Learning the Algebraic Method of Solving Problems

Learning the Algebraic Method ofSolving Problems

Kaye Stacey

Mollie MacGregor

University of Melbourne, Melbourne, Australia

This study of students' attempts to formulate and solve algebra word problems shows that the logic

underlying algebraic problem solving methods is little understood. Students' prior experiences

with solving problems in arithmetic gives them a compulsion to calculate which is manifested in

the meaning they give to `̀ the unknown'' and how they use letters, their interpretation of what an

equation is, and the methods they choose to solve equations. At every stage of the process of

solving problems by algebra, students were deflected from the algebraic path by reverting to

thinking grounded in arithmetic problem solving methods.

Learning to solve problems using algebra is hard. It is well known that students often have

difficulty in formulating algebraic equations to represent the information given in word

problems and that it is hard to learn the ways in which the symbols must be manipulated to

get solutions, even for simple equations. Many factors that contribute to making this a

difficult task have been identified. For example, several research studies have identified

contextual and grammatical features of word problems that affect students' success in

solving them (see, for example, Bednarz & Janvier, 1996; Chiappini & Lemut, 1991;

Kirshner, McDonald, Awtry, & Gray, 1991; Lewis & Mayer, 1987). In this article, we look

at simple problems that avoid these known complicating features as far as possible.

We propose that a major reason for difficulty is not understanding the basic logic of

solving a problem by algebra. Students' prior experiences with solving problems in

arithmetic gives them a compulsion to calculate which is manifested in

1. the meaning they give to `̀ the unknown'';

2. their interpretation of what an equation is;

3. the methods they choose to solve equations.

We will show that, at every stage of the process of solving problems by algebra,

students may be deflected from the algebraic path by reverting to thinking grounded in

arithmetic problem solving methods. As a result, a very large number of different routes to

a solution of even simple problems are observed.

149

Direct all correspondence to: Mollie MacGregor, Department of Science and Mathematics Education,

University of Melbourne, Parkville 3052, Australia; E-mail: [email protected]

JOURNAL OF MATHEMATICAL BEHAVIOR, 18 (2), 149±167 ISSN 0732-0213.Copyright C 2000 Elsevier Science Inc. All rights of reproduction in any form reserved.JMB

In the section Arithmetic and Algebra: Processes and Structures of this article, we will

discuss a cluster of concepts and distinctions made by mathematics educators describing

the differences between arithmetic and algebraic thinking. In the Method, Sample and

Results section of the article, we present details of a study of word problem solving by 900

students and we summarize their performance. In the section Multiplicity of Routes

from Problem to Solution we focus attention on the many strategiesÐalgebraic and

non-algebraicÐthat students used to solve the problems. In the next section, Obstacles

to Successful Use of Algebraic Route, we use interview data to explain how thinking

based in arithmetic affected the meanings students gave to unknowns and how they

wrote equations. In the final section, Implications for Teaching, we discuss the

implications of our findings for curriculum planners and teachers.

1. ARITHMETIC AND ALGEBRA: PROCESSES AND STRUCTURES

1.1. Comprehension of Word Problems and Formulation of Equations

There have been many studies of the processes of comprehension of word problems

(Just & Carpenter, 1989, 1992; Mayer, Lewis, & Hegarty, 1992; Nathan, Kintsch, &

Young, 1992). Formulating equations is considered by Chaiklin (1989) and others to be

part of the comprehension process. However, as shown in this article, there is no easy

transition from comprehending a problem to formulating an equation or set of equa-

tionsÐin fact it is a major site of difficulty that operates differently in solving problems

arithmetically or algebraically. Students in our study comprehended the problems but most

did not formulate equations. Instead, they tried to use a sequence of calculations to reach

the answer (behavior also noted by Arzarello, 1991). Furthermore, as will be demon-

strated later, students use of `̀ the unknown'' in equations also reflected the primacy given

to calculating.

Our explanation of the several phenomena to be reported in this article is that they all

revolve around the cognitive discontinuities involved in the shift from using methods of

arithmetic to algebraic reasoning. Various aspects of this shift have been described by

mathematics educators, who have made a cluster of related distinctions. Firstly the change

from calculating with numbers to operating with the unknown has been described as a

`̀ cognitive gap'' (Herscovics & Linchevski, 1994) or `̀ cut-point'' and `̀ didactic cut''

(Filloy & Rojano, 1989) between arithmetic and algebra. Filloy and Rojano (1989)

claimed that solving an equation with the variable appearing only once is not really an

`̀ algebraic'' process, since every stage involves only calculating with known numbers. For

them, truly algebraic operating with the unknown is first required in solving equations

where the variable appears on both sides. They explain the greater difficulty that students

have in solving equations with variables on both sides by this different cognitive demand.

A second discontinuity in the transition from `̀ arithmetic thinking'' to `̀ algebraic

thinking'' has been the focus of much theoretical discussion. In the context of generalized

arithmetic, distinction has been made between two ways of interpreting an algebraic

expression or equation, called `̀ procedural'' (Kieran, 1992) or its equivalent term

`̀ operational'' (Sfard, 1991), and `̀ structural'' (Kieran, 1992; Sfard, 1987) or its equiva-

lent term `̀ conceptual'' (Hiebert & Lefevre, 1986). For example, x + 5 can be interpreted

as the procedure or operation `̀ add 5 to x''; it can also be interpreted as an object or

STACEY AND MACGREGOR150

concept in a mathematical structure (e.g., in the ratio (x + 5)/(x ÿ 3)). The alternative

conceptions are also referred to as `̀ process and object'' (Sfard, 1991), or `̀ process

and product'' (Tall & Thomas, 1991) or `̀ process and concept'' (Gray & Tall, 1991).

Some authors associate procedural or operational thinking with arithmetic, and

structural thinking with algebra. For example, in her comprehensive review of algebra

research, Kieran (1992) uses the term `̀ procedural'' to refer to `̀ arithmetic operations

carried out on numbers to yield numbers'' (p. 392). A view of x + 5 as an instruction to

add 5 to particular numbers or quantities is procedural in this sense.

Other authors making the procedural/structural distinction have not exclusively

associated it with arithmetic and algebra, but have pointed out how mathematical

concepts (including numbers and algebraic equations) are created through a process of

`̀ reification'' (Sfard, 1991), also called `̀ encapsulation'' by translators of Piaget's

work. Mathematical objects begin as operations on other objects (for example, x + 5

begins as an instruction to add 5 to numbers) but become objects in their own right.

Gray and Tall (1991) refer to the result as a `̀ procept'': a fluid notion which holds

understanding as both a process carried out and the resulting concept or mental object.

They point out the importance of `̀ proceptual thought'': the ability to switch between

the two interpretations.

A third distinction, which is directly associated with the findings in this article, has

received less attention. It contrasts the ways in which problems are solved using arithmetic

and algebra. In arithmetic, problems are solved by calculating with known numbers and

working towards the answer. The basic logic of the algebraic method is different:

Relationships are described using both knowns and unknowns in a similar way and then

equivalent relationships are derived from them artfully until an answer to the problem is

obtained. The historically relatively recent formalization of this `̀ method of analysis'' by

Viete and Descartes is described by Charbonneau (1996).

Charbonneau quotes a description of the algebraic method of problem solving by

Descartes who pointed out:

. the problem is first supposed solved;

. names are given to all unknown quantities;

. no distinction is made between knowns and unknowns;

. the relations between the known and unknowns are found and a single quantity is

expressed in two ways, giving an equation.

Much of the behavior reported in this study can be explained by the difficulty that

students have in reconciling algebraic and arithmetic approaches to solving problems.

It reflects students' attachment to the belief that problems are solved by direct

calculation. Our data show that this belief, firmly rooted in everyday experience and

primary grade mathematics, accounts for much of the difficulty experienced by

students in their early algebra learning. The compulsion to calculate prevents students

looking for, selecting, and naming the appropriate unknown or unknowns, and

prevents them focussing on what Kieran (1992) and others have called `̀ forward

operations'' in formulating an equation. It prevents some students from attempting an

algebraic approach, and deflects others away from an algebraic method that they

have started.

LEARNING THE ALGEBRAIC METHOD OF SOLVING PROBLEMS 151

2. METHOD, SAMPLE, AND RESULTS

We designed a set of problems for use by teachers in 12 secondary schools. Although the

schools were not randomly selected, we consider them to represent a range of typical

schools in the large city where we work. We collected written solutions from approxi-

mately 900 students, and we carried out one-to-one interviews with 30 individuals. The

students involved were aged 13±16, mainly in their third or fourth year of algebra

learning. The interviews gave us insights into students' reasoning and explained much of

the thinking underlying the written solutions that we collected from the main sample. Four

of the problemsÐthose which we use as examples in this articleÐare shown in Fig. 1.

They eliminate as far as possible the features documented in the literature that increase

problem difficulty. They are representative of problems used in the early years of algebra

learning in Australian schools, having the following characteristics:

. the problems are straightforward applications of linear equations that students have

learned to solve,

. addition is the only operation not explicitly mentioned,

. the situations described are familiar and easy to comprehend,

. difficult concepts and vocabulary are avoided (with the possible exceptions of

`̀ equation'' and `̀ perimeter''),

. all operations are carried out on positive whole numbers,

. no derived quantities (e.g., rates, speeds) are involved,

The first three problems lead to equations with the unknown on only one side. These

(Filloy et al's `̀ arithmetic equations'') were predicted to be easy to solve. The fourth

problem, NUMBER, was expected to be harder to solve, since the unknown occurs on

both sides of the `̀ equals'' sign, but easier to translate into an equation because it can be

translated phrase-by-phrase.

Most of the traditional word-problems of school algebra cannot be directly

translated from words to symbols in this way but inferences must be made from the

information given (Kirshner et al., 1991; Stacey & MacGregor, 1995). For example,

formulating an equation for BUS requires the inference that `̀ If you add the three daily

distances, you get 1,410 km'' (stated as a procedure) or `̀ The sum of the three

distances is equal to 1,410 km'' (stated structurally). Either statement (procedural or

structural) can be translated to an equation of the form a + b + c = 1,410. Thus, an

equation that apparently represents the structure of relationships may also represent

a procedure.

In Table 1 we present the results for students in their fourth year of algebra learning

(Year 10), who comprised the bulk of the sample. The table shows success rates for (1)

writing any correct equation and (2) finding the correct answer by any method. The results

for the smaller number of younger students are not presented but show a similar pattern

with lower success rates.

As Table 1 shows, about one-third of the students were able to write correct equations

for all four problems, including NUMBER. Many of the equations were not used to get the

answers, and some students wrote the equations after they had worked out the answers. As

predicted, students were able to get correct numerical answers for the first three problems


by a variety of methods often not algebraic, and about two thirds were successful.

NUMBER was more difficult.

There was no evidence that students were choosing the most appropriate method for

solving the problems and were therefore only showing their algebra skills when it was

essential. Almost no students, for example, did the first three problems by logical

reasoning but used algebra on harder problems such as NUMBER where it has

definite advantages.

It was evident that students in particular classes had been well trained in the setting up

and solving of equations. In these classes, most students used conventional formats and

manipulation procedures, and their solutions were concise and usually correct. Students in

other classes and schools used little or no algebraic symbolism. Algebraic thinking does

not develop spontaneously (Herscovics & Linchevski, 1994) and students are unlikely to

switch from an arithmetic approach unless they are specifically taught (Filloy & Rojano,

1989; Stacey & MacGregor, 1997). In the rest of this article we try to explain more

precisely what it is that they need to learn.

FIGURE 1. Four problems, referred to in this article as TRIANGLE, MARK, BUS, and NUMBER.


3. MULTIPLICITY OF ROUTES FROM PROBLEM TO SOLUTION

The first major observation of this project was the great variety of methods that students

used. The routes they used are summarized in Fig. 2. The second major observation was

that very few students followed the complete algebraic route (see Fig. 2, left-hand-side),

even when alternative routes were very difficult or time consuming.

As Fig. 2 shows, some students made no attempt to use any algebra. They used

logical arithmetic reasoning or trial and error. Examples will be given below. Others

began along the algebraic route but then switched to an arithmetic method (see the many

routes out of `̀ Try algebra'' and `̀ Write equation'' in Fig. 2). Our impression is that

students were deflected from the fully algebraic route because they were still bound by

the arithmetic way of solving problems. Students want to work with knowns, not

unknowns. They have a compulsion to calculate and they do not realize the value or

logic of Viete's and Descartes' `̀ analytic method'' for describing relationships and

manipulating them.

3.1. Non-Algebraic Routes: Arithmetic Reasoning

Despite prominent instructions on the test papers to use algebra, many students in our

large sample chose non-algebraic routes right from the start (moving down the right-hand

side of Fig. 2). More often than not they arrived at correct answers for the easy problems

but not for harder problems in the test such as NUMBER.

Many students solved all the easy problems by arithmetic reasoning. For TRIANGLE,

for example, they could see from a logical analysis of the situation how to get directly

from the measure of the perimeter to the measure of the unknown sideÐsubtract 14 from

44, then divide by 3. They implicitly know what Kieran's (1992) `̀ forward operations'' are

(that 14 has been added to get 44) and they know that there are three of the unknown

lengths contributing to the perimeter, but they use the `̀ backward operations'' (in this case

subtracting and dividing) to move directly to the solution, calculating from known

numbers at every stage.

For problems like MARK and BUS, the form of arithmetic reasoning is influenced by

the student's initial perception of the structure of the situation (Stacey & MacGregor,

1995). The perception of the MARK problem as the division of a whole into parts leads to

the procedure `̀ Divide 47 by 2 to give equal shares, and then adjust up or down by half of

$5.'' So Mark gets (47/2) + (5/2) and Jan gets (47/2) ÿ (5/2). An alternative perception,

perhaps also associated with the way in which actual money might be distributed,

TABLE 1. Percentages for Equation/s Correct and for Answer Correct by Any Method,

for Students in Year 10

Year

TRIANGLE MARK BUS NUMBER

Equation

(%)

Answer

(%)

Equation

(%)

Answer

(%)

Equation

(%)

Answer

(%)

Equation

(%)

Answer

(%)

10 38 63 30 73 32 60 30 17

Note: MARK and BUS were attempted by 524 students in nine schools. TRIANGLE was attempted by 188

students in three schools. NUMBER was attempted by 180 students in four schools.


FIG

UR

E2

.R

oute

sfr

om

apro

ble

mst

atem

ent

toth

eso

luti

on.


suggests `̀ Take out $5 first, give it to Mark, then share the rest equally.'' Both these

methods were used by the students in our sample. However, with one exception, students

who tried the division method for BUSÐwhere the whole is divided into three parts

instead of only twoÐfound arithmetic reasoning too difficult and switched to the

alternative `̀ subtract-and-share'' strategy (see Fig. 3) or to trial and error or gave up

the attempt.

Arithmetic reasoning cannot be used effectively for NUMBER, essentially because the

problem corresponds to a situation where the unknown is on both sides of the equation. An

earlier item in the test (not shown in Fig. 1) was a number-story problem modeled by an

equation with only one occurrence of the unknown. Many students had solved it by

working backwards, using inverse operations, often aided by a diagram or flowchart (for

example, see Anna's solution in Fig. 4). Students generally also tried the same method for

NUMBER, but then gave up or switched to trial and error. The mental set to directly

calculate an answer prevented access to the required algebraic solution for NUMBER.

3.2. Non-Algebraic Routes: Trial and Error

Many students found the answers to the problems solely by trial and error processes

(i.e., by testing numbers in the statement of the problem) but, as Fig. 2 shows, trial and

error methods were also used by students who had followed the algebraic route as far as

writing an equation. These students understand that the solution of an equation is a number

which when substituted into both sides makes the equation true. However, they have not

moved away from the security of working with known values. Whereas students using the

logical reasoning approach work with the inverse operations, the trial and error approach

works directly with the `̀ forward operations'' inherent in the problem situation. Kieran

(1988) therefore distinguishes these two approaches as `̀ algebraic'' and `̀ arithmetic,''

respectively, again highlighting the difficulty with lack of agreed terminology in this area.

Trial and error was carried out in three ways:

. Random: Guessing answers in no particular order and hoping to guess the correct

answer sooner or later by chance.

FIGURE 3. Two solutions to BUS using logical arithmetic reasoning.


. Sequential: Trialing numbers in sequence, as in a spreadsheet, hoping that the correct

answer will be reached.

. Guess-check-improve: Carrying out one or more trials, using the result of each trial to

choose the next one.

Only the third version of trial and error was sufficiently powerful to solve NUMBER

because the answer (1.5) is not a whole number and hence often not guessed or trialed. As

one student wrote, `̀ It does not actually work. It always seems to be 0.5 off.'' The inability

of students to conduct trial and error on an appropriate range of numbers was a major

cause of the low success rate for this problem.

3.3. Writing Formulas: The Superficially Algebraic Route

A large proportion of students who were counted in Table 1 as writing correct equations

in fact wrote them from the very beginning as formulas; for example, x = [1,410 ÿ (85 +

125)]/3 or x = [c ÿ (a + b)]/3 for BUS. This formula describes a sequence of calculations;

it shows how to work out the answer from known information. As a student explained, `̀ It

seems easy that way. You don't have to rearrange it for working out.'' This behavior has

also been noted by Teppo and Esty (1994) and Chiappini and Lemut (1991) where it is

referred to as an `̀ a posteriori procedure.'' Although disguised as algebra, the formula

represents the same reasoning as the solutions by arithmetic reasoning in Fig. 3.

Students who wrote formulas used a letter (usually x) to label the unknown quantity to

be worked out but they were not using an algebraic approach. Therefore, in Fig. 2 `̀ Write

formula for solution'' is shown outside the algebra route. This is in accordance with

Janvier (1996) who points out the fundamental distinctions between a formula and an

equation: using a formula requires calculating with given numbers whereas using an

equation involves operating with unknown numbers; the formula defines a procedure for

FIGURE 4. Three correct methods used to solve 3x + 14 = 44.


computing whereas the equation specifies the structure of an equality among variables.

Cortes, Vergnaud, and Kavafian (1990, p. 28) observed that when an equation is written as

ax + b = c, its `̀ tool-like characteristic'' is not visible to students. On the other hand, when

it is written as a formula (e.g., x = (c ÿ b) � a) they can see how to use it. The concept of

an equation as a statement about equality, rather than as a formula, is crucial to the use of

an algebraic method for solving problems. Indeed the perception of an equation as a

formula, that is, as a rule telling you what to do, does not require any knowledge of algebra

(Janvier, 1996) except its syntax. It does not conflict with students' compulsion to

calculate first.

3.4. Using Algebra; Writing the Equation

The first evidence of a student's decision to use algebra is the use of a letter. As we

explain later, some students use letters only to denote the results of calculations to be

performed whereas some used them algebraically to denote an unknown quantity. Some

students went no farther than this along the algebraic route (see Fig. 2, row 3). Some got

as far as describing relationships correctly (e.g., x, x + 85, x + 125) before abandoning

algebra (see (Fig. 2, Row 3, Box 2)). Others wrote a complete equation (e.g., x + x + 85

+ x + 125 = 1,410) which they might solve by a variety of methods (see Fig. 2, Row 5)

or, in surprisingly many cases, leave unused and returned to the original problem

statement for either logical arithmetic reasoning or a trial and error strategy (see (Fig. 2,

Rows 4 and 5)). These students did not know that writing the equation was useful for

solving the problem.

3.5. Using Algebra: Solving the Equation

If students see solving the equation as useful for solving the problem, they have the

choice of three methods they have been taught (see choices after `̀ Write equation'' in

Fig. 2):

. reverse operations on a flowchart (see Anna's solution in Fig. 4);

. use trial and error with a choice of three approaches, as described above (see Brian's

solution in Fig. 4);

. manipulate the symbols, including those for the unknowns, in a chain of deductive

reasoning (see Carla's solution in Fig. 4).

The first two of these methods are more closely related to arithmetic and

calculating at every stage with known values than they are to algebraic thinking

and working with unknowns. Equation solving by these two methods is `̀ doing

algebra'' only in the sense that the immediate problem to solve is represented with

algebraic symbolism.

In order to use the algebraic method students need to understand the equivalence of

left-hand and right-hand expressions structurally, as well as the logical principles

underlying algebraic problem solving solutions. Some students perceived an equation

simply as a narrativeÐa sequence of operations that produces a numerical answer. For

example, in TRIANGLE they interpret 2x + x + 14 = 44 only as a way of saying that


when the three side lengths are added, the answer is 44 cm. These students did not know

how to deal with equations like the one arising from NUMBER.

The full algebraic route, that is, formulating an equation by representing essential

relationships in the problem and then solving it by manipulating the symbols, was used by

a small minority of the students in our sample, concentrated in three of the nine schools. In

these schools, success rates were high: for example approximately 70±80 percent solved

BUS algebraically and correctly.

4. OBSTACLES TO SUCCESSFUL USE OF THE ALGEBRAIC ROUTE

Students in our sample demonstrated their capabilities in comprehension and in solving

problems by non-algebraic methods. However, many of them had not learned how

algebra can be used to solve problems. A few students made errors in using algebraic

syntax (e.g., x + 85x + 125x = 1,410 instead of 3x + 85 + 125 = 1,410 for BUS) when

formulating equations or solving them, but errors of this nature did not account for most

of the unsuccessful attempts to use algebra. Interviews with students, illustrated in the

examples below, shed light on two common ways in which arithmetic thinking interferes

with the formulation of equations. Because students do not understand the logic of

solving problems by algebra, they see calculation as the priority. Then the compulsion to

calculate interferes with what they think an equation is and also with what they think an

unknown is.

4.1. Why Write an Equation?

When students are given a problem in mathematics, they want to work out the answer.

Nearly all the 30 students we interviewed (drawn from three schools) felt compelled to

start `̀ figuring out the answer'' as soon as they had read a problem. They were surprised

and puzzled when we asked them to write an equation before finding the answer. Several

made comments such as `̀ Don't you want me to work it out?'' and `̀ I can't write the

equation until I have worked it out.'' The latter comment explains the writing of formulas

and descriptions of the operations required, such as 44 ÿ 14 = 30 � 3 = x. When the

interviewer explained how to construct an equation by representing information given in

the problem, students found it hard to accept the utility of this approach. One student

exclaimed, `̀ That's just another way of writing the question!'' and another said, `̀ Is that

all!'' During the interviews, a few students learned how to represent given information as

an equation. Their initial inability to write an equation without working out the answer

appeared to stem from lack of experience and a wrong expectation of what they should be

doing. However, other students seemed unable to switch from their calculating approach.

The basic logic of algebraic problem solving by representing forward operations using the

unknown as if the problem had already been solved seemed to be an insurmountable

obstacle for them.

The separation in students' minds between getting answers and `̀ doing algebra''

was clearly evident in the interview with Neil. Neil is a 14-year old who easily solved

the MARK problem mentally by working out equal shares and adjusting one up and

one down until the two amounts differed by 5. He then spent a long time on written

calculations, trying to work BUS the same way, but insightfully recognizing that the


problem was harder `̀ because you have three [quantities to find] this time.'' When the

interviewer suggested doing it an entirely different way that would be easy, Neil

remembered algebra. `̀ Oh yes! You put x for Day 1, and then for Day 2 you do x

plus 85 and you put x plus 125 for Day 3. And that's right!'' He wrote his three

correct expressions in a vertical column, and seemed satisfied that he had found the

required `̀ solution by algebra.'' However, Neil did not see that these expressions could

be used to help him find the numerical answer to the problem. When he was asked

how he would work out x he immediately went back to his arithmetic method of

dividing and adjusting, believing that this was the only strategy that would lead him

to the answer.

4.2. Multiple or Shifting Referents for `̀ The Unknown''

The orientation of students' thinking to arithmetic problem solving is also apparent in

the way that many of them used unknowns. Operating on the unknown with understanding

requires recognizing that it has a particular referent. Our data show that some students do

not know which quantity or quantities in a problem could or should be symbolized as the

unknown, or which quantity a chosen symbol refers to. If x stands for some unknown

quantity or quantities not clearly definedÐsome thing or things with multiple or shifting

valuesÐthen it makes no sense to operate on it using rules that apply to numbers. Many

students seemed to use x to stand for the quantity that is currently being calculated, a

consequence of thinking based in arithmetic.

Below we present excerpts from interviews with students working on TRIANGLE or

MARK to show how different uses of a letter to symbolize `̀ the unknown'' and different

ideas of `̀ the unknown'' affected their attempts. All of these students understood the

problems well. In the interviews, we saw several modes of letter use:

. x refers to different quantities in one equation.

. x refers to different quantities at different stages.

. x is a general label for any unknown quantity or a combination of quantities.

Evidence for the first of these modes has been reported by Booth (1987), Fujii (1993),

Stacey and MacGregor (1997), and Ursini (1990). In Fujii's study, for example, 6th and

8th grade students accepted that if x + x + x = 12, then the first x could be 2 and the

other x's could be 5. As far as we are aware, the other two modes we have listed above

have not been documented. These modes seem particularly related to thinking based on

the problem solving methods of arithmetic, not algebra, and we illustrate them below

with interview extracts.

Dean thinks of x as signifying the total of all unknown quantities. For TRIANGLE, he

uses arithmetic reasoning to work out that x is 10 but writes down his solution as x = 30.

He then says that is wrong, and writes x = 10 � 3 as his final answer to the problem. He

explains, `̀ It is three lots of 10.'' He seems to think that x should represent everything that

was not explicitly given in the data, although he knows that the side labeled x cm is 10 cm

long and the side labeled 2x cm is 20 cm long.

Dean's thinking may be the reason why several students in the large sample wrote x +

85 + 125 = 1,410 as their equation for the BUS problem. (Note: The most common nearly


correct equation for BUS was not this one but x + 85 + x + 125 = 1,410, which we believe

is a stack error (Davis, 1984) with a quite different origin. The initial x being given in the

question statement is simply forgotten.)

Joel has in mind multiple and shifting referents for x in the MARK problem. He writes

the correct expression x + 5 for Mark's amount. Then he writes x + 5 = 47 and the

interviewer queries him.

1. I: What does it say?

2. J: You've got one starting number and you add 5 and get 47.

3. I: This [indicating 47] is the total amount, this

4. [indicating 5] is the extra five, so what is this x?

5. [indicating x in the equation x + 5 = 47]

6. J: The amount they both get. The amount that Jan gets. I

7. just like to keep the three of them, 47 dollars, x, and 5

8. dollars more, and make something out of them.

Although Joel has written the correct expression for Mark's money in terms of x, he

then sees the unknown x as `̀ the amount they both get'' (line 6). He also sees it as Jan's

amount (line 6). When asked to explain his equation, he does not relate it back to the

problem situation but interprets what he has written as a narrative about numbersÐa

sequence of events (line 2). His equation is written and solved for the interviewer, and not

seen by Joel as useful for obtaining a solution to the problem. When he solves it and

obtains x = 42, he says that Jan's amount is $42. He has lost connection with the meaning

of the problem, which he initially understood. His thinking seems to move back and forth

between x as the amount to be shared equally ($42) and x as the value of each of the final

shares ($21 and $26).

Les also uses x to refer to different quantities in MARK. He uses mathematical notation

informally to try to keep track of his thinking, using x to refer to any quantity he wants to

work out. His first equation for Item 2 has an x that is the first thing he wants to work out

(`̀ what is left,'' lines 1 and 7 below), but then he talks about `̀ sharing two x's'' (line 3).

Les knows how to work out the answer by keeping in touch with the situation, but does not

know how to write down his procedure. He finally writes a description of the steps in his

calculation, trying to make it look like algebra (line 11). To begin, Les writes the equation

5 + x = 47 and explains what it means.

1. L: x is what is left out of $47 if you take 5 off it.

2. I: What might the x be? [indicating x in his equation]

3. L: Say she gets $22 and he gets $27. They are sharing two x's.

4. I: What are the two x's?

5. L: Unknowns . . . they are two different numbers, 22 and 27.

6. I: So what is this x? [pointing to his equation 5 + x = 47]

7. L: I thought that was left over from $47, so it's $42.

The interviewer points out that he has three meanings for x, which makes the equation

hard to solve. Les, therefore, decides to use x and y for Jan's amount and Mark's amount.

He writes x + 5 + y = 47, and explains what he is going to do.


8. L: He's got 5 more than her, so you take 5 off. Would you

9. minus 47 from 5 plus x plus y?

It appears that he wants to subtract 47 from both sides of his new equation,

remembering a method he has been taught. The interviewer wants to see how he will

do this (line 10), but Les reverts to arithmetic thinking (lines 11, 12).

10. I: OK, try that.

11. L: [writes 47ÿ5 + x + y] Then minus 5 is out of the

12. way, so you split it in half. If you take the 5 off,

13. then you've just got two unknowns, 21 and 26.

The logic of Les's reasoning is sound, but he does not know how to express it clearly in

words (lines 11, 12) or how to write down his solution procedure. Like Joel, he has

thought of x as standing for 42, 21 and 26, that is, for any quantity that was unknown and

was to be worked out. Les has used the equation as a personal shorthand, temporarily

jogging his memory of what the numbers and quantities are.

Tim begins with x standing for Jan's amount. He then writes x + 5 correctly for Mark's

amount, but extends it to make x + 5 = x, saying that the x after the equals sign is `̀ Jan's

x.'' The interviewer queries him about the meaning of the other x.

1. I: So what is this x? [points to the first x in x + 5 = x]

2. T: That's Mark's x.

3. I: And why do we add 5 to it?

4. T: Because Mark has 5 more dollars than Jan. No, that's

5. not right, it should be Jan's x plus 5 equals Mark's x.

After writing the correct expression x + 5 for Mark's amount, Tim perceives the x as

now `̀ belonging to Mark'' (line 2). He explains that the 5 added to it shows that `̀ Mark

has 5 more'' (line 4). This errorÐassociating the larger of two variables in an equation

with the added quantityÐhas been observed and explained in other studies of algebra

learners (MacGregor & Stacey, 1993).

6. I: Could you write an equation to say that Mark and Jan

7. have $47 in total?

The interviewer explains that to write an equation you don't have to work out a

numerical answer first. Tim now thinks he should write what he would do to work out the

answer (line 8).

8. T: x divided by half equals x [writes x � (1/2) = x]

9. I: So you take the money and you halve it. It that what you mean?

10. T: Yes.

In line 8 Tim is writing x to mean `̀ some total amount of money'' and then x again to

mean `̀ half the money.'' For him, x � (1/2) = x makes sense because he knows, at least


momentarily, what each x refers to. He wants to share the $47 equally first, and denotes

this sharing incorrectly as �1/2.

During the interview, he has used x to mean `̀ Jan's amount'' (line 5), `̀ the total'' (line

8), and `̀ half the total'' (line 8). Although he recognizes that if Jan has $x then Mark has $

(x + 5), he is not sure whether the x in (x + 5) is `̀ Jan's x'' or `̀ Mark's x.'' Since (x + 5)

represents Mark's money, he first thinks, not unreasonably, that the x in it should be

`̀ Mark's x'' (line 2). He later uses x to mean any unknown quantity he is thinking of.

Tim's uses of both letters and equations exhibit all modes. His equations (x � (1/2) = x

and x + 5 = x) are action statements about how to get from one unknown quantity to

another. `̀ Equations'' such as these, which he and many other students wrote, represented

steps in their reasoning but were not useful except as an informal reminder of the most

recent idea they had in mind.

4.3. The Effect of Arithmetic Thinking on Equation and Symbol Use

Both the written responses and the interviews have provided examples to illustrate how

students' prior experiences with solving problems in arithmetic gives them a compulsion

to calculate which is manifested in the meaning they give to `̀ the unknown,'' their

interpretations of what an equation is, and how they solve equations.

The interviews show students who have failed to distinguish the different notions of

unknown quantities in arithmetic and algebra. Arithmetic use as exhibited by Joel, Les and

Tim sees the unknown as the thing that is going to be found from a calculation. For this

reason, they changed the referent of x at different stages of their thinking. Joel and Les

each used at least three referents within a minute or so. As the solution process continued,

they used the one letter to stand for the many different unknown quantities that are present

in even the simple situations portrayed in the problems. All the interviews also show how

the total of all the unknown quantities (the whole unknown length in TRIANGLE or the

whole unknown money ($42) in MARK) is also seen as `̀ the unknown.'' In the light of

this, the standard instruction given by teachers to let a letter stand for `̀ the unknown''

seems particularly inappropriate and highlights the technical knowledge underlying how

teachers automatically classify problems as having a certain number of unknowns.

The equations that students wrote show that they have three different perceptions of

what an equation is:

. a formula for working out the answer;

. a narrative describing operations yielding a result;

. a description of essential relationships.

All these are useful in certain contexts. From mathematics and other subjects such as

science, students are familiar with equations as formulas, that is as instructions for

calculating. They know how to use a formula by substituting values for variables. Many

students in our study who were willing to write equations wrote them as formulas for

direct calculation.

The other common perception, besides `̀ equation as formula'' is `̀ equation as

narrative''Ða story of what has happened. Students were able to use this interpretation

for NUMBER and translate the narrative into mathematical symbols. In all of the schools


where this item was used, some students (10% of the sample) narrated the sequence of

actions only, as n � 8 ÿ 3 � 3. More than one-third of the equationsÐwe have counted

them as correctÐwere written as n � 8 ÿ 3 � 3 = n � 2, in which each action is

represented (i.e., you multiply n by 8, then you take away 3, then . . . etc.). Students were

unable to manipulate the equation in this form, possibly because its format did not provide

a familiar visual stimulus. The small number of students who wrote (8n ÿ 3)/3 = 2n were

more likely to be successful.

The notion of an equation as a narrativeÐa story of `̀ what would happen''Ðunderlies

the successful formulation of many equations other than number stories. For example, we

see this approach in the interview with Joel, who said `̀ You've got one starting number

and you add 5 and get 47.'' In Joel's case, however, the `̀ starting number'' was `̀ the

amount they both get'' instead of the conventional `̀ amount that Jan gets.''

For some problems (e.g., TRIANGLE) the narrative and the description of relation-

ships produce the same equation (x + 2x + 14 = 44). The formula and the narrative

both encode actions in a temporal sequence. For example, `̀ To work out x, you

subtract 14 from 44 and divide by 3'' becomes the formula x = (44ÿ14)/(3), and `̀ If

you add x, 2x, and 14 together, you get 44'' becomes x + 2x + 14 = 44. The

description of relationships specifies a situation of equivalence, but, as we have seen

earlier in this article, it may be constructed by translating a narrative. The concept of

an equation as a statement about equality, rather than as a formula, is crucial to the use

of an algebraic method for solving problems.

5. IMPLICATIONS FOR TEACHING

Algebra is a language for expressing mathematical information, but it is also a set of

extremely powerful problem solving methods. These methods transform the solution of

large classes of problems from intellectual challenges or puzzles to routine tasks. Our

research has shown that in many schools the majority of students have not learned how to

use algebra in this way.

In our study we observed great variation between schools and classes. It is clear, from our

sample of Australian students, that different teachers emphasize different problem-solving

strategies (see Chiappini & Lemut, 1991, for similar variability in Italian students). Under

the guise of teaching algebra, some teachers promote non-algebraic methods because they

believe they are easier for students. By doing so they fail to provide opportunities for students

to learn more powerful mathematical methods.

Jennings and Dunne (1996), referring to shortcomings of teaching methods popular in

Britain, have pointed out that an emphasis on responding to students' needs (as perceived

by the teacher) makes teachers reduce the complexity of tasks and encourage informal

methods. Our data indicate that, in certain schools in our sample, teachers had taken this

approach. Instead of using simple problems merely to illustrate methods essential for

harder problems, and moving students on to problems that require algebra, teachers had

allowed students to continue solving simple problems using intuitive, non-algebraic

methods. However, the substantial variation in success rates in this study and others

(Stacey & MacGregor, 1997) indicate that routines for solving word problems can be

taught effectively.


From the student's point of view, using algebra to solve a problem is an extra

difficulty imposed by teachers for no obvious purpose. This attitude is reasonable given

that most of the problems that appear in our commonly used textbooks can be readily

solved by arithmetic reasoning or trial and error. To appreciate the value of algebra as a

problem-solving tool, students must work on problems that they can't easily solve

without algebra. Trivial problems (e.g., MARK and BUS) should be used only to

introduce and demonstrate the algebraic method. It has been suggested that it is not easy

to devise problems requiring algebraic solutions that are within students' capability

(Bednarz & Dufour-Janvier, 1994; Panizza, Sadovsky, & Sessa, 1996). Certainly, such

problems will be more complicated to describe in words than others (e.g., comparing

costs of two car rental companies with linear relationships between distance traveled and

charge, instead of finding charges for one company). The practical solution probably lies

in using one such setting for several questions, rather than constantly changing context

as we did with our test items.

This article has set out to show how students have to learn a new logic of thinking

for solving problems with algebra. Several aspects of learning this new logic have

been documented in the past, but the way in which aspects of arithmetic problem

solving operate on use of unknowns, types of equations and the way they are solved

has not been shown before. Patterns of thinking based in previous arithmetic practices

deflect students from the algebraic path at every step. Algebra can be taught so that

the arithmetic aspects (solving numerically, writing formulas, interpreting equations

only as narratives, using unknowns to stand for things to be calculated) are

paramount, but this approach gives students little of the power of the algebraic

method of solving problems.

REFERENCES

Arzarello, Ferdinando. (1991). Procedural and relational aspects of algebraic thinking. In Fluvia

Furinghetti (Ed.), Proceedings of the Fifteenth International Conference for the Psychology of

Mathematics Education (Vol. I, pp. 80±87). Assisi, Italy: PME.

Bednarz, Nadine, & Dufour-Janvier, Bernadette. (1994). The emergence and development of algebra

in a problem solving context: A problem analysis. In J. P. da Ponte & J. F. Matos (Eds.),

Proceedings of the Eighteenth International Conference for the Psychology of Mathematics

Education (Vol. II, pp. 64±71). Lisbon, Portugal: PME.

Bednarz, Nadine, & Janvier, Bernadette. (1996). Emergence and development of algebra as a

problem-solving tool: Continuities and discontinuities with arithmetic. In Nadine Bednarz,

Carolyn Kieran, & Lesley Lee (Eds.), Approaches to algebra. Perspectives for research and

teaching (pp. 115±136). Dordrecht: Kluwer.

Booth, Lesley. (1987). Equations revisited. In J. C. Bergeron, N. Herscovics, & C. Kieran (Eds.),

Proceedings of the Eleventh International Conference for the Psychology of Mathematics

Education (Vol. I, pp. 282±288). Montreal, Canada: PME.

Chaiklin, Seth. (1989). Cognitive studies of algebra problem solving and learning. In Sigred Wagner

& Carolyn Kieran (Eds.), Research issues in the learning and teaching of algebra (pp. 93±

114). Reston: NCTM.

Charbonneau, Louis. (1996). From Euclid to Descartes: Algebra and its relation to geometry. In

Nadine Bednarz, Carolyn Kieran, & Lesley Lee (Eds.), Approaches to algebra. Perspectives

for research and teaching (pp. 15±37). Dordrecht: Kluwer.


Chiappini, Giampaolo, & Lemut, Enrica. (1991). Construction and interpretation of algebraic mod-

els. In Fluvia Furinghetti (Ed.), Proceedings of the Fifteenth International Conference for the

Psychology of Mathematics Education (Vol. I, pp. 199±206). Assisi, Italy: PME.

Cortes, Anibal, Vergnaud, Gerard, & Kavafian, Nelly. (1990). From arithmetic to algebra: Negotiat-

ing a jump in the learning process. In George Booker, Paul Cobb, & Teresa de Mendicuti

(Eds.), Proceedings of the Fourteenth International Conference for the Psychology of Mathe-

matics Education (Vol. II, pp. 27±34). Mexico: PME.

Davis, Robert B. (1984). Learning mathematics: The cognitive science approach to mathematics

education. London: Croom Helm.

Filloy, Eugenio, & Rojano, Teresa. (1989). Solving equations: The transition from arithmetic to

algebra. For the Learning of Mathematics, 9(2), 19±25.

Fujii, Toshiakira. (1993). A clinical interview on children's understanding and misconceptions of

literal symbols in school mathematics. In I. Hirabayashi, N. Nohda, K. Shigematsu, & Fou-Lai

Lin (Eds.), Proceedings of the Seventeenth International Conference for the Psychology of

Mathematics Education (pp. 173±180). Tsukuba, Japan: PME.

Gray, Eddie, & Tall, David. (1991). Duality, ambiguity and flexibility in successful mathematical

thinking. In Fluvia Furinghetti (Ed.), Proceedings of the Fifteenth International Conference

for the Psychology of Mathematics Education (Vol. II, pp. 72±79). Assisi, Italy: PME.

Herscovics, Nicolas, & Linchevski, Liora. (1994). A cognitive gap between arithmetic and algebra.

Educational Studies in Mathematics, 27, 59±78.

Hiebert, James, & Lefevre, P. (1986). Conceptual and procedural knowledge in mathematics: An

introductory analysis. In James Hiebert (Ed.), Conceptual and procedural knowledge: The

case of mathematics (pp. 1±27). Hillsdale: Erlbaum.

Janvier, Claude. (1996). Modeling and the initiation into algebra. In Nadine Bednarz,

Carolyn Kieran, & Lesley Lee (Eds.), Approaches to algebra. Perspectives for research

and teaching (pp. 225±236). Dordrecht: Kluwer.

Jennings, Sue, & Dunne, Richard. (1996). A critical appraisal of the national curriculum by

comparison with the French experience. Teaching Mathematics and Its Applications, 15(2),

49±55.

Just, Marcel A., & Carpenter, Patricia A. (1989). The role of working memory in language com-

prehension. In D. Klahr & K. Kotovsky (Eds.), Complex information processing (pp. 31±68).

Hillsdale: Erlbaum.

Just, Marcel A., & Carpenter, Patricia A. (1992). A capacity theory of comprehension: Individual

differences in working memory. Psychological Review, 99(1), 122±149.

Kieran, Carolyn. (1988). Two different approaches among algebra learners. In A. F. Coxford & A. P.

Shulte (Eds.), The ideas of algebra. K-12 (pp. 91±96). Reston: NCTM.

Kieran, Carolyn. (1992). The learning and teaching of school algebra. In Douglas Grouws

(Ed.), Handbook of research on mathematics teaching and learning (pp. 390± 419).

New York: MacMillan.

Kirshner, David, McDonald, Janet, Awtry, Thomas, & Gray, Elizabeth. (1991). The cognitivist

caricature of mathematical thinking: The case of the students and professors problem. Paper

presented at the 13th Annual Conference of the North American Chapter of the International

Group for the Psychology of Mathematics Education, Blacksburg, Va.

Lewis, A. B., & Mayer, Richard E. (1987). Students' miscomprehension of relational statements in

arithmetic word problems. Journal of Educational Psychology, 79(4), 363±371.

MacGregor, Mollie, & Stacey, Kaye. (1993). Cognitive models underlying students' formulation of

simple linear equations. Journal for Research in Mathematics Education, 24(3), 217±232.

Mayer, Richard E., Lewis, A. B., & Hegarty, M. (1992). Mathematical misunderstanding: Qualita-

tive reasoning about quantitative problems. In J. Campbell (Ed.), The nature and origins of

mathematical skills (pp. 137±153). Amsterdam: North Holland.


Nathan, M. J., Kintsch, Walter, & Young, E. (1992). A theory of algebra word-problem comprehen-

sion and its implication for the design of learning environments. Cognition and Instruction,

9(4), 329±389.

Panizza, Mabel, Sadovsky, Patricia, & Sessa, Carmen. (1996). The first algebraic learning: The

failure of success. In Luis Puig & Angel GutieÂrrez (Eds.), Proceedings of the Twentieth

Conference of the International Group for the Psychology of Mathematics Education (Vol.

4, pp. 107±114). Valencia, Spain: PME.

Sfard, Anna. (1987). Two conceptions of mathematical notions: Operational and structural. In

J. C. Bergeron, Nicolas Herscovics, & Carolyn Kieran (Eds.), Proceedings of the Eleventh

International Conference for the Psychology of Mathematics Education (Vol. III, pp. 162±

169). MontreÂal: UniversiteÂ de MontreÂal.

Sfard, Anna. (1991). On the dual nature of mathematical conceptions: Reflections on processes and

objects as different sides of the same coin. Educational Studies in Mathematics, 22, 1±36.

Stacey, Kaye, & MacGregor, Mollie. (1995). The influence of problem representation on algebraic

equation writing and solution strategies. In Luciano Meira & David Carraher (Eds.), Pro-

ceedings of the Nineteenth International Conference for the Psychology of Mathematics

Education (Vol. 2, pp. 90±97). Recife, Brazil: Universidade Federal de Pernambuco, PME.

Stacey, Kaye, & MacGregor, Mollie. (1997). Multiple referents and shifting meanings of unknowns

in students' use of algebra. In Erkki Pehkonen (Ed.), Proceedings of the Twenty-First Inter-

national Conference for the Psychology of Mathematics Education (Vol. IV, pp. 190±197).

Finland: University of Helsinki, PME.

Tall, David, & Thomas, Michael. (1991). Encouraging versatile thinking in algebra using the com-

puter. Educational Studies in Mathematics, 22, 125±147.

Teppo, Anne R., & Esty, Warren W. (1994). Problem solving using arithmetic and algebraic thinking.

Paper presented at the Annual Conference of the North American Chapter of the International

Group for the Psychology of Mathematics Education.

Ursini, Sonia. (1990). Generalization processes in elementary algebra: Interpretation and symboliza-

tion. In George Booker, Paul Cobb, & Teresa de Mendicuti (Eds.), Proceedings of the Four-

teenth International Conference for the Psychology of Mathematics Education (Vol. II, pp.

149±156). Mexico: PME.


Learning the Algebraic Method of Solving Problems

Documents

Transcript of Learning the Algebraic Method of Solving Problems