The road to inverse and multiplyAuthor(s): THOMAS BATESSource: The Arithmetic Teacher, Vol. 15, No. 4 (APRIL 1968), pp. 347-354Published by: National Council of Teachers of MathematicsStable URL: http://www.jstor.org/stable/41185777 .

Accessed: 16/06/2014 07:30

Your use of the JSTOR archive indicates your acceptance of the Terms & Conditions of Use, available at .http://www.jstor.org/page/info/about/policies/terms.jsp

.JSTOR is a not-for-profit service that helps scholars, researchers, and students discover, use, and build upon a wide range ofcontent in a trusted digital archive. We use information technology and tools to increase productivity and facilitate new formsof scholarship. For more information about JSTOR, please contact support@jstor.org.

.

National Council of Teachers of Mathematics is collaborating with JSTOR to digitize, preserve and extendaccess to The Arithmetic Teacher.

http://www.jstor.org

This content downloaded from 62.122.73.226 on Mon, 16 Jun 2014 07:30:55 AMAll use subject to JSTOR Terms and Conditions

The road to inverse and multiply

THOMAS BATES The University of British Columbia, Vancouver, British Columbia

Professor Bates* s paper is adapted from an address that he presented at the Vancouver Meeting of the National Council of Teachers of Mathematics that was held at the University of British Columbia,

'W hether we are pupils or teachers, in most mathematics lessons we are constantly shifting our thought back and forth between what we generally regard as the "real" world of physical objects and the "nonreal" world of mathematical abstractions. The shift is often quite unconscious because the dichotomy between the two worlds is not always clear-cut, and this seems particularly to be the case in arithmetic lessons. In arithmetic the two worlds seem to be separated (and joined) by an ill-defined and neb- ulous region which I like to think of as a "patterns of action" zone, a zone into which from one side we project mental images of physical things and mental images of sensory- motor actions, and from the other side mental images of mathematical structure. In this zone there is often a merging of physical world language and mathematical lan- guage, a sort of lingua franca, as when we say - and presumably also imagine - "How many twos are there in six?" I think it is in this zone, rather than in the world of phys- ical things or in the world of abstract mathematical structure that most arithmetic be- comes understood by children.

The following sketch may illustrate the model that I have in mind:

WORLD OF WORLD OF PHYSICAL THINGS | MATHEMATICAL ABSTRACTIONS

I Physical Model -« -f- *■ Mathematical Model | » i

Physical actions -* Patterns -► Operations i ' 1 governed by -« of - ► governed by R t feasibility | •* Action -► definitions, BvDass г / relations, yy

(v4 | principles I ' 1

. I . rJ i ' . 1 Physical procedure ^ Mathematical .

| (Technique) ** ^ 1 ^ procedure (Algorithm)!

i Concrete rationale or abstract rationale?

In common with several others who are concerned with education in mathematics, I think that there is a tendency in some of the recent programs of arithmetic instruc-

April 1968 347

This content downloaded from 62.122.73.226 on Mon, 16 Jun 2014 07:30:55 AMAll use subject to JSTOR Terms and Conditions

tion, particularly at the level of the intermediate grades, to base the supposedly mean- ingful explanation of why we do what we do, too far over on the right of the above model, after insufficient experience with explanations which are appropriate to the "pat- terns of action zone." The result of such a tendency could be that pupils are forced into what, on the model, I have called the "rote bypass."

It is true that physical devices of one sort or another are often used in the devel- opments of the primary abstract concepts of cardinality, order, and the operations of arithmetic in the system of counting numbers. But when it comes to dealing in the intermediate grades with the numbers we represent by fractions, there seems often to be a halfhearted and apologetic attempt at "objectification" followed by a sudden leap into abstraction. Particularly does this seem to be the case in division using fractions. The rationales used suggest to me, in many cases, a considerable assumption that the ideas that we have, as teachers, about isomorphism and "imbeddedness," and our in- tuitive feelings about the way we expect numbers to "behave," will be experienced by children. How often we reason by analogy as in: "Since we know that for whole numbers . . . , therefore it seems reasonable to suppose that for the numbers repre- sented by fractions . . ."; how often we appeal to a sense of structural extension from one number system into another, as in: "Now it would be nice if we could do this with fractional numbers as well as with whole numbers. ..." I suspect that when we do this sort of thing most children fail to see, except at the most superficial level of ac- quiring a convenient mnemonic, that such essentially post hoc and retrospective rea- soning is even pertinent, let alone convincing.

Most of the concepts needed for an understanding of division using fractions are what we may call "secondary" concepts - generalizations which are based upon previ- ous abstractions, that is, on "primary" concepts. This raises a further issue. There are those who claim that physical devices do not - indeed cannot - help the formation of secondary concepts. There are those who claim that they can and do. I do not know of any powerful evidence that either side is correct, and at present the two claims seem to be declarations of faith. My own declaration of faith is that when we teach division using fractions, something is needed to bridge the gap between the physical world of tangible objects and the abstract secondary concepts of reciprocal, multiplicative in- verse, and inverse operations; and that otherwise we might as well give children a state- ment of division applied to rational numbers by definition.

General case or special cases? There are those also who, basing their case on a field theory of learning, claim

that wherever possible general cases should be presented right at the start. Others main- tain, rightly I think, that without an all-round retreat to the "unit-skills" position of the past, in something such as division using fractions children understand better by meeting one special case first, then another, then another, and so on. I doubt, however, that there is an optimal ordered sequence of special cases. The position here seems rather to be this: that certain special cases have a variety of plausible prob- lem models in the world of physical things, and have (therefore ?) plausible physical teaching aids to assist the development of the required generalizations, whereas it is commonly agreed that it is not possible to invent a variety of plausible real world problems at the intermediate grade level for such things as 5/7 -f- 3/4.

The sensory aid we call a "number line" looks as if it ought to assist in the plau- sible derivation of an algorithm for quotitive interpretations where fractions are involved; by the same token it looks as if it would be of little use, except for estimating a prob- able answer or checking an already obtained answer, in partitive interpretations. "Frac-

348 The Arithmetic Teacher

This content downloaded from 62.122.73.226 on Mon, 16 Jun 2014 07:30:55 AMAll use subject to JSTOR Terms and Conditions

tion plates" and sketches of rectangular regions look as if they ought to assist in the plausible derivation of an algorithm for partitioning, but of course partitioning makes no sense as an interpretation unless we consider special cases of a/b -f- c/d in which c/d corresponds to a whole number. It may be the better practice, as I believe it is, to treat special cases in some circumstances, but I think that the value judgment probably de- pends more on the particular physical aids we decide to use, and on the particular algorithms we would like children to "discover," than on any particular theory of learning.

If and when we do decide to deal with the general case of a/b -f- c/d - and this may be either before or after we deal with sundry special cases - then as I see it, since I have yet to meet a plausible rationale based on physical devices, there are three ap- proaches we can take. We can develop a rationale in the "patterns of action" zone, we can develop a rationale in the zone of mathematical structure, or we can abandon all reasons (except the pragmatic one "because it works") and deliberately teach in the "rote bypass."

In the following section are suggested some algorithm developments in each of the three categories above. "PA" refers to "patterns of action"; "MS" refers to "mathe- matical structure"; "D" refers to "definition." For each one I have used the same exam- ple, namely 5/7 -5- 3/4.*

The rationale for each PA algorithm is written on the left since it leans toward the physical world; by the same token the rationale for each MS algorithm is written on the right.

"Pattern of Action" Algorithms PA 1. Interpretation. - How much

/3' ^ "" ^ *" 5 ^ ' 3 _ ~~ of thisf - jean we get

"" ^ ^ *" 7

^ ' 4

_ ~~

out of this f - ) ? D of - = -

Well, out of 1 we can get ~^^ ^ 4

° 3 _

4 3 ^

3 °

4 _ "

- of - ; so out of only ^^

-ofl,we ^^^

can expect to get only- .. Ъ_ о£Л of Ъ' Ъ_ f { 7 7 '3 4/ 7 as much. ^^

And, as we have seen before, __ ^ |-of-)of- = - in many such cases, we can ' 3/ ^ '

"group" this in another way.

♦It is arguable that such an example, in which all terms are relatively prime one with another, and in which a/b < c/d is no less a special case than some of the more usual special cases. This is true, of course. There is no general case except the algebraic one.

April 1968 349

This content downloaded from 62.122.73.226 on Mon, 16 Jun 2014 07:30:55 AMAll use subject to JSTOR Terms and Conditions

5 4 3 5 So the replacement for the frame is - of - ~* D of - = - which и- и л а л, know 73v 47 which и- и we already л а л, know we can '

54 '53__54 compute as-X-. 7^4~D~7 3

Clearly the above development could be modified in the light of pupils' familiarity with (1) the relation between the physical world "of" and the mathematical "x" and (2) associativity of multiplication of rational numbers.

PA 2. Interpretation. - 5 sevenths «» »» - -r- - = □

separated into groups of 3 fourths. ^ 5 Í- U 3 Í- '= D

5 objects separated into ^s^ groups, 3 objects to a group ^s^ ,- ^ we can visualize and symbol- "y

"*" ~£

~ a

ize; but the objects have to • , xr 5X4 3X7

be equivalent. • , xr How can we ^ ■+ = О

7X4 4x7 get equivalence between these

objects? 20 21 - -4- - = О 28 28 20 twenty-eighths separated

into groups of 21 twenty- ^ / 1 ' / 1 ' eighths? The model we have 2°'28/ '28/

D

used before, for 20 objects -^_ separated into groups of 21 . 20 ^- 21 = D

20 , ^ "^ * "

on suggests - of , a group.

^ "^ ™ on _ D

We can't get a complete group of 21 out of only 20. 5 3 20

20 _ -s- - = □ = - In fact we can get only - 7 4 21

Z* 1

of a complete group. This leads eventually to the "smoothed"

.,_ 5 3 5x4 a*rthn7+4!S773'

.,_

PA 3. If special cases leading toward this algorithm have been examined first - cases where the terms of the dividend are integral multiples of the corresponding terms of the divisor - then the renaming of 5/7 as p/q such that we can "get 3 out of p a whole number of times," and "4 out of q a whole number of

350 The Arithmetic Teacher

This content downloaded from 62.122.73.226 on Mon, 16 Jun 2014 07:30:55 AMAll use subject to JSTOR Terms and Conditions

times" seems to present a plausible rationale. Otherwise we could think as follows:

3 5 3 D Interpretation. - How much of - * - 4- - = -

4 7 4 Л can we get out of - ? Suppose

□ it turns out to be - where O D 3 5

л - x - = - ^* Л 4 7 and A can be replaced by whole ^^

numbers. Then - of - = - * Л 4 7 DX3_5

- "~**0*0^*^ A X 4 7 But if D X 3 gives 5, how can -"

□ be replaced by a whole number? ^S* A , .- A „ л . - , A ^^ ^ D X 3 5 5x3 And A , if .- Л A X „ 4 л gives

. 7, -

how , can Л A ^s^ ^s* ̂ = - = i,i i , *< ^^ Л X 4 7 7x3

be replaced i,i

by a whole i number? , ^^ '^^

и . .r t. ^^S^^^L^ D X 3 5 5x4 и How can we justify . .r

chancing t. the ^^ - ̂̂ ^ = - =

^S^^ -

Л x 3 7 7x4 5? or the 7? or both? so that we "^"^^ - ~^

, - ̂ ^ - D x 3 5 5x4x3 can get out a whole number , of = - = A с Л х 4 7 7x3x4

3's or 4's? A с ^y^

wu- и rui ^^ D x 3 (5x4) ХЗ wu- Which и we сап regroup, rui for whole *< = Л x 4 (7X3)X4

numbers, as we have seen many 5 3 D 5x4

times. - -r- - = - = 7 4 Л 7x3

This leads eventually to the "smoothed" f .,5 3 5X4

algorithm f --=-- = • 7 4 7X3

I think that it would readily be conceded that if we wanted children to do much "discovery" and we wanted algorithms such as PA 2 or PA 3 to be the outcome, then it would be more promising to deal with certain special cases first, since these special cases contain broad hints for the general case. On the other hand, for PA 1 it would not seem to matter so much.

Mathematical Structure Algorithms MS 1. 5 3

7 ■*"

4 = D * fl-5-b= (ax k) + (b xk)

5 ^ 3 _ 5 X

4 -~"^^

7 ^ '

4 _ ~

7 X

3

April 1968 351

This content downloaded from 62.122.73.226 on Mon, 16 Jun 2014 07:30:55 AMAll use subject to JSTOR Terms and Conditions

MS 2. Basically the same as MS 1 but with complex fraction notation

5 3 , * - -i- - = D a + b , = - 7 4

^^^ Z>

5 3 7 ^ « _ ^ X к

7 4 з" ^ * b X к

4 ^^ 5-XK

5^3 ' _ ~ 7 7

' 4

~ 3 - x К 4 5 v 4 n p q

5 3 7 4 7 = "> - X - = 1

7 '

4 3 4 ► - x - 4 3

5 _ 3 _ 5 4 7

_ ' 4

_ ~7

X 3

In some programs it has become the practice to use one or another of the above algorithms at the intermediate grade level. Here are instances of the danger of our hind- sight being mistaken for intuition and foresight. It is sobering to consider the question, "How do we know that a + b and a/b are names for the same number when a and b are names of rational numbers?" If we allow ourselves the naive answer, "By definition of the symbol x/y," and allow similar answers for similar questions about these two algo- rithms, then I suspect that many of our pupils will be in the "rote bypass."

MS 3.

There are many different names £^3_/£x4'^/3 ' 7' ^

" for the multiplicative identity. 7*4 '7 4/

' '4 1 ) ^^^_^^^ ' ^

53_5x4 ~ "" 3x7 ' ^ а с ~ _ a X с

7 '

4 ~

Г~28 ""

28 - - ^--- r^_

* ~d ~

b X d 5 3 [5X4 28] 13 X 1 2в'

r^_ а с (a p' I с р' ?

5 + ì 3 = Vir- [5X4 x t) 28] + Ъг-

13 X 1

х-) 2в' - ъ

а + с а = [ьх1) (a p' +

[ахд) I с р'

5 ^ 3 _ (5X4)X28 ^ (3X7)X28 ^^ a c a x c l'A 28X1

• 28X1 -

bXď =

~bin 5 _^ 3 _ (5x4)x28 (3X7)X28 ^--^^ 7

" 4 ~. 1X28

"~ Б<28 " a X b = b X a

- - ' - = (5 X 4 x ?A^ /3 X 7 28^^- axe _ а с -

7 - ' -

4 =

' l x

28/ ^ 1 28/ b X d~ _

bX d 5 3 "~^^^^ - -s- - = 5 x 4 4-3x7 Multiplicative identity

352 The Arithmetic Teacher

This content downloaded from 62.122.73.226 on Mon, 16 Jun 2014 07:30:55 AMAll use subject to JSTOR Terms and Conditions

MS 4.

5 3 _ e . Closure ' 4 / ^ Inverse operations 5 e 3 = - x - 7 /4 "--

а с _ e a _ e с

^ " - ч^ e e

5 4 (e з' 4 - P q л -x-=(-x-)x- - • - = л 1 7 3 '/ 4/ 3

- """ - g p

5 4 e /З 4' """ - -- __ j , 7 3 / '4 3/ ^^^^^^_ 5 4^ ^ >.^^^ Associate multiplication 7

x з"

~" 7 - - ______^^ >.^^^ "^ multiplicative identity

5 3 5 4 ^ x - y and z - y => * = z

- -г- - = - x - ^ " 7 4 7 3

MS 5.

5 3 e 5 e 3 л Closure 7 4 ^ 7 j 4 y Inverse operations

^ 5 _ 3^ 5 3e Commutative multiplication 7

"" "47 7

~" ^7 of rational numbers

Equivalence defined on <=> 5(4/) = 7(3e) elements within the same

class

« v (5 X 4)/ n = v (7 X 3), y Associative multiplication v n v y of integers

5x4 e Equivalence defined on 7 x 3

~~ 7 elements within the same class

5 . 3 _ 5 X 4 - ~^~ . - _ - - - Transitivity and symmetry 747x3

It will be noticed that the language used in the rationales for MS 1 through MS 5 has tended to lean progressively further towards the right of the model. It will be no- ticed also that there are affinities between certain PA algorithms and certain MS algo- rithms - between, say, PA 1 and MS 4, or PA 2 and MS 3 - and this might have implications for a "spiral" development.

And finally some rote algorithms. Here the "actions" have no basis in the world of the physical referents of the numerals; they are actions performed on the numerals themselves, and hence do not qualify for consideration in the "patterns of action" zone. The ones I have chosen to give are simple, and are therefore easy to remember - and forget. At the risk of laboring the point, we should note that any of the MS algorithms could degenerate into a similar, but more involved juggling of symbols.

April 1968 353

This content downloaded from 62.122.73.226 on Mon, 16 Jun 2014 07:30:55 AMAll use subject to JSTOR Terms and Conditions

Rote Algorithms - "by Definition"

Dl. 534X5 D 2. 53 - -г- - = _-h- = 4x5-f-7x3 7 4 3x7 74

-х- -ж- D3. 5^3_5 Х

4 7

" 4

~ 7

Х 3

Inverse and multiply All of the above treatments lead eventually, of course, to what we may regard as

the ultimate "algorithm," namely, 5 3 5 /3V1 5 4 _ - _ = _ x (- ) = -х-. 7 4 7 '4/ 7 3

However, the intermediate thought of translating division by n into multiplication by the multiplicative inverse of n, depends on a concept which I suspect is not acquirable, at present, by many children of elementary school age-

Conclusion

To recapitulate, I have suggested that we can teach division using fractions inten- tionally by rote or we can at least attempt to teach it with understanding. To teach it with understanding, we can make the rationale lean toward the world of physical objects, we can make it lean toward mathematical structure, or we can use a combination of ideas, allowing intuition, induction, and carefully guided "discovery" to play a part, and allowing analogy between physical and mathematical models and between one mathemat- ical model (the system of whole numbers) and another (the system of rational num- bers). I have suggested that there is, in some of the algorithms which try to give under- standing, by implication if not by direct assertion, reference to an isomorphism between whole numbers and a subset of the rational numbers. I have suggested that there are cer- tain ideas which we have, that were probably post-factum to our own understanding of the concepts we are trying to promote. We "know" that for all rational numbers RI, R2, R3, Rl = R2 implies Rl X R3 = R2 X R3, whereas the child we are teaching may "know" only that for all whole numbers Wl, W2, W3, in a finite set which is within the limits imposed by his experience, Wl - W2 implies that Wl X W3 = W2 X W3. There is a great temptation for us to believe that a child will project his Wl x W3 = W2 X W3 into our Rl X R3 = R2 X R3, and that even if he does not do it without our assistance, once he has become convinced of its plausibility, then in any situation in which he would readily be motivated to change Wl = W2 into Wl X W3 = W2 X W3, he would be just as readily motivated to change Rl = R2 into Rl X R3 = R2 X R3.

I hope that eventually children do arrive in the "rote bypass," but preferably after understanding, and after considerable experience in the "patterns of action" zone. If they are to arrive there via rote memorization disguised as learning with understanding, then I would rather that they arrived there more painlessly.

354 The Arithmetic Teacher

This content downloaded from 62.122.73.226 on Mon, 16 Jun 2014 07:30:55 AMAll use subject to JSTOR Terms and Conditions