** ^1F*. * * * * * 0 **4°64 41" IF (14 * 44 V) *10

13 0 0 +1 * '14 r-1 b ril *0 0 4. ni * 0 1i) W 0 *4) 0 irl 114 +4 W 'N Oki 4 1r0 44 .n.,41 r4. 1riLrC*il tr1 0 44 W 41 41' W)44 4-1 r1 .0' 4.) 0 *

13 4) 4) F.: t3'14) 4' 11;4 .C1

Ai ..ct w tyl. 0t1 0 4.6 U 0 14

or 4.4 k$4 0 1:1) III U a 0 L.; r4

44 `1:71 t.)* V/ 41 0 ri 0 0 ,.; *CI 0 (U Ci

111 () 014'(1 "41.)6.1 11)14 ;I:IfOrl

0 4 4 fil I4 ,1 4+ zILA Irl '.4 04) C)4 VI A 11.1 VI 44 ,1

* .14 lig 14 171 ri r rt, rn*$1O VP r-4

OsIII lei- CI W will 4.i 8.4 Dog 44 41°1 01 41" 49'4144

*it* eri,v01,1.4414.:4,,r4w itill,,r0i 1.,1'1.4::':

ri F1 111 +4 00 01 44 11 9.H, 0 14 ,C1 .. 14 'Pi 11/ 414/ Ir1 V I %I y 14 01 1:71 Q' 4) .Q' 11J 41 40

* L'410' 0'44* 411) 44 41 tr it t.) es, 'rn * 0 4 0 'V ni

V) VI 04 41 *

4) CI 0 1 0 11) : &I , 41411 11) 441 1'410 14 14 r0 0.60) '111 .4 ** 0) 44 At 4.) 44 *

A 1 1 ) 4 V gel XI 01 0 A l ' 7 4 - 1 1 1 1 1 1 1 0 0 0 . 1 0 4 4 0 * ri 4* CD 't), 0.141 *II Cl MI :14 0 01r4 111 VI (1) rg Pi 4,1 . 44 44 44 * Mr 10 43 0J 1111 0 ft*W 14 04 0 a 11.1 4.4 W 44 'rd 1.1 CPA V) * 44,64 's IV 1.4r4 U'' 41 al It4 U VI 40 14 0 417, 14 4:4 0 0 0 * N

0,4 RI 114 * (V Cl U .C1 U Ii1' 410/1 01 Nr1 1444 1:4' ...O. II 1.1-** '0 I.) RI 1-1 4.41 0 64-4'*

44 Cl 44 0 46 '1 004 +4 411 14r1 ri 0410 * 61+ H. VI (6,1, at U 41.'444 M U Q 44 1 P 01 40 ` 40 CI .4 44 rs 10'4 est) * cl 411 10 1,0:.114 IQ 4:::) 111

41 Q 1144 4 a U) * 44 (i'44 0 II 44 1,4 41 4.1 0) 4) * 13) -*A' 41'

ON 141

4)9104 0)+1 4. 0,0 +4

41 4) 44 4 a 0 V) a 04 1.1 girl .4' 'r' * r4 .0' In - to *t) to u) vel 10 GI to on

4.1 41 VI , VT a 01 $.4, * Om 4) 44 Cl trir *A 4 k I-1 4 ' 43 tli 44 43 0 * r0 14,A, , 0 W 10' *

XI 44 1.4 744 0 0 +6 016Q +6 1+ 0 RI ta U) 41 13 V) V al * RI' 44 +4 r044 11 13 41 *'10 +1 13 0 r (/) (4) 01 11i.1 )4 0 ,, 0:1 0 X Al' in .14 cl 4.! a -6 6r1,`H

NII +1 8 11 4.' r?, * Q 1 4 T.)ite 104 44 t)(1) 49 19. r.4.16

WC j0 410f * 14 W ttl U 0$ ** 0.1 Cl 0 14 .11 CI 44

11 *I 174 Itt 4 al ro 14 i4 04 Vi 0 "1 a a oa 4441 ri, 'e. in 44 ri 04; tl) U .44 4-6 +6 14 0 44 (4 0 44

* r0 0 W V 00 k21 44 03 ** 0 1/1 M 41 0'1,1 0 0 *

.1.1.1 'II?) 11413 ."(.4 r4 ili Id ,'.%' 4 U vi ni ,I9-131 w 4) r4 1410 40 t1/ to. a 11 (1) = 0 * U 14

44ti1 r4 41 44 *

it- r4 *+6 41 0 4r1 111 fii N .4-1 Cl !I. fitt 40 ,.4 61 11r$ W 4 1 0 44 * a ii) tli 4J co 0.0 0 *111 (1 orl iv r C1041411 14'6 *01: 0 (V 01 *,,n1 0 4) 4) 441 * 11 Al 4-4 0 1.4 441, 44 .0 *'M W 01 41 I r 0 (U 'V 41 1 14 .C1 .0 Cl',, 1 0 XI 00 114 VI * 4-1',4, Cl 4." 44 *4' 0 41 4) Or, 44 ri -Pi' 411 r4 (4 r4 N W 10 4. 0 (I) 4" or; 14 tn it 'U 0 r0 0 Di'l-U 44 *

.1-1 4/ 01 fl' w 41 IP = 01 Cl r 14 .r1 .4 N1 4 'ri 1.1

4.$ 14R1 P. 14 i-I 10) 4. 44 .$ or4 Cl ......

Ir 111 ,Cl 01 U'1(1 ,r110 V) 4,1#:14P $.411,1 ot.) lo,41 NW

1.1ITI 11A * 04 III 'ri Cl 0 10 0 *

1tt ,,ini (U /P 'I- IA (11 144 C) CI 114 :4 C) 01 U tn Q P (I n1* M 0 AS W U 0 )!, 0) *'

a (V $1-6 U M IA 10 !'q,44, W tii1.1 03 C 0 V in i11.4 U RI 1.,14,1,04r1 U 41,111- U ,t41 ;U.1, 4.4q 1406

* 14 00 (4 4$ .0'** >044 0 Cl, W 12144 *

1.64 .r1 444 111 orl 44410 1.4 If) r) Prl 06 +1 0 WI Ai 44 46 V)* 0. (16 t'si 41,r-1 4) ft40 COL *14.) Ar 14, R1 A *

, 0) Cl 111 lik in A .CI (U .4 .r4 0 VI 414- IV 4-4 IN Cl A t1 0, 43 *'t,1") ,Ili 0 01 01 CC' +4 1.) ri tY1 11. 04 111 0 F4 rl 0 04 'Q .r4 : rt3)31' .4r-i nE0 Cji1L r0 +4 DI

4E 11 rt c 44 .4 14 0 14 4) 4.' 0 44 0 41 0 01 84 * 1401) 01-1 4.1.1*141 00. V) 14 1.1 17;4 14 AI :1 i li IP al 't, N 0.1 Ul 4.) i Q WA" ,,o4

00 11 0,1 '0 0 44 1,6 44 14 ') U* 414-4M 44. 11 11) 4) 44

'OP 44 W 4) *4 '76 01 it?" L7 14 t",n * 0,,.r1 44 H. W ,L: KV*I V ni ao "I 44 A I , Q V) 0 114 '11 AI o t i 0 o (il 1 W 4 614PI WO 01,1 $1 11 ri /64 t.) 1.14 0 4) X' .1-1' 41 V) 41 In .0 0 .r. 4) 01 : U" pPeti Wtil 11:1 .Cq E 4) tr)VI 11 E'i r" 1.4 1-0 0'11 WI * 04 2144 tot GI (.4 144 14 0 '14 447` $11 (.4 46 41

V U .r1 PI $.1 0 op nj 4* As n$ ,'.), jt..4.t.i) (-) 64 64.1*

M 41 '10.? 0 ) i t 1 0 1,1 Il, 41 4 / * V , 44 4 ) *I:1 '44 44 PI *$4 l'i 04 .14 II 44 0 4-1 V 14 * .4 0 44 r-4 0 )..) *14 ,C4 (t 0 * M. g 4, rt 14 4-1 41 ?i*01 0 Cnorl 0 44 4 1+ N * 0.

,r) f.+) A .14 0 114' 4.4 I 1,4 (1 VI.0 'U 1-$1 ri .11'*

* a V) ,0 441 or4 .141 A) *63 101 ....t 1.1 61 t) U 4.1 U 45 on * 0 ''4,'14 'U W q-1 MI *N V $ . 4 ' 1 "1 Ill 4-6 orl 0 14 * U 0 0 0 44 0 W *174 14 1 1 0 14 1 4 1, IP 4.' 144 44 r1 4 4) 4 C: 1.1 * 0 1 44 r0 0) -c r4 .14AR

E 1 ill 04 ri, u A; 4 o Ili A 4 .114 0 4) * (l 1-1 .1:1 0 ,01 4$ 0 -4 *It 01 4I he. I I I 1 fl: 40 ri 1111 Al 44 114 N 44 1.Ify Q 1,40 1-11 I 4 * 43 0 I-1 4.) lao Mt*I )0 44 *1 ID I-4 1,64 4) 14 411, * 44 04 VI 04*

r" 141 4.4 101 11 U ,.1 El NI .0 '.41 Al '.1 4' r4 AI 4) 144 4-4fl C-) 111 1 I I' (r) P1 II) 0 el ill 44 oj .V1 1,0 ,14 04 421 ..4

* n$ 0 4) 64.1 4r1 0 a*-"N (4 44 /4 0 1-1 V) *

41 L4 1.4 r4., Ca 14 1)1 )4 )4 14 41 U '01 i 'ri 4.4 0 04 Cl) 1 * *ILI pig ir 0, 1 141 (-74 Prel 441 r4 II PI 111 04 n1 1111 4) '(U "" .

44 44 * * * * 41 * '}4 *

us vs* feast Of motila_ta,tagoCatesou auturami

ficiSsObad.

- .411. itiNt airy_ ft .4Z, foist

RD' MATHDIATICS AND' COGNITIVE OEVELOPtiEllt - Pi I's 4, 4.1.44.-:ru t4Nrog

:_ -; 41.4, -Or *VIVI(

M. _P. ByeCalgary Board of Education

SECOND TRANSMOUNTAIN REGIONAL CONFERENCEINTERNATIONAL REJ-WING ASSOCIATION

CALGARY CONVENTION CENTRE

PALLISER HOTELCALGARY, ALBERTA

NOVEVIElt 13-15, 1975

4.11,cSn. ANt,.0.7:711?-(10--ft t4146 "IASI: kV

M. P. Bye

AZOICAVCA"Icre. 11:1404N416A.14)1A1 Ai,

-.1 174 VirA'Olg 4disnelp intas,4_I '4-11110-

flourSKSII ts'41 ricrwrowl

There are canraspects to reading. Amongst then are the following:

- mord recognition

vocalizing the word (oral reading)

understanding individual words

- understanding groups of words

- understanding sentences and groups of sentences

- remembering

It is with the understanding part that thiS Paper deals. I contend

that a student nay "know" or may "recognize" or "underttand"

dual words and be able to "vocalize" words yet not comprehend- what a

group of -words or .the sentence(s) is saying. We may apply a word

recognition test, a vocabulary test or soMe other tests to a .olathema-

tics passage and the results may indicate the reading level is very

low = at least two or three years below the grade leVel (age level)

the students for which the book is designed, In' many instances the

students nay be reading in other content fields at a much higher read-

ing level. Yet in the mathematicS content area these students l'ay just

not 'be able to get the meaning from the passage. The question of course

is Wily?

AUhy night a student who is reading at a level consistent v h his are

in literature, social studies, etc. not read at a comparable level in

mathematics - at least in nathenatical material which by many tests nay

be rated comparable to or lower than that of the student's level? There

are a mumber of reasons which I will present. I'll pass quickly over a

number so that I can go Into some depth and specifics on one that cuts

across many of-the others.

Mathenatics passages often (1)*

1- are conceptually packed

2. have a high content deasity factor

3. require other eye movements than left to right (requiresvertical novement, regressive eye movement, circular eyemovement, from word to chart to word movement)

4. requires a rate adjustment

. requires multiple readings (to grasp the total idea, tonote the sequence or order; to relate two or more signifi-cant ideas, to find key question, to determine the operation or'process necessary, to conceptualize or generalize)

6: use symbolic devices such - as graphs, charts, diagraIs, andmathematical symbols

7:' are heavily laden with its own technical language which Isvery precise.;10ften commonyords are used with specialexact meaningS-e.g. function..

The reason with whicithis paper deals is:

Mathematics is an abstract subject. It is usually presented in a Very

symbolic and-,.generalized form. Its language is .highly specific. Read-,

ing in mathematics often requires a higher level of cognitive and con-

ceptual development than the students have- achieved.

In order to put this all in a logical framework a rsodel of cognitive .

deVelopment is necessary-. 1 shall use a modified- Plaget

(I)* Utimerals In brackets indicate the references at the end ofthe article.

3.

Proponents of Piaget generally agree that a child, in developing a

concept, goes through certain stages. lbese stages are not discreet

but rather blend or meld into one another to form a continuum.

Now about the stages:

The order of the stages is- fixed.

2. The rate of progression through the stages isnot -fixed-- nor can-they-be tied to-any-chrome--logical ages. Where this is done - it is onlyfor referencing.

3. The movement along the continuum can be alteredby certain factors - teaching, environment,maturation, etc.

Now for more details:

Different authorities give different names for the stages and often

break the continuum into different stages. A general breakdown lists

these four stages:

1. SENSORI 7 MOTOR STAGE (From .0 to 2 years)

- preverbal - though language does startto develop

- direct motor action

- ;begins to learn to coordinate perceptive and motor

functions -= see thing with reference to what they

do with It

2. -PREOPERATIONAL :STAGE (2 to 7 years)

- objeCts take on qualitatiVe features

- permanent

nom permanent

- tends -to=egocentuism,

tendency to see things from one point of

viry - his own

tendency to see One-feature (elation). to

-.the exclusion of others 4;_

3. um OPERATIONAL STAGE (7- to 12 years)

- Piaget describes operation as a mental act - an inter-

nalized action.

The concrete aspect comes from the fact that the

actions (mental or internalized) had their starting

point as Solae real system of objects and relations -

that the person could see, feel, hear, etc. (senses)

or coutd,imagine.

Concrete, - in the-sense it's real - involves no-assumptions

The child can-now lotus on more than one attribute

dt a-tine - two, then Maybe multiple ones

begins to see -other person'S point of view

This is the stage where the classic operations are developed

combinativity

reversibility

coimutativity

- identity

associativity

For example:

The operation of conservation Is not a unitary developlent.

Conservation of discreet quantities - (number) develops

before continuous quantity (substance)

Conservation of weight (force of gravity) and of volume

(space occupied) follow yin that order.

Fo_:;MAL PnITATION (12 onward)

characterize by full, logical thinking

- can now deal with s t-zents that are not known or

supposed to be true

then

either.... or

either or both.... or

a;pb, b)Pc a)Pc

Let me take some exaiples of how e. concept would-develop - or

rather the ManifeStatiofts of the development or a concept.

Given a set -of -beads

- Sensori-Motor hit, throw

- Preoperational

- Concrete operational

Formal operational

Another Example:_

Equivalent Rate Pairs

2

3 9

23 = a49 98

or

only present when seen

red, round, thick ) one) feature

yellow, square, thin) ,at a time

red and round and thin ) two ory Mote at

yellow and and- a- -tine

either the-red-Or the square. or

the thick ones-

neither the ted- nor thxtound ones

a- - x xolve-for x

x 2

x --- 4 x2 - 16

9(concrete)

solve for y (formal)

3rd Example:

An ideal car travels due east for-2 hours-at 60 mph

returns at 30 mp_

What is the average speed? 40 mph or 45- mph

You a. ll know two things: d = it and r = d

average sumNo of number

(concrete(generalizaticin)

(concrete ??)

In Sumtary

Piaget maintains a child moves along a continuum .-= the continuum

representing the blending Of the stages of develop-pent for each

concept.

While it is known that some students may enter the formal operational

stage at the age of 12, many will not do so until the late teens and

some never will. IL has been reported from one piece of research that

82Z of the 8th and 9th grade students are still in the concrete opera

tional- stage while 50Z of the university freshman classes are still in_

the concrete generalftation stage (1). Yet many -of the matheMatiCal

terms are used as though-every child' is in-the formal operation stage.

So What does all this- mean ? If all I have ,claimed it true, -whit do we

do, about it?

I believe there are two things we must do:

(1) diagnote more specifically the weaknesses than in the past

(2) deVist Ways and meant (a) to'bring about greater cognitivedevelopMent (b) to- build-more word and word group associa=tionsto faCilitate greatercognitive_develepient

Let us loolcat wayS of diagnosing weakness. I Will deal with diagnosing

the lack of interpretation of the- printed symbol because of a lack of

I)

VE

0

H

1.

1

FV

L.

0P

E

A

r rn

Ueda me.

eig h

Subs i-ance.

on Le. r h

cognitive development in the conceptual area represented by the symbols.

We Jill st;,3"; with two easy words: some, all

Example 1 (3)

Place in front of a student (age 5-9) red squares and blue squares

and circles.

Ask the question: "Are all the squares red?"

The -reply typical of a 5-6 year-old is "No." When pressed -the-child- mayreply "No, because-there are some red -circles too."

Let us considet extending this. Take the two statements:

Dogs are animals (Squares are rectangles)Animals arc dogs (Rectangles are squares)-

What do we Mean? In one- case we mean- ail. dogs are aniMali, and lit the

.

other We-mean some animals are dogs. We expect ,children- to make these

subtle-interpretations.

Example 2 14)

Place iii_front bf child drawings Or _photographs ofteese,- ducks and

rabbits (I have Changed this-exaMple ftem Poodles, dogi and horses,

colis, sheep and, cats, to geese, ducks .and rabbits = since I can- draw

theM better.):

Ask the students (age 5-9) these questions:

"Is every goose some kind of bird (animal)?" "Why?"

"Are .lL geese some kind of bird (animal)?" "Why?"

Then"Is every bird (animal) some kind of goose?" "Why?"

"Are all birds (animals) some kind of geese?" "Why?"

.

.Typical replies indicate that students realize geese are birds and geese

are animals. But they fail to realize that all birds (animals) are not

geese. Yet when pressed they will identify a bird (animal) that Is not a

goose.

With older children similar situations,can be structured:

Is every triangle some kind of polygon? Why?

Is every polygon some kind of triangle? Why?

5

Is every natural nUMbet some kind-of a rational number? Why?

Is every rational ,number some kind of natural number? Why?

rti

The concept underlying the two words 'some' and 'all' is that of in-

elusion. It is not an easy mathematical concept. It n Ids to be

developed over a long period of time starting during the concrete opera-

tional state. Consider the situation of the three geese, two ducks and

four rabbits.

Example 3 3)

Place pictures of three geese, two ducks and four rabbits in front

of the child.

Then ask these questions:

"Are there more geese or are there more ducks?" "Why?"

"Are there more geese Gr are there more birds?" "Why?"

"Are there more geese or are there more animals ?" "Why?".

Typical replies indicate that some students will not be able to accept the

two classifications = geese and, birds. They reply =

"There are-More geese beCause there are-only two ducks.'

Interviewer: "What did t ask you?"

Student: "Are there_Mor geese or more ducks?"

Interviewer: "Are-there-more geese or are here-Mote birds?"

Student: "There are more geese -becauSe there.Are only two-ducks."

.

Inclusion is reflected in Mathematics through the grades

Are there more

squares or more

rectangles in the

set shown?

Are there more

natural numbers

or more rational

numbers in the'set

-shown?

Other words we use casually in mathematics and expect students to auto-/P.

matically have a grasp of are those which indicate relationships.

Example 4 (3)

Each student is given these incomplete sentences and asked to

complete them (typical reply in brackets)

Peter went away even though.._.. (he went to the country)

It's not yet night even though....(it's still day)

The an fell from his bicycle because....(he broke his are)

Fernand lost his pen so....(he found it again)

I did an errand yesterday because....(I went on my bike)

Instances where we use these terms in mathematics are:

8/3is not an answer even though 8 10

8 is not an answer because it is not a counting number

8 is not an answer we do not include it in the/3

solution set

,

We have a tendency in mathematics to put simple words together into very

difficult questions.

(5)

-Each student is shown a set of shapes consisting of large ttiangles,

circles and squares ea0 in red; -blue and green. Sea J- ones of each

colot and. shape were alsc in the set. The shapes ire-lettered A to

(Simpler versions can --he given to younger children).,

Blue ; Blue$ H

0reenN

,--

GreenN.

/Blue\

Green 16;4 -fi;

(1

;;W

I/ -0

Questions asked are of this type:

1. (a) Write the letters of all the shapes that are not circles:

(b) Write the letters of all the-shapes that are_ not blue circles..

(c) Write the letters of all thecircles.

2. (a) Write the letters of all theand red.

(b) Write the letters of all theand green.

3. (a) Write the letters of all thered.

(b) Write the letters of all thenor green.

4. (a) Write the letters of all theand red nor big and green.

5--. (a) Write the letters of all the

shapes that are =not small green-

shapes that are not-both big

.

shapes that are not both small

shapes that ate neither big nor

shapes that are neither snail

shapes that are-neither small

shapes that are not yellow,.

Our preliminary research has indicated-that many students in high

school had problems with the questions 3(a) and 4. Of one sample of

318 students only 212 of the grade 10's, 34% of the grade 11's and-

342 of the trade 12's got all the qUestionS tOrrect.

nearly 802 of the grade 104t,-65Z-Of the grade ire and 12'S h44-

difficulty with question -4; Apparently students were unable to-keep

track of four attributes at one tiMe, i belieWe this is not a reading

priblem in the phy_Ical sense 'but rather a, probleiSof laCksf-cOgnitiVe

development to be able to properly respond to the written words.

10

1 A.

A final example of simple words that contain a difficult concept .ts

the if....then.,.. sentences. We all know that muCa of the logic

underlying mathematics hinges on this sentence structure.

Example 6 (5)

A picture of two girls is shown to the students.

Jean Cotol

The following statement is presented:

"If Carol goes for a walk, then Jean always goes, with het."-

The following questions are asked:

(a) Would it be possible for Carol to go for a walk, and Jean

to stay at home?

(b) Would-it be possible for Carol to go for a walk and for

Jean-to-go for a, walk?

JO Would it be possible fOr Carol to stay at home and for

Jean to stay at hone

(d) Would it be possible for Jean to go for a 'walk, and for

Caro] to stay at -home?

(e) Would it be possible for Carol to stay at home, and for Jean

to go- for a walk?

Our test results indicate -that many students in the secondary school

have difficulty with questions c, d and e. They usually reply "Jean

and Carol must always go out for a 91k, together."

jig

.

1

So far in this paper I have only presented a sample of the problems

connected with words. There is the problem of interpreting symbols

as well. it is a compound one, first the symbol needs to be trans-

lated into words (a difficult enough task in itself), second, the words

need to be interjreted. ( symbol -word .concept). Many students be-

come familiar enough with symbols to omit the word stage (symbol*con-

cept) but often in gaining this familiarity the word stage is used. Many

students do not progress to the symbol-koncept stage.

Three instances with reference to symbols will be illustrated. The

first involves = (equals). Steffe, (6) points out that young children

he worked with would write 3 + 4 = 7 but would not write 7 = 3 + 4.

The reason given by the children paraphrased here, is that equals always

follows an operation - a. doing-. Is this built into our teaching or is it

inherent in the stages Of .a child's development? We are not certain.

The second example involves basic concepts of union and intersection Of

sets.

12,3.41 t 12,4,53

2,3,45 10,4;5, = 2,4

Using a test devised by Bliss (5) ve tested, a sample of high, school stu-

dents -and found many studentt had a cemnOtatiOnal facility with union and

_,--

intersection,-yet did not have a -usable conceptual Understanding_of the two..

Finally., Collis (7) has indicated that

8 x 3= 3 x-L is of lower cognitive--level than 7 4- A.= 7

8 + 4 4 = C.,:s is of lower level than 4283 + 517 - 517 =

a ,b- = 2a L reqUiteS a coMparitiVely high -level of cognitive

.development..

In summary, have presented a few- samples which ten toIndicate simple

words often convey high level concepts requiring high level cognitive

development. These samples are only a few of the many we have. I have

15

3.

tried to point that there is a definite need to not only teach stu-

dents words but we must teach students the ideas being represented by

the words. We must not assume that because a student does not perform

adequately in response to a passage in a mathematics book that it is a

reading problem in the physical sense - it may be a cognitive develop-

pent problem.

I feel we are making progress in identifying the specific nature of

some of the weaknesses (objective 1). We nust now attack the second

objective - that of indicating ways and means to overcome the-weaknesses.

We have ideas but at this point we have no experimental results to back

them up. For what they are-worth, let ne put forth a couple of Ideas

related to what I have been saying.

I think we must find ways to build word-neaning associations and especial-

ly word groups - meaning associations. Perhaps symbol groups - weaning

associations will follow note naturally.

Example 7 (8)

Read: + b

Four ways to read this are:

+ b

48 + b

b

"eight squared plus b"

"the square of eight plus b"

"b- is added to the square of eight"

"b is added to eight squared"

Many varied experiences will more likely develop the proper associations

than just using one of the four.

A problem that -symbol n present is that a student may not be able -to pro,-

nounce the word(s) represented. There are no hints. He can't use the

rules of his phonetics class. To establish a ,phoneme-grapheme relation to

...)11(x)dx 1

one has to recall the following words as spoken by the teacher: the

integral of the function of x with respect to x. -The student can at

least read aload the word fors but *ay have no clue as to how to read

the synbolic form. Utilizing symbols only when necessary or only after

students have adequate facility with them-will aid the weak reader in

mathematics. Drill in symbols Is necessary.

Mathematics teachers7along with othersllike to quote Lewis Carroll in

Through The Looking Glass where Humpty-Dumpty says to Alice "'Then I use

a word it means just what I choose it to mean - neither more nor less."

In mathematics words are used with special meanings (we pay them extra -

said Humpty-Dumpty). These special meanings often 7' though not always -

are related to the common meaning. Take the word "associative" as In

The Associative Property. We can illustrate the relation this way:

nathestatics example

(8 + 9) + 3 = 8 + (9 + 3) Jerry and Harry joined Mary.

Jerry joined Harry and Mary.

To sum up, we mdst provide students with a broader set of experiences

centred about the concepts in which weaknesses in reading -are reflected.

These broader experiences will provide for greater cognitive develOPIent

in the conceptual area hence the words the Students read will have more

specific and deeper meanings. This aspect of teaching reading is more of

a concern to the mathematics teacher while the mechanics and other aspects

of reading not mentioned here maybe of equal concern to both the teacher of

mathematics and the teacher of reading.

1r4

17.

REFERENCES

1. Earp, N. Wesley, "Procedures for Teaching 'Reading in Mathematics."The Arithmetic leacher Vol. 17, No.7 (Nov.1970): 575-579.

2. Sorry, at the time of writing I have been unable to find the titleof thisresearch.

3. Duckworth, Eleanor, "Language And Thought". Piaget in The Classroom,Ed. Milton Schwebel and Jane Raph, Basic Books, Inc.New Tork,1973.

4. Checking Up I, Nuffield Foundation, W 4 R Chambers, Edinburgh, 1970,18-19.

5. Checking Up 3 - Unpublished manuscript obtained from Joan Bliss(author), Nuffield Foundations (1973).

6. Steffe, Leslie, University of Georgia (Oral presentation, NationalCouncil leachers of Mathematics Convention, Denver, Colo. 1975.

7. Collis, Kevin F., "A Study of Concrete and Formal Reasoning 14 SchoolNathematics," Australian Journal of Psychology, Vol-23, No.3,(1971):239-296.

8. Hater, Mary Ann, Robert B. Kane and Mary Ann Byrne, "Building ReadingSkills in the Mathematics Class." The Arithmetic Teacher Vol.21,No.8 (Dec.1974)i 662-668.

16