On the Shoulders of Giants New Approaches to Numeracy (1990) / Chapter Skim
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Pages 61-94
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From page 61... ...
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-~)
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From page 62... ...
m~l arid dais sv=P" 50$~- :~0 p~0 0: ~: t5.~$ =~ti0~ is t~ =~: anthmetic probiems or some aigebmic equations. The core of elemenmr:y acid middIc schoo!
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From page 63... ...
{~ ~0 p88~7 hit ~5 th=e rout:~e Pi mills :~s - n ~ p~quistte ~r emotive u~se of ma~atics. Howev=- me emergence 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:.
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From page 64... ...
schoo! ~~:s ~ n~ deal Or ',<~ ~ N~Y c 'it ~2—~ i vew sm~ n~= ill ~~ -~d ~~—mlb 9~-¢ =~ Of the~ =~*
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From page 65... ...
wc~ ~~ ~~ :~ s~ =~a ts ~e ex~ n :~ qua~e ~~s ~~o n~ ew~ =~ A: p=~ and pm—s:~ t~. ^~ As hi abbeys been us M' - l,{ =~s h~ Amp.
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From page 66... ...
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~*
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From page 67... ...
:~ - - ~~e to: ~~ ~e n~t I ~~ - w ~~' ~ it seems m~ to ~ t~ Al u~ fires ~~ of . ~ ~S :impo=m to m~ to ~~' as.
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From page 68... ...
? # I: (~ time line, test scOms)
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From page 69... ...
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)
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From page 70... ...
. 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$~*
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From page 71... ...
i 1 1 1 ~1 1 1 ! i 1 ~nGERE ~ T~ ~~ior of ~n~mcn~1 ly~s of ~l~ions among gad =n ~ seen mod Dailey ~~ lyrics INS.
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From page 72... ...
Converts van~n—~ one variable incrust mthout limo, another ~r3~= so~ :~= - be-. -art rim:—as =e vanity increases -~i~:xndy, the other incomes and ~~es tn some =~l~—le Sapped vary—~ -~e wn~e inch Other charges :n .
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From page 73... ...
mathematics is ~ sweet that deals mth =~ con~ts th~ are Ad f:* ~:m pattems art o~.
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From page 74... ...
For =~1e, =ndit:~ns on ~rs~dbles w~h ;~5 ~x ~ 9: ~ 3 870 9~;~10 common in a1~a and its appli~. The so1~:~s can be Teen in ~ similar symbolic ~7 but :t has become almo~ as common to display the ~~1ts on ~ number 1~:ne Ah (figure 51.
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From page 75... ...
h:~i=~. ~~s it is : tant fur mathematics ~~s to b~-e ade~m ~ ii Chic mpre$~ns 1~:~dv and to u~d the =~s among sum - ~^ Chin and numenc~ ~~s of the =~e ideas.
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From page 76... ...
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.
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From page 77... ...
~.S ~~ AyB] _ DS C Co.~d =~31 =~.~s of ~~c exp slons a~ ~'i.~g to be ~ w~ u~!
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From page 78... ...
c~ an-d ~~ ~~ :~ and =~s hew cIea~y ~n Over routine aSpe=~s of both Ire a~ manipulation of guantit~i~ ~i:~tio-~—the two key mmponen~ of pmcedu:~ kales The =~k of t=~:~:~ng t50$C :~W £~S iHtO D0W ~15 ~t =~8 pOSOS ~ cntlcal psvehm 10~al Question conceming the interim—t~ mncept:ual and g7m ce~ age ~~ ma:the~s educators worm tt~ cxtens~ use of =~:~r and computer toots with cO=~ding de~:~.s 0[ t~iOl~g in 5~;i5' Witt U~609~i~6 ~0~t O: 00~8t UO6Ct-X dim pmheien~ in solid ~~-r~lems, and Ability to iea:m new ads vanced mathematics. lPhe I: of undLe~nding and skill in mathematics has been stud:icd and de1~d ~~v ~r :many vea~ but both re:~d enthusIasm :~n the pa~ dCC30Cx ~ :re~:t meta-~sis of over 70 reseawL
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From page 79... ...
Den if mangoes take over ~e bum of co~Mtion, it remains ~~:~-~t for use~ of those machines to plan =:~t operati=s and to m~:~et r~ts inwIll~- Planning c~ions—uires mund undersmn~g of the meanings of oper~:~s—of ~e Rustics of a=:~ns that co=~nd to v~uS anthmetIc ~mlions~ In~ te~ion of wsults requires judgment about the likelib~ that the machine o-~t is cornet or thM an e=~r m~v Me been ma~ fin d~a ent~' ~~= of operations' of* machine pe~- (Development of nu - = wnse is discussed :n d~i!
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From page 80... ...
~ ~~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:~!
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From page 81... ...
-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!
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From page 82... ...
~~:~= ~m N~:~y sea:.
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From page 83... ...
and i2- ~t ns ne ~~or 23 Act?
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From page 84... ...
in : =~ cKp~s If::: e~c~v—= ply ~ written =-a m—mation of :~le i* Thew allocations Brand f~¢~e -add ~~;~3~g mast c m~ dimples ~ dam comma aims ted beg parallel ~~ ~~mm ~ mid {I Woo assay hat Nary p~om~al ~h complex -decedents can ~ wr3~:~n ~ a p - ~= ~ i~ Of: (film which the: senses ~ ~ may - coned)
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From page 85... ...
In ~~- the t:~:~1 nu - ~m alone aw mnumemus 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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From page 86... ...
u:~:~. S~10 pew n:~ equsti~s bLe ~ ~ ~ ~ ~ or ~ ~ ~ ~ I: ~ :~ no m~ ~~ - ~ bump ~~ m~d w~ ~ memoir s~ns ~ thew equanons, and to ~]
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From page 87... ...
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 $: like matrices that ~~: to ob~ ~~ tommies th~ our nal~ intuition tells us are tme was x dramatic step :n the d~hopment of modem mathematics.
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From page 88... ...
t~ =~uW * is ~ ~v m =~ ~ lnterm in At, ~~ of ~~r v~- Ad, I973 harm Hi; He ~~~ "~ :~O mo~ -A ~at:m=s ~ac mm~e to Oe11ew t~= ~ ~tb0~ |
From page 89... ...
In m=v - : appl:~:s of m~hema~s one thi;nk$ ie~ abom ~~g specthe ~~d pmblems and cOnc=~s in~d on cO=~l,~ and ana^~nR ma: mo~s of the broom wit Me As stem am:: pm - ~ of sch=l mMbem~= us~V :~ nu men~ l:n~;~on and ~ s^e question th~ =n be =~e—by ~ n~.~l I ~ by~ mining an equating Outside s~, prow i0= Ins It t~ m~ or extm 3$ =~ :~fi~6 g~054 In ~ mathematical modeling approach, Me h~t st~ is tO i mI~ want va~- ~ n=t is ~ - ~* , in m:~e fo~ Ian~' relatIom ~~ represent caus¢~e~= connections -~ fame wn ~ ~~.
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From page 90... ...
brim ~.l .$ 8~0 =~6 ¢7~5 ~7 ~~t co:~s of ~ unit and ~ n:~er—the :~r of whole and partial cop:es of the unit ne.~d tO exa=~v -A the measured o~" The m~ m~cs student who unde:* ~3 nds this pnnciple—as ~ ~~:M pmpe~y Of manY ~~t m~s—:~s =~uired pmdu~:~e t:~t imp e connectlon between ~: s:~S and Amp; m.:~50 The unit and cown~ prope~es of ~~nt explain qu:te cIea~y W58t ;5 50~ indicated Fir 8~ ~~18t mommy moreover, the atta~:t of units tO measummen:tS =n ~ explo~ed to Bide £ :~ reasoning about Scie~c pnnc191650 I~ mamas.
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From page 91... ...
~~ tinges dass~ p~ and c-~ts aw d=~:V aMi—Te. Buff it is incr=~V =~n that emetic quandmt:~ ~~.~g in the socials a:~d human sciences requires undemanding of a~ of number t~t pe~.T nexio~ T=~.~.~n or new m~s m- new*
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From page 92... ...
- w s~ us:~ ~c, ~1, =d ~~c represents Perk Ash ex~ ~ approx~^ cay -~th~ metm =d a—~c :~ b~ vanous ~?
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From page 93... ...
20. Spa Jab ad Sim~^i~l, judilb AEON in Salon 1O Marc mucous: Rag ad impli~ons.- ~ ~~ ~~~ ~ ~ ^~^ 5 {19S3)
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