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-

) --':t.::.,,:. , .

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tea~hkB .~

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~ome"ot, . ~e mo~t

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in

teaohlfig'lIIi'th~atlo~

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<gr.de·8 ~enlJ.an?-:.n~ ~grade:~.~g~_~,Olas8.~8~, :,~.:wor k ed.•~\h

': ; , ~hl_~~-elg~'; t.ea~~·~~s:, 1fh~ sr~~ ~1T~:~~ ._1nVOiV.ed'.w1~t~ '~h~

··' ·te.c~~

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^-. ..:..., -'7 :.:"

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th:8'

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. D1ttiou1tl11S.with8uoh,~plos"asplaoe-valu.e .'. or k-wi t h-.

'~the~~ ~Udl ' P~1" ~~t;:.nd _e~e~l!I" ~d vc'~ull:es

^{·.e r e}

:~p~~~t1~,

~: ·to· P.rtl~~:~~. ~,~ ^_. ~~~ei~·; ^... ~: ' ^{::_:' .: ",:'} ,:~;:;:('; ^{: '." F. ' ·} ·: "'"" ". <~

',"',- :',:.:""',,' :: :_~e~~ ona. f'O~"~~,d.l,~nc.~~t:les,'r~~ed,fl'!Jm,ve~, ^.

- :.• ; AITt': .:;2~:::~1:· .·~::~:~~;f:·::;;::::·:;th .

.:-.•o~prob1em~,;.h.U~~ ,.,"" _,:~!1ng:IIu1.~1,P:7~ C.~1~.~. ~1imb~r..'

"tact!mastered.~^{or ten}'g1~en·. s.,one_relso~^{'for the'}

/I ^{' ~ ' ~ ~ug8eat10~}

^{such 1lI8'D...}

~':~re. ter

^useot

m~lPU1.t1Te ~. ~

, t~s~e01.f'1lB~~ltt1on"^{sn e h}aLallo.1l.1na.-thlLUS"c'--~e:-I-

I of'

m~;lPl111at;ion f'~ot

cardll tor 1lork with

dl"181on._SOllle~~ _~~ ~

, sugg estionswere.verye:rpltolt1nthatthey outlined st ep-

. ~ , by -step prooeduresCor de.lingwithcertain probl.emareas. -

~ .';;

" 1 7. '.

i

.•..\

i

,

.

.

^'

. tzhe

.rl.~e~· ~: giollt~~"

^tor,'

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^arid

,~:::::~'0n ~. r . . o" T~ "·oon~o;:,"!# r.~t~th1~. ' , ',

. " He 1sde eply indebtedto thepr lna lpa1 s"as s l s tan,t

··. · lt~~f

^.~reola~lOn,18"e:z::p~~8sitd~_~ the',prorel~~~nal

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t

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.toDr;'G!1n',)lOoldrll!ae,·Dr•.Lio n el Pere l r a -Mena.Gzlll and:H8V.r.

.: / .'

IE:.

'TABLE II.

"'"

~ ~ISTOFTABLEE! ; ; . ( "").

~~her.- ~er1~Ce

^"an.d

~~ep8rei"!o~

^_.

Indi ca ted,onth~first:QU.estlonna lrec,.

,"' - .~ -." ·~.1 ' .."

MOl!t __D1f't'l~u1t'nl~.lC8~B",Seleoted ,bt,

~:aO~I!I~~ , ':_:~' '\. " >/-~i" .) I< ::. '.

'.~e8sof Dl ff1oultJ at,varl ous'1

\Gr ade,Levels,••:~_.~.;•

.-t ..

;:·I~',..

/;' . itf~ I' ,

··' .: 1 : . .. "

... ~

^.....

'/0 .

. ,

PAGE

.~

'/., .28

29

'".

...

"

·A:t . ~he C.l.Olll!I ' o-r th~ '1975_ d~hoOl" ^{7ell',- '} 1.t·~lU!I

, . /'., ,:: ' , " . .. -"'." "" .-J,' :-/.

/'4·e~1d,ed.b1the~1n1atra:tlon:.andstaftot' BrotherRloe :.;;:,", ".','_',:', .... ,- ::,-.',.',- ,~,. -.,.~ r.. , ~_. ..-'

~_g,~ BO~~~ ~_~,.s~. ~~hrt.~.IJ: , ~: '_:._;~~t, ' :~I!IV.,._:~~.the~. ~. ~,~

^{,: :}

. ::~: ·;:: ~ ?~:.r:;·~:::::; ::r.:r:;: t ~::·~.t · ·

:':'''','Fe~~~<~9_76.;,.~~^__.a_:,~~por~.~,p~e8~ted._~·~~,·tl!lfo~~rll^..

';'",'. ,

,~e~~ 2;1'·,~~76. ',;~~ ~~. ~~li-'~~~~t.~O~8 · ~~th~r'

.

_Bello.8.',r'ePo;f1i~.',~t.:.~lle::o~ord:1n"t~f;n,;'.bll~~~~1ori~

a~~d-~~o'de~~bD~~ '~l~b~ '~~·~,.hlg~ ,~~~oi"~~u~~,,~~, ,

^.

. 'readilyaohleTad,.-,thesameoould,no t bel!Ia ldabo,ut,pro gr lllll

-: , ' ,\ . .: ", :.:. .;' . ..:'

.

'coord ination,b etw.e en;:Brot her"B ioe Hi gh, Sohooland 1tl!

·t~eder· i1~h~o~a

;-:'.··-at,'

" pa~lok '~ " ~i. , ~O'h001, B~.

^.

Bo~avM t~~8' S~hool:" ,aJd Ho~' ~a. ,~~hooi. ·

^Brother'

Be],lc:>_~-~fl'rerr&d,

:to

-'t hiS"

g roup.'

ee._t h e

' B ra 't her

R1.oerall1ily"

·~t,; ~~·~f8".~_d- r~~~:~~~'~_.:'~t",~.~,p~gr~_, .~~'_8U~jeot

oOO~'1Unationfrom k1n~erg~ten',to:gr.e4~.e1evel1,be ".

~'renbw.1i'-p~~~"~p·ted;^8.nd·1m

p

i'emen t ed. BelI~gellt~

that

~~par~erit··h~ .,~t; B~~e~

^{Blo eR1ghSoho'o l}pl 'Q" .'

:1:" " ',:' " -, ' -,". .' ' , ' , ',

ma10rrole,1rl,th1.s:'l'leoea u1'1','endeaVour.'

. I .• :.

"~hi, "p~j~t, -_then ." e~lv~:rrt;~'·~:ro1;;her-Bello"S I

eV:U~~lo~· '~~ '~ep~:~"~~~_:.:th~· ~~r~t ~~t~.P : t4,,~ open~g

up~aTen~e'ot.~~miiun1~at.l0h.• ltli··the'~-eao~e';t's_c)t ra:-'

, ..

'mmODUCTI ONr

.r

1... \ . " .... . / :~{.. , ^{• .} ma t hemat1c 8in

t~ el~m'en~8n}30ho~e

^{thB}feed; into}

'Bro,the r me e HighSch o ol,' :rn'tlle'.eOUl"Be of·t he pn?Jeo t, thelnveetls stor,

Plat~em~t1C:s De~)ll rtm:nt

^Head,et

Br~thel' · .

I,' ', ' ". . ,'

R;1ceHighsO.!'1OO,l ,.becee e8Cq~81n'te,(\..~1theve.:ytea~in

l

gr a de S' fourto e1ght 1n,th~.^three~!eeder" 'Sohool'~B ~Jr'

, ,.. I ·' "

BS,w1th'·t hei r prlnn lp. 8 1e an d vlce-pr1.nolp81s. Ttfe

',.

, ',~1~ve8t~g8tor

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,.i·1~P ~.~V~1?~:;:1~.~,:thi,:~Y-n,~n~, ~l~f~r~t,

~

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~@~f~4~2~r~J[~~'

problems endldees,with-ool leagu es.-1n~he:Bro therRlc e Fo mli,'ofSOhoO l;','" ,

' ,I ' ''"

J ,

: J'_.r' ., '

.Ste tementof'the Problem

:, '.'/.",":'-..,' . . ,_,' , ;. c- . . DUri ngthe late..r1ftles".th er e1I9S.9 tremendous "'

~" ,

'.

,,' , '.

'

"

ups urS,e.Of,,~nt~re~'t~~:d~.rC?J~C~SIl s~ed. _~t·t~e.·~~~v.e~ent o:f"msthemst1cs elfu"C'!.t1ori.,rn'our,8cho~,8. Ne.,lJla,thematlc~

i>r:ogrsma_{.-'_ " i_},w_.'er e_'.

~;S:I.;ne~. e'xP~j.1m·~te~(.,t'th

_{'• '.' " .. - ',. ' , , ._•}

- ~d

_-~

'1ntrod~c"e4

_{• I. .}

·

'·~~o.^the.aCh~Ol^.•,~J:8.~em8 ~~.~~erto~e~~~:Btr.dehbl '~. ."

.und e~8ta:ndmathemati c s.rather~:t,hsn,Just~'~,learn-.l:t

'Iff . t'ot~; ~ter: '~I/~U~~t8

^{bec; ; i}tnv?ivbdllitti

'~he

^.

~8nlpl;llat!.on·~r,.m~ter181s.~nd"t~8¢~~rs

.tire

enCOUr8ged~t:O^'

use'8Ul_~l!d d'i8~o~eri

^{'i n}

~~~r .~ 'h~i~' 8 t'u~m=lts··8ct.1v~i, ·

f'ormui~.te~ttie.8'tr~ct~eoi",J!l8'the:m~'tlcs.··

In '·oi-dsr

to

br 1J1g n ~:'d:~V~l~p~~nt~ ~: m8~~em8'11~a': .to'· th~ ,elem·~t.17

^.school

s,tuden,t •

~tllDe,. to~ICB

^"ere_.

'~Ol'ed :

^do"n'1nto ':boli,ar grades

- , .~. .'

'.

- . "" --

,':~

.; '.; ,' :" .v.. :.;,.• < .t"'W ;i" ':;" ' ; "': ''''Ci.' ,;.R;;· :

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,." ··t ..•. . •''.

. ~d:

^1%1'ol!;,me

0':88. 'dri~~ ' 8%~r018~~_:-'1'~r"~c~~i~1ng ~", ~.~

,

oanpu~~~ "k111~' ';ere'~

de.:.'-.Jlhas1zeei';"-

~l~e':th~;e" •.

^;,8

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w;rem.or~ C~Ut10.~ _~_ ·ao.o.~t~th~ .::--=- .,'~

. ,.m eri t s.

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IItllthellla~l~a-,IIIpat~~e~t' .t~t'.~-~'~ra·',.ertt~·· ·'.

a,pp81'enth:un~,1~8tand1:Ag~ ~jOl~.~e· ._8~J eo,~~~o~e,. /' .·}Jo1(~.r,·-BBone"lIould,-ezpeot,t:Penew mathfllll.Qt1oB

····. :::=:,:::'): ::~r:~:;{.~.A:;r::.;~:~.~~e:i:~'\;:- . . :, . . . :I~; ^{. '}

'::;:··::~:~: ~":':~ 1:dC:~~:~:~ L1:: ;' • . . .. ~ j ;. ;

^•

.~raot1:o e,til.,o~putatl.on~-'e1l;~;l1a ,,',~l?the1}.~1r.lou1.t1

,"'~~:~'ved:'.~o,or·-;~~1ng?k~-i~,'~.h~.~h~~ ~l!~;~.81~~~~_:a~.;

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..h~dr~o.eto.le~1Zlg t~~ ltl0nal.·~B:th!d.t1o/J.~d''OOl1t1zlU~,- ,'

.:J~.reeeprobi~B.1l:tll~~'Dlode~'~;a,the!Jlat i~8 :,A~o.'

~1th' t~~ 'ii:~ 'to~;&~, ~d-:~e .. -,:met~~ :~~~~Ol-'~~"' ';~

^thnew,

.' _." ., ~, " "' . '--'

ma~h,eIQat~os,•.m¥,",t:eaonerliJ l~ek~t!te..,neces.8~' p:.ep~~t;on_ to.te~oh the~8~'bjeot sat1.e ia~tOr1l1:"·-.'inIElMit1on-to.t:l'leait' ,

:;:~~::~ .. ~~. ::;t:;:d~::.t::: ~::~::;~n.; ..::.

>: . ~::t::~:i~:;:~ ··;!:;:~:~~~:·:!.:t~~: .

" t~basB_.ten.th~' :010&~e,:pro~ert;r._.~~Pla:e.~iders-:..

, . . ; \;. ;:~:2;~ :·:::::~~n~~;;:~:i~~:o1::;~:~::' ~ ' :

\ . . -. ,At,Br9therBi0~',ll1gh Bci,hO.o~.-:~e,.~th~~t~os '," .

\

te~~~ero, ~~,t~o~. ~~~. ,8~/le >,~~,:,~~1:_ .8,~en~.~:o~_e·'::~' ~~~~':::' ^{: , ' . :- ,:}

!'sc hoo1:,W1 t ho n thav1na:mas t er lid.DlIltl.y o t.thitbasio'o·oonepts·,: -.';

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,..

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t.

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and"8k;'il~",d1'-'-~1~~~~~ChO01'Ilathematl-oa;:'

It

-APpe~ed-.

;.~~ ; : ~~.~~'~o~~" ~:~~ ~~'::~o~oo~':'~:Mc~en:~er~ " ~!,~ ^I'

:e,;parleno~^::dlttlcu,ltl es'1n-wOrk1nS:wlth _th~.^new''-''

;.' ~ '" ','m~.th~a~-lo~::~~gr~~.~,: :!_I~:':'.~· : t~~ht" ~t" :~ber_e

^_

~~ght: b~'

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^{. .'}

. ~~1 : ,.. ^{";i that : , ren} ':~uSh,

^{th13:Srades'.i thout}

b~ d~Eiit\flt"il- '·'.'

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'.(.':/1.. .

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;1

·~I ·

2•.

ht, ':1n

theteaohers'OP1nio~,~e^theIl&~

" " , ': '-, : • :" . • •.•.0', • .

9'

reas ons t~r t,~e8~ ~1tflC.a1:t.l$8__?I ' _ .: . •

I j; ^{'b i:}

ohaaroo~^'tastedteobn'lq't~llsa'^ban' ieao he.re

.,I,:,/~"'810Ped- tor:,~T,e~~: ~e$~.:~.~~~·D~.~1~8:;' ^{I .:" .}

. _.', ,4. Are.~thereanJ'trmda.wi t h d1t:1DUltl.eethrough

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~"tour:Q.~l!ltlon8·posed in this·.ilttldy pointedto ,threeIll.in83."l!lu::dr

:cio.b.~~tratl0n. _

^'lhey'

.er~,

^.the

Cilrrl~uiuml 'the 1nstruo~1onl

^the:

b~b,~~

^:llhtle:th18

S~dT' de~t ::;,;a~~·· .~·t~';11>~t·~Cu1t-i'~~" oo~'~ot~ ' .l~~. tb~

' O~i:~U1~~

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tw.~ .:~.~~: t~~~ '~~~~ ~e~o~-~

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d1tf1.2ul!le~endBugg~~t1on8tor'.~eac.h1ng.teohnljlues~so,::

f.~~~ed_':,-to::, ~~~X~~ :'~Vol~~-:~'e~'~1n'8" -~d:' -~~~~~~tl0~~ .

"t heor i es ,;

Con8equ'~tJ.7-/-.i'e:;'1t;"01'·the l'1terature:~u oar r l ed

~~t ~·i~st\.o_:d.~te~i.i'e··the

^.•

~~ ~d

direotionsof

. .. . . ' .. . . ...I

thene~ math~lLtlos,," l ~!, re~p eo tto-currl oulUlll •.les rn 1l1g, and-in s t rUot i on. :It.wilei·felt,that-t his'1J1t Ol1lst lonwould

br~'.1nta-focusth~ rttlonsl~etor.tb.e:-m~th~at'l~S

~uoa~lo~ p~sr~' ri~' .'1n

^..

'.~ e .·~ ·Newt~~;n~d 8~hoO~8.

B~co'n~I__100ktnS"'a~:: ~hS-.~~Ile_:.,thi;,~e"ars";8Of.'ourriCu1W1i .

'le~~;

_{.i ,;}

:

_.•

and··:1n8:u~tion

_{• .........} ^•_C"..•^.

an -ait~p~·1f88 1l1~'.

_{'. ...;... .. . •}

·~

_{. .}

' ~ete~1ne

_{• ..}^.

~ro~",t~e~l~z:ature,..~~ ~U'~l~ult1'8Sarose in:~eaC~1n(t~s-""

~e.:·math_~.ii08'-,1nthe:~lemint8rySOhO~18-~ ,

'.'

,~:.:., ' .~~ith:>i\e\rer~~.,.~:~~~~~.~ :.~j._ttl~~,r,~f_tle~_ .~~~ .

~i%~ie~.

.on

-th~, ~teilch~,

or

,~ t~eIlIat~~,..{N~o~18 ,,~_9~~)•

.the"re"

,d~,.e1~ped-

a

-PhenomM:o~'t ha t'

b 'eoU'e ' ;mo n as-

the._"~e.

'. i -' . . . . . . .. . ' .

llIathematios-,

~~ul~ 'e:Z:Pla1n lt1!1l1.e~~ .

indeed, ltlfOuld be very dU'fl0ult'to

.~Jv~ ·a p~e:o.l.se e~18nat10D

^{of·the'newor}

modern matheio.atlos. SOwever. Bome1.nslgtl'ts to the sUbjeot.

~fu~ Prob~b1Jo

^{be gained}

:rro~.

8 revlew of some,o f the

~1llI.s

pr~8en~e~{1n the'earl1e~stages'811d the11ew'd ire ot i ons 1n

.

^{',., \ . '}

mathemat101J.eduoationwhlohrollowed.

. .

.

^. ' .'.

....;:'; ·CUrr i oUl UDi:".._

!"

,-them~:' ' ~e~ '(1960,' p'~ ~:h Jl~~tatn'ed' ~,t'- o~,

^s'choois

~

~~~~, pre~~o.~, ~ear~ b1:'PCI~tp?~~: t~e teaO~1ng.ot

^"

m8l11important8~bjeotson the groundetha:tthe1'aretoo,.

dirfi~ttlt. ~ " pont1il~1p8'

^the

.~gUll~t B;~er 'tlger1,-rstated'l

.Webeg~

with . ;th e

hYP'cith~iethat'&n7:8~b'eot' .oan-be"' t aught'ef f,eo t t vel;r 1n some1ll.telleo:ttially

honest.form· :toany, ohl1-d·at811T,staa'eor . d~ve pment.:,It,ts a.bold h¥potbestsandaD

eee tia3.on e m,·t h inking about'th e nature of'.

o 0ulum •.Noev1denee·ex1BtB~,to·_oontradlot. "

1t,'oonslderableeVidence.aebeing amassedthat.' supports,'1.t,(P o)))~ . .

htbb'(1959)'~aS'"'.pporentlyoonvtncedtha't^-the

,:,,1 ,. " ,',I.",,' .._.---'--"-'-:... ,,__....,. ch lld should,b e tntroQ.uoed'td"somesophistioated"

.

;math~~'tl~~

^{!it an}

'e~l'~e~ ' ~g~'. ·, ·~h~. ~~1dl .

Ve'need':experlen~'es1ii··~ur:8~t1:·tOietl~'c i as se:s

':.=~~:m;~~~~~;~C~il:m~e:O~~~, ~~:n~t/~ '-<

:

'i.~sp~ee,'.~.d.e~~fl'ences.,.~1chwl11,.lend~.

8,

p. 1eO )! ,~,?

,.~,~dd1t1~n ~_d·_m~lt~~.1.108tl~r.~ 0d. ·~~~ .-~1.~~O!~~.~! ',I~he

^..,'

ohlldrea.ofthe ooJllDutatlv,enature of add1t1on-and .

~ult1Pl;o~tl~D.

^{,·.I t}

~~~o:'suggea l;-;d

^the

~e

^{CIt .·bo:!:'&sa}

p18oehoi~er, en: ~~auotlod ; ·to :.~arte~l·en· oo.o~in~te8.I.\b~

^.

8&r11UB~of-8e~',:~d,thee'oon~e~t__;Of.~ot i.o~.';'Lhe,·report.

~~eoolllillendfll;l :'~;Wt~e~~'h:,'C.~ld-.ih8V8 '~~un~f ~'ppor~;tJ.' -~

m.n1P~.t8_·8ult8bi8 phis10~1·,o~j'~~·~'.

. , . :

,

· "'.' C"br1~g' ~~por;

. _ ; . tedthat,ingr••••

. , .. ····1 · . ' , I . ' . ,.

, ! 'J

th8lllulves to't hehlblt8otllathemaUoll1 tti1hk 1.ngi.order, _swHole~d-lt8parts.,'end the oOllbinatlon'of wholestomake.anothel' .hole~ !nsodo1nS,"let~.ke8Pp1earl1"1n mind the oritiosl pointsot development,or' mathem,tlo81underat8nd1.rl8(P.'1)7 ). '.

. _{~UlIl 'Ad;oo~t~- ~t ~he}

-Cemlir l M e.

Con 08,~ , . _.':. .

Inan stteuiptto'deeorlbethekindof ollX'rlculua :

,' ~Vb~:~:ed -~~ ta~·:a~~~.t~s ;·o:~:~-t~~,-·~;4~4 .~;.t~em.~10'~ .'._ '

.:..-.:::,()

progr~"'~'~ .b~ler.lOo;.t _t,h~' c8m_i~1~~ ,: ~~rer~Ce .~r· 19~j.

( . . · ft:lt ~~~:t;.::~,j:S:~~j:;:~1!~~~:; 1~or~~

.~r~l..d,~eOOlll~':_b:',.~h~., .,.e~,~~:19~~~';"~lf',d~,~~,_~,~1a~~_.'8.~~.:

"" ; "" . ,

~llIa1n tlJ 1ned_tha t 1~ 11~8 ~S!lbleto move·to p l oa d?1U1·t hro ue:h,

the grades'by'en'slI'~rg~nlz8 t:lonOf.the.8~bJ e~t~.tt~r

and .

'by

8b,.indonm~~-

^bfdrlil'

rci~; drlli'8r8~~ -

^(A4ier;

19~61

"',,':,:,

"

i

W~~dgo'b~o~d'thie'~ori'OlU~ l~~',,:',hOwn er,'. nd- . IIrgue',that'the1no~u8Icn'ot8uoh..·.dvanoed· ,

- . ::~;p~t'6rii;~d:::-:~~:o~~o~o::~~;t~~ult :

,

1

,I

·to t eaoh, but sota.}.l,:r••klllll iteasler.to.~e. oh'-

~':e'i=~ r:e~;rm:::,:::l:~l~~ti:~d .'. ,: ':: ' ;

un~.r7~~.t.,~e~~!~dren ~~e. ·~.l~arn:',~P~ ~8,~l./

' . ,

- 'In-.n~ttemptto·exPl,a.~'I.to" paren~.,~. ~~ern lIl.themStiOS,

r~~·1l1t .. t~le.rn1ng '

p.etronlll'·U?66),.

S . Jld'

~sent1a.li ~

^..

th~.: ,:1t

^lJIlpl1es'

·a, ~e'crg&rl.izat1tl

^:.

.. , .,:n

^f

d.~~~:~:i.~:'l;::;'~;.: . ~t;..e~~~:::o~~wa~~n:,~

^....t,fi'.

.;

~::~~o:et~::~~ko~,~~~=e~:~~.,:~»:o"ep.t,

^,',",'

.', 'lIla t h' a tl CIl'88'".hole~. Ita.••inobJectlve d' , pr1m1U7 ooncern 18 to nave"t he stlldent Wldei8tand .vby hedoeswhathe does. Ita: 8.Pproaoh1s kDeot

~~:~:~' ~='::2 ')~

three,'tosix, Pup,:,~.'l.~.mabout the~utatltllll"

~s80·C~IIt1ve,.~~'dl.,~Z:~~,~~J~,:bWS.an~·th~~~~~e~,o,~~.' slghednUlllberll,,Othe;o toplos to;be t::*uantinol uded r~t1o.nal^andlrratlontll'~~berS I p~~ ooOrd~.te~- ^,.,/

:,eotOr~~ . trut~ t.ble~ -'" O~.~· s~~~ces

^{.an d}

.the:-soi~tl0n ·~'t

'.~,~.~~l,~,eq~:~lon~:,

~~'~ln~ {'th'th~.".~eof ^... :~ti ^{•• •.} r J.~t ^.. t:' )ih"'~·

, ~o~ ot SO~~~10~.,;1iit;o: ,th~:lO~~~ lP'adeB~. ,

^•.

~t 'dld·'.th~~~e.1f

·::~:.t~:~t.~~~< '~~"~.~: ,~~~ve \e~.~~g~;,~.' ~~. \f~d .~~"',~:,~,~' -':

:;!-w,sr~,~III';ro t!'lllemO:l,A.u.onc.t.1.ota,:8n~'.\P~,l,~~on, .,:"-:'

"'.\ ,...- learn1l1s:~1:un~~B~Il1tts:. ~e.~~1 t~t,thi~•• •.tebe .

,':f

.8oooinpi1shed'1I8Sbr'

.truoturlng

^{"eneoontent/so}

' - that

- . ,," ..: ,' ."': ' .r. .'.'.'"' ...-' ; . ~'-

re6Urrl.n8".Jl~tterns.w.er eU8,~.to. un l fY theIIIl~Jeo t ~··Ad~ er

I

^I

1:" " /"

,,,,,l

'><. '

.:.10

m~thod.,.,'1;bU8 I"&no 1;.h~rped.gog1cal.pilro~Oh'rrom:muah: eri~1~'t 1Jaes'tuisn;:~~h~e:V:ed',re8peotab'll~

t1.,

.:)~ornot

o~

^,

~~,:,w~~1~~,~:"'a,~8e:8't?~ ''th18' .me.~~d '~: ^h1e:

t~'t~~Ok,'~~', t~1 ' ,b~~:·~~~

^..

~e.~.~~~r8

^{:t braUSh.}

~', ~e& h4T~

~8en..,gu,1d1ng,p~p~l8

t;o

^d1aoo!er.things,t orth8lla.e~,!8~

\" ~, ,'.. " ',''"' " ,"" , "

.Bes1deathe,&IIlPhla18on·the,8truct~e

.

·~a~.h:em~'~~~~·.', t~~',~tt~~'~:.::'~'.~~a"~~~~:~~~';~·~d~g,' ·~l~~,

^..'

ene~~,~ed~

.,'

~e

',: ,..

, .d1.eO"&~~"

^{, "}

ap~ro·.oh.-.

^{, . ' ,}

' ~ .

^.7he.

,m~~', ~~a~h~~

J~~eS:"8nd coxto~~

^{(1970, p.49)noted}

thl~

^&8

earl;r'aa'the 1114-1930".8.:Brownel l"had bean ac1'i'ooaUng an

emp~1a

^on

undejfabmd1ng

8&the ke;r

to'le~rn1ng

' i ' " . . '

tllithemat108", ;,Bro1mel~'8ad:lo!,,'th~",._8n~.~,1.nd1.rLgw~~e,.

aeoep'teno&--w1thin,t h&'t r u eworkot,the

Jew

math._Ues, .

~~~'~~t~' '.~'..

" " , ., . ' I ' .

.

, .. " . .... . ..

^{. '}

^,

.J'~

^thel:lllUolt ••Wee n r:(1964)-

n otJ

l

. .

' "

' " ^'.

I ttin"therefor etelt-ttut t the dea1gn or lillg1ft.the'prob l e- lIequenc8e 1IlTol Ytns:

comb1nlltlonsof routineteCM.lquuend

~:i~~::i.:~~~:::r: ~:r~e:i:et; - or

develo~ent.(p. 209). .

.", -.. .

. -,BlCImsr£

. ~: ".:,".: < .

_,'th e pu.rPose o•

~

^{..'l' :.}

^{;';:: ..:..1',,,}

t_..

th18 ^{-; : "} ~llect1on ^{", . . " ..}

^{0. •118.-,to:.br1ng. • •tnto_:·....;.\\" •}

..::;

;::~~t~~.r::':;i:~:r.::'.~::::~? ~;~~~i'l::;.::;r::~ ..

.: ;11t~r8:ur~,.•b~~,t{;,~he.l l~~.and:d1re~ t.1o~a·~f ~he..rie,'

:: ~ , ~ ^.. 'th~_~~t~~~ · h~gtu._~.~~t~:d d.~~i~:~~'.··~~~i .0.' m~th~~~.~Cs

. . educ ation thstMerged during.theputtwen t,.7e8;a~·..'D'Iese' trends

~8tg~

b;r oon s.l d e r ing:thre easpe ota-o f'the,

~~~

._

m.th~8.t·lC8 1 c:u.riCPlu:n. 1~_8~1~ "

^an";

l~lItr:-'~~l~~. , '

, - ' , '

; ..• t .· .·;.:~·

i l ' .

. I . . :~the '::r::a, ,~t,currr~~~

""

^:rendW~8

^{Jto "" , .}

more or ttte·r.e. l~•• thematicsdo wninto't he lowergrades. .' .

=::~~:Ti~i4: .,~1tiE~ ~ .. . ....•. . ;

,

I·~~the;:t~n.· ~~~'·1~~~1~.~:

^{,. '.,';- ., . .}

.... or·the.contentwaa1ntended.'tOun1,fythelJub'j ectand, thus ,

? .. ^~.~. l·t .~~i~/to· '.er.: ' ,; . .. : ' '...: ,.1 . :: .. , " . +.'1 ':": ,~e : ~~,tr~,~:t l<l~.81 ~tec~1Q.ueL·~~ t. 'fa8. Btro~.g~1

^.

.'. advocated. Jlss'thedhcOVBrT'meth od; J'Hl!Ir,B,theBtu~entwaa

.::;,':~'I'~ ^, ~. ~~.': ~Ot1VB~~ ,:n~~~~~d

^..

"it~.:,th~ ,~.\iP.~.{~'i'~on Of, ^." '~e~ia~~ ,

, ;.. tdea lg",ed

to

help h1JDdi s cove r certalnfecta for'fi1JlJ..se l (.-.

.. ,. . .' . r/.. .. · . .... I '·..··· . ' . ' .. .. .;

; f . ^{.. 1; , ., '. '- / ; , • )}

,~;.:~:;,:;~&r~·::;:·:··::'!;:f'.-:;~·_:.-,:' :~;;.f;:..n.,·~l;'~~ .~, ,~:,

^{. ;..;}

._:.~

^{:.. \-}

.

,.' , ',' .:" .-',' , " ', ; .

. ..;"mg g1ns·

'(1972 ): ,OatT1ed'out·an ~:Vlul t1g·. ;1on

Pf "

thee:fte~ts^ot non-de01ll81i:tum'~at10n1nstruot!on~' ^He

.oonoludedI

,,.

\' lotll

: \ U:~ ^as .~,~~ ...'.",. an~ ~. ~.o:tJono ^{ore•• '}

i~

mathematios

.~~: r.e~?e.o \~ ~ O~l_~ul" ~ rnlns;

^alJd

4 trllot l on , the

1Dve8t1~8to;r

ag81ft turned-Ito the

~l~erature.

Ris pr1mAr7

81m\ .8

to

deterlll~e

^what

^{I ...}

I r '

^A I

~

^{thematloseduoatorsoonsldered}to be prob ella ar1aing 1n e

.\ \:1.8.

_\, ourrloulUlll. However,Ihe\also_{\ J}

80ug~0l1t

^' _•

d SCUl!ls10bSordiffioulties1I1th thelearn in~

'\ ' ,

\ . .

~.~~~.~o~}.~~:r,l~^':o~';'.~O.de~^..~~.~h\:"tl~~,^{",.:,.', ",}"" "~

t ..

. . (i .·, l"il1~. e'. 1nV8.elg.to.·. Prol.?~ .e~",:.e .. _~g.e ..

: ~t:·;:.::.:;o:~t:.to; ^{• •} ::;"::~!t: ...::;: '

of

~~~ ~lt'er8~.e· r~vealed~~~r th~· ·~~~'t'P~~ ;";~·~.~:~~ :,,;;;'.-'\

_t.!'1a'wer e-mor e sensral,lDnat:ure. ~~m~.re.g~e.r1l1 ~the. '

"

,_' . .r. !

d11'tOult,tN,rep?r~ed-b er e1DVO~'v~l:arn.~an~~.~ tru_otl~~,.

I . .--:-c~,c"--'

^{',. ue, ..}

f~' •••• • ,.0".'0,0...10U1.... ,ffl?t

^e'.". ,. ' : ,o..'

" .

~

' _. I

(_ . Non-neo1JllIil'NUIIlers1;1obSt8~~;8''- -

.'lklPihg,to1.norease·the'ch-1ld'&'tUld ers t arntt.n8:-'

oi

our:base

,t~"~umb~r:8y~t:em, '~·. oi1rr~.~~~:· ~~va~r~

^I,.

~tro~uoltd nttmber '&7st~s other'~\as ~ t:en . ;~hr

^:(1966,

.

'p·~''- 84) .~1nta1ned· t~t~ "Wh:le',1~:'8~~ed i.~'k;_a ~~~o~le

.a,pprosch;·.uper1enoewd'P~rlng_1t W~Dl t 1fOrk~•

·'I fread,inli-sk11lsinstruotion-t e.

giVen

~ur1lig mathematicslnstrtlotton.oer tain readtng skills learnedduring-·thedevl'l1opmentalread ingper tod 11"111be,reinforoedendlearn ing .Ul,more l1kely

..,.: ' , ' .

teachersI

bloOk'~'shestigge~ted,ocours beeeuee't hei8nguagl'l:"8 enten~l'l.

:- . ... . ., .. ' .:I . ' . . .. .

struoture,lUld'wcabu1&r1become',Iloreoomple:z:atthatll'lvel

- -__• " " :••11 " ", -',' • '.

and'thedtrt1o_ult;ywi t h problem solving-es s en t1 D:llr

Orig1nlit~~

^'as'

~ ~ro~1:~"l~h' ;~~

^•.'

Sh~f~~" on~, inte;e~;t1ngsugse8'tlon,

^{. :} I

0 ,.

·~~ /~ " " - ",-,-, ' · :.' nr ¹ .~ ^{' : , .,. ...} ~' ': :; < '__ ,,"-,--",3 :

i r

~ 1 _~

'!hedata:t'or'lt~1,8'BtudJ'8uSs:eat.that,tbe .

1ra.plem.entatlon;~tncn-deo1ma l numeration as 8.'Vehiole ot,l1nStruotloninelementarJ • mathematics OUl'l'toUlwil seemlJ·-:to.haTe

.ec

effect otlmathl!llllatle:altmdl'lr.l!ltandtnS"or base ten

(P.296). \ .. I .:- ... .'

.

~' .

. ,'!he Naull'alAdvlBOryCOu1ttee,,9nKathematlcs

~ ?a:t16n

^...

nl~_~~~-E>'.\ ~e~,~?S)

^,

8q~~';".:: ' , .

,,\ "Besearch.'on

&;-lthm~tio

difficuitles,M.10ultg.

'ohlldr en,lI ho " ffthatinadequate UIlders,tandlng

ot,our ,deo1malpl.oe~lue-numeratlon;_81stell,....

18':one'or,the·iDlost -cOllUllon barr~ersto'1JD.proved~ oomput at i on·,(p.',19)._," •

' ·~B~~o/6 .~~.~~_~~e~.>':~·_~~-",.>::.~,: ~ _:',~>~ .. : ..: :" . '}

" ,,'·In.lI.~de~.J!p.)h~'tlo~ p~,~~,e^•._~(en·o,h~~en,:~

the

pr·ti~ '~r~e~ " ~"~~~d~Oed" to·:·~:id '·~~bl6118 . :

~~~~r, ~e~p'1t~14 ' e~_~- ~~dUC~1~n a gr~~'~':nUlllb,e;-

^of

8tud~t~^{ttaTetroubf,El•.}~t~w0rP-'pro bl ema rl,8ht,up th:fough the school.system. "..,

' . Hennl!Y(1971.,P.,224)'ma~~l.nedthAt althftlgh

~~ldrenwork wit~~8 to1'1problems-

f1'om'

k1nde~88rtenon, the

a~1?~~ bl,Ocki-,~~th .~~~lem 'B~l.~,' ap,~eh8_~,t ,

approzlmately---thefOlirthgradelevel.";'lh18stUDl.bl1ng

,- " " " , . ' "

.4

oneproblemmentl~ned^in.theliteratUr e.hioh18

q . ' . . '

I . .'

notspeoifi ca lly III currioulumpro bl em'Is thatof/t eac he r preparation. aoweverf:the,investigatorinoluded it~.

.~.th1~t1on,bSOQuas1~1sa probl"e1ll1!h1oh"a8d1reotlY

rs.latedto_the ohange.Ul'.the.CIU':r10u!'um.··' ,'1henew'prosr~e, .reqU1~ed teacher~.~h~we~e t~1.11.arr·ith·'~~1r"co~t~tand' structureinoi-der to ensureth8tthe mathemat1cstaught 1nthe

sCho'~lB: lf~ c'ons'~~~er:t

^withthephllos opbJ'of

th~

new

~ath~at1.C" .mo~SDI~tL,~,~·' ,~~i~~'h' (t961IBtat~

^itl

.ADl-o'Btl8erioua'~probl~ It'~81'andstUl·1s •

.£

.

~:~~k~:i1:;::8:;;b~ ::;~:c~~: . . ~,~h

ohalleng1ng and excit1ngc],assroom_situations.

Inadd1t1on---tobe1nga good teaohe r. this-- requi res substantia1.8ubjs ot.-matter .R1as t er,yt'

~~:~t~~~ (:~~~92r:^{lIlost,oasss.}~'ool~6ge ~ .

Teao~~rs

ⁱⁿthe

~lementlU'1" SOhoo~s

still

J o n~t

TeacherPreparation

. . .

be seenesrunot1onal(p.22S).

Onem"ghtwe1l8~kl ls'theab1l1ty'tosolve prOblem sagift thati0n.l;r o~rta 1b mat~em.~70a1lJ'talented stu~enta.PO~8esSor.aret~e~8- :t'a~tt80hnlqu~that teac hers,.~~ UB~~^{deve l op.problem}~olv~/l!l:kl1l8¹¹¹

:ch1 'ld reni.i" A,' ,'re v le. of

th~ ~'lt;er~tUreShd"resss;chbl early '.

1n.d1~~~ed: t~t .'t~~r.e;, ~e ',.'t~'~~~q~~~.': ~~l~.~le: to·_ te8~.~~r8

^.

. .:' wh ich

:hev e

p~~~ sti~~8SSfUl '~ r08t8~~', Probl~1801~

~k;~i:8 .' t~r- .ex·~~l~, -POlla·'?(~.9~1:i-. nilim~':

^(1970')

~ , ~o'.~e~

^.

Ci'972) ;, ~81~; Be~r8-"'~ BI0~~O.ri,' (;9"5)~ , !~~e1r~':' -':'

"en dOd 'tl~7S) ~

-:

i -

,.'-.

i

I

....

.

. '"

15

receive the assistance ne08SS817 to prepare them to teaoh the new mathematics pro'grams eftect1vely. Green (1976 ) carrled out a studyto Identitybl"Oadareas0: concern 1:n teaohing eJ.emen.tary mathematics. Fromhis stud7 1.l1volT1ltg

;~.a.pEro%1m.~t~~1700element/~sohool tell~hers. G~~~

",'?"e oncl uded ,

.e ;

^1. :m.l!lllentarT-!'!choolteaohershaveV8r7 r eai

and _

genuine'-

ece ceree,_

and".8llTunanswered

. :~::iO::t~=:~:::.o~~g/~~rn, el~.~.tary . / . '/

t;.,_ ~:~~tC~~lp=::;8oi_e:;t~g: ~;~p

~t each1ng,use,ot:_t~tb o_okB_-endmaterials, and

)~nt~;s~t~=~~g:~::n~ie~i~ c~~e~

and inservice tt'81n1n8programsareno t meeting lD.any.ot the needs-of,e1 emen t u ,y sohool teachers and.teacher trainees (P.1(2)"

. .... ....••...•. . :. . . .•.•. . / 1 .

.

•

~

.

/

\

:0 &;:::ei fo~h~~t'o;~:ttl;:i-~~~: :~~~C:e8 ).

st er n e 4rtthmet l o:·,.B;v:this term,-Im.ean arithmetio,as.the pw::e eeaenee-ofJ;l.UIllb er . taught withlittle regardfor the ability ofchlldrm to

learn

it01;<t or,thesooi dpurpOSeS,:tob e serv¢. . 'Ibls'd.8t?-ge r .,howe'u r , istUll,ikelyto eeecee-a reallty,for reec eearea"vaile.~le wi,thinthe-e enec f e to counteraotit(p.44).

statedI. •n

' I

,:. j

II I

: ~ - ' .~~",~

)

'.:Inthis'C:Ol1!l,e~'t1on, also: ; l t l s'rt~nSense'.to '

':;~~~:~.~;~~/~~=~;~~:m~:r~e:en~:~:~~~

^{fUtd "}

numerals are names'orsymbols fOr theabstract,

:~~~~ep~ral~~:,:;:~::~~l;n(~i;:)~ ln

^.

" ':'Will~ugbbY(1968,~.

i»

doubt~'that'S:UOha

,cons i s t ent"

an~

^oareruldistinotion

'b~tlreen

,ob j ec t s and.-their names really-e er vee au~lefUl^purpose.

Hen~(1971),s ugges t edthat qUite often,t~e

• " I

souroe oraChild's dlfrtoultles with arithmetic .1s his .Febio'

"~1966.)

_"

' reBC~~"

_••' ', 4 " "

~ the di~t1notl~n

, ' . ^this

"a;y',

16 One a1m of,t h e'nell' proJeots'lI'as,t ha t.,t he

. " "/ : ' , :, . '..

m.thematics tlttlght1nthe schools be:,~ oo rreo t.. lIl.8thematics.

ConsequMtiy. most.n e. progrsm/t

re~o~end~

prec1ee use of mat~em8.t108.l l~elind.s;yJl~I!..

Pol'

empIe, the distinotion between ,,"umber (the ldea)andnumeral'(the~e orsymbol for the id'ea) "a,s·vert popularf~rquite some

1nabiHt7to'read,t he,notation. Hes~tedl 'Ihqs71llbols"thatseuee inmathematto8 to

::~::~:;~:b:e~~t~~:ena:: ,~ :e:~:~ ~

various relations dr,eeeneaee, effioient, end usable. But 1f a ohild.cannot,understand the meaning of these s;ymbols, they S97'little .toh1m(p • .3'7).'

\ aeneen,(1974 ,p. 2.36)maintained'thatthe .

exceBs,~'ve us~

^of

~bois

1s a sou'rce"'orthe dift'1cultT

ohildren often- have

_ith

sets1n elementary .!!lathwtlcs.0

'!he'NACO~lE'B~port(1975, p, 20) statedt!i8.t.In..

reee,~ost teac_h~I'Bhave:re~~to a'~UCh.mor e modest

. I " ,

, .

"

l

:(

'\

\

Bo.teach er oan .nordto ~aked1soo!f1rrher slngl ell.thod~ r-teach1ns. 1hereare-ra~ts thatfI,ve17 ohildlIIustknow.andbr 1ng1Dg.this : 1l'ltorllll~.tlollto .·ohild's.ttentlOn~9-01"equlre . .. . d1r~ct1ve~~eaoh1ns~ro~.ed~e; (po.2.~~.. ..:'. ' . :'

. ~

^{.'1'here1s avon a}

~or. ~ha~ 'an

^•.

O~1Y1t7

^'.111.

~

^.

beccee~endIn'lts e l:f' w1ththe li t ud en t obl.lv1ousor,··thJ. .

t'aot.. t~~\f1: 1P 'S.~POS.~

^{to be}

~erlen~1n8 '1i1d'1gtits ~d

^.:

.~aw~ oono11lll1o~a: M:ler(1966 )thererore,empha s1zed.

v- ~

. ....

~.'-.

- " - - ---"--

- - - ^-- .'

/ .

elIP~18

^on

~boi.

^mid

>'~re018 10~- ~~ e~re8810~\ '

''l'hat•tbe·"unda e

.~PMa;1I

^_

~n

e"to;".i.

.;proaoh :8:

stW a Tltal ta.ue1nel ecaen ta%7.sohoOlaath.atl08was

fndlCiat~.

^bT

SaPP-?Ort

U976t"t' "'-.0 , ",' . ' : ." . . . :•

':thema th__ti c s currioul~'8bo~dba~e17 ." .

. ~~~~;:. th;n ~r=~gr~~:_::l=o~:C:~~: .

experlaen t l,-to,trf .o ut"manlp ulat l Y8 denc. s"I.,"

and-t o leAm'.tooopewithreall1te:'d taatl0ftS.,.";;-.•

. ~~~t~: ~~~ .::-~~~;·~/:r:t~~lDl~ .~

,",I

_'"

:

:~~~;:;:~p~rmr."g·'_~~IlJ;~~~.~~,;~~~~·~·~8 '·"''' . .

.'.

..

' . - ."," ', -,

.

"~'." , "

~~t).u~.t1on

^'.

..Probie!!l~

with

th eDisoov erY

A ppbiao'. " .' \ . ' ".

ModemmatbMl a Uee_{" ,}'

~·coDra8:e. . · "

^{t ••}

oh~~.

^to'

UBc~ ..

^-the'

. ... .•

^,'.

d18oover,r·approac h. Indeect .IUlDJ' teacherswhOhaTe

w.ed·..

·.t~8. lIethod 8~O&~~8tu117

^{oan get}

ven"

^e%olt"ed

.boU~

^'ttas.:

. . ' .- J ' ;' '. " , .

one.ll e thodot'teachlng. Bowevet".as1Il8n7eduo~torshI1~e

Obll~rTOd:~;ther.^Arep~bl.ecia "1th.1t~^.fJa: bfl?..l1tI'e'-stu~~~t8 I~pr esenJ;oo " " ..otl!~tles~llD.d.-~deas'_~~~

n o:

~t:fI.·:"·.;·.

~t,.the]'.1lldlsDOver th~ 1nt~ded. OU~fI.o C?rle'(1964 ) .

I .

~ .

, I

.

, '

._;.:

'.18,

. ,'

'''m:l!Icov err'intheclas~~olD. m~tbe guldeeld-lsoovery, and

.ott":

^,t

eho;"d~' ooop;'r.t'~. d'e~ov.r,y··(p: '84>. '

I'

'?per~tl 't;he :tea~her :~oui.d ~Je

^ttiat"

^{the - aoti Vity}

do~s.notbecom'~^:0:i:'D~'tn:,ltS~lr.b~ ~a;1t1ng^{uSe of}

pehaV~~uf.~Abjec~~Ve8

^...

:'~'

^{SUch a}

p·r.~O~~·C6

^would

rore:e. th.~

ir

;_~oft~_~."

~ !?1~Jt~L~~~~

tor'.,~1tf~r~t: 8~~~:t.:,~~_1.~.Qe~~_~.~e.,re~o~~..::",,:,:"

.~~:{~·. :j~i:: ::~:~;io~i:iitf:.t.:t::g:j:~ ··

: ' 8n;-~~k16:~1~ ine~~~ '-1n..··d~.~1.ng '~1

^th

the m~re 'oap.b~~

^i"

• ' , '•.'-:: ".:-'; '1 ,'.." ,,": " ", .... " .. ,.-, "

,s tu dents and'.toemphaslzeo,?lllPutatl~hal8kll1 andprobl~s

~·r·'Jv~~·.l1re.·andspeolal1zed"'~ad~a^with

ihe

"si6~'~r'

. , . . . . " { ',

~ "

..

~'ff.~:r,en~_ ~r:t.t~t"

^-rof:

d~.ff,er.'~nt _~.l~_l~ . g~~S

^-.

c~~m'i.i,

^not

re~O~~ded~

^by""the"

'e~ly ~~cat'e~· '~f

"-:

m~~e~· ~a~~~~~~s~<'~'~proh~~ co~eo~" ~"~¥~at~:~~ :·~d

19

To..prov1d~^{to r'}'·~d.l:Vld~ d~ff~~~Op~.Johil~on (196 7..p~ 188)8usge~ ted's.'91Ir1ety ofaPprqaches,for . .

t~aOh~m8them.~tl~S:' 'lh~y' .~re l "

¹⁾

" ~~~~atort

^{les s ona;'}

2.

' L.e~~' gatn.asi' ,>'.~ ' ~~~11~V1;~ p~es~~~~o.ns.;":.~:

4• .Cl8psea..dlr~otedby,t~epUJjllsl 5. '.Dl a ooyery l es s onsl

6~ El1rlcbment lessonS,·~7; indlv1d~lzed1.nstruot), on. . .Cop1ng

.~~,h: t~e, 'varY'~ ' ~1:1ty ?~ve18

^of ,tuden"'. tbin.'1,','.,arl>'1ndlcat,edln tb,~1t~tUr.·"

J'

:~px-o~iem1flth>irhl,oh^{:t he}~lem:entU7'^scttoc1'ma t hema t t os .t eaob er ts'd~f-.1nltelY':raoed; 'KhO~ ~ha,t^..mater1alto

;". i...

end'"""t

to.m~~ ,";d·Jc'~:"";' · ;to ^{, ' A'Q} t~a.~u;.,

preparationfor an,.,vo o8't l onthat ueee

mathematics.,"! •

W-::~~:U;:B~'t~:~ ~;.e~~~~~;;::t108

ci t izen s h i p.

5! Anynormalindividual can appr eoi ate eca e, at least,of thebeauty and pOwer or mathematios, '/and thisappreoiation18 an important partor a

clnl lzedperson'sculturalbaokgrolind (pp. 61 6-

'617) . ~ . '

!,' _{-- ~} 8d'Vl:?~at;8 " o~

the.neW',mat,helllstl'CB-

~rogrsn::s

sugges;ed.'t ha t

'th~ ele.m~tei7 BOh~,?~', t·~~~he~.·O~~d. p~vlde

.

', ~or_ .d1fr~r~~ , a~ll~ty".:'le~~;~~ ' 1h ~o ' ~8J'.s . < _~e;~f1~st ~~.'

-b1 se~e.ot1ng app~p~J,.t;,m~t~rl~i' . f~~ 'a~<Oe?ta.b1:, "~o~~ent

~eQ8.,

.'lhe

:seoond'.lIaB, to,v~.teaeh1.nS:oPPrOsohes'·alid

...-met"h Odol ogy•

. .

G\~;' .(1961'

ma:nta1nod'

. 1 ' , I '

Provld1nga d1Cf erentiat-ed curr iculum for individual chlldren,ou:r1es-wi thitthe responslb1l1ty oflllek~'appropriate seleotions orma terlalsandteaoh~methodaBu1tablenot only.1'or d11'ferent'age 1eTe18'bu.talso.,for d1f :feren tma 1;url t1_, ,l enls'. t thesame .ese (P. 21 ) .'

", .:~

.

.' .,

.

20 strategIes_, presents the

.

teacherWl~h

.

.qul te8ohallenge~d aneedrorlataor aselstance.

.. -.

:

\

.

'Ihel1teratur~abo ut the dirr1culties~^teaoh1n6

I

oe l emen 'tar7 l!IchOollllathem atlc8.then,ror the BOst pu tdoes

no~ ' 1n~1Te'

^{apeoinod}1r.nc ultl6s}

w~th

^{the'content area.}

~ere· la~ ·ho.ever. :qu.lte

^a

d1scU8~10n

^or

Prob,i~

1ri-'mo r e geber ai

·~!,rms .

^{;'.,,, /. ' " "}

.W·1thr.e·8P~.pt:^to curri oulumi'one spec1f1c{areaof

d;lr;I~t4'trnlen'~l~n~d lf~

^..the

·zi~~-.deo:1m~ num~~tlon'S;y~tem

^•

. 'Ch ildren ; r@~aren'tl:r,'^are~V1ng.'ltroUbie~ork~^.in-ot~~r

bases ,_. and , con_I trBrJ'_,

to ,w~t

_{. ,}

~1l8

^expeoted',_. ^{llt does},not l!IeeID .to~provework wIthbas.eten •.Another,speoinc dtr:rlcult:r

inth.earea or currloulUlD 1.nvolve s.worli~wIthword. problems. '!hlseeee e ee lIo reserIousat the

grad"e

rour ievel

I ', ' .

and 1s perha,ps related,to·rea d1.ng prqblems,. cee jlroblem that'16

d1're~tl)-

re1'a t ed.

'to~ CUrrlculUII f~ Iteac~~r

pre.parat1on. ~'..moT1ng moreSOPh1s#~eat~~.. the:llatlcs down 1ntothe lovergrades .'!Iproblem~se ~pr epa r1ng· 7eaCher~^..

wh?,

,ror the mos t~8rt. ^are'not·ma t hM'a.tlcs major e:

,In·the·areaof learning . eproblem:arosew1th the

8t~,oture,

^of,the

~n~~t.'

^Children

llPP8ren~7_¥d · mo~e.

d1ffloult/b~6aUse ·t·he,.

.f OUnd\ ;'too

r~mal '

^{and,"perhaPs ,}

~

^.

. ' :1

wer e no tready ror

S Uih

apr e ol se treatm ent.. ,

:'".:'::,:. / . , cee problem re;ated to ins truotlonwas w1th the

:~ . ~. . _'. _{~_.} - -- 7 _ .. _{~_.} ^{- I}

I '>t~~,,, r~

..

~,~<-,.':...-...'..."~...,,,;~t.t'~1-~~.;t....~-...·~:'•.;:::•..:#...'.:.~b~ .~. ~.;,·'.. ;.. . -'.•.•: •.~..•.~....,: .

~-~-....:.-.

- - - - - - - -

^{.., ..- .,}

---

^.-...,

21

; j . ...

I

tesohers1II'Ould beoome

\

80 involved inthe aot1v1t1es that

,th~y w~~~.

^j

io~t_

^{ab.outwh'atwas}

~'be_A1800ver~.

^{Anothe r}

1nBtruO~l~a1 ~~\em

involved 8l1.owing

~d1V1dua-1

^{- - -}

d1ft:er~oes. '!heorlg1/lalview was tha:t al.l students of_.,r. .

~~ abl~~tleB

cO}11d"'tumdle the BalD.e

oon ten~

^{1f the -....}

teach~a,pproaah was var1ed. 'Ih1sapproaoh, apparently.

1S slowly~e1ng'rejected.

I . .

~UBIa rev iew,'or .the:r:lt~~tur,'a~out elem~ta:7

s~hoolmath'ematlc~:hlg'hl1ght8

th,

ls~~es ~d probi~8that

. ~:~=:~::.~::;p::::1:;~:i::·~.t:~t::·:o:! . ...

ofdep8~ture^of ~hlS.siu_~~S!l^to~~nm~e ~he^more8p~oH'lo

I

teaohingandll!iarri 1n..s :probleID8 associated

.

^,With

. .

elementary school mathematioriin'grades four

to

eight~thr.essJ.?eolf'10 schools .

I

I

I

'.-;_._'.;'-'--,-

..

~ :,.,

....

^{; .}

CBAPT!B III

i~

I ' : .. ' ^{" ~}

^{- As .}..t'on;" in Chaptar

I.'

tOls projeot or'gineted _ froll are~ollllD.endatlonthat lladh departaentinBrotherRloe

.HighSchOolrshould,wor kacreolo~el"';lth the teaohers-in .-1 the

Ithr~e

^{'f e e dersOhools}

~

's e.. Patrick's Ball,s c hool ,'·st.:

I. ^l. ~n.v~'~u.z:e;B 'SO~lt and' ,.~1~ ,cro8~ ·:S6hoO;; ·

^''lbe

r~d7 ' " ^.

,lnvoi

ve_~

^"the

dit~lC~1 ^tles'... 8~·~lated, ~l

^ththe'-

t'~aoli1ng ~d .' le.rri~'g-Of..'Dii~h~~;'C8 -1n: th~~'e '-~8rtl~~lk :elem'~~;""~

^,

so~oi". ,~~1n\_:: '~·e.'~~l~' O~j~o~;'~e ' w~'" ~'_~~e ~, ~. b'e~~lnS'

st worklng-,With .tb~~~;~~~~r^S,OhOOl,t~.~h~~-^on8Ubj~ct.

ooord ln . t l on . 'lbelnt~~t1.SJ1wasto'lIl~euse

or'

^the

~xpe~tlseandexPerlenoeO!t~eeeeenere in an attrp'ttoE findanSlleX's for'th e tollowing questions I

1."What topios at speclt'lcgrade level.·ses to eeue e themosttrouble forstudents?

2 ".~t,in~hl!lteaohere"op1nlons, ar ethem.in

,reason~^for these·.dltf'10Ult1~81

, •.~t,,_Ol~~8room'te,ted'tecim~ques.haTe .teachers dev:eloped. tor overoollwg-t hes e.difflcultles?

. 4.

" Ar~ th~re

any trends-'. l t h

d~rf:1oultlSS' lth:toU8h

the grades? ' .

( re

'ea~{ 1

septeirlbs; ,

th~ ~~estlgltor .~st

^.ith

each cf·thet~esl!lchoolprlnolpall!1 IUld 1!I:1J11a.1nec1..ths

propos~ ~roJsot; As~"ur~·

^hlm

^or:,

^t.heir

·ps.rson~l

~ooper•.t·lon,Itheprinoipals sr.rigSdm.eetlngs J

With.th.ell'

resp eotl n st ar t s etwhlo h h.e_gain expla 1ned:thepro J e o t

" ' J <

andBoughttheoOOile r a t i o n IIIld epprovstat'theteaohers . All01' theteaoherswere 'YOl7op en'snd.'reoeptivetoth e study and.deelled,e ag er tocoop erate.,

·''!h e1nv e stl g Bta r then proce eded togo outto the aOhoo~ s:on,oe.a .e ~ tor,'!periodot,app1"Ox1lll at elya1xmonths ."

FromSep~Sa.b er

ee

Pebruar,y'he

B Peat

olletull4ay inIla e h

' 8 chO01_ en~; ~h~:~_~e. · :1n ~otBh.o.n'~ ,:_· - "," "-'::", ":',", .. " , .:

~1ng ·'th~~e ·~181 U1 . ···~~

^{hel d}

· ~i'ol'll.i·. d;8~USSIJ~

:::2:;:::r:,::l'-;~~~m:.~:;1o~·:~:~:·:~::; , ~~~.t

8ChOOl.~

^•.

'The 1pVe8tl~~tor Ob~efl'~_ "~lV'~ ·~ifterent 'gr8~~

^J;:.-"

tour01&888a. sev en si-Bdetl v e s'" t en gr ad ea1:r..~8,'e ig h t gr , de sevenJana'nin agradee1ghbl. Be.ork·~'lI'lt~'thlrt;r- ^\

.:Illg'h t te'aohersWho aro'>dlriotl1

~volvod

^{w1t hthe}

te8~h1ng. :

of,elemen~soboolll.theutlC~.

InJ&nuarr.·1 9.11" . ques t ionnair ew.s pr esented'·to

.,-

^{' .- ,, ' .';'; ',' .'}

teachersat",e.oh-grtld~level (BeeAppendix J). F1r"st or~lt

, ' " :.. -" - ', ' ' .

thequestlonnalreaak8d'tor thetollow1ng ,teacher

1n~orm.t'lon·l .

^the

n;':tt';~"~r:~ea~8 teach~~_ i1~them8t1eBI,

^·the

n\1lllb e ror years teaoh1llg lIlath emat l es at the-statedgrade

leV"~l i

^_

the

number'ot

pr~re8~~~!l.~

^.

eO~8~

.'(onesemester').

~

teaeh1nBmat~elIl"tiosl ^{sndthe numb}<&:z:-ot,Unlversitloour~~s

.(O~~

^{:s emestrrJ.1.n}

1Il.~'~h.~.tlCl!lt

^"

'.~:'

^{-. .-. .}

Seeond17, theques t i onn a i re11 s t ed.th etopi cstor

! ", ". " . ':' .. ,... - -'

eac h'grlde len3-tobe taugh t dum gtbe,ear~an4asked

I " ,

24 TNIohere'werealso uked tora t e th8le topios ona eoale or 'o neto riTe1ndi oat1ngtheltegr ee ord1tf1cult1thq

expeot edpupil.s'to e~erlenoe1Ii thea c htopio. '!herating:

s~ale

^{used.as118.rono".sI}

I ,

j 1 - nodlrtlouJ.tt ,

2-'

m~~er.te

dl f nol11t7- ol.eered"<m;ve n

qU.16~ry,

J-'

se~loU8'

d'U f iou ltY,

-:'U8~'1T '~~ en

'c a r e.ot', .

'. ~' - ' : ~m.. s'e'~1~U8 :;d~-rncU1~

^..•

:-so~e '

^neve,r

:~r8BP

^at.

, ", ' ", ,I "

this lenl, ' .

~'- ·,~~~r~ e"~'lrn~\1\t~·:~::~o~t:dO ' :n~t .~~8'P. a' ~th:a .

loV.,'" " , ..

, I' ,I ,f,' :'. '

.Am.e~,rat1ng"t'or.e.•chtop i 0was,Ob~81ned., 'I!'1eD

:b7,.l]S~^.,Ill.eflllratln"g"gX"~.jt-uth.in'or ~11B'1to three•.the 1nvest1gatorIll,de.ali s t or ,peoitlc'tciPi Q"".. at'"e a oh grade leTel Whi~,h^{';'OUld.pr esen t,}th~lllol!lt'dl1r~oulty ,uiteacli.lfg•

, " InJoI8~h , i9'l7~',.aecond ,qUes tio riria1r e'vas,

, , " I, '

,deliveredtoeaoh teaoher.(See APpendb::B). 1hls

~~eatl0nne.1~e l1~t~

" ',: '

th~

.....'"^mos"^,t"

dirt1~t11t',~PiOl!l

, ', ,-0;1, ' : ,

1o~~a~h.1gr8de

.

l~,Vl!ll '8S:,d~,term~~,b1',,~he^t1rst,~I1,e~:tlo~.lr^{e,' Tea chers',}

~~~l!I ~Sked"~ s'el~ct' the ,: ~ 'III~S,~ ., ~Ubies~~ ,toP~OB ~~

thenexp18.':h'~ha.t',.i!l~,eOlfl0aa~~~ts o~..e&,Oh~r,~hel,thr e e . toP1C~^.~~~e.stu~.~~.^tt:loJAO~tf;tro~~,l.e. 5~C~n.d17't"aOhei'S

·...-·.ereasked;.~try tOeXp181n,'whT the setoPiosJfere themost

d1f t1 oul t . llnaJ.11 1teaohers'lisre

:as ked

to suggest

:, ' ' i '",. :' ',':' ,' ,., ': '."' "

.teOhnf que;.,:fO~,~~~l1ng'wi t h't.he.se,'~roUb~~l!lOlll etop~e~.

Fro;m'th~COlllp;~.te4~~~ond que8 t~onnalre, th.e.

, "

,, ' ,

^.

.~~e9tlgatoj,~~s tedall'the,',troub~esOllletopios'~ent1onedby

I.

I .

the teaohers at eaoh grade level1ntheir'three ohoioes.

AnattemptW8Bthenmad eto t1Jldaoonsensu~ISto the specifioaspeots of the d1t'floult;r with •• ch topio and the lls 1n r eas ob s for the diffioulty. SAlllPle,sorIs.uggeil t ed tellohlng.teohniques.e~eael eo t ed for eeen grade level. '

r~a:i)j

^{•.'tJi e}

;1nv~~:1s~:tor ,oom.:P8~,~ .dlrr~Ou.:!:,'~le8

^{,,8Jro SBijrade}

levels1nordertodeterm11l ePoss1ble'trends.

I

2S

I I;

.!

I

RESULTS CHAPT!Jl.IV

, ,1.

' . " .

.

Sohool,St.'Bonaventure's Sohool,andRolJ'cros ssoh ool;

Plr~t.

^Table

~ 1nOllJdel!l'''8o~e t~o~al -1nf~rm~tlon

on thepr e p8,r a t :1onAnd"exp e~~ ano eorthetea c h er s~TOlved 1n'the~tUd1,.._ Then, TableII'pres ents·t he·lis t 9!",to pics

t· ', .

foreae~8l'ld e le'Ve11l~lchrecei v ed ad1ft1oUl~1ra ting e;re~terthBnor ;eq 1l81to thre~011the·f' l r stquestionnaire..

Sincethe inv:,s t1g~ tor~Oul.dnot"deal.d eq~tl!lly^{w1th.11 of} the.dif f'l c ultl esobtainedf'rolll.the first'questi cnIlllllre,he

8Sk~

^{'t e acher s onthe}

.'~ec~.rid q~8Stionnaire

^to

s ~ieot

^:rrolll

^{I '}

theetated11st the threemoptdf!l1ou1ttopics.-.,In this

••,., there

~8I!l '~ SI18ht'red~iilt~0J1

^inth'entllD.be; 01'toPiCS"to - ,de 81111tho Ai ltoplc~ seleo~edf"or ea~hgrad eleve~onthe,

sec ondquJtlo:nnalre81088~SO l1st~ .inTabl.eon. Next;

", '.,"

~, _ 'Ibl~ 's~otron :~~t~j~l( ^{to " 'answe. r}

^the

^{8bo,:e .}

questions t'orthreeap8oif1cellDentu;r.s chool s. It .pre s ent s a reportof

~h~

d.8t8obt.t1nedtromt:he

.·te~~-herll

^o't

Illatheutlcs"in grades tour toei gh t 1nSt.Pa triok' s Ball Wh at do theteaohe rsIdentltJlUI the most dlf:flou.1t conceptsInd8k~lls^involved in ' teach/rigmadem

--- "

. :"' ,

.1!I1 em en~l"_8~oOl.:-mlthell. tl 0 8 ? Whatrea sons do ^the;yhaveto off~rIUr~llhy t~8d1f~lou1t1e8,occ~ 1 What,sugges t.l ons

or

teOhn.lque~do theY·havetootterfor_.d~81.1ng~ 1th thes'e dlf t'lcul t l e S?:.:"Are thereany trends'aoross'thevariousgrad e

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11 ^".~..

27 taking e~chgrade 1e'Yel s~8t'8 tellteach.topic se l eo tod on theeee eeaqUe8t1 onn e1re 18d.i scussed.111te rasorspecIfics. re81ons, and.suggested teach lne:techrllques. Fine llI.~bl e

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.IIrhlgh:18:ht8the eeeura- ee eeofd.1fr loul tle8.0~118varlol1S .gr.de1e"81& whi ch 1; dl sculS ed 1naomede t s ll. '

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28

Teaoher E:r;perlenoe lUld Prepa.ratlonInd loatedOn.theFlrflt

Questi onnaire" .

GradeLevel POUR lIV,E BIX BIMJ' EIGHt Number of Teaohers

Rea pond 1ll.g .S 7 '8 S 7

Aver Q8e NUIIlberof Years

.~.h1ilg,Math~tl0"s Ij 13.3 8 7 4.4'

Av erag e'NtlIIlber~tYears ^'.

I

T~h 1n g Ma~(lIDatlo 8,a~

8 ,.1.

'l'hlll GradeLevel •• 4 4 J

Average NumbfIX'of' Profe8s1ona1COurses (on.

Sea;'elter)10Teaching

1.8 1;8 _1.6,'

Msthemat lo8 1.9 L

Av era g e NUlllberof t1nlnrs ltrceuesee(0'"

1.8

'S emes t er)int'.athem.aeiee 2. 1 2.6 4.6 1.7

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