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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~

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~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

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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 peBins 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~dsu~ some eXcltIng - possib:~.lities. ~~8m6~Y

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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~ ~..~sm =~.~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 et~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.

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wc~ ~~ ~~ :~ s~ =~a ts ~e ex~ n :~ qua~e ~~s ~~o n~ ew~ =~ A: p=~ and pms:~ 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.

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:: ~ ~~Y 4~r ~~ ~~ ~ ~ ~~= ~r te~ ~~t ~ qu~w tn ~ Arcs : ~e e~ce ~~= waste bet of ~ on human; eognu~+ $Wilem ~s ~ Io~ her on ~~- en~an=~=ch~: 1~m ~ Sal Is the pay shim years he- ~n an unp~-~:~kt~:~ Swabs yang Mopes ~ - un~ ~~f ~~r In- and ~~= ~ ~;~. *as a Cheek Uses ~ a - airing nch m~ mo the item-: ~ hum~ comity ~~men:t farei: ~~* pnnc~, a~ ~s m~ ~ -I ~ pee ~ learns ~~ Bask saws :~ pa ~r admix ~~= ~~t Use of Cum=~a and ::~ctio=! appmach~ in skew Ate; ANAL cow~ Me Ace of rapidly Malay demand ~r brie =d coil enn6c apphc~*on of qu=tiwli`~ sew wit powers new technics = that s~n tho~ sets has prom~ed IBM of goals for schoo] m-~- ~ Rap: ~e dde of ~ 1982 :~ of ~~e Confb:~e Bo=d of the Al ~9 ~~ ~ add! =~ "What Anti abilities ~ be fUndamen~1 in the ~~um of m~W ~~- De~e extenstv-e :~ d~ wer the pa~ de= th=e is as yet no consensus on ~ p-~t mur$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 '

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:~ - - ~~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 :ipectin wall un~ ~ mo~ W(~ 1f ~ ls :~d ~ un~ of shed ~~ of Or ~~ ~ :~n ~~e am the ~ ~: Whim Id~ =d ~~s :have ewd ~ 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 sizeto 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\+

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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 thater. 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

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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 $hiD:i~, S~:~n my Await comp-~' Aid, or recovering ~~ ~7 ~ Pi miss s=e cha~' AL ~~, or u~ of ~ rate fin :~ =~5 =~' =~' t=0 ~ i=.' ~~ ~~ ~0 ~ 7 or :~venng ~ ~~r. W~e mmemat1~s and t~ ~ qu~n ~e me~g of these He and -~e shed -~mpletene~ or ~07 it 5~5 cen ~= totemic to sib an~es Wit belp= 1nstru=~= -on the ,n,~,m,,~ ~ of wpa~g ~~S to u~ nu~ e~y to e p - ~~+ Em: Of th~e dI~t ~~: ~~ ~~= ~~ :~0 h~t t~ =~ =~ts ~ ~ q=~w - ~ }~ 5~ ~7 ~0 t6880~i~ ~5 ~37 ~ If; r Axed= an6 ~ cowe~= ~~en phenomena ~nd Aim that ,~ - x~5 esw=~ ~~. Each God' - ~ Bier 'ax ~~: ~ ~t ~ A h== w~ have ~x~- ~~m ~ mi=~s am0~ o~ - s co~ng ~ Fathom ~ t~ Abe;*~~* ~ un de=~d ~~s :~g press ~~s need Se experience m X Bile -is must ce~ acqulm c~^e skis 1n d~ng h many ~i6c u~ of n~ - s' Rev To need to ~~e ~ ~~r Ape: on plies that number u~ h~ in mmm-ofn. Dew is c1~ ~~m re~ in m~hemat:= e~on that u ~~g ~,~: s~.~! pmpe~:~s of.a mathematical -I ~.Cilitat-= r~n ~ the ~~m =d a-~li=~n to n~ sl.~^ -- .4-- .4.- ~'001 ma~:= ~^ tnererO~fm' emphasIze the It's t~ died ent Apes off :;~r -en - is se - e as mode~ ~ :~ o - ~ng$ 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

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-~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 ~~ ~ -

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~ 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.

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=~A ~~ ~ N~:~y ~e h=~ * ~~n thecmm - ls one :~.f m~ ~~: * in : =~ cKp~s If::: e~c~v= ply ~ written =-a mmation 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 complx 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

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'' - - ~ :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 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 ~do06 th~ ~~ ad the holesis 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 ono~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

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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 am 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 TOi ~~! ~~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

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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~;rf~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~ertems 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

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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~ 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

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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 =~eby ~ 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 numcad vanities are m=~umments~ Thus the theo~ and tech:~-~e of measure:~t play i- roles ::n

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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:~erthe :~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 pnncipleas ~ ~~: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

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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 ~~ ~ ~~sthese ; 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 aMiTe. 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

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^~ ~ . 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^~. usoo] 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 - enems 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: OCR for page 61
: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 ^ . . . . .

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:~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