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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$~0 pecple ham -6~
~2 ~~ :~h sublime and sinme~m numenc~ - ~= ne~ of p~ :n ~~s h~ p:~=d the ma~ema~ c~ :n :~.s o~f pro~ ~ ~~r m^~ ~ ~l ~s ~ ~~ate~yj$ 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 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
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.
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.
:: ~ ~~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 '
:~ - - ~~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 $~:~# ~0 0~5 0t 8~iti0~#7 5~-~, Ii 8~6 d:~ isle co-~-~d directly to operations such as io:~;~£, com~ panng~ or pa~:~tion:~g of o~s that numbe~n measure. Other lmp-~t con.~s such as wiocitV, accelemtion~ a~d de:~sit\+
68 }~ ?~ In ~ ~~:?~ ~ ~~ at?= ~~ - em ~ :~ 5~, 5= t~ 3~ ~$~ -~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
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
-~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 familiaruation~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 pI~ too 1-~ ~:~r =~s t~ m=~h problem con~-~. ~ ~ ts ~Iv part of the A- phase of solving, -isthe 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 Heon 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 $~-t - ~.~v tO thet ~~ the use of I~:~=l wn~es with a~c syn~ seen as x: ~ 4~x ~ 2~~: is =~.~:~g b~ -A Graphing ~p~n Bile t=~1 pl=e -ad nu~is =d Cleric expressions am the most lmpon=t sy - ~:~c ~~s ~r recording q=~e if tiO~, ~~> Ot~Ct :~tiO~ tO~$ 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 Iere ~ ~ 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~ ~~ntm ~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*~2A3~ ~ 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 ~~;$?r~1 :~ ~~ oafs for 6~ding the =~t =mmo-n -divisor of :~o intend, the Si - ~e ~ ~ , ~ ~ ~ ~ .. of F;~:~? :s onto one of mam-+ al~hms ~r :~mi~ pyre Am e~ ~ t:b ~ ~ ~ ~ ~ Eli ~ f o~ ula i ~ ~ ~ ~ ~ ~ mam alacrity ~ a? ~ ~ I?. {~g Brim 09~8ti0~?~ 3~6 t6070 876 60~8 0 8~5 ~t 50~g svstems of linear ~~ti00.~? 8~6 Haiti -
~8 I.. ~ ~~ D~ Id app~n of ~~s =e Ray ~ t~ h~= ~~f Tin ~~Ct t~ =n ~ AID to ~w ~~ :~ ~ =~e Is. l~ ~5 0t ~39~: ~~= ~~t ~ ~e p~ of mathem~:~s ~~m lbe ~~ :~ 118 used in bu~:~d ~~= ~:~ s~:~::~:~= med:~n p~= and :~ ~ :~e ~~e ~~ ~ :~ na~ of ~e~ ale makes t~ =~y p~ed ~r car exe~.~. his ~ ~s ~~r tmpli~-s ~r wh~ cumcula" am saga agony ~ is ci: ~~ Stamens importance AL bm~ a~ic~ili~ to ment ;~ elusion ::n elemental Air secondary ~~! mi! ~ain3:y h~ been I =d ~~e al :n -I = ];~r aDd mm:~r share Inexp=~ =~ ~ perch : 9~' ~~, 8~( 0~th=5 t5= ~ t=~ l~ ~~- Th~ =~ t=~ Bier ~~ ~~ ~~ ~= s ~~ ~~ pm§~= I.~. eXecutIng ~ Chin 0~m Bale they ~ led ~at s~! I~: 1n ~ ~ ~ t~ same time ~ Ieam~ of :~ a; has ~rr~i:~d in impolite ~r Chow m=~i~, :t has become ~ mow ~~> ta:nt ~r eve~ne Gil qu=tit~:ve ark to hew :~! un~ of tbe a: po:~t ~ view 9-~ To- be an I u~ of Comgute - ~ed a~s, it is- usei~) to Idea su~ at~uas v ec=~ and - In as ~~! ~ ~~damen~l m ca:! conce~s like I a:nd wcu:rsion th~ =e too I:~e app:3~ in traditional c-~. In sho~ t:h£, ~~mic ~~t of my ta~ o~n ~ v¢~ distant ap~:~:~ceen ;; and com:~;rs take Over routine sy~c pmceduws. This new con~i.~n requlms Ida. reconsidemi~= of ~~s ~~r qu.= m.~.:~tics. c~ an-d ~~ ~~ :~ and =~s hew cIea~y ~n Over routine aSpe=~s of both Ire a~ manipulation of guantit~i~ ~i:~tio-~the two key mmponen~ of pmcedu:~ kales The =~k of t=~:~:~ng t50$C :~W £~S iHtO D0W ~15 ~t =~8 pOSOS ~ cntlcal psvehm 10~al Question conceming the interimt~ 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 $~e time* It is ce~inI>Y ~e =~! tssue raised ~ the *~:~t of t~' in school ma~ics. while ~:~*~e I:s considered ~~e Concem1~ ~ kits of ~~:~g ~~:ti:~n in schO~ m~s ~ tra~:~ ~s to mncep~ and p010m so~ri:~, there is no di=~eM a~ut the id of de - Ming student. ach1~t ln ~ wn~y of in:. mal a~s of quantitative r=~ni.~' to ~~- - at moist numb~ I. Den if mangoes take over ~e bum of co~Mtion, it remains ~~:~-~t for use~ of those machines to plan =:~t opera- ti=s and to m~:~et r~ts inwIll~- Planning c~ionsuires mund undersmn~g of the meanings of oper~:~sof ~e Rustics of a=:~ns that co=~nd to v~uS anthmetIc ~mlions~ In~ te~ion of wsults requires judgment about the likelib~ that the machine o-~t is cornet or thM an e=~r m~v Me been ma~ fin d~a ent~' ~~= of operations' of* machine pe~- (Development of nu - = wnse is discussed :n d~i! in the Feb=~ 1989 issue of ~e /~0 ~~,$ 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 coumm~ 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 Ito ~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~ reamds ~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~ ~~sBe - .~= 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 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 ~$e =t between ~ W:O mllonal =~ there :. th~ ~1~7 t00= ~ ~~ I: ~~= ~ 5~ 33 0~ =~t mst. On the other hand' ~r a~* mt:~1 ~~ ~ and b~ there is an -~:~r ~ such ~~t na ~ lo this p(~, my:= tt:0 =ti00~ ~5 into an mimed -~:~d fiery While the -~tio~*s a~ -:: of m~i~ num~= am si~fi~:~dv -I compl¢x than intend, the den~ and A: pr-~ies of ~ combine to Iav the groundwork ~r p~n l:n m=~ement4~ ~~ei~ ~~t ~ unit -of an\Y desl,70d Aid:: =:n ~ ~~ ~~ =~: ~ ~~ of ~~ Fix ~ ~ i: Rm Id: The n=~! *m- intends' and rational numbers p;~e ~~ mat sv510~s to m~) the st=~s -of ma~ pra~. 9,~e reasoning tasks. But unresolved que~s Ado as long as 2000 v~s
'' - - ~ :O nG#~E ~ ~ : ~ ~ mIon~ n~ - lmc :~e ~ ~ of i=~h ~ ~s ~ hole' ~~ thew :~s no a: n~r equal to ~. 8~ m~e tt q~e c~ that ~ m~ n~ =e :~m t~ ~~ ~~ in ~~r so ~e pmof -God ~~e is TO mIo~ w~r - ~~ :is ~ (or 3-' or 5, or ~er mt~ ~t ts =t ~ pe~ ~~> is an ~c in=:~s in ~e - ~~! n~= ~~= s). god. n.~.ers =e =~d as m~= -a ~~.~c Wrest , 4 , ~ ~ ~e .-. ~~m ~~s ~~t there =e line ~~s ~b no rationed mesons. ¢~ are "~- -ash not ve~ b~ holes) in n~r h~ th~ has o~y ramps -~ooT<i~e ~~* ~O =~ :: C3~ ~ CXt0~66 IO ~ V~> O w~$ 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 §~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 ~ $i~i§~-~t step in the d~-~:t of quantitative co:~s and methods in another fundamental sen=- bile tt:0 ~:~l ~~, it 8~6 I numbers are each if I-- sets
5 94 ~~$ 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
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~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
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 =~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
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
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
^~ ~ . 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: <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