A. Bell ALGEBRA LEARNING RESEARCH AND THE ... learning research and the curriculum 131 In the...

Rend. Sem. Mat. Univ. Poi. Torino Voi. 52,2(1994)

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A. Bell

ALGEBRA LEARNING RESEARCH AND THE CURRICULUM

Abstract. The learning of algebra involves developing strategies for forming and solving equations, and working with functions and formulae; as well as using algebraic language. Research is quoted which shows extensive inability to employ these general algebraic strategies. Problems and methods of approach aimed at developing the strategies are discussed and exemplified.

Introduction

We shall begin by considering the aims and objectives of the school algebra course.

We shall then review the research evidence on students' performance and proceed to some

suggestions for curriculum modifications which might be helpful; including some remarks

on outstanding research questions.

Aims of School Algebra

We shall give a few examples of tasks to focus the discussion.

EXAMPLE 1. Generalising (i) Show that the sum of a number of four digits and the number formed by reversing

the digits is always divistole by 11. (ii) The greatest and least of four consecutive numbers are multiplied together; so also

are the middle pair. Show that the difFerence of the two products is always 2.

EXAMPLE 2. Forming and Solving Equations (i) A boat rows a certain distance upstream at 2 mph, stops for an hour, and returns at

4 mph. The total time is 3.5 hours. What is the distance each way? (ii) Same questions with speeds 2.6 mph and 4.1 mph.

EXAMPLE 3. Functions and Formulae (Modelling) (i) The cost per hour of running a ship is a fixed amount of Ca together with a variable

amount of jCbV3 which depends upon the speed (V miles per hour) of the ship. The total cost of a journey of d miles at uniform speed of V miles per hour is CC. Prove

128 A. Bell

that

C = d(a/V + bV2).

Given that a = 3.3, V = 15, d = 3000 and C = 1300, calculate b correct to two significant figures.

EXAMPLE 4. Manipulating

Show that, if 2s = a -f b + e,

(i) s(s — o) + (s — b)(s — e) = bc

/..v s2 — (s — a)2 _b + c W (5 _ c)2 _ ( s _ ò)2 - IZTC

EXAMPLE 5. Functions from Data

PIZZA PRICES - ' •

Size Diameter of Pizza Piate Cost

Mini 20 cm $4.00 Small 25 cm $5.00 Medium 27.5 cm $7.50 Large 30.5 cm $8.60 Family 38 cm $10.50

Explore the relations between diameter and cost, and discuss what is good value.

The first of these examples requires the representation of the problem algebraically, using knowledge of place value, as

1000fl + 1006 + lOc + d + (lOOOd + lOOc + 106 + a)

which leads, in 2 steps, to

1001(a + d) + 110(ò + c)

which is easily verified to be a multiple of 11.

The second part of the question involves expressing the consecutive numbers as n, n -f 1, n + 2, n + 3, leading to

n(n + 3) - (n + l)(n + 2)

which on multiplication shows the difference of 2, independent of n.

This problem thus demands formulation in algebraic terms, some manipulation, and finally interpretation of the conclusion. This is a typical example of the use of algebra to establish generalisations - except that the full process includes the discovery of the generalisation, not simply its proof.

The second example is intended in a similar way to typify the process of forming and solving an equation to obtain a value as the solution to a problem. The equation is .

Algebra learning research and the curriculum 129

d/2 4- rf/4 = 5/2, leading to rf = 31/3, which can be checked, the two times being 5/3 and 5/6, adding to 5/2. (The second version is given simply to show how harder numbers influence the feasibility of attempting. numerical rather than algebraic solutions: the first version might possibly be attempted without algebra.)

Example 3 is one kind of question in which functional relationships are being used in a situation with some links to the real world. A fuller treatment of this might start By hypothesizing the form of the cost function, perhaps working from some real or plausible data; the determination of the Constant might well be part of a practically important modelling task.

A more valid task in the field of functions might be Example 5; here some hypotheses about the functional relationship between diarrieter, area, volume and cost need to be generated and tested against the data.

Example 4 is included to raise the question of this quite different kind of algebraic objective which is sometimes taken to be the main or only goal. I suggest that such activity should be subsidiary to the other more global tasks which generate the need for manipulative facility.

Algebraic Language

The uses of algebraic symbolism and algebraic thinking go beyond the types discussed here; for example, tasks in geometry and trigonometry often require the use of algebraic representation and processing, as when the angle size of an n-sided regular polygon is expressed as (2 - 4/n) right angles, or 23.4 = h sin 15° has to be rearranged. Indeed, a moment's thought about its role in co-ordinate geometry, in calculus, in the expression of physical laws, and in quantitative design problems of every kind, makes it clear that the mastery of the algebraic language is indispensable to further mathematical work, and that uncertainty or error in its use is as much of a handicap at these levels as is slowness or uncertainty in handling numbers at an earlier stage. Also important is reading an expression or formula so as to recognise how one variable in it depends on another (eg. y = x + 6, or y = ax2, or V = irr2h, if r is doubled what happens to V if h is fixed, or to h if V is fixed?).

These ali exemplify the wider use of algebra as a language for the expression of mathematical statements. But mathematical languages such as algebra are unique in having transformation rules by which implications may be revealed by symbol manipulation, without conscious reasoning with the represented concepts.

Indeed, the analogy between naturai language and algebraic language may be taken farther, in that language has a number of characteristie modes of use - for description,

130 A. Bell

expréssion, reasoned argument, persuasion - each of which need to be experienced and developed by the learner. These may be compared with the three modes of algebraic use illustrateci by the examples above and which we have taken as our curriculum framework.

Students need to become competent in the handling of these whole processes, and not just in the symbol-manipulations represented by Example 4. Moreover, it appears that conceptual obstacles arise less frequently and are easier to overcome when the work is embedded in appropriate meaningful activities of these kinds. This claim will be discussed more fully in the last main section of this paperi, The section following here will review the evidence from research regarding the nature and extent of diffìculties which students experience in the learning of algebra in existing courses.

Research on Algebra Learning

It is generally accepted that students' failures in algebra are extensive and of two kinds. First, there are characteristic errors in notation or manipulation such as 32 = 6,4(n + 5) = An + 5, or (x + 8)(x -f 2) =. 8/2, or x - 5 = 7 giving x = 2; many of these related to incorrect responses to perceptual cues in the expressions (Saad, 1960; Carry, Lewis and Bernard, 1980). Secondly, there are more global conceptual breakdowns, such as the failure to appreciate the signifìcance of checking the solution of an equation, thus regarding the aim of the task as the performance of the solution process, rather than the obtaining of a vai uè of a; which makes the equation true ( Lee & Wheeler, 1987). Some research attention has also been focused on the modes of interpretation of the letter symbol - an object, evaluated as a number, as specifìc unknowns, generalised number or variable (Kuchemann, 1981).

Manipulation errors are discussed extensively in a study of university students by Carry, Lewis and Bernard (1980). These will not be considered further her, except to note that they could be expected to be dealt with by the methods of experiment, cognitive confìict and discussion developed in the Diagnostic Teaching Project (e.g. Bell, et al, 1985; Bell, 1993b).

The more global conceptual breakdowns include the unawareness of how to use algebra to fomulate, to test and to prove a generalisation, how to form an equation from a situation and to interpret its solution; inability to work with an expréssion or equation syntactically; unawareness of how to identify the functional relationships obtained in a formula, or in a set of data.

These abilities are, I suggest, the cruciai acquisitions in the fìeld of algebra at school and early college level, but receive relatively little emphasis in many curricula in comparison with the practice of manipulative skills.


In the remainder of thìs paper I shall review the existing research hearing on known conceptual obstacles in this field; then discuss some curriculum responses to this diagnosis, concluding with remarks about outstanding research questions.

Research on General Algebraic Strategies

Generalising

Generalising was the focus of a study by Lee and Wheeler (1987). It concerned the algebraic thinking of students aged about 15, focusing on their conceptions of generalisation and justifìcation. Three hundred and fifty four students were tested, using 12 problems, and 25 interviews were conducted. The problems demanded the use of algebraic language in reasoning about situations.

The first problem used was:

Is the statement = - defìnitely true? 2x + 1 + 7 8 J y

•possibly true? never true?

Say how you know.

The general conclusion here was 'defìnitely true', but for two distinct reasons -either the 2x's were cancelled, or cross multiplying led to an equation which was solved or at least appeared to be soluble. Those who cancelled were asked to check by putting a numerical value for x, but no-one was able to use this to recognise the truth of the equation for # = 0 but for no other value. Hence the concept of an equation-like statement as potentially true for some values of x, but not ali, and thàt these might be identified by direct checking, was absent. The predisposition was towards applying learnt procedures for cancelling the fraction or for solving the equation.

Similarly, another question asked

What are the mairi differences between the following two algebraic, statements?

(a?-l)'(a; + 2) = 4

(a>-'l)(a? + 2) = x2 + x-2 '•

Both statements were treated primarily as equations to solve; no student remarked on the fact that one was an identity and the other true for (two) particular values of x.

The dominance of manipulation over reasoning was shown also in the following item:

132 A. Bell

Suppose you worked out the following equation as shown below and carne to the conclusion that 20 = 4.

+ . S - T - = 4 2-a? 2 + x

5(2 + x) + 5(2 - a:) = 4 5(2+ x + 2 - x) = 4

5(4) = 4 20 = 4

How do you explain this result?

Here, over naif the students appeared to accept the validity of the line 20 = 4; more than naif of these accepted the entire work, the remainder indicating a problem elsewhere. Only about 20% indicated, implicitly or explicitly, that 20 = 4 was unacceptàble. Some were non-plussed by the non-appearance of a value for x at the end, and wondered whether x was perhaps 20, or 4, or both. None of them thought to test 4, or 20, or any other number by substitution.

The inability to read and use algebraic symbolism in such a way as to work with its meaning was shown in the Lee & Wheeler study by a sequence of questions on odd and even numbers. These were

Show, using algebra, that the sum of two consecutive numbers (te. numbers that follow each other) is always an odd number

HA

The sum of two consecutive number is always an odd number. The product of two consecutive numbers is always an even number. Are these two statements true? If they are can you show why? (Note: consecutive numbers are numbers that follow each other)

IIB

The product of two consecutive whole numbers is an even number. Is this

IIC

The sum of two consecutive number is always an odd number. Can you show why?


ANSWER A ANSWER B ANSWER C

5 + 6 = 11 30 + 31 = 61 176 + 177 = 353

Thése three varied examples show that no matter what numbers you use, if you add them they will be odd.

5 + 6 = 11 30 + 31 = 61 176 + 177 = 353

You're always adding an even number and an odd which always gives òdd.

Which answer do you think is best? Explain why.

5 + 6 = 11 30 + 31 = 61 176 + 177 = 353

Let x and re + 1 be any two consecutive numbers. Their sum is x + (x + 1) = 2x + 1. 2x is even (a multiple of 2). 2x + 1 is odd (even +1).

IID

(These different forms of question were given to different students.) In these

problems students did not generally use algebra; even when requested to do so, they tended

not to base their conclusions on the algebra but rather on their own numerical checks, and

to give verbal arguments, in the style of ANSWER B. Some chose C as the best, but largely

because they felt it looked more 'proper' since it used algebra, not because they had any

sense that it was a more powerful or more complete argument.

Thus, the centrai purpose of algebra was perceived by these students as the

performance of some manipulation; its use as a mode of expression of some generalisations

allowing discussion of the conditions of its truth, or as the basis of an argument, was absent.

Another study of typical 15 year-olds' ability to explain and justify generalisations, also showed a low level of application of algebra. One of the tasks required derivation and explanation of the fact that if the same number is added to 10, and subtracted from 10, and the two results added, the answer is always 20. Only 3 of 41 students even attempted to represent the situation algebraically; the remainder used verbal explanations, often failing to distinguisi! data from conclusion (Bell, 1976).

Formìng and Solving Equations

The forming of equations was the subject of a study by Galvin and Bell (1977).

Here there showed a strong tendency to write the arithmetical calculations required to

solve the problem. Writing an equation to represent the problem, and then working with

the equation instead of with the originai problem, was a major obstacle. (Perhaps this is

the most important difference between arithmetic and algebra). The authors say (page ii):

It appeared that equations represented a quite distinctive form of expression which

was unlikely to be adopted by the pupils spontaneously unless they both recognised

134 A. Bell

that this expression led to the possibility of algorithms for solution, and also that

they had had some practice in the solution of such equations resulting in the solution

of the corresponding problem, that is they needed to know that equations could be

manipulated and solved in order to decide that it was worthwhile trying to formulate

the problem in this way.

The suggestions nere would therefore be that equation solving should be practised before equation-forming; but this carries its own hazards of meaninglessness. A pedagogical solution i s to begin equation-forming in situations where the degree of abstraction is reduced; see the sections below (e.g. p. 50).

One problem used said, 'the tail of a fish weights 4 kilograms; the head weighs as much as the tail and half of the body; the body weighs as much as the head and the tail together; how much does the fish weigh?' The first boy's formulation of this said, 'tail = 4, head = 4 + x — y, body = y + 4' and below this *x = half body, y = head'. The boy then proceeded to solve by trial substitution of various numbers for x,y, the head and the body. There were too many variables in the expression to allow a proper algebraic solution and it appears necessary that the pupil should understand that the most economical solution would be obtained if the minimum number of variables was introduced. The other question which rose was whether it was more advantageous to use suggestive letters, such as h for head and b for body, or conventional algebraic ones, like x and y. It appeared that the use of suggestive letter facilitated the initial writing of equations, but was less helpful at the

h subsequent stage of manipulation. In proceeding from h = - + 4 and b = h + 4 to the

combined equation b = - + 4 + 4, it is necessary to stop thinking about h as head, b as body, and to think of them as symbols which are manipulated and moved around according to algebraic rules. Another example gave the length of various combinatiòns of lengths of a race track. The first length plus the second was 100; the third plus the fourth 200 and there were two further equations. The boy who wrote 'lst + 2nd = 100; 3rd + 4th = 200' immediately wrote 'total length of track = 100 + 200 = 300' but was quite unable to subtract a pair of equations to give the difference between two sections of track. Here the use of suggestive notation seemed to have led to one particular composition and to have inhibited others.

The conditions influencing choice of an arithmetic or algebraic method for solving problems for which both approaches are feasible has been extensively studied very recently. Lins (1992) used a set of problems to identify and distinguish different approaches, with the aim of characterising algebraic thinking, and distinguishing it from analogical thinking, this to be independent of whether or not algebraic symbolism was employed. Analogical thinking worked in the field of the part-whole relation, including multiplication and division


by positive integers, while algebraic thinking displayed an awareness of working in a full numerical semantic field (ie. the field of rational numbers). Some of the problems and solutions are shown.

Question 1. At the right you have a sketch of wooden blocks. A long block and a short block measure 162 cm

altogether. A short blocks measures 28 cm less than a long blocks.

What is the length of each individuai block? (Explain how you solved the problem and why you did it that way)

Question 2. Mr Sweetmann and his family have to drive 261 miles to get from London to Leeds. At a certain point they decided to stop for lunch. After lunch they stili had to drive four times as much as they had already driven. How much did they drive before lunch? And after lunch? (Explain how you solved the problem and how you knew what to do)

Question 3. From a tank fìlled with 745 litre of water. 17 buckets of water were taken. Now there are only 626 litres of water in the tank. How many litres does a bucket hold? (Explain how you solved the problem and why you did it that way)

Question 4. / S \ ^ Z j S » l1**fr George throws away 11 bricks and Sam throws away 5 bricks Now they are balanced

What is the weight of one brick? (Explain how you solved the problem and why you did it that way)

Question 5. I am thinking of a "secret number". I will only teli you that

(6 x secret no.) + 165 = 63

The question is: Which is my secret number? (Explain how you solved the problem and why you did it that way)

Question 6. Maggie and Sandra went to records sale.

136 A. Bell

Maggie took 67 pounds with her, and Sandra took 85 pounds with her (a lot of money!!) Sandra spent four times as mudi money as Maggie spent. As a result, when they left the shop both of them had the same amount of money. How much did each of them spend in the sale?

(Explain how you solved the problem and why you did it that way)

A Gànadian group at CIRADE (Radford, 1992; Bednarz et al, unpubhshed) has

analysed typesof solution to similar problems. A typical example follows (Radford, 1992).

588 passengers must travel on 2 trains. One train has only 16-seat cars, the other has

only 12-seat cars. The train with 16 seat cars will have 8 more cars than the other.

How many cars will be in each train?

(Bednarz, Radford, Janvier, Lepàge, CIRADE)

Typical solutions might proceed as follows:

ARITHMETIC SOLUTION

8 more cars —> 8 x 16 more passengers =128

588-128 = 460

1 car on each train holds 16 + 12 = 28 people

16

28.) 460

28

180

168

12

17 cars needed on each train

17 + 8 = 25

17 x 12 = 204

25 x 16 = 400

ALGEBRAIC SOLUTION x cars of 12 seats —» 12# passengers x + 8 . . . 16 . . . 16(x + 8) 12z + 16(a; + 8) = 588 28»+ 128 =.588 28» = 460 x = 16.4

Algebra leamìng research and the curriculum 137

Note that, in both arithmetic modes of solution, at each step the quantities calculated

are meaningful in relation to the problem; the solver gains steadily increasing knowledge

of the quantitative details. In the algebraic solution, however, once the equation has been

formed using the semantic details of the problem, the transformatiorts lèading to the solution

are governed by syntactic considerations, internai to the symbolic system, and are unrelated

to the seirjantics of the problem - until the final interpretation is made.

Bednarz et al further remark that the trial and adjustment solution is in fact dose

to the algebraic method, instead of supposing there are (say) 8 cars on the first train, we

suppose there are x cars.

These researchers aim to characterise problems which tend to move students towards algebraic methods. They note that in the above problem relations between the key quantities are given (16, 12 places per wagon), but some of the key quantities themselves are unknown (number of wagons). Such problems they cali dìsconnected. This problem is schematised as follows

Q L 4 rfiCùH+tc

4*T» *««s / Wdc.cn

d6j>! f l c« / w a ^ n

To add to this discussion we may contribute the fish problem (see above), which

may be solved by trial but is very difficult by 'semantic' arithmetic methods - but is of a

different type from the train question. Would this also be classed as dìsconnected?

Functions and Formulae

Parts of the Lee and Wheeler study touched on Functions, in looking for the ability

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138 A. Bell

to recognise and express a functional relatiònshìp in sequences of numbers, in dot patterns

and in tables of values.

The drawing on the left represents a set of overlapping rectangles

The first contains 2 dots v

The second contains 6 dots The third contains 12 dots The fourth contains 20 dots

How many dots in the fifth rectangle? How many dots in the hundredth rectangle?

How do you know? How many dots in the nth rectangle?

How do you know?

IVA

• • • • • • • • •

• • • • • • • • • IVB

Suppose the above sequence of dot-triangles is continued according to the same rule, how many dots will there be in (i) the 5th triangle (ii) the lOOth triangle (Hi) thenth triangle?

X y X y - 1 - ì - 1 - 6 - 1 0 - 1 - 3

0 ì 0 0 1 2 1 3 2 3 2 6 3 4 3 9

X y - 5 - 3 - 1

1 3 5

IVC

In each ofthe above tables, what is the rule which tells you the value ofy ifyou know the value of x?

Can you express the rule as a formula, beginning 'y =' ?

A sequence of numbers begins 1 4 7 10 13 16 IVD If it continues in the same regular way, what will be the 7th nmber in the sequence?

the lOOth? the nth?

Algebra leamìng research and the curriculum 139

The generation of possible variables and relations in practical situations -mathematical modelling - is a strong interest amongst curriculum developers, but not very much research exists which documents students' difficulties in perceiving the relevance of algebra to such situations, nor their ability to see how to make the appropriate application. However, material designed to display and evaluate these modelling abilities was developed by Treilibs et al (1980), using problems such as a request to advise the management of a largè supermarket, which is trying to estimate how many of the checkout tills should be operating at any given time. As well as tests of ability to solve the total problems, subtests were given on Generating Variables, Selecting Variables, Generating Relationships, Selecting Relationships and Specifying Questions. On a sample of able 17 year olds, ali the tests (except Selecting Relationships) had low correlations with mathematical attainment as measured by school tests and teacher ratings, indicating that the development of the ability to apply algebra in this way is an area of neglect in the school curriculum.

Conceptual Obstacles

Firth (1974) showed that many 14 year olds students responded to a request to 'add 15 to rr' by asserting that they could not do so until told the value of x; they could not accept the 'unclosed' x + 15 as an answer. However, subsequent work has shown that this need not be a serious problem, given work in suitable contexts. For example, Sutherland (1992) has shown that 10 year old pupils working with a computer spreadsheet can use such expressions; see also the later part of this paper. Another item used by myself (Bell and Low, 1982), following Collis (1975), was

1 1 - 6 = 7 - 1 1

This showed two kinds of error. At one level, the response was '5 ' , indicating a fixation on the early notion that = means 'makes'. (Aspects of this directional approach to the equals sign have been shown to persist even to college level (Kieran, 1981; Mevarech & Yitschak,1983; quoted in Kieran, 1990)). At another level, the response was '6', showing the persistence of an assumption of commutativity of subtraction deriving from experience with naturai numbers. Each of these errors has been shown to be part of a substantial complex of misconceptions.

The so-called students and professors problem (Clement, 1982) has received much attention in the research literature over the last ten years.

In these studies, several groups of university engineering students were presented with the problem:

Write an equation using the variables S and P to represent the following statement: "There are six times as many students as professors at this university. Use S for the

140 A. Bell

number of students and P for the number of professors".

It was found that 37% of the students answered incorrectly; and of these, 68%

represented the problem as 6S = P. A related result was reported by Mevarech and Yitschak

(1983), who found that 38% of the 150 college students they tested answered that, in the

equation 3fc = m, k is greater than m.

MacGregor (1992) also studied this phenomenon; in a test related to her work, I

found some 40% of secondary school students selecting the reversed response to a similar

additive questioni

I have m dollars and you have k dollars. I have $6 more than you. Which equation must be true?

6/c = m 6ra = k fc + 6 = ra m + 6 = A: 6 — m = k

(A smaller minority selected the commuted 6 — m = k).

The first and most obvious explanation offered for these errors was that of

transliteration from the verbal form: six students to every professor => 65 = P. However,

subsequent work showed that though this was true in some cases, the error occurred in

translation to algebra from tabular or diagrammatic presentations. In these cases it appears

to relate to an associative perception of the statement 65 = P, the larger number S being

associateti'with... the 6. Wollman (1983) and MacGregor (1990) have separately conducted

teachirtg experiments which show that improvement results from generating numbers to fit

the equation, thinking which is the larger quantity, and expréssing the relationship verbally

in different ways.

The incorrect direct transliteration of verbal statements into algebra, in which Za 4-46

means 3 apples and 4 bananas (irreverently known as 'fruit-salad algebra') has shown to be

a serious problem (Galvin & Bell 1977, Kiichemann 1981), and is of course éncouraged by

some algebra texts. In Kiichemann's test, only 10% of 14 year olds correctly symbolised

the following problem, as 5ò + 6r — 90, while 17% gave b + r = 90:

Blue pencils cost 5 pence each and red pencils cost 6 pence each. I buy some blue and some red pencils and altogether it costs me 90 pence.

If 6 is the number of blue pencils bought and if r is the number of red pencils bought, what can you write down about 6 and r?

Misconceptions regarding the commutativity of the subtraction and (particularly) the division operation, and their notations, have been widely documented. (Brown 1981, Bell et al 1984, 1989). At this point I shall only remark that whereas in an arithmetic problem confusion or error in the order of a division operation, e.g 7.2 -i- 3 for 3 -r 7.2, may be recognised from the size of the answer and corrected, in considering the rearrangements of


D = S x T to T = D/S or S/D the error is more disastrous. In a question asking which of six such rearrangements (of the verbal formulae) were also true, only 6% of secondary school students chose the correct two. (Bell & Onslow, 1989).

Breakdowns of the equation concept at a higher level have been studied by Filloy and Rojano (1985), (see also Kieran, 1990). They showed that whereas equations containing a single occurrence of the variable (such as Zx — 5 = 22) could often be solved intuitively or by 'backtracking' or 'unpacking', those of type 3x—5 = 2a;+22 or indeed, 3x—5 = 22 — 2x needed a more detached concept of equation.

Some algebraic errors appear initially to be more technical failures, or memory lapses. Examples are adding fractions by adding both numerators and denominators; false cancelling of part of a term; manipulating directed numbers by the implicit rule 'drop sign while operating' (giving e.g —11 — 6 = —(11 — 6) = —5); treating indices as multipliers (32 = 6). However, these ali arise from the failure to make important conceptual distinctions and hence need to be treated by appeal to meanings rather than simple reassertion of a correct rule.

Some Survey Results

Some specific results are as follows (from the British national survey of 15 year olds).

1. Evaluating d3 when d = 3; (44%; 16% gave 9, 25% other, 15% omit). 2. (a) I am x years old. Peter is two years older. What is his age? (49% gave x + 2, 10%

gave 2x.) (b) Similar question for 1 more, 3 less, twice (ali about 45% correct.) (e) Represent the number which is n bigger than 3: (27% correct; 24% n 3.)

3. (a) What is x if 2x + 7 = 45 (73% correct) (b) 2x + Sx + Ax = 2x + 21, what is xl\ (50% correct). (e) If A — L x B tells us how to work out A, what formula tells us how to work out L?

(39% correct).

Any manipulation beyond these levels tended to have facility 30% with some 25% or more omissions.

Àn Approach to the Algebra Curriculum: A Response to the Diagnosis

With this background of research on the current state of understanding and misunderstanding of algebra, I offer suggestions for a curriculum and a pedagogy aimed at dealing with these problems, together with remarks concerning outstanding questions. In general, the approach is to learn the algebraic language in a way similar to that in which the mother tongue is learnt, that is by using it in order to communicate, with oneself and

142 A. Bell

with others, in the course of activities defìned by the three main modes of activity already

described:

1. Generalising.

2.. Forming and Solving Equations.

3. Working with Functions and Formulae.

As we have indicated, each of these strands embodies its own characteristic purposes,

and its own set of general strategies and procedures for achieving them. There is, of

course, some overlap, but fluent use of the language and methods of algebra certainly

includes the ability to handle each of these types of situation confìdently. One might

make an analogy with the various uses of language for description, instruction, expression,

discussion, persuasion and so on. (The now-common recognition of language competence

as including speaking, listening, reading and writing can also suggest illuminating ways of

thinking about mathematical activity).

I propose also a pedagogy consisting of alternation between broad activities

embodying the purposes of algebraic activity (and in particular, of these three strands)

and more focused activities aimed at particular concepts or skills. We have called this

Diagnostic Teaching. These focused activities, consisting mainly of provoking cognitive

conflict and of intensive discussion of criticai conceptual points, have been shown to be

strikingly more effective for longer terni retention than more usuai methods (Bell et al, 1985;

Bell, 1993a, b). In this paper we shall consider the broad activities aimed at developing

strategie abilities to handle the three strands we nave been discussing. Our claim is that

these provide a context in which the specifìc misconceptions can be more easily dealt with.

Generalising

What is to be learned about generalising is the process of exploring a given situation

for patterns and relationships, organising the data systematically, recognising the relations

and expressing them verbally and symbolically, and seeking explanation and appropriate

kinds of justification of proof according to level.

Examples include, at the earlier stages, relations such as are illustrated by 9 = 4+5 =

3 + 6 = 2 + 7 . . . ; 9 + 5 = 14 -+ 19 + 5 = 24....; the digit patterns in 5 ,10,15,20. . . and in

9,18,27,36. . . ; through various patterns in the addition and multiplication tables, Pascal's

triangle and spirai patterns. The same process applies to the study of the number field itself,

with the recognition and expression of the general laws of associativity, commutativity,

inverses and so on.

Divisibility properties, such as that the sum of any three consecutive numbers, the

sum is divisible by 3, provide good examples of the power of algebra to prove general


properties. Here the method of expressing the number following n a s n + 1 will arise; but this does not appear to be a serious obstacle when it is dealt with in context. Similarly, expressing the number with digits x,y as lOx + y is needed for a certain class of problems; two problems of this kind are mentioned in the initial examples above. Learning to read expressions for meaning, e.g. 3(n + 1) as a multiple of 3, is a key part of this work.

Treatihg statements precisely

This is one of the conceptual obstacles in the leaning of generalising. Pupils' tendency to ignore the need to insert brackets in expressions was explained by Booth (1984) as arising from fact that they knew what they meant, so didn't need to bother. I have seen pupils interpreting 8 -f- 0.5 as 4, the division and the 0.5 being used though not co-ordinated as we would wish; and writing 'when you divide by numbers less than 1 it usually gets bigger!' Some tasks aimed at developing the awareness of the possibility and need to treat a statement precisely (or 'literally') are shown. (One might draw an analogy with the way in which the working of resolutions at formai meetings is treated very seriously when important agreements between people are being recorded. Legai language is a similar case.)

Further examples follow, related to the table of counting numbers 1 — 100, the multiplication table and the calendar.

DAYS AND DATES —~

Today is the 3rd of October and it's my birthday on the 23rd. Today is a Monday. So my birthday will be on a Sunday! Great for a party!

Peter

Of emme. If you go 20 days on, it's always one day eariier in the week. ~ — "̂m—

fe IH • '/ ' Rachel

<• 1 b Rachel right? Explaìa 2 What happens if you go forwafd

(a) lOdays; (b) 30days; (e) 40days: (d) 100days?

3 Expuda the rule which gorenu these problems.

ti

144 A. Bell

These generalisations in Days and Dates appear not to have been previously noticed by many 11-12 year olds; so making this link with a possibly everyday situation is a good way of encouraging awareness of mathematica! relations in the real world.

LINE PATTERNS

Take a copy ofMixcd Tablet.

Look al PATTERN PLAY on pago 4

Rnd UM box showing

14

21

28

Notice Inai 14+ 2 8 - 4 2

and2x21-42 ^ H e ^

Try this other box in the tanta veniale" Un*

28 top 35 middle 42 bottom

b it itili trae that T • B - 2M ?

Try a few more, in the same line.

Try tome similar boxes in other iines.

Try hori7onial boxes alta' Does L

left R

righi 2M?

middle

<* Try Iines of 4 and 5 numbers. Write what you find. Use lettera if you wish.

CORNERS AND MIDDLES

Cut out a frame to puf

over this table.

1 i i i i •nr i 1 Il II 11 t 4 i i il IIIM • II 21 a » 1 1 i il n iiln 124 1? 11 a 11

DOLJLJUULiJLJLJLlULIJ

DELJULJLILJLJLJLJUIi] i Minia B|M n N|ll Jl 7? M i sitila «ilm si Min m i n i titsiN|«lM|nln|ii!n|ii!iii inpFini'.MiiLHJLJLjmjuj ii a ni M| MI nini nini lumini a » »l«iMiii|Mmi»Kin,«ii44

* •

• ^ 8 7 8 12 14 16 11 21 24

ai 20

20 25

11 24 30

(the B number

B-A =

the A number) < C-B

(the C number

.A 6 C ile P

MH l.

the B number)

Check that this is true in both the boxes shown.

Check other places on the table.

So B - A » C - B is always truè

• ConsiderA+D*F + I

This is never true. Check this.

• Consider E » ?2A

This is sometima true. Check this.

• With your partner, make a list of 6 relations and challenge

another paid to decide whether they are always, never or

sometimes true. But sure of your answers first.

Try to get some of each type.

Line Patterns and Corners and Mìddles have proved in our trials to be good ways of getting pupils into using algebraic language in situations where it forms a naturai means of communication. Note that checking, and the possibility of a relation being true always, sometimes or never, are built in also.

Some more extended individuai explorations of these situations are shown; these were from 13-14 year olds, but could be used earlier with faster pupils.

Julia is exploring patterns found by making L shaped boxes around some numbers in an old calendar. Notice how she has generalised the pattern with algebra in two different ways.

Julia compares the sums of numbers in the boxes, the difference in the numerical case is 44. We might expect another numerical example at this point, but instead she introduces an x for the number in the centrai celi, and obtains the difference as (4x -f 22) - (4x - 22) = 44, showing that the pattern is true everywhere on the square.


ânsgrà^r

\Mr. To\u ^ A

Tue

fr.

I.

3

15

ir 5

»/

12. 13

7 1f

t ^ * ^

ia

X I 2<*

L*3

Xf

24

3 /

vV>ae*#»- c«rnx\n rvmoena \p

v*«. cêtsôx- a n à W ^ a.oà

To"? \ VA Wto sv^\a S x S - f i f e 3.01 cVktfSe lSea«i. Co^NNo\f%Ìôfft ©V t \ o r v W s . 7 l / ^ i n l T L - \ V \ V \Y L

74- -S£> - * * .

— l l ^ + * T X * 7 * • * * « + X ' f l 3LX

QtX f X ^ W ^ - x - XX> = 4*. . i v * <Mt<

* T*l U H *

it */5

t H

X t 7 t x U t i ^ X<-2. - Vx + to

tfrX*+5<*5-(<|.x*io> r U - O^CUA.

These students were encouraged to experiment and find their own such patterns for different shapes on the calendar; and to include some which did not give a regular pattern. The second example is one in which the non-generality of the pattern is shown by the algebra; the difference x — 3 means, as the student says,

"the answer isn't always 13, but, in the case I took, x — 3 = 13. So the difference

will vary, depending on where the shape is".

146 A. Bell

These non-general relations show" more vividly than the general ones how the use

of a letter as a generalised number demonstrates the generality or non-generality of the

proposed pattern.

rfall*" fc'V

: €éuh group. 1h&&, comi fa 5-3 <îd èL co'M a

IÓS+-I ^C re

2cJf<\

t

x-f-'s-

fìofuA\ ) cMeJ tw iboo ^roapS Up and jèak

ÛrSîA* ^ ZZC4XI

*1his means

Algebra leaming research and the curriculum 147

A more ambitious pattern, using multiplication, is that of James.

P r o v i t i toOubii P\\ai.hrq. ^Q^£±.

r^v-.IKp*w fcVvA ^ u ^ e r S - »-» H/va. tu«»o fcs>P C o r r i e r i

C o r ^ f t r i to f t t^ve/ " . X t^\a^> É-'v̂ cA "cê. dUfferci*^ Cfl. Vaet-i^»

6̂»)i yyjpsm rao 7 & « 8 0 » &CgO 77<cO 6 O S O

\ &&o -t o, 0» o

bowU t^s*. tU\jz. ^ A W U r—ejctnokêî. «»s above.-

laULUr*1 ^-M r i r^ J

; (>ca - l u x - >uo)- (â+t*jc)

C X tws*U>-f

In

= » 5 3 - 0 t 1 uo .

z \fc&C

Cor>CJ

148 A. Bell

This is a point at which the broad pattern-finding activity was alternated with the

more focused activity, in this case discussing how to multiply brackets. Throughout, periods

of exploratory work alternated with lessòns on specific manipulations. Subtracting brackets

was discussed at an earlier point. These laws were approached first in an experimental

mode, using cognitive conflict, expressions such as 13 — (6 — 2) being compared with

1 3 - 6 - 2 and the discrepancy discussed, before giving a lot of exercises for practising

the correct translation. Similarly, products such as 17 x 13 were discussed to lead into

(a + 6)(c + cfj;

The pedagogical principles here are that when the x is introduced it is clear that it stands for the number in a certain celi of the table; and the purpose of the manipulation is to establish the generality of the pattern. Thus we avoid the incomprehension which often arises when manipulative exercises are given in the absence of the meaning-giving formulation and interpretation stages. A similar rationale exists in the spread sheet situation (Sutherland, 1992).

Arìthmagons

This is another situation rich in possibilities for generalisation. We get

generalised numbers in a concrete setting, and also into recognising equivalent

without recourse to prescribed rules.

In Arìthmagons, the first task is to

fili the circles A, B, C, D so that the

number in each square is equal to the sum

of the numbers in the adjacent circles.

Students will find different solu-

tions and these are tabulated on the board:

A B CD

into using

equations,

2 5 1 8

1 6 2 7

3 4 0 9

0 7 3 6

After some consideration it is usually clear that, if we are working with non-negative

integers, 0, 1, 2 and 3 are the only possibilities for A and C, so that this is indeed a full

set. Looking for a generalisation, we see that the number of these is F + 1 or, in general,

one more than the smallest of E, F, G, H. Next, students are asked what relations they can

see between the columns of the table which are true for every row - for example, B - C =

4. A list something Hke the following will emerge.


B -C = 4.(1) A + B = l (4) A + C = 3(7) D = A + 6(2) J + . B + C + D = 16 (5) £ - 4 = C (8) C - D = 9(3) C + 4 = JB (6)

Note that we are now using A, B, C, D to denote any member of the column with that label; these are generalised numbers, in a concrete setting.

We can then solve questions such as "Is this the complete set?" The fact that B - C = 4, C + 4 = B and B - 4 = C are essentially duplicates of the same information may be noted; we may deléte two of these. The absence of B + D - 13 (9) may be noted. Then the possibility of obtaining some of these from others may be raised; for example (3) and (5) entail (4). (1) and (3) maybe seen to imply (9); and so on. A minimal set may be found from which the whole set of relations may be generated. In this way, a very important idea may be met in a quite elementary setting; and also, an intuitive familiarity is built up with the possible transformations of a given additive relationship. Later, a systematic look at such a set will lead to the identification of formai rules which summarise the possible transformations of the given additive relationship.

Forming and Solving Equations

The earliest examples may be missing number problems such as8 + ? =11 or? -1 -5. The corresponding verbal or situational problems at this stage are not normally solved by representing the problem symbolically and transforming it, but by a mental transformation into the solution form. Thus a problem asking how many marbles Jane had at first, if she gave away 7 and had 5 left, would be answered directly by adding 7 and 5. As later work progresses to more complex problems, involving first one operation, then several, the same tension arises between those which can be resolved mentally and those which require symbolic representation of one or more equations which are then solved by manipulation. Learning solution methods for the different types of equation is important here.

At a later stage, this process is centrai to design problems, where there are sizes or quantities to be chosen, and there are constraints to be met. An example is the design of a box of a given shape, to nave a given volume. A more complex case is the general transportation problem where various quantities of goods are to be carried from a number of sources to a number of destinations, the cost of each journey being known; this may involve forming and solving a set of linear equations. The same general process is at the core of many of the problems and puzzles which have fascinated men throughout the history of mathematics. Here are two early ones:

150 A. Bell

Baker 1568 AD:

One man demanded of another in the morning what o'clok it was y other made him

this answer: yf you doo adde (sayth hee) the 1/4 of the howers which be past midnight

with the 2/3 of the howers which are to come until noone, you shall haue the just

hower, that is to saye, you shall know what o'clok it is.

Calandri 1491 A.D:

A lion can eat a sheep in one day, a leopard can eat it in two days, and a wolf in

three days. How long will it take ali three?

The first of these leads to the equation

I/Ah + 2/3(12 -h) = h -

and the somewhat unsatisfying solution 5 11/17. The second may be solved by considering

the reciprocals of the rates given i.e a leopards daily meal is 1/2 of a sheep, a wolf's is 1/3

of a sheep, so in a day ali three eat 1 5/6 sheep or one sheep in 6/11 day.

(Bath filling problems and problems of electric resistances in parallel have the same structure; they require what we might cali the reciprocai method, or the 'harmonic method' by analogy with the harmonic mean).

These two problems together perhaps illustrate how algebraic symbolism becomes

advantageous for more complex problems. They also remind us of the motivating power

of a good puzzle, which we should aim to retain in school work.

Situations encouraging forming and solving equations

Children's first introduction to the notion of extracting an unknown number from a statement expressing constraints may well be through missing addend questions. Later, steps towards more systematic equation forming and solving may be taken by setting up diagrammatic situations containing hidden numbers.

These have the advantage that (a) they are self-checking, and (b) the unknowns have

an obvious concrete existence, thus avoiding difficulties associated with having to decide

what quantity to denote by x and how to translate from verbal information into symbolic

statements. Arithmagons (see above) provide one example; others follow. These ali provide

ways of familiarising students with symbolic methods in easy contexts, so that the adoption

of algebraic rather than arithmetic methods for harder problems is accepted more readily.

Other examples of diagrammatic situations

Other examples of possible situations are Pyramids and Number Routes. In Pyramids

the construction rule is that a lower number is the sum of the two adjacent ones above it.


21

5 X 4

5+x x+4

21

5 + a;-f:r + 4 = 21

2z-f9 = 21

2z = 12

x = 6

Put in 6 for x\ it is seen to be correct

The mode of writing the equation may be suggested by the teacher but is normally readily adopted. The collection of terms and solution of the equation may similarly be the subject of discussion but is normally adopted as self-evidently good by the pupils. By changing the construction rule from 'A + B' to A + 2B, and then to A - B, more difficult manipulations can be made to arise. These would need dealing with by focused discussions, with experiment to show the validity of laws 2 ( A + B) = 2A + 2B and A - (B - C) = A -B + C.

Number Routes is another type of diagram which can act as a concrete setting for equations.

( 5 x 7 ) - l l l = : F

(S x 4) = F .

So (S x 7) - 111 = S x 4

or 7S - 1 1 1 = 45

So 35 = 111

and S = 111/3 = 37

Check: 7 x 37 = 259

259 - 111 = 148

37 x 4 = 148

Number Routes can be made harder by, for example, having addition (or subtraction) before multiplication, which gives rise to the distributive law, and by making longer paths

152 A. Bell

and more complex networks. Students can be asked to make up their own examples, and to pitch thern at a level of difficulty suitable for themselves.

Forming Equations from Verbal Problems

The activities discussed nere nave had letters embedded in thern from the outset. ,, We now need to considér cases where equations have to be formed from verbal situations. Students (aged about 13) were given two generic examples, which were discussed, then asked to make up (in groups) similar problems to be solved by another group. The first two problems given were the following.

1. There are two piles of stones. The second has 19 more stones than the first. There

are 133 stones altogether. Find the number in each pile.

2. 3 piles; the first has 5 less than the third, and the second has 15 more than the third.

There are 31 altogether.

Students were asked to solve the first problem, no method being specified.

Most solved the first numerically; setting, correctly, 57 and 76 or, wrongly, 47 1/2 and 66 1/2. I then showed thern the algebraic method, taking x for the first pile, x + 19 for the second, obtaining 2x -f 19 = 133 and solving to give x = 57, and 76 for the two piles. They were then asked to take x for the second pile.

This gave x - l§ -{- x — 133, # = 76, and the same numbers 57 and 76 for the two piles. The aim here was to display the possibility of different x assignments, and to observe the different expressions and equations which resulted, and to note the appearance of the same solution for the size of the two piles; and thus to get some insight into the relation between the algebra and the problem.

Following this they were asked to work, in groups of three, at solving the next question, taking in turn each of the three piles as x\ and to compare their results. On the following day each group was asked to make up and solve 3 similar problems, two easy and one hard, to be attempted by another group.

This led to a lot of insight into the way different x-assignments affected the expressions, turning -f into — and multiples into fractions. It also led to an unexpected degree of richness in the problem statements. As well as four bean bags and the number of pupils in three rivai schools, we had

"A nuclear scientist must complete 4 experiments to save the world, and has 23 days

to do thern in. The first will take twice as long as the second ..."

To conclude, we may say that the main initial difficulties lay in expressing relations such as 'pile 3 has 15 more stones than pile 2', when pile 2 was x, making pile 3 'x +


15'; and more so when pile 3 was x, needing a reversai to make pile 2 'x - 15'. '10 less than x + 15' was another step up in difficulty. However, although this was observed as a serious obstacle for some students in the early lessons, on being offered the answer, they soon picked up the idea, and in the school examination quéstion on this work, no student failed to formulate àn equation, though there were some 'reversal' mistakes.

Intuitive and visual approaches to algebraic language

The research quoted above shows a strong need for activitiés in which algebraic expressions and equations are treated by common sense and normal understanding rather than by a number of special rules. 'Educating the intuititions' describes one aspect of such an approach. Arithmagons (above) provides one example in which correct transformations of equations are apprehended directly, as 'saying the same thing in a different way'. The following situation provides another.

Giving Clues

This is another generic situation which leads naturally to the intuitive recógnition of equivalent equations. It also provides a body of experience on which the study of sets of linear equations can be built.

(L-Z)/M = N 1 3 M - L = 1 2 4N-M = 2 3

These equations were put on the board, with the statement that the teacher had chosen three numbers, denoted them by L, M and N, and written down these three clues by which they might be found. The game was for each person to try to find the numbers (by trial and error or any other method). When a person had found them and had checked that they satisfied ali three equations, she was not to say what they were, but was to make up a new due and offer it to be added to the list.

It took some 5 minutes for the first person to find the numbers, and after that clues carne along quite quickly. As more clues were added to the list, these provided additional help to those who had not yet found the numbers.

L = 2M + N . 4

N-L = 2M 5

L - 2M = 1 6

2M = L-N 7

N + L = 3M 8

2L = 4M + 2AT 9

154 A. Bell

L = M2 + N , 10

L-AN = 1 11 L + M + N = 8 12

L = 5N 13

L-AN = N 14

2N = M 15

L-N = W 16 The discussion of this list included, as well as identifying duplicates, considering which

were good or useful clues - that is those containing just two of the letters. One letter clues

were rejected as 'give-aways'.

Subsequent work led to considering ways of starting with such a set and having

a general routine for fìnding the hidden numbers. This consisted of combining equations

so as to get a pair both of which contain the same letters (such as (11) and (13)), then

combining or comparing them. The method of 'matching' consisted of working towards a

pair such as

x + 4y = 9

x -f 6y = 15

from which 2y = 6 can be seen intuitively.

Tvît are ÌCYML ftiut f v*q ,n a bcujl

1 ba/ionoj and l.cofflic.hùvii (±1 maqgotS 1 ca<Y0t a/i4 2 Qfpl*$ favo. 37 mafflotS HÂÌ Aaa/iM rv\aôtf cfa&j tiach Ouit M i M ,

S a x l b - ^ « I b + C - > 7

e + . l a - V i

Ac*d 'hjjttKflr- SQ tó-U + le - \ \1

We dose this section by giving

some examples of problems made up

by pupils; they indicate the experimental

approach which the pupils had, and their

sense of control over the situation.

This is an interesting example,

and successfully solved. But note the

attraction for adding ali the equations,

which might not have been helpful had

the second equation been different.

* CIAC! *

2tQ -\T\ q rS


Making up problems gives the pupils a strong sense of ownership of the mathematics, and an outlet for creativity and humour. It also has a diagnostic value, in showing the teacher what aspects of the situation have been recognised by the pupils.

y

And 2 cdwfùi ì&«c?lfl rW tt emù fot HI/VI éfeOO •

Uè KAA«) & IWQ/I wno Ov«J dai/i b^ roa</ <*Ho haa 2f»»S m*té of-jo/tf. He had enfi s'rtna't, rW only o/^ (Wia(j AAC* it.onfy.Gttt torvi £]oò

H: carneo Rum cw to y?u, Ucw rv»uch cloiln *acK ttoro GaSt ?

I. p4lSi-2c -4oo 2 IffìrCzZco

f + 2S-UC :boo-2f-r StC -3oO ^ *-f-t lS*C =200

Working with Functions and Formulae

In work on Functions and Formulae, the typical situation is either a practical situation which generates a sequence of numbers in some way, for example, a row of squares made with matchsticks, or polygons of increasing size made on a peg board; or a set of practical or experimental data, such as prices of a ferry crossing for cars of different lengths. The usuai task would be to determine the rule defining the function to express it verbally (and later, symbolically) so as to be able to interpolate and extrapolate from the data actually given to predict for other cases. The shape of the corresponding graph might be of interest; comparisons being made between the shape of the graphs obtained and those of known functions. When an organised set of sequences is studied it becomes clear that their difference patterns play an important part in recognising and classifying the different types. Thus one gathers knowledge of a variety of kinds of function, linear, square, inverse, exponential, wave (sine or cos), together with the appropriate algebraic formulae. Equation solving to determine values comes in as a part of this study.

156 A. Bell

Reading and writing algebraic formulae

We quote some questions which càn forni starting points for focused discussion on

this point.

1. If three numbers, A, B and C are related so that A x B = C, in what other ways can the relations be expressed?

2. If A = L x B how do you get L if you know A and B? 3. If V = 7rr2/i, and h is doubled while r is fixed, what happens to V? Similarly, if r is

doubled while K is fixed? And if r is doubled but V kept fixed, what must happen to hi

4. Which of the following formulae do you think is most likely to be correct? Underline your answer. Give reasons. (k, a and b are constants)

Force needed on pedals to ride bike with speed v mph. F = kv F = kv2 F = k/v F^k/v2

Pulì on the earth on a satellite at height h. P = kh P = kh2 F = k/h F = k/h2

Number of marbles of diameter d in a kilogram. N = kd N = k/d N = k/d2 N = k/d3

Stopping distance of a car with speed v mph. D — av + b/v2 D — av

These questions were included in a recent study of our own, they were very poorly

done by otherwise competent students aged 12-14. In other questions the pupils were asked

to construct and select valid rearrangements of the formulae C = VR (for current, voltage

and resistance), S = DT (Speed, distance and time) an A = B — C (three numbers); the

last was given (in another section of the test) also using the fraction bar A = — (written

vertically, B over C). Later in the test, they were asked to say which most closely described

their way of thinking about these questions:

Look back at the question on C, V, R. Tick the one or two of the following statements which is closest to the way you were thinking

a I though which would be the biggest number, so the others would be divided into it.

b I tried some actual numbers

e I just remembered the formulae

d I thought the one on top of the right hand side would go underneath on the

other side.

Of the school pupils, aged 14-16, about 40% reported trying actual numbers,a and

a further 25% thinking which was the big number. 20% just remembered the formulae


and only 10% reported thinking of physieal movements of the symbols. But of the two samples of 'experts' - teachers or prospective teachers in training - the great majority reported thinking of physieal movements,a and none considered numbers or sizes. It was also noticeable that, for the school pupils, the fraction bar form was slightly harder than the division sign, whereas some of the experts reported imagining the latter changed into a fraction bar form for working with (Bell and Malone, 1992).

Whàt would be valuable here is a study of the process by which conscious working with the meanings of the expressions in terms of numbers develops into fluent manipulation of the symbols as objects. We know that breakdowns often occur in this process, for example, that 'change the side, change the sign' is often misapplied. The questions are (a) whether more conscious discussion of the visual aspeets of the symbol movements can be used to help develop a more refined and correct awareness, (b) how far well-chosen trial numbers can assist the transition, (e) how experts monitor their manipulations, particularly when 'chunks' rather than single letters, are the objects (as when V = 7rr2h becomes h = V7(7rr2)).

Visual aspeets of algebraic manipulation may be of more importance than has been recently recognised. Collis (1974) experimented with 14 year olds' responses to items such as the following:

Find the relation between x and y (or x,y and z) if:

(1) x + b = b (2) x + 5 = a 2/ + 5 = 6 y + z + 5 = a

It is clear that in these cases, the perceptual aspeets of the presentation support the

correct responses. But in an item such as

x- 15 = 281

x - 1 6 = ?

the perception needs to be slightly more sophisticated. Collis identified matching and

substituting as fundamental algebraic operations, and material using and developing these

was incorporated in some experimental curriculum materials (Bell, Wigley and Rooke,

1978).

Larkin (1989) developed a system of nested blocks to represent questions such as

6 - 3(# 4- 5) = 20; otherwise there have been few attempts to work consciously on the

perceptual features of algebraic expressions.

Substituting - or imagining chunks as single entities - was also a feature of mathematical thinking discussed by Krutetskii (1976) - for example, the recognition. of

158 A. Bell

Studies of these phenomena would compare cases where (a) the visual aspects

support correct comprehension, (b) where they do not, (e) where the correct response

can be visually conditioned such as with software producing a continuously running screen

display such as the following:

s = t + 1

s — 1 = (pause) Show t

s — 2 = (pause) Show t — 1

Compare the existing programs AUTOSUM, AUTOFRAC (ITMA/Longman, 1983).

REFERENCES

BELL A.W, 1976, The Learning of General Mathematica! Strategies. Shell Centre for Mathematical Education, University of Nottingham.

BELL A.W., Low B.C., 1982, Additive Problems in Everyday Situations, Shell Centre for Mathematical Education, University of Nottingham.

BELL A.W., FISCHBEIN E., GREER B., 1984, Choice ofoperation in verbal arithmetic problems: the effecis ofnumbersize, problem structure and context, Educational Studies in Mathematics, 15, 129-147.

BELL A.W., ONSLOW B., 1987, Multiplicative Structures; development of the Understanding of Rates/Intensive Quantities, Proceedings of Eleventh International Conference for the Psychology of Mathematics Education, 2, 275-281.

BELL A.W., ONSLOW B., PRATT K., PURDY D., SWAN M.B., 1985, Dìagnostic Teaching: Teaching far Long Term Learning, Report of ESRC project 8491/1, Shell Centre for Mathematical Education, University of Nottingham.

BELL A.W., GREER B,, GRIMISON L., MANGAN C , 1989, Children's Performance on Multiplicative Word Problems: Elements of a Descriptive Theory, Journal for Research in Mathematics Education 20, (5), 434-449.

BELL A.W., 1993a (in press), Principles of Teaching Design, Educational Studies in Mathematics. BELL A.W., 1993b (in press), Some Experiments in Diagnostic Teaching, Educational Studies in

Mathematics. BROWN, 1981, Number Operations, in: K. Hart (ed.), Children's Understanding of Mathematics

11-16, London: John Murray, 102-119. CARRY L., LEWIS R., BERNARD J., 1980, A Psychology ofEquation Solving, University of Austin,

Texas. CLEMENT J., 1982, Algebra word problem solutions: Thought processes underlying a common

misconception. Journal for Research in Mathematics Education, 12, 16-30. COLLIS K.F., 1975, Concrete and Formai Operations in School Mathematics, Hawthorn, Victoria:

Australian Council for Educational Research. FILLOY E., ROJANO T., 1985, Obstructions to the acquisitìon of dementai algebraic concepts and

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Alan BELL

Shell Centre for Mathematical Education

University Park, Nottingham NG7 2RD, England.

e-mail: rszab® unicorn.nott.ac.uk

Lavoro pervenuto in redazione il 2/8/93

http://unicorn.nott.ac.uk

A. Bell ALGEBRA LEARNING RESEARCH AND THE ... learning research and the curriculum 131 In the...

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