National Academies Press: OpenBook
Suggested Citation:"Front Matter." National Research Council. 1990. On the Shoulders of Giants: New Approaches to Numeracy. Washington, DC: The National Academies Press. doi: 10.17226/1532.
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Suggested Citation:"Front Matter." National Research Council. 1990. On the Shoulders of Giants: New Approaches to Numeracy. Washington, DC: The National Academies Press. doi: 10.17226/1532.
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Suggested Citation:"Front Matter." National Research Council. 1990. On the Shoulders of Giants: New Approaches to Numeracy. Washington, DC: The National Academies Press. doi: 10.17226/1532.
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Suggested Citation:"Front Matter." National Research Council. 1990. On the Shoulders of Giants: New Approaches to Numeracy. Washington, DC: The National Academies Press. doi: 10.17226/1532.
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Suggested Citation:"Front Matter." National Research Council. 1990. On the Shoulders of Giants: New Approaches to Numeracy. Washington, DC: The National Academies Press. doi: 10.17226/1532.
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Suggested Citation:"Front Matter." National Research Council. 1990. On the Shoulders of Giants: New Approaches to Numeracy. Washington, DC: The National Academies Press. doi: 10.17226/1532.
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Suggested Citation:"Front Matter." National Research Council. 1990. On the Shoulders of Giants: New Approaches to Numeracy. Washington, DC: The National Academies Press. doi: 10.17226/1532.
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Page viii Cite
Suggested Citation:"Front Matter." National Research Council. 1990. On the Shoulders of Giants: New Approaches to Numeracy. Washington, DC: The National Academies Press. doi: 10.17226/1532.
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On the Shoulders of GIANTS . New Approaches to Numeracy LYNN ARTHUR STEEN Editor Mathematical Sciences Education Board National Research Council NATIONAL ACADEMY PRESS Washington, D.C. 1990

NA~ ACADEMY s 210~1 ~~t~ Hague, ~~ ~ ~~+ ~ 2~18 ma: Me awl ~ ~ ~e s laws ~ ~ Ivy ~ ~ ~ ~ ~~ N~ ~~ ~D, ~_= ~ ~~ ~~~ my Shims. ~ ~~ amp IBID. ~ ^ _ of Aim. ~ ~~ ~^ -~i~ Is. ~-~ -_ you ~^my's~ owing baud ~~ ~ ~~v_l. Bug ma ~~ ~- - an_ ~ ~e ~~{ ~ Co^D ~ ~~ ~'~P~ mug gay may ~ Is ~ ~ ~~ my ~ ^~ in Brig ale -# ~ ~ ~~ -~g amiss. ~ ail ~~ jowly ~ ~ aims ~ he ~~ of Am. a. ~ ~= ad Rag a. ^~ ~ - ~ via am. ~=s~^y, of ~ ~~ b Mail .. .. . . ~ gamy Saw Amen ~ ~ amid in T9S! ~ a cunning cow ~~v ~ ~se~e~ ~~ ~ Was ~~- ~ is Memo ma alas Adam ~ ~1 -~ ~ ~ leas ~ Boa =~= dandy ~ B_ ~ ~ N~ _ Coral. D~bl~=nl,p~lic^n,~ddi.~=~is~=sup~bya~^m~ C=~0~-Y~.~ ~~~a-~don~es-~y~ hat ~( vies of Li~b~^ ~ ~~ C ~in-^b-1- ~~ of Eta: =~ Is ~ Buggy / Lag ~r Sun, Ace; ~~~ ages ~~ I, Resorb gull. ~ cm . . ^ Atlas big As ad ~x. 1SB~ 0~2~-8 a) 1. ~~tiC~, ~ 1~- Sags. 1~. Sag. Lao ala, 1941- ~a~atic~ Views Mucus So~. QA13.~53 1~ 513'.~107~20 TIN ~~ Cub anvil (u.S.). ~15 ^~;Acom~-~s(~csu~=ofa~ug~ut-sbe~d5~).W~n -~-~m~me~i-ils~e.~$di~vi~s~10f~i~ns~ionals-(intot~0 sym~os.C=~d~=om=B=~hol~=omp~natB~~nLOive~i~ mission fir Lilly caution ~ Is of this ask ~ educational paws bat ~01 ~E sat m~ ~ agog weir of ~ -~n m~CS1 10 ~C Cation gamy lass, 2101 Constituted Event, ~NW. W~b~on. ~ 2~18. Calm of dais =~ may ~ pumas Cam ~e Ions Ac~cmy Is, 2101 Constitution avant, DOW, W~b~~n, ~ 2~18. IS_ F~~i~,Sep~m~l~ S~~d~ntin~.JAnu~l~ =~,D~~1994

Preface ~ ~ v ~ v ~ v ~ ~ ~ v ~ ~ en v ~ ~ v ~ ~ Today's headlines are filled with reports of illiteracy, innumeracy, and other signs of educational decay. Tomorrow's schools can be filled with evidence of renaissance if we begin now to till the soil for effective education—in mathematics, in science, and in all subjects. This vol- ume offers five visions of mathematics suitable for tomorrow's schools— visions rooted in imagination, in mathematics, and in science. Ideas in this volume can provide fertile soil for new approaches to tomorrow's numerary. Forces created by computers, applications, demographics, and schools themselves are changing profoundly the way mathematics is practiced, the way it is taught, and the way it is learned. Even as we work to make incremental change in today's schools, we must think also about more significant change that will be possible, indeed inevitable, in the future. For this reason the Mathematical Sciences Education Board (MSEB) decided that one of today's priorities is to stimulate imaginative thinking about tomorrow's curriculum. In this volume readers will find a vision of the richness of mathe- matics expressed through five vignettes that illustrate different possible strands of school mathematics. These papers expand on the theme of mathematics as the language and science of patterns and are introduced by a brief essay that highlights interconnections and common ideas. The authors were asked to explore ideas with deep roots in the mathematical sciences without concern for limitations of present schools or curricula. They do, however, suggest through numerous imaginative examples how . . .

it can be ~~ - m ln~ ~~d : ~7 ~e papem ~n chic v~e =e infolded ~ ~ ~~e ~ n~e c Me Arcades to m^~cs cum=la in the ~= -I ~e wIume iced ~ =d ~ ~ Hi. ~.~e ~ ~~S edu=~n mmulated bY ~ senes of -I - ~~. EM <I ~ Re~t lo ~e ~~n on ~e ^ kIhemat'cs ~~n —~~ ~m and ~~~ ~~r m~ .. m Taken mother th=e—heat~ns pawn ~ =:~nt and u=nt wsmn that ~~.-d ~p the Un~d Sm~ ~= eXcellence tO ~~-~:ics :~. Althou~ ~ exa-~s are -~d -in Die volume' ~~y ~ =~m not the oni? 6~e po~:ill:ti~. Appmpn~e cumcula for the twenty tm cenm~ w:~! necessarily Ike ~ m~ variety of ~~, reDe~:~g bow the~ Cad spe~m of ~~ sc:~s am ~ indivi&a) Aim Of i~!—oo} ~~+ ~ o~r ~=e themes -A ~ Me ~ ,. ~ =~a 6~t 85 -~5 ~ - ~t is p0~$ t:0 S~e d~: of n~ ~ im~i.~ p-~s th~ - ~t the wmi~ty =d uses -A mathem=~" Ah. =~h = - in Die wI~e Is the .~. ork ~ one ~~' e=h haS bene61~d enormously from advIce arm cntiq-~e pmv:~ 'by ~~,~ a- Bern! the vo~me was developed under the auspices Of tM I9:~9 MSEB Cumcul~ Comm~e :~ by Henry o. ~~' m~ -tired assI=.m wce p=~! of ~~} ~~-~l,=~s Res~. Other members ~f this A~ Com:~:~:=e :~uded Aid E;Dis' Jr. of ~t Alley ~~' A.~w M. ~~n of Howard ~:Yersi~, Mmin 13* kai 0{ p=~ ~~iVCtSitV7 ~10 ~~) 0[ Chaste Hall, I ~~ ~ t56 S~ ~~\ 0[ ~~ York at Bu~' I:~= Singer of the M:~ns :~:~:e of 1~7 8~0 = I: ~~ ah- I 05 ~],08,~. ~~= ii 0086~0 mum air the c - n ~r helping shape the Plume at ~s i:- and Or kee;ping-~ on Ala. 5- `~ding 13~- wage e$~lis:~ bit the [I to review draDs off the es=~s ~ the volume p:3~e for the o~rewi~ erg One ~r each main es=v, and One ~ =amine the i:nk$ w:th acid ence. ~r "~- the Bonding Board. cO:~si~d of I~= singer and Hi. Ii ~: I ~6 ~~ 0: t6:8 l0~: S::~0i~

If:: it:: At:: *w ~ ~:~f ~~ ~~:. f:m ~~:~:~q _ ~.~ ~~ ~~ ~~ ~7~ ~~ .': ::: .~:-:::~: .:: :~::A - .': ~ ~ _ . ._,':._. : ~ .::::::::::::~::_~:.:::::::::::::: ~ ~~ ~~mones~ Hand: ~~ ~~:~ ~~m ~~ K ha in, ~l K ~t ~ 5~ ~ ~ 5 — —~ ~ ~~t ~ ~ 8~ ~t ^~=m ot tbe Un~ty ot ~~ ~=d ~~ce~ 0t 0~: ELK ~ ^~ 8~ ~ - ~l l~ ~ city ~~ ~ lo: K ~ ~ ~ 8~m ~ of 3 ~ :~0 ~:~3d ~~ p im It ~8 m Bba~h Previewed tM~:~:~s - ~~ By, mdmd Chapman ~~ot~:~eD ~~' M=~= ~X' pm~mr ~ ~ K 'l ~ ~~l ~ ~~ ~~ ~ ~ 4 ~ ~ - ~~ bra ~ Be~. M; mpr~vem:~s :n th~s—ume ~ due ~y to the day -amp =d ~ :~= otth~ ~~d ~~ Boar Crews ~ ~ ~ 7 ~t ~ =~ - ~~t ~0 :~ did ~ ~w—he ~e Ice embed ty ~~r rev~:, ~ fire we ~ ~~:~ mte~) ~r the~ ~~, ~E wspon~ ~ - ~e pO];~S ~ Or- ~~ I.~ thIS~ ~rOlUm.6 ~S ~~ ~ -~- P~tion of ~~s wiu.~e -~.~= ~e 6~t ~e ~ ~e ~~ of ~e MSEB to expre~ to the nation ~ new v1S:on -I mathematms -e:, of how ~ meres - Ad. =~ =n ew~e ~ meet ~ Ch~s o.f the next ;~:illenn:~:~. m~t - m the beginn:~-~f MSE:B in -I ~> mer M:~:B—~r Schism H:~:~) ~ ~e Casio C'f M1ssoun at Citv took up the d:i~uk! c~ ~ f6;~cing ma:~ci=s a~ ma emetics edu00~ to th:~ t~er ~~ pOssIble new $~Is8~<iS ~r ~ ~~emancs cumculum. She challenged ~! of us on the MSE3 to ~k om :~ that :~ ~ more ~~te to -ant -mmpu41er ~ than 3~;:~ti - Good ~~S Am;; We have ::~d - m preppies In editions ~r whom =~-~on w~ the pnmam pu~ow ~mathem~- Th:~-~ is the ~~t newt! of Sh:~ik~s =~e :n emphasi;ri~ the :~= o: motion cu~r bum in the eme~g pmetice of mathematics. C~i.~-~n and produ~:~n derails have ~b=n ;. m.~d h~ t~e ~ a: ~:'~b . ~ *. Hi ~ ~ ~ #I ~ ~ W w :~D sTa~ }~o nTST ~ mamla b~c anc nOw' bY ~~h Ho - ~~. 5~0i~ t68~5 3~0 6~6 IBM ~:~7 wh0 550960~6 moth -~:~:ing good bumor all technical aspects of :~:~=io-n mom the ;~;~:itial p1 ~~.~ =~4~.~85 to Find—a:Is of am: cop 06itln87 8~6 pt06~Cti0~. Thanks also are due Jana CodSeY - ~~e tenacity and patience wew :in va:~e in co11~ng the ma~ i]~4ti0~5 t07 t50 OK Much of t~ mmputer~d a:~k ~s provided by Thomas Banch-~fl: and \rt(1 ~Q03707 with specie:! suppod fmm David-e Ce—~e, ~ ~~te student at 13~n C:~. by: thmu~t -the man~ dm~s office

V1 ^~-,~- ~' ~~ ~ . ~ dingy erase ~~ ~ ~~ m=~d~tb emcee ~1 tab / 1~:l~ and Is n~s~ 10 enable Me and ~1 3~ be paged by ~1. d~~nic mews. S1. oaf Colas

::~: PI / 1 ~ A-= gun / ~ ~ Tromp F. age QUA01# / James I. ~ UNCEASING / 95 Owed S. ~~ - / I39 ~~o~c Seneca C ^OOR /183 lan Steads IDES / 223 / 21

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What mathematics should be learned by today's young people as well as tomorrow's workforce? On the Shoulders of Giants is a vision of richness of mathematics expressed in essays on change, dimension, quantity, shape, and uncertainty, each of which illustrate fundamental strands for school mathematics. These essays expand on the idea of mathematics as the language and science of patterns, allowing us to realize the importance of providing hands-on experience and the development of a curriculum that will enable students to apply their knowledge to diverse numerical problems.

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