Processing Load and the Use of Concrete Representations and Strategies for Solving Linear Equations

JMB JOURNAL OF MATHEMATICAL BEHAVIOR, 16 (4), 379-397 ISSN 0364-0213. Copyright 0 1997 Ablex Publishing Corp. All rights of reproduction in any form reserved.

Processing Load and the Use of Concrete Representations and Strategies for

Solving Linear Equations

GILLIAN BOULTON-LEWIS,TOM COOPER,

BILLATWEH,HITENDRAPILLAY,LYNNWILSS,AND SUEMUTCH

Queensland Universit;v of Technology

This paper is a report of’students’ responses to instruction which was based on the use of concrete

representations to solve linear equations. The sample consisted of 2 1 Grade 8 students from a mid-

dle-class suburban state secondary school with a reputation for high academic standards and inno-

vative mathematics teaching. The students were interviewed before and after instruction.

Interviews and classroom interactions were observed and videotaped. A qualitative analysis of the

responses revealed that students did not use the materials in solving problems. The increased pro-

cessing load caused by concrete representations is hypothesised as a reason,

The conventional wisdom of mathematics teaching has long been that understanding is

assisted by the use of concrete representations, that is, materials which can be physically

manipulated. Teachers are faced with the task of providing experiences which enable stu-

dents to abstract concepts in order to develop links between materials used and the mathe-

matical concept (Queensland Department of Education, 1991). The belief is that students

can construct a mental representation from the concrete representation as long as the

manipulative material is sufficiently isomorphic to the mathematical concept or action.

Many theories of mathematics education are based on the use of concrete representations

as manipulatives for making connections with mathematical ideas.

The triadic model (Payne & Rathmell, 1975) for learning number related modelling

with materials to language and symbols. Hiebert (1988) stressed the importance of making

connections “between the written marks on paper and the quantities or actions they repre-

sent” (p. 336) and preserving the “relevant properties when connecting referent to symbol”

(p. 338). Bloomer and Carlson (1993) stated that the abstract stage followed understanding

of the concrete and connecting stages. There is a growing body of evidence, however,

which shows that concrete materials often fail to produce the expected positive outcomes

(Boulton-Lewis, 1993b; Hart, 1989; Sowell, 1989). It is proposed that the difficulties expe-

rienced with concrete representations and children’s apparent reluctance to use them are

due to the processing load that their use initially entails.

Direct all corrqmndence to: Gillian Boulton-Lewis, School of Learning and Development, Faculty of Educa-

tion, Queensland University of Technology. Victoria Park Rd., Kelvin Grove. Queensland, Australia, 4059

<[email protected]>.

379

3x0 BOULTON-LEWIS ET AL.

COGNITIVE LOAD THEORIES

Contemporary cognitive research in early mathematics learning has assessed the effect of

the processing load imposed by the use of concrete representations and procedures used in teaching selected aspects of early mathematics such as: place value, addition, subtraction, and measurement (Boulton-Lewis, 1993a; 1993b; Boulton-Lewis & Tait, 1994; Boulton- Lewis, Wilss, & Mutch, 1966). This research accounted for use of alternative mental strategies in terms of extraneous cognitive load imposed by strategies involving concrete representations and teachers’ lack of awareness of this load. Boulton-Lewis (1993a, 1993b) acknowledged the usefulness of concrete analogs (materials) in reducing learning effort, mediating transfer between tasks and situations, and indirectly facilitating transition to higher levels of abstraction. She suggested that if the analogs were not well understood, and imposed an additional processing load, they could be a hindrance to understanding. In earlier research she observed that concrete representations often failed to produce expected results, arguing that this was due to the extra processing load required to map the manipulative activities into a mental model (Boulton-Lewis, 1991).

In line with this, Hart (1989) found that there appeared to be a gap between the use of concrete representations and symbolic mathematical language that may possibly be due to uncertainty when using materials. Children apparently preferred to calculate mentally than to use materials to solve problems. Boulton-Lewis and Halford (1992) proposed two main and interrelated explanations for the difficulties that children experience with early mathematics learning. The first is that if students do not know representations, symbols, and procedures well enough, these will impose a load greater than students can process. The second is that if teachers are not aware of the processing load, they may make the task either more difficult, meaningless, or actually mathematically incorrect by the strategies

they choose. Cognitive load has also been considered in relation to different instructional

approaches. Cooper and Sweller (1987) and Sweller and Low (1992) have argued that the cognitive load imposed by the means-end problem solving strategy interferes with novices’ learning of algebraic procedures. They contended that well constructed worked examples, which reduce cognitive processes such as split attention and redundancy, impose less cognitive load. As a result, students have more cognitive resources to direct to learning the solution procedures.

THE CONSTRUCTION OF ALGEBRA KNOWLEDGE

It is no longer satisfactory to equate algebra with techniques such as variable manipulation, simplification, and solving for unknowns, or with topics in secondary texts such as linear equations and quadratics. Kieran (1992), on the basis of the psychological model proposed by Sfard (1991), distinguished between procedural and structural conceptions of algebra. Procedural refers to arithmetic operations carried out on numbers to yield numbers, such as solving 2x + 5 = 11. Structural refers to operations carried out in algebraic expressions, such as simplifying 3x + y + 8x. The consequence of this distinction for algebra is that it is seen as emerging from the operations of arithmetic. This has implications for learning and teaching. The generalised conceptions that are algebra must be constructed in terms of the students’ prior knowledge of arithmetic and then extended into higher order abstraction.

CONCRETE REPRESENTATIONS FOR LINEAR EQUATIONS 381

Prior concepts include arithmetic and its symbols, operations, and laws. Further requisite

knowledge includes particular concepts related to the forms of expression in algebra such as variable, expression, equation, equality, and equivalence (Leitzel, 1989; Kieran, 1989).

Achievement rates in algebra have been poor (Brown, Carpenter, Kouba, Lindquist, Sil-

ver, & Swafford, 1988). For example, secondary students often seem unable to apply basic algebra concepts and skills in problem-solving situations and do not appear to understand many of the structures underlying these concepts and skills. Difficulties in learning algebra

have long been documented (Thomdike, Cobb, Orleans, Symonds, Wald, & Woodyard,

1923), particularly when a distinction is drawn between performance and understanding as outcomes of instruction (Rosnick & Clements, 1980). Booth (1988) categorised algebra

errors as including the non-numerical nature of algebra answers and misconceptions con-

cerning the meaning of letters and variables. Herscovics and Kieran (1980) identified difficulties with equations and the equals sign. Instruction does not seem to be bridging the

gap between arithmetic and algebra, particularly in developing meaning for variables and for the equals sign. In many cases algebra learning and teaching have been reduced to a col- lection of meaningless rituals (Davis, 1988).

CONCRETE REPRESENTATIONS AND ALGEBRA

Current curriculum approaches, such as those described by Quinlan, Low, Sawyer, and White (1993), entail teachers using concrete and other representations to introduce con-

cepts such as variables in algebra. However, because algebra knowledge quickly becomes abstract, manipulative techniques can become artificially complex (Thompson, 1988). The implications from cognitive theory described above are that such approaches may be inef-

fective unless cognitive load is considered. The reasons for this can be seen when the

knowledge needed to solve the equation 2x + 5 = 13 is analyzed.

1. The student has to recognise that these symbols represent an equation which includes

a variable as an unknown and involves two operations, the “times 2” (with a variable) and the “plus 5.”

2. The student has to determine whether to use an inverse or balance approach in han- dling the equals sign and to identify the sequence “plus 5” and then “times 2.”

3. Operations have to be carried out correctly to determine the value of the variable. The

difficulty for students is that they have to integrate knowledge of:

a. symbols, numbers, and variables,

b. basic computations,

c. arithmetic laws for individual operations and sequences of operations,

d. mathematical meaning of equals, and

e. mathematical meaning and laws of operations on variables.

It can be seen that this is a complex process with the potential to cause cognitive overload unless all or most items are well known. It is posited that the use of concrete materials in addition to this, unless they are well understood, will increase the processing load further. If one believes that algebra emerges from arithmetic operations, then it may be more effec-

382 BOULTON-LEWIS ET AL.

tive to ensure this knowledge is well established and to work from that base to variables and operations on them.

RATIONALE FOR THE STUDY

Recent research has addressed the use ofconcrete representations in teaching algebra (Booth, 1987; MacGregor & Stacey, 1995; Thompson, 1988) but has failed to consider the processing load that is inherent in the process. In addressing this issue, in this research, the effect of instruction with concrete representations on student performance in solving linear equations was investigated. The first analysis of the results was given in Atweh, Boulton-Lewis, and Cooper (1994), with a further analysis in Boulton-Lewis, Cooper, and Atweh (1995).

The aim of this paper is to (a) analyse in detail students’ understanding of variable and equation, (b) document students’ strategies for solving linear equations, and (c)discuss these findings in relation to the cognitive load imposed by the use of concrete representations.

METHOD

Participants

Twenty-one students from a grade 8 class participated in this study. The majority of these students were 13 years old. They attended a middle-class suburban state secondary school with a reputation for high academic standards and innovative mathematics teaching.

Procedure

The students were interviewed individually before a period of instruction using concrete materials (pre-interview) and one month later (post-interview). Interviews consisted of tasks for: meaning of a variable, variable representation, solution of linear equations, and the use of concrete and pictorial representations. These interviews took about 30 minutes and were videotaped.

Tasks

A card depicting 2x + 3 was presented to students who were asked what the variable within this expression meant. For variable representation, 2x + 3 was presented again and students were asked to show this using materials. Students were then shown 2x + 5 = 17 on a card and asked to represent this equation with materials and to solve it using any means they wished. Initial strategies used in equation solution were noted. During post-interview students were asked if they thought materials were useful in solving equations.

Materials

Students were provided with materials (cups, counters, and sticks) based on those used during classroom instruction. Pencil and paper were also provided. If the students did not use the concrete representations voluntarily, they were asked to do so.


TABLE 1. Concrete and Pictorial Representations of Variables and Units Used in the Lessons

Description Explanation Concrete

Pictorial

Representation Symbol

White cup Variable (unknown)

Yellow Cup Negative Variable (unknown) $368 VW/y -jx

Green Counter One unit

Gs 9% 3

Yellow counter Negative one unit 0000 -4

Y Y Y Y

Instruction

Prior to this study students completed a unit on variables as generalisations through pat- terning and use of other representations. The concept of a variable as an unknown quantity was explored and linear equations and their solutions were investigated. The unit of work relevant to this study was covered in five lessons which were videotaped and transcribed. Instruction was based on use of representations as summarized in Table 1.

Rules guiding these representations were introduced throughout the lessons. These were: x is an unknown quantity (variable), the white cup is unknown (variable), yellow cup is negative of unknown (variable), green counters are units, yellow counters are negative units, subtraction is the same as adding a negative, in 2x the missing operator is multiplica- tion, within expressions and equations cups must have the same value.

The contents of the lessons were sequenced as follows:

Writing Algebraic Expressions. “I had a bag of marbles and I lost half of them” as x + 2 or x/2, with x as the unknown number of marbles. Modelling Mathematical Expressions. The teacher represented the expression 2x + 4 concretely, then pictorially on the board (see Figure 1). Students were set examples to represent pictorially; those who experienced difficulty with this task were encouraged to use the concrete materials that were available at the front of the room. Solving Equations Concretely. The teacher placed a stick on the table to separate the left (LHS) and right-hand sides (RHS). For 2x + 3 = 7,2 cups and 3 green counters were

FIGURE 1. Pictorial Representation of 2r + 4


LIIS RHS

FIGURE 2. Pictorial Representation of tr + 3 = 7

placed on the LHS and 7 green counters were placed on the RHS. The technique of upsetting and restoring balance was shown by taking 3 counters from each side. Sharing the remaining counters equally between cups showed that x = 2.

Solving Equations Diagrammutically. The above equation was then drawn on the board as shown in Figure 2. Maintaining the balance was depicted by crossing out

symbols.

Solving Equutions: A Short-cut! The teacher referred to the diagram depicted in Fig- ure 2 and explained to the students what he referred to as a “short-cut” method for solv-

ing equations. This is summarised in Table 2.

RESULTS

All interviews were analysed qualitatively, in line with Richard’s and Richard’s (1994) belief that qualitative research involves “recognition of categories in the data, generation of ideas about them, and exploration of meanings” (p. 446). The analysis, carried out using the computer program Non-numerical Unstructured Data Indexing Searching and Theory- building (NUD.IST) version 3.04, revealed categories which are described and delineated below, along with interview extracts that illustrate each category.

Data from pre and post-interviews were compared for variable meaning and variable representation for 2x + 3; and for equation representation, initial solution strategies for an

Short-Cut

TABLE 2. Teacher’s Explanation of the “Short-Cut” Method

Teacher’s Explanation

2x + 3 = 7 “Instead of going through this model process we can represent this with short-hand mathematics.”

2x +A= 7 “If you add 3 and take 3 you are left with nothing there.”

46

2r = 4 “We’re left with 2 cups (points to LHS] and 4 counters [points to RHS].”

2.x = 2 x 2 “We pair them off.. rearranging them. Two in that cup and 2 in that cup. What we’ve really done is

divide those 4 counters into two lots of 2. We can see that is two counters per cup. The front two

numbers are the same so the second number of the product must also be the same.”

x=2 “x must equal 2. That’s the short-cut.”


equation, and material use for equation solution for 2x + 5 = 17. Students perceptions of

material use are also presented in this paper.

Variable Meaning

Three types of variable meaning were identified for 2x + 3. These were any number, unknown number, and object, as shown below.

l Any Number. The interviewer asked a student what the x meant. The student replied, “To stand in as a number, and like, you can use them in rules, so it is used

instead of a number really and it can be used for any number. It stands in for any

number at all.”

l Unknown Number. The interviewer asked, in reference to variables shown, “Do you know what x stands for?” The student said, “They are used to symbolise an unknown

number.”

l Object. The interviewer asked, “So what do these [variables] stand for?’ The student replied, “Well, things you are using . like sticks.”

Pre-interview responses revealed the majority of students (11) believed the variable

meant any number, while eight students said it was an unknown number. One student referred to the variable as standing for objects, while another student could not verbalize what the variable meant. Post-interview responses revealed a change as only seven students regarded the variable as any number while the majority of students (14) now per-

ceived the variable as an unknown number.

Variable Representation

Students exhibited the following representations for 2x + 3.

Productive Representations. The following representations were productive in that

they could have been used to find the solution to a linear equation.

l Cups. A student stated, “There’s two cups with the unknown amount in them and it’s three counters on the outside.”

l Mixed Objects. A student placed two sticks and three green counters on the table and stated, “That’s two x’s [pointed to the two sticks] and that’s 3 [pointed to the three blue

counters] .”

l Different Coloured Counters. A student held up one yellow counter and said, “Okay, one of these equals lx.” She then placed two yellow counters on the table and then

added three green counters, saying, “And this equals the 3.”

l Same Object. A student placed two sticks and three sticks on the table in two distinct groups and stated “So, 2x + 3.” The interviewer asked, “So this is the 2x and this is the +3?” The student said, “Yes.” The interviewer held up one of the sticks from the group of two and asked how much it was worth. The student replied, “lx.” The interviewer held up one of the sticks from the group of three. The student said, “1.”

386 BOULTON-LEWIS ET AL

Unproductive Representations . . . The following representations might have lead to

error and were therefore considered unproductive.

Concrete Value. A student put 2 blue counters in a bag and 3 outside the bag. The

interviewer asked, “So is this 2x [pointing to 2 blue counters in the bag] plus 3 [pointing

to 3 outside]?’ The student replied, “Yes!” The interviewer then said, “Oh, that’s 3. So

each of these is worth?’ The student stated, “One.” Finally the interviewer asked, “So

what would x be?” and the student replied, “One.”

Literal. A student placed two sticks on the table and said, “That’s 2.” He then placed

one green counter on the table next to the sticks and said, “... and that’s the missing

number, plus 3 [placed 3 yellow counters next to the green counter].”

An analysis of variable representation for this expression revealed that for pre-inter-

view only eight of the 21 students used productive representations. This had increased

to 15 students by post-interview. Of the eight students whose representations were pro-

ductive during pre-interview, four chose to use mixed objects (i.e., counters and sticks),

three used different coloured counters, and one student used the same object but quali-

fied the value of each use. Thirteen students represented the variable during pre-inter-

view unproductively. Of these five gave the variable a concrete value which was

usually one. This occurred because students would represent the variable with the same

object as the other numbers in the expression and when asked what each object was

worth would reply, “One.” Four students represented the variable literally, which

involved using objects not only to represent the numbers in the expression, but also to

represent x. Some students even used crossed sticks to show the addition sign. Four stu-

dents could not represent it at all.

By post-interview, four students were able to use cups as taught in class, three students

used different coloured counters to differentiate the variable from the number, and five

used mixed objects. Three students used the same object for both variable and number, but

again qualified their difference in use. Of the six students whose representations were

unproductive, four represented the variable literally and two could not represent it at all.

A comparison of representations from pre to post-interviews, by student, showed eight

students had improved. Initially these students had either represented the variable literally,

given it a concrete value, or could not represent it. By post-interview, however, four could

represent the variable correctly as taught with cups, one used different coloured counters,

two used mixed objects, and one used the same object but qualified its value.

Of the seven students who could represent the variable for both pre and post-interviews,

six used the same representations on both occasions. These were mixed objects (3), differ-

ent coloured counters (2) and same object with qualification (1). The remaining student

showed the variable with mixed objects initially while later in the year used the same object

and qualified the value.

Equation Representation

The following representations were evidenced for 2x + 5 = 17.


Productive Representations. The following representations were productive in that

they could have been used to find the solution to a linear equation.

l Cups. A student placed two empty cups on the table and said, “x is equal to one white

cup and then 2x there.” The student added 5 yellow counters, and stated, “Then you add

5 there,” and finally added 17 yellow counters saying, “Then that’s equal to 17.”

l Mixed Objects. A student made a group of 17 sticks and said, “That’s 17 stripes.” The

student then made a group of 5 blue counters as he stated, “Then I have to have 5.”

Finally, the student added 2 green counters and said, “Then 2x, they resemble . . that’s

lx, that’s 2x.”

Unproductive Representations. The following representations might have lead to error

and were therefore considered unproductive.

Incomplete. A student placed two cups on the table, put counters in each, placed five

green counters near the two cups on the table (but didn’t put out 17 counters to complete

the equation) and stated “Okay, equals 17 and there’s 12 in both of these cups.” The

interviewer asked about the 17 to which the student replied, “There should be six in

each cup, so the 12 plus the 5 is the 17.”

Literal. A student placed 2 sticks on the table and said, “So there’s 2x and x has to be

different.” The student then placed a pile of green counters next to the sticks and stated

“So there’s two [pointed to the 2 sticks] and there’s the x [pointed to the green

counters], and plus 5 [placed 5 sticks beside the group of counters] equals 17 which

should be in the red as well.”

Only two students showed a productive representation of the equation 2x + 5 = 17 dur-

ing pre-interview, with both using mixed objects. Nineteen students could not represent the

equation productively with materials during pre-interview. Of these, nine students had no

idea how to use the materials, six showed an incomplete representation where only one side

of the equation was represented, and four displayed the equation literally, i.e., representing

2x as 2 counters and one cup or one object.

For post-interview students used similar methods. Although 17 students still could not

represent the equation correctly, eight showed an incomplete representation, three repre-

sented the equation literally, and six could not represent it at all. For the productive repre-

sentations, cups were used once and mixed objects were used three times.

A comparison of pre to post-interviews, by student, showed that three students moved

from an unproductive representation to a productive representation. Two of these students

initially represented the equation literally but used either cups or mixed objects produc-

tively for the post-interview; the other student went from no representation to productive

use of mixed objects. Only one student was productive for both interviews, using mixed

objects each time. It is interesting to note that the other student who could represent the

equation productively with mixed objects for the pre-interview, showed an unproductive

representation for the post-interview. A total of fifteen students either could not or would

not represent the equation on either occasion.


Initial Strategies Used in Solving the Equation

Productive and unproductive strategies used by the students for 2x + 5 = 17 are shown

below.

Productive Strategies. The following representations were productive in that they could

have been used to find the solution to a linear equation.

l Inverse Mental. A student said, “I sort of took 5 away from 17, which was 12, and then 12 divided by 2, which equals 6.”

l Systematic Trial and Error. A student said “Six!” The interviewer asked, “How did

you work that out?’ to which the student replied, “Well, I started off with--I went 2 times 1 equals 2 plus 5. It doesn’t equal so many. Then I did that till I got up to 6. I went

2 times 6 plus 5 equals 17.” The interviewer then asked, “Did you try all the ones in

between? Did you try 3,4, 5, and 6?” The student replied “Yes.”

l Material, Counting Up. A student stated, “Six. I got the 5 [5 blue counters], then I fig- ured out-1 counted up to 17 with counters, but then I counted how many counters there

were and divided it by 2.”

Unproductive Strategies. The following representations might have led to error and

were therefore considered unproductive.

l Mental. A student said “Ten.” The interviewer asked, “How did you get ten?” to which the student replied, “Because 2 and 5 is 7, and x is 10, so it adds up to 17.”

l Material and Mental. A student placed two yellow counters on the desk and said, “There’d be 2, then X, then the plus sign, then the 5 [5 counters]. Then you’d get, like

the 7, those two equal the 7. Then you’ve got your answer on the other side. x would be

5. I plussed 2 times 5 was 10, and then you add the 2 and the 5, and that gives you the 17.”

l Paper. A student wrote 2x + 5 = 17 and said, “Well I’ve got the 2x plus 5 and then the-you’ve got the 5, that equals 17. Then you’ll have to find the x, which is, the x rep-

resents 10, because there’s the 2 and the 5, that makes 7, plus x which is 10, equals 17.”

The student then completed writing 2x + 5 = 17, x = 10.

Analysis of pre-interview strategies revealed that 14 students could solve the equation

correctly with 13 of these employing arithmetic mental strategies. Of these, ten chose to

use an inverse mental strategy which involved reversing the operations in the equation,

such as subtracting 5 from 17 to get 12 and dividing this number by 2 to get 6 for their

answer. Another three employed a systematic trial and error approach to find x. Only one

student used materials to arrive at a correct answer; however, this student did not use a bal-

ance approach. He counted up from 5 using counters, found that 12 were needed to reach

17, then divided the 12 into two groups to get 6 for the answer. Seven students could not solve the equation productively during the pre-interview. Six of these students calculated

mentally but lacked knowledge of concatenation (Herscovics & Linchevski, 1994), inter-

preting 2x as 2 plus x (MacGregor & Stacey, 1993). Another student used a combination of

materials and a mental calculation to obtain 5 as the value of x. This student had repre-


sented the left side of the equation literally but did not represent the 17. He calculated that

2 and 5 were 7 and said that x must equal 5. By the post-interview, students used only two productive strategies for initial equation

solution. These were the inverse mental strategy (14) and the systematic trial and error approach (1) described above. Five students used the unproductive mental approach of adding the 2 and 5 then taking this from 17 to get 10 for x. One student wrote this unproductive approach on paper.

A comparison of pre- and post-interview strategies, by student, showed that 13 students were productive on both occasions. Twelve of these used the inverse mental strategy and one moved from using materials and counting to the inverse mental strategy by post-interview. Only two students showed improvement in moving from the unproductive mental strategy to the productive inverse mental strategy by post-interview. Five students were unproductive on both occasions and one student who had employed the systematic trial and error approach initially actually employed the unproductive mental strategy for the post- interview.

The “Short-cut” Method

With the instruction given, it would seem logical that some students would use a productive balance mental strategy, as this was the strategy taught to the students as the “short- cut” method by the teacher. However, no students used this method in either pre- or post- interviews. The students stayed with the inverse mental strategy that many of them had used prior to instruction. This could be due to the fact that the “short-cut” method was only introduced briefly towards the end instruction.

Material Use for Solving the Equation

Two types of material use for 2~ + 5 = 17 were identified. Examples of these are given below.

Generate. The students used the balance perspective to generate the answer.

l Balance. A student placed 2 white cups on the table and said, “Hang on.. . x is equal to one white cup. And then 2x there.” The student placed 5 yellow counters beside this, saying, “Then you add 5 there.” Seventeen yellow counters were then put out, “That’s equal to 17.” The student then took 5 yellow counters away from both sides and said, “You take away 5 from there, so there’s no more counters, [then] 5 from 17 counters. So 12 divided by 2 equals 6.”

Illustrate. Operations used in mental strategies were applied with materials, sometimes in an incomplete way.

l Fitting the Answer. A student placed two empty cups on the table, put 6 counters in each cup and said, “You’ve got two cups with x amount of counters inside. We’ll put 6 in there.” Five green counters were placed on the table outside the cups, “Then you add 5 outside and all the counters equal 17, so now you know that there’s 6 counters inside the cup.”

390 BOULTON-LEWIS ET AL

l Following the Mentul. A student mentally calculated x to be 6. When asked to use the materials she made a group of 17 counters, removed 5 and divided the remaining counters into 2 equal groups. She said, “These 17, take away 5, and divide that by 2.”

l Reversing. A student placed a group of 17 sticks on the table, removed 5 of them and split the remaining sticks into 2 equal groups, describing this as “So you’ll have 17, I think-yes 17, then you take 5 away which is done in reverse and then you divide it by 2 which leaves 6.”

Only one student generated an answer with materials during the pre-interview. This student represented the equation literally, but ignored the object used to represent x and used the other objects to generate the answer. During post-interview, four students generated answers using the balance perspective. One was the generative student from the pre-interview who continued to use mixed objects to represent the equation and generate the answer. Of the other three students, one represented the equation literally but was still able to generate an answer, another used mixed objects to represent the equation and generate an answer and one student used cups correctly to represent the equation and generate the answer. Some students started with two groups of 6 counters, added 5 and got 17. Other students used the inverse method of starting with 17 counters, subtracting 5 and dividing by 2 (without using cups). Six students during pre-interview and ten students during post- interview illustrated their answers. One student was an exception during the post-interview. She used mixed objects to represent the equation but went on to illustrate the answer. Fourteen students during pre-interview and seven student in post-interview could not or would not use the materials to solve the equation.

Students’ Perceptions of Material Use

After solving the equation during post-interview, students were asked if they thought using materials had been helpful. Three students responded favourably with one student replying, “Instead of having to use your head, you can use either the counters to keep the number on the desk and then you can, after you’ve used them, you just write it down on your paper.”

Eight students felt that materials were no use at all. This was evidenced through responses such as, “I’m no good at it” (using the materials), “No, I can do them in my head,” and “No. Well, I find it a bit easier to sort of work it out on paper, because sometimes the things [materials] confuse you a bit. I find it easier to just write it straight down on paper.” Using materials some of the time was an option that ten students preferred. Most of these responses evidenced prior working out, either mentally or with pen and paper, before using materials. For example, “Now I can do them in my head and I work it out” and “When I can’t do them in my head [I use materials],” while another student answered simply, “I usually just do it on paper.”

DISCUSSION

In this paper, the effect of instruction with concrete representations on students’ understanding of expressions and linear equations and strategies in solving such equations has been explored. The focus has been on two factors: the students’ preferred mode of thinking (their understanding of variable and initial strategy for solving a linear equation) and the


TABLE 3. Preferred and Directed Responses for Variable Meaning and Representation, and Equation Representation and Solution Strategies, Incorporating Productivity

Pre-Interview Responses (n = 21) Post-Interview Responses (n = 21)

Preferred Mode of Thinking Variable Meaning (2x + 3)

Any number

Unknown number

Objects

No idea

Equation Solution - Initial Strategies (2x + 5 = 17)

Productive: Inverse mental

Systematic trial and error

Material - counted up

Unproductive: Mental calculation

Material/mental calculation

Directed Mode of Thinking

Variable Representation (2x + 3)

Productive: Mixed objects

Different colored counters

Same object

Unproductive: Concrete value

Literal

Not represented

Equation Representation (2X- + 5 = 17)

Productive: Mixed objects

Unproductive: Incomplete

Literal

Not represented

Equation Solution - Material Strategies (2x + 5 = 17)

Generate

Illustrate

Not represented

Students’ Responses to -

“Do you Think Using Materials has been Helpful?’

II

8

1

1

21

10

3

I

6

I

21

4

3

1

5

4

4

21

2

h

4

9

21

1

6

I4

21

Any number

Unknown number

Inverse mental

Systematic trial and error

Mental calculation

Paper

Mixed objects

Different colored counters

Same object

cups

Literal

Not represented

Mixed objects

cups

Incomplete

Literal

Not represented

Generate

Illustrate

Not represented

Yes

Some of the time

No

7

I4

21

14

1

5

I

21

5

3

3

4

4

2

21

3

I

8

3

6

21

4

IO

7

21

3

10

8

21

students’ directed mode of thinking (their understanding of how to represent variable and

equation with concrete material and solve a linear equation with the same material, when directed to do so). Students’ responses have been described and categorized under two sec-

tions, productive (likely to lead to a successful solution of an equation) and unproductive

(unlikely to lead to a successful solution of an equation). The findings of the interviews in

terms of these categories are discussed in relation to the lessons given and the literature and


are summarized in Table 3. This is presented in two sections, preferred and directed. An

analysis of these responses revealed changes in students’ perceptions of algebraic concepts

from pre- to post-interview. The discussion delineates these changes and explores causes

and reasons.

Changes in Preferred Thinking Mode

According to Usiskin (1988) there are two early conceptions of variable for algebra:

variable as a generalisation of arithmetic (any number) and variable as an unknown in an

equation (unknown number). The first of these conceptions is evident in expressions such as 3x + 2 and the second in equations such as 3x + 2 = 11. It is evident from the responses

for variable meaning during post-interview that there was a change for many students from

“any number” to “unknown number;” a change which corresponds with the correct mean-

ing of a variable in an equation. This would be expected following lessons focusing on

equations.

The preferred strategies for solving the linear equation were mostly productive and evi-

denced very little use of materials for pre and post-interviews. In fact, during post-inter-

view, students gave only limited support to the usefulness of materials. This lack of support

was obvious in the fact that the strategies employed most successfully were inverse mental. Cups and counters were used to teach the balance model, where the same operations are

completed on both sides of the equation. The mental symbolic “short-cut” method was also

based on the balance model. Therefore, it would be expected that if the cups, counters, and the “short-cut” method, were the basis of student learning then students’ mental models

would be isomorphic with the material use and thus be based on a balance strategy. How-

ever, this was not the case. The students predominantly used an inverse approach.

Changes in Directed Thinking Mode

With respect to students’ directed mode of thinking, there was an increase in productivity for variable representation (2x + 3) by post-interview. This increase, however, did not

reflect the representation that the students were taught (cups and counters) as appropriate for equation solution but rather the representation for expressions (different coloured counters or mixed objects). Equation representation on the other hand was mostly unpro-

ductive (incomplete or literal); in fact many students did not attempt representation at all.

When students were specifically requested to use materials to solve the equation, very few could generate an answer. Again concrete materials were seldom used and of these only

one student used cups.

When given an equation to represent, the students’ initial response was to solve mentally, using the inverse model. It appeared that the students were actually solving the equa-

tion as they read it out. The mental aspect of this situation completely overrode any desire to use materials. Any insistence of material use resulted in an illustration of this mental

process. In fact for one student, it seemed that use of materials to represent the equation had become an obstacle. This student moved from a productive representation pre-interview to

an unproductive representation post-interview. It seems likely that this is an unwanted side

effect of the use of concrete materials.


Concrete Representation and Algebra

If concrete materials are used meaningfully to solve linear equations then it seems rea-

sonable to expect that the use of these materials would be generative, that is begin with the

representation and then work forward to the solution, and the representation of the variable

and equation would be productive, that is lead to a solution. As well, it would be expected

that the solution method should relate to the strategy most in harmony with material use,

that is “maintaining the balance.” Responses to the tasks, as shown in Table 3, show little

evidence of meaningful concrete understanding, particularly with cups and counters. Dur-

ing post-interview, only four students could represent a variable in an expression with cups

and counters, only one student could represent an equation with cups and counters, and

only four students could use materials to generate a solution.

These results appear to be contrary to the theories of Payne and Rathmell (1975) Hie-

bert (1988), and Bloomer and Carlson (1993) in that students’ responses indicated little

connection between concrete and symbolic representations. The results appear more to

reflect the findings of Hart (1989) in that a gap exists between concrete and symbolic rep-

resentation. The question is, why does such a gap exist? In response to this question several

factors might be considered in light of our findings.

If concrete materials are used to represent simple concepts and procedures that can as

easily be developed without the use of materials, students may avoid using them. The

majority of students in this study could visualise and solve simple linear equations without

the need to resort to concrete representation. For these simple equations the inverse mental

solution was found to be simpler and hence preferred by these students.

Unless the explicit function of concrete materials is made clear to students, use of them

to develop new concepts and procedures will be futile. According to Sowell (1989), stu-

dents’ mathematical attitudes and achievement are enhanced through long-term use of con-

crete materials when instructed by teachers who are knowledgable about their use. One

might argue that cups and counters are familiar objects, however their use in the represen-

tation of unknowns and numbers, particularly the use of different coloured cups and

counters to distinguish positive and negative values, will not be familiar. Thus, use of these

materials to represent algebraic expressions meant acquiring new rules while at the same

time learning algebraic concepts and procedures; a process that may increase cognitive

load.

Furthermore, use of cups and counters to solve an equation using the balance method-

performing the same operation on both sides-imposes the load of conceptualising the procedures required. This load was evidenced by the fact that only four students used the bal-

ance method successfully during post-interview; most preferred the inverse mental

approach to solve the equation. Another reason for preferring the inverse mental approach

may have been that these students viewed the equality symbol as denoting where to put an

answer, or as a stimulus to do something. Had they developed the correct notion of equality

as equivalence they may have been more productive in using materials to practise the balance method.

The students’ responses seem to support Thompson’s (1988) contention that the abstractness of algebra makes manipulative techniques artificially complex. The use of

symbolic techniques and the inverse mental strategy appeared to be much easier for the stu-


dents. Contrary to this, use of cups and counters seemed to make tasks complex, which in turn affected the processing necessary to carry out algebraic representations and solutions.

Cognitive Load

The reason for the gap between concrete and symbolic representation can be explained in terms of cognitive load. A linear equation such as 2x + 5 = 17 appears to be a simple task when viewed in terms of an inverse mental solution, the numbers are low and there are only two operations to invert, but it becomes complex when relying upon concrete materials. It seems that the observation of Boulton-Lewis (1993a, 1993b) might be appropriate here, in that if concrete analogs are not well understood they could be a hindrance to understanding, therefore adding to the processing load. The difficulty of integrating a material solution into a complex algebraic procedure is not surprising if the solution to 2x - 3 = 5 is analysed. The student has to put two cups (recognising a “2x” as two x’s) and three yellow counters (recognising that “- 3” is “+ -3”) on the left and 5 green counters on the right. The student then has to add three green counters to both sides to remove the -3 and recognise that dividing by two is partitioning the eight green counters into two sets of four counters. The difficulty for students is that they have to integrate knowledge of mathematical laws and relations, and the “laws” of representations of variables and numbers by cups and counters and hold all this in working memory while they solve the equation. This imposes a signif- icant processing load much greater than mentally visualising 2x - 3 = 5 as multiplying by 2 and subtracting 3 and then reversing these operations to find the solution (i.e., adding 3 and dividing by 2). In addition to the above reasons, the students’ negative perceptions of the usefulness of concrete representations should not be overlooked. These perceptions appear to support the contention of Boulton-Lewis and Halford (1992) that using materials to solve equations increases the processing load, particularly when an answer has already

been arrived at mentally. Sweller and Low (1992) have argued that requiring novice students to solve algebraic

problems using a means-end problem solving strategy increases cognitive load and that well-constructed worked examples are superior. Although the instructional method com- monly used across the five lessons was to go through examples, the material solution to equations was presented in a problem solving environment. The students were solving the equation in a “means-end” manner, working backwards from the end to determine what moves had to be made. The mental inverse “rules” allowed the equation to be solved by

working forwards, with much less cognitive load.

The Construction of Algebra Knowledge

Kieran (1992) argued that algebra knowledge develop sfrom procedural to structural. It seems obvious that solving a linear equation (like 2x + 3 = 9) by the inverse mental strategy is a procedural activity, as operations are carried out on numbers to yield numbers. The expression 2x + 3 is not seen as an expression but as a series of operations. However, using cups and counters on the same equation appears to have structural tendencies. This is because the cups and counters appear to require comprehension of the expression 2x + 3 and a solution strategy in which operations are performed on expressions (e.g., 2x + 3 - 3 and 2x/2). Similarly using a balance mental strategy has structural tendencies because the


expression 2x + 3 has operations performed on it. That is it is perceived, albeit mentally, as

a total expression. In terms of structural knowledge, the cups and counters solution and the balance mental strategy are significantly more difficult than the inverse mental solution and tend to lead to cognitive overload.

Kieran (1989) also argued that algebra knowledge has to be constructed from prior knowledge, particularly arithmetic. Table 3 indicates that over 50% of students were using the inverse mental and trial and error strategies in pre-interview. Previously successful strategies tend to be defended by students and it is difficult for new approaches to replace this knowledge. These new approaches are quickly discarded, particularly if they appear more difficult, hence it is not surprising that so few would use materials after instruction.

CONCLUSIONS

The results in this study are unequivocal at one level-the students did not use the procedures taught to them for the concrete representations. Only one of the 2 1 students used cups and counters correctly to represent the equation. During post-interview, no students used materials voluntarily. When asked directly to use them, only four of the 21 students could use the materials to generate an answer.

Why was this? When given the choice students preferred the mental approach (predominantly inverse) which met their needs simply and effectively. As a result most uses of concrete representations were illustrative because mental strategies overrode any knowledge of the generative use of materials.

We have already discussed reasons for this, however the most persuasive reason appears to lie within the processing load associated with concrete representations (Boulton-Lewis, 1991) and transferring understandings from arithmetic to algebra (Halford & Boulton- Lewis, 1992). Even during instruction, very few students used materials. Although we believe that concrete representations might have a place during the instruction of some algebraic concepts, if used explicitly and unambiguously by the teacher, to establish the notion of variable, it is evident from this research that students did not want to use concrete representations themselves, preferring a mental approach. In fact, use of concrete representations by students was counterproductive.

The matter of cognitive load needs further investigation. To allow its effect to be discussed unambiguously, there should be research into the prerequisite knowledge students have. Mapping the effect of instruction on the prior knowledge required more insight into students’ beliefs as well as mental and pictorial representations. There is a need to follow students and the instruction given for a greater length of time.

REFERENCES

Atweh, William, Boulton-Lewis, Gillian M., & Cooper, Tom J. (1994). Representations, strategies,

cognitive load, and effective learning and teaching of linear equation solution procedures. In

the Proceedings of the Seventeenth Annual Conference of the Mathematics Education. Lis-

more, NSW: Research Group in Australasia.

Bloomer, Anne M., & Carlson, Phyllis A. (I 993). Activity math: Using manipu/ative.s in the classroom. Menlo Park, CA: Addison-Wesley.


Booth, Lesley R. (1987). Equations revisited. In J. C. Bergeron, Nicolas Herscovics, & Carolyn Kie-

ran (Eds.), Proceedings of Psychology of Mathematics Education XI, Vol. 1 (pp. 282-288).

Montreal.

Booth, Lesley R. (1988). Children’s difficulties in beginning algebra. In Arthur F. Coxford & Albert

P. Shulte (Eds.), The ideas of algebra, K-12 (pp. 20-32). Reston, VA: National Council of Teachers of Mathematics,

Boulton-Lewis, Gillian M. (1991). The values and limitations of using concrete representations to

teach mathematics to young children. In the Proceedings ofthe ICMI-China Regional Confer-

ence on Mathematical Education, Beijing, August 5-8.

Boulton-Lewis, Gillian M. (1993a). An assessment of the processing load of some strategies and rep-

resentations for subtraction used by teachers and young children. Journal of Mathematical

Behavior, 12(4), 387-409.

Boulton-Lewis, Gillian M. (1993b). Young children’s representations and strategies for subtraction.

British Journal of Educational Psychology, 64,45 I-456.

Boulton-Lewis, Gillian M., Cooper, Tom, J., & Atweh, William. (I 995). Concrete representations

and strategies for solving linear equations. In the Proceedings ofthe Eighteenth Annual Con-

ference of the Mathematics Education Research Group in Australasia, Darwin, NT.

Boulton-Lewis, Gillian M., & Halford, Graeme S. (1992). The processing loads of young children’s

and teachers’ representations of place value and implications for teaching. Mathematics Edu-

cation Research Journal, 4(I), l-23.

Boulton-Lewis, Gillian M., & Tait, Kathleen. (1994). Young children’s representations and strategies

for addition. British Journal of Educational Psychology, 64, 23 l-242.

Boulton-Lewis, Gillian M., Wilss, Lynn A., & Mutch, Susan J. (1996). An analysis of young chil-

dren’s strategies and use of devices for length measurement. The Journal of Mathematical

Behavior, 15(3), 329-347.

Brown, Catherine A., Carpenter, Thomas P., Kouba, Vicky. L., Lindquist, Mary M., Silver, Edward

A., & Swafford, Jane 0. (1988). Secondary school results for the fourth NAEP mathematics

assessment: Algebra, geometry, mathematics methods and attitudes. Mathematics Teacher,

81,337-347,396.

Cooper, Graham, & Sweller, John. (I 987). The effects of schema acquisition and rule automation on

mathematical problem-solving transfer. Journal of Educational Psychology, 79, 347-362.

Davis, Robert B. (1988). The interplay of algebra, geometry and logic. Journal of Mathematical

Behavior, 7, 9-28.

Halford, Graeme S., & Boulton-Lewis, Gillian M. (1992). Value and limitations of analogs in teach-

ing mathematics. In Andreas Demetriou, Anastasia Efklides, & Michael Shayer (Eds.), Neo-

Piagetian theories of cognitive development: Implications and applications for education (pp.

183-209). London: Routledge.

Hart, Kath. (1989). There is little connection. In Paul Ernest (Ed.), Mathematics teaching: The state

of the art (pp. 1388142). New York: Falmer Press.

Hiebert, James (1988). A theory of developing competence with written mathematical symbols. Edu-

cational Studies in Mathematics, 19, 333-355.

Herscovics, Nicolas, & Kieran, Carolyn. (1980). Constructing meaning for the concept of equation.

The Mathematics Teacher, 73(8), 572-580.

Herscpvics, Nicolas, & Linchevsci, Liora. (I 994). A cognitive gap between arithmetic and algebra.

Educational Studies in Mathematics, 27(l), 59-78.

Kieran, Carolyn. (1989). The early learning of algebra: A structural perspective. In Sigrid Wagner &

Carolyn Kieran (Eds.), Research issues in the learning and teaching of algebra (pp.33-56).

Reston. VA: Erlbaum & National Council of Teachers of Mathematics.


Kierdn, Carolyn. (1992). The learning and teaching of school algebra. In Douglas A. Grouws (Ed.), The handbook of research on mathematics teaching and learning (pp. 390-419). New York: MacMillan & National Council of Teachers of Mathematics.

Leitzel, Joan R. (1989). Critical considerations for the future of algebraic instruction. In Sigrid Wag- ner & Carolyn Kieran (Eds.), Research issues in the learning and teaching of algebra (pp. 25- 32). Reston, VA: Erlbaum & National Council of Teachers of Mathematics.

MacGregor, Mollie, & Stacey, Kaye. (I 995). What is x? The Australian Mathematics Teacher, 49(4),

28-30. Payne, Joseph N., & Rathmell, Edward, C. (1975). Number and numeration. In Joseph N. Payne

(Ed.), Mathematics learning in early childhood (37th Yearbook, pp. 125-l 60). Reston, VA: National Council of Teachers of Mathematics.

Queensland Department of Education. (1991). Years I to 10 mathematics sourcebook: Activities for

teaching mathematics in year 3. Brisbane: Queensland Government Printer. Quinlan, Cyril., Low, B., Sawyer, T., & White, P. (1993). A concrete approach to algebra. Sydney,

Australia: Mathematical Association of New South Wales. Richards, Thomas J., & Richards, Lyn. (I 994). Using computers in qualitative analysis. In Norman

K. Denzin & Yvonna S. Lincoln, (Eds.), Handbook of qualitative research (pp. 445-462). Berkeley: Sage.

Rosnick, Peter, & Clements, John. (I 980). Learning without understanding: the effect of tutoring strategies on algebra misconceptions. Journal of Mathematical Behavior, 3(l), 3-27.

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, l-36.

Sowell, Evelyn J. (I 989). Effects of manipulative materials in mathematics instruction. Journal for

Research in Mathematics Education, 20(5), 498-505. Sweller, John, & Low, Renae. (1992). Some cognitive factors relevant to mathematics instruction.

Mathematics Education Research Journal, 4, 83-94. Thompson, Frances M. (I 988). Algebraic instruction for the younger child. In Arthur Coxford &

Albert P. Schulte (Eds.), Ideas of algebra, K-12 (I 988 Yearbook, pp. 69-77). Reston, VA: National Council of Teachers of Mathematics.

Thorndike, Edward L., Cobb, Margaret V., Orleans, Jacob S., Symonds, Percival M., Wald, Elva, & Woodyard, Ella. (1923). The psychology of algebra. New York: MacMillan.

Usiskin, Zalman. (I 988). Conceptions of school algebra and uses of variables. In Arthur Coxford & Albert P. Schulte (Eds.), Ideas of algebra, K-12 (1988 Yearbook, pp. 8-19). Reston, VA: National Council of Teachers of Mathematics.

Processing Load and the Use of Concrete Representations and Strategies for Solving Linear Equations

Documents

Transcript of Processing Load and the Use of Concrete Representations and Strategies for Solving Linear Equations