National Academies Press: OpenBook
« Previous: Dimension
Suggested Citation:"Quantity." National Research Council. 1990. On the Shoulders of Giants: New Approaches to Numeracy. Washington, DC: The National Academies Press. doi: 10.17226/1532.
×
Page 61
Suggested Citation:"Quantity." National Research Council. 1990. On the Shoulders of Giants: New Approaches to Numeracy. Washington, DC: The National Academies Press. doi: 10.17226/1532.
×
Page 62
Suggested Citation:"Quantity." National Research Council. 1990. On the Shoulders of Giants: New Approaches to Numeracy. Washington, DC: The National Academies Press. doi: 10.17226/1532.
×
Page 63
Suggested Citation:"Quantity." National Research Council. 1990. On the Shoulders of Giants: New Approaches to Numeracy. Washington, DC: The National Academies Press. doi: 10.17226/1532.
×
Page 64
Suggested Citation:"Quantity." National Research Council. 1990. On the Shoulders of Giants: New Approaches to Numeracy. Washington, DC: The National Academies Press. doi: 10.17226/1532.
×
Page 65
Suggested Citation:"Quantity." National Research Council. 1990. On the Shoulders of Giants: New Approaches to Numeracy. Washington, DC: The National Academies Press. doi: 10.17226/1532.
×
Page 66
Suggested Citation:"Quantity." National Research Council. 1990. On the Shoulders of Giants: New Approaches to Numeracy. Washington, DC: The National Academies Press. doi: 10.17226/1532.
×
Page 67
Suggested Citation:"Quantity." National Research Council. 1990. On the Shoulders of Giants: New Approaches to Numeracy. Washington, DC: The National Academies Press. doi: 10.17226/1532.
×
Page 68
Suggested Citation:"Quantity." National Research Council. 1990. On the Shoulders of Giants: New Approaches to Numeracy. Washington, DC: The National Academies Press. doi: 10.17226/1532.
×
Page 69
Suggested Citation:"Quantity." National Research Council. 1990. On the Shoulders of Giants: New Approaches to Numeracy. Washington, DC: The National Academies Press. doi: 10.17226/1532.
×
Page 70
Suggested Citation:"Quantity." National Research Council. 1990. On the Shoulders of Giants: New Approaches to Numeracy. Washington, DC: The National Academies Press. doi: 10.17226/1532.
×
Page 71
Suggested Citation:"Quantity." National Research Council. 1990. On the Shoulders of Giants: New Approaches to Numeracy. Washington, DC: The National Academies Press. doi: 10.17226/1532.
×
Page 72
Suggested Citation:"Quantity." National Research Council. 1990. On the Shoulders of Giants: New Approaches to Numeracy. Washington, DC: The National Academies Press. doi: 10.17226/1532.
×
Page 73
Suggested Citation:"Quantity." National Research Council. 1990. On the Shoulders of Giants: New Approaches to Numeracy. Washington, DC: The National Academies Press. doi: 10.17226/1532.
×
Page 74
Suggested Citation:"Quantity." National Research Council. 1990. On the Shoulders of Giants: New Approaches to Numeracy. Washington, DC: The National Academies Press. doi: 10.17226/1532.
×
Page 75
Suggested Citation:"Quantity." National Research Council. 1990. On the Shoulders of Giants: New Approaches to Numeracy. Washington, DC: The National Academies Press. doi: 10.17226/1532.
×
Page 76
Suggested Citation:"Quantity." National Research Council. 1990. On the Shoulders of Giants: New Approaches to Numeracy. Washington, DC: The National Academies Press. doi: 10.17226/1532.
×
Page 77
Suggested Citation:"Quantity." National Research Council. 1990. On the Shoulders of Giants: New Approaches to Numeracy. Washington, DC: The National Academies Press. doi: 10.17226/1532.
×
Page 78
Suggested Citation:"Quantity." National Research Council. 1990. On the Shoulders of Giants: New Approaches to Numeracy. Washington, DC: The National Academies Press. doi: 10.17226/1532.
×
Page 79
Suggested Citation:"Quantity." National Research Council. 1990. On the Shoulders of Giants: New Approaches to Numeracy. Washington, DC: The National Academies Press. doi: 10.17226/1532.
×
Page 80
Suggested Citation:"Quantity." National Research Council. 1990. On the Shoulders of Giants: New Approaches to Numeracy. Washington, DC: The National Academies Press. doi: 10.17226/1532.
×
Page 81
Suggested Citation:"Quantity." National Research Council. 1990. On the Shoulders of Giants: New Approaches to Numeracy. Washington, DC: The National Academies Press. doi: 10.17226/1532.
×
Page 82
Suggested Citation:"Quantity." National Research Council. 1990. On the Shoulders of Giants: New Approaches to Numeracy. Washington, DC: The National Academies Press. doi: 10.17226/1532.
×
Page 83
Suggested Citation:"Quantity." National Research Council. 1990. On the Shoulders of Giants: New Approaches to Numeracy. Washington, DC: The National Academies Press. doi: 10.17226/1532.
×
Page 84
Suggested Citation:"Quantity." National Research Council. 1990. On the Shoulders of Giants: New Approaches to Numeracy. Washington, DC: The National Academies Press. doi: 10.17226/1532.
×
Page 85
Suggested Citation:"Quantity." National Research Council. 1990. On the Shoulders of Giants: New Approaches to Numeracy. Washington, DC: The National Academies Press. doi: 10.17226/1532.
×
Page 86
Suggested Citation:"Quantity." National Research Council. 1990. On the Shoulders of Giants: New Approaches to Numeracy. Washington, DC: The National Academies Press. doi: 10.17226/1532.
×
Page 87
Suggested Citation:"Quantity." National Research Council. 1990. On the Shoulders of Giants: New Approaches to Numeracy. Washington, DC: The National Academies Press. doi: 10.17226/1532.
×
Page 88
Suggested Citation:"Quantity." National Research Council. 1990. On the Shoulders of Giants: New Approaches to Numeracy. Washington, DC: The National Academies Press. doi: 10.17226/1532.
×
Page 89
Suggested Citation:"Quantity." National Research Council. 1990. On the Shoulders of Giants: New Approaches to Numeracy. Washington, DC: The National Academies Press. doi: 10.17226/1532.
×
Page 90
Suggested Citation:"Quantity." National Research Council. 1990. On the Shoulders of Giants: New Approaches to Numeracy. Washington, DC: The National Academies Press. doi: 10.17226/1532.
×
Page 91
Suggested Citation:"Quantity." National Research Council. 1990. On the Shoulders of Giants: New Approaches to Numeracy. Washington, DC: The National Academies Press. doi: 10.17226/1532.
×
Page 92
Suggested Citation:"Quantity." National Research Council. 1990. On the Shoulders of Giants: New Approaches to Numeracy. Washington, DC: The National Academies Press. doi: 10.17226/1532.
×
Page 93
Suggested Citation:"Quantity." National Research Council. 1990. On the Shoulders of Giants: New Approaches to Numeracy. Washington, DC: The National Academies Press. doi: 10.17226/1532.
×
Page 94

Below is the uncorrected machine-read text of this chapter, intended to provide our own search engines and external engines with highly rich, chapter-representative searchable text of each book. Because it is UNCORRECTED material, please consider the following text as a useful but insufficient proxy for the authoritative book pages.

off ~5 T=~r I. * ,~ ~ } O~ 0f ~e pnnc:^ ~~ :n human :~:~-~} devei~t ts our desire to ma~ wnse of the -I and bib- wodds in wh=h we Ilve. ~ sea~ h)~ ~~s ~~.~ that explain or w=~t =n diction, and we bib theorist that moist p=-~= ~ ~~. In ~~' —MAY -Is of t~ pa~ or ~~£~St of the ~~, prominent ~~= include quant-~-~e a~* ien~' area, and wiume of I :~d masses, 3~6 0~$ t0~' 6~-~v =6 p=~6 0: 0ur Am sp~=K p0~' ~nbutions^* an~ ~~t =~5 0[ ~~' 05 p[~7 t:~ 8~6 pi- - ~~$ 00~$ 3~4 pt0~5 0{ 000 Chic 8~ ~7 ~5 :~t K . —r=~:w obse—~~= ba\Y6 not~ th~ pa~ms in O~ects c~ be mo ~ ~ ~ a~ ~ an ex=~tiOn to ~V as ~~d Ke~n once Ivy when Tofu ~an ~~um what -act aw Sp~Dg ~OW $ome~:~:~g ;a be: ~: ::~ ~~ ~ at 0n vow ~ expw~ 11 {n numbe~% Vour I, is of ~ meager and ~~#~ ~ ~- ~ ~ ~ CX~g¢~3 1-g~ tO Sir thai I} ~UM68f ~ SetO=~5 Of maths matins are i~:~=ble tools for ma~g wn~ of the world in which The human faw-~tion with nu - ~= to 3~ {~6 in = -~$ of wh:msi-~) or supe=~:~us numerology. From the O~c P~s to Niacin Oardne(s fictional ]3~. Matnx$~0 pecple ham -6~

~2 ~~ :~h sublime and sinme~m numenc~ - ~= ne~ of p~ :n ~~s h~ p:~=d the ma~ema~ c~ :n :~.s o~f pro~ ~ ~~r m^~ ~ ~l ~s ~ ~~ate~yj$ tt-~= ~ ~5 5~ ~~ 85 ~ t~ of art -I entemns~m ~~+ to I .$ IN IN ~~.~ Given the :~dac~! m~ of: qu=~;w re$amn~g :n apph~m of ma~ ~ ~ ~ ~e mnate ~= Anacin to ~~' ~ ts t Lansing ~~ ~~r =~-~pts a~ spas few ~ =re o~f school mathemat:~- I:n ~ m~t grades ~ Cared =~ on ~ ma~ {~ p=h d=~ tO Wyeth i p~S of ~~nc t - ~~r m~ =:~ding c~l u~ ~ ts - - to- so~ qu=~e problems =d make ~~d beds ChDdmn Ieam man~ w~s to describe q~ dam and :relai~onsh~s using K s. to p~ emit and al~c o~s =d to exe~e thow plans ust~ emotive pro- cedu=~' and to ~~m q:~ntitadve mfd~ation, to Maw i- =d to te~ the co=~= for r=~nab:~. The skills Am. - ~ ~~e ~ ~ mn~d ln ~e ~~c of Yar~om nu - er ~~s and ~ the It of ~~c w=~ in to e;~taw ~~+ ~e publ~c =~$~.~5 - ~0 ~~ ~$~5 ~ - = Amman names (whole ~$' ~~ decimals), emus u~ more ureas terms (inward=, ~i031~$ [~{ ~~. Swamps of thmr a~, ~~e nu - ~r systems a~ wel~= pans of mat5ematics$ am 5~ ~~n ~t in whom ~r centuries. E~ enced ~~ :have—~~—c~s cTewr str~= ~r ~~ng ~~t s~! in m~-~g Editions p - 10~. ~~* So- it is enn~y rea- sona;b-~ to ask "~at can be n~ and exc,1~ a~ut teaching q=~ ntat.i~ reamni~" Su~si:n~, the an:~r ought to be' "~ ~~t :~hi~:~- Schoo] anthm~ic and a~bm have al~s been dom:~:~ed b~ the ~~] of tmini~ students ~ manipulate nu*m~l arid dais sv=P" 50$~- :~0 p~0 0: ~: t5.~$ =~ti0~ is t~ =~: anthmetic prob- iems or some aigebmic equations. The core of elemenmr:y acid middIc schoo! mathematics fe~s addition* s~tion, :~lication, and division of whole ;~s am Bastions' the cow of seco;~ school

to~ em: now dl~' ~~S aU ~~:~s or~w ~~d 1n wh~l :~+ ~~e ~ of ~~:e man1~; tO A :~:; :~ a~ ~~e =uat~ns. :3 cove~ similar operations ~ polynomial, mtIon~, and ex Hi =~.~. {~ ~0 p88~7 hit ~5 th=e rout:~e Pi mills :~s - n ~ p~quistte ~r emotive u~se of ma~atics. Howev=- me emer- gence of i: elect-~-~c =~s and com~ has ch=~d the condition Fox Tt is now ~ut ~ ~ ~v=rs ~~ce the technol~' of trans:~$' pnn~ =~uits~ =d bill chips h=t ma~ hand-~d c~m available On the mass cons~r m=~- Rap~ ~ :n electronics has ~~ produced Red w~he ~~om that a:~:thm~c on ~~ that can be entered a~ di$~-lay~ in decimal' common f:~:icn' or ex~:l Ernst ManY =~=s also haN ~ ~~e button submutln'~ ~r Aide A:. ~~iO'ns and pe—Bins cOmmon statistics ~~ions* Pm=~mable Cal=~=S o~r :~e powerful civilities, incIu~g Age Ail m~ipu- Iation' and matrix o~ns (~ee F:~re i)- Each of these ma i=] procedu=s is ~~-~;~e :~n mom powerful and sophisticated ~~ Ad. p~am.s tha,l mn on desktop- A: ~~w why wail~-~e in Schools" envlS100~d—su~ some eXcltIng - possib:~.lities. ~~8m6~Y

schoo! ~~:s ~ n~ deal Or ',<~ ~ N~Y c 'it ~2—~ i vew sm~ n~= ill ~~ -~d ~~—mlb 9~-¢ =~ Of the~ =~*~e co ~5 0n those ~~. M}~e If: ~~s c~ :~:~::~h : ::: :: :~:: :: ~ ~ que~s ~~t vanity, ~~, =d ~~s east - :m arc i~ :~ be~= ~~y -~ ~ Amp. ~ ,~::~.~ eX ~ =~- J~ ~e w0~6 0~0 0t ~7 ~~ ~0 ~5 0~ -art Hi and co~ for fa~ aM '' '...,.., ,..,,,.. Ale, ~ =~a ~ y~ to ch=~ S:~fi~iln~tO ~~= In mn~ dnIc:~. ~ ~ ~ . ~ ~ .~ :~ Ail. C~ and =~ ~ at.: h~ a, pm~d ~t -en -~e ,~w of ma- n~ m~ ~~.~ ~~= t=~.' mathemat1~ Ci=$ C=* ~h t~ p~S t~ ~~t ~~ ~~ mentors ~~ m~ ~..~s—m =~.~s with ~~ ~.3~: wria~.~. The expenmen~ m~an c=~ few berm cause An: ~ ~~ :n ~ - $~ - I.. of t~ t:.~e ~~d ~ "~- ~ * ,5 = ~5 =~6 not ~ 60~0 ~ 8~ ~ ~ ~ - ~ ~~, ~~ -~¢ =~s ~~ eme~ would ~r have been w=- ~e ex~r:~! -~a ~ ma~ema6= In ~ w~~ =~z—~ Id ~~ ~ ~ ~~ to rev=! b~ Is =~:vana:~. Me ~t— mat ~~ -~:r venEc^~n rema::~s -I ~~f b~ mason~ - m axmm~l: ~~ns H~—er =~tS a~ =~-~ I ated ~ new 7~e between 0~em~fi~di~ and th-~:~em~. ~w of calcul~= and compute~ for mathemat1~! w~ has ~so :~d to ~ dramatic :~e in interest ~n aunt - :~c methods and re- sults~ Many of the ~~ a~ most t-~=i~! re=~s of :~tiC$ -I that ~~;~-= t~ I- of In: hit :;;: pmp- C~$ O: SO1~ti:~S tO imp I YCt t6~0 ~~:C t6~S =-6 their p~ QUl~te I* ~ve *no c1ue as to bow one mi~t e—t~IV =~= the prom-~d ~~- :~emat:~ -~ranes of :~lid =~d prOw that ~~:e ls no ~~t p:~:e ~~r and ~~ an-v n=~: nu:~r ~~r can ~ ~~d uniquely 1~o ~ pmdu~ of pn~s Sut ~~ticlans wording tod~ ~~ d~ =~t :~! to Nimbi=! and t:hecreti~ pmblems poster b~ the need to const=ct Ia~ po m-es and to Id the pi ~~tio-~s of Ia~e =~si:~e numbed. The s=~-~h for edicts and e~cie*~t al~thms that w:~! guide come puter :~ws h~ become ~ cent=: a~t o~f 501b pure and agg3led Hal =~h :~n our t-~olog~inte:~e wood.

wc~ ~~ ~~ :~ s~ =~a ts ~e ex~ n :~ qua~e ~~s ~~o n~ ew~ =~ A: p=~ and pm—s:~ t~. ^~ As hi abbeys been us M' - l,{ =~s h~ Amp. ~~r pmd~ em. rimmed ~ A. ~ell £agr~n ~ ~ 61 ~ ~ A i~ · ~ K i ~ ~ mn60= 35 weL -= 6~int= ~ K t~ mpiex - 5 0; t~ - V8~S~* ~ viSe and me~m mt^~ m^~s ~r Brig ph=~= - ~~e no S= m~s'=~.. Sx~I'es am ~ aro~d uS~ figure used ~o ~~ =~* =~ ~p := K 8~6 tO ~-0~ :~-~S tO Y=iO~S =Ci~ p:~. ~~W = 5t ~ (~t 0~ seVem! h~ Same Ce~*~ ~ I've ~~a Aim Air. ~ £` K 8- 0: ¢~5 50 =~6 )~ t~6 =~ mean:~: wa~ The =~:~;r pn~ ~ndex -is -uwd to =~cu:1ate =~—K0~?£~.g, l0 creases ~ - Ace ~:~+ ~~S =d ~ I** of other =~W s~m How Can, in.~n be~ be m Pi~rt`~= O~ ~—ad teams ::n 6~= p~d sta*~:~::caLv to sw ho is bash :n part to —ermine fair I: ~~ 68~3 8~6 t~ ~ - t.0 =~t t6~ :~ - K mo~ ac=*,{~:~9 .* i ~ d: ~ 501el re=~—K:, - - StC=$ p! OCC55 til~i:~S Ot ~~l Imp Filet OSi~*g ~8 aims CO=~:~tiO~ ~-~tWOf~$ t~t 37-C p=~6 8~ C= and unaut:~d Ace lIow can wcum systems be Ovine and used i:. ~~h of t~ Or =d ma~ oth=s of similar complex~ and si~.~.~e - ~~.~e to I.. to or=nize~ mani.~* =.-d :inte=~t I ~ in~ti=.+ sk-u - t~ ~~n a~s or anthm~c and a~a or in solstice of tm~iti=~ t~s ~ W~ pmblems is: not :~:~Y lnsu:~:~t pre:~.~:~n ~r th~ tasks' it is i~:hy id* Q u~ t: ta*tivei ~ i: temte wung peo ~ ~ ~ ne c~ ~ ~: ex idle ~ :: ~y to ogle nt tt~ cnt ~ =: Mali ~s in n - e ~ situ=! on ~ $ ~ eXpress ~ hese re rations in e~ ctI~ —I,, Styli ~7 tO =0 co~mp-~ng too]s to pmcess i~:~$ 3~6 to >~t t6~ Ye.?* of t~?8 ~~ti0~* :~0 ~~:yi:~g =~l Amp? use6 :n his my: —en exte~d ~~ *: and Notions :o ma:~' I:~r alg~' and the ;~-met:c of con~e cIasses The Ail com:~! ~~:~s e>;~-~d ~~ 68~^K6:~6 -* t i.?* 33~8 t350S7 3*~0 0~, ~~; *i.

:: ~ ~~Y 4~r ~~ ~~ ~ ~ ~~= ~r te~ ~~t ~ qu~w tn ~ Arcs : ~e e~ce ~~= waste bet of ~— on human; eognu~+ $Wile—m ~s ~ Io~ her on ~~- en~an=~=ch~: 1~m ~ Sal Is the pay shim years he- ~n an unp~-~:~kt~:~ Swabs yang Mopes ~ - un~ ~~f ~~r In- and ~~= ~ ~;~. *as a Cheek Uses ~ a - airing nch m~ mo the item-: ~ hum~ comity ~~men:t fare—i: ~~* pnnc~, a~ ~s m~ ~ -I ~ pee ~ learns ~~ Bask saws :~ pa ~r admix ~~= ~~t Use of Cum=~a and ::~ctio=! appmach~ in skew Ate; ANAL cow~ Me Ace of rapidly Malay demand ~r brie =d coil enn6c apphc~*on of qu=tiwli`~ sew wit powers new technics = that s~n tho~ sets has prom~ed IBM of goals for schoo] m-~- ~ Rap: ~e dde of ~ 1982 :~ of ~~e Confb:~e Bo=d of the Al ~9 ~~ ~ add! =~ "What Anti abilities ~ be fUndamen~1 in the ~~um of m~W ~~- De~e extenstv-e :~ d~ wer the pa~ de= th=e is as yet no consensus on ~ p-~t mur$e of cha~e, =d most w idea sumSt.s th~ wh - ~s h=e nOt mowd tOwa~ any m~ ~~ in ~ ma:~e branch of m~h~tics such ~ Biter them- analysis, or a1~ :~=V art co-~S =d o=rmions can* be pmsen ID ;) =~ ; ~~= Ot ~~t i66~_8 four 6050160~$ 8~6 3~;~S~ ~ which e~ mbe:r far =d principle ~~ Ace But this r:~- 0~$ 6~i6~t 0~*~*i78ti0~ 0; (~ =~ it 0*~> ~6 hn~ pro~t of ~ Hi pro~s in - :~h Al ides were uw<1 i:nform~y long ~~re they become formal de5~ns and check rems. ~*~h~' practical wowing Edge Ski :~e than ~ city to =cite or derive £~] pnn=~. It regU*ires ~e Ability to :~ize quantitadw :~tion:~s in ~ broad ~e of concrete situate t:~s ~ well as the technical s~11s to ~~t =d r=~n ~~t those ~ , A hat ~ In Hi ~~ut school mathematics maw mathematicians and tea~s b^e Rued that the best guide is ~ cu:~m that :~s the meandering he pa~ but ethic numen=! techniques have de veloped. Others su=st that we shou:id capitalize on st:~=l items that have A: at the end o-~t path' to pmv:~e ~~r children ~ more efficient ~Y to d - ~~p num~r Ace and. tech.nigues. There is little '

:~ - - ~~e to: ~~ ~e n~t I ~~ - w ~~' ~ it seems m~ to ~ t~ Al u~ fires ~~ of . ~ ~S :impo=m to m~ to ~~' as. qu:~V as. possum' ~~ ~~m t~ ~r A. sent~ng =6 :~g amp ~—cal Jata. ~~ that :i—pectin wall un~ ~ mo~ W(~ 1f ~ ls :~d ~ un~ of shed ~~ of Or ~~ ~ :~n ~~e am the ~ ~: Whim Id~ =d ~~s :have ew—d ~ time. :~e~ m~ ~~nW )~ 00~37 I- ~~l p=~, 8~0 ~5 0t ~0 ~~= ~~:~ 3~t ~~ ~ t=~t p0.~l 0: ~e ~~ ~ ~ 5~t K ~~s ~~ :~ Am ideas =d metho~ ~~y Cop* In se~g ~r ~ ~~ of ~~:~! ~~r =:~s ~ Oe dewIoped in s~! ~~, ~ ls ~~! tO b=n ~~h ~ simple question How ~ n~rs used? In common sw~es su~ as daily' ~~, c~' in~tio~*~. *A or h~d bu~, o~e will find ~ :~g list of si~ions :n w~h ~~ ~~ ~ vi=} mie. ~~' s~l :n gu=~ve r=~ng is ~ cn:~! prerequisite ~r ~~s in my- sciennhc, te~' or bu~S o~:~:~' =.d ~e ll~ of ~vs Wat n:~rs are used in thow Mds his bo~ :~ and diverse. Wagner of cum.~:~a are ~~ fm=~d ~ the ch 05 Magi. I. t5~ ~~ p=~0 $~5 [~r ~ p - ~m sow s~=i=s ~~ Fit waSo~Iv ~ce outside whoop However' ~ warch f.~: commOn fe~-~s tn q~tim~e t*~:~g He. - ~ that th - -v: ~ ~ gmuped into ~ ~w cate~." one com~n an - Is of nu - = uses shows that - -ace examp:~e :~= one of thr~ basic tasks. . A. To use operations of arithmetic to :: aims size—to an~r ~~s like "~ many?- or"~w much9- 2- OR~.~:~. To use ~~s to in.diC~ position in ~ ~~= mth the relano~s of "~er shave o~r "*less th~ ~ 3- :~- ~0 p7~7']~0 ii i~.3 ~t - ~5 in ~ I lllustratiC'ns of thew di - ~~t Msks ~~d in. TV life Here a w~e pa~icul~ exa:~*~3 Sten~d m:~t tasks l~Wi:~.g :~s I: as lemh~ awa, volum~, mass, a~d time all em:~' numb=s to- ::~dicate $~:~# ~0 0~5 0t 8~iti0~#7 5~-~, Ii 8~6 d:~ isle co-~-~d directly to operations such as io:~;~£, com~ panng~ or pa~:~tion:~g of o~s that numbe~n measure. Other lmp-~t con.~s such as wiocitV, accelemtion~ a~d de:~sit\+

68 }~ ?~ In ~ ~~:?~ ~ ~~ at?= ~~ - em ~ :~ 5~, 5= t~ 3~ ~$~ -~erwe~ by -speranon.s on `m :~ - east As cu~ -it ~ ~ Hey Be: o~n ~~ d: Infers # ~~ ~e o - r tn At. ~~ ~ ~~,~ serum: ~~ - o -em - = - ~ have Owen :~m~s Am: -- I: the hardier of arrival bonds- the coder ~ =~ w~+ I.'' Dim ~~ I? ~~ ~ - S airy used~ :~ ~~ ~~. ]t ~?5 ~0 ~~= t0: ~ 0t =~7 ~6 ~~, ~~ station ~~ ;~ ~to ester ~~ want ate. ?. l66 ~5 ~ ~ 8~ ]~0 ~~ :~ ~4 ~~ t~6 0~ ~ 0= Eve ~sta~ Em :~ ~~ I~ ~~r WIt5~t ~~r ink ~= ~ te ~ TItt ~ ~ dImn~ tetw~ te~S 1n that—er. In analyzing ~~= of ~~= each possible ouwo~ to =~ Imps b~n ~ =d :! ~ As pmb~- Event ~ -amp maim i~> ~an went 3: =~?~ ~ ~ ~~) ~ ing~er~thep~ Aft ~~' )~/ 8~6 3 =e d~-~' p(A ~ B) sh - d -~1 p(A) ~ p(~* Tn ~~s s:~ t~on the as ir~= ~ m~m -~1 - Brood. ~ thme atmn of unlon ~r dI~ oint - ~~ 00~?5 t~ ~¢ 8~0 ~ m~-'~ numbe~. a? The um~ of ~~.c reams ~~y haYe numbers Or each :~vidual pl~- Bib ~ =~= $~tim-= :~:~e an assigned fashion, ~thmet~c ions or ~~:~ns Irving ~ seldom Eve am ?si~ifi?~t :~ ~~n ~^ ~,~ NEW u=d mIe~ 'as I~ - ~i?5 t=~V 0{ U~ 0[ ~75 Aims 560= t00 0~?5 to men t:~" But it owes the first Am Awed ~ - Ok for or:ganizmS t6? - p~= -of q~timtw-e r~g ~~s ~~o manageable ~i:lies— ~ to 6~6 51~fi? 8~t ~^ ~.~#~ t6? ~:~.~t?~?~ ~{ I ~.?~?~.??.~ ski:11~, and 3~li~ti0~# ~ ~ = ~ ~ +~ a, -or ~,41L JO ~ ~~~ Wish I re6~t ~e taxonoN'=n netp revem ~ omn tea-~-~= and ~~:ts the ex-~ntia1 mot mea:~:~ng 0t ~~, t~ ~~$ iffy: 0~ t:~0 [~$t ~ ~~l 35 t56 tt00?~. ~r j ust ~:~t p~?~?0 Unison and Bell have proposed ~ mO t31~66 8~75i5 0[ ~~} k~65 0t ~~? t I I- SUIT SiX d;~t uses of bile numbe~ - ~?5 ~: die collections (~p ulations) ~ MC38LL[OS ~r con~us quant:ti~ (~:~- l-~67 =~-~' R&liC CO~:~S (6iSCOU~, P(Ob~b~1i~iCS7 =~ $?? # I: (~ time line, test scOms), ~ Codes (:hi~wav, ,;~1~' pmduct m-1 numbest' and

c~:~Y :~9 - Id ~-~-~s (~ ~ A ~ ~~. \ parable! mxcno~ movers ~ th~ ~~s ~ As can If. {,0 O=:~s, -an ^~s t~ nu:~-~: ~sc~e~ dnion models putting t~er or $hiD:i~, S~:~n my Await comp-~' Aid, or recovering ~~ ~7 ~ Pi miss s=e cha~' AL ~~, or u~ of ~ rate fin :~ =~5 =~' =~' t=0 ~ i=.' ~~ ~~ ~0 ~ 7 or :~venng ~ ~~r. W~e mmemat1~s and t~ ~ qu~n ~e me~g of these He and -~e shed -~mpletene~ or ~07 it 5~5 ce—n ~= totemic to sib an~es Wit belp—= 1nstru=~= -on the ,n,~,m,,~ ~ of wpa~g ~~S to u~ nu~ e~y to e p - ~~+ Em: Of th~e dI~t ~~: ~~ ~~= ~~ :~0 h~t t~ =~ =~ts ~ ~ q=~w - ~ }~ 5~ ~7 ~0 t6880~i~ ~5 ~37 ~ If; r Axed= an6 ~ cowe~= ~~en phenomena ~nd Aim that ,~ - x~5 esw=~ ~~. Each God' - ~ Bier 'ax ~~: ~ ~t ~ A h== w~ have ~x~- ~~m ~ mi=~s am0~ o~ - s co~ng ~ Fathom ~ t~ Abe;*—~~* ~ un de=~d ~~s :~g press ~~s need Se experience m X Bile -is must ce~ acqulm c~^e skis 1n d—~ng h many ~i6c u~ of n~ - s' Rev To need to ~~e ~ ~~r Ape: on plies that number u~ h~ in mmm-ofn. Dew is c1~ ~~—m re~ in m~hemat:= e~on that u ~~g ~,~: s~.~! pmpe~:~s of.a mathematical -I ~.Cilitat-= r~n ~ the ~~m =d a-~li=~n to n~ sl.~^ -- .4-- .4.- ~'001 ma~:= ~^ tnererO~fm' emphasIze the It's t~ died ent Apes off :;~r -en - is se - e as mode~ ~ :~ o - ~ng$ and coding, to~ther with ~e ~ th In q:~w s:~:ion~- ~i~S afnd Relations FIemenm~ -yes of number focus on descn-~tions and IBM conw cem:ing specie quan~timt:ve fact~he cO~ -of ~ caned ba~ pnced at ~ ~ r I) Maxi Of ~ 5~d Oh: if i ) 0 ~~t iO-~g 3~d 30 ~~t V$ir]-d6 O.t thO By: s~ed of ~ car-~at tr^~is 300 m::~-~-s :n ~ hours. Ma~0Y of -~ C~:$; t69UitCd For me teaks IS By 3: CC~1 8~d ~~C t8-~:k

-~pAP~ ~ N~y ~ ~ quantl=~w :~=so:~:~s to v :=at p - o- ~ =m—~~' ~ Use ~~ ~ h~v ~~d 1n ~~ ::: of :~ :~d relate compu~:~^ 7~ ~~ ~~= Is a relation amok `O Or m=e varying q-~^ r example, ~ AS tl~ pa~s .~. dep~ cf w~r tn ~ tI~ pool m~s and decrease :~n ~ per;~c ~~. a- As b=k s~ rates :3~e, ~ m:~ =~: o~ ~ 5~d m=~Iv depos1t ~m inc~es tf a =~uence of ~~ haw ~~s :~' 2, 37- 4, 5' ~ ~ ~ ~ t~ 3~5 0~: th~ 13~ 4, 9~ :~, 25~ .~. W: my- ~=a~e of baw ~ =d he~t h, t:~: penmeter ~ :~$ 2t ~ :~ K e key mathemati=l odes required ~~o reamn ~~t such p=~:~:s are e =m =~s of elements ~~. wn~, :: =~$ ~;~i0~, bitt - 3~6 ~ 0{ DISK {n school ma~e''m=~ Aft ~5 590~6 ~ ~t '6~ 0{ t~ ~~g ~~t card ~ Ietter names I* =~own numbe~ ~ wK~ ~3t pl~ ~~:ns on thow ~~* `~bm In~=~;~ ~~ on ~~! pm=~.~ ~r t:~:~g sm - ~~c ~~s and so~-~g equa ons ~ kind the hidden val~ of the van~. But ~~e skills are onTy ~ 5,=~! pm of the pOwer t~: ~~a ~~o- vtdes" Tn ea~ of the examples ~~' and in cou:~s ot~r s:~:~r p—-ems, t~ concept~—= of the matter its undemadi~ relations Ems ~g 5~ 9~= ~ ti65 ~0 ~5 0~ K ~0 ~~) 0~ 05 ~ ~ stu=~s most un:~d is not ~~y `~a Ic~: s~:~g ~r ~ ~~ orphan ~~n va*~;e in an ~ua~#" it mu~ aim :~ ins a5~-t ~~s as m. ~,~tit:~s t~t ctange as t~e SItU~$ lO W61~5 t~V OCC~t ~~*~- Is are no! mual: s=~nt by themwIVes~ but only :n =~ Idiom to o~r ^~I0SK In ~ mallsti:c applic~s of ~~m the ~~,~.~l =~g t85:\ t~ ~.~t t~ 5~6 ~ ~~ 0; Kid twit 5~i55:~5 0 particular =~:ion~ but to analv~e the wIat:on between ~ and ~ `~r ~l x+= The ~~t uSe~! air ideas ~r thinking ~~. :~s of th1s w~ is the concept of ~~:~n ~ .~ . .... . ~ ~ .. .. . .~ ~ ~ ~ ~ ~ .. ~ ~ ^'~ . ~ I .~ ~ . `~ ~.~ a: K ~ _ ~ _ . . I c ~~op uncemIanol~ :equl~ to: cnec:t~e appllcatlon ct alge 6:~' st:~:~s need to encoun:~: ~~ 8.~,3I,~r;~ ~ ~~ ~,=~':,~ ~: ~~ Eli ~~;~:~d 16v .~:~;~:~.~.s am:- v~:~.~q TheV mod co:m.~able un der$~*nding of relational phr~es Such as ~ depends O~ it? ~ :5 Mention of it?- 07 053~ i~ ~ ~5 ~~ ~ -

~ 1 ~ ~1~ b ~ Icon Carrion ^ ~ ~ . . e ~yc`~c Y=l~10~ d Consuming ~~ ................................................................................................... ~ _ ,.,. _ .......... ~ . i 1 1 1 ~1 1 1 ! i 1 - ~nGERE ~ T~ ~~ior of ~n~mcn~1 ly~s of ~l~ions among gad =n ~ seen mod Dailey ~~ lyrics INS. Gum {a} =d {b) if dim a~ i~ Flagons, (~) and {d) Cow ac=~ and conning variation, and {~} and {6 ill~ralc cyclic find slapped vadalion. Virtually all vadalion actually ob=~ed is ~ combination of 1~ basic lye.

~2 . ~ N~:Y d.e - en; ~ mpe - ~e of cntena ~r c~ and -toning b~ st in ~~> 0~= ~ He's ~~e ~~ A~ion ~~ ~~e A~t of ~:1 S~StS ~t ~~s shoddy =~e to ~ I~ ~ Meowing Bind :. amo~ w~= :~ ~~e 2~. 46 [}~ =d l~e ~~n - = one ~~ mem~, =~= also tncr~s :~:= d=~) at ~ simIlar me:. * 'A=~ - v~ o~ ~~e t~s ~~, nd tn~ ~ an into r~. Converts van~n—~ one variable incrust mthout limo, another ~r3~= so~ :~= - be-. -art rim:—as =e vanity increases -~i~:xndy, the other incomes and ~~es tn some =~l~—le Sapped vary—~ -~e wn~e inch Other charges :n . :C i6= t~6 iO~g p=~S O: - ~~ Bali-= Ot [~tiO~$ :iS tmc~ of ~ I:. ~~on of ~:~! Mmi~es in ~~ parenth~ dimwit ~~s ~~s application of ~~.! daunting methods to :new pmblems. Wnh -~he ~~us of ~~bm Are at vary a6~$ am ~ions' equat:~s ~ inequalities can be u=d to ~~t specify =~tLitions~ :~f ~e hate of ~ pr~e is ~ ~:~:n of its time in digest wish m~ h(~) ~ ~2 ~ 88t, the equation ~116~: ~ It ~ ~ asks when the projectile is at bawd I-~-~1 :~e ~~e 3~* if the population -of ~ county (in millions) is ~ If- of lime miep(~) in t20~20~)' the inegu~i~ I2Q(2;~ ~ 200 asks w~n the population will ~~ below 900 million (~e Fat 4~. -* . . ~ . ~ Ab: . * A w] course' :~.~g angst q=.~.~.e relat.~-s as tun~,10~s enCou~ es mooning that =:~ds ~~nd familiar—uation~ed gue~i-~-ns t~ I: 0t =~5 0{ (~7 Chime and minima, and owed! t=~s. n:~E ~ The ~~d pam~e t=~ be comes v131~c 1n ~ =:~h of to h¢~! of ~ pr~:~e ~ tun*~n of )~s t)~e l;) clew . ~ ~ . ~ : .

FIGURE 4. ¢e =~= ~~ =~e r=~ . ~~a :~y Whole Few -quell= =e ~t goners =~=d ~ to s~! ~a, t~e-can ~ no ~~t -I =e :~t -I a~ slt~n . ~~c -his maim.. -I 6~m ~ep in effective problem -~:~ng ls to Maize ~e p—I~ too 1-~ ~:~r =~s t~ m=~h problem con~-~. ~ ~ ts ~Iv part of the A- phase of solving, -is—the con~ de~n of - ~t Is kn~. Pr - ~= solving ~~o Hi -~.~e of new ::~:~ation that ~s new infix. {n mathem~i~ th~ tn~= 1~y rell~s on ~~c - ~~.~s ~ Bill =d m=-~- elating i: a~, in q=~w Ire ~ pmcedu~ ~r :* ~~" Wcent an~ of mathematl~ =~ - t5:s kin] 0: I- a5 p~/ ~0~$ ~ 00~St ~ t5C = ' - ~~! knOwied~ - -A to - ., ldeaS-*,4 PrOC~! -e In-~:~es te~nigues .- to :~nt ::~ - nation =d to exec=e :*- ~~t I- :.- to. ~~:~c numen~ pmb:~s Fo~! mathematics is ~ sweet that deals mth =~ con~ts th~ are Ad f:*~:m pattems art o~. Butt :~*~cians have also i 3, - dO~ Of C~ ~ tO if; ~S Of fCp ~ l2~8 1d~03S 1~n COO ~ ~~r ~~ :s ~ All of ~~s that co~v tn~: Amid ln I' aOd com;~t ~~. :~-~:n of :~=s se~s as an aid to -~ and as ~ medium r -~:~nicat~* 1~* mat:~-~-~tics ~e repmwn:~ti-~:~s become ob~ ~ ~~ ~~ ~ ~~ ~~ ~~ ~~ ~ 7 o~en ~—e as uw~! models o[ unantiClpated patties ~:~ti0~3Y concrete sit_

ea ~~ en~s emCi=t re~n of: hers ~s the place ~e system of num~+ :Eve~ w36~e near has un:qw reed ~n ~e Standard b~e I He—on system, and rational ~~ =n ~ Use using ~~racucns or as ~- of who~ num~+ These ~~ systems- ~ sum remand ~ p~e ^ue systems mth East b~' espy ~ ~= Few the ~~e base has obvious- advances Or ~ particle= burl White the ~~e—~ Em ~s m~ for =~d today:, ~ Dim ~~t m^~= t~` :t as- ~—r~ben~g that the evolution of such ~ powered Resend wheme wok ~ ~~ long nme. -A s=s in the r~d of ~y Mempotam:= ma~=cs ~t a base 60 numeration system' and ~ Ever symbols, was undlersto~ =d used. H~ ~e place Bale =~W eluded ~ ~thtm~s in their voider em~ It was not u~T lI:~u m^~¢~s of the eight cent~ ~ how ~ use ~ (;~) ~ ~ place Alder thm the ~~on of place - = nOudon ~s =~.~. He sec=d m~r mL in ~~ng Emend :in: - nation is to mI~:~ps ~at are ~e ~~r all ~~' ~ By numbe~- or for cem~n Ados ~~ He fall m^em~! con~ C=s -Al are wn~, NncnOns, a~ relations. We nowY ro~W t:~ use Ime~ to Ante variables and ~ wrhe :~= ~~r ~;~=ions mI=~- At ~in:, :t is worth ~~g the the histonca:! devel~ ment of Add. - ,,~ Anemic n~:n is a. :1~g $~-—t - ~.~v tO the—t ~~ the use of I~:~=l wn~es with a~c syn~ seen as x: ~ 4~x ~ 2~~: is =~.~:~g b~ -A Graphing ~p~n Bile t=~1 pl=e -ad nu~is =d Cleric expressions am the most lmpon=t sy - ~:~c ~~s ~r recording q=~e if tiO~, ~~> Ot~Ct :~tiO~ tO~$ 8= 1~ If- USC~ ~C = ~~r are those that identify num~bem w:~h ~~s :n ~ go: line or pairs of numbers With :~ in the p1~. For =~1e, =ndit:~ns on ~rs~dbles w~h ;~5 ~x ~ 9: ~ 3 870 9~;~10 common in a1~a and its appli~. The so1~:~s can be Teen in ~ similar symbolic ~7 but :t has become almo~ as common to display the ~~1ts on ~ number 1~:ne Ah (figure 51. A1~ chic rep~- semati.~n is ce~ainIV not as =~t or computation~v use~] as the ~~:~c ve=~, it conveys Akin ~ tot~ p:~e of ~e guan~-e condition. The use of visual Aide to display ~ =~tion among quan~ Bill ~8 is It him - ~~ 0~0 ~6 :5 ~ ~~

A: AS ~4 -3 why w ~ ~ ~ 7- ~ ~ ~ ~ ~ :, - ~ l~ ~~ - - ~~ ~~ ~ ~~ - 6n A ~ nGURE )~ I. ~~= 0t ~C '1~$ t6~ ~ ¢x in 2: ~ 3. ~ ~ Here~s ~ c~on ex~3~ The ws~on of ~ pl~on ~h 4~h ~~ ln an: :~e ~~g ~ 3:~O Am ls ~n ~ the ~~n I—ere ~ ~ ume mea=:wd ~'n ~~* ~e =~= pluck. po~s ts Am; ~~d by- ~ M=~n ~h (:~m 6~* u~ = :~me ~~' ~:~ V:~ im~ of ~ rel~on b~n t~ vam :- :s ~~t pamcul~y emcee ~ ~ =~;~nal a~' but :t does m~ ~e s~.~ant pen~ pa~m in piston motion :~n ~ w~ the ~r T~ ~~nt—m ~e sy~:c few. ThiC ~~ 0f ~~t i.~S 3,~d If ~~S ~8 3, -war ~~: mathematical wchn~-~- ~w~' the amen of ~= I at CO=~: SO~tO ~~$ =~ ~ ~~C i: O~ ~C == O; producing ~phs and thus on their ugliness. It :is ~t po:~e ~ produce g=~S q=~y and acCur~ ~th thou As a~ ~~> it ~a d~ - m ~ienuf~ Ammo or - m I~ data bases th ~ Comput6= - e maw ~~le As ~ =~iC mphir dI~-s are b~:ng common =d :~si~y so~?h:~i=~. ~~s it is : tant fur mathematics ~~s to b~-e ade~m ~ ii Chic mpre$~ns 1~:~dv and to u~d the =~s among sum - ~^ Chin and numenc~ ~~s of the =~e ideas. The~e has been ~t An:: ~~.t the potential I: of ustng ~0 Its =~:C =~$ 8S 8~ fit it It ~~' eady expenm~ Shave w~d the ~= that the mesons provided b~ ~5 ar~ ~~t ~6 ts¢ ~~ng eame= ~ ca~iv as m ~0 be e>;~' wh:~e the effects ~ sc~e and ~e Aced Viking mn~w inhe:~t :n =~r d~:1~s c~e su~pnsing ~~ miwon=-tions. ~ . .... 61~6 ~ :~¢ =~n 0t ~ IDA )$ p:~:~6 ~ ~ 5,A~A~ g=~' ~tA:~6 (~\ ~ Ce~a in k.; ndS of ln ~~:~t lon mo~ c~.~( t}~v than S~d a,)~c ~035^

76 ^~ ~ ~_~r C==ian Was ~ numerics and Cubic p81~S ~ only lbe awes fig ~ impala of vim -= alas ~r qu~^ _ ~ bu_ning~t~o~ ~ Is find nether invades ma new 1~gu~ for ~p^nti~ MOODS gab inter I quad e and ~1i~ Mu. In some ^~> ma ~ s Ural dialog qu Ike accost ~~-pi~ Is or Lang ~~ link ~~ us Sable. Tool Melons such use =1~ ~ urn lo odorize ~d~l~ ~e-~ i^~1~ ~~ Ember- of Sale -~ baleen o^ of a gab. we laid of ex~ data alas includes m2~ coder nag ad emu Or wing ~1 I in gays 1~ =~ waning quirk ~nci~ly~ and early. we use of c~pules ~ prepuce Hose disks is Ming ~~ prim in ~ am of sbDl~d m~bem~i~. we of lag pd~dp~1 mans for using Compaq Colic Ems lo ^- p~ss Hellions Song Climb wales is the balms eggnog of Plum 1~ ~11 pager of mug nuked or Cues #~ ~ singe s~b~ic sentence. Hoar 1~ ~l~clion Equip- 10 Ada ~11 lions of ago 10 acyclic ales also maces 1be idiot in oboe loss Cede to ma Henna used. 6~un~el~ computer tools Mae dimly ad Cloning gab 1~ day IS easy. exa=plc, 1be Mince eq=1i~ ^+1 = 1~1^ ~ ~ = 5000> d~bos lbc balance of ~ $5000 can a 12~ i~ that is aid paid back in monthly prowls of $~. far mod apple 1be gamut Intc~1 Pa Balsam $ $~5 00 $50 ~ $395.= ~$460i. $~5.~ $46.03 $398.95 $42~06.05 $~ $42.^ $~2.94 $3$03J I $=i.~ $36.03 S406.97 $3396.14 $~X $S3s96 $41 1.~ $2985J0 ~5 ~1~15 =~ $. $2: 70 S419.30 $2150 65 $~5 ~ $21.51 $423.49 $1727 16 $1~ ~ $1~ as ~ $12.99 $432 01 $867.43 $~5.~ $s.67 $436.33 $43 Jo $445.~ $4 31 $~.69 ($9 59) nOURE 7. Spat ~p=~l~n office brand ma $50* loan a 12~ inane 1ba1 is being paid back in movably ~y~ments of $44i.

actual =~m in the dog:= v~e of th~ ~~n and the I: of `, A ~ ~ int070~ 'it ~6 :~ ~ ~ 3~> hi ~~ - Si~0 ~~ ~ 8$ t63t S~ i~ ~~tC 7- Of cou=~ const=~-~-n of Air Are =~s some ~:~v exp=~ relations in the ~~Im ~~ th~ has be=me Rare -: Sp.~+ T:n this =~e the ~:~.= ~~ - h ~.~g inches -~.~ Payment I:- JPnnmpal _~ Iffy i 1 445 ~ D Ol*~2—A3~ ~ D~3 __ A3: ~ ~ O:'l*~.S ~~ AyB] _ DS C Co.~d =~31 =~.~s of ~~c exp slons a~ ~'i.~g to be ~ w~ u~! mo) 1n my: pmblem s^:~ng. ~r :~ to pre~re the prev10us examw' we =~—~e ~ pro:~e monthIV p~t ~ em. Succ~e ~~' -act ~ using the -Gus c~:tion~ ~~:~. But ~=e ~~ns a:~-~o sewe a~s ~ en- Jam the mnc=~e wodd ~ anthmet:c mason~g to th6 m0= ~stract wo~d Of ~~-m and =~:~ts that begin -mar Fu.~-~' the w~ of relaxed =.~s comes ~~! circle when compute -~fi:~g t - ~s ~ ~d to 6~-d symbolic mies th~ t! pattems i~ .~s of ~~ -ma. The secon.d m~r aspect of p~: ~~ consists of tech nigu:~s =~v re~d t:o ~ a~ ~r u~g I:: ::~ fo~ma:~on to solve problems. An al~:~hm is ~ "~eci~ly~def~ed se~ q~e of =.~s telling how to p~e specified ou:~t :~:ion fro-m ~~n input :: in ~ fiche number ~ stead. ~3 D~-~ng stu-~t sk:11 in -~on -of mathe~ati~ a~thms has alw<~-~d Id =~a at both eiem-~-~ —and semnd~~ {~. The m~ pr-~:mi:~t a~:~s have been pm=~s ~r adding* sub~:~;~, made an~ 0~ w-~e num~' common ::~- tiOn.~, and decimals,~, along math the patois! -I on ~~-~l and sari- exCreSsi~-~n alRebra. - ^— But those are only the mo~ ba s:c and familiar am~g ~ w~ ]~' ~ mut~e my: tools. Em 8150~= ~f CX8~0 3S O~y O~.0 0f ~~;$?r~1 :~ ~~ oafs for 6~ding the =~t =mmo-n -divisor of :~o intend, the Si - ~e ~ ~ , ~ ~ ~ ~ .. of F;~:~? :s onto one of mam-+ al~hms ~r :~mi~ pyre Am e~ ~ t:b ~ ~ ~ ~ ~ Eli ~ f o~ ula i ~ ~ ~ ~ ~ ~ mam alacrity ~ a? ~ ~ I?. {~g Brim 09~8ti0~?~ 3~6 t6070 876 60~8 0 8~5 ~t 50~g svstems of linear ~~ti00.~? 8~6 Haiti -

~8 I.. ~ ~~ D~ Id app~n of ~~s =e Ray ~ t~ h~= ~~f Tin ~~Ct t~ =n ~ AID to ~w ~~ :~ ~ =~e Is. l~ ~5 0t ~39~: ~~= ~~t ~ ~e p~ of mathem~:~s ~~m lbe ~~ :~ 118 used in bu~:~d ~~= ~:~ s~:~::~:~= med:~n p~= and :~ ~ :~e ~~e ~~ ~ :~ na~ of ~e~ ale makes t~ =~y p~ed ~r car exe~.~. his ~ ~s ~~r tmpli~-s ~r wh~ cumcula" am saga agony ~ is ci: ~~ Stamens importance AL bm~ a~ic~ili~ to ment ;~ elusion ::n elemental Air secondary ~~! mi! ~ain3:y h~ been I =d ~~e al :n -I = ];~r aDd mm:~r share Inexp=~ =~ ~ perch : 9~' ~~, 8~( 0~th=5 t5= ~ t=~ l~ ~~- Th~ =~ t=~ Bier ~~ ~~ ~~ ~= s ~~ ~~ pm§~= I.~. eXecutIng ~ Chin 0~m Bale they ~ led ~at s~! I~: 1n ~ ~ ~ t~ same time ~ Ieam~ of :~ a; has ~rr~i:~d in impolite ~r Chow m=~i~, :t has become ~ mow ~~> ta:nt ~r eve~ne Gil qu=tit~:ve ark to hew :~! un~ of tbe a: po:~t ~ view 9-~ To- be an I u~ of Comgute - ~ed a~s, it is- usei~) to Idea su~ at~u—as v ec=~ and - In as ~~! ~ ~~damen~l m ca:! conce~s like I a:nd wcu:rsion th~ =e too I:~e app:3~ in traditional c-~. In sho~ t:h£, ~~mic ~~t of my ta~ o~n ~ v¢~ distant ap~:~:~ce—en ;; and com:~;rs take Over routine sy~c pmceduws. This new con~i.~n requlms Ida. reconsidemi~= of ~~s ~~r qu.= m.~.:~tics. c~ an-d ~~ ~~ :~ and =~s hew cIea~y ~n Over routine aSpe=~s of both Ire a~ manipulation of guantit~i~ ~i:~tio-~—the two key mmponen~ of pmcedu:~ kales The =~k of t=~:~:~ng t50$C :~W £~S iHtO D0W ~15 ~t =~8 pOSOS ~ cntlcal psvehm 10~al Question conceming the interim—t~ mncept:ual and g7m ce~ age ~~ ma:the~s educators worm tt~ cxtens~ use of =~:~r and computer toots with cO=~ding de~:~.s 0[ t~iOl~g in 5~;i5' Witt U~609~i~6 ~0~t O: 00~8t UO6Ct-X dim pmheien~ in solid ~~-r~lems, and Ability to iea:m new ads vanced mathematics. lPhe I: of undLe~nding and skill in mathematics has been stud:icd and de1~d ~~v ~r :many vea~ but both re:~d enthu- sIasm :~n the pa~ dCC30Cx ~ :re~:t meta-~sis of over 70 reseawL

:~m Id: ~$~3 =~ ~~t ~= O~ O; ~~= =D C~C St~t conce~ unpin problem Cling and abodes Away maths ~~:~ws withy a;~nt h~ to a~is~io~n ~f traditional skied Mo~ d ~~ ln ~:= su=~s st~ :~s ~~e thew ls ! of Ark ~ In* on His ~~' ~e pnnmp~ wpomd. s are ~~m Demana =d mu . lD ~~ - ~ of ~~ e~s the Mc~= of* COm p=~r w~ used to complement inch ~ t.rad:tion~ ambm~c ah. ~~ ~ skid ~ ~ w ~ . ~ A A . ~ ~ . . . wnat remalns ~ open =c w~ tmpon=: Amp :s ~o d~e ~M =;~.~s of I* ~~ - ;~r::ne3~s ,~ wh studen~ ~ ta~ to red more I ~ technol~l hchp with ~~ Muir =d ~~lw ~nipu*~:~- it seems sa~ to ~—th~ ~e d~=e *A =~t I 0~:~*~ ~ ~~l t~8 . _. combine ~r $~e time* It is ce~inI>Y ~e =~! tssue raised ~ the *~:~t of t~' in school ma~ics. while ~:~*~e I:s considered ~~e Concem1~ ~ kits of ~~:~g ~~:ti:~n in schO~ m~s ~ tra~:~ ~s to mncep~ and p010m so~ri:~, there is no di=~eM a~ut the id of de - Ming student. ach1~t ln ~ wn~y of in:. mal a~s of quantitative r=~ni.~' to ~~- - at moist numb~ I. Den if mangoes take over ~e bum of co~Mtion, it remains ~~:~-~t for use~ of those machines to plan =:~t opera- ti=s and to m~:~et r~ts inwIll~- Planning c~ions—uires mund undersmn~g of the meanings of oper~:~s—of ~e Rustics of a=:~ns that co=~nd to v~uS anthmetIc ~mlions~ In~ te~ion of wsults requires judgment about the likelib~ that the machine o-~t is cornet or thM an e=~r m~v Me been ma~ fin d~a ent~' ~~= of operations' of* machine pe~- (Development of nu - = wnse is discussed :n d~i! in the Feb=~ 1989 issue of ~e /~0 ~~,$ 6~7 in t56 bitt t~ 80~) The= 870 tw0 ~~ kinds of I: :~d to wst numerics remits for reason~. First ~s ~ bared ~owIed~ of q:~tities in ~e real ~~- is ~e p~tion* of the United States closer to 20 million' 200: million, ~ ~ billions Is the Deed of an m~e ~~r to tOO 1000, At* IU,0-~U k:~ome~s per hour? ~~t are appm~imate permit {~5 ~r ~ 581~5 are ~ 08r i0=, th6 tip 3t ~ ant 0: 8~5 0: ~ m~r Iea=e 15~l hitter' White this so~ of information aft pm o: ~~} mat5~tics7 it is an in- 580~p for :~t of arithmetic applied to real :~rob;~* The awed co~:~t of computat:~: number sense :s the acidic to make quick o:~of~itude approximations. As

~~ w*~*~£7 c~a~r pm~ an exa~ an~, ~ :-s :~ :- to~ check t ~ ~~d m=~ are "~e ~ ~] pa~ ~ Thy me=~ ~r it - ~8 t: ~ 343 ~ 257 ~ ~:54 :s apt t~r ~~ 85 :x 2:~-~ :~< t~:~- b~:~-~- ~ =~:~e tm::~g i:n :~ En ~ ~a~:~- bm :~n 00;~le wph-~on Of ~~= ~ue ~60~=~=6 =~_a ~ ~~t mMa ~= the~ - ~.~.~.~ :: who~ art sIn~ :~= 31~ been, =~e I.. ~ 10 ]~ ~~dc =d co—umm~ e~ ~~:r~e ~ Act. =W ~~e ~~ wal~ c~ cumc~m ~~, ~ e fact ~~ s~.~s ~r ~~s impO - m but Ion~d to~+ There ~~s Boom =~v ~ com~e :~! s~! requ:~ - to deal e~y with symbolic expresMons and ~~c I—to ~i- ~ and ~ ~ ~~) 0~3 =~$ i ~ ~ ar~ are no! aS Mi:y ~~. ^~ r~.~e =t of ~~h ~r -I ~~ ~~ ~~ ~~ ~ ~~ ~ ~~ ~~ ~~ K ~ Ability to s=~ an ~~:C CXp~= to m~e rOu~ 01~es ~ the ~~:~.s that wY=I~d emerge in :~c Or =~c repr-~ ~ .F ample ~~:n f t ~ ~ in ~ ~ ~ ~ ~ ~ 5~ ~ wnse =~ld s~ch the ~ph of this ~:~on 3nd re~ze that ,~ ~ ot on ical:l ~ l~n ~ magi ng mth smog values of ~~) ~ ~~w ~ and :~v 'ncre~i~g # >~K ~ ~ (~ ~ 05 ~ 0: ~0 ~ K ~-~-t to make ln~d compa~s of o~n of: m.~.it~e ~ ~~ ~~ ~~ ~~ ~~ ~~ ~ # K ~ ~ $ 8~6 ~ K ~.i~ skill, ~ bn~ bet~en number and symbol sen=' pl~s ~ image tant role in ~ u~ the co~! c~16x ltv ~ ~ a1~:~:~.~s Or mathemat:~! ad in~=i-~;~g tasks that are the Cheat of computer science. Abl.~y to wan ~ ~~ of function values or ~ gape or ~ :~ pr~ ve~: s~d cond:~:~, to :~-e:~tif>' tile :~3~v form of an 30~( =~0 t53t 0XptCSSO5 t~0 399~]:~10 DISK ~0 0~= p:07 gi\~ t50 following t~~ ~ st-~t with ~:~ sen~ could p:~ ~ ~~ =~ ~: ~~ ~~g function is :~v to be o: the fo~ i<~) #a- ~/ ~ ~ bitt ~ Dim: ~ 5 and ~ ~~t 50:~. sales ~ :~: ~~ 20 ¢~s fix,) 510 67: 825 30 40 ~~ 960 ~tOO -WHO

-81 ~ Abdity to IBM ~~c ~~=l=s ~ p-~= the ~~ of ~:6 ~~ 07$ 85 it 3~.~ 0~.~i=' t~ i~= ~0 ~~ and ire t~ brim bat it bas b~ Marl comer. ~ Incas, ~ ^~t Shored w.~e ~~ m~t thI - ~g 16~ the pmd~ of :~;~= and -~a~¢ '~s mI! be ~ chic ~~:~. ^~W m- bins vamp of several e~L:~nt Arm- might she.= ~ saw se~ Bomb ~~ ~~ ~ aide t~ -~e ~~ed ~~ -of ~ pclyr~ ream—ds ~nforrnation ~~:t ns zeroes but m~s ve~ ~~ Ion off Ivan bw. eS or in:~- ~ e~s ~~s how =~: ~ and Alpha ic -~: tools can be med effectively ~ bilk smdent ii ab algebraic -~mbol:-c forms. N~ th~e even of more ~~ em] ~l wn~ Bins an imp r:~h ta~ on the pMb tO n~ appma~s ~r de - ~~ continual am pmeedu~ College -of quan.~;~;~. NUMBER SYSTlEMS For ~ == mans ~~s mathem~ic-s is ~ -ark IoowI~ connected ~ ~ ~ ~ ~ d; ~~;~ ~r07d p=~,5~- HOW~~~ it; is imp: to r-~-~r th~ the -pique ~er ~ math-~ati=: con~ s depends on ~~:~t meaning Chin lies at the hean of a~ spe=~c em~+ Reaming the fundamentals of any branch of ma~mat~ ics should :inciude r~nnio-n of those I- ~1 pnn-ciples ~~t Inn the relat~-~s amen its =~s and I* For number 87~5 ~ rain 3~) Mimi of b~ and po~} ideas Ire e ~;~w of each svst-~. W~n one --I back frorn specific details' Pies ~em ~ Hebraic and tOpol~ pm=~= of numbers* -I principles -am ~ used lo de* ~ ~ x~ 0,~r ~s,t,~s Ed to Aide the mat between ~~! ~~ms an-d If I q=~ problems. In the hi~:~! de - ~~t of =~r systems' the p~si-~n be- n *with the natum! number Ext-~s over ma~ centunes am ~=ions, then n+~ numbers' se? finality; ~ rigorous, chamctenzax* dial 0: t=: ~~. ~= ~ I ~34: the end of the t*~ti~h cent:~ :t :s poss:~e to organize al:! those st=~:~s mom t~e t~ '.4

~~:~= ~m N~:~y sea:. wsw~ ~ ls the onK c~ c—~d 6016* rat nu~ system His ~ Abet :~d of R" Be inter ~~r Deem I~: is the ~sm~t nng ~ ~ `t : ~ i~ ~~ ~~£ Said ~ :~S ~C S~::~500t 0{ ~ th~ ;~5 ~6 ~~:~w :~ a~ ~ ~ cio~ ~ ~~ ~6 ~ ~ A ~ t5C tC~6 ~-~ that lS DISK O ~~ ~~S, th=O four ~~ =~:n ~ Ace; d~ of~ Breath about ~~- Him the ~ no*—t~ Is ~ s~ m~ ~ b~ ~~ns a. b* o~r wi=~- ~= ~e o~s are com:~e a ~~, ~ ~~mn d~= ~r ad~' t two l~y el~, olive ~~r ~~n =d the other :~r m*~hipll.~' arid th~ ~e opemi=s 1:~ct m~ she order ~**elaUcn 1n fir win. 06~ =~' However o~= :: probers of ~e ~~ ~~*~ be:r system that aw n~ ~ apparent - ~m etch Pomade ha: t~$* Here aw Aim ~~es ~n ~e a - ~c and =~ p~i65 0{ ~ I- 57~$ ~~ ~t ~0 ~ ~ ~~ c§~ tO'*~t ~m both oure and a~# ~ ulu*~, in pro lI*~:m we ~*~ to me mo~ SULLY, helps d - ~~.~ ~~t l.~t Wo- the n=~= of nu~= ~ nu~ ~~. while st:~ts should emit ~m sch~ ma~:~cs with nch ~- 009~ ~6 p~-~i \~$ :t )~5 8~0 i=~*~t t~t ~ ~~¢ some sent of the Ho:* pn:~s t~: provide I~! cohe:~e to :. ~~. |~0 ~~*~.~*~*l 3*~6iti~, =~ip:~$ 8~6 0~: 5~8 0t t~3 netuM' numbers and i~* are based on s - ~~) slmpT pnnc~* ~~t is she pnnc:~e of finite i:~tion* if ~ # ~ set Of natu~ numbel*s th~ con=~s i, a~d if ~ con taints the :: ~ ~ ~ when:~ j! :~s tbe nu~r ~ ~~n 3] con:~s all~ the natural :~" ~ n15 p:~V 1:~s t~t t~e natul*~t n:~:~s (~d their eXt:~*io:n to all in*:~) ~:~ ~ :~:e set' ~ sequin:= of equally ~~d elements w:tb :no :~umber between any integer ~ and its successor ~ ~ I* They pr - ~~e a* set of tales ~r o~:~g ~~s in anY prOces~S that =n be Viewed aS oCcum~ in ~ sequence of d:.~e stops ~:e Mite induction pr i.31ciple )s used to Afire co:~:~s '*writ h ink pammet:~' ~~ 3~< tO I p! OpOS3;~iC~S th;~t in . ~~:

c::~:~Y 8:3 that ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ (2n ~ ~ ~ ~ in- ~r ! n' on-e 0~.S ~~n the pnn=~e of h~.~.e ., ~ ~ 4t ~ be We =t of numbers ~~r ~~ch ~e ~~ As- tme. S-ince ~ ~ ~~ ~ If* suppo=~.~cM ~~.~ ~ ~-~ ~~+~+~2F:~ ~t :~s~t i~ S=3 +.* —(~k~ 1) HAZE i) ~ =~< ~ (ik- ~ I)~^ so the ~::~n :s Ago- t=e ~r ~ ~ I. Hence ~ ~ ~ ~~o Ace- - to M. S~ It Meows—= the Arc: of ~~te indigos that ^V =~ns all the :~tu~ numbers~ ~ ~e ~:~ Smug ~ t=e ~r all n. Me method of ~~.~f bY ma~ i.~n is uSed ~~ out ma~s, p~-~ng specl~ p~ ~ c~ ~~ns Ix~ ~ to 0~: tt00~= ~ ~~ 5000~ p8~: t=~t ~ ~ roof techn q e- co~er sc ence wL d ~ am th : c ~ cesses aw to :~:~l ~~:s of s~;~. Bile n.~:~. ~~= and tnt~ ^= the ~~.~e oven ~~= — — # ~ — ~ . — .; I: bY the p:~le ot- 1~:~' t~re ts one mtlc~ He between the t~ ~~s—Be - .~= of ~~.i.~.ive in ~r int~* ~: A— 1,~ ~ thC~ iS 8O t~t~: ~~ SU~ ~8t ~ ~ ~~ ~ ~1S ma~ the i: ::~:~- an adze ~~' im:~= that s~;~:n :~s d.~. ~~r ~: Awed p== of image' and shows ~~ ~~- ~~= of the Mom a: ~ ~ ~ iS has ~ ~=e sofut:~on in I. Oh ~e additive ~~s of ~` and ~ are extmmeW =~r . an~ 085~? t~ ~~ ~~ I: 8~ 6~\ i~i0*~ 05 ~~*~ ~5 an6 >~= tol~6 =~ct =~= Heisting (~* Since ~6 i~? c=~:n no :~:ipli=~we item (~= the m~! c=~s of :: and ~ ~ ), 6~?~: l3 ~ ~6 0~n :n —~ 3~d i, 3~4 =~ equations 0: ~0 ~~ ax ~ have no integer sol:~:~. ~~;f. t6~6 )5 ~0 8?i=~0 pattem su~~ng Chin multIpll-~on eq~ns (~: *id derisions) are solvable. The :~r 24 :s d:~e by 9, S. 4' 6, S. and i2- ~t ns ne ~~or 23 Act? ~0 ~t ~5 ~6 25 685 0~\ 0~0 p~Ct ~~Ot ~ se! of 24 ~~s =~ be pa~:itioned into equal sub=~s in six d:i~t ma t~t ~ 3~t 05 23 cannot be Idiom in ~v 5?~*~ Act, # W~0 8~6 6~:5i00 05 Iffy 8~ g0~6 5> tW~ Id prope~:~* The ~~: th=~m of arithmetic *I that any positive :~r =n be written ~ ~ pmdu*ct of pnme I it ¢~17 one waV~ The d.~Nrisio-n ~.~:~m guarantees that Or an~ I irk tepees ~ and ~ t:~e are unique intent= ~ and ~ =~h that ~ as b? I ~ p~0ipi63 3~ 05 0~;~Us ~ ~ :~:~! s:~:~e in :~e thec~+ of numbe~ and, :n :~e ~~:~! ~:~' In a.~a

=~A ~~ ~ N~:~y ~e h=~ * ~~n thecmm - ls one :~.f m~ ~~: * in : =~ cKp~s If::: e~c~v—= ply ~ written =-a m—mation of :~le i* Thew allocations Brand f~¢~e -add ~~;~3~g mast c m~ dimples ~ dam comma aims ted beg parallel ~~ ~~mm ~ mid {I Woo assay hat Nary p~om~al ~h complex -decedents can ~ wr3~:~n ~ a p - ~= ~ i~ Of: (film which the: senses ~ ~ may - coned). ~e d~ws:on Writhe ~$, ~of mu=~' ~~e ~ me tam~:~:= ~= for i~ *sky of balm ::* =d Amass ~ ~ : to ~e Careen I-~r mec.~.~m of pompoms a~. ll p~ me e~ *e anthm=e ~=n=~ for ~ mt~ d b' :a -# ~ Cod ~~) ~f am o~ i:~a ~ :~b ~r mme k. He ~~:~e =~s =d 601~ I: aim= - ha. ~s Meow .~ pm~ u=~..m -all ~ ~ =~5 ~—~6—~37 I art impo~t ap~:~s 3n cOmp~ ~~, In ~~ =~ in t~-n a~ 0~ of busses ~~ g~ ln~ ~~*~ 'N~ The ~~ nu~ mtem t~ Endues elements ~:~;mi~ ~ ~0 ~~* 0{ i- ~~t ~~ - ~-~ t) it 0[ =~' ~0 -=~er ~~em Q. ME ~ ~ ~ ~ ~ :~:0 j05.~0 - ~t ~5 ~~ 8~: ~0 If t~ ~ ~t 00~o - ~~t has ~ :~phcatwe inverses and ~~ ~~;~r emotion of ~ ~~ rx ~ ~ ~ h~ ~ unigu*e solution ~ rational ~ s' =*6 ~ (~: ~~= t)- ~~$ tt)5 ~0 p~t is ~~ 3t ~0 arm 05 ~44 Be Award oMenng of mt:~1 numbers makes Gem ~—$e =t— between ~ W:O mllonal =~ there :. th~ ~1~7 t00= ~ ~~ I: ~~= ~ 5~ 33 0~ =~t mst. On the other hand' ~r a~* mt:~1 ~~ ~ and b~ there is an -~:~r ~ such ~~t na ~ lo this p(~, my:= tt:0 =ti00~ ~5 into an mimed -~:~d fiery While the -~tio~*s a~ -:: of m~i~ num~= am si~fi~:~dv -I compl¢x than intend, the den~ and A: pr-~ies of ~ combine to Iav the groundwork ~r p~n l:n m=~ement4~ ~~ei~ ~~t ~ unit -of an\Y desl,70d Aid:: =:n ~ ~~ ~~ =~: ~ ~~ of ~~ Fix ~ ~ i: Rm Id: The n=~! *m- intends' and rational numbers p;~e ~~ mat sv510~s to m~) the st=~s -of ma~ pra~. 9,~e reasoning tasks. But unresolved que~s Ado as long as 2000 v~s

'' - - ~ :O nG#~E ~ ~ : ~ ~ mIon~ n~ - lmc :~e ~ ~ of i=~h ~ ~s ~ hole' ~~ thew :~s no a: n~r equal to ~. 8~ m~e tt q~e c~ that ~ m~ n~ =e :~m t~ ~~ ~~ in ~~r so ~e pmof -God ~~e is TO mIo~ w~r - ~~ :is ~ (or 3-' or 5, or ~—er mt~ ~t ts =t ~ pe—~ ~~> is an ~c in=:~s in ~e - ~~! n~= ~~= s). god. n.~.—ers =e =~d as m~= -a ~~.~c Wrest , 4— , ~ ~ ~e .-. ~~m ~~s ~~t there =e line ~~s ~b no rationed mesons. ¢~ are "~- -ash not ve~ b~ holes) in n~r h~ th~ has o~y ramps -~ooT<i~e ~~* ~O =~ :: C3~ ~ CXt0~66 IO ~ V~> O w~$ tO =~ eiements t~= 5~t mme -of - w toies an~ t63t ~1 S~c ~~w or geometn-c3~ds The-~onQ(43-~{a~b4 at b~c>7[Ot ~~ ~~e ls an o~ held, ~ ~ sul~e deEn~i=: of addition ~ ~h i. BACK an~ inve~5 ~: ~0 0~\ ~~ ordered §~d—o06 th~ ~~ ad the holes—is the =~ number w~em R. {t is an fired 6~d ~n which every It—~~ ~~ is ~~ {~= ~0 :has Ie=t -I ~~d in R. A key Am of number - ~~, -~e ~ I: ~ ~~e mic for: R' is ~~t ~~ such complete ordered field must be isom-~bm to R. S1:~e the mal I- s=m OnI) to 611. minim holes on the AN A: At ,f'A:~: · ~ ~ . A rational n~r iliac' =~ :~.= dI~s 'between ~e ~ number helds =e ~nui=Iv Su~nsin~ Firs' ~e IVY mbO~ n~er lS the solution of ~ s:~e ec-~n ax ~ ~ where ~ aM ~ =-e intents, there are transcen~t :~-~ ~~em (like ~ and ~) th~ =e not moue ti0~5 05 8~ 5~t p0~ ~~. ~~$ - ~0 ~0 =~l = =n ~ pla=d in on—o~ne =~ce faith the natur Imps and are thus munmbW :~fi:nim (a s~g Ash that was not com~ly unde=~ Anti:] -~v in this centu~37 this :s not tme ~r the real :~. In ~~- the t:~:~1 nu - ~m alone aw m- numemus than the al~aic :~:~hose that an~ as solutions to rational aig~b:~:c equations bile this IaSt result was prm~n at I= u~ ve~ ci - ~~r ramming w:~h transEn~e ~~ numbed them are bill Abbe outstanding gumptions About the chants ter of specific rea:! numbers The real numbed bovine ~ $i~i§~-~t step in the d~-~:t of quantitative co:~s and methods in another fundamental sen=- bile tt:0 ~:~l ~~, it 8~6 I numbers are each if I-- sets

5 94 ~~$ t~ ttt p:~C (~ - d tO- mm^x nu:~, m~ .x O: ~~£~= 1O ~C pi3~C ~~g 866~:~O of Complcx num~~ ~'ult1:~n Is more Compll=~—the =~X~.~t'~5 Of the v~= mult191y as ex~~ but the an~s gS ^~.= ~ N~- of ~~, thelr p:~r u~ Is ~ =~t oMer and co~ hnhe se of ~~ ~:~- - e real num~ ~~ the e~ m=~W desk— =d re~on ~~ut 1~te and l~i:~mal pawn. T~ alone soon ramps de—~~m of the- Imp of Stir cO~ they p~w'~'~e bn~. ~ =.aW:.s of motion =d c~. ~e 0~ns:~n few Marion! to~ =~ =~m~na~s mIut~ - of ala - - simple =d sl=fic=t a - ~~e cqu~ns.: Wt ~ Ie~S an away s1~ ntfi~ =~on of ~~ equat~s =~! u:~:~. S~10 pew n:~ equsti~s bLe ~ ~ ~ ~ ~ or ~ ~ ~ ~ I: ~ :~ no m~ ~~ - ~ bump ~~ m~d w~ ~ memoir s~ns ~ thew equanons, and to ~] ~~. ~~s in ~~, ls ~e mmpl=~ nu~ C. ~e i. nudged mnstW~ the ~~ po~e field c:~;~ of add. - : requl=d mot of x, ~ ~ ~ ~ Rem~y the eX~n ~ ~] ~b Is s:~e eq=tion ~ - s wiudons ~ ~ mber ~~mi~ equatmm =,6 opens ~ n~ st=~= of ma~l p~S ~ ~~ti-~. dew =~x n:~= ~ be - ~0 :~n ~e ~ .a ~ ~ ~ ~ - ~ ~ mplex numbers are Mimi bY oMewd pal~ of r~ nu - ~~: ~~ the :~ num~ ~ ~ oMeM 1:n on~e =~.~e wlm ~ polnts ot ~ ilne the c.,0~^ numbed co=~d to po~:~s of ~ two~dimen~-~! - he =d am not I~:~y o~-~. Thus Ims c: sample o~r :~:~: seem to promise ~ much :~e compli=~ !~ :~n ~ than :n the re~ numb~ or t~r 5~. 80~5 :t 6~ 81~ ~15 85—:~;. lte =~e between complex =~ am points :n the plane opens ~ :~! m*~on between ~e an:~c and a—m of ~ and the Emmett of sh~s and tr:~ ions :n the pl~ (~e ~~e 9~. The compl~ nu~ i: some num,5~,5 on~v d=~ as ~~ ~ ma~:~=s who c~d not ~~t the wSsibU:ty Of ~ n~ ~:~.~* I* ~ ~~w ps=~ u=~1 as m~- els of many TO—i ~~! ~~h~V510~ I * ~= t~~0 D~- Of altC=i3ti~8 ...... ~V.~'.'.'.'.'.~'.'.'.'.'.'.'.' '.'.'.'.'.'.~'.~'.'.V.~'.'.~'.'.~'.'.'.'.'.'.'.' ~ :.: :.:.:.:.:.:.:.~:.:.:.:.:.:.:.:~.:.:.:.:.: ~ : i } ~ :~ rig- ' ~ ~ /'' '"''"'< `'St'' '"'""" """'a""""""'"' .= ~ i' : ~~ . . ~~ ~ it,, ~~ i: : : : : :: : ::,~r ~ i' 2 .~ . . ~~ ~~ . .~ : : :.:.:.:.:.:.:.:.~:.:.:.:.:.:.:.:.:.:.:.:.:.:.:.:. ..:.:.:.:.:.:.:~.:.:.:.:.:.:.:.~:.:.:.:.:.:.:.: :.:.: :.: :.:.:.: ~ ~ i' ~ ' . ~ ~~ . ~~ : ~ : ~ : : ~— ' ~ Ai ~ PI

c:~m g? e~ :~= to. ~ n~ Of mr ~~r an al~e mng. ~~ - ~~o seme ~ ~~;~ Abram que=~n of :~= mathe~ ~~ ~~ mi~ of Agree ~ has exa~y ~ Bier ~~. ~:US -Ed wI~! equ=~n ~ at I~ one and at ~~t n. amid Co~x ~~- New Number ~ems 0~: S~ O t~ p=~CipiC$ Ot OU~—Stems ~= ~- fam:~ =~- Hen :: of ~ Me n:~ cent Aid ~~t t56 :~ ~~: ~ ~ t56 ~6 00=kP~m 0~0—~67 Bang =~ pmo~ ~ Owss a~d o - = ~~t ~e co=.~ nu~r ~ Igeb · Ib ~ Cd7 t~ed~the~;—r<>f~U~)e SY~ ~5 (~* ~6 ~ :~' i~ 50~ 50~$ 8~ =~0 ~~' t~ Eve- of ~~w her ~~s is ~ no me=s finished. ~r Sampler s)~ce Heir l~on ~n the m~ =~ the a - m of m~ teas ~~e an id t~l Or fig ~~m complex nume~ ~a. ~ matnx is ~ kind of su~3~' within certain ~~ms of maraca, ~e operations of ad~on and mulUpl~- non baw ~~c p=-~s ve~ so to ~~e of ~ re~ nu - ~m e ~~ pmmln=t =~:~n is the ~= that matn~x muh.ipl~= ts :~ve—~ ~ that has ma~Y i. mn.seguenc~ in ~e :~ of 1100~r ~~a Mam:~ are pame~^ use~! ~r d=~g =~X S~s of:~tit~ ~M su~ aS tho~ that comp ~ ¢e appli~on of mm:}~:ing ~ Anti - e :~ :~s aims ulated Eve of m~tic~ ~~s :n Bother d~:;~;ion of bOth A:: and theoretic interns. Despite t~= Emily = mama and ids speed, comput~ wo~ =t lath the ~mi:~- far n~er—tems wch as I, Q' or i?, bm :n 6~te ~mxi:mat~ns of those ~ ~~s - ~~e ~~S is limited b) the ~ili.~V of =~r languages to :~t numbers with ~~.~ ~ finite nu - ~r of positional pl~* These "~,' rabbis of n~er Alms do not obey the conditional nmctural paddies of =~hers Ash as asmciat~v of . Thus it seems imps that ~~ts exte~ them- ~~dy to :~e the ~~ propeni~ of those fiche - ~~s ~~t underlie ~ much of thelr actu~ qualitative ~~- me dlSC~r~- Of ~UI~Ot~likC =~iCal $: like matrices that ~~: to ob~ ~~ tommies th~ our nal~ intuition tells us are tme was x dramatic step :n the d~hopment of modem mathematics. Contempom~ algebra on~ed in an atw~ to prm~de ~ theory to explain the stmctural dies of vanous number systems. 1n the Ian 150 vea~ al~a has ~~d ~ nch a~ of abstract theories t~t spnng Am stu~ of ~~e inherent ~ ~ va,~s ope:373110~s and

IV~ 4~ :w N~:~Y s on sets~ A- hi shin ~ ~~ of me* ~~.s ~ p~ ~ n1.~g i.. ~~.~ ~ h^e ~~ sh~ ~~ th1s insect m~! ~m Deny has item I. ~~.~s ~0~ ~~p57 =~, :fi~, im3~s Act. ;~ mon.~., CALL T~ m~ ~~e -led Briny ~ ~~= ~~, thev a*= n~ used r=~ as t=Is r ~~h on Len :~:s -of A* am ~~= A .,~ ., .~ ~ ~ ~ ~ ~ , a. ~ I'm: , *am me mICOl6 O! t~ =~uW *is ~ ~v m =~ ~ lnterm in At, ~~ of ~~r v~- Ad, I973 harm Hi; He ~~~ "~ :~O mo~ -A ~at:- m=s ~ac mm~e to Oe11ew t~= ~ ~tb0~<s sho~ ~ ~^ Did - = t50~ ~5 0{ 50t 8~6 It ~~, ~ and Mac~e ~ =~r A-—:~h ~ Hi- -~e d~ r=~on of Flog ~5y W~= Z1~ much of w~ ~ - m ~~d ~e ~~ 0~5 0t It ~~' =6 $~g.*- }~ s~! m~ -Gus -of ~e 196~Os =~—the ~~.~;~es of o~ Arc amused simile= ~ ~~:M = ~~s ~~, :n I as ~h as ~ (resin -of b~n a aim MAY ~e ~~= =~omatic pm:nt ~ V:eW s~s much Iess Mali. ~ ~ ~~e to cl~r mathe.~ w~ of* edu=~. ~~e t50~$ t60~6 ~~£ 00~] pi ~8t ]~C ~ ~6 66= 0 ~~Ct 5V— ~s and amoral Thev pr~e ~~:t o=ni~ion -~* - = ~ be an impenetrable m=e of s=ciSc Ants and techn:~s, and this In: Zurich :~s as useful! ~r ~~;~s as for pm~-ici~ m*~them=:~# *I it Cams wise ~ =-~*~:~m p:~*~s to identify and build from such pnnci:ples as Why plan sch=t 0~#. bchoo! mathematics: must d:~p ln ~~s an u~d^ b8$~C p~:~10$, I;— id t00~:~i4~, 8~:6 ~~> id ~~. 6~t the ullima~ teM o:~-~! Am: is 0~-~ it enables stud~s to ~~V -~ei.r I.- ~ Solve i;. qua*~ti~:~e pmuems:. ¢e v to some pr~:~-ems is not o~ the mo~ impo~:t gem -of whool mathematics but ~$O ttC =~St IBM C6~-=tiO~ Hi* e A- "~M problems st:~:~s te=~r :n the b=~s of mathem=~s ~-~:~s of all ages~ The Joey h=t step in I: wo~ on* problems is to tden:ti~ in pmb::tem s:~;~ns con=~-~at are $~-~:iv aims ~~*~: to those of number As* T=dition~ approaches to oh's msk can be soned into two broad cI=~$* The p=~atic approach :~-~:ps Att ~ Or ~:f ~~! (bawd ~~1~V tOUt AWAY Me ~s to prm'-~e stude-~$ faith ~~c guideline ~r

each ~—~a sperm Beam or organza, ~~:~n ~~ -I emblems' a ~~ for I :kev art: twe ~~s imp ~~:~:c ~~ns at .a~h =~s to An. ~~s to =e ~~e bl (~t 56~) t~t fit t~ p - .~5 :~ ~~ 6~t ~5 aid; emus ~~r so ~ ~= ne316elK appma~ ts ~~nSt~ e~we ~~g ~~s ~ ~~t =d mns~e model p~*~$~ - ~~. pro~ :~s ~~e t~ n~e ~e (~) wp~= "~ WORM- ~ - ~07 8~ 0~ -~8 e -of oMin,8W I=~:~-~m ~ t~ imo arm a:~s ~ ~ de~e 0.~ ~ ~e mber ~~, - ~e Tea ~~ h~! Exeunt= Agent ~ JP ~ ~ ~ ~t h~ Am. w~ dI:~ tO d~p their fac~y tn ~e kind of m~w *~ ~ ~~=t th~ ls :~=d ~—I-0Y 0se :~= e~v in ~i6c mu=~-~- Recent ~~ to d~p ~ I:;: - 90~ on p~.~-~e~s she arm: but n~ ~ -~*~e r=~s >0 wade the search continues ~r e~e i0= 50~' tt0~ :5 ~~ :~# St~i66~t :~ ~~.~g ~ ~~ ~ ~~t ~e natum o't qu=~e 9' - ~~S th~. In m=v - : appl:~:s of m~hema~s one thi;nk$ ie~ abom ~~g specthe ~~d pmblems and cOnc=~s in~d on cO=~l,~ and ana^~nR ma: mo~s of the broom wit Me As stem am:: pm - ~ of sch=l mMbem~= us~V :~ nu men~ l:n~;~on and ~ s^e question th~ =n be =~e—by ~ n~.~l I ~ by~ mining an equating Outside s~, prow i0= Ins It t~ m~ or extm 3$ =~ :~fi~6 g~054 In ~ mathematical modeling approach, Me h~t st~ is tO i mI~ want va~- ~ n=t is ~ - ~*, in m:~e fo~ Ian~' relatIom ~~ represent caus¢~e~= connections -~ fame wn ~ ~~. Specific q-~tions =n then be posed ~n terms of input or om~t V3~ 0: ~~} pt0~05 0[ t60 =~]~g t0~84 ~V, computer t~ 8~t ~0 4~5 hit numencat ~ 67. s-~mbo! methods. _ · ~s. ... . ~ I, ~ The mo~ common s-~es of num—cad vanities are m=~umments~ Thus the theo~ and tech:~-~e of measure:~t play i- roles ::n

I: ~ ~~Y :~ Stems. ~ anth~c of numb m~, me me t~ t~e a: ~d Em feat of ~~t mathema:~s— ~ th ~~ mw~r -ads ~~ mter~ce~n mathemancs~ am hs appl~3~.s :s not ~ remarks ~~ amid :n ma- pmtm~1 mea~ ~ ~ s~! mathem~i=: ~~ ~ ~ ~-$: ~~, ~ - ~ 0[ ~~ ~^ it ~5 ~: :~ ~ that ~ me studen~ ironing ~ me'a~'=t imps bnef - pos~ ~ ~ ~ st=da~ ~ns ~r 1~ =d then pm=)= Amy= fb;r pen~, ama, am wIume based ~ thow A; Has .k~=~::'~ ~ ~~] ~f(~) ~ b~ ~ h - ~] ~[~ ~ PI,Wi~C )s f~ :x ~h ~ heed or: f~ ~ r: ~ h]' =d so On. w~ = be=~e ~~c p~e In tM ~ of the I=~n ~ h=~. d to this ~~ ~~h to ~~m ~~v Avid =d ~ ~d Concephons of Ie~h, ama' and Volume. n of area =d pen~r ~ ~ Is- =~n e~r on student a,~n,ls. The com,m,0n, "*~;,1.~" ~~ by unth1,~g ~ ~~s, ~~ss of any wording 1n the pr~em Stawment, is that lf there =e two- Ou:~m at~d to ~ - of ~ ~~.~:~d ~~e one mult;~:~s them' :f there a~ nu - ~~n on each s:ide of ~ I one ~0 0~S O~ tO~S ~~O i=~= $~S :~l p=~6 to- ~~ )~7 ~~ t50 I ~~ 0t t08} ~-~57 t~6 ~ mulat~ve e~ts of e=~n :n combinations of :~s, and ~e n to imps sham ~~t oc=r ~ so m=v P~M appl: cations or to- the ~~s =d Acts t;h~ are ~ndemen~ :~n ~~. ~~= [~7 5~ t68~= 0: ~0 avant ~ stmctura) 51 l~s ~~ un~di:e mo~ appll:~:~s of measurement A! the hea~ of am- ~~nt pmcess 15 ~ ~~:~g that 3SSi~S numb~ ~ objects. The mapping a~:s ~~e ~ ~ s~e despond unit. Other Jets are then cowmd t~v copies of the unit. 4e choice of unit elem=t is a~itm.~*~, but once the cholCe is made' it provides the ~:~ ~. brim ~.l .$ 8~0 =~6 ¢7~5 ~7 ~~t co:~s of ~ unit and ~ n:~er—the :~r of whole and partial cop:es of the unit ne.~d tO exa=~v -A the measured o~" The m~ m~cs student who unde:*~3 nds this pnnciple—as ~ ~~:M pmpe~y Of manY ~~t m~s—:~s =~uired pmdu~:~e t:~t imp e connectlon between ~: s:~S and Amp; m.:~50 The unit and cown~ prope~es of ~~nt explain qu:te cIea~y W58t ;5 50~ indicated Fir 8~ ~~18t mommy moreover, the atta~:t of units tO measummen:tS =n ~ explo~ed to Bide £ :~ reasoning about Scie~c pnnc191650 I~ mamas. sciences quantll-~w ~~:~.~g Is ~~d by ~ we:11~d al~m of qua:~:~s =~.~:~:v

c~-~Y :~! =~d "~ an~" in ~ ~~:d ~ ~~c :~ tmn ~ pe - ~~ =t hat 0~ 0~ 6~t 0~ ~0 ~8 85 ~~. }t t56 em :r~t :ls ~ number Gore Bite are ~~—~r t~ p—~, ~e ~~ Is ~~ wppo~ to ~e ~~ of me: Bite ~t h^e b~ -I ~~e th1s atte~n to ~~s aS as ~~ in m-~-~nt is ~ ~ com~n 1n m~ = t smen=:~. :~t :~ stm~. ~~ a~ thow ~~ : c mth h~ ~~ :~ake- ~e C=~= between ~~ m - emancs and ~S ~~:~= 18~! ~ ~ ^4 a. ~ ~ ~ ~ .. i. i. |= m=~ =c pm=~:= o! m=~*~:~e =~s :~n the ~~ ~~d haw:a ~ h10~+ Aft m - ~~= =d tts te=~- However, jabs as- ~~' chasms mat~ma6=l :~ have been ~~- eralized and ~~d :~n =~-~' ~~t h~ been aid to ;~t u~ thmu~ut tM soc~ me~+ While t~ Chic idea :s ~e met ~~ to ~~ ~ ~~s—these ; su~ o~ :~y ~~ ~~.~s ~~t able vew ~~:t - m measu=s of lentil- ~ and ~~me ~~litl=t S=~ti:~S 8:~6 S=~S 6~O 6081~6 ~ Y=iC~ O: ~*~ s~s of item o~r Owes all social s~tions* Eco~= ~~e d Or S~ ~~35~S Of =~tS =d benefits to quant;~ 09~5 :~ 6~8i0 maH~. Pw. cholo~sts and Ads use ~ wst a~ of :~$ deserve mIt~S a~d ~~s of lndlV~.~. st=~i=.~s m~ sum p~e caus~-ef~et wI~io:~s am~ Gary ~~t ~~ of s:~c va;~. ~ =~u =~~ nutted operat~.~~ =~ -I a~ u~d to- model ~~.i6~t st~1 properties of snuationS* ~~ tinges dass~ p~ and c-~ts aw d=~:V aMi—Te. Buff it is incr=~V =~n that emetic quandmt:~ ~~.~g in the socials a:~d human sciences requires undemanding of a~ of number t~t pe~.T nexio~ T=~.~.~n or new m~s m- new* situations* Wit qu*~io:n the most i:: ~~! of whoo! mat:~:~s is tO d~\rClOp ~(~115' mill tO t=~ int;0lU~tN with quantitative ilk formation. ~e mathematical con - -ale techn:~' and pnncip:~s that :~! 9,=~titative aspects of expene~e are proving bY st=~s o~f :~u:~er sV~' ~-m, and meaSu=:*~nt that hew 10:Og been the bean of school cumcula How~: the Age of elect*~*ni.c calcu baton and co:~= ~ powe~t mols ~r Anti am ma;nipu- i8ti~ gU8~tit3ti~ ~ iG~*~tiO~ ~~:S Ct~6 t:~6itiO~] It ~: Inst=~ti:~n in th.~e su~.~- Tt no- Ion~r =,&~s Sen~ to dewte 18= ~n:ons of the school ~mculum to training students in anthmetic or al~miC al~s that can be ,~d quickly and accuratelwN by

^~ ~ . Morose a~d I: C~" ~ Icy ~~ I: m to cOm08~= ~ also: Ied to ~ dmmat1c tn~e :~ ~e ~~ of: urns w ~ -—~~:w ~~ng ls bew :a^~. ¢us—oo] m~= mun p~: ~~s to ~e - ~~::krmMe~e of :~ - er' ~m, Bad art. 1.n If: :~d c~ days—~ -of -. ~~le ~c~:~^ ~ -I arm. ~r ~e ~~ ~ q~ ~~ in ~~e it ~ at;, ~~ p ~~; ~~t K Un~nd f~1 :~es of nut - en—ems a~ ~e ma~ ~ ~~e m - small -is =d re~ Is dons ~n - ~~ Hey are e~:~^ De~e add lnt=m Hi:* - w s~ us:~ ~c, ~1, =d ~~c represents Perk Ash ex~ ~ approx~^ cay -~th~ metm =d a—~c :~ b~ vanous ~?~0 =~5—= *ilk * 3,~) t~ ~~ or =m pates y **era and a1~c -elks to so~ nom mutate and On~.~ ~~ pr~:~. -I: ~ K. . * ~e who~ experience litany to ~-~ -~e general ~~11s and unde~ =~ :~*~t be nch in :.. to ex;.~.~e # ~ pO—~t gOS~ SitU~iOOS &S ~~ 85 I~ t~6 ~~$ t53t ~~i*~tC :m,8~.~ td=~-~ied in ~~ settIn~. It.~FERENC~ I'm =~-G # ^~ ^~.~0 ~' t~ AM =~ *at ~ ~ ~ t=~¢ A $~ >~/ # 4:. W8~.~., =~ ^~:~ ~~K ~K. It'd ,~r~= Of K ~ 9:~9 Be, ~ )~n ~ - ~~¢ hiMOw Of alg - ~+ Ii, ~3Kil0~6K.~ I'm \ ~ 7~~—`#6e art *- ~~: >~< ~~{ ~~on =,. ~~t ~ SKIN ~0 ~( ~ ~ 0: ~~t ¢= 0[ ~~* 3: :~, ~ ~ ~ ~ ~ ~ ~ * - ~ 60~ K.~ ~~# ~~*~t t:~ l~ 80~. /~.~ ~~: ~~ 8:0 ~ ~ 97')~ 760-~82 ~ *~)K~' ~5 ~ X [~6 - ~= 0/ t~6' ~~{ ~~ ~~ Am ~^ ~~[ I9:~; 87 K85 - .. {~ t~ ~~£~> ~~ ~*'0~. *~ YorkS ~ ~ K =K.~.~ ~ ~ 59. ~ oaV:~s Harold ~ ¢e ~h t~:~ cf Co mputa Ace: <, ~ 5~ 0 :f': A: K . e: hi. ~~ .~ . 3 ~ H3 } ~ e - WAX A~b ur F . (~ S~ ~ } X 0~j/*~#~ ~= i07 [~$ .~= -~$~#~.^ If 7~< -it :~7A~. wash * *can ~ :~c A Id* C~Kn=) 0: ~~ ~ ~ ~3 t~ IDA ~ 969 ~ ~ ~ — ~ ~ ~ A ~~, p5K.~; p ~ {~ t0= of ~~ I W3$A~ ~ ~~K ~ 0C A ~ 3~* 311~) ASSO( t |~ O: <AK=C ~ {~+ ~ ~ ~ ~ A

:9s 8. ~a, Wok and #1~, Am. -~-i~ mew fits ~ pus m^~ Em ~~- In X, Abut OF ~ ~1~ ~ P. (Eds.~: ^ Amp, ^7~ 7~ #~ ~~e ~ ~on, Ha: Amid Cough act, 19S8. 9. fly, ~~= ~ ~~ Q~ ~~ ^e O~ ~ ~~ ^^ Am. ^- YA: Riot ^~ of ~~ of ~^ 1 9S4. 10: ~^ ~rid. ~ ~~ ~ ~~ ~6 But, BY: P~mc~b~ ~ 198i 1 1. ~~~ at. ^~ ~ /~ ~ [~- ~ Beg ~> a: D. ~ ~~ 1977. 12 ~ ~~ ~ ~~e bitt ~nu~ ad numerals ~ DIN Ha, augur E: (Ed-): ~ ~~, ~ ad-- ~: ^~ ~~4 aims, ~ ~~i-~ aunt ~ ~~ of ~i=, 1969, 1~36. 13. Hemb~ ~ ad ~^ gad ~ of ~~~ ~~ in ~U~ m^e^ valid ^ ~~yd$,S ~~ ~_^ ~ ~~_- ^,17~19S6~8~9. 14. dial, Jag (Ed ). ^~ ~ ~ ~~_~ ~ Hillock. a: ~~e tam ace, 19S6 15. Hi^^ Jump ad IBM, an a.. ^~# ^~ ~- ~ ' ~ ma. Rag, V^: ~ili^il guano of ~~ of ~_11~, 1988. 16. Hogan, Hilde 4~hi~ nags Ant e ~2 ~~^ SO (~1989). ~11. 17. ~ii, avant a. ~~ ^~ ^~: ~~ ~~w ad, a: ~~= ~~ 19B5 18. gut, lamp. daylily slang ~ gabs ~~ pace A p~limina~ an~y- sis.- 1~ ~~an, GO a~ an. Ruben ala: ~ ~~e ~~ / ~~ ~~ ~~> 1986' 1 14~120. 19 ~^ Jim=. ~~C Elliot INS ad m~1i=.~ in Liar, C1~ a.: - f~ ~ {~ ~~ ~~ ~~_~ Pillar, a: ~ m ~ 1987, 19-26. 20. Spa Jab ad Sim~^i~l, judilb AEON in Salon 1O Marc mucous: Rag ad impli~ons.- ~ ~~ ~~~ ~ ~ ^~^ 5 {19S3), 63- Hi. alter, aria tumor and: Me Me of m~u~cnl Elision.- /~- ~^ 36 (1989~6. 22. gaily, Jan W. ~ ~ ^~ y~ zig ^~ ~~ ~1~. Saigon, a: Batik trillion of Amedca, 1989. 23. auto. aged E. -~dlbmic Skiing ad Remain 1binki~. _ 92 (1~), 16181. 24. ash, Right a.. ^~- ~~ ~~r ~~~ ~ ~_- act. ~ a: ~ 1973. 25 ~^ E11. ~ ~~ ~~ ~ ~~ ~~ Bag. MA: Bight 1987. 26. Knauer, Gabon B. to meddles O1 antic Mama ~ ~ ~~= ~ 77 ( 19S4). 43~35. 27. Rebill, Jobn. ~Wba1 is a roar numb ]~ ~~) 79 (1972). 74~754. 28. ~~> Abo R. ~c matbemali~ a~ ~~b~ mullions of me. ln ah, Ricb~ a.: ^~ ~~ ~~r ~~~ ~ ~_- act. ~ ~ _ 1# 1~ 21 ^ . . . . .

:~4 ^~w A:~= TO: ~~' 29. ~~ Hen~ O~ ). ~e m~ w~s c~m :~12* ~~ ~s ~ I; and - ~~ :s not.- W~ng~n, OC. ~~e ~~d oi t~ ~~ - :~ ~~ ;~- :: 30: ~4 ~= - ~ ~~~ l~ ~~ ~~= ~~# ~ ~ ~~ ~~ and an A.~t'ed ~ ~~' m~ Ma~ - A~ w=~ ~ ~~? ~ ~ SS A. ~ ~~ ;~. "~e aspms of ~~- Image, MA~ ~sach~ts 32- ¢~$ ~ ~~ (~ ~~- - ~~ t~ ~~ ~ ~ ~' Em. MaCm~n =d it-? ~ 89 ~ ~ ~ 894~ As- I- ~ ~ ~~ Em- A~ An~ A: Hancock of ~~ ~ /~= spy t~ - ~~t ~ ~~.~#,. ~ 0t ~~: i~ 983 ~ '4 - - ~~ ~~ - ~ m:~= of p~ :~' ~ I: - Ma~ ~ - s ~r mea~nt ~ Amm^~ ~~ mnth~ 75 ~ ~ 968~' ~ it S~ ~ >s. - '#, ~~- "~ =~ ~ p - ~ 4'~' p~ {~- p~ 5~'(_ ~ ~ K ~ ~~ · ~ ~ ~ =~= ~~: =/ ~~ ~ ~ .( ~ 96 % ~ 22 ~ ~ 25:6

Next: Uncertainty »
On the Shoulders of Giants: New Approaches to Numeracy Get This Book
×
Buy Paperback | $34.95
MyNAP members save 10% online.
Login or Register to save!
Download Free PDF

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.

  1. ×

    Welcome to OpenBook!

    You're looking at OpenBook, NAP.edu's online reading room since 1999. Based on feedback from you, our users, we've made some improvements that make it easier than ever to read thousands of publications on our website.

    Do you want to take a quick tour of the OpenBook's features?

    No Thanks Take a Tour »
  2. ×

    Show this book's table of contents, where you can jump to any chapter by name.

    « Back Next »
  3. ×

    ...or use these buttons to go back to the previous chapter or skip to the next one.

    « Back Next »
  4. ×

    Jump up to the previous page or down to the next one. Also, you can type in a page number and press Enter to go directly to that page in the book.

    « Back Next »
  5. ×

    To search the entire text of this book, type in your search term here and press Enter.

    « Back Next »
  6. ×

    Share a link to this book page on your preferred social network or via email.

    « Back Next »
  7. ×

    View our suggested citation for this chapter.

    « Back Next »
  8. ×

    Ready to take your reading offline? Click here to buy this book in print or download it as a free PDF, if available.

    « Back Next »
Stay Connected!