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OCR for page 61
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$~0 pecple ham
-6~
OCR for page 61
~2
~~ :~h sublime and sinme~m numenc~ - ~=
ne~ of p~ :n ~~s h~ p:~=d the ma~ema~ c~ :n
:~.s o~f pro~ ~ ~~r m^~ ~ ~l ~s ~
~~ate~yj$ 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
OCR for page 61
to~ em: now dl~' ~~S
aU ~~:~s or~w ~~d 1n
wh~l :~+ ~~e ~
of ~~:e man1~; tO A
:~:; :~ a~ ~~e =uat~ns.
:3 cove~ similar operations ~ polynomial, mtIon~, and ex
Hi =~.~.
{~ ~0 p88~7 hit ~5 th=e rout:~e Pi mills :~s - n
~ p~quistte ~r emotive u~se of ma~atics. Howev=- me emer-
gence of i: elect-~-~c =~s and com~ has ch=~d
the condition Fox Tt is now ~ut ~ ~ ~v=rs ~~ce the technol~'
of trans:~$' pnn~ =~uits~ =d bill chips h=t ma~ hand-~d
c~m available On the mass cons~r m=~- Rap~ ~ :n
electronics has ~~ produced Red w~he ~~om that
a:~:thm~c on ~~ that can be entered a~ di$~-lay~ in
decimal' common f:~:icn' or ex~:l Ernst ManY =~=s also
haN ~ ~~e button submutln'~ ~r Aide A:. ~~iO'ns and
pe—Bins cOmmon statistics ~~ions* Pm=~mable Cal=~=S
o~r :~e powerful civilities, incIu~g Age Ail m~ipu-
Iation' and matrix o~ns (~ee F:~re i)- Each of these ma
i=] procedu=s is ~~-~;~e :~n mom powerful and sophisticated ~~
Ad. p~am.s tha,l mn on desktop- A: ~~w why wail~-~e
in Schools"
envlS100~d—su~ some eXcltIng
- possib:~.lities. ~~8m6~Y
OCR for page 61
schoo! ~~:s ~ n~ deal
Or ',<~ ~ N~Y
c 'it ~2—~ i
vew sm~ n~= ill ~~ -~d ~~—mlb
9~-¢ =~ Of the~ =~*~e co
~5 0n those ~~.
M}~e If: ~~s c~ :~:~::~h
: ::: :: :~:: :: ~ ~
que~s ~~t vanity, ~~, =d ~~s east - :m arc
i~ :~ be~= ~~y -~ ~ Amp. ~ ,~::~.~ eX
~ =~- J~ ~e w0~6 0~0 0t ~7 ~~ ~0 ~5 0~ -art
Hi and co~ for fa~ aM '' '...,.., ,..,,,.. Ale, ~
=~a ~ y~ to ch=~ S:~fi~iln~tO ~~= In mn~
dnIc:~.
~ ~ ~ . ~ ~ .~ :~ Ail.
C~ and =~ ~ at.: h~ a, pm~d ~t -en -~e
,~w of ma- n~ m~ ~~.~ ~~= t=~.' mathemat1~
Ci=$ C=* ~h t~ p~S t~ ~~t ~~ ~~ mentors ~~ m~
~..~s—m =~.~s with ~~ ~.3~: wria~.~. The
expenmen~ m~an c=~ few berm cause An: ~ ~~ :n ~
- $~ - I.. of t~ t:.~e ~~d ~ "~- ~ * ,5
= ~5 =~6 not ~ 60~0 ~ 8~ ~ ~ ~ - ~
~~, ~~ -~¢ =~s ~~ eme~ would ~r have been w=- ~e
ex~r:~! -~a ~ ma~ema6= In ~ w~~ =~z—~ Id ~~
~ ~ ~~ to rev=! b~ Is =~:vana:~. Me ~t—
mat ~~ -~:r venEc^~n rema::~s -I ~~f b~ mason~ - m
axmm~l: ~~ns H~—er =~tS a~ =~-~ I
ated ~ new 7~e between 0~em~fi~di~ and th-~:~em~.
~w of calcul~= and compute~ for mathemat1~! w~ has ~so
:~d to ~ dramatic :~e in interest ~n aunt - :~c methods and re-
sults~ Many of the ~~ a~ most t-~=i~! re=~s of :~tiC$
-I that ~~;~-= t~ I- of In: hit :;;: pmp-
C~$ O: SO1~ti:~S tO imp I YCt t6~0 ~~:C t6~S =-6
their p~ QUl~te I* ~ve *no c1ue as to bow one mi~t e—t~IV
=~= the prom-~d ~~- :~emat:~ -~ranes of :~lid
=~d prOw that ~~:e ls no ~~t p:~:e ~~r and ~~ an-v n=~:
nu:~r ~~r can ~ ~~d uniquely 1~o ~ pmdu~ of pn~s
Sut ~~ticlans wording tod~ ~~ d~ =~t :~! to Nimbi=!
and t:hecreti~ pmblems poster b~ the need to const=ct Ia~ po m-es
and to Id the pi ~~tio-~s of Ia~e =~si:~e numbed.
The s=~-~h for edicts and e~cie*~t al~thms that w:~! guide come
puter :~ws h~ become ~ cent=: a~t o~f 501b pure and agg3led
Hal =~h :~n our t-~olog~inte:~e wood.
OCR for page 61
wc~ ~~ ~~ :~ s~ =~a ts ~e ex~
n :~ qua~e ~~s ~~o n~ ew~ =~ A:
p=~ and pm—s:~ t~. ^~ As hi abbeys been us
M' - l,{ =~s h~ Amp. ~~r pmd~ em. rimmed ~ A. ~ell
£agr~n ~ ~ 61 ~ ~ A i~ · ~ K
i ~ ~ mn60= 35 weL -= 6~int= ~
K t~ mpiex - 5 0; t~ - V8~S~* ~
viSe and me~m mt^~ m^~s ~r Brig ph=~= - ~~e
no S= m~s'=~.. Sx~I'es am ~ aro~d uS~
figure used ~o ~~ =~* =~ ~p := K
8~6 tO ~-0~ :~-~S tO Y=iO~S =Ci~ p:~. ~~W =
5t ~ (~t 0~
seVem! h~ Same Ce~*~ ~ I've ~~a Aim Air.
~ £` K 8- 0: ¢~5 50 =~6 )~ t~6 =~ mean:~:
wa~
The =~:~;r pn~ ~ndex -is -uwd to =~cu:1ate =~—K0~?£~.g, l0
creases ~ - Ace ~:~+ ~~S =d ~ I** of other =~W
s~m How Can, in.~n be~ be m
Pi~rt`~= O~ ~—ad teams ::n 6~=
p~d sta*~:~::caLv to sw ho is bash :n part to —ermine fair
I: ~~ 68~3 8~6 t~ ~ - t.0 =~t t6~ :~ - K
mo~ ac=*,{~:~9
.* i ~ d: ~ 501el re=~—K:,
- - StC=$ p! OCC55 til~i:~S Ot ~~l Imp Filet OSi~*g ~8
aims CO=~:~tiO~ ~-~tWOf~$ t~t 37-C p=~6 8~ C=
and unaut:~d Ace lIow can wcum systems be Ovine
and used i:.
~~h of t~ Or =d ma~ oth=s of similar complex~ and
si~.~.~e - ~~.~e to I.. to or=nize~ mani.~* =.-d :inte=~t
I ~ in~ti=.+
sk-u - t~ ~~n a~s or
anthm~c and a~a or in solstice of tm~iti=~ t~s ~ W~
pmblems is: not :~:~Y lnsu:~:~t pre:~.~:~n ~r th~ tasks' it is i~:hy
id*
Q u~ t: ta*tivei ~ i: temte wung peo ~ ~ ~ ne c~ ~ ~: ex idle ~ :: ~y to ogle nt tt~
cnt ~ =: Mali ~s in n - e ~ situ=! on ~ $ ~ eXpress ~ hese re rations in e~ ctI~
—I,, Styli ~7 tO =0 co~mp-~ng too]s to pmcess i~:~$ 3~6 to
>~t t6~ Ye.?* of t~?8 ~~ti0~* :~0 ~~:yi:~g =~l
Amp? use6 :n his my: —en exte~d ~~ *: and Notions
:o ma:~' I:~r alg~' and the ;~-met:c of con~e cIasses
The Ail com:~! ~~:~s e>;~-~d ~~ 68~^K6:~6 -* t
i.?* 33~8 t350S7 3*~0 0~, ~~; *i.
OCR for page 61
:: ~ ~~Y
4~r ~~ ~~ ~ ~ ~~= ~r te~ ~~t ~ qu~w tn
~ Arcs : ~e e~ce ~~= waste bet of ~—
on human; eognu~+ $Wile—m ~s ~ Io~ her on ~~-
en~an=~=ch~: 1~m ~ Sal Is the pay
shim years he- ~n an unp~-~:~kt~:~ Swabs
yang Mopes ~ - un~ ~~f ~~r In- and ~~= ~
~;~. *as a Cheek Uses ~ a - airing nch m~ mo
the item-: ~ hum~ comity ~~men:t fare—i: ~~*
pnnc~, a~ ~s m~ ~ -I ~ pee ~ learns ~~ Bask
saws :~ pa ~r admix ~~= ~~t Use of Cum=~a
and ::~ctio=! appmach~ in skew Ate;
ANAL cow~
Me Ace of rapidly Malay demand ~r brie =d coil
enn6c apphc~*on of qu=tiwli`~ sew wit powers new technics
= that s~n tho~ sets has prom~ed IBM of goals for
schoo] m-~- ~ Rap: ~e dde of ~ 1982 :~ of ~~e
Confb:~e Bo=d of the Al ~9 ~~ ~ add! =~
"What Anti abilities ~ be fUndamen~1 in the ~~um of m~W
~~- De~e extenstv-e :~ d~ wer the pa~ de=
th=e is as yet no consensus on ~ p-~t mur$e of cha~e, =d most w
idea sumSt.s th~ wh - ~s h=e nOt mowd tOwa~ any m~ ~~
in ~ ma:~e branch of m~h~tics such ~ Biter them- analysis,
or a1~ :~=V art co-~S =d o=rmions can* be pmsen
ID ;) =~ ; ~~= Ot ~~t i66~_8 four 6050160~$ 8~6 3~;~S~
~ which e~ mbe:r far =d principle ~~ Ace But this r:~-
0~$ 6~i6~t 0~*~*i78ti0~ 0; (~ =~ it 0*~> ~6
hn~ pro~t of ~ Hi pro~s in - :~h Al ides were
uw<1 i:nform~y long ~~re they become formal de5~ns and check
rems. ~*~h~' practical wowing Edge Ski :~e than ~
city to =cite or derive £~] pnn=~. It regU*ires ~e Ability to
:~ize quantitadw :~tion:~s in ~ broad ~e of concrete situate
t:~s ~ well as the technical s~11s to ~~t =d r=~n ~~t those
~ , A hat ~
In Hi ~~ut school mathematics maw mathematicians and
tea~s b^e Rued that the best guide is ~ cu:~m that :~s
the meandering he pa~ but ethic numen=! techniques have de
veloped. Others su=st that we shou:id capitalize on st:~=l items
that have A: at the end o-~t path' to pmv:~e ~~r children ~ more
efficient ~Y to d - ~~p num~r Ace and. tech.nigues. There is little
'
OCR for page 61
:~ - - ~~e to: ~~ ~e n~t I ~~ - w ~~' ~
it seems m~ to ~ t~ Al u~ fires ~~ of
. ~ ~S :impo=m to m~ to
~~' as. qu:~V as. possum' ~~ ~~m t~ ~r A.
sent~ng =6 :~g amp ~—cal Jata. ~~ that :i—pectin wall
un~ ~ mo~ W(~ 1f ~ ls :~d ~ un~ of
shed ~~ of Or ~~ ~ :~n ~~e am the ~ ~:
Whim Id~ =d ~~s :have ew—d ~ time. :~e~ m~ ~~nW
)~ 00~37 I- ~~l p=~, 8~0 ~5 0t ~0 ~~=
~~:~ 3~t ~~ ~ t=~t p0.~l 0: ~e ~~ ~ ~ 5~t K
~~s ~~ :~ Am ideas =d metho~ ~~y Cop*
In se~g ~r ~ ~~ of ~~:~! ~~r =:~s ~ Oe
dewIoped in s~! ~~, ~ ls ~~! tO b=n ~~h ~ simple
question How ~ n~rs used? In common sw~es su~ as daily'
~~, c~' in~tio~*~. *A or h~d bu~, o~e
will find ~ :~g list of si~ions :n w~h ~~ ~~ ~ vi=} mie.
~~' s~l :n gu=~ve r=~ng is ~ cn:~! prerequisite ~r
~~s in my- sciennhc, te~' or bu~S o~:~:~' =.d ~e ll~
of ~vs Wat n:~rs are used in thow Mds his bo~ :~ and diverse.
Wagner of cum.~:~a are ~~ fm=~d ~ the ch
05 Magi. I. t5~ ~~ p=~0 $~5 [~r ~ p - ~m sow
s~=i=s ~~ Fit waSo~Iv ~ce outside whoop However' ~ warch
f.~: commOn fe~-~s tn q~tim~e t*~:~g He. - ~ that th - -v: ~
~ gmuped into ~ ~w cate~." one com~n an - Is of nu - = uses
shows that - -ace examp:~e :~= one of thr~ basic tasks.
. A. To use operations of arithmetic to :: aims
size—to an~r ~~s like "~ many?- or"~w much9-
2- OR~.~:~. To use ~~s to in.diC~ position in ~ ~~=
mth the relano~s of "~er shave o~r "*less th~ ~
3- :~- ~0 p7~7']~0 ii i~.3 ~t - ~5 in ~ I
lllustratiC'ns of thew di - ~~t Msks ~~d in. TV life Here a
w~e pa~icul~ exa:~*~3
Sten~d m:~t tasks l~Wi:~.g :~s I: as lemh~
awa, volum~, mass, a~d time all em:~' numb=s to- ::~dicate
$~:~# ~0 0~5 0t 8~iti0~#7 5~-~, Ii 8~6
d:~ isle co-~-~d directly to operations such as io:~;~£, com~
panng~ or pa~:~tion:~g of o~s that numbe~n measure. Other
lmp-~t con.~s such as wiocitV, accelemtion~ a~d de:~sit\+
OCR for page 61
68
}~ ?~ In ~ ~~:?~ ~
~~ at?= ~~ - em ~ :~ 5~, 5= t~ 3~ ~$~ -~erwe~
by -speranon.s on `m :~ - east
As cu~ -it ~ ~ Hey Be: o~n ~~ d: Infers
# ~~ ~e o - r tn At. ~~ ~ ~~,~ serum: ~~ - o
-em - = - ~ have Owen :~m~s Am: -- I:
the hardier of arrival bonds- the coder ~ =~ w~+
I.'' Dim ~~ I? ~~ ~ - S airy used~ :~ ~~ ~~.
]t ~?5 ~0 ~~= t0: ~ 0t =~7 ~6 ~~, ~~
station ~~ ;~ ~to ester ~~ want ate.
?. l66 ~5 ~ ~ 8~ ]~0 ~~ :~ ~4 ~~ t~6 0~
~ 0= Eve ~sta~ Em :~ ~~ I~ ~~r
WIt5~t ~~r ink ~= ~ te ~ TItt ~ ~
dImn~ tetw~ te~S 1n that—er.
In analyzing ~~= of ~~= each possible ouwo~ to =~
Imps b~n ~ =d :! ~ As pmb~- Event ~ -amp
maim i~> ~an went 3: =~?~ ~ ~ ~~) ~
ing~er~thep~ Aft ~~' )~/ 8~6 3
=e d~-~' p(A ~ B) sh - d -~1 p(A) ~ p(~* Tn ~~s s:~
t~on the as ir~= ~ m~m -~1 - Brood. ~ thme
atmn of unlon ~r dI~ oint - ~~ 00~?5 t~ ~¢ 8~0 ~
m~-'~ numbe~.
a?
The um~ of ~~.c reams ~~y haYe numbers Or each
:~vidual pl~- Bib ~ =~= $~tim-= :~:~e an
assigned fashion, ~thmet~c ions or ~~:~ns Irving
~ seldom Eve am ?si~ifi?~t :~ ~~n ~^
~,~ NEW u=d mIe~ 'as I~ -
~i?5 t=~V 0{ U~ 0[ ~75 Aims 560= t00 0~?5 to men
t:~" But it owes the first Am Awed ~ - Ok for or:ganizmS t6? -
p~= -of q~timtw-e r~g ~~s ~~o manageable ~i:lies—
~ to 6~6 51~fi? 8~t ~^ ~.~#~ t6? ~:~.~t?~?~ ~{ I ~.?~?~.??.~
ski:11~, and 3~li~ti0~# ~
~ = ~ ~ +~ a, -or ~,41L JO ~ ~~~
Wish I re6~t ~e taxonoN'=n
netp revem ~ omn tea-~-~= and ~~:ts the ex-~ntia1 mot mea:~:~ng
0t ~~, t~ ~~$ iffy: 0~ t:~0 [~$t ~ ~~l 35 t56 tt00?~.
~r j ust ~:~t p~?~?0 Unison and Bell have proposed ~ mO
t31~66 8~75i5 0[ ~~} k~65 0t ~~? t I I- SUIT SiX
d;~t uses of bile numbe~ -
~?5 ~: die collections (~p ulations) ~
MC38LL[OS ~r con~us quant:ti~ (~:~- l-~67 =~-~'
R&liC CO~:~S (6iSCOU~, P(Ob~b~1i~iCS7 =~ $??
# I: (~ time line, test scOms),
~ Codes (:hi~wav, ,;~1~' pmduct m-1 numbest' and
OCR for page 61
c~:~Y
:~9
- Id ~-~-~s (~ ~ A ~ ~~.
\ parable! mxcno~ movers ~ th~ ~~s ~ As can
If. {,0 O=:~s, -an ^~s t~ nu:~-~: ~sc~e~
dnion models putting t~er or $hiD:i~,
S~:~n my Await comp-~' Aid, or recovering
~~ ~7
~ Pi miss s=e cha~' AL ~~, or u~ of ~ rate
fin :~ =~5 =~' =~' t=0 ~ i=.' ~~ ~~ ~0 ~ 7
or :~venng ~ ~~r.
W~e mmemat1~s and t~ ~ qu~n ~e me~g of
these He and -~e shed -~mpletene~ or ~07 it 5~5
ce—n ~= totemic to sib an~es Wit belp—= 1nstru=~= -on the
,n,~,m,,~ ~ of wpa~g ~~S to u~ nu~ e~y to
e p - ~~+ Em: Of th~e dI~t ~~: ~~ ~~= ~~ :~0
h~t t~ =~ =~ts ~ ~ q=~w - ~
}~ 5~ ~7 ~0 t6880~i~ ~5 ~37 ~ If;
r Axed= an6 ~ cowe~= ~~en phenomena ~nd Aim
that ,~ - x~5 esw=~ ~~. Each God' - ~ Bier 'ax ~~: ~
~t ~ A h== w~ have ~x~- ~~m ~ mi=~s
am0~ o~ - s co~ng ~ Fathom ~ t~ Abe;*—~~* ~ un
de=~d ~~s :~g press ~~s need Se experience m
X
Bile -is must ce~ acqulm c~^e skis 1n d—~ng
h many ~i6c u~ of n~ - s' Rev To need to ~~e ~ ~~r
Ape: on plies that number u~ h~ in mmm-ofn. Dew
is c1~ ~~—m re~ in m~hemat:= e~on that u
~~g ~,~: s~.~! pmpe~:~s of.a mathematical -I
~.Cilitat-= r~n ~ the ~~m =d a-~li=~n to n~ sl.~^
-- .4-- .4.-
~'001 ma~:= ~^ tnererO~fm' emphasIze the It's t~ died
ent Apes off :;~r -en - is se - e as mode~ ~ :~ o - ~ng$
and coding, to~ther with ~e ~ th
In q:~w s:~:ion~-
~i~S afnd Relations
FIemenm~ -yes of number focus on descn-~tions and IBM conw
cem:ing specie quan~timt:ve fact~he cO~ -of ~ caned ba~ pnced at
~ ~ r I) Maxi Of ~ 5~d Oh: if i ) 0 ~~t iO-~g 3~d 30 ~~t V$ir]-d6 O.t thO
By: s~ed of ~ car-~at tr^~is 300 m::~-~-s :n ~ hours. Ma~0Y of -~
C~:$; t69UitCd For me teaks IS By 3: CC~1 8~d ~~C t8-~:k
OCR for page 61
-~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$~*nding of relational phr~es Such as ~ depends O~ it? ~ :5
Mention of it?- 07 053~ i~ ~ ~5 ~~ ~ -
OCR for page 61
~ 1
~ ~1~ b
~ Icon Carrion
^ ~ ~ . .
e ~yc`~c Y=l~10~
d Consuming ~~
...................................................................................................
~ _ ,.,. _ ..........
~ . i 1 1 1 ~1 1 1 ! i 1
-
~nGERE ~ T~ ~~ior of ~n~mcn~1 ly~s of ~l~ions
among gad =n ~ seen mod Dailey ~~ lyrics INS.
Gum {a} =d {b) if dim a~ i~ Flagons, (~)
and {d) Cow ac=~ and conning variation, and {~} and
{6 ill~ralc cyclic find slapped vadalion. Virtually all vadalion
actually ob=~ed is ~ combination of 1~ basic lye.
OCR for page 61
=~A ~~ ~ N~:~y
~e h=~ * ~~n thecmm - ls one :~.f m~ ~~:
* in : =~ cKp~s If:::
e~c~v—= ply ~ written =-a m—mation of :~le i*
Thew allocations Brand f~¢~e -add ~~;~3~g mast c
m~ dimples ~ dam comma aims ted beg parallel ~~
~~mm ~ mid {I Woo assay hat Nary p~om~al ~h
complex -decedents can ~ wr3~:~n ~ a p - ~= ~ i~ Of: (film
which the: senses ~ ~ may - coned).
~e d~ws:on Writhe ~$, ~of mu=~' ~~e ~ me tam~:~:= ~=
for i~ *sky of balm ::* =d Amass ~ ~ : to ~e
Careen I-~r mec.~.~m of pompoms a~. ll p~ me e~
*e anthm=e ~=n=~ for ~ mt~
d b' :a -# ~ Cod ~~) ~f am o~ i:~a ~ :~b ~r mme k. He ~~:~e
=~s =d 601~ I: aim= - ha. ~s Meow .~ pm~ u=~..m
-all ~ ~ =~5 ~—~6—~37 I art
impo~t ap~:~s 3n cOmp~ ~~, In ~~ =~ in
t~-n a~ 0~ of busses ~~ g~ ln~
~~*~ 'N~
The ~~ nu~ mtem t~ Endues elements ~:~;mi~ ~
~0 ~~* 0{ i- ~~t ~~ - ~-~ t) it 0[ =~' ~0
-=~er ~~em Q. ME ~ ~ ~ ~ ~
:~:0 j05.~0 - ~t ~5 ~~ 8~: ~0 If t~ ~
~t 00~o - ~~t has ~ :~phcatwe inverses and ~~ ~~;~r
emotion of ~ ~~ rx ~ ~ ~ h~ ~ unigu*e solution ~ rational ~ s'
=*6 ~ (~: ~~= t)- ~~$ tt)5 ~0 p~t is ~~ 3t ~0
arm 05 ~44
Be Award oMenng of mt:~1 numbers makes Gem ~—$e =t—
between ~ W:O mllonal =~ there :. th~
~1~7 t00= ~ ~~ I: ~~= ~ 5~ 33 0~ =~t
mst. On the other hand' ~r a~* mt:~1 ~~ ~ and b~ there is an
-~:~r ~ such ~~t na ~ lo this p(~, my:= tt:0 =ti00~ ~5
into an mimed -~:~d fiery While the -~tio~*s a~ -:: of
m~i~ num~= am si~fi~:~dv -I compl¢x than intend, the den~
and A: pr-~ies of ~ combine to Iav the groundwork
~r p~n l:n m=~ement4~ ~~ei~ ~~t ~ unit -of an\Y desl,70d
Aid:: =:n ~ ~~ ~~ =~: ~ ~~ of ~~ Fix ~ ~
i:
Rm Id:
The n=~! *m- intends' and rational numbers p;~e ~~
mat sv510~s to m~) the st=~s -of ma~ pra~. 9,~e
reasoning tasks. But unresolved que~s Ado as long as 2000 v~s
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''
- - ~
:O
nG#~E ~ ~ : ~ ~ mIon~ n~ - lmc
:~e ~ ~ of i=~h ~ ~s ~ hole' ~~ thew :~s
no a: n~r equal to ~.
8~ m~e tt q~e c~ that ~ m~ n~ =e :~m t~ ~~ ~~
in ~~r so ~e pmof -God ~~e is TO mIo~ w~r -
~~ :is ~ (or 3-' or 5, or ~—er mt~ ~t ts =t ~ pe—~ ~~>
is an ~c in=:~s in ~e - ~~! n~= ~~=
s). god. n.~.—ers =e =~d as m~= -a ~~.~c Wrest
, 4— , ~ ~
~e .-. ~~m ~~s ~~t there =e line ~~s ~b no
rationed mesons. ¢~ are "~- -ash not ve~ b~ holes) in
n~r h~ th~ has o~y ramps -~ooT
5 94 ~~$ t~ ttt p:~C (~ -
d tO- mm^x nu:~, m~ .x
O: ~~£~= 1O ~C pi3~C ~~g 866~:~O
of Complcx num~~ ~'ult1:~n Is
more Compll=~—the =~X~.~t'~5 Of the
v~= mult191y as ex~~ but the an~s
gS
^~.= ~ N~-
of ~~, thelr p:~r u~ Is ~ =~t oMer and co~ hnhe se
of ~~ ~:~- - e real num~ ~~ the e~ m=~W
desk— =d re~on ~~ut 1~te and l~i:~mal pawn.
T~ alone soon ramps de—~~m of the- Imp of Stir
cO~ they p~w'~'~e bn~. ~ =.aW:.s of motion =d c~.
~e 0~ns:~n few Marion! to~ =~ =~m~na~s mIut~ - of ala - -
simple =d sl=fic=t a - ~~e cqu~ns.: Wt ~ Ie~S an away s1~
ntfi~ =~on of ~~ equat~s =~! u:~:~. S~10 pew
n:~ equsti~s bLe ~ ~ ~ ~ ~ or ~ ~ ~ ~ I: ~ :~ no m~ ~~ - ~
bump ~~ m~d w~ ~ memoir s~ns ~ thew equanons,
and to ~] ~~. ~~s in ~~, ls ~e mmpl=~ nu~ C.
~e i. nudged mnstW~ the ~~ po~e field c:~;~ of
add.
- : requl=d mot of x, ~ ~ ~ ~ Rem~y the eX~n ~ ~] ~b
Is s:~e eq=tion ~ - s wiudons ~ ~ mber ~~mi~ equatmm
=,6 opens ~ n~ st=~= of ma~l p~S ~ ~~ti-~.
dew =~x n:~= ~ be - ~0 :~n ~e ~ .a ~ ~ ~ ~ - ~ ~
mplex numbers are Mimi bY
oMewd pal~ of r~ nu - ~~: ~~ the :~ num~ ~ ~ oMeM
1:n on~e =~.~e wlm ~ polnts ot ~ ilne the c.,0~^
numbed co=~d to po~:~s of ~ two~dimen~-~! - he =d am not
I~:~y o~-~. Thus Ims c: sample o~r :~:~: seem to promise ~
much :~e compli=~ !~ :~n ~ than :n the re~ numb~ or t~r
5~. 80~5 :t 6~ 81~ ~15 85—:~;. lte =~e
between complex =~ am points :n the plane opens ~ :~!
m*~on between ~e an:~c and a—m of ~ and the Emmett
of sh~s and tr:~ ions :n the pl~ (~e ~~e 9~.
The compl~ nu~ i: some num,5~,5 on~v d=~ as
~~ ~ ma~:~=s who c~d not ~~t the wSsibU:ty
Of ~ n~ ~:~.~* I* ~ ~~w ps=~ u=~1 as m~-
els of many TO—i ~~! ~~h~V510~ I * ~= t~~0 D~- Of altC=i3ti~8
...... ~V.~'.'.'.'.'.~'.'.'.'.'.'.'.' '.'.'.'.'.'.~'.~'.'.V.~'.'.~'.'.~'.'.'.'.'.'.'.' ~ :.:
:.:.:.:.:.:.:.~:.:.:.:.:.:.:.:~.:.:.:.:.: ~ : i }
~ :~ rig- ' ~ ~
/'' '"''"'< `'St'' '"'""" """'a""""""'"'
.= ~
i' : ~~ . . ~~ ~ it,, ~~
i: : : : : :: : ::,~r ~
i' 2 .~ . . ~~ ~~ . .~
: : :.:.:.:.:.:.:.:.~:.:.:.:.:.:.:.:.:.:.:.:.:.:.:.:. ..:.:.:.:.:.:.:~.:.:.:.:.:.:.:.~:.:.:.:.:.:.:.: :.:.: :.: :.:.:.: ~ ~
i' ~ ' . ~ ~~ .
~~ : ~ : ~ : :
~— ' ~ Ai ~ PI
c:~m
g?
e~ :~= to. ~ n~ Of mr ~~r an al~e mng. ~~ - ~~o seme
~ ~~;~ Abram que=~n of :~= mathe~ ~~ ~~
mi~ of Agree ~ has exa~y ~ Bier ~~. ~:US -Ed wI~!
equ=~n ~ at I~ one and at ~~t n. amid Co~x ~~-
New Number ~ems
0~: S~ O t~ p=~CipiC$ Ot OU~—Stems ~= ~-
fam:~ =~- Hen :: of ~ Me n:~ cent
Aid ~~t t56 :~ ~~: ~ ~ t56 ~6 00=kP~m 0~0—~67
Bang =~ pmo~ ~ Owss a~d o - = ~~t ~e co=.~ nu~r
~ Igeb · Ib ~ Cd7 t~ed~the~;—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~er—tems wch as I, Q' or i?, bm :n 6~te ~mxi:mat~ns of
those ~ ~~s - ~~e ~~S is limited b) the ~ili.~V of =~r
languages to :~t numbers with ~~.~ ~ finite nu - ~r of positional
pl~* These "~,' rabbis of n~er Alms do not obey the
conditional nmctural paddies of =~hers Ash as asmciat~v of
. Thus it seems imps that ~~ts exte~ them- ~~dy to
:~e the ~~ propeni~ of those fiche - ~~s ~~t underlie ~
much of thelr actu~ qualitative ~~-
me dlSC~r~- Of ~UI~Ot~likC =~iCal $: like matrices that
~~: to ob~ ~~ tommies th~ our nal~ intuition tells us are
tme was x dramatic step :n the d~hopment of modem mathematics.
Contempom~ algebra on~ed in an atw~ to prm~de ~ theory to
explain the stmctural dies of vanous number systems. 1n the
Ian 150 vea~ al~a has ~~d ~ nch a~ of abstract theories
t~t spnng Am stu~ of ~~e inherent ~ ~
va,~s ope:373110~s and
IV~ 4~ :w N~:~Y
s on sets~ A- hi shin ~ ~~ of
me* ~~.s ~ p~ ~ n1.~g i.. ~~.~ ~
h^e ~~ sh~ ~~ th1s insect m~! ~m Deny
has item I. ~~.~s
~0~ ~~p57 =~, :fi~,
im3~s Act. ;~ mon.~., CALL T~ m~ ~~e -led
Briny ~ ~~= ~~, thev a*= n~ used r=~ as t=Is
r ~~h on Len :~:s -of A* am ~~=
A
.,~ ., .~ ~ ~ ~ ~ ~ , a. ~
I'm: ,
*am me mICOl6 O! t~ =~uW *is ~ ~v m
=~ ~ lnterm in At, ~~ of ~~r v~- Ad,
I973 harm Hi; He ~~~ "~ :~O mo~ -A ~at:-
m=s ~ac mm~e to Oe11ew t~= ~ ~tb0~ id ~~. 6~t
the ullima~ teM o:~-~! Am: is 0~-~ it enables stud~s
to ~~V -~ei.r I.- ~ Solve i;. qua*~ti~:~e pmuems:. ¢e
v to some pr~:~-ems is not o~ the mo~ impo~:t gem -of whool
mathematics but ~$O ttC =~St IBM C6~-=tiO~ Hi*
e A- "~M problems st:~:~s te=~r :n the b=~s of mathem=~s
~-~:~s of all ages~ The Joey h=t step in I: wo~ on* problems
is to tden:ti~ in pmb::tem s:~;~ns con=~-~at are $~-~:iv aims
~~*~: to those of number As*
T=dition~ approaches to oh's msk
can be soned into two broad cI=~$* The p=~atic approach :~-~:ps
Att ~ Or ~:f ~~! (bawd ~~1~V tOUt
AWAY Me
~s to prm'-~e stude-~$ faith ~~c guideline ~r
each ~—~a sperm Beam or organza, ~~:~n ~~
-I emblems' a ~~ for I :kev art:
twe ~~s imp ~~:~:c ~~ns at
.a~h =~s to An. ~~s to =e ~~e bl
(~t 56~) t~t fit t~ p - .~5 :~ ~~ 6~t ~5
aid; emus ~~r so ~ ~= ne316elK appma~ ts ~~nSt~ e~we
~~g ~~s ~ ~~t =d mns~e model
p~*~$~ - ~~.
pro~ :~s ~~e t~ n~e
~e (~) wp~= "~ WORM- ~
- ~07 8~ 0~ -~8
e -of oMin,8W I=~:~-~m ~ t~ imo arm
a:~s ~ ~ de~e 0.~ ~ ~e mber ~~, - ~e
Tea ~~ h~! Exeunt= Agent ~ JP ~ ~
~ ~t h~ Am. w~ dI:~ tO d~p their fac~y tn ~e kind
of m~w *~ ~ ~~=t th~ ls :~=d ~—I-0Y 0se
:~= e~v in ~i6c mu=~-~- Recent ~~ to d~p ~
I:;: - 90~ on p~.~-~e~s she arm:
but n~ ~ -~*~e r=~s >0
wade the search continues ~r e~e
i0= 50~' tt0~ :5 ~~ :~# St~i66~t :~ ~~.~g ~ ~~
~ ~~t ~e natum o't qu=~e 9' - ~~S th~. In m=v
- : appl:~:s of m~hema~s one thi;nk$ ie~ abom ~~g
specthe ~~d pmblems and cOnc=~s in~d on cO=~l,~
and ana^~nR ma: mo~s of the broom wit Me As
stem am:: pm - ~ of sch=l mMbem~= us~V :~ nu
men~ l:n~;~on and ~ s^e question th~ =n be =~e—by ~
n~.~l I ~ by~ mining an equating Outside s~, prow
i0= Ins It t~ m~ or extm
3$ =~ :~fi~6 g~054
In ~ mathematical modeling approach, Me h~t st~ is tO i mI~
want va~- ~ n=t is ~ - ~*, in m:~e fo~ Ian~'
relatIom ~~ represent caus¢~e~= connections -~ fame wn
~ ~~. Specific q-~tions =n then be posed ~n terms of input or om~t
V3~ 0: ~~} pt0~05 0[ t60 =~]~g t0~84 ~V, computer
t~ 8~t ~0 4~5 hit numencat ~ 67.
s-~mbo! methods.
_
· ~s. ... . ~ I, ~
The mo~ common s-~es of num—cad vanities are m=~umments~
Thus the theo~ and tech:~-~e of measure:~t play i- roles ::n
I: ~ ~~Y
:~ Stems. ~
anth~c of numb m~, me
me t~ t~e a: ~d Em feat of ~~t mathema:~s—
~ th ~~ mw~r -ads ~~ mter~ce~n
mathemancs~ am hs appl~3~.s :s not ~ remarks ~~ amid :n
ma- pmtm~1 mea~ ~ ~ s~! mathem~i=: ~~ ~
~ ~-$: ~~, ~ - ~ 0[ ~~ ~^ it ~5 ~: :~ ~
that ~ me studen~ ironing ~ me'a~'=t imps bnef -
pos~ ~ ~ ~ st=da~ ~ns ~r 1~ =d then pm=)=
Amy= fb;r pen~, ama, am wIume based ~ thow A; Has
.k~=~::'~ ~ ~~] ~f(~) ~ b~ ~ h - ~] ~[~ ~ PI,Wi~C
)s f~ :x ~h ~ heed or: f~ ~ r: ~ h]' =d so On. w~ =
be=~e ~~c p~e In tM ~ of the I=~n ~ h=~.
d to this ~~ ~~h to ~~m ~~v
Avid =d ~ ~d Concephons of Ie~h, ama' and Volume.
n of area =d pen~r ~ ~ Is- =~n e~r on
student a,~n,ls. The com,m,0n, "*~;,1.~" ~~ by unth1,~g ~
~~s, ~~ss of any wording 1n the pr~em Stawment, is that lf
there =e two- Ou:~m at~d to ~ - of ~ ~~.~:~d ~~e one
mult;~:~s them' :f there a~ nu - ~~n on each s:ide of ~ I one
~0 0~S O~ tO~S ~~O i=~= $~S :~l p=~6 to- ~~
)~7 ~~ t50 I ~~ 0t t08} ~-~57 t~6 ~
mulat~ve e~ts of e=~n :n combinations of :~s, and ~e
n to imps sham ~~t oc=r ~ so m=v P~M appl:
cations or to- the ~~s =d Acts t;h~ are ~ndemen~ :~n ~~.
~~= [~7 5~ t68~= 0: ~0 avant ~ stmctura) 51
l~s ~~ un~di:e mo~ appll:~:~s of measurement
A! the hea~ of am- ~~nt pmcess 15 ~ ~~:~g that 3SSi~S
numb~ ~ objects. The mapping a~:s ~~e ~ ~ s~e despond
unit. Other Jets are then cowmd t~v copies of the unit. 4e choice
of unit elem=t is a~itm.~*~, but once the cholCe is made' it provides the
~:~ ~. brim ~.l .$ 8~0 =~6 ¢7~5 ~7 ~~t
co:~s of ~ unit and ~ n:~er—the :~r of whole and partial cop:es
of the unit ne.~d tO exa=~v -A the measured o~" The m~
m~cs student who unde:*~3 nds this pnnciple—as ~ ~~:M pmpe~y Of
manY ~~t m~s—:~s =~uired pmdu~:~e t:~t imp
e connectlon between ~: s:~S and Amp; m.:~50
The unit and cown~ prope~es of ~~nt explain qu:te cIea~y
W58t ;5 50~ indicated Fir 8~ ~~18t mommy moreover,
the atta~:t of units tO measummen:tS =n ~ explo~ed to Bide £
:~ reasoning about Scie~c pnnc191650 I~ mamas. sciences quantll-~w
~~:~.~g Is ~~d by ~ we:11~d al~m of qua:~:~s =~.~:~:v
c~-~Y
:~!
=~d "~ an~" in ~ ~~:d ~ ~~c :~
tmn ~ pe - ~~ =t hat 0~ 0~ 6~t 0~ ~0 ~8 85 ~~. }t t56
em :r~t :ls ~ number Gore Bite are ~~—~r t~ p—~, ~e
~~ Is ~~ wppo~ to ~e ~~ of me:
Bite ~t h^e b~ -I ~~e th1s atte~n to ~~s aS
as ~~ in m-~-~nt is ~ ~ com~n 1n m~ = t
smen=:~. :~t :~ stm~. ~~ a~ thow ~~ : c
mth h~ ~~ :~ake- ~e C=~= between ~~ m - emancs
and ~S ~~:~= 18~!
~ ~ ^4 a. ~ ~ ~ ~
.. i. i.
|= m=~ =c pm=~:= o! m=~*~:~e =~s :~n the
~~ ~~d haw:a ~ h10~+ Aft m - ~~= =d tts te=~-
However, jabs as- ~~' chasms mat~ma6=l :~ have been ~~-
eralized and ~~d :~n =~-~' ~~t h~ been aid
to ;~t u~ thmu~ut tM soc~ me~+ While t~ Chic idea
:s ~e met ~~ to ~~ ~ ~~s—these ;
su~ o~ :~y ~~ ~~.~s ~~t able vew ~~:t - m
measu=s of lentil- ~ and ~~me
~~litl=t S=~ti:~S 8:~6 S=~S 6~O 6081~6 ~ Y=iC~ O: ~*~
s~s of item o~r Owes all social s~tions* Eco~= ~~e d
Or S~ ~~35~S Of =~tS =d benefits to quant;~ 09~5 :~ 6~8i0
maH~. Pw. cholo~sts and Ads use ~ wst a~ of :~$
deserve mIt~S a~d ~~s of lndlV~.~. st=~i=.~s m~
sum p~e caus~-ef~et wI~io:~s am~ Gary ~~t ~~ of
s:~c va;~. ~ =~u =~~ nutted operat~.~~ =~ -I
a~ u~d to- model ~~.i6~t st~1 properties of snuationS* ~~
tinges dass~ p~ and c-~ts aw d=~:V aMi—Te. Buff it is
incr=~V =~n that emetic quandmt:~ ~~.~g in the socials
a:~d human sciences requires undemanding of a~ of number t~t
pe~.T nexio~ T=~.~.~n or new m~s m- new* situations*
Wit qu*~io:n the most i:: ~~! of whoo! mat:~:~s is
tO d~\rClOp ~(~115' mill tO t=~ int;0lU~tN with quantitative ilk
formation. ~e mathematical con - -ale techn:~' and pnncip:~s that
:~! 9,=~titative aspects of expene~e are proving bY st=~s o~f
:~u:~er sV~' ~-m, and meaSu=:*~nt that hew 10:Og been the
bean of school cumcula How~: the Age of elect*~*ni.c calcu
baton and co:~= ~ powe~t mols ~r Anti am ma;nipu-
i8ti~ gU8~tit3ti~ ~ iG~*~tiO~ ~~:S Ct~6 t:~6itiO~] It ~:
Inst=~ti:~n in th.~e su~.~- Tt no- Ion~r =,&~s Sen~ to dewte 18=
~n:ons of the school ~mculum to training students in anthmetic or
al~miC al~s that can be ,~d quickly and accuratelwN by
^~ ~ .
Morose a~d I: C~" ~ Icy ~~ I: m
to cOm08~= ~ also: Ied to ~ dmmat1c tn~e :~ ~e ~~ of:
urns w ~ -—~~:w ~~ng ls bew :a^~. ¢us—oo]
m~= mun p~: ~~s to ~e - ~~::krmMe~e of :~ - er'
~m, Bad art. 1.n If: :~d c~ days—~ -of
-. ~~le ~c~:~^
~ -I arm. ~r ~e ~~ ~ q~ ~~ in ~~e
it ~ at;, ~~ p ~~; ~~t K
Un~nd f~1 :~es of nut - en—ems a~ ~e
ma~ ~ ~~e m - small -is =d re~ Is
dons ~n - ~~ Hey are e~:~^
De~e add lnt=m Hi:* - w s~ us:~ ~c,
~1, =d ~~c represents
Perk Ash ex~ ~ approx~^ cay -~th~
metm =d a—~c :~ b~ vanous ~?~0 =~5—=
*ilk * 3,~) t~ ~~ or =m
pates
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=~ :~*~t be nch in :.. to ex;.~.~e # ~
pO—~t gOS~ SitU~iOOS &S ~~ 85 I~ t~6 ~~$ t53t ~~i*~tC
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~ oaV:~s Harold ~ ¢e ~h t~:~ cf Co mputa Ace: <,
~ 5~ 0 :f': A: K . e: hi. ~~ .~ .
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A ~~, p5K.~; p ~ {~ t0= of ~~ I W3$A~ ~ ~~K ~ 0C A ~ 3~* 311~) ASSO( t
|~ O:
:9s
8. ~a, Wok and #1~, Am. -~-i~ mew fits ~ pus
m^~ Em ~~- In X, Abut OF ~ ~1~ ~ P. (Eds.~: ^
Amp, ^7~ 7~ #~ ~~e ~ ~on, Ha: Amid Cough
act, 19S8.
9. fly, ~~= ~ ~~ Q~ ~~ ^e O~ ~ ~~ ^^
Am. ^- YA: Riot ^~ of ~~ of ~^ 1 9S4.
10: ~^ ~rid. ~ ~~ ~ ~~ ~6 But, BY: P~mc~b~
~ 198i
1 1. ~~~ at. ^~ ~ /~ ~ [~- ~ Beg ~> a: D.
~ ~~ 1977.
12 ~ ~~ ~ ~~e bitt ~nu~ ad numerals ~ DIN Ha, augur
E: (Ed-): ~ ~~, ~ ad-- ~: ^~ ~~4
aims, ~ ~~i-~ aunt ~ ~~ of ~i=, 1969,
1~36.
13. Hemb~ ~ ad ~^ gad ~ of ~~~ ~~ in ~U~
m^e^ valid ^ ~~yd$,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
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