Problems, these declarations sped the solution)J
60 225 :M
.96 .096(along, by curtailing potential solution paths that Schoenfeld knew to be)J
60 243 :M
.78 .078(unproductive. But to do so, he implicitly asked students to accept his role as the)J
60 261 :M
1.598 .16(mathematical leader and decision-maker.)J
60 291 :M
.911 .091(How then do these teaching moves fit with the overall goal of creating a)J
60 309 :M
1.511 .151(mathematical community in the classroom? More generally, how do mathematics)J
60 327 :M
1.268 .127(educators deal with mathematical authority, balance their informed authority)J
60 345 :M
.824 .082(against emerging student autonomy, and support students\325 growth toward a)J
60 363 :M
1.678 .168(powerful and independent mathematical competence?)J
60 393 :M
.985 .099(The first step toward a resolution is to acknowledge the mismatch: the solution of)J
60 411 :M
.953 .095(the inscribed square was inconsistent with some of the overarching goals of the)J
60 429 :M
.759 .076(course. Schoenfeld\325s directed solution does not easily square with the ideas of a)J
60 447 :M
1.515 .152(classroom mathematical community pursuing its own solutions, and his)J
60 465 :M
1.033 .103(statements about what was possible mathematically are not consistent with the)J
60 483 :M
1.052 .105(methods of public justification employed by the professional community. )J
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1.108 .111(On the)J
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1.193 .119(other hand, the Overview points to evidence)J
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.142 .014( )J
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1.409 .141(that Schoenfeld\325s teaching has)J
60 519 :M
.963 .096(moved students substantively toward his declared long-term goals, and our)J
60 537 :M
.589 .059(observations suggest this was the case for the 1990 class as well. Students became)J
60 555 :M
1.092 .109(more proficient problem solvers; they learned to use heuristics productively; they)J
60 573 :M
.739 .074(interacted as a community of problem solvers; and they accepted the task of)J
60 591 :M
1.185 .118(judging and nudging)J
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.112 .011( )J
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.868 .087(each other\325s ideas and arguments. So the question must be)J
60 609 :M
1.069 .107(restated as, \322Why was the direction not counterproductive to the long-term)J
60 627 :M
.146(goals?\323)A
60 657 :M
1.003 .1(Our view is that there were more important goals for Schoenfeld early in the)J
60 675 :M
.897 .09(course and that these may have required his exercising a leadership role in)J
60 693 :M
.943 .094(deciding some issues for the class. He must convince students that they have)J
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1.089 .109(important things to learn in the course and that he will support them in that)J
60 69 :M
.639 .064(effort. If he did not strongly guide students\325 problem solving in the early days, they)J
60 87 :M
.931 .093(could easily flounder, pursue too many deadends, and come to question the entire)J
60 105 :M
.943 .094(enterprise. Instead, Schoenfeld made sure that they struggled enough to realize)J
60 123 :M
.784 .078(that they could not solve the problem easily but could be successful with his)J
60 141 :M
.935 .093(direction and the proper tools. These experiences were part of the transition)J
60 159 :M
1.302 .13(toward more independent problem finding, problem solving, and justification. In)J
60 177 :M
1.097 .11(short, his directive instruction gave priority to some goals over others.)J
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1.156 .116(Our goal in emphasizing the complex relationship between instructional goals and)J
60 225 :M
.916 .092(teaching practice in this segment has not been to question or endorse the)J
60 243 :M
.988 .099(optimality of Schoenfeld\325s decisions. Rather, we have shaped the analysis to)J
60 261 :M
1.283 .128(illustrate the interaction of ambitious educational goals, detailed instructional)J
60 279 :M
1.151 .115(plans, teaching moves, and students\325 contributions. One main lesson is that)J
60 297 :M
1.376 .138(innovative teaching oriented toward ambitious, non-traditional goals can embrace)J
60 315 :M
1.364 .136(both traditional and non-traditional elements. The achievement of such goals may)J
60 333 :M
1.259 .126(depend as much on the traditional elements as the non-traditional and on skillful)J
60 351 :M
1.082 .108(balancing of short-term objectives and quite different long-term goals. This)J
60 369 :M
1.349 .135(conclusion, we believe, undermines simple descriptions and explanations of)J
60 387 :M
1.259 .126(successful non-traditional mathematics teaching. We need to look more closely to)J
60 405 :M
.848 .085(understand what works in these settings and why.)J
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-.099(We acknowledge that Schoenfeld did take some explicit actions toward building the classroom community and)A
60 687 :M
-.029(shifting the locus of authority in the first week, e.g., his statement to the class about standards for written and oral)A
60 696 :M
-.112(arguments before Devon presented his constructive argument and his statement about asking nasty questions to the)A
60 705 :M
-.02(class afterward. Our argument is that this was not his primary goal in the first week.)A
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-.082(Practicing mathematical communication:)A
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-.199(Using heuristics with the magic square)A
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.508 .051(Cathy Kessel)J
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1.735 .174(The language is not alive except to those who use it.)J
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.121 .012(\(Thurston, 1994, p. 167\))J
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.808 .081(Schoenfeld \(1991\) gives an account of a classroom discussion of the magic square)J
60 227 :M
.975 .097(problem. Here is the version of the third problem he gave to his students in the)J
60 245 :M
(class.)S
96 281 :M
.392 .039(Can you place the numbers 1, 2, 3, 4, 5, 6, 7, 8, and 9 in the box below, so that)J
96 299 :M
.914 .091(when you are all done, the sum of each row, each column, and each)J
96 317 :M
.713 .071(diagonal is the same? This is called a magic square.)J
112 339 1 1 rF
112 339 1 1 rF
113 339 17 1 rF
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131 377 17 1 rF
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112 396 1 1 rF
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60 423 :M
.867 .087(In his account of the discussion of this problem Schoenfeld describes briefly how)J
60 441 :M
1.054 .105(the problem, though trivial, can be used to illustrate many heuristics and other)J
60 459 :M
1.348 .135(important aspects of mathematical thinking: Establishing Subgoals, Working)J
60 477 :M
.996 .1(Backwards, Exploiting Symmetry, Working Forwards, using systematic generating)J
60 495 :M
.841 .084(procedures, focusing on key points for leverage, exploiting extreme cases, solving a)J
60 513 :M
.794 .079(problem in more than one way, and using a problem as a springboard for further)J
60 531 :M
.866 .087(mathematics. At the end of his account he says that an important aspect of the)J
60 549 :M
.954 .095(discussion, the classroom dynamics which \322reflected the dynamics of real)J
60 567 :M
1.024 .102(mathematical exploration\323 was not described. One might wonder how a classroom)J
60 585 :M
1.028 .103(discussion could reflect the dynamics of mathematical exploration and how such a)J
60 603 :M
.71 .071(discussion could happen on the second day of a course. The goal of this section is)J
60 621 :M
.847 .085(to examine the classroom dynamics of another magic square discussion, led by)J
60 648 :M
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-.052(I would like to thank Alan Schoenfeld for the many ways in which he helped to make this article possible. His)A
60 669 :M
-.055(support, and that of the Functions Group and the School of Education at the University of California have helped me)A
60 678 :M
-.064(to learn about and do research in education. For comments, criticisms, and encouragement as this article slowly)A
60 687 :M
-.022(evolved, my thanks go to: Margaret Carlock, Marisa Castellano, Judith Epstein, and Jean Lave; the )A
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-.035(RCME)A
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-.022( editors:)A
60 696 :M
-.044(Ed Dubinsky and Jim Kaput; the )A
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-.067(RCME)A
f1_9 sf
-.044( reviewers: Barbara Pence, Beth Warren, and one anonymous reviewer;)A
60 705 :M
-.062(Mary Barnes, Sue Helme, and Derek Holton; and my co-authors: Abraham Arcavi, Luciano Meira, and Jack Smith.)A
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.838 .084(Schoenfeld in the fall of 1990, and to consider some of the features of)J
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1.061 .106(mathematical practice it reflected.)J
60 99 :M
.896 .09(Several aspects of this discussion are striking. Very little was written on the)J
60 117 :M
.67 .067(blackboard. What did appear were heuristics, diagrams, an equation, and a)J
60 135 :M
1.04 .104(question, rather than the line by line theorems and proofs of traditional upper)J
60 153 :M
1.122 .112(division courses or the line by line theorems, examples, and solutions of lower)J
60 171 :M
1.938 .194(division courses.)J
60 201 :M
.907 .091(The kind of speaking in Schoenfeld\325s classroom also differed from that of a)J
60 219 :M
1.369 .137(traditional class. For example, in one seventeen-minute segment of the whole-)J
60 237 :M
.688 .069(class discussion of the magic square, though the teacher was at the front of the class)J
60 255 :M
.982 .098(and the students were not working in groups or presenting solutions, there were)J
60 273 :M
1.425 .143(fourteen utterances\321questions, comments, suggestions about the mathematics at)J
60 291 :M
.791 .079(hand\321made by students. One would expect only questions in a traditional class)J
60 309 :M
.549 .055(and very few at that.)J
60 339 :M
1.067 .107(These differences suggest that communication is an important feature of this class)J
60 357 :M
1.386 .139(and that research on language and communication might illuminate some of the)J
60 375 :M
1.055 .105(reasons why Schoenfeld has chosen to conduct his class in this manner. In this)J
60 393 :M
.979 .098(section I use some analytic frameworks from sociolinguistics to describe classroom)J
60 411 :M
1.291 .129(communication and to compare it with that of research mathematicians. I have)J
60 429 :M
.859 .086(been an undergraduate, graduate student, and faculty member at various)J
60 447 :M
.8 .08(mathematics departments across the U.S., and I draw on this experience as well as)J
60 465 :M
.706 .071(on written accounts to describe the teaching and research practices of)J
60 483 :M
.421(mathematicians.)A
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-.046(Differences in speaking and writing)A
60 543 :M
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.693 .069(At the beginning of the first day of a typical upper division undergraduate)J
60 561 :M
.967 .097(mathematics class, the professor writes her or his name, office location, and office)J
60 579 :M
.859 .086(hours on the board. After a brief statement about determination of course grades,)J
60 597 :M
1.217 .122(she or he begins to lecture, starting with definitions and notational conventions)J
60 615 :M
.958 .096(and perhaps reaching some new material midway through the first class. There are)J
60 633 :M
.828 .083(few, perhaps no, questions from the students, which will be true for the rest of the)J
60 651 :M
1.013 .101(term. The professor writes almost everything on the blackboard and almost)J
60 669 :M
1.182 .118(everything has one of the following labels: theorem, lemma, corollary, proof,)J
60 687 :M
1.233 .123(example, axiom, conjecture, definition, notation. One exception is pictures or)J
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1.017 .102(diagrams, another \(in applied mathematics classes\) is applications. With such a lot)J
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.718 .072(of writing to do, blackboards, chalk, and erasers become extremely important.)J
60 87 :M
.822 .082(Professors become skilled at arranging their writing on the board, not erasing)J
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.999 .1(theorems or diagrams until they won\325t need to refer to them again. \(An)J
60 123 :M
.975 .097(inattentive professor may also cover one sliding board with another so quickly)J
60 141 :M
.873 .087(that students can\325t finish copying the writing on the board. Schoenfeld did this on)J
60 159 :M
.563 .056(the first day of class in his enactment of the typical calculus professor, as described)J
60 177 :M
1.206 .121(in Meira\325s section.\))J
60 207 :M
.77 .077(The main focus of the typical mathematics class is the blackboard and students\325)J
60 225 :M
1.144 .114(main activity is taking notes, and following the lecture. As in written mathematics)J
60 243 :M
1.167 .117(the statement of results tends to be impersonal. Names occur mainly in important)J
60 261 :M
1.28 .128(theorems, definitions or axioms, e.g, Stokes\325 Theorem, Green\325s Theorem, G\232del\325s)J
60 279 :M
1.405 .141(Incompleteness Theorem, the Zermelo-Frankel axioms of set theory, the Peano)J
60 297 :M
1.051 .105(Postulates, Noetherian ring, Abelian group. The time at which the object was)J
60 315 :M
.809 .081(constructed isn\325t often mentioned. \(Later, as students reach the edges of)J
60 333 :M
1.061 .106(mathematical knowledge in graduate school, names and dates appear with much)J
60 351 :M
.482 .048(greater frequency.)J
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.843 .084(The classroom language of the typical mathematics professor reflects the way)J
60 399 :M
.911 .091(mathematics is presented in writing. \322Assume the following holds . . . ,\323 \322it)J
60 417 :M
.506 .051(follows easily that . . . ,\323 \322it is not the case that . . . \323 are frequent phrases \(for more)J
60 435 :M
.829 .083(examples see Pimm, 1987\). Rotman \(1993, p. 7\) describes written mathematics as)J
60 453 :M
1.454 .145(\322riddled with )J
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.332(imperatives)A
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1.525 .152(, with commands and exhortations such as \324multiply)J
60 471 :M
.679 .068(items in )J
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.377(w)A
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.804 .08(,\325 \324integrate )J
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.261(x)A
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.677 .068(,\325 \324prove )J
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.926 .093(,\325 \324enumerate )J
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1.151 .115(\325\323 and \322completely without)J
60 489 :M
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.33(indexical)A
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1.666 .167( expressions, those fundamental and universal elements of natural)J
60 507 :M
.447 .045(languages whereby such terms as \324I,\325 \324you,\325 \324here,\325 \324this,\325 as well as tensed verbs, tie)J
60 525 :M
.922 .092(the meaning of messages to the physical context of their utterance.\323 For example,)J
60 543 :M
.757 .076(the magic square problem could have been stated in the following way: \322Place the)J
60 561 :M
.808 .081(numbers 1 through 9 in the boxes to the right so that the sum of each row,)J
60 579 :M
.903 .09(column, and diagonal is the same. Such an arrangement is called a magic square.\323)J
60 597 :M
1.308 .131(This imperative statement doesn\325t mention where and when the action of placing)J
60 615 :M
.853 .085(numbers in boxes is occurring, nor who is acting.)J
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-.121(This description is not meant as a condemnation. Some of my happiest and most instructive hours have been spent)A
60 696 :M
-.064(in such courses. I appreciate well-tended blackboards and good chalk and my spoken language reflects written)A
60 705 :M
-.101(mathematics. At the end of this section I suggest some reasons why not all students are so fortunate.)A
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.617 .062(Chafe and Danielewizc \(1987\) characterize a style with few indexical expressions as)J
60 69 :M
.594 .059(\322detached,\323 \322show[ing] an interest in ideas that are not tied to specific people,)J
60 87 :M
.749 .075(times, and places, but which are abstract and timeless\323 and which avoids)J
60 105 :M
1.125 .113(mentioning concrete doers. They note that this style predominates in writing and)J
60 123 :M
.873 .087(suggest a reason for it\321\322for writers, the audience is usually unseen, and often)J
60 141 :M
.864 .086(unknown\323 \(p. 19\). This detached style of speaking and writing about mathematics)J
60 159 :M
.915 .092(suggests to listeners and readers that mathematics is independent of time and)J
60 177 :M
1.068 .107(place. This is consistent with the epistemology \(a mixture of formalism and)J
60 195 :M
.924 .092(Platonism\) held by many mathematicians \(Davis, 1986; Davis & Hersch, 1986;)J
60 213 :M
.735 .074(Ernest, 1991; Fauvel, 1988\). It is also consistent with the epistemology held by many)J
60 231 :M
1.298 .13(high school students that learning mathematics is mostly memorizing facts)J
60 249 :M
.932 .093(\(National Center for Educational Statistics, 1993\) and that the ideas of mathematics)J
60 267 :M
.785 .078(have always been true and will always be true and were discovered \(not invented\))J
60 285 :M
.828 .083(by mathematicians \(Clarke, Wallbridge, & Fraser, 1992\). And it is also consistent)J
60 303 :M
1.067 .107(with the mathematics classroom experiences of undergraduates \(Mura, 1995\).)J
60 333 :M
.874 .087(In contrast, note Schoenfeld\325s wording of the magic square problem which begins,)J
60 351 :M
.927 .093(\322Can you place\323 rather than \322Place the numbers.\323 We shall see many)J
60 369 :M
.671 .067(indexicals\321\322I,\323 \322we,\323 \322you,\323 \322Devon\325s question,\323 \322what Jeff guessed last)J
60 387 :M
.976 .098(time\323\321in Schoenfeld\325s speech. Chafe and Danielewizc call a style with many)J
60 405 :M
1.067 .107(indexicals \322involved\323 and note that it is characteristic of, though not limited to,)J
60 423 :M
.915 .092(spoken language. They also note, \322In most spoken language an audience is not)J
60 441 :M
.641 .064(only physically present, but has the ability to respond with language of its own\323)J
60 459 :M
.69 .069(\(pp. 18-19\). This suggests that, in addition to acknowledging their presence, an)J
60 477 :M
1.313 .131(involved style may invite listeners to respond.)J
60 507 :M
.924 .092(And Schoenfeld does want his listeners to respond. Because a main goal of his)J
60 525 :M
.992 .099(course is the creation of a \322mathematical community,\323 one of his goals for the first)J
60 543 :M
.43 .043(days of the course is to get the students to talk: \322Clearly what I need to do is begin)J
60 561 :M
.65 .065(pulling things from [the students] because part of what the course is supposed to)J
60 579 :M
.715 .072(do is turn things over to them\323 \(audio taped discussion, May 22, 1991\).)J
60 609 :M
1.372 .137(Schoenfeld\325s language is not only involved, but informal and non-technical with)J
60 627 :M
.999 .1(occasional shifts in style to language that reflects written mathematics. Such shifts)J
60 645 :M
1.025 .103(are known as code-variation. Saville-Troike \(1989\) defines codes as \322different)J
60 663 :M
.893 .089(languages, or quite different varieties of the same language\323 and code-variation as)J
60 681 :M
1.029 .103(a change in code within a speech event. She notes that code-variation may serve)J
60 699 :M
.941 .094(many different purposes, depending on context. Here, Schoenfeld\325s shift to)J
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.959 .096(language reflecting written mathematics serves to display his group affiliation as)J
60 69 :M
1.009 .101(well as to help the students to become familiar with mathematical language. Just)J
60 87 :M
.792 .079(as patients don\325t have faith in a doctor who \322doesn\325t talk like one\323 despite the)J
60 105 :M
1.381 .138(miscommunication that can occur when doctors use technical language \(Saville-)J
60 123 :M
.906 .091(Troike, 1989\), students may not have faith in a professor who doesn\325t talk like one.)J
60 141 :M
.953 .095(Schoenfeld must talk a fine line between being understandable, approachable, and)J
60 159 :M
1.176 .118(interested in students\325 contributions, and maintaining his status as the)J
60 177 :M
1.292 .129(knowledgeable member of the mathematics community depicted in his)J
60 195 :M
.984 .098(monologue at the beginning of the first class session. His informal language may)J
60 213 :M
.62 .062(suggest to students that it is acceptable if they reply in the same manner. It also)J
60 231 :M
.875 .087(contrasts with and emphasizes the few technical words he does use: names of)J
60 249 :M
1.4 .14(heuristics \(which he sometimes labels \322jargon\323\) and mathematical terms.)J
60 279 :M
1.065 .107(As with spoken language, there are differences in how much Schoenfeld writes,)J
60 297 :M
.676 .068(what he writes, and the way he uses writing. This difference suggests a different)J
60 315 :M
1.042 .104(emphasis, and a different view of what is important in this classroom. Not only)J
60 333 :M
1.024 .102(does Schoenfeld write considerably less than a traditional mathematics professor,)J
60 351 :M
1.059 .106(but when and what he writes are different. Both mathematics and heuristics)J
60 369 :M
.65 .065(appear on the blackboard as one would expect in a class on heuristics and their use.)J
60 387 :M
.693 .069(The diagrams in this class would not be seen in a textbook or an article, they are)J
60 405 :M
1.003 .1(used to work with, rather than to illustrate. They are altered throughout the)J
60 423 :M
.905 .091(course of the discussion and erased only when the discussion is over. This use of)J
60 441 :M
.886 .089(diagrams allows Schoenfeld to avoid technical language as well as to make the)J
60 459 :M
1.088 .109(problem more immediate and his descriptions more direct. The other blackboard)J
60 477 :M
1.042 .104(writings are names and brief descriptions of heuristics. The connection between)J
60 495 :M
.914 .091(mathematics and heuristics is not recorded on the blackboard, it is made through)J
60 513 :M
1.089 .109(questioning, and interaction with the class. Formal proofs aren\325t given, instead,)J
60 531 :M
.971 .097(their genesis is enacted in the classroom discussion. Later in the term, students)J
60 549 :M
1.125 .113(will present their own conjectures and proofs.)J
60 579 :M
1.131 .113(Not only are there differences between the informal, involved style of his)J
60 597 :M
.792 .079(language and the detached style of a traditional professor of mathematics, there are)J
60 615 :M
.825 .082(also differences in the content of Schoenfeld\325s language and its mode of use. Its)J
60 633 :M
.967 .097(content includes what is traditionally thought of as the subject matter of)J
60 651 :M
.943 .094(mathematics classes, but is also )J
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.126 .013( )J
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1.019 .102(math, about how math gets done, about)J
60 669 :M
.738 .074(revealing \322the tools of the trade,\323 and about learning that trade. This suggests a)J
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1.456 .146(different emphasis\321not only is mathematical content important, but how one)J
60 69 :M
.98 .098(does mathematics is a legitimate topic of classroom discussion.)J
60 99 :M
1.149 .115(As Arcavi points out in the overview, Schoenfeld has two different modes of)J
60 117 :M
1.157 .116(communicating with the class as a whole: presenting and involving the class in)J
60 135 :M
1.037 .104(discussion. As we have seen in the previous sections and shall see in this section,)J
60 153 :M
1.261 .126(Schoenfeld uses these different modes of communication for different purposes.)J
60 171 :M
1.035 .104(In general, Schoenfeld presents heuristics, either as an explanation of a)J
60 189 :M
.793 .079(mathematical suggestion given by a student, a name for a process that\325s just been)J
60 207 :M
1.061 .106(illustrated, or to give direction to the solution of a mathematics problem. In the)J
60 225 :M
1.041 .104(latter case, the heuristic is frequently instantiated in the mathematical context at)J
60 243 :M
.696 .07(hand by the students.)J
60 273 :M
.966 .097(Schoenfeld gets the students to instantiate heuristics by a method of questioning)J
60 291 :M
.958 .096(similar to that described by P\227lya \(1973\) in )J
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.878 .088(How to Solve It)J
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.843 .084(The teacher\325s method of questioning . . . is essentially this: Begin with a)J
96 345 :M
.716 .072(general question or suggestion on our list [of heuristics], and, if necessary,)J
96 363 :M
.769 .077(come down gradually to more specific and concrete questions or suggestions)J
96 381 :M
.759 .076(till you reach one which elicits a response in the student\325s mind. . . . It is)J
96 399 :M
1.065 .107(important, however, that the suggestions from which we start should be)J
96 417 :M
.78 .078(simple, natural, and general, and that their list be short. . . . The suggestions)J
96 435 :M
.769 .077(must be general, applicable not only to the present problem but to problems)J
96 453 :M
.655 .066(of all sorts. . . . The list must be short in order that the questions may be)J
96 471 :M
.76 .076(often repeated, unartificially, and under varying circumstances. . . . It is)J
96 489 :M
.641 .064(necessary to come down gradually to specific suggestions, in order that the)J
96 507 :M
1.093 .109(student may have as great a )J
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1.105 .111(share of the work)J
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.964 .096( as possible. . . . Our method)J
96 525 :M
.77 .077(admits a certain elasticity and variation, it admits various approaches . . . it)J
96 543 :M
.95 .095(can and should be applied so that questions asked by the teacher )J
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1.367 .137(could have)J
96 561 :M
2.167 .217(occurred to the student himself. )J
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3.27 .327(\(pp. 20-21\))J
60 591 :M
1.031 .103(Schoenfeld has taught his class many times before. Though he does not know his)J
60 609 :M
.73 .073(students well at the beginning of the course, as in the cases of the telescoping series)J
60 627 :M
.865 .086(and the inscribed square, he knows most of the responses students will make to)J
60 645 :M
.813 .081(his questions about the magic square. Students are not completely predictable)J
60 663 :M
1.238 .124(though and Schoenfeld\325s management of the discussion also had opportunistic)J
60 681 :M
.915 .091(elements \(Hayes-Roth & Hayes-Roth, 1979; Schoenfeld et al., 1992\). Students\325)J
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.665 .066(questions and suggestions, both predictable and unexpected, were used to serve)J
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.738 .074(goals of the discussion and of the course.)J
60 99 :M
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.473 .047(The nature of the problem)J
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.836 .084(Both the magic square problem and the way in which it is used are important)J
60 147 :M
.984 .098(elements of this discussion. The magic square was the third problem on the sheet)J
60 165 :M
.738 .074(given to the students and the third to be discussed. One difference between the)J
60 183 :M
.762 .076(magic square problem and those preceding it is that its solution is indubitable \(it)J
60 201 :M
.662 .066(can be checked by a simple calculation\), and easy to reach. Unlike the problem of)J
60 219 :M
.562 .056(inscribing a square in a triangle, it is easy to solve, and couldn\325t be used to)J
60 237 :M
.904 .09(convince the students that they needed to learn heuristics. It may even suggest to)J
60 255 :M
1.178 .118(students that they have been using \322raw\323 heuristics\321heuristic tendencies that)J
60 273 :M
.96 .096(need refinement before they are likely to be consistently useful \(Silver, 1985\).)J
60 303 :M
1.053 .105(However, the magic square serves as an excellent vehicle for the introduction and)J
60 321 :M
1.553 .155(illustration of heuristics. Because the mathematics involved is elementary,)J
60 339 :M
.69 .069(students can discuss it without the fear of displaying ignorance\321it\325s easy to talk)J
60 357 :M
.71 .071(about \(Schoenfeld, audio taped discussion, May 22, 1991\). Because there is no need)J
60 375 :M
.741 .074(to focus on getting a solution, students can focus on the process of arriving at a)J
60 393 :M
.987 .099(solution. What follows is an account of the discussion of the magic square.)J
60 411 :M
.948 .095(Annotations and interpretations in brackets are interspersed. Italics is used in two)J
60 429 :M
.786 .079(different ways in the transcript: Italicized phrases were both spoken and written on)J
60 447 :M
.592 .059(the blackboard; single words in italics indicate words emphasized by the speaker.)J
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.089 .009(The discussion of the magic square)J
60 521 :M
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.958 .096(Schoenfeld asks for volunteers to present the magic square problem. Jeff)J
60 539 :M
.983 .098(volunteers and presents his group\325s solution. After Jeff sits down, Schoenfeld goes)J
60 557 :M
.66 .066(back to the board, acknowledges the solution \(\322the answer speaks for itself\323\) and)J
60 575 :M
.808 .081(indicates a transition to another activity \322What I want to do is play with this a)J
60 593 :M
.594 .059(little bit. First of all it\325s not a problem you want to do by )J
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.22(pure)A
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.702 .07( trial and error.\323 He)J
60 611 :M
.975 .098(then gives a standard combinatorial argument to show how many different ways)J
60 629 :M
.575 .058(there are of filling a 3 by 3 grid if one places the digits from 1 through 9 randomly)J
60 647 :M
.555 .056(in its cells. He notes, \322There are 9 ways that you can stick any number, say [points)J
60 665 :M
.49 .049(to top left square of grid], in this square, 8 in that one [points to top middle square])J
60 683 :M
.728 .073(after you\325ve used one, 7 in the next one\323 and so on. This yields 9! ways to fill in the)J
60 701 :M
.592 .059(3 by 3 grid\321but, he observes, the magic square has eight-fold symmetry, so there)J
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.431 .043( = 9 x 7 x 6 x 5 x 4 x 3 x 2 x 1 non-equivalent ways one might fill it. Then,)J
60 75 :M
.856 .086(saying intermediate products aloud, he quickly calculates the result.)J
60 105 :M
1.16 .116([Professors teaching undergraduate courses don\325t often do computations involving)J
60 123 :M
.818 .082(large numbers in front of their classes. One reason for doing it here might be to)J
60 141 :M
1.026 .103(emphasize the improbability of obtaining a solution by random search. This will)J
60 159 :M
.932 .093(contrast with the solutions obtained by the use of heuristics that will follow.)J
60 177 :M
.97 .097(Another might be to display mathematical expertise to those who consider quick)J
60 195 :M
.914 .091(calculation a sign of expertise. A more subtle message which this calculation may)J
60 213 :M
.897 .09(convey is that none of the students used pure trial and error. They all solved the)J
60 231 :M
.131(problem.])A
60 261 :M
1.319 .132(During his calculation Schoenfeld mentioned the eight-fold symmetry of the)J
60 279 :M
.914 .091(square. Devon asks if there were any solutions to the magic square not equivalent)J
60 297 :M
.845 .085(using symmetry. Schoenfeld replies \322That\325s a good question, let\325s leave that as)J
60 315 :M
.98 .098(something to look at\323 and writes the question on the sideboard where it remains)J
60 333 :M
.845 .085(for the rest of the discussion.)J
60 363 :M
.962 .096([Schoenfeld\325s knowledge of the magic square and its different solutions allows him)J
60 381 :M
.848 .085(to make this response, knowing that the question will be answered before the class)J
60 399 :M
.943 .094(ends. His action serves several purposes: It legitimizes the student\325s question)J
60 417 :M
1.125 .113(without immediately changing the flow of the activity, begins a community)J
60 435 :M
1.17 .117(history and gives an example of mathematical practice\321questions are important,)J
60 453 :M
.89 .089(they may not be immediately answered, but one may later note as Schoenfeld will)J
60 471 :M
.609 .061(do, that a particular question has been answered by a proof or construction.])J
60 501 :M
.607 .061(Schoenfeld says \322So if you don\325t want to do it by trial and error then what you)J
60 519 :M
.631 .063(really want to do is look for ways to reduce the number of things you\325ve got to)J
60 537 :M
.692 .069(consider\323 and summarizes Jeff\325s presentation of group\325s work as a strong appeal to)J
60 555 :M
.603 .06(symmetry \322 . . . if you make those two guesses, 5 is in the center and 15 is the sum)J
60 573 :M
.793 .079(then you don\325t have too much trial and error to do before you get there. And that\325s)J
60 591 :M
.424 .042(a good sane way to go about doing the problem.\323 [Here the students may be)J
60 609 :M
.866 .087(reassured, they all solved the problem, and their solutions weren\325t gotten by pure)J
60 627 :M
.89 .089(trial and error. The \322two guesses\323 will reappear as instantiations of heuristics,)J
60 645 :M
.672 .067(again suggesting that the students may be using \322raw heuristics\323 which can be)J
60 663 :M
.123(refined.])A
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-.044(First \(re\)solution: By establishing subgoals and working backwards)A
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1.72 .172(First subgoal: What is the sum?)J
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1.985 .198( So far heuristics haven\325t been mentioned.)J
60 111 :M
.704 .07(Schoenfeld shifts the focus to heuristics, says \322What I want to do is ask a couple of)J
60 129 :M
.799 .08(questions that illustrate some of P\227lya\325s strategies and use the answers to make)J
60 147 :M
.727 .073(progress on this problem again so we\325re going to revisit the problem a little bit.\323)J
60 165 :M
.542 .054(He erases the board and states \322We\325re back to the beginning, we want to place the)J
60 183 :M
.647 .065(digits from 1 to 9 into this [the empty grid he has just drawn] so that the sum of)J
60 201 :M
.998 .1(each row, column, and diagonal is the same.\323 Now he introduces a heuristic.)J
60 219 :M
1.016 .102([Note the shift in the meaning of \322you\323 in Schoenfeld\325s first utterance. At the)J
60 237 :M
.889 .089(beginning \322you\323 is an unspecified person, perhaps one of the listeners. At the end)J
60 255 :M
.633 .063(it is Richard.])J
55 277 72 18 rC
95 291 :M
.462(AHS:)A
gR
gS 0 0 552 730 rC
132 291 :M
f1_12 sf
2.213 .221(The first question is generic: )J
f2_12 sf
2.66 .266(What piece of information would)J
132 309 :M
2.038 .204(make the problem easier to solve?)J
f1_12 sf
1.734 .173( [He turns to face the class.])J
132 327 :M
.563 .056(That\325s a really broad generic question. But you\325re facing a)J
132 345 :M
.514 .051(problem, it\325s posed in a particular way. Now you can ask)J
132 363 :M
1.029 .103(yourself is there some piece of information, some bit of)J
132 381 :M
.746 .075(knowledge, so that if you just had that, the problem would be)J
132 399 :M
.862 .086(significantly easier to solve? [To Richard] And you\325re nodding)J
132 417 :M
.757 .076(your head yes, what would it be? [Schoenfeld moves closer to)J
132 435 :M
.141(Richard.])A
55 439 72 18 rC
78 453 :M
.303(Richard:)A
gR
gS 0 0 552 730 rC
132 453 :M
f1_12 sf
.654 .065(Just to look around for the sum of the triples . . . and add the)J
132 471 :M
1.393 .139(three smallest numbers for the minimum, the three largest for)J
132 489 :M
.793 .079(the [inaudible].)J
55 493 72 18 rC
95 507 :M
.462(AHS:)A
gR
gS 0 0 552 730 rC
132 507 :M
f1_12 sf
.663 .066(OK. So the key piece of information is, or certainly a key piece)J
132 525 :M
1.045 .104(of information is: this says that the sum of each row, column,)J
132 543 :M
.697 .07(and diagonal should be the same, it would be awfully nice to)J
132 561 :M
.842 .084(know what that number is, so what is the sum? [He writes)J
132 579 :M
.789 .079(\322What is the sum?\323 on the board.] And we had a suggestion)J
132 597 :M
.926 .093(about how to think about this that I\325ll mention in a second. Let)J
132 615 :M
.89 .089(me throw some more jargon at you. This is called, as simple as)J
132 633 :M
.821 .082(it seems, in other contexts it\325s a little bit more complicated, and)J
132 651 :M
1.694 .169(worth having a name, )J
f2_12 sf
2.979 .298(Establishing Subgoals)J
f1_12 sf
(.)S
60 681 :M
1.133 .113(Now Schoenfeld starts on the work of answering the question What is the sum?)J
60 699 :M
.773 .077(by noting easy upper and lower bounds on the magic number\321it must be less)J
endp
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60 51 :M
f1_12 sf
.768 .077(than the sum of the largest three numbers in the magic square, 9, 8, and 7; and)J
60 69 :M
.76 .076(larger than the smallest three 3, 2, and 1.)J
60 99 :M
.445 .044(He invites a response from the class by saying [33] \322Is there anything else I can say)J
60 117 :M
.812 .081(about that sum?\323 Gary)J
185 114 :M
f4_8 sf
(1)S
189 114 :M
(2)S
193 117 :M
f1_12 sf
.887 .089( responds. He seems to assume that Schoenfeld is)J
60 135 :M
.643 .064(considering a magic square with 1, 2, and 3 in a row, column, or diagonal because)J
60 153 :M
.554 .055(he says \322You can narrow it even closer because if you used 1, 2, 3 in a single)J
60 171 :M
.834 .083(column, row, or diagonal then you know that you\325re going to be building)J
60 189 :M
.752 .075(something even larger, because 2 and 3 for instance are already gone so you have)J
60 207 :M
.352 .035(to use 4, 5, and 6.\323 [Here Gary responds to Schoenfeld\325s use of \322I\323 by using \322you\323)J
60 225 :M
.913 .091(both of which suggest that Schoenfeld is engaged in doing mathematics, rather)J
60 243 :M
.563 .056(than presenting a finished product. Gary\325s use of \322you\323 also suggests a collegial)J
60 261 :M
1.736 .174(relationship with Schoenfeld.])J
60 291 :M
1.089 .109(Rather than clarifying his earlier statements, Schoenfeld rephrases Gary\325s)J
60 309 :M
.741 .074(response: \322OK, so in some sense the very least I can get for a sum if somewhere)J
60 327 :M
.696 .07(I\325ve used 1, 2, and 3 in a row, that might try this row, that row, or something like)J
60 345 :M
.745 .075(that, the 3\325s going to be involved in another sum, and that\325s going to use at least 4)J
60 363 :M
.531 .053(and 5.\323 He writes in the empty grid)J
170 385 17 19 rC
174 400 :M
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169 386 1 18 rF
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205 386 1 18 rF
223 386 1 18 rF
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f1_12 sf
(2)S
gR
gS 0 0 552 730 rC
169 404 1 1 rF
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188 404 17 1 rF
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223 405 1 18 rF
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174 438 :M
f1_12 sf
(3)S
gR
gS 188 423 17 19 rC
192 438 :M
f1_12 sf
(4)S
gR
gS 206 423 17 19 rC
210 438 :M
f1_12 sf
(5)S
gR
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169 423 1 1 rF
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188 423 17 1 rF
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223 424 1 18 rF
223 442 1 1 rF
223 442 1 1 rF
60 469 :M
f1_12 sf
.533 .053(saying, \322And if that uses 4 and 5 . . . [his voice trails off and he pauses] What else)J
60 487 :M
.384 .038(can I say? [pause] This says that there\325s going to be one sum that\325s at )J
f2_12 sf
.363 .036(least )J
f1_12 sf
.666 .067(12. [pause])J
60 505 :M
.64 .064(Can you say anything else?\323)J
60 535 :M
.894 .089(During the next utterances the square on the blackboard undergoes the following)J
60 553 :M
.634 .063(changes \(an arrow indicates that the square to its left has been altered to yield the)J
60 571 :M
.735 .074(square on its right\):)J
170 593 17 19 rC
174 608 :M
(1)S
gR
gS 260 593 17 19 rC
264 608 :M
f1_12 sf
(1)S
gR
gS 296 593 17 19 rC
300 608 :M
f1_12 sf
(7)S
gR
gS 350 593 17 19 rC
354 608 :M
f1_12 sf
(1)S
gR
gS 368 593 17 19 rC
372 608 :M
f1_12 sf
(6)S
gR
gS 386 593 17 19 rC
390 608 :M
f1_12 sf
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gR
gS 0 0 552 730 rC
169 593 1 1 rF
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223 593 1 1 rF
223 593 1 1 rF
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259 593 1 1 rF
260 593 17 1 rF
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296 593 17 1 rF
313 593 1 1 rF
313 593 1 1 rF
349 593 1 1 rF
349 593 1 1 rF
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386 593 17 1 rF
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403 593 1 1 rF
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205 594 1 18 rF
223 594 1 18 rF
259 594 1 18 rF
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295 594 1 18 rF
313 594 1 18 rF
349 594 1 18 rF
367 594 1 18 rF
385 594 1 18 rF
403 594 1 18 rF
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f1_12 sf
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249 631 :M
psb
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174 646 :M
f1_12 sf
(3)S
gR
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f1_12 sf
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gS 206 631 17 19 rC
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f1_12 sf
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gR
gS 260 631 17 19 rC
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f1_12 sf
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f1_12 sf
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gS 296 631 17 19 rC
300 646 :M
f1_12 sf
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gR
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60 693 :M
f4_12 sf
( )S
60 690.48 -.48 .48 204.48 690 .48 60 690 @a
60 702 :M
f4_8 sf
(1)S
64 702 :M
(2)S
68 705 :M
f1_9 sf
-.078(This student did not stay long in the course. He does not appear in our overview of students.)A
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f1_12 sf
.484 .048(If you actually wanted to build it this way then you\325d go up on)J
132 73 :M
.686 .069(the right with 6, and 7 next.)J
55 77 72 18 rC
95 91 :M
.462(AHS:)A
gR
gS 0 0 552 730 rC
132 91 :M
f1_12 sf
.624 .062(Well, that\325s good, you go 3, 6, and 7. Is the argument now that)J
132 109 :M
.709 .071(every sum has to be at least 16? That\325s what it looks like we just)J
132 127 :M
.73 .073(proved, right? No matter what magic square you draw, you\325re)J
132 145 :M
.365 .036(going to get one sum that\325s going to add up to 16? [pause])J
55 149 72 18 rC
89 163 :M
.299(Diane:)A
gR
gS 127 149 360 18 rC
132 163 :M
f1_12 sf
.442 .044(No, because you could put the 3, 6, and 7 after the 1 [inaudible].)J
gR
gS 55 167 72 18 rC
95 181 :M
f1_12 sf
.462(AHS:)A
gR
gS 0 0 552 730 rC
132 181 :M
f1_12 sf
.623 .062(So the claim is, well I could put the 6 and 7 after the 1, that)J
132 199 :M
.542 .054(gives me a 14, but then I\325ve got to use an 8 and that says now)J
132 217 :M
.316 .032(I\325ve got a proof that I get at least a 16\321a 17. [pause] What\325s)J
132 235 :M
.655 .066(happening here? [pause] We already saw that there\325s a magic)J
132 253 :M
.713 .071(square with a 15, but it looks like we just proved that you\325ve)J
132 271 :M
.529 .053(got to get an 18. [pause] What\325s happening?)J
55 275 72 18 rC
96 289 :M
-.141(Gary:)A
gR
gS 0 0 552 730 rC
132 289 :M
f1_12 sf
.857 .086(Well, we know that we can\325t have 1, 2, and 3 in the same line)J
132 307 :M
.455 .046(anyway because we can\325t construct a magic square from it.)J
55 311 72 18 rC
95 325 :M
.462(AHS:)A
gR
gS 0 0 552 730 rC
132 325 :M
f1_12 sf
.848 .085([confidently and quickly in contrast to his previous utterances])J
132 343 :M
.547 .055(OK. What we just showed is if you start with a 1, 2, and 3 in a)J
132 361 :M
.696 .07(row then you\325re going to get some fairly large sums, that)J
132 379 :M
.526 .053(doesn\325t mean that every sum has to be that way. OK. [Erases)J
132 397 :M
.537 .054(square.] So the sums are going to be larger than 6.)J
60 427 :M
.678 .068([Gary\325s line of inquiry, trying 1, 2, and 3 in the same column to begin a solution to)J
60 445 :M
.656 .066(the magic square, was not quickly curtailed as in the inscribed square discussion,)J
60 463 :M
.683 .068(though it also does not lead to a solution. In fact Schoenfeld encourages Gary by)J
60 481 :M
.81 .081(writing his suggestions on the board and asking \322What else can I say?\323 \(though in)J
60 499 :M
.926 .093(a rather uncertain tone of voice\). This path is curtailed in an obvious sense by)J
60 517 :M
.907 .091(Schoenfeld, he erases the magic square and changes the subject. However, in)J
60 535 :M
.883 .088(contrast to the dead-ends in the inscribed square discussion, a student is involved)J
60 553 :M
.692 .069(as a collaborator in this action; Gary has noted that the assumption that 1, 2, and 3)J
60 571 :M
.674 .067(are in the same column of a magic square can\325t be true.])J
60 601 :M
.35 .035(He asks \322Is there any other way we can get a handle [on this] besides good)J
60 619 :M
.52 .052(guessing? And I don\325t at all, want to put good guessing down, a symmetry guess is)J
60 637 :M
.615 .061(an excellent way to go. Is there any other way we might get a handle on what this)J
60 655 :M
.872 .087(might be?\323 Devon responds that the sum of the three rows of the magic square)J
60 673 :M
.85 .085(must be equal to the sum of the numbers from 1 to 9. He then shows, using the)J
60 691 :M
.888 .089(grid, how its use might give rise to Devon\325s answer, and continues to show how)J
endp
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f1_12 sf
.781 .078(the observation yields a proof that the magic number is 15. The subgoal has been)J
60 69 :M
.325(achieved.)A
60 99 :M
.929 .093([Devon\325s suggestion has provided a \322natural\323 way to introduce Working)J
60 117 :M
1.063 .106(Backwards and to give an example of illuminating the source of a mathematical)J
60 135 :M
.753 .075(idea\321showing that Devon\325s suggestion is not a \322rabbit pulled out of a hat.\323)J
60 153 :M
1.027 .103(Schoenfeld\325s use of this suggestion to introduce Working Backwards, a heuristic)J
60 171 :M
.729 .073(which he would bring up during the magic square discussion in any case, is an)J
60 189 :M
.638 .064(example of the opportunism described by Hayes-Roth and Hayes-Roth \(1979\).])J
60 219 :M
f2_12 sf
1.946 .195(Second subgoal: What goes in the center? )J
f1_12 sf
2.061 .206(This is a natural moment to again)J
60 237 :M
.846 .085(invoke Establishing Subgoals since finding the center of the magic square is a)J
60 255 :M
.842 .084(useful next step. Schoenfeld pulls down the board with Establishing Subgoals and)J
60 273 :M
.583 .058(erases all but Establishing Subgoals. [Here is an example of traditional blackboard)J
60 291 :M
.73 .073(expertise and evidence of Schoenfeld\325s plan for this discussion.] He says,)J
96 327 :M
.92 .092(Since I have this statement, Establishing Subgoals, in a nice box on the)J
96 345 :M
.704 .07(board, why don\325t I take advantage of it again. We now know that the sum of)J
96 363 :M
.853 .085(each row, column, and diagonal is supposed to be 15. What\325s the next major)J
96 381 :M
1.112 .111(piece of information that would help me make significant progress on this)J
96 399 :M
.387(problem?)A
60 429 :M
.803 .08(He again uses P\227lya\325s method of questioning. Student 1)J
366 426 :M
f4_8 sf
(1)S
370 426 :M
(3)S
374 429 :M
f1_12 sf
.945 .094( responds \322What goes in)J
60 447 :M
1.015 .102(the center.\323 Schoenfeld answers \322Yeah. What goes in the center\323 and presents)J
60 465 :M
1.132 .113(another heuristic, Consider Extreme Cases. He then gives its mathematical)J
60 483 :M
1.638 .164(instantiation in this context.)J
55 505 72 18 rC
95 519 :M
.462(AHS:)A
gR
gS 0 0 552 730 rC
132 519 :M
f1_12 sf
.625 .063(So let\325s ask an extreme case, can 9 go in the center of the)J
132 537 :M
.486 .049(square? That\325s as extreme as you can get. [He writes 9 in the)J
132 555 :M
.561 .056(center of the square.])J
55 559 72 18 rC
69 573 :M
1.363 .136(Student 1:)J
gR
gS 0 0 552 730 rC
132 573 :M
f1_12 sf
.74(No.)A
55 577 72 18 rC
95 591 :M
.462(AHS:)A
gR
gS 0 0 552 730 rC
132 591 :M
f1_12 sf
2.693 .269(Why not?)J
55 595 72 18 rC
69 609 :M
1.363 .136(Student 1:)J
gR
gS 0 0 552 730 rC
132 609 :M
f1_12 sf
.65 .065(You run out of numbers that you can add pairs of to 9.)J
60 675 :M
f4_12 sf
( )S
60 672.48 -.48 .48 204.48 672 .48 60 672 @a
60 684 :M
f4_8 sf
(1)S
64 684 :M
(3)S
68 687 :M
f1_9 sf
-.052(This student was invisible to the camera and can\325t be identified with certainty. His voice appeared to be coming)A
60 696 :M
-.191(from the right side of the room.)A
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f1_12 sf
.763 .076(If the magic number is 15, that raises a serious problem,)J
132 73 :M
.477 .048(where\325s 8 going to go? If I put an 8 there [he writes 8 in upper)J
132 91 :M
.61 .061(left corner] I need a \3202 over there and I ain\325t got none. If I put)J
132 109 :M
.649 .065(an 8 there [the upper middle square], I need a \3202 over here and)J
132 127 :M
.401 .04(so on. OK? So 9 )J
f2_12 sf
.527 .053(can\325t )J
f1_12 sf
.541 .054(go in the center. [He erases 9.])J
60 157 :M
.836 .084([Here the code-shift from \322that raises a serious problem\323 to the attention-getting \322I)J
60 175 :M
.765 .077(ain\325t got none\323 emphasizes the reason why 9 can\325t go in the center of the square)J
60 193 :M
1.024 .102(and mirrors the student\325s rather awkward sentence. Writing and erasing serve to)J
60 211 :M
.744 .074(dramatize what Schoenfeld is saying and to display his reasoning.])J
60 241 :M
.569 .057(He continues more and more quickly through the cases of 8, 7, 6, gets to 5, says)J
60 259 :M
.726 .073(\322Maybe. How about the other extreme?,\323 writing 1 in the center of the square. He)J
60 277 :M
1.644 .164(continues and eliminates the remaining possibilities.)J
60 307 :M
1.001 .1(The subgoal of finding the center has been achieved. Schoenfeld doesn\325t point this)J
60 325 :M
.718 .072(out explicitly but makes the transition to the next activity by saying,)J
96 361 :M
.827 .083(Having gotten that far we could consider some trial and error. But we ought)J
96 379 :M
.716 .072(to at least take advantage of symmetry to see how much trial and error we)J
96 397 :M
.815 .081(really have to do. So let me ask the question, how many different places are)J
96 415 :M
.62 .062(there that we might stick a 1? There are really only two different places . . .)J
96 433 :M
.289 .029([corner and side pocket].)J
60 463 :M
1.157 .116(He then explicitly shows the symmetry he has mentioned several times, using)J
60 481 :M
.695 .07(hand gestures accompanied by his verbal description of rotating the board. [The)J
60 499 :M
.609 .061(gesture of rotating the board is an example of a deictic \(McNeill, 1992\). It is a visual)J
60 517 :M
1.238 .124(analogue of Schoenfeld\325s involved language: he, not some undescribed)J
60 535 :M
.878 .088(mechanism, is the rotater of the square. His gestures also allow him to give a)J
60 553 :M
1.199 .12(definition of symmetry without using technical language.])J
96 589 :M
.708 .071(There are really only two different places. If I had a solution with a 1 over)J
96 607 :M
.795 .079(here [writes 1 in the upper left corner] then\321and all the rest of these were)J
96 625 :M
.853 .085(filled in, I could take that solution [puts his right hand, crooked left, over)J
96 643 :M
.595 .059(the center of the grid and straightens his wrist], take the board, and rotate it)J
96 661 :M
.562 .056(90 degrees [he puts his right hand above the grid, his left below, and rotates)J
96 679 :M
.696 .07(them about the center of the grid so that the left hand ends above the grid],)J
96 697 :M
.815 .082(that gives me a solution with 1 over here [points to upper right corner]. Or)J
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.991 .099(equivalently, if I had a solution with 1 in the corner over here [points with)J
96 69 :M
.765 .077(his left hand to upper right corner], rotating it that way [his left hand moves)J
96 87 :M
.843 .084(up and to the right as he reverses his previous rotation gesture] gives me a)J
96 105 :M
.908 .091(solution with the 1 over here [points to the 1 in upper left corner]. Same for)J
96 123 :M
.828 .083(the other corners [points to the two lower corners]. So a solution with a 1 in)J
96 141 :M
.916 .092(the corner is equivalent to, or generates a solution with a 1 in any other)J
96 159 :M
.771 .077(corner. Similarly for 1 in a side pocket. That generates any of these [pointing)J
96 177 :M
.653 .065(to each side pocket in turn].)J
60 207 :M
1.065 .107(Another heuristic is quickly noted \(it\325s just been illustrated\); Schoenfeld writes)J
60 225 :M
.783 .078(Exploit Symmetry, saying \322That\325s another strategy that comes in handy\323 and)J
60 243 :M
.742 .074(returns to the work at hand. He writes 1 in the corner of the square, notes that 9)J
60 261 :M
.797 .08(must go in the corner diagonally opposite, and discusses the placement of 2. Using)J
60 279 :M
.926 .093(symmetry one need only check three places. Schoenfeld indicates each and shows)J
60 297 :M
.798 .08(that no matter where 2 is placed some row, column, or diagonal will not add up to)J
60 315 :M
.849 .085(15. He concludes \322So what I\325ve just showed is there\325s no solution with a 1 in the)J
60 333 :M
.612 .061(corner. That leaves us a 1 in the side.\323 He erases 1 and writes 1 in the side pocket,)J
60 351 :M
.699 .07(discusses the placing of 2 and finishes the solution. [Looking at the case where 1 is)J
60 369 :M
1.036 .104(in the corner first makes the discussion smoother since this case doesn\325t hold.)J
60 387 :M
1.176 .118(Such a choice is usual in both classroom and professional mathematical)J
60 405 :M
.184(presentations.])A
60 435 :M
.922 .092(Now he summarizes, \322What we\325ve proved along the way that the 1 has to go in)J
60 453 :M
.706 .071(the side pocket, the 2 has to go in one of the two bottom positions opposite, and)J
60 471 :M
.899 .09(the rest is forced, so the answer is that\325s the only solution modulo symmetry,)J
60 489 :M
1.119 .112(which answers Devon\325s question.\323 [A mathematician who solves a problem posed)J
60 507 :M
.576 .058(by person X will, especially if X is famous, frequently say in an account of the)J
60 525 :M
.824 .082(solution \322This answers a question of X.\323 The episode of Devon\325s posing of a)J
60 543 :M
1.003 .1(problem to its solution outlines in miniature the way a problem is posed and)J
60 561 :M
1.672 .167(solved amongst professional mathematicians.])J
60 591 :M
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.222 .022(Second \(re\)solution: By working forward)J
60 621 :M
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1.051 .105(Schoenfeld asks the ritual question \322Are we done?\323 and Jeff replies \322We\325re never)J
60 639 :M
1.145 .114(done.\323 \(The students are beginning to internalize the new classroom rituals.\))J
60 657 :M
1.085 .108(Schoenfeld replies \322You\325re learning\323 and makes the transition to a new activity)J
60 675 :M
.639 .064(\322What I want to do is to go back to this problem in an entirely different way,\323)J
60 693 :M
.909 .091(summarizing the approach used before, erasing the board, then giving a)J
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.903 .09(description of the new approach which he\325s termed \(1991\) Working Forward. Here)J
60 69 :M
.866 .087(he doesn\325t label it, just describes and gives an instantiation of this)J
60 87 :M
1.038 .104(approach\321listing triples whose sum is 15.)J
60 117 :M
.859 .086(Now he initiates the students\325 participation by asking for triples. Different students)J
60 135 :M
1.09 .109(call out responses hastened by Schoenfeld\325s \322Any more?\323 or \322Another one?\323)J
60 153 :M
.573 .057(which follows quickly after he writes each triple on the board. He lists 159, 294, 258,)J
60 171 :M
.381 .038(168, 357, and 195, says \322Oops, we got that already\323 and crosses it out. They continue)J
60 189 :M
.673 .067(456, 762, and stop. Schoenfeld says \322Are we done, is that all of them?\323 A student)J
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.913 .091(produces 834 and Schoenfeld asks \322Are there any more?\323 No one replies.)J
60 237 :M
1.043 .104(Schoenfeld tells the class, \322This is now something like the 142nd time I\325ve used)J
60 255 :M
.958 .096(this particular problem, 142nd time I\325ve asked this particular question, \322Are there)J
60 273 :M
.745 .075(any more?,\323 and I get to ask the same next question for the 142nd time: How the)J
60 291 :M
.868 .087(hell would you know? You\325ve sort of generated them randomly, so you got a)J
60 309 :M
.975 .098(whole bunch of them, but you might of caught them all and you might not.\323)J
60 339 :M
1.026 .103([Here again code-shifting serves several goals: \322hell\323 emphasizes the seriousness)J
60 357 :M
1.026 .103(of the students\325 dilemma, and \322generated\323 and \322random\323 suggest mathematical)J
60 375 :M
1.047 .105(affiliation. As he says to the students, Schoenfeld has seen classes implement his)J
60 393 :M
.881 .088(suggestion of listing triples of numbers unsystematically before. Here again, in)J
60 411 :M
.736 .074(contrast to the inscribed square discussion he\325s allowed, in fact encouraged, the)J
60 429 :M
.83 .083(class to follow a path which will not easily lead to a solution. The nature of the)J
60 447 :M
.932 .093(magic square makes this dead-end more quickly reached and more obvious than)J
60 465 :M
.825 .083(those occurring in the inscribed square discussion. One reason for doing this is to)J
60 483 :M
.837 .084(show the students that they\325re in need of his teaching as well as the heuristic, Be)J
60 501 :M
1.044 .104(Systematic. Another is to illustrate the issue of control, the students don\325t know)J
60 519 :M
1.004 .1(how to implement his suggestion in such a way that they know when they\325ve)J
60 537 :M
1.004 .1(achieved it.])J
60 567 :M
1.016 .102(He mentions the strategy whose omission led the class into its predicament,)J
60 585 :M
.806 .081(writing \322IT HELPS TO BE SYSTEMATIC!\323 on the side board. He summarizes the)J
60 603 :M
.674 .067(difficulty, pointing to the crossed out triple 195 which serves as a record of the)J
60 621 :M
.648 .065(students\325 activity and suggests a way to instantiate the strategy\321listing the triples)J
60 639 :M
.852 .085(in increasing order and beginning the list with all the triples that start with 1.)J
60 684 :M
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-.057(It was difficult to identify the students who participated in this segment, but possible to tell that different students)A
60 705 :M
.04 .004(were calling out triples.)J
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.776 .078(To start this new path he erases the unsystematic list of triples from the board and)J
60 69 :M
.702 .07(starts the new list with 1 5 9. The class calls out the rest of the triples. Schoenfeld)J
60 87 :M
.962 .096(points out the connection between the triples and the magic square,)J
96 123 :M
.5 .05(So we\325ve got a total of eight triples, . . . , that\325s nice, because there are eight)J
96 141 :M
.994 .099(rows, columns and diagonals. Now what was the most important square? In)J
96 159 :M
1.019 .102(the magic square? The middle. How many sums was that square involved)J
96 177 :M
.812 .081(in? [Here he uses the empty magic square to calculate, drawing horizontal,)J
96 195 :M
.929 .093(vertical, and diagonal lines through the middle square to show that it is)J
96 213 :M
.917 .092(involved in four sums.] How many digits appear four times? Only the 5,)J
96 231 :M
.771 .077(that\325s the only digit that appears four times. So if there\325s a solution: Guess)J
96 249 :M
.776 .078(what, this is a completely independent proof, 5 has to go in the center)J
96 267 :M
.446 .045(square. [He writes 5 in the center.])J
60 297 :M
.854 .085(He uses this idea to show where the other numbers in the magic square must be)J
60 315 :M
.845 .085(placed\321numbers which appear in only two of the listed triples must go in \322side)J
60 333 :M
.857 .086(pockets\323 and numbers which appear in three of the listed triples must go in)J
60 351 :M
.262(corners.)A
60 381 :M
.99 .099(The finding of two solutions has been enacted.)J
96 417 :M
.487 .049(Now we\325ve beat it to death. Are we done? [He pauses and looks at the class.])J
96 435 :M
.596 .06(Of course not, because so far we\325ve only solved the problem I gave you. If)J
96 453 :M
1.081 .108(that\325s how mathematics progressed, mathematics wouldn\325t progress.)J
96 471 :M
1.078 .108(Solving known problems is not what mathematicians get paid for nor is it)J
96 489 :M
1.216 .122(anything they have fun doing.)J
60 519 :M
1.214 .121(Schoenfeld\325s closing statement illustrates some themes of the course, that)J
60 537 :M
1.203 .12(problems may have multiple solutions and that solving a given problem is only)J
60 555 :M
.765 .077(the beginning \(problems can be generalized, extended, etc.\). This is one of the)J
60 573 :M
.757 .076(aspects of the course that reflects mathematical practice \(Kitcher, 1984\). After)J
60 591 :M
1.091 .109(Schoenfeld\325s statement, students suggested extensions and generalizations of the)J
60 609 :M
.339 .034(magic square. In session 3, the class discussed ways of generating 3 by 3 magic)J
60 627 :M
.908 .091(squares with entries other than the numbers from 1 to 9. In session 4, after finding)J
60 645 :M
.864 .086(there are no non-trivial 2 by 2 magic squares, the students conjectured that there is)J
60 663 :M
.915 .091(no even-dimensional magic square, Mitch discussed a procedure for generating a 5)J
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.434 .043(by 5 magic square, Christina)J
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.45 .045( described a procedure for generating one of odd)J
60 69 :M
.749 .075(dimension, and Devon showed that the magic number of a 3 by 3 magic square is)J
60 87 :M
.703 .07(always divisible by 3 and that the number in the center is always one third of the)J
60 105 :M
.616 .062(magic number. This was followed by a discussion of what a magic cube might be.)J
60 123 :M
1.184 .118(In session 9 Schoenfeld provided a counterexample to the conjecture that no non-)J
60 141 :M
.993 .099(trivial magic squares of even dimension exist by showing the students an)J
60 159 :M
1.02 .102(engraving of D\237rer\325s )J
f2_12 sf
.247(Melancholia)A
f1_12 sf
.776 .078( which depicts a 4 by 4 magic square.)J
214 203 :M
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.229 .023(Why teach a class this way?)J
60 233 :M
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.94 .094(Implicit in the preceding description is the question of why Schoenfeld chooses to)J
60 251 :M
.788 .079(conduct his class in the manner he does. A proof that the magic square has a)J
60 269 :M
1.183 .118(solution that is unique modulo symmetry could have been given in far less time.)J
60 287 :M
.792 .079(Why do it this way?)J
60 317 :M
.711 .071(I\325ll begin with the issue of blackboard writing. The writing that appeared on the)J
60 335 :M
.696 .07(blackboard was devoted to names of heuristics, diagrams, an equation, and a)J
60 353 :M
.866 .087(question. One might consider Schoenfeld\325s blackboard writing to be in conflict)J
60 371 :M
.936 .094(with traditional mathematical practice, since it differs greatly from the kind of)J
60 389 :M
.913 .091(writing seen in textbooks and articles and in other mathematics classrooms. It was,)J
60 407 :M
.447 .045(as Schoenfeld says, \322sparse and sloppy\323 \(audio taped discussion, March 8, 1991\))J
60 425 :M
.866 .087(while that of textbooks, articles, and traditional mathematics classes is profuse and)J
60 443 :M
.094(precise.)A
60 473 :M
1.027 .103(Certainly writing is an important part of mathematical discourse. However, its)J
60 491 :M
1.075 .108(relationship with the way mathematics is done is not obvious to those who aren\325t)J
60 509 :M
1.497 .15(mathematicians. Mathematicians\325 descriptions of mathematics show that writing)J
60 527 :M
.871 .087(is but one way of communicating mathematically. Davis and Hersh\325s \(1986\) Ideal)J
60 545 :M
1.161 .116(Mathematician communicates results to fellow experts \322in a casual shorthand\323 but)J
60 563 :M
1.073 .107(in published writings \322follows an unbreakable convention: to conceal any sign)J
60 581 :M
.901 .09(that the author or the intended reader is a human being.\323 Stewart \(1993\) points out)J
60 599 :M
.299(that,)A
96 635 :M
1.256 .126(Much of mathematics is communicated by informal discussions over coffee,)J
96 653 :M
1.037 .104(seminars, lectures, and other media that do not produce permanent records.)J
96 671 :M
1.575 .158(When important mathematical ideas are \322in the air,\323 other mathematicians)J
60 693 :M
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-.052(Christina was a computer science major who entered the class in the third session.)A
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.794 .079(get to hear of them by these informal routes, long before anything appears)J
96 69 :M
1.023 .102(in a technical journal. \(p. 121\))J
60 99 :M
1.4 .14(Thurston\325s description of mathematical communication gives a sense of the)J
60 117 :M
1.398 .14(differences between spoken informal mathematics and formal written)J
60 135 :M
.397(mathematics.)A
96 171 :M
1.204 .12(One-on-one, people use wide channels of communication that go far)J
96 189 :M
.866 .087(beyond formal mathematical language. They use gestures, they draw)J
96 207 :M
.559 .056(pictures and diagrams, they make sound effects and use body language.)J
96 225 :M
.936 .094(Communication is more likely to be two-way, so that people can concentrate)J
96 243 :M
.894 .089(on what needs the most attention. . . . In talks, people are more inhibited)J
96 261 :M
.905 .09(and more formal. . . . In papers, people are still more formal. Writers)J
96 279 :M
.592 .059(translate their ideas into symbols and logic, and readers try to translate back.)J
96 297 :M
-.015(\(1994, p. 166\))A
60 327 :M
.494 .049(The description of the genesis of a proof by De Millo et al. \(1986\) suggests that)J
60 345 :M
1.016 .102(written mathematics is the end of a long process which begins with informal)J
60 363 :M
.536(communication.)A
96 399 :M
.691 .069(In its first incarnation, a proof is a spoken message, or at most a sketch on a)J
96 417 :M
.627 .063(chalkboard or a paper napkin. That spoken stage is the first filter for a proof.)J
96 435 :M
.892 .089(If it generates no excitement or belief among his friends, the wise)J
96 453 :M
1.035 .104(mathematician reconsiders it. But if they find it tolerably interesting and)J
96 471 :M
.699 .07(believable, he writes it up. After it has circulated in draft for a while, if it still)J
96 489 :M
.878 .088(seems plausible, he does a polished version and submits it for publication. If)J
96 507 :M
.787 .079(the referees also find it attractive and convincing, it gets published so it can)J
96 525 :M
.108 .011(be read by a wider audience. \(p. 272\))J
60 555 :M
.876 .088(This aspect of mathematics is generally hidden from students \(Rogers, 1992\). One)J
60 573 :M
.745 .075(doesn\325t often see the genesis of a proof in a classroom, instead one sees the end-)J
60 591 :M
.54 .054(product of the process described above, presented in detached language that erases)J
60 609 :M
.965 .097(its author and origins. Such classroom experiences help to explain why students\325)J
60 627 :M
1.089 .109(ideas about the nature of mathematics are sometimes so very different from those)J
60 645 :M
1.372 .137(of mathematicians. Some students may not even believe that mathematics is done)J
60 663 :M
.748 .075(by human beings \(Belenky et al., 1986\) just as some city children used to believe)J
60 681 :M
1.062 .106(that milk grows in bottles. Work in cognitive science has shown that students\325)J
60 699 :M
.886 .089(beliefs about the nature of a subject may have profound effects on their learning of)J
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1.041 .104(it \(McLeod, 1992\). De-emphasizing writing and formal mathematics, not only)J
60 69 :M
.76 .076(reflects mathematical practice, but may also change students\325 beliefs about the)J
60 87 :M
1.205 .12(nature and the doing of mathematics.)J
60 117 :M
1.375 .138(De-emphasizing writing and formal mathematics may have other consequences)J
60 135 :M
.898 .09(for students\325 learning. Thurston suggests that familiarity with the ideas of a)J
60 153 :M
.783 .078(subfield of mathematics may need to precede the ability to recognize the same)J
60 171 :M
1.22 .122(ideas in written form.)J
96 207 :M
.867 .087(People familiar with ways of doing things in a subfield recognize various)J
96 225 :M
1.162 .116(patterns of statements or formulas as idioms or circumlocutions for certain)J
96 243 :M
.905 .09(concepts or mental images. But to people not already familiar with what\325s)J
96 261 :M
1.123 .112(going on the same patterns are not very illuminating; they are often even)J
96 279 :M
.771 .077(misleading. The language is not alive except to those who use it. \(p. 167\))J
60 309 :M
1.004 .1(In general, students of mathematics, like those new to a subfield of mathematics,)J
60 327 :M
1.009 .101(are not already familiar with what\325s going on. Mathematicians who want to learn)J
60 345 :M
.61 .061(about a subfield usually ask what the ideas, questions, and objects of that subfield)J
60 363 :M
1.082 .108(are. Unlike mathematicians, students may not know to ask those questions and to)J
60 381 :M
1.543 .154(look for idioms and circumlocutions in written mathematics. When they)J
60 399 :M
1.06 .106(encounter written mathematics they may be focused on its form rather than its)J
60 417 :M
.891 .089(meaning; reading each line of a proof, rather than trying to understand the ideas)J
60 435 :M
.874 .087(behind it. More importantly, they may not have any sense that such ideas exist.)J
60 465 :M
1.027 .103(This suggests that an emphasis on formal written mathematics causes difficulty for)J
60 483 :M
1.052 .105(students, both from a cognitive and a metacognitive perspective. Students don\325t)J
60 501 :M
1.173 .117(appear to perceive and interpret formal mathematics as mathematicians do.)J
480 498 :M
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.985 .098(Their beliefs about the way mathematics is done and hence how they should learn)J
60 537 :M
1.122 .112(mathematics are derived from presentations of finished products. It seems)J
60 555 :M
.931 .093(unlikely that their beliefs could be changed by seeing even more formal)J
60 573 :M
1.218 .122(mathematics, particularly since the students\325 means of interpreting that formal)J
60 591 :M
.934 .093(mathematics would have to be addressed at the same time. Biographies of many)J
60 609 :M
1.468 .147(mathematicians suggest that informal mathematical experiences, often occurring)J
60 627 :M
1.202 .12(outside of the classroom, were an important factor in their mathematical)J
60 645 :M
1.027 .103(development \(see for instance, Albers & Alexanderson, 1985; Albers,)J
60 663 :M
1.032 .103(Alexanderson, & Reid, 1990; Hersh & John-Steiner, 1993; Ulam, 1976; Weil, 1992\).)J
60 684 :M
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-.105(In cognitive science terms, students don\325t appear to have schemata for formal mathematics similar to those of)A
60 705 :M
-.167(mathematicians.)A
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.971 .097(Thurston\325s statement, \322The language is not alive except to those who use it\323 and)J
60 69 :M
1.041 .104(Schoenfeld\325s \(1994\) statement, \322When mathematics is taught as dry, disembodied,)J
60 87 :M
.668 .067(knowledge to be received, it is learned \(and forgotten or not used\) in that way\323)J
60 105 :M
1.255 .126(outline extreme cases which might illuminate the problem of how to teach)J
60 123 :M
.951 .095(students mathematics. What I have tried to suggest in analyzing the language of)J
60 141 :M
1.001 .1(Schoenfeld\325s classroom discussion, is that the discussion is an example of)J
60 159 :M
1.273 .127(embodying mathematics\321presenting it as a particular kind of communication to)J
60 177 :M
.666 .067(be used by all the people in that classroom, rather than as knowledge to be learned.)J
60 195 :M
1.088 .109(The complexity inherent in the word \322communication\323 is suggested by listing)J
60 213 :M
1.368 .137(some of its components \(Saville-Troike, 1989\): linguistic knowledge; interaction)J
60 231 :M
1.146 .115(skills \(this includes perception of salient features and norms of interpretation\);)J
60 249 :M
1.102 .11(cultural knowledge \(this includes values, attitudes, and schemata\). In this view,)J
60 267 :M
1.216 .122(transmission of knowledge and skills is just one aspect of communication.)J
60 285 :M
1.641 .164(Similarly, communication among mathematicians is not restricted to formal)J
60 303 :M
1.023 .102(writing, it includes other methods: informal writing, as well as talking, gesturing.)J
60 321 :M
1.214 .121(Moreover, values, attitudes, and schemata are an important part of mathematical)J
60 339 :M
1.21 .121(communication. The work of Schoenfeld and others suggests that these other)J
60 357 :M
1.276 .128(aspects of mathematical communication play an important role in students\325)J
60 375 :M
.94 .094(learning of mathematics. Schoenfeld\325s classroom suggests that such aspects of)J
60 393 :M
1.158 .116(mathematical communication may be taught inside as well as outside the)J
60 411 :M
1.354 .135(classroom and thus, unlike me and many other mathematicians, students need)J
60 429 :M
1.145 .114(not wait until they begin doing research to start communicating mathematically.)J
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-.326(Concluding discussion)A
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.744 .074(In this section we synthesize our analyses of the early stages of Schoenfeld\325s)J
60 113 :M
1.104 .11(problem solving course and offer some implications. We began this article with an)J
60 131 :M
1.239 .124(illustration of some long-term goals of Schoenfeld\325s problem solving course: That)J
60 149 :M
1.182 .118(the class become a \322mathematical community\323 advancing and defending)J
60 167 :M
1.089 .109(conjectures and proofs on mathematical grounds; and that the locus of authority)J
60 185 :M
1.189 .119(be the \322mathematical community,\323 not the teacher. Because students\325 experiences)J
60 203 :M
1.134 .113(in mathematics classrooms are, in general, very different from those of the)J
60 221 :M
.888 .089(community he wishes to create, achieving these goals is not easy and the path)J
60 239 :M
.76 .076(from the beginning of the course to a microcosm \322of selected aspects of)J
60 257 :M
.982 .098(mathematical practice and culture\323 \(Schoenfeld, 1994, p. 66\) is not obvious. Rather)J
60 275 :M
.878 .088(than examining its later stages when its beginnings were likely to be invisible, we)J
60 293 :M
.803 .08(focused on the course at its inception. Our initial question was: How does)J
60 311 :M
1.152 .115(Schoenfeld create a community of problem solvers where undergraduates learn to)J
60 329 :M
1.119 .112(think and )J
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.35(do)A
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1.453 .145( mathematics, when their past experience in mathematics has mainly)J
60 347 :M
1.442 .144(involved listening, writing notes, and learning procedures?)J
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.811 .081(Short-term goals)J
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1.017 .102(After twelve or more years of schooling, undergraduates usually have well-)J
60 425 :M
1.033 .103(developed expectations about how mathematics classes will run and how)J
60 443 :M
1.074 .107(mathematics teachers will behave. Instructors of courses that differ from these)J
60 461 :M
1.105 .111(expectations often find that students question their competence, the value of the)J
60 479 :M
.633 .063(course, or what they are expected to do in the class. Because Schoenfeld\325s course is)J
60 497 :M
.915 .091(an elective, if students decide he is not competent, that the course is not of value,)J
60 515 :M
.659 .066(or they don\325t understand what they will be asked to do, they may well leave the)J
60 533 :M
.827 .083(course. The students who stay in the course will need to understand what they are)J
60 551 :M
1 .1(expected to do. Schoenfeld\325s path to achieving a \322classroom mathematical)J
60 569 :M
1.37 .137(community\323 includes the short-term goals of:)J
96 605 :M
.93 .093(\245 establishing his \322credentials\323;)J
96 623 :M
.92 .092(\245 showing the students that heuristics are an important part of)J
105 641 :M
.397(mathematics;)A
96 659 :M
.684 .068(\245 giving the students a sense of what the course is about;)J
96 677 :M
1.327 .133(\245 communicating his expectations for classroom behavior.)J
420 674 :M
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-.012(Or as Schoenfeld put it \(audiotaped discussion, May 22, 1991\) \322letting them know what they\325re in for.\323)A
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.546 .055(The first two goals are related to an ancient pedagogical problem \(cf. Plato\325s)J
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.259(Protagoras)A
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.907 .091(\): How can a student ignorant of a subject judge whether or not)J
60 99 :M
.736 .074(someone is capable of teaching it? The last two goals address a similar problem:)J
60 117 :M
.912 .091(How can students be asked to do mathematics \(in some ways\) like mathematicians)J
60 135 :M
1.351 .135(if they have no idea how mathematicians do mathematics?)J
60 165 :M
1.36 .136(Schoenfeld\325s solution includes illustration and enactment. Here we use)J
60 183 :M
1.128 .113(\322enactment\323 in a somewhat theatrical sense. His introduction to the course and)J
60 201 :M
.984 .098(his treatment of the telescoping series portrayed him, though by different means,)J
60 219 :M
1.252 .125(as both a member and critic of the mathematical community. The introduction)J
60 237 :M
1.128 .113(was a monologue not involving the students. In contrast the caricature of the)J
60 255 :M
.969 .097(\322typical calculus professor\323 enacted the distinction between how mathematics is)J
60 273 :M
1.023 .102(presented in classrooms and how it is done by mathematicians using heuristics.)J
60 291 :M
1.079 .108(Rather than telling the students about the drawbacks of traditional mathematics)J
60 309 :M
1.19 .119(teaching, Schoenfeld depicted them in his caricature, then modeled the solution of)J
60 327 :M
1.072 .107(a mathematician. During the discussion of the next two problems the students)J
60 345 :M
1.264 .126(responded with traditional behaviors and then, with Schoenfeld\325s prompting,)J
60 363 :M
.967 .097(enacted some of the mathematical behaviors that he was trying to establish.)J
60 393 :M
.888 .089(The discussion of the inscribed square illustrated the power of heuristics and the)J
60 411 :M
.927 .093(skill required to use them successfully. Students will, on average, not succeed in)J
60 429 :M
.744 .074(showing that a square can be inscribed in an arbitrary triangle, whether or not the)J
60 447 :M
.647 .065(heuristic Try an Easier Related Problem is suggested, so they will consider it a)J
60 465 :M
.922 .092(difficult problem. Because that difficult problem will yield, when an appropriate)J
60 483 :M
.781 .078(heuristic is suggested and its use scaffolded \(Collins, Brown, & Newman, 1989\), the)J
60 501 :M
.849 .085(inscribed square problem serves to show the power of heuristics in obtaining a)J
60 519 :M
1.067 .107(solution\321as well as the skill required to use them successfully. Allowing students)J
60 537 :M
.626 .063(to struggle may be an essential part of this process, both in showing the power of)J
60 555 :M
1.071 .107(heuristics and Schoenfeld\325s ability to teach them. Students often aren\325t conscious)J
60 573 :M
1.036 .104(of the important role that non-traditional teachers\325 suggestions and questions play)J
60 591 :M
1.019 .102(in their progress toward a solution and sometimes conclude that such teachers)J
60 609 :M
1.364 .136(don\325t know very much mathematics\321otherwise they would tell them the answer.)J
60 627 :M
1.235 .124(The first student presentation gave Schoenfeld an opportunity to mention a)J
60 645 :M
1.106 .111(traditional student behavior, looking to the teacher for approval, and to have one)J
60 663 :M
1.039 .104(of his expectations for classroom behavior enacted, that students not look to him)J
60 681 :M
.74 .074(for approval. This was a step toward the long-term goal of shifting the locus of)J
60 699 :M
.713 .071(authority away from the teacher and having the class, aided by Schoenfeld\325s \322nasty)J
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.947 .095(questions,\323 develop its own standards of correctness. The presentation also)J
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1.172 .117(allowed the enactment of another expectation, that of \322not just beautiful finished)J
60 87 :M
.942 .094(solutions but also explanations of how and why.\323)J
60 117 :M
.859 .086(The discussion of the magic square served different goals: illustrating uses of)J
60 135 :M
1.056 .106(heuristics and the theme of multiple solutions, and as a vehicle to engage students)J
60 153 :M
.782 .078(in discussion. It showed that different heuristics can be used singly or in)J
60 171 :M
1.013 .101(combination to solve the same problem. \(In contrast, the inscribed square showed)J
60 189 :M
.806 .081(that the same heuristic can be used in different ways on the same problem, a)J
60 207 :M
1.103 .11(different illustration of the theme of multiple solutions.\) Because it is easy to)J
60 225 :M
.833 .083(solve, the magic square could not be used to illustrate the power of heuristics in)J
60 243 :M
.896 .09(obtaining a solution. Instead it allowed students to focus on the use of heuristics.)J
60 261 :M
.851 .085(Students find generalizations of the magic square easy,)J
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1.317 .132( hence Schoenfeld could)J
60 279 :M
.889 .089(and did use it to have students enact another long-term goal of the course: That)J
60 297 :M
1.086 .109(students take problems and make them their own by extension or generalization.)J
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-.335(Pedagogy)A
60 357 :M
.541 .054(Planning, direction, and authority)J
60 387 :M
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1.058 .106(The sequence of problems and associated activities \(Schoenfeld\325s introduction to)J
60 405 :M
1.28 .128(the course, lectures, reflective presentations, student presentations, small-group)J
60 423 :M
.8 .08(work, and whole-class discussions\) give an overall structure for the first days of)J
60 441 :M
1.105 .11(the course. The problems are not chosen to cover content in the traditional sense,)J
60 459 :M
.928 .093(but to make certain points about heuristics and the course. Schoenfeld has used)J
60 477 :M
.523 .052(the telescoping series, inscribed square, and magic square for years. He was thus)J
60 495 :M
.926 .093(familiar with probable student responses to each in the contexts that he provides.)J
60 513 :M
.777 .078(For example, he knew what students are likely to do with the inscribed square)J
60 531 :M
1.007 .101(without heuristics, and with the heuristics Try an Easier Related Problem and)J
60 549 :M
.991 .099(Relax a Condition. In this sense he controlled the class in the same way someone)J
60 567 :M
.926 .093(who digs a ditch controls the water flowing through it: The overall structure for)J
60 585 :M
1.129 .113(the course channeled students\325 \322natural\323 responses in directions that served many)J
60 603 :M
.908 .091(of Schoenfeld\325s goals. At some points \(for example in the discussion of \322easier)J
60 621 :M
.955 .096(related problems\323 for the inscribed square\) without additional direction students)J
60 639 :M
1.42 .142(might have become entangled in a fruitless exploration\321an authentic)J
60 657 :M
1.011 .101(mathematical experience, but one which was not likely to encourage students to)J
60 684 :M
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.016 .002(\322Easy\323 like \322difficult\323 has an operational definition in this context\321if, on average, students readily suggest)J
60 705 :M
-.05(generalizations of this problem, then generalizations of the problem can be said to be easy.)A
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.977 .098(stay in the course or have confidence in its teacher. At such points Schoenfeld)J
60 69 :M
.882 .088(used his authority as a teacher and mathematician to guide the flow of the)J
60 87 :M
.921 .092(discussion. However, a goal of the course is that the teacher not remain the sole)J
60 105 :M
1.085 .108(authority. Schoenfeld\325s delegation of authority to the students during Devon\325s)J
60 123 :M
.78 .078(presentation of his solution for the inscribed square problem was a step toward)J
60 141 :M
.962 .096(satisfying this long-term goal.)J
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.137(Opportunism)A
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1.096 .11(Within the structure imposed by the sequence of problems, the heuristics)J
60 219 :M
1.081 .108(illustrated by the problems, and the activities surrounding them there is room for)J
60 237 :M
.923 .092(opportunism \(Hayes-Roth & Hayes-Roth, 1979; Schoenfeld et al., 1992\).)J
60 255 :M
1.052 .105(Schoenfeld\325s knowledge of the problems \(and of mathematics\) allowed him to take)J
60 273 :M
.896 .09(advantage of student remarks such as Devon\325s suggestion about a solution for the)J
60 291 :M
.626 .063(magic square. Here the analogy might be to a navigator who knows how to get to a)J
60 309 :M
.761 .076(particular location in any event, but is able to take advantage of an unexpected)J
60 327 :M
.941 .094(wind not only to arrive, but arrive sooner. In this case, Devon\325s suggestion)J
60 345 :M
1.034 .103(provided a context not only for discussing Working Backwards which Schoenfeld)J
60 363 :M
.571 .057(would do in any case \(Schoenfeld, 1991\), but showing it as a possible source for)J
60 381 :M
.854 .085(Devon\325s suggestion. This satisfied the goal of discussing Working Backwards and)J
60 399 :M
1 .1(additional goals: incorporating student suggestions and again illustrating the)J
60 417 :M
1.168 .117(notion of \322not just beautiful finished solutions but also explanations of how and)J
60 435 :M
.085(why.\323)A
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.959 .096(Discourse and communication)J
60 495 :M
f1_12 sf
1.155 .115(Schoenfeld also used P\227lya\325s method of questioning to involve the students in)J
60 513 :M
.999 .1(using heuristics. This method of questioning has social and cognitive aspects. On)J
60 531 :M
.928 .093(the one hand, Schoenfeld was asking for a response from the students which got)J
60 549 :M
1.068 .107(them talking, helping to begin the community he wished to establish. On the)J
60 567 :M
1.008 .101(other, the method of questioning scaffolded the students\325 applications of heuristics)J
60 585 :M
1.31 .131(to particular cases. Other features of classroom communication \(involved)J
60 603 :M
.835 .084(language, involved gestures, informal blackboard writings\) suggested that the class)J
60 621 :M
1.179 .118(was doing mathematics rather than being presented with mathematics.)J
60 651 :M
1.306 .131(In summary, we suggest that important elements in achieving the short-term)J
60 669 :M
.478 .048(goals for the first days of the course were:)J
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.89 .089(\245 the sequence of problems Schoenfeld used;)J
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.719 .072(\245 his knowledge of probable student responses to the problems;)J
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.916 .092(\245 his knowledge of possible solutions to the problems and the heuristics)J
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1.429 .143(that generate them;)J
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.81 .081(\245 his use and delegation of authority;)J
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1.088 .109(\245 his patterns of written and oral communication, and classroom discourse.)J
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.064(Implications)A
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1.006 .101(We will not venture to draw universal implications from a study of two days in)J
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.914 .091(one classroom. Nor is our intent to prescribe a teaching method. Instead, we hope)J
60 237 :M
1.213 .121(this example will help to illuminate the difficult task of teaching mathematics.)J
60 267 :M
1.339 .134(Teaching is sometimes dichotomized as either transmission or discovery.)J
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.632 .063( In the)J
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.833 .083(language of calculus reform \(e.g., )J
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1.284 .128(UME Trends)J
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.634 .063(, 1995\) a teacher is either a \322sage on)J
60 303 :M
.799 .08(the stage\323 or \322guide on the side.\323 Because it contains elements of both,)J
60 321 :M
1.323 .132(Schoenfeld\325s teaching provides a counterexample to this notion. The form)J
60 339 :M
.983 .098(\(though not always the content\) of his presentations to the class contains)J
60 357 :M
.897 .09(traditional elements such as lecturing and blackboard expertise, but he combines)J
60 375 :M
1.26 .126(these with non-traditional elements such as questioning and student work in)J
60 393 :M
1.082 .108(groups. Furthermore, our analyses show that characterizing teaching in terms of)J
60 411 :M
.837 .084(use and frequency of methods such as lecture, small-group work, and whole-class)J
60 429 :M
1.037 .104(discussion is inadequate because such characterizations omit the complex)J
60 447 :M
1.03 .103(interaction between curriculum and pedagogy.)J
60 477 :M
1.226 .123(The problem solving course also counters the notion that a curriculum must be)J
60 495 :M
.771 .077(composed of individual strategies which are learned and practiced separately.)J
60 513 :M
.986 .099(Traditional algebra and calculus courses do just that\321and instructors find to their)J
60 531 :M
.813 .081(dismay that students know the strategies, but may not know when to apply them.)J
60 549 :M
1.1 .11(However, instructors who simply change course curricula without addressing)J
60 567 :M
.859 .086(student beliefs and expectations often find their students bewildered or resistant)J
60 585 :M
.57 .057(\(see e.g., Cipra, 1995; Culotta, 1992\). In turn, instructors often react by returning to)J
60 603 :M
.949 .095(traditional practices, and thus the status quo is maintained.)J
60 633 :M
.84 .084(The curriculum and pedagogy of Schoenfeld\325s problem solving course suggests a)J
60 651 :M
1.081 .108(way to alter unmathematical student habits\321that they be enacted, mentioned, and)J
60 664 :M
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-.074(What we have labeled here \322knowledge of probable student responses to the problems\323 and \322knowledge of)A
60 685 :M
-.06(possible solutions and the heuristics that generate them\323 is related to Shulman\325s notions of pedagogical content)A
60 694 :M
-.103(knowledge and subject matter knowledge \(Fennema & Franke, 1992; Shulman, 1987\).)A
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-.065(We thank Barbara Pence for reminding us of this.)A
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.916 .092(revised. For example, the unsystematic listing of the triples in the magic square or)J
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.986 .099(the student looking to the teacher for approval were situations in which typical)J
60 87 :M
.776 .078(student behavior occurred \(and was expected to occur\), was commented upon, and)J
60 105 :M
1.005 .1(an alternative enacted. Such situations can be engineered in other courses as they)J
60 123 :M
.846 .085(are in the problem solving course. Here is a brief sketch of an example: Students)J
60 141 :M
.421 .042(can be asked to work a problem that can be solved by a strategy that has just been)J
60 159 :M
.811 .081(taught, then asked to work a problem that is superficially similar but which can\325t)J
60 177 :M
.736 .074(be solved using the same strategy. Students\325 usual response is to try the most)J
60 195 :M
.913 .091(recently taught strategy. The instructor is then provided with an opportunity to)J
60 213 :M
.979 .098(mention that an important part of knowing a strategy is the recognition of the)J
60 231 :M
1.057 .106(situations in which it can and can not be used\321and to comment on the)J
60 249 :M
.895 .089(expectation that problems given in class are to be solved using the material that)J
60 267 :M
.753 .075(has been most recently taught. As with all curricular and pedagogical changes, this)J
60 285 :M
.848 .085(one would probably require several cycles of trial and refinement.)J
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1.608 .161(Final commentary)J
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.535 .053(P\227lya wrote in 1963:)J
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.966 .097(Everybody demands that the high school should impart to the students not)J
96 399 :M
1.582 .158(only information in mathematics but know-how, independence, originality,)J
96 417 :M
.834 .083(creativity. Yet almost nobody asks these beautiful things for the)J
96 435 :M
1.049 .105(mathematics teacher\321is it not remarkable? . . . Here, in my opinion, is the)J
96 453 :M
.787 .079(worst gap in the subject matter knowledge of the high school teacher: he [or)J
96 471 :M
.584 .058(she] has no experience of active mathematical work. . . . \(P\227lya, 1981, p. 113\))J
60 501 :M
1.114 .111(Current reforms in precollege education make the experience of active)J
60 519 :M
1.221 .122(mathematical work even more necessary for teachers now than in 1963. Moreover,)J
60 537 :M
.87 .087(studies suggest that prospective mathematicians, as well as prospective teachers,)J
60 555 :M
.763 .076(benefit from such an experience \(Tucker, 1995\). But it is still the case that few)J
60 573 :M
.897 .09(undergraduate courses offer students the opportunity to do, rather than ingest,)J
60 591 :M
1.151 .115(mathematics \(Tucker, 1995\). Instructors have little opportunity to observe such)J
60 609 :M
1.013 .101(courses and those who do may have little time in which to make sense of their)J
60 627 :M
.966 .097(curriculum and pedagogy. We hope in this article to have provided a useful)J
60 645 :M
.872 .087(substitute for a visit to one such course.)J
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2.352 .235(Advances in Instructional Psychology)J
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2.612 .261(nineteenth-century Germany)J
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1.508 .151( \(pp. 121-160\). New York: Oxford University Press.)J
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2.148 .215(, MAA Notes no. 38. Washington, DC:)J
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1.588 .159(The Mathematical Association of America.)J
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1.963 .196(Ulam, Stanislaw M. \(1976\). )J
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2.324 .232(Adventures of a mathematician)J
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2.061 .206(. New York: Scribner.)J
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2.34 .234(University of Michigan \(1993\). )J
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2.034 .203(The new calculus at the University of Michigan)J
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1.237 .124(progress report. )J
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1.763 .176(Ann Arbor, MI: University of Michigan.)J
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2.136 .214(Weil, Andre \(1992\). )J
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2.433 .243(The apprenticeship of a mathematician)J
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2.343 .234( \(Jennifer Gage, Trans.\).)J
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.404 .04(Appendix A: Schoenfeld\325s introduction to the course)J
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.567 .057(OK . . . Let me give you a little sense of what the course is about\321a little bit of)J
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1.035 .103(problems, talking about them, doing mathematics. This is one of the rare courses)J
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1.479 .148(in the department\321in the country\321where you actually )J
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.342(do)A
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1.441 .144( mathematics from the)J
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1.13 .113(some neat stuff about problem solving.)J
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1.026 .103(mathematician, had finished my degree, was a topologist and measure theorist)J
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.745 .074(How to Solve It)J
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.844 .084( written by George P\227lya in 1947. P\227lya is\321was one of)J
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1.282 .128(the eminent mathematicians of this century\321probably one of the ten, fifteen)J
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.965 .097(greatest mathematicians of the 20th century. When he was about sixty, he)J
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.998 .1(some things that would help other people to do and learn about mathematics.\323 So)J
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1.241 .124(he wrote this little thing called )J
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1.191 .119(How to Solve It)J
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.96 .096(, in which he did a lot of)J
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.829 .083(introspection, said basically: \322You know, there are some things that seem to be)J
60 447 :M
1.064 .106(productive ways to solve problems, for me, for other mathematicians\321ways of)J
60 465 :M
.662 .066(thinking that we\325ve picked up, tricks of the trade that enable us to be really)J
60 483 :M
.723 .072(successful at solving problems. And they help us a lot. Maybe if I wrote them)J
60 501 :M
.988 .099(down\321shared them\321it would make life a little bit easier for other people as)J
60 519 :M
.687 .069(well.\323 He went on, stayed in that business for another thirty or so years,)J
60 537 :M
1.851 .185(productively until his mid-nineties.)J
60 567 :M
.685 .068(I read the book in 1974 when I was a very young mathematician, and had a very)J
60 585 :M
.384 .038(funny reaction to it. I started out, read a few pages . . . and he said,)J
60 603 :M
.76 .076(\322Mathematicians do this.\323 I read a few more . . . he said, \322Mathematicians do this.\323)J
60 621 :M
.863 .086(And I started to smile. \322Hot damn, I must be a real mathematician\321I do all the)J
60 639 :M
.345 .035(things P\227lya says they do!\323 Then got pissed off, and said, \322Hey, wait a sec. You)J
60 657 :M
.948 .095(know, here I am. I finished an undergraduate career. I went through an entire)J
60 675 :M
.878 .088(career as a graduate student. I\325m a young professional. )J
f2_12 sf
.34(Now)A
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.667 .067( for the first time I\325m)J
60 693 :M
.662 .066(reading about these tricks of the trade. Why didn\325t they tell me when I was a)J
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.995 .1(freshman, and save me the trouble of discovering all of them for myself? Maybe)J
60 69 :M
.805 .081(it\325s a version of the medieval trial by gauntlet: the only people we want are the)J
60 87 :M
.801 .08(ones who succeed without knowing the rules.\323 I don\325t know . . .)J
60 117 :M
.597 .06(So I asked around, and I asked some of the people who prepared people for)J
60 135 :M
1.086 .109(problem solving competitions. There is a thing called the Putnam exam that a lot)J
60 153 :M
.744 .074(of people study for\321if you do well on it you\325re guaranteed admission to the)J
60 171 :M
.811 .081(graduate school of your choice. I asked people who were in mathematics)J
60 189 :M
.97 .097(education. And the uniform response I got was:)J
60 219 :M
.614 .061(Every mathematician I talked to said, \322Yup, P\227lya is absolutely right. My guts tell)J
60 237 :M
.837 .084(me he is right. I do the things he writes about.\323 And every problem solving coach,)J
60 255 :M
.962 .096(and everyone I talked to who was involved in getting students to solve problems)J
60 273 :M
.573 .057(better, said, \322You know, it\325s a strange thing. I\325ve never been able to use P\227lya\325s)J
60 291 :M
.513 .051(ideas in such a way that my students actually wound up being better at it. So I don\325t)J
60 309 :M
1.546 .155(use them very much anymore.\323)J
60 339 :M
1.033 .103(That was the intellectual dilemma that in the mid-1970\325s got me to turn to)J
60 357 :M
.81 .081(problem solving and got me to focus on it, as the main thing that I would do for)J
60 375 :M
.673 .067(the next 15 years. \324Cause on the one hand, I believe in the ideas that P\227lya had.)J
60 393 :M
.825 .082(And on the other hand I believe the people who said, \322As constituted, P\227lya\325s)J
60 411 :M
.659 .066(ideas don\325t work.\323 So what I\325ve been involved in largely for the past 15 years is)J
60 429 :M
.832 .083(figuring out how to make those ideas work\321figuring out what it is that it takes to)J
60 447 :M
.838 .084(use the kind of problem solving strategies that he talks about, effectively; and)J
60 465 :M
.833 .083(through the years building and changing and modifying this course so that it)J
60 483 :M
.651 .065(works. And the one thing that I can guarantee you is: It does work. By the end of)J
60 501 :M
1.044 .104(this course you will have an arsenal of problem solving tools and techniques that)J
60 519 :M
1.008 .101(will enable you to be much more successful, not only in solving problems that)J
60 537 :M
1.042 .104(you\325ve been shown how to solve, but also at encountering new things and making)J
60 555 :M
1.309 .131(sense of them\321which is something that your math courses don\325t normally train)J
60 573 :M
.619 .062(you how to do.)J
60 603 :M
.614 .061(I\325ll tell you about the ideal goals for a course like this and then, what I actually did)J
60 621 :M
.685 .068(as evidence of what you can expect to be in for; and then I\325ll tell what the structure)J
60 639 :M
.648 .065(of the course will be; and then I\325ll stop talking and we\325ll do what we should do,)J
60 657 :M
.971 .097(which is get on to solving problems.)J
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.865 .086(The goal of this course is to give you enough experience and exposure to solving)J
60 69 :M
.791 .079(problems and learning about the tools and techniques of the trade so that you walk)J
60 87 :M
.699 .07(out of this course a far more resourceful and better problem solver, . . . again, at)J
60 105 :M
.862 .086(not only at dealing with the kinds of things I\325ve shown you to deal with in the)J
60 123 :M
1.089 .109(course, but also when you encounter something new\321having at your disposal a)J
60 141 :M
.741 .074(set of techniques that will enable you to make progress on and make sense of a)J
60 159 :M
1.147 .115(problem that you haven\325t been shown how to solve.)J
60 189 :M
.825 .083(Here\325s the ideal test for the course. I\325ve been in problem solving for fifteen, twenty)J
60 207 :M
1.094 .109(years. There are other people who have massive reputations for such things.)J
60 225 :M
.957 .096(There\325s a guy named Paul Halmos who used to be editor of the )J
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.368(American)A
60 243 :M
2.774 .277(Mathematical Monthly)J
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1.651 .165(, who\325s been writing about problem solving forever. The)J
60 261 :M
.722 .072(kind of thing you might want to do is say to Halmos, \322Hey, look. Schoenfeld\325s)J
60 279 :M
.883 .088(gonna teach his problem solving course. Here are the backgrounds of his students.)J
60 297 :M
.668 .067(Here are the kind of people you can expect to see in the course. What we\325d like you)J
60 315 :M
.668 .067(to do is make up two tests. Make up a matched pretest and posttest, which in some)J
60 333 :M
.855 .086(sense are identical in content. And, he won\325t know what\325s in your tests; you won\325t)J
60 351 :M
.732 .073(know what\325s in his course. If he really does what he says he does, then his students)J
60 369 :M
.715 .072(should do far better on the posttest than they did on the pretest. And other kids)J
60 387 :M
.945 .094(taking, say math H50A, or analysis, or Riemann surfaces, shouldn\325t really show)J
60 405 :M
1.034 .103(any performance difference.\323 )J
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.617 .062( would be the sort of iron-clad test that I did)J
60 423 :M
1.55 .155(something in this course.)J
60 453 :M
.8 .08(I never had the nerve to do that [giggles from the class, Schoenfeld smiles]. But I\325ll)J
60 471 :M
.958 .096(tell you what I did\321which was worse\321some two or three versions of the course)J
60 489 :M
.791 .079(ago. I gave an in-class final exam. \(Now I actually prefer, although rules require in-)J
60 507 :M
.666 .067(class exams . . . what you\325ll be doing is a couple of take-homes, for the mid-term)J
60 525 :M
1.06 .106(and final, just a )J
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1.804 .18(pro forma)J
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1.682 .168( simple in-class written exam.\))J
60 555 :M
.772 .077(I gave an in-class final, and there were three parts to the final exam. The first part)J
60 573 :M
.724 .072(was problems like the problems we solved in class. No surprise, you expect people)J
60 591 :M
.619 .062(to do well on those. The second part was problems that could be solved by the)J
60 609 :M
.86 .086(methods that we used in class\321but ones for which if you looked at them you)J
60 627 :M
.901 .09(couldn\325t recognize that they had obvious features similar to the ones that we\325d)J
60 645 :M
.493 .049(studied in class. So yes, you had the tools and techniques, but you had to be pretty)J
60 663 :M
.63 .063(clever about recognizing that they were appropriate. And, the class did pretty well)J
60 681 :M
.731 .073(on those too. Part three of the final exam . . . There\325s a collection of books called the)J
60 699 :M
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2.792 .279(Hungarian Problem Books)J
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2.027 .203( which have some of the nastiest mathematical)J
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.931 .093(problems known to man and woman. I went through those, and as soon as I)J
60 69 :M
.801 .08(found a problem I couldn\325t make any sense of, whatsoever, I put it on the final. \(I)J
60 87 :M
.855 .085(know that makes you feel good.\) [Laughter from class, Schoenfeld smiles.] The)J
60 105 :M
.697 .07(class did spectacularly well, and actually wound up solving some problems I)J
60 123 :M
.757 .076(didn\325t, OK\321which is pretty good proof that amazing things happen in the class.)J
60 141 :M
.733 .073(And they happen, I think, because we\325re serious about really doing)J
60 159 :M
.949 .095(mathematics\321which is the name of the game. So let me tell you a bit about what\325s)J
60 177 :M
1.676 .168(gonna happen.)J
60 207 :M
.565 .056(Most days I\325m going to walk in \(today being only a slight exception in the)J
60 225 :M
.708 .071(sequence\), and hand out a bunch of problems. I\325ve got enough here to probably)J
60 243 :M
.549 .055(keep us busy for two days or so. And what you\325re seeing here is unusual, because)J
60 261 :M
.594 .059(you won\325t be seated in rows watching me talk. Instead you\325re going to break into)J
60 279 :M
.811 .081(groups of three or four or five, and work on problems together. As you\325re working)J
60 297 :M
1.27 .127(on them, I\325ll circulate through the room, occasionally make comments about the)J
60 315 :M
.635 .063(kinds of things you\325re doing, respond to questions from you. But, by and large, I\325ll)J
60 333 :M
.904 .09(just nudge you to keep working on the problems.)J
60 363 :M
.638 .064(Then at some point I\325ll call us to order as a group, and we\325ll start discussing the)J
60 381 :M
.822 .082(things that you\325ve done, and talk about the things that you\325ve pushed and why;)J
60 399 :M
.963 .096(what\325s been successful, what hasn\325t. I\325ll mention a variety of specific mathematical)J
60 417 :M
.827 .083(techniques as we go through the problems. Many of the problems are chosen so)J
60 435 :M
.948 .095(that they illustrate useful techniques. So you\325ll work on one for a while; may or)J
60 453 :M
.683 .068(may not make some progress; and then we\325ll talk about it. And as we talk about it)J
60 471 :M
.782 .078(what I\325ll do is indicate some of the problem solving strategies that I know, and that)J
60 489 :M
.97 .097(are in the literature, that might help you make progress on this problem, and)J
60 507 :M
.744 .074(progress on other problems. And we\325ll use those strategies as a means of)J
60 525 :M
.813 .081(bootstrapping our way into the problems.)J
60 555 :M
.867 .087(The course is pretty wide-open. I\325ve taught it now seven or eight times, every)J
60 573 :M
.76 .076(other year, thereabouts. And every year the course is different, because it turns out)J
60 591 :M
.857 .086(to be a creature of the people in it. Everything mathematical is fair game in here,)J
60 609 :M
.65 .065(which means that you\325ll find if we get turned on by a problem, we\325ll push it. If we)J
60 627 :M
1.053 .105(see interesting things, we\325ll pursue that particular domain of mathematics for a)J
60 645 :M
1.124 .112(while. The bottom line is: I\325m happy when we\325re doing real mathematics. What)J
60 663 :M
.732 .073(that means may not be clear to you now but it will become increasingly clear)J
60 681 :M
.77 .077(during the semester . . . and this for me is the course I most love teaching \324cause)J
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.951 .095(it\325s the greatest fun and the one that is most involving for both me and my)J
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.214(students.)A
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.623 .062(As I said, what\325s going to happen is: Most days I\325ll hand out problems. We\325ll work)J
60 117 :M
.886 .089(on them in class. Some of the problems can be solved fairly fast and some of them)J
60 135 :M
1.21 .121(merely serve as introductions to more conjectures and more problems. Other)J
60 153 :M
.702 .07(problems may be things that we visit for two or three days\321of classes\321maybe)J
60 171 :M
.9 .09(even a week or two as we do something, find something interesting in it, but don\325t)J
60 189 :M
.491 .049(make enough progress as a group; so I\325ll say, \322Fine, let\325s get back to it next time\323)J
60 207 :M
.653 .065(and we\325ll keep working on the problem over a period of days. So what we do the)J
60 225 :M
1.121 .112(vast majority of times in here is just do and talk mathematics. And learn some)J
60 243 :M
.397(mathematics.)A
60 273 :M
.757 .076(The grading for the course. Well, a week or two into the class I\325ll give you the)J
60 291 :M
.623 .062(opportunity to write out a problem or two for me so that I can get a sense of the)J
60 309 :M
.613 .061(kind of writing you do, and give you some feedback on the kind of writing I expect.)J
60 327 :M
.788 .079(The first main thing we do is: about half-way through the course I\325ll give you a)J
60 345 :M
.756 .076(two-week take-home. It\325ll consist of about ten problems and they will occupy you)J
60 363 :M
.819 .082(for a long time. But you\325ll make progress on them and you\325ll do reasonably well)J
60 381 :M
1.177 .118(on them. And then, the final. Again, the department formally requires me to give)J
60 399 :M
.915 .091(an in-class final, so I usually wind up giving a one-problem in-class final to meet)J
60 417 :M
.842 .084(the rules and regulations. That\325s about ten percent of the final exam grade. The rest)J
60 435 :M
.987 .099(of it is another take-home that you\325ll have two weeks to work on. There are some)J
60 453 :M
1.16 .116(funny rules, which are that:)J
60 483 :M
1.103 .11(What counts is not simply the answer, what counts is doing mathematics. And)J
60 501 :M
.816 .082(that means, among other things, if you can find two different ways to solve a)J
60 519 :M
.624 .062(problem, you\325ll get twice as much credit for it. If you can extend the problem and)J
60 537 :M
.885 .088(generalize it and make it your own, you\325ll get even more. The bottom line is, I\325d)J
60 555 :M
.977 .098(like to have you doing some mathematics and I will do everything I)J
60 573 :M
.905 .091(can\321including using grading\321as a device for having you do that. Grades turn out)J
60 591 :M
.827 .083(to be pretty much of a non-issue in the course. What usually happens is, people get)J
60 609 :M
.717 .072(sucked into it. You get out of the course what you put into it, basically\321that)J
60 627 :M
.998 .1(becomes clear. If you haven\325t done much through the semester, you\325ll find you\325re)J
60 645 :M
.817 .082(not ready to do terribly well on the midterm and final; if you have, you\325ll find that)J
60 663 :M
.834 .083(you\325ll do fairly well. Anyone who kicks in and just participates actively during the)J
60 681 :M
.905 .091(semester \(it\325s obvious\) and no one who\325s done that has ever gotten less than C+.)J
60 699 :M
.689 .069(Typically, the grades have been mostly As and Bs because people have done very)J
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.733 .073(well on what\325s demonstrably good mathematics. So we can say more about grading)J
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.85 .085(when the time comes, but it really will turn out to be a non-issue. Today what I\325m)J
60 87 :M
.909 .091(going to do, is just now, stop talking\321that will make us all feel better\321and then)J
60 105 :M
.731 .073(hand out a bunch of problems. They\325re especially chosen for the first day, to make)J
60 123 :M
1.006 .101(a couple of points\321rhetorical points about problem solving strategies. I will make)J
60 141 :M
.919 .092(those points clearer after you\325ve worked on the problems for a while. But I think)J
60 159 :M
.553 .055(designed to give you a sense of what the rest of the semester is going to be like. OK.)J
60 177 :M
.477 .048(Anybody got any questions? [Pauses, no questions.])J
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.695 .069(OK. Then what you ought to do is break into groups of three or four or)J
60 225 :M
.541 .054(thereabouts; and you should be prepared to work together . . .)J
60 255 :M
.835 .084([Refers to the fact that someone is videotaping the class.] Oh, the man with the)J
60 273 :M
.733 .073(camera. As I said, this is\321this has been part of my own enterprise now for about)J
60 291 :M
.804 .08(15 years. And over the years the course has developed and grown in interesting)J
60 309 :M
.744 .074(ways. And one of the things that I like to do is make sense of what happens in the)J
60 327 :M
.485 .048(course myself. I often write about it as part of my research, as well as part of my)J
60 345 :M
.835 .084(teaching. \(They go hand in hand because my research is about understanding the)J
60 363 :M
1.243 .124(nature of mathematical thinking and using that understanding to help build)J
60 381 :M
.769 .077(courses like this.\) So that camera is a record for me of what\325s happened in the)J
60 399 :M
.775 .078(course, so that I can reflect on it, in the hope of making sense of it and making it)J
60 417 :M
.815 .081(better for the next round of students.)J
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