The arithmetic connection

LESLEY LEE AND DAVID WHEELER

T H E A R I T H M E T I C C O N N E C T I O N

ABSTRACT. From test and interview data obtained during an investigation into Grade 10 students' conceptions of algebraic generalisation and justification, we have extracted evidence of the extent to which these students have coordinated the "worlds" of arithmetic and algebra, and can move freely between them. The data show more dissociation than we expected, even among students who were successful at standard algebraic tasks. Conceiv- ing algebra as "generalised arithmetic" may obscure the many genuine obstacles that the learner has to overcome in moving from fluent performance in arithmetic to fluent performance in algebra while achieving and maintaining a smooth coordination of both modes of action.

Historically, algebra grew out of arithmetic and

it ought so to grow afresh for each individual.

(Mathemat ica l Association, The Teaching of Algebra in Schools, p. 5)

Our investigations into the algebraic thinking of high school students

show that the connect ion between the algebra and arithmetic in their

minds is not always as direct and transparent as the quoted precept might

suggest. In t roducing algebra to beginning students as "generalised arith-

metic" may be a sensible strategy but there are distinct pedagogical

difficulties to be faced if it is adopted. Our account o f some of the

misconceptions that can occur at the ari thmetic/algebra interface may give

some pointers to the character o f these pedagogical difficulties.

Fo r this article we have drawn on the results o f an open-ended test

administered to 350 Grade 10 students (ages 15-16), on interviews with 25

of these students, as well as some interviews f rom a pilot project under-

taken two years ago. Our main study was designed to examine students '

conceptions o f generalisation and justification. Preliminary findings have

been reported in Lee and Wheeler (1986), and a fuller report is available

in Lee and Wheeler (1987). Here, in this article, we focus on what our

data tell us about the coordinat ion o f arithmetic and algebra in the

students ' minds as this is indicated by the extent to which they seem able

to move into and out of each o f these worlds at will. In general we find serious lapses in their coordinat ion o f these two worlds though of course

not all students exhibit or express the same degree o f dissociation between

them.

Educational Studies in Mathematics 20 (1989) 41-54. �9 1989 by Kluwer Academic Publishers.

42 LESLEY LEE AND DAVID WHEELER

MOVING FROM ALGEBRA INTO ARITHMETIC

One of the four t~pes of questions we used in testing and interviewing students confronted the student with algebraic statements such as:

2 x + 1 1 1 1 1 - - - " and (aZ+b2)3=a6+b 6. 2 x + 1 + 7 8' 6n 3n 3n'

We asked in each case "Is the statement definitely true? possibly true? never true? Say how you know." There are two types of response to the request to "say how you know". For the first statement, about equal numbers of students solved the equation to get x = 0 or quoted a rule concerning the cancelling of the 2x's. In the next, students either took the left hand side of the statement and manipulated it, correctly or incorrectly, to see if it could be made to look like the right hand side, or they quoted an algebraic rule or law (often of their own invention) which they felt covered the situation. Half of the students given the third statement justified its " t ruth" by producing an exponent law and a third of these recognised there would be "mixed terms". Of the 268 students given one or other of these problems only 10 made any attempt to "check with numbers" and only one of these was responding to the third question involving exponents where the production of a single counter-example would have been much more efficient than the cubing of aZ+ b 2.

The rule-bound approach of these students can be illustrated by looking at responses to this third statement. In defence of the truth of the exponential statement we were offered some very elaborate explanations such as this written one given by a student from an accelerated group:

"This statement is definitely true. There are several laws in dealing with exponents. And the one that applies here is you multiply the number (outside the bracket) with those exponents inside the bracket. You don' t add them like you normally do. If you had an example like a 2 + a 3, you add them so you get a 5 but the bracket tells us to multiply."

or this oral defence of the truth of the statement:

"Well the numbers and letter or letters that the number is attached to is timesed by the number outside the bracket. Since there is a plus sign it takes each number separately. Let's say the equation read as follows ( a 6 + b 6 ) 3 then the answer would read a~8+ b j8 but if the equation read (a4b4) 4 then the answer would read a16b16, without the plus the numbers

are sort of pulled together. The reason ! believe this is true is because if you took (a 2 + b2) 3 and added each one like this (a 2 + b2)(a 2 + b2)(a 2 + b 2)

THE ARITHMETIC CONNECTION 43

then by adding each a-equation in each bracket your answer would read a 2 + a 2 + a 2 which equals a 6 and the same goes for b. It all depends on the

sign and the number outside the brackets."

Other students were content to say "It 's a rule" or to write a 2x3 + b 2x3.

Rules were also produced to justify a s + b s and a 8 + b s.

And yet students gave some hints that they see these "rules" as somewhat arbitrary. One student who had given the rule which would make (a 2 + b2) 3 = a 8 + b 8 said: "On the other hand l could forget grade 8math and

it could be that when you have a number inside the bracket you multiply it by

the outer n u m b e r . . . " which would give a 6 + b 6. Others said, "It 's definitely true, from what we have learned in school but it is just a theory like everything

in math"; "The statement is definitely true i f I were to base my answer on

today's mathematics." One student indicated that if the expansion were done by finding the product of (a 2 + b 2) (a 2 + b 2) (a 2 + b 2) there would be mixed

terms, but if it were done by multiplying exponents, a 2• 3+ b2• 3, then the

statement as given would be true. The one student who did substitute numbers also succeeded in demon-

strating the truth of the statement. Using a = 1 and b = 2 he wrote ( 1 2 + 2 2 ) 3 = ( 1 + 4 ) 3 = 1 + 6 4 = 6 5 . In spite of the use of numerals the

student did not move into arithmetic since the 1 and the 4 were not added before cubing. One might say that he too remained in the world of algebra. In fact all the students who worked this question did so from a strictly algebraic perspective in spite of the definite advantages of looking at it from

an arithmetic perspective where a single numeric example could immediately settle the matter.

In the interviews, when students produced an erroneous "rule" conclusion we asked them to try a few numbers. One might say we pushed them into the world of arithmetic. The substitution request was often made quite specific in that simple values were suggested for the variables. Where students had declared algebraic equality, substitution with numbers produced inequality. This did not always generate the expected disequilibrium in the students. Some students when questioned about this indicated that they did not really expect the same result in arithmetic. A student doing the ( a 2 + b2) 3 problem in our pilot project explained that numbers and letters

behaved differently and so could be expected to give different results. In most of the interviews, however, we told students that both their results could not be correct and in essence forced them to choose between their algebra and their arithmetic. Most in this case voted for the result given by their arithmetic and suspected that in the algebra they had used the "wrong rule".


In the case of the statement

2 x + 1 1

2 x + 1 + 7 8 '

we asked all three interview students who cancelled the 2x's to check their response for a certain value of x. This led to general confusion. One said "It 's not the same when you substitute a number", but confronted with the possibilities of crossing out or not, he tended to favor crossing out. On the other hand he did waver, considering, "This is a whole equation so I can't

just take part of it and cancel out." He realized that he did work with the number example and the algebra in different ways. When asked "Which is the right way to test that one?", he replied, "That's where I 'm always stuck

in math." Another student who expected her substitution x = 1 to work out, immediately questioned her algebraic cancelling when it didn't. She said, " I guess I shouldn't have taken the 2x and 2x out for some reason", but didn't really know why. She admitted that in arithmetic you can't cancel

the 2's (she was putting x = 1) "because of the rest of the stuff around it. It gets in the way". In her second attempt at the problem she cancelled the 2x + 1 and referring to her previous attempt, said " I forgot to cancel the 1

in the first try". When a second substitution attempt did not work out she eventually remembered that " I can only cancel with multiplying". The third student also suspected that cancelling the 2x's was the problem after two

numerical examples didn't work out. She too could not say why and decided to abandon the problem.

Several points are worth underlining here. Firstly, not one of these students (like their test counterparts) had established a "substitution" reflex to check their work. Secondly, when a check was imposed on them it did not seem to serve as a corrective tool. Rather it placed them in a dilemma and seemed to force them toward a choice between their algebraic and arithmetic behaviors. Two of the three interviewees seemed to trust their arithmetic behavior slightly more but could not use it to resolve the dilemma or to throw any light on their algebra. The third seemed to be used to living with arithmetic/algebra contradictions but leaned more toward algebraic behavior.

We see here another illustration of how algebra and arithmetic are two dissociated worlds for these students. They do not spontaneously think of going from algebra into arithmetic and when they are pushed to do so their algebra is not instructed by their arithmetic as one would suppose it ought to be if they perceived algebra as generalised arithmetic.


A problem of a type similar to the above three was worded slightly

differently. Students were given an erroneous algebraic development of

5/(2 - x) + 5/(2 + x) = 4 which led to the conclusion that 20 = 4 and asked

to "explain this result". Only a quarter of the students we tested indicated

either explicitly or implicitly that 20 cannot be equal to 4:

"As far as I know in mathematics 20 cannot be equal to 4. Maybe

5/(2 - x) + 5(2 + x) :/: 4."

"The answer is fine. It works. But it is not logical. 20 cannot = 4. Plain

numbers can' t equal other plain numbers!"

A third of the students went along entirely with the erroneous algebraic

development and its conclusion, 20 = 4:

"Well by following all the proper procedures to get to the result of four

should explain everything for instance 5 • 4 = 20 which is true. But if you

want step by step explanation how you got to th i sanswer well then,

step.

1) find lowest common denominator which is ( 2 - x)(2 + x) 2) then multiply them together with 5.

3) You'll end up with 5(2 + x + 2 - x)

4) which then equals 5(4) = 4

Result 20 = 4."

"Then 5(4) 5 • 4 is equal to 20 which shows your value for 4. In this case

4 is used as a variable not using its original meaning."

And an eighth of the students felt that one cannot stop at 20 = 4 but should

either simplify this:

"The answer will be 4 - 20 or 2 0 - 4"

"Dividing by 4, 5 = 1"

"0 = - 16" "Yes it could be 20 = 4 or 1/5 depending on the sort of answer asked for."

or introduce an x into the answer.

"Results should be x = 4." "x = 4 or x = 20"

As in the previous problems, students gave a justification by rule for the algebraic development. That these "rules" could lead to a result which is nonsense in arithmetic did not appear to be a problem for the majority of these students. For many the main problem with "the answer" was that


they did not get a value of x. It was the unexpected algebraic result that led many students to question the algebraic development, although very few were able to identify the error. Once again students behaved as though algebra were a closed system untroubled by arithmetic.

The general acceptance of 20 = 4 was not the only manifestation of something having gone wrong arithmetically. On looking at the arithmetical work done in this series of four problems, one suspects that some regression had taken place. One does not expect even grade seven students to exhibit some of the arithmetical behaviour of these grade 10 students. As we have remarked before, some students did not even seem to see the possibility of effecting simple sums and differences of whole numbers: for example, (1 + 4) 3 was written as 1 + 64, (4 + 9) 3 as 43 + 93, and the (8 - 1) in the expression 2 • (8 - 1) was not read as 7. Some work with fractions was also strange. A student who was putting n = 1/4 in the expression 1/6n - 1/3n = l/3n wrote the following:

1 1 1 1 1 1 6(�88 3(�88188 6�88 3�88

6 ~ - 3�88 = 3�88

Examples also occurred of the "classical" error of subtracting fractions by subtracting their numerators and denominators separately. One student we interviewed was asked to do 1/6 - 1/3. Subtracting numerators and denominators she got 0/3. Similarly she reduced 1 / 4 - 1/2 to 0/2. Then looking at her algebraic work (where 0 had been replaced by 1) she changed the numerators in both cases to 1 because "0 somethings still has to be

something". Yet when given 1 / 3 - 1/6 she proceeded to find a common denominator and correctly carried out the subtraction. Asked why she used a different method here, she said "3 minus 6, I can't do it". Zero is a number that takes on a mysterious realm of meanings for these algebra students. The above student felt that when 0 occurred in a numerator it ought to be replaced by 1. Many students did not feel 0 is a full-fledged candidate for a solution to an equation. Some interpreted a 0 solution as the null set or as an exception: "No because for 0 it would be true because 0 makes everything 0", "0 would be the only exception", "x belongs to the null set except when x is 0". One student used 0/4 in a long substitution attempt without ever treating it as 0 and another said that an x cannot be equal to 0. Not only did these students move with great difficulty from algebra into arithmetic in these problems but their arithmetic appeared to have been disturbed by their algebra.

THE A R I T H M E T I C C O N N E C T I O N 47

MOVING FROM A R I T H M E T I C INTO A L G E BRA

If we look at another series of problems which required students to move

in the opposite direction, from arithmetic into algebra, we see that very

similar problems occurred. We refer to three problems, one involving

consecutive numbers,

The sum of two consecutive numbers is always an odd number. The product of two consecutive numbers is always an even number. Are these two statements true? I f they are can you show why?

and two others of the "pick-a-number" type,

A girl multiplies a number by 5 and then adds 12. She then subtracts her starting number and divides the result by 4. She notices that the answer she

gets is 3 more than the number she started with. She says, " I think that would

happen, whatever number I started with." Is she right?Explain carefully why your answer is right.

Choose any number between 1 and I0. Add it to 10 and write down the

answer. Take the first number away from 10 and write down the answer. Add your two answers. What results do you get? Will the result be the same for all starting numbers? Explain why your answer is right. (See Note.)

These problems did not explicitly request the students "to use algebra"

although students were aware that this was an algebra test. Three-quarters

of the 352 students who were given one or two of these problems on their

test did not use any algebra at all. In the consecutive numbers question, for

example, 78 of 118 students avoided any algebraic work, most giving one

or two examples as a demonstration with about half of these giving some

even/odd arguments. Typical examples of this were:

"Well: 2 + 3 = 5,/ Also: 2 • 3 = 6,/

3 + 4 = 7 , / 3 • 12,/

4 + 5 = 9 , / 4 x 5 = 2 0 , /

5 + 6 = l i d 5 x 6 = 3 0 ` / "

"Let 1 + 2 be the two consecutive numbers. - this adds up to 3 - odd.

2 x 1 - let's use the same numbers.

= 2 this is an even number.

Yes these two statements are true Using the same numbers 1 and 2.

You can see added = 3 odd

multiplied = 2 even."


When this problem was worded "Show, using algebra, that the sum of two consecutive numbers is always an odd number" 27% still avoided algebra, giving examples again. And when we examine the work of the 25% and 73% respectively who did in fact write some algebraic symbols (generally x and x + 1) on their papers, we see that very few of these students actually carried out the complete algebraic demonstration. For example,

"x + (x + 1) = x(x + 1) =

2 + (2 + 1) = 2(2 + 1) = 4 + 2 = 6,/even 2 + 3 = 5,/odd 3(3 + 1) = 12,/even" 3 + ( 3 + 1 ) = 3 + 4 = 7,/odd

In fact only 10% of the students successfully used algebra to demonstrate the odd-ness of the sum of consecutive numbers.

Results were similar on the other two questions. Only 44 of the 116 students given the "girl question" used some algebra and half of these used their algebraic formulation only to create examples. When the same question was asked in the form "Using algebra, show that she is right", a third of the students still wrote no algebraic statements. Very few students actually carried out a complete algebraic demonstration but relied instead on a few examples. The "choose a number" question elicited the lowest algebraic response rate with only 9 of 118 students even attempting an algebraic explanation.

The series of problems asked the students to establish some fact about numbers. We expected of course that students would move from' the arithmetic number situation into the algebraic in order to establish the arithmetic generalisation. We found that the students tended to stay in the arithmetic mode when the problem involved numbers. In fact their reluctance to leave the arithmetic mode here was almost as strong as their reluctance to abandon the algebraic mode in the first series of questions.

Students do not appear to see algebra as generalised arithmetic. Indeed the question arises whether or not they believe that arithmetic can be generalised. There were indications in both the interviews and tests that they do not. For instance many of the students who did use some algebra in this series of questions, when asked whether their algebraic work demonstrated the truth of the proposition, responded negatively or indicated that examples would be preferable. One student, when confronted with a complete algebraic proof said that if she were explaining the problem to a friend she would "solve it" with algebra and then "prove it by examples". Another student who had been pushed and prodded through an

THE A R I T H M E T I C C O N N E C T I O N 49

algebraic demonstration said " / t w o r k s . . . 'Cause I tried it with a number".

The algebraic work was often considered on a par with a single numeric example or, in a few cases, rather less reliable.

Even when students believed generalisation was possible they did not see algebra as a tool for establishing such a generalised statement about numbers. Comments such as:

"This statement could possibly be true because with numbers there is great variations in the way that you go about solving any math question. Numbers have different possibilities."

"Numbers do go on and on and we can't check them all."

"I can't go on trying every single number to find o u t . . , numbers can go on

to a certain extent."

"You would never think or realize that you can have statements that are always true no matter what numbers you take."

certainly make one wonder whether these students believe that generalisation is possible.

V A R I A T I O N S IN THE E X T E N T OF A R I T H M E T I C / A L G E B R A

D I S S O C I A T I O N

There were of course considerable individual differences in the degree of

algebra/arithmetic dissociation exhibited by the students; many of these are evident in the examples given above. Those students who were quite willing to accept and who even expected different answers in their algebraic and arithmetic solutions to a problem could be said to have the highest degree of algebra/arithmetic dissociation. An example from our pilot study is typical of this. (See Note.)

5

J c 2

Asked for the area of the rectangular shape with sides 5 and c + 2, the student replied "10c".

S: I just took length times width and I took 2 times 5 is 10 and I just put 10c whatever c is for the variable c.


I: Okay, um, let's suppose, just for the sake of argument, that c is equal to 4. Okay? Now, what would your answer give if c is 4?

S: I 'd put 30. I: Good. Uh, how did you get 30? S: Well 4 and 2 is 6 and 6 times 5 is 30. I: Okay. Now does that agree with what you would give if you gave that

answer with c equal to 4? Forget the figure. What would that (points to 10c) be if c was equal to 4?

S: It would be 40. I: Um, so one way you get 30 and the other way you get 40. S: Yeah. I: Do you want to change y o u r . . , if you want to have another go at an

answer then you can. S: No.

Students such as the one above felt no need to justify or correct incoher- ences that arose when they were asked to check their algebra with numbers. Nor did they see any link between their arithmetic work in the second question series and the algebraic solution that was either imposed or done for them. One student for example after having been dragged through an

arduous algebraic demonstration was asked whether it meant that the generalisation was now true. Totally ignoring the algebra she replied

"Probably, i f you pick a number out o f anywhere and they do work out well,

there's a major chance that all the other numbers are going to work out as

well."

Other students seemed to expect some degree of algebra/arithmetic coherence. In the first series of problems they appeared slightly uncomfort- able or confused when algebra and arithmetic gave two different results and wondered whether one or the other was wrong. For instance, all three students we interviewed who had cancelled the 2x's in the problem involving

2 x + l 1

2 x + 1 + 7 8

immediately realised something was wrong when they were asked to substitute a number for x: "Oops", " I guess I shouldn't have taken the 2x

and 2x out for some reason", " l t ' s not the same when you substitute a

n u m b e r . . . That's where I 'm always stuck in math". Similarly, in the problem which concluded with 20 = 4, 24 of the 89 students given this question indicated either explicitly or implicitly that 20 r 4 and 13 of these embarked on a check of the algebra in an attempt to find the error.


A third group of students immediately rejected their algebraic work when

a numeric substitution did not work out. A couple even suggested that algebra could be checked by trying some numbers. One student checked the statement (a 2 + b2) 3 = a 6 + b 6 by substituting a = 3 and b = 4 and con-

cluded "This statement is never true, also the left number is always greater than the right number. And here is proof" He then proceeded with an algebraic demonstration. A few students also used an algebraic demonstration in the second series of problems: six students out of 118 did use algebra in the problem "Choose any number between 1 and 1 0 . . . " and felt that the fact that (10 + x ) + ( 1 0 - x ) = 20 explained the constant result. This last group exhibited the least degree of algebra/arithmetic dissociation.

CONCLUDING DISCUSSION

The tendency of some students to justify algebraic statements by appeal to a rule rather than to their experience of the behaviour of numbers reminds us of the 19th century debate in Britain about the nature of algebra. Was algebra "universal arithmetic" and therefore governed by the known behaviour of quantitative arithmetic or was it a "symbolic system" with essentially arbitrary rules? (Pycior, 1984). The proper historical answer is that algebra is not in any straightforward sense either of these alternatives, but just as most of the British mathematicians in the second quarter of the 19th century seemed temporarily unable to transcend the opposing posi- tions in order to resolve them, so many of the students we examined seemed to be at an either~or stage of development in which their arithmetic and algebraic behaviours were not at all comfortably integrated. The question we cannot yet answer, and which is obviously very important to try to answer, is whether any of these students - all, many, or some of them - will in the natural course of events achieve the coordination they lacked at the time we examined them. Our guess is that those who have already decided that algebra just doesn't make any sense are unwittingly shielding them- selves from the intellectual conflict that might push their understanding a step further.

Davis et al. (1978, p. 127) point out that students doing algebra rarely apply a "check with numbers" strategy even when recommended to adopt it. Our analysis shows that the strategy presupposes a reasonably clear understanding of the connection between the worlds of arithmetic and algebra, and that many of the students we examined were not yet clear about the relationship. And, indeed, there are some technical difficulties that everyone has to resolve. First and foremost is the striking difference


between writing and manipulating expressions in algebra and writing and manipulating expressions in arithmetic. In spite of the use of common (operational) signs, what one actually does in the two cases is very different, so different that one cannot be surprised if students do not immediately spot the connection. An algebraic expression may perhaps be transformable into equivalent forms, but its value cannot be computed. The same expression with numerical values substituted for the letters is immediately computable and "collapses", losing all its individual character, into a single numeral. The pedagogy of algebra (in so far as it exists) seems to have nothing to offer to help students grasp the arithmetic/algebra connection that underlies all these differences between two modes of symbolic behaviour.

Indeed, some pedagogical devices produce a miasmic fog of their own. It is not unknown in traditional algebra teaching to offer numerical instances - of combining arithmetic fractions, say - as clues to the processes to be used in combining algebraic fractions. The device, of course, is intended to serve as a reminder, as a structural model for the algebraic procedure, not as a validation of the procedure, but one must sympathise with students who are confused about the purpose of exploiting the parallelism. Does it not quite strongly suggest that an algebraic generalisation may be established from numerical instances?

The parallelism works because, in a quite strong sense, the algebra of combining fractions is already present in the arithmetic of combining fractions. There is no question here of the algebraic form generalising the arithmetical procedure: the arithmetical procedure is already fully generalised. Indeed, as with such items as the commutative and distributive laws, say, the algebraic form is only the record of a generalisation which is already known. A difficulty for students is to appreciate the purpose of writing down such familiar information about numbers in such a formal way.

The arithmetic-algebra connection is decidedly not plain and simple. Combine the above obscurities with the asymmetry embedded in the use of numerical substitution as a means of validation - one numerical substitution may disprove an algebraic statement whereas no finite number of numerical substitutions can prove it - and one has a situation which might seem designed to confuse everyone but the angels.

The peculiar arithmetical behaviours we observed in some of the students make us think of Filloy and Rojano's (1984) suggestion that the challenge of dealing with the syntax of a new and unfamiliar (algebraic) language tends to destabilise students' semantic control of the familiar (arithmetical)

T H E A R I T H M E T I C C O N N E C T I O N 53

language. We did not pinpoint this issue in our research so we are not

really in a position to confirm or deny this intriguing hypothesis. It does, however, seem plausible to us that the shift from arithmetical language to a formal algebraic language that needs to be coordinated in complex ways with the first language could well cause temporary lapses in attention to meaning that would give rise to aberrant arithmetical behaviours. All of our subjects would certainly have identified 20 = 4 as an invalid statement if it had appeared on its own or in an arithmetical context, but setting it at the tail end of a sequence of algebraic statements disturbed the students' normal interpretations. Those students who did try to dissolve their residual unease looked without exception for syntactical "solutions" - e.g. by treating the statement as an equation and dividing both sides, moving one of the numbers to the other side of the equals sign, etc.

It surprises us that these aberrant behaviours are still present in stu-

dents who have been learning algebra for at least two years. Lapses which might be anticipated in the work of students just embarking on algebra (as with those studied by Filloy and Rojano) now seem to us more serious because they are still occurring after students have attained more surface fluency in the standard algebraic routines.

The picture our data yields shows the track leading from arithmetic to algebra to be littered with procedural, linguistic, conceptual and epistemo- logical obstacles. It is tempting to describe high school algebra as it is unveiled in our research as a disaster area - an impression that would probably be confirmed by reading reports of the CSMS and SESM researches (Hart, 1981; Booth, 1984) or the writings of Stella Baruk (1977, 1985), say. But the point we would stress in coming to the end of this brief exploration is that the obstacles are real and not all trivial. The students we worked with, in most cases, were neither lazy nor foolish; their teachers, in most cases, were neither lazy nor foolish either. At the end of the research we have an enhanced appreciation of the difficulties that have to be overcome in bridging the worlds of arithmetic and algebra, and perhaps a greater dismay at the inability of traditional pedagogy to give teachers or students much help in overcom- ing them.

N O T E

The "add and take" problem comes from Alan Bell's (1978) doctoral study, 'The learning of general mathematical strategies', and the rectangle question from materials produced by the CSMS Mathematics Team. Sources for the other problems cannot be acknowledged as we are no longer sure where we first encountered them.


REFERENCES

Baruk, S.: 1977, Fabrice ou I'kcole des math~matiques, 6ditions du Seuil, Paris. Baruk, S.: 1985, L'dge du capitaine: de rerreur en mathkmatiques, ~ditions du Seuil, Paris. Booth, L. R.: 1984, Algebra: Children's Strategies and Errors, NFER-Nelson, Windsor,

Berkshire. Davis, R. B., E. Jokusch, and C. McKnight: 1978, 'Cognitive processes in learning algebra',

Journal of Children's Mathematical Behavior 2(1), 10-320. Filloy, E. and T. Rojano: 1984, 'La aparici6n del lenguaje arithm6tico-algebraico', L'Edu-

cazione Matematica V, 278-306. Hart, K: 1981, Children's Understanding of Mathematics, John Murray, London. Lee, L. and D. Wheeler: 1986, 'High school students' conception of justification in algebra',

Proceedings of the Eighth Annual Meeting of the N. American Chapter of the International Group for the Psychology of Mathematics Education, East Lansing, Michigan.

Lee, L. and D. Wheeler: 1987, 'Algebraic thinking in high school students: their conceptions of generalisation and justification', Research report, Concordia University, Montreal.

Mathematical Association: 1945, The Teaching of Algebra in Schools, London, G. Bell and Sons. (The report was first written in 1929.)

Pycior, H. M.: 1984, 'Internalism, externalism and beyond: 19th century British algebra', Historia Mathematica 11, 424-441.

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The arithmetic connection