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Pattern
~~~~~~~~~~~~~~~~~~~~
LYNN ARTHUR STEEN
"He just saw further than the rest of us." The subject of this remark,
cyberneticist Norbert Wiener, is one of many exceptional scientists who
broke the bonds of tradition to create entirely new domains for math-
ematicians to explore. Seeing and revealing hidden patterns are what
mathematicians do best. Each major discovery opens new areas rich
with potential for further exploration. In the last century alone, the
number of mathematical disciplines has grown at an exponential rate;
examples include the ideas of Georg Cantor on transfinite sets, Sonja
Kovalevsky on differential equations, Alan Turing on computability,
Emmy Noether on abstract algebra, and, most recently, Benoit Mandel-
brot on fractals.
To the public these new domains of mathematics are terra incognita.
Mathematics, in the common lay view, is a static discipline based on
formulas taught in the school subjects of arithmetic, geometry, algebra,
and calculus. But outside public view, mathematics continues to grow
at a rapid rate, spreading into new fields and spawning new applications.
The guide to this growth is not calculation and formulas but an open-
ended search for pattern.
Mathematics has traditionally been described as the science of num-
ber and shape. The school emphasis on arithmetic and geometry is
deeply rooted in this centuries-old perspective. But as the territory ex-
plored by mathematicians has expanded into group theory and statis-
tics, into optimization and control theory- the historic boundaries of
mathematics have all but disappeared. So have the boundaries of its

OCR for page 1

N~:~= ~ ~~y
. no Io~ j~ the I~= of - ~~s ~ ~~:~,
em IS :~ an e~ tO~ ~~r ba~ ~~g mm
scions, ~ m~e
~.~ .. .~ ~ ~ ~ ~ . ~ . ~ ~
When v:~ed in this ~ X=~~ ~ s=-
. -A ls n~ j~n about num~ and sh~ but ~ pat
and ones of ~ ~^ ~~r
=e but ~ of m~ media :n—Itch ;~emanc~ans ~~. Act~
paws is n~ him by who cad :~ ~ mth ~e—, chews
ni:~th~ :3aathe~n=;~ ~~s Dim Chaws am Po~care had
<~end more ~ mung m~ Am.- mimes- m.
"T see' Is ~~s
had two Ad: meanings.* to~ perceive with ~e = =d to uMe~d
why ~e ml:~. ~r centu:~.~ the m.~d ~S ~~d the ~ ln the
hiera—i ~ ma:them~ic~ pr=~~ today the b~e :~s b~g =~d
ma—mercies 6nd ~~ ~~s to =e pattems, both with the -~Ye and
h the m~.
C~ in the p~e of Hi forces w~.~= o~f m
C=8ti~ ~~3tiOO. ~Ot ]~:St CO=~' ~t ~~O ~~ ~~:i~iOO$ 3~6
:~ew themes ~ ~ ~
6~' It 3~ ~~" ~5—0 Wl~l Die 8~6 W~ U5iOg
~~ as ~ mutate too] :~ ~ Ieam ~ defeat ~~em~:~s than
the~ fbr~the~. Scandal sow paid-=, mo~ ~ tra~ns ~~ are
:. ~~ A ~ ~ ^~6 ~5 ~~ At,: to
mathematical needs of ~e—Aria =~
Amp: in ~ prompt mcord of ma~em~ic~ e~n elm
pr—id strong ~~es ~r cha~. Indeed' since new deVelopm=:ts amid
on ~~l pnnci:~., It {~s pl=~, as m=~ ~~ arm. su~
gusts th~ one -A few ~ on r - ~~g ~;~h to dm - ~~d
nda~nen~$ Ace em:~i~ on :ref6~s bash on chan~s ~n ~e
Am practice of Mathews ~tics. ~ ~~
.. .. ........ .. ~ . ~ ~ . ~ ~ ~ ~ ~
tUOt:~C .£ for ~~g ba-
sac cu~n=~a w~es tne mmom ot tte pa~—the traditional sch~l
:. if c~y ~~t and well Ieamed' prov~s sound prepa
mt:~n bo~h for ~e wodd of wo~ and for a~c`~d By in mathe:~:~
imps ta~d 501ds
The key :~e for mathe:~ti-~s education :~s not whether to t=~h ~~-
dam=~s but Inch Of; to teach and how to teach t50=K
Changes in ~e practice of mathematics do alter the balen~ of pnon
ties among the many top~s that are important for numera - .~" C3~s
~n society, :n techno1~' in ^~s—among othe~—will hew great inn
pa~ on what mI:t be possible in schoo:! ma:~-s in the :~:t Hi-.
M! of th=e changes ~~} a~t the fundament~s of school mathematics.
~ develop Hi new ~~$ C~, OH~ must attempt
to ~~e the :~.~:~ti=) needs of tomorrow ~ students. lt ls the

OCR for page 1

PATTERN
present and future practice of mathematics at work, in science, in
research—that should shape education in mathematics. To prepare ef-
fective mathematics curricula for the future, we must look to patterns
in the mathematics of today to project, as best we can, just what is and
what is not truly fundamental.
FUNDAMENTAL MATHEMATICS
School tradition has it that arithmetic, measurement, algebra, and
a smattering of geometry represent the fundamentals of mathematics.
But there is much more to the root system of mathematics- deep ideas
that nourish the growing branches of mathematics. One can think of
specific mathematical structures:
Numbers
Algorithms
Ratios
or attributes:
.
Linear
Periodic
· Symmetric
Continuous
or actions:
· Represent
Control
Prove
Discover
Apply
or abstractions:
· Symbols
· Infinity
· Optimization
· Logic
or attitudes:
· Wonder
· Meaning
or behaviors:
.
Shapes
Functions
Data
Random
Maximum
Approximate
Smooth
· Model
· Experiment
· Classify
· Visualize
· Compute
Equivalence
Change
Similarity
Recursion
Beauty
Reality
Motion
· Chaos
Resonance
Iteration
Stability
Convergence
Bifurcation
Oscillation

OCR for page 1

4
off dicholomi^:
Disc- as. ~~ndsuous
Fights infinite
at Tic as. ~istebti~
vs. ~e~i~ni~ic
~ ,
~ ~xaclv~s Pimple
^~ Ha.\ cat
Fee Dale ~~ mumble Be ~-l.exi~ of Rectums lb~
supper m~bemat~.
^^_^^ an ^~.^.^ a... .
~~ a_ ~ ~ ~ ~~
u~ ~ti-~ll~ that ~# Aging lhem Me r sit lop a Si~i,C=1
b~1i~ idea ham inky Loons of eye chilled ~ the
1b~ so Ed Age and~on into SC~li5C or Hem
. ^ Be ma
counter glib visuals a1I of Fed ~~ distal Yes ad ideas.
liaditionaI school matbematics picks very fag ~ _ (e.g., Melba
nicety> geometry, aIgebna) and an~I~eslb~m bodzon~]ly to form the
cunic~um: [~1 arilbxnelic:1ben simple ~gebra~lben ~eomel~y~lben
more Bra and h~ily=-as i~fil bare 1bc epitome of ~~atbemal~ic~
kno~ded~e--~ulus. ThL~yer

PA TTERN
s
effect will be to develop among children diverse mathematical insight
in many different roots of mathematics.
FIVE SAMPLES
This volume offers five examples of the developmental power of deep
mathematical ideas: dimension, quantity, uncertainty, shape, and
change. Each chapter explores a rich variety of patterns that can be
introduced to children at various stages of school, especially at the
youngest ages when unfettered curiosity remains high. Those who de-
velop curricula will find in these essays many valuable new options for
school mathematics. Those who help determine education policy will
see in these essays examples of new standards for excellence. And ev-
eryone who is a parent will find in these essays numerous examples of
important and effective mathematics that could excite the imagination
of their children.
Each chapter is written by a distinguished scholar who explains in
everyday language how fundamental ideas with deep roots in the math-
ematical sciences could blossom in schools of the future. Although
not constrained by particular details of present curricula, each essay
is faithful to the development of mathematical ideas from childhood to
adulthood. In expressing these very different strands of mathematical
thought, the authors illustrate ideals of how mathematical ideas should
be developed in children.
In contrast to much present school mathematics, these strands are
alive with action: pouring water to compare volumes, playing with pen-
dulums to explore dynamics, counting candy colors to grasp variation,
building kaleidoscopes to explore symmetry. Much mathematics can
be learned informally by such activities long before children reach the
point of understanding algebraic formulas. Early experiences with such
patterns as volume, similarity, size, and randomness prepare students
both for scientific investigations and for more formal and logically pre-
cise mathematics. Then when a careful demonstration emerges in class
some years later, a student who has benefited from substantial early in-
formal mathematical experiences can say with honest pleasure "Now I
see why that's true."
CONNECTIONS
The essays in this volume are written by five different authors on five
distinct topics. Despite differences in topic, style, and approach, these
essays have in common the lineage of mathematics: each is connected
in myriad ways to the family of mathematical sciences. Thus it should

OCR for page 1

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v 5~0x In- in many examples IO TOlS VOtU=0 8~G :15 0-~6
of the most mpi-~W ~~g ~~.s of ma~! and w:iend6c =~
search. The first step- in ~ta an~`s- ::s the visual d:~' of data to
search ~r h:~n pattems. C)~s of v~s ~e`s pmv~ hi: d: -
play of relations a~ f6~ion`~, they aw m~:v used t:hro~ghout science
~ ray the behavior of one ~r3~alb-lC (C 2.
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map ma~ ~e us-cc ~ometr:c ~~es suc~ as :~ction to rep:~-
sent I s=~s on ~ two~li~sional canvass or sheet of
paper. ~w computer graphics auto~e these processes and iet us
explore as well the p~s of shapes in hi~dimensio:~! space.
Leami~ to- v~e ~~tical pa*~s e;~ll`~t~~
OCR for page 1~~

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mhem~ M~ ~~ ~ ~ sore ~~=s ~ Ad?
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liti=^
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sights
t sm~ the t~e of Newton h~ m~cs chid as much
~s in recent ~. M~ - 7n I~ ~~ ~ the m~n of c~
puters' ~e nMu:re and pi of mathemadcs have been Amy
Do- 57 fit 0~:~57 t00~57 appli~ons, and m~. ~
the wI~e of Gaines em that en~d the Newtonian revolut~, to-
6~:~$ CO=~: C5~—t=~ItiOO~ V!~S 8~6 ~~=S ~~:~
of deeuh.~ held values.
~S it did t~= C0~$ 880 1~ t5C tt=SitlO~
trom =~= promS to—~~an ~~' mathemat~ Once a~n
is un - :~ing ~ ~~ rec=~tion of pmeedu~ ~~*
~~es of Emmy ~~ a~ :~n the r=~ ch i~re
of :mathemati~ and :n pm~! =~lica:tions of mathem~i=l methods.
:~+ =e ~~:n in the es=~s in this volu:~*

~~
OCR for page 1~~

Ns~ ~ :
A 1s n~ ~~ SmCe ~W :~.
fit phmomena :~:~:~:~m ~~ ::
~ I :is n~ ~~ a Her Stack aIm~ a~ me=s
I: .~ O - ~~ ~~
~ ~ .~
.~ . .
~ ~~n ~~ ~.e ~~ of ~~, .~ :~ cat
Yis~ ~esem~= ~~;~ ;~s ~ amen = :~=en
—~ ~^—an~c approaches. ~ ~~ -
~ ~~e~.=~ - s
B~ Ala;: ma~ d:~m errands o:E mat~emat~^ ~ =~:n
. . ~ . ~ . ~ Hi. ..
~;ve on ~~=—cures =d And 1~S" Returns m=~ts
(~, ~~ ~7 Amy id: ^~n to ~ -me me
in order to un-~M =~7 =~ 0~5 (~.~, ~~'
, pmw) Deal -ids th~ one must develop -in o~r to ~ math~
em:~ ~:~, c=~s =d Al ~ tb.e nwns =d ve~s of the
Togae of math~+
~~:t bum=s do Oh the tankage of m~ ~s ~ desk p~
tems. Anti is an I; s=~:~= ~~ seeks to u~d
—e~ k~nd of p~em—patwms t~ oc=r in nature, pa=ms metered
bra the h~n m:~' =d wen panems ~~d by o~r patw=s To
~ =~i~77 Brim ~~ t~ 0~6 t~ ~ ~ ~
jam to t5~r o~ ]~ ~~= which Chow 08~ =C ~7
re~anty, a~ i:
~e es=~s -it fit vol:~e prmr~ide f~e -~d == swdI~ ~~t eX~
employ h~ thus can ~ done. Ot~r w~ =~ld jug as city h=~e
deS~d ~~e or ten -A examples. The ~~s and a~= listed
ow are replete m:th additional e=:~es of n~ -: ideas.
Wh~ At: ::n the stu~ of Am :~s :~t so much which -4
ular strands -one explores, but ~e p=~ce :n these Ads -of gait::
eXa.~s of su~cien:t v~y aM dash to :~ pa.~. :~Y encour~
a~ng ~~:ts to explore patt-~ms that hwe proven thm: po~r and
s~iEca3~, we o~r them broad shoulder= - m -which thev mI! see
~ 8~ ^~ ~~ ~ ~ ~ ~ A .~ are. =.~t t:~: ace. ~ p~~ ~ ~ /~ /~—
W~4 ~~' MA~ Bitk~,"$0r B0~+ ~ 985-
~, ~3, id, :~6 5~< ~~ 4~ Oaf ^,$~c pmss~ 'i 988*
. Sa msI—— ~ ~ lc~Xe) F. ~ D=~p Hacf al ~n smem ~ N~ YO~ ~ NYX ~ =~m te
~ 84~\ pt(~6 0t 81 i 565~^ {~e - ~~n ~~. ~ Y0~^ Oaf ~~}

~~
OCR for page 1~~

hi:- I:
: :: :
~9
~5~ ~~=ed~ Am ~~#
#_1^,
6~is<~FbOlp~s~J~.~B ,~n~s~s~s~ ~s~sss~s~s~s~
I, : ~ ~~ ~B~966 ~~ ~ ~~ ~~ SO Sag ~~ IS :S~ ~ ~ ~ ,~.
.~S~73~E~D~i~PbJ~#~d~H~b~s~ ~~S~_~^
~8~^~S~ I
~9,i~^'~~ ~s~ ~^ I< bitt
I: Imp ~~ APE, 1~989~. ~~ Emit' ~~ ~~ E~:~i~E~:~ ~~ ~:~:~ ~~SS~:~SS~S~;~:~S~ IS ~~S~S~
A ~ o ~ ~ ~ s ~ ~ ~ ~ ~ ~ ~ ~ ~ i ~ l ~ < ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ * ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ i _ , ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ _
If ~~ A ~ Ii ~~ ~~ ~~<~ss~s~s~ ~~ ~~s Am ~~
~1~1,~ ,~- -1~s~,~^~S
4- E~ ~~ ~~- &~ {~) 1~986. ~~ ~ ~~ ~~ ~~ ~~ ~~ as
1
13. ~~ ~~ ~~ ~ ~# #: ~~, 1988.
4. ~ ~ ~~, ~ ~613 ~ /S ~~( ~ ~~/ ~ ~~ ~~ ^ I
15. IS: ~~- ~ ~ ~~ ~~ ~
~~ ~ ~~~ ~ ~ ~ ,~ 198. : ~ ~ :: ~ ~ ~~ ~ E
16. OR ~ ~ ~ J~ ~~~
I: Vita P 1 ASP.
17. On. Aim. ~ ~~ ~~ ~~ ~ I: Cat ADAM
~ 1971.
~.s.s ~ : ~ s: s
... .
... .
: :: :::E
ESSsSs :SS :~::: HESS :SSSSESSESSSSS S SSS:SESSSE~ So :Es~: ::SESS~ESSS SSS~:E:ESSSS::SSSSS~ ~~E~S~ I age
:~S~: S ::S~ S :~:E :.:: SS~S'~E~ ~~ SE~S~E~ :: US: ~~:'~ss:~:~ :E
' ' 'a
gum, _1 ~~ ~ ~ ~~ I. ~~ ~ ~ SAW. Inn ~~ Aft,
19. Runty, ~ E ^e ^~ ~~: ~ ~# ~ ~~# ~# ~3,
20. I, Our. 4~g He Ante ~ ~k ~ ~e=~ ~ ~^ LS
~~ ^~7 ~~r ~~ ^~ Divot, I: N~ ~^
I, 19~> 117-162.
21 tier, Philip Of ~~/ ~~ Ne~ ~ Ad: ^
ad, 1983.
22. ad, Boas. ^~_# ~~ ~~ ~ ~r ~~ ^~ Ha. Ha: Okay
Unive~i~ PI 1985.
23. . ~~. E. I: SO
~~ 1985
24. And Ad. ~ ~~ ~~ ~~# ~ ^^ ^~ ~~ Ha.
ha: _ 1985.
. ma, Debug. #~ ~~ ^~ ^~ ~~ ~~. amid, Ah:
Id, 1976
26. ~andelb~, Benoi1 B. ^~ ^~ ~~~ ~^ ~~ ad, ma: W.~. F~-
ma, 1982.
. Mat, David S. If ^~ ~~ ^~# ~~ ~~ few I,
at: ~ H. ~~ 1983
28. godson. Pbilip and Moron, Pby~li$. ^~ ~ an. ~~w at, at: ~ienliSc
~~ ~^ 19S2.
Dawn, Into a~ Ricb~ler. ales a. ~ ~~ ~~ :~ ~^
~~7 #~ ~~ ark, BY: Sprin~^rle~ 1 986.
30. Prime, ~einz

10
#~ ~ _
31. ~ am. ^~1~ ~~ ~^ ~-
. ~ ~ ~ ~ ~
3~2. ~ ~~,~ Jo. ~~ ~~ ^~ I
Nc~ ~ ad: ~bdd~ I 197~-
33. ~~ ~~ ~# ~~ ~~ ~ ^e ~~ ~~ ^~ ~ ~ ~~~
age, ~ ~~ 1982.
34. Rat ^~. ^~ ~- ^~ ~~ ~~ ~~e~ age,
~ ~ 1^ ~~, 1984.
35; ma, ~~ ad ~~ ~ ~~. ~ ~~ Arm an:
Unfair ~ ~~_c-~ Pa 19~77.
36 ~~ ^~ ~ Raven, ~~- I ~~ ~^ ~ ~ ~~;
~~# ~ ^~ an, a: ~ Unfit Pa, ~981~.
37. as, -n Agog. ~~# ~ ~ #~ ~- ~~ a:
-~ 1978
38. ~< ~~ ~~ me ~= ~ P_~- 240 (29 #~ 198~8}, 6 I 1-
616.
39. ~~ H. ~~= ^~ ~^
~~ ha. #: ~ U~ an, 1983
~ - - ~ - ^ ~ ·- .
41.
~$, ales b. ^~ '~ ~~~ Bang ~ ~~ ~~ & Cam, 1974~.
Aim, #. ~~ ~~_~ ~ ~ Ad: 0~ polyp,
1987.
42. ~~ lag ~ ~ ~ ~7 ~ ~~ ~^ ago: ~11,
1989.
43. ---
= Jag M.> al ~. (a.). ~1~^ ~ ~~ ~ 76~ aim, ^~ ace.
1989.
44. and ~~ a. ^~ ^~ ~~ ~l^zi~ ala. chit, an:
Oasis Pa, 1983.
Inning, ~~ j. ~~ ~# r4~ ~^ ~ age. ~on,
~ align ^~ ~ ~~= ~ Me~li=. 1975.

~~
~~