Visual and nonvisual processes in grade 6 students' mathematical problem solving.

M. A. (Ken) Clements 

Abstract. This paper reports an investigation into the problem-solving methods employed by three Grade 6 students who, over the course of a school year, worked on a variety of mathematics problems. One of the students tended to solve problems in a visual manner; the second preferred a more verbal/nonvisual approach; and the third tended to use both visual and nonvisual strategies. Over the school year, the three students moved toward more nonvisual, verbal/analytic forms of reasoning as task familiarity increased.

In the late 1970s and early 1980s, psychologists and education researchers interested in mathematics problem-solving strategies sought to classify problem solvers as belonging to one of three categories: 1) visualizers, who had a preference for holistic approaches involving extensive use of visual methods; 2) nonvisualizers or verbalizers, who had a preference for more verbal approaches; and 3) those who tended to use both visual and nonvisual methods (Burden & Coulson, 1981; Krutetskii, 1976; Richardson, 1977; Lean & Clements, 1981; Presmeg, 1985, 1986; Quinn, 1984; Sheckels & Eliot, 1983; Suwarsono, 1982).

Around the same time, Pylyshyn (1973, 1979, 1981) questioned the existence of visual images as "pictures in the mind," claiming that what people called "visual" images were, in fact, complex sequences of linguistic representations and relationships. Other writers (some of them attached to various post-structuralist schools of thought) have been decidedly unhappy with the very idea of problem representation. Nevertheless, the notion that many people do have a preference for visual (or nonvisual) approaches to solving problems is worthy of further investigation.

Many great scholars have maintained (see Clements, 1981-1982) that they did (or did not) prefer to use visual modes of thought, at least in the initial stages of thinking about possible ways of tackling a problem. It is also the case that the framework of mathematical abilities provided by Krutetskii (1976), as well as the findings of Lean and Clements (1981), Suwarsono (1982), and Presmeg (1985, 1986), provided support for the view that learners do develop preferences for tackling mathematics problems using either visual or nonvisual (more verbal) approaches. This paper reports an investigation into the ways three students from a Grade 6 class in New South Wales (Australia) tackled elementary mathematics problems. It reveals that within the space of a year, their preferences for using visual or nonvisual methods on the same or similar problems tended to move in the direction of being more nonvisual.

Some studies (e.g., Clements & Del Campo, 1989; Suwarsono, 1982) show that students taught in a visual way tend to learn to use visual methods. Clements and Del Campo (1989) demonstrated that the type of learning experience in which a student is engaged is likely to influence not only the extent and quality of visual imagery evoked, but also the role of the imagery in the student's cognitive schema. Depending on the problems, and on a problem solver's existing schema with respect to the problems, visual reasoning may be more efficient (or, indeed, Less efficient) than reasoning that employs more verbal/nonvisual modes of thinking (Lean & Clements, 1981).

Visual and spatial thinking are mental activities that are not easily studied. Gardner (1983), for instance, pointed out that children find it difficult to verbalize their visual or spatial knowledge. A child's spatial knowledge may not develop, however, at the same rate as his or her ability to represent the knowledge via a symbolic code. In the present investigation, a variety of assessment procedures were used in an attempt to overcome this problem. In particular, students investigated in this study were encouraged to introspect (i.e., "think aloud") when solving problems.

With the notable exceptions of Krutetskii (1976), who worked with students such as Sonya L and Volodya L over a period of years, and Presmeg (1985), who followed the mathematical thinking of Grade 12 students over an 8-month period, earlier scholars who contrasted visual and nonvisual (verbal) problem-solving approaches did not trace students' thinking processes over time (e.g., Pylyshyn, 1973; Sheckels & Eliot, 1983; Suwarsono, 1982). Usually, students were required to complete a set of activities or mathematics problems, and to report either introspectively or retrospectively on the method or approach they had used to complete the task. Neither a student's specific and general knowledge of the task, nor the problem's complexity or novelty, was taken into consideration.

By contrast, the present study investigated each student's understanding and knowledge of a particular problem, and analyzed the methods employed to complete a task. Because the same students were monitored and assessed as their problem-solving abilities and domainspecific knowledge developed over a period of time, a more holistic analysis of the use and development of visual and nonvisual/verbal thinking processes was possible.

Method

This one-year study investigated the problem-solving methods employed by three Grade 6 students as they worked individually on a variety of mathematics problems. During the study, the three students regularly participated in special problem-solving sessions that were especially convened for them (and at which they and the first author were usually the only persons present). In these sessions, the students attempted routine and novel mathematics word problems, spatial and measurement problems, problems involving computer simulations, and picture and puzzle problems. Sometimes the students were expected to complete activities individually, but on other occasions they worked cooperatively. In this paper, four of the problems the children solved on an individual basis are reported.

After each session, the children were asked to reflect, both verbally and in writing, on the difficulty of activities and on the methods they had used when attempting to solve problems. The sessions were audiotaped, and audio and written responses were filed and subsequently analyzed in terms of 1) problem type, 2) knowledge and understandings required to solve the problems, 3) ease with which tasks were completed, and 4) aspects of problemsolving strategies employed.

Selection of the Participants

The criterion employed for the selection of the three case-study students was in line with Krutetskii's (1976) contention that many people solve mathematical problems using one of three modes--verbal-analytical, visual, and harmonic. (1) Although the three case-study students had performed at about the same level in the Australian Primary Mathematics Competition (see Table 1), they appeared to have quite different verbal/visual preferences for solving similar mathematical word problems.

Entries in Table 1 refer to mathematics problem-solving performance measurements and "visuality" scores gained by the students when, early in the school year, they were administered the Mathematical Processing Instrument (MPI) developed by Suwarsono (1982). The "visuality scores" were derived from the students' retrospective reports on the methods they used when tackling the MPI word problems (see Appendix).

Suwarsono employed item-response-theory (IRT) techniques to develop the set of mathematics problems for his MPI. He claimed that the MPI could locate both problem solvers and problems on a single unidimensional verbalizer-visualizer scale. The MPI could measure not only a problem solver's preference for using visual (or nonvisual/verbal) approaches to problem solving, but also the visuality of a problem (i.e., the extent to which that problem was likely to be processed by visual methods, or nonvisual methods, or some combination of these).

It can be seen from Table 1 that one student (Jayne) preferred a verbal, nonvisual approach when solving mathematical problems; another (Matthew) tended to use both visual and nonvisual strategies; and the third (Robyn) preferred to solve mathematical problems in a visual manner.

Over the course of the school year, the first author systematically made entries in a journal on matters relating to the problem-solving approaches and performances of the three case study students.

Description of Case Study Participants

Jayne. Jayne was one of the most capable students the first author had ever taught. The following journal entry outlined the manner in which Jayne approached tasks.

Jayne appears to be systematic in her approach to solving open-ended problems. At times she seems to work around tasks "in her head," although she usually sequences ideas in a rather individual manner on paper. While it is often difficult to get children in the class to [monitor their own thinking and] "show working" when solving difficult problems, Jayne tends to write down most of her thoughts. This is usually achieved in a somewhat untidy fashion, with scribblings emerging from all over the page. While it is often difficult for me to understand her sequencing, her systematic approach is evident in her description of what she has done. Her capacity to describe her thought processes (what's going on in her head) is outstanding. In fact, her written and verbal vocabulary are better than for any other child of her age I have taught.

Jayne was a gifted reader and writer, with a vivid imagination and an excellent grasp of English. She loved reading and would seize any opportunity to read. Given the choice of working on mathematics problem-solving tasks or reading, she invariably chose the latter.

Mathematics is about numbers and how numbers may or may not relate to each other. When I write I can go off into a different world, be a different person and create hypothetical worlds. You can't hide with maths, because sooner or latter you have to be right or wrong. Mostly I like maths because you can work around problems to get something right, but sometimes you get blocked and no matter how hard you try you can't find the answer.

Jayne felt that she was not very good at mathematics, although it is likely that this belief derived from her propensity to compare this aspect of her schoolwork with her capacity to read and write so well. During the school year she gained distinctions in two writing competitions, but also received a high credit in a state-wide mathematics competition during the year, being placed in the top 15% of participants in the state. Nevertheless, her lack of confidence in herself as a mathematics student often seemed to be a key factor in her inability to solve novel problems, and was particularly evident when she was asked to solve unfamiliar problems.

Jayne regarded mathematics as a subject that demanded correct answers. She told the first author in an interview:

The methods that Jayne employed to solve mathematics problems reflected her beliefs about the nature of mathematics. Although task complexity or novelty would often influence the manner in which the three students approached mathematical tasks, personal ideas or beliefs about the nature of mathematics, and what was expected when one did mathematics, also contributed to solution design.

Matthew. Matthew's competitive spirit characterized many of his endeavors, especially his participation in state swimming and athletics meets. He was more confident and comfortable about his own mathematical capacities than the other two case-study students. Matthew's desire to solve difficult problems was particularly noticeable.

The following journal extract demonstrates the first author's perception of Matthew's enthusiasm for solving challenging problems.

When preparing for the week's lessons I was fascinated by a mathematical problem I had recently come across in my professional reading. The problem required the nine natural numbers 1, 2, ..., 8, 9 to be placed on a 3 x 3 grid, with each number being used only once. Each row of three numbers (including the two diagonals) had to total the same number. This relatively complex magic square was given to the entire class. Anxious to find out how many children would be able to complete the task, I presented the problem to the children on Monday afternoon. After the usual cries of "this will be easy" the children realized that there was more to the problem than they had first anticipated. After 20 minutes of frustration the bell signaled an end to the day. Several children were still working on the problem as they packed their bags. For some, the motivation to complete the task remained quite high. [Later] I went back into the classroom to tidy up and pen a journal entry. Matthew stormed into the room and exclaimed: "I think I have it." He had solved the problem 15 minutes after the bell had rung. (Journal Extract)

Matthew was regarded as a "gifted" student. (2) His capacity to solve novel problems was more advanced than might be expected for a Grade 6 child, and he employed a diverse range of methods when tackling mathematical tasks.

For questions that challenged him, Matthew was task persistent. With such questions, his ability to conjecture, synthesize, and evaluate were often impressive, and he was prepared to use his own time in his efforts to develop satisfactory solutions.

Robyn. Robyn, the third case-study student, tended to contribute less than either Jayne or Matthew in class discussions and open-ended learning situations. On some occasions, Robyn felt that she should not have been selected for the "special" activities associated with the study, believing that she was academically inferior to the other two students. She tended to be anxious when asked to perform difficult mathematical tasks, and felt inhibited by the presence of the other two pupils.

The accuracy of Robyn's self-assessment should not be taken for granted. The first author (her teacher) felt that her doubts about herself had more to do with the other two students' strong personalities than with any differences between their mathematical capacities and Robyn's. In fact, Robyn's performance on the Australian Primary Mathematics Competition (see Table 1) provided support for the first author's view that she was a mathematically capable student.

Throughout the study, it was evident that Robyn was likely to approach mathematical tasks in qualitatively different ways from the other two students. This is suggested in the following transcript, taken from an audiotaped interview, which also draws attention to Robyn's lack of confidence when working with the other two students. In this instance, the three students were discussing a velocity and time problem from a software program. (The interviewer was the first author.)

Int: It seemed like you really enjoyed working on the computer today?

Robyn: It was great, although parts of it were frustrating.

Int: What do you mean?

Robyn: Well, Matthew is able to work things out very quickly and is able to go through screens before I get a chance to work things out.

Int: What does Jayne do?

Robyn: Sometimes she tells Matthew he is wrong, or an idiot, but most times they work things out together really well.

Int: How do they involve you in the task?

Robyn: We usually take it in turns and try to solve it [the tasks] together.

Int: So, you are an important part of the group?

Robyn: Not really, because they are able to think things through quicker than me, although they do ask me my opinion about things.

Int: Do you ever find that you can do things the other two can't do?

Robyn: Only sometimes, usually on some easy problem when Jayne wants to go one way and Matthew another. Sometimes I surprise myself, then they get the idea and work out the difficult parts.

Robyn felt that she had little impact on the decisions made in the joint problem solving activities. In fact, however, the first author noticed that often it was Robyn who provided the stimulus needed to approach problems in productively different ways.

Methods of Analysis

Data reported and analyzed in this paper were part of a larger data set in which the case-study students, and other 6th-grade students, attempted a wide range of problem tasks over a school year (Lowrie, 1996b). This paper focuses on the casestudy students' responses to four tasks--the "swimming pool," "garden," "turf," and "proportion" problems--which will be elaborated upon in the next section. Here, it suffices to note that although no diagrams were given or requested in the presentation of these four tasks, the first three problems were, on Suwarsono's (1982) scale, high visuality problems. The proportion problem would be considered balanced in the sense that it was likely to be solved using either visual or nonvisual approaches. Two of the tasks (the swimming pool and garden problems) employed in the present study were administered to the case-study students early in the school year, while the third and fourth problems were administered at the end of the school year.

The turf problem, like the garden problem, contained the idea of an "annulus" surrounding a central region. During the year, the class that contained the three case-study students was taught to solve similar annulus-type problems. The class teacher (i.e., the first author) tended to emphasize the need for a balanced approach to problem solving; he stressed the need for students to draw a clear diagram, and then to reason from the diagram. Given this teaching emphasis, the issue of whether any or all of the case-study students would employ the same problem-solving approach on the turf problem as they had employed on the garden problem, conducted earlier in the year, was of theoretical interest.

The proportion problem was administered to the children because students of this age were likely to use a range of visual and nonvisual approaches when attempting to solve ratio and proportion problems. The children had not specifically been taught to solve these problems at school (in fact, such problems were not usually encountered until high school). Since this problem may have been novel to some of the students, it was included in the investigation as a means of measuring whether task difficulty and task familiarity contribute to a student's selection of particular methods.

The Research Questions

This investigation sought answers to the following questions:

1. At the end of the school year, did the case-study students still use the same methods or approaches when attempting to solve similar mathematical problems as they had at the start of the year?

2. Was there a relationship between a student's preference for using visual (or nonvisual/verbal) approaches to problem solving and that student's tendency to continue to use (or not continue to use) the same method (or a similar method) for solving a mathematical problem of a certain type?

3. To what extent does task difficulty and task familiarity contribute to a student's selection of either a visual or nonvisual method for solving a problem?

Changes in Students' Problem-Solving Processing Over a School Year

The Manner in Which Students Approached Problem-Solving Tasks

Whenever any or all of the case-study students were required to complete a task, the researchers attempted to analyze the processes the students employed and the understandings they demonstrated in their attempts to construct "mathematical meaning." On most occasions, the three students were given the same task or problem to complete, and this usually allowed comparison of the students' processing and written responses.

When working individually on the same problem, the children often employed different strategies. If they were asked to work together, their problem-solving methods and approaches often fused together, and became complementary. Jayne and Matthew displayed a healthy rivalry, but Robyn was the most willing to persevere when it appeared that a problem could not be solved.

Early in the school year, each of the three case-study participants was observed individually solving a set of mathematics word problems. This enabled us to not only identify and document the knowledge and procedures that each child tended to use, but also make conjectures concerning any similarities and differences in the way they constructed mathematical ideas.

Responses to the Swimming Pool Problem

The commentaries in Table 2 were based on notes taken by the first author when, early in the school year, the three students were attempting to solve a swimming pool problem. The commentaries draw attention to their different problem-solving strategies, which were apparent even when they were attempting to solve the same problem. An analysis of the different strategies used by the three children on the swimming pool problem indicated that the students' declarative and procedural knowledge in relation to the problem influenced the manner in which they solved the problem (see Mayer, 1992).

The most noticeable difference among the approaches used by the children was Robyn's desire to draw a picture of the swimming pool. Although Matthew did not draw a picture, he more than likely related the problem to something he was good at, namely swimming (indicated by the positive expression that appeared on his face as he was reading the problem). For Jayne, who also did not draw a diagram, the numerical dimensions of the pool (which she gained from the problem statement), and information relating to the calculation of volume, and the relationships between volume and capacity (which she retrieved from long-term memory), were what was relevant. It was not necessary to take into account the fact that a swimming pool was mentioned in the problem statement.

Robyn did not solve the problem successfully. Like Matthew and Jayne, she was aware that the first step in the procedure was to calculate the volume of the pool, but her assumption that 1,500 [m.sup.3] corresponded to 1,500 L was incorrect. Except for that, her method was appropriate. As would be expected of someone with a preference for visual strategies, she felt the need to construct a diagrammatic representation of the task situation, and her subsequent thinking was strongly influenced by that diagram.

Jayne's and Matthew's strategies for converting a volume measure to a capacity measure differed. The first symbols that Jayne placed on her page (1 [cm.sup.3] = 1 mL) seemed to lock her into a complex set of calculations, and she had to perform several conversions before she could state the correct answer. Her "known" conversion relation, which she retrieved from long-term memory, was 1 [cm.sup.3] = 1 mL. Using that relationship, she needed to convert cubic meters to liters and then to relate her calculations to the pool's volume of 1,500 [m.sup.3]. Matthew, on the other hand, began with 1 [m.sup.3] = 1,000 L, and from then on his calculations were relatively easy.

Although it became obvious that the three children preferred to employ different strategies to solve the same mathematics problems, the issue of whether these methods or strategies were stable or changed over time is of interest to the classroom teacher, and was something that the first author continued to monitor throughout the year.

Responses to the Garden Problem

During the first term of the school year, the students were presented with the following problem (hereafter termed the garden problem):

A lady wanted to put a fence around a garden so that her dog could not eat her tomatoes. Her garden was 4 m wide and 5 m long. She had a cement path 1 m wide around the entire garden. A fence was then to be positioned around the path. How many meters of fence would she need?

This was a novel problem for the three students to solve, and each considered it to be relatively difficult. Interestingly, each student attempted to solve the problem by drawing a picture of a garden (see Figures 1, 2, and 3). It was apparent that the children had not encountered this type of problem before and relied on a pictorial representation of a garden with a path around it to help them construct an understanding of how the problem might be approached.

An Analysis of Jayne's Response to the Garden Problem. Initially, Jayne attempted to solve this problem "in her head," but then, rather reluctantly, she opted to draw a diagram. When asked why she had chosen to draw a picture, Jayne stated that she "realized that the problem had a trick to it," and that "sometimes I mix up perimeter and area problems, and that's why I checked with a picture."

After drawing the picture, Jayne used symbols to summarize the steps she would take to solve the problem on paper (4 m + 2 m = 6 m; 5 m + 2 m = 7 m; 7 m x 2 = 14 m; 6 m x 2 = 12 m; 12 + 14 = 26).

The following interview transcript, audiotaped soon after Jayne completed this task, illustrates how uncertainty led her to modify her strategy.

Int: When you first began to work the problem out, did you have an image of a garden in your mind?

Jayne: No, not really.

Int: What were you thinking then?

Jayne: Well, I realized it was a perimeter problem, so I just added one to each side and worked out the perimeter.

Int: Then you weren't sure if you were right or wrong?

Jayne: That's when I drew the diagram.

Int: After you drew the diagram of the garden, did you then think of any garden in particular?

Jayne: We have several gardens at home, and have large slabs of cement around them, so I suppose I thought about them.

This transcript is entirely consistent with information in Table 1, which suggested that Jayne had a strong preference for nonvisual thinking. Her first reaction to the problem was to classify it as being of a certain type. Then, when she was not sure if her initial thoughts were appropriate, she decided to draw a picture. Jayne's decision to draw a picture was not the result of her wanting to externalize a visual image. Rather, it was an expression of her desire for guidance on whether the task was a "perimeter" or an "area" problem. Jayne had misinterpreted perimeter and area concepts in the past, and she had learned to approach tasks such as this one with caution.

In her initial attempt to solve the problem, Jayne focused on the type of mathematical ideas presented in the task without constructing an image of a garden either on paper or in her mind. This changed, however, after she drew the diagram. It is possible that before she drew her diagram, Jayne constructed an image of her own garden to develop an understanding of what the question was asking her to do. It appears that Jayne was "folding back" (a term used by Pine & Kieren, 1991) from a less visual to a more visual representation of the problem.

An Analysis of Matthew's Response to the Garden Problem. Matthew's initial approach to solving the problem was to draw a diagram representing the garden and to mark on it the length and width of its borders (see Figure 2). He then proceeded to draw a representation of the one meterwide concrete path around the outside of the first diagram, and to change the labels for length from 5 m to 7 m and for width from 4 m to 6 m. He then wrote the following statement: 14 + 12 = 26.

A debriefing session was conducted about 30 minutes after Matthew had completed the problem. The following extract from this audiotaped session provides a self-report on his problem-solving strategy:

I read the problem and thought about my pop's garden. You know him, the one that plays golf at the Grange. Well, he has three gardens like that at his house. I drew a diagram of the garden, then put the path around the outside of it. I added the lengths to the widths and got 26 meters. It's right, isn't it?

When Matthew was asked if thinking about his pop's garden had helped him solve the problem, he stated: "No, not really, it just made the problem more interesting." The transcript is consistent with Table 1, which shows that Matthew did not have a strong preference for visual or more verbal strategies. He seemed to be happy to try a bit of both.

Although Matthew felt that visualizing the problem did not help him address some of its important elements, the images that he evoked might have helped him internalize the process of adding one meter to the length of each side of the garden bed.

An Analysis of Robyn's Response to the Garden Problem. The outstanding feature of Robyn's attempt to solve the garden problem was the fact that she did not overtly use any mathematical algorithms or symbols in her working. A diagram showing a garden with a surrounding pathway enabled Robyn to "count around" the perimeter of the picture and thereby "count out" a perimeter of 26 m. The diagram was a scaled representation of the actual problem (in this case, the scale she used was 1 cm to 1 m). (3) Interestingly, Robyn made pencil marks around the outside of the picture, indicating that she used the pencil as a pointer to calculate the perimeter by moving around the outside of the diagram. She did not write any numeral on the page to represent the length or width of the garden or paved area. All that she wrote was her answer, "26 m." It was obvious that Robyn had relied almost entirely on her diagram to solve this problem.

Robyn's diagram seemed to be a diagrammatic representation of a holistic visual image, and her subsequent reasoning relied heavily on the diagram. The data appear to be entirely consistent with the classification in Table 1 of Robyn as a person with a strong preference for visual thinking.

Responses to the Turf Problem

Near the end of the school year, the three case-study children were asked to solve the turf problem. This was a "rectangular annulus" problem that, although similar to the garden problem, required the students to calculate an area rather than a perimeter. The strategies used by the students to solve the problem differed considerably from the approaches they had used at the beginning of the year.

The turf problem stated:

A husband and wife wanted to turf their backyard (put grass squares down). Before purchasing the turf, they had an inground pool put in their backyard. The pool was 3 m wide and 5 m long. Sensibly they also paved an area 1 m wide around the pool. If turf costs $10 per square meter, how much would it have cost to turf the backyard (150 [m.sup.2] in total) once the pool and the paving were finished?

When reading the following analyses of Jayne's, Matthew's, and Robyn's responses to the turf problem, the reader should recall that at the start of the school year all three students had drawn pictures to help them process the information for the garden problem (although Jayne had merely used the diagram to check if her initial non-visual solution was correct).

An Analysis of Jayne's Response to the Turf Problem. Jayne drew a diagram (see Figure 4) to represent the pool and the paved area. Asked why she had drawn a diagram to solve this particular problem, Jayne responded, "To make sure that you add two meters to each side and not one."

Jayne's response corresponded to her response to the garden problem ("add 2 extra meters on each side for path, then add the totals"--see Figure 1). Once again, Jayne used the diagram as a check on her strategy of adding 2 extra meters that she had developed before she drew the diagram.

An Analysis of Matthew's Response to the Turf Problem. Early in the school year Matthew had begun his attempt at solving the garden problem by drawing a diagram. For the turf problem, however, he did not draw a diagram. He tackled the turf problem in a nonvisual way, reaching a correct solution after carrying out a sequence of calculations.

The following statement by Matthew throws some light on whether he had a visual image of the problem in his "mind's-eye":

After I read through the problem, I realized that it was one of those where you had to increase the length and width of both the top and bottom of the first object. That meant that the 3 m changed to 5 m and the 5 m went up to 7. I then worked out the area of the pool, subtracted it from the total area and multiplied it by $10 to get the answer.

This suggests that Matthew did not visualize a picture of a garden or paved area when attempting to solve the problem. Instead, he remembered a similar problem that employed the same type of "trick." He had now become familiar with problems like the garden problem and the turf problem, and he had no need for a diagram.

This interpretation is consistent with the classification (in Table 1) of Matthew as someone who did not have a strong preference for either visual or nonvisual thinking. Although he often drew diagrams or imagined the "problem scene" when attempting to solve mathematical problems, such strategies were not employed in this instance, mainly because he found them unnecessary in this circumstance.

An Analysis of Robyn's Response to the Turf Problem. When first confronted with the turf problem, Robyn did not draw a diagram, but rather began to calculate the area of the pool (5 m x 3 m = 15 [m.sup.2]). Then, in an effort to account for the 1 meter-wide paved strip surrounding the pool, she (incorrectly) added 1 [m.sup.2] to 15 [m.sup.2], to get 16 [m.sup.2]. She concluded her written solution with the following statements:

150 [m.sup.2] - 16 [m.sup.2] = 134 [m.sup.2], and 134 [m.sup.2] x $10 = $1,340.

However, Robyn was uncomfortable, both with her approach to the problem and with her solution, and when asked if she would like to attempt the problem again, she immediately drew a diagram. Figure 5 is a reduced version of her diagram. In her actual diagram, the outside large square measured 15 cm by 15 cm, and the inside rectangles measured 3 cm by 5 cm and 5 cm by 7 cm.

Interestingly, Robyn's attempt to draw the backyard to scale yielded an incorrect response (she drew a 15 cm x 15 cm square, which did not correspond to the given 150 [m.sup.2]). Nevertheless, once her diagram was in front of her, Robyn appeared to be more comfortable with the problem, and she proceeded to make the calculations shown in Figure 5.

Although Robyn's calculations for her second attempt at the turf problem (in Figure 5) demonstrated that she understood the processes involved in completing the remainder of the task, the act of drawing the situation to scale was an important part of her solution. Once she had appropriately represented the inside rectangles (representing the pool surrounded by the paved area), Robyn was able to calculate the length and width dimensions of the inner rectangles by the cognitive act of adding 1 meter to the dimensions of the pool at both its sides and both its ends. She had not been able to do that without the diagram.

Significantly, the two calculations shown in Figure 5 (150 [m.sup.2] - 35 [m.sup.2] - 115 [m.sup.2] and 115 x $10 $1150) were done after the diagram had been completed. The manner in which Robyn constructed the diagram is described in the following extract.

I needed to draw a backyard that was 150 [m.sup.2], so I drew a 150 square centimeter shape. I then realized my first mistake, because I didn't go all the way around the pool with a 1-meter path. I drew the pool to scale, making sure a 1-meter path could fit around the outside of the pool. I then drew a line to represent the path. I then knew that the length of the shape was 7 m and the width was now 5 m. I then worked out the problem.

Robyn's attempts to solve the garden and turf problems (Figures 3 and 5) suggests that her approach to solving annulus problems of this type had changed over the year. She did the garden problem in a manner consistent with what might be expected of a Grade 6 student with a preference for processing mathematics problems in a visual way (see Table 2). Yet, she initially tackled the turf problem in an essentially nonvisual way. An interesting question is why this change occurred over the school year.

When asked why she had attempted to solve the turf problem without drawing a picture or diagram, Robyn replied: "Because I have done problems like this before, and thought I knew what to do." This response is consistent with comments made by other students in the class containing the three case-study students (see Lowrie, 1996b). Robyn appeared to have acquired the belief, during the year, that it should not be necessary to draw a picture to represent problems like the turf problem. Such drawings were only needed if "you don't understand what the problem is actually about."

Earlier in the year, however, when attempting to answer the 20 largely unfamiliar questions on the Suwarsono Instrument, Robyn consistently employed visual approaches (see Table 1). This suggests that Robyn preferred to rely on visual processing to interpret novel problems, but that she switched to using more nonvisual approaches when she thought she had become more familiar with, or thought she was expected to have become more familiar with, the same kind of problem. This is consistent with findings of Dreyfus (1991), Lowrie (1996a), Lowrie and Kay (2001), and Lowrie and Hill (1996).

Responses to the Proportion Problem

At the end of the school year, the three case-study children were asked to solve the proportion problem. This problem contained ratio and proportion understandings, with a problem structure that was quite different from that of the other three problems reported in the study. This task had a balanced visuality measure in the sense that it was as likely to be solved visually as it was nonvisually.

The strategies these students used to solve the problem were consistent with their approaches from the beginning of the year, and consequently reflected the visuality measures obtained from the MPI. Thus, Jayne used nonvisual methods and employed Robyn's visual methods to solve the problem. Matthew, who tended to use a combination of visual and nonvisual methods to solve word problems at the beginning of the year, solved the problem in a nonvisual maner. Jayne and Matthew were able to generate a correct solution, whereas Robyn was not.

The proportion problem stated:

A farmer had five storage containers that were used to store wheat collected from the year's harvest. The capacity of the largest container was four times that of the smallest container. The second largest container stored three times as much wheat as the smallest container while the other two containers stored twice as much wheat as the smallest container. How much wheat would be stored in each container if the farmer wanted to store 72 kg of wheat?

This was a relatively difficult problem for the three children to solve. It was evident that they had not encountered problems of this type in the past. Thus, the novelty factor of this problem was very high.

An Analysis of Jayne's Response to the Proportion Problem. After Jayne had finished reading the problem, she immediately wrote down 10 in her workbook. She then wrote the numbers 20, 20, 30, and 40 horizontally across the page before calculating the total (120) in her head. She also wrote this number in her workbook (circling the number). After a short time, Jayne generated the following sequence--5, 10, 10, 15, and 20--before circling the calculated total of 60. Almost immediately, Jayne wrote the number 7 on her page and continued the sequence before circling a third calculated total of 84. By her next attempt Jayne had solved the problem. The following extract from this audiotaped session provides a self-report on her problem-solving strategy: I knew that I had to work out a pattern, so I had to use my tables (her multiplication table facts]. I worked out that if the smallest container had 10 kilos you would need 120. This was too much. My next pick was too small but it was closer. I knew I was on the rig ht track because it was something in between. I kept getting closer until I got it...I didn't take that long but it wasn't easy.

The fact that Jayne knew that the task required her to "work out a pattern" and "use my tables" suggested that she had developed foundation concepts associated with proportion. Her understanding of these concepts was reflected in the somewhat weak "trial and error" approach she used to solve the problem. On the other hand, the problem did not take her very long to solve. Her approach was quite analytic and she did not rely on any visual methods to solve the task. Thus, Jayne's understanding of proportion influenced the problem solving method she used to solve the problem. She realized that her nonvisual approach was working, and so she persisted until she generated a solution with which she was happy.

An Analysis of Matthew's Response to the Proportion Problem. He tackled the proportion problem in a nonvisual way, reaching a correct solution after carrying out a sequence of calculations. He generated a solution very quickly, using a more efficient approach than Jayne's to solve the problem.

The farmer had to divide the wheat up into five containers but the containers were not equal because some held more wheat than the others. Altogether there were 12 parts, instead of five, so one part was 6, not 14, remainder 2 [72/12, not 72/5]. In most problems you have to find out equal parts, but this is different. Once you work out what one part is, you have to double or triple that number and so on for the larger containers. If all the parts add up to the total, you know you are right.

Matthew was able to recognize that this task was different from the part-whole relationship problems he was accustomed to solving. In this case, he has used a nonvisual method in a manner consistent with his solution to the turf problem. That is, he has generated a series of calculations to produce an efficient solution path. It appears to be the case that Matthew uses nonvisual methods when he fully understands what is involved in solving the problem. Although Matthew had indicated that he had not solved problems like the proportion problem in the past, it was evident that his knowledge of proportion was sound.

An Analysis of Robyn's Response to the Proportion Problem. After reading the problem, Robyn began to represent the task visually in her workbook. Previously, she had suggested that you should draw diagrams when you do not know what the problem is about. It was apparent that Robyn was trying to interpret the problem through her visual representation of the "problem scene." She drew five objects to represent the containers, and placed a number related to the size of the container in each figure (see Figure 6). Interestingly, the containers were different sizes and related to the proportions described in the problem.

Robyn then wrote the number 72 on her page and began to put lines under each container. Robyn described this one-to-one correspondence approach as "sharing out the kilos of wheat into each container." She had only placed 12 strokes on her page when she stopped and commented that "the problem was too hard." Although encouraged to continue, Robyn indicated that the way she was attempting to solve the problem was incorrect, and added that "I don't know any formulas for these problems." If Robyn had continued with the visual method she had employed, she may have generated a correct solution, but it was evident that she considered the method to be inappropriate for this problem. This may have been because it was taking Robyn a considerable amount of time to complete the problem. Her beliefs about what constituted an efficient strategy may have also influenced her decision not to proceed with the method she had employed.

Conclusions and Conjectures

The case studies permitted an investigation into the mental processes employed by three Grade 6 students as they attempted to solve a variety of mathematics problems. The analysis presented in this paper suggests that during the period of the study all three case-study students moved towards more nonvisual, verbal reasoning as the novelty factor of problems diminished. Once the students felt that they possessed the appropriate schema or conceptual knowledge to link problem-solving strategies to previous experiences or understandings, they appeared to want to solve mathematics tasks in nonvisual ways. This was particularly the case with Robyn, whose preferred problem-solving approach at the start of the school year was visual.

Robyn's tendency to adopt less visual methods during the school year may have been the result of a desire to imitate strategies that had been used by "significant others" (in this case, Jayne and Matthew). Ironically, because Robyn did not always fully understand the methods used by these "significant others," her tendency towards adopting their strategies did not necessarily enhance the quality of her individual problem-solving attempts. With the turf and proportion problems, for example, Robyn tried, without success, to use a nonvisual strategy. She had more success with these problems when she reverted to a visual method.

Early in the year, Matthew displayed a willingness to use both visual and nonvisual approaches to solving the garden problem, but later in the year his approach to the turf problem was decidedly nonvisual, even though the turf problem is, according to the classification based on Suwarsono's (1982) MPI instrument, a high visuality task. Matthew's move towards a more nonvisual approach can probably be explained by his having become more familiar with annulus-type problems during the year, and by his desire to solve problems in an "efficient" manner. Both task difficulty and task familiarity seemed to contribute to Matthew's selection of the approaches he used when attempting to solve problems.

Jayne started the school as the case-study student who had the most nonvisual approach to problem-solving (see her responses to the swimming pool problem and the garden problem), and her responses to the turf and proportion problems suggest that little had changed during the year.

Students who had attempted to solve a problem in a visual manner early in the year often employed, or attempted to employ, nonvisual methods on a similar task later in the year. Successful nonvisual processing may suggest that a student has a more complete understanding of a particular problem. On the other hand, it could testify to a student's desire to solve problems in a clear, succinct manner.

The following conjectures relating to the methods or approaches used by students when attempting to solve a variety of mathematics problems were supported by the data in the study:

1. Students tended to change their approaches to solving mathematics problems from visual to less visual, more verbal/analytic methods as they become more familiar with tasks.

2. Although the approaches used by students when solving mathematics problems can provide information on their capacity to cope with other apparently related problem-solving situations, there can be no guarantee that useful transfer to these other situations will occur. In the present study, for example, Robyn became confused when she tried to use a method employed by her peers on an annulus problem that was not structurally the same as apparently related problems covered in class. If a student applies a method in a routine way, and cannot demonstrate an understanding of relationships between elements in a problem situation, then the student may not be able to work meaningfully on apparently similar, but nonetheless slightly different, tasks.

3. Although the strategies that students employ when they are attempting to solve mathematics problems are influenced by task complexity, these strategies can also reflect the students' recent educational experiences, interests, beliefs, values, and personalities.

Practical Implications for Practitioners

The findings of the research suggest that more capable elementary students should be introduced to symbolic strategies, including algebraic representations, once they have demonstrated the capacity to solve such problems in a visual manner. As classroom teachers recognize that particular individuals are able to identify the specific elements within a class of problems (including links between content and appropriate strategies), forms of nonvisual representation should be encouraged. Nevertheless, it is also important to recognize that movement from visual to nonvisual solution strategies is not always hierarchical--with a range of affective factors contributing to an individual's method of representation. Classroom teachers also should be conscious of presenting and modeling solution strategies in a variety of visual and nonvisual forms so that students are presented with a repertoire of powerful solution methods. In order to model these modes of representation in a balanced manner, it would be worthwhile to place an emphasis on visual approaches when students are introduced to new topics, and to increase symbolic modes of representation once students are more familiar with the mathematical content contained in the tasks.

Footnotes

(1.) Note that in this section the names used for the three students are not the students' real names.

(2.) A gifted student being someone who demonstrates superior talent in more than one area of the curriculum.

(3.) The children worked in grid books with 1 [cm.sup.2] unit markings.

References

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Clements, M. A. (1981-1982). Visual imagery and school mathematics. For the Learning of Mathematics, 2(2), 3-9, and 2(3), 33-39. (Published in two parts)

Clements, M. A., & Del Campo, G. (1989). Linking verbal knowledge, visual images, and episodes for mathematical learning. Focus on Learning Problems in Mathematics, 11(12), 25-33.

Dreyfus, T. (1991). On the status of visual reasoning in mathematics and mathematics education. Proceedings of the Fifteenth Annual Conference of the International Group for the Psychology of Mathematics Education, 1, 33-48.

Gardner, H. (1983). Frames of mind. New York: Basic Books.

Krutetskii, V. A. (1976). The psychology of mathematical abilities in schoolchildren. Chicago: University of Chicago Press.

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Lowrie, T. (1996a). Higher-order thinking in co-operative problem-solving situations: A case study. Australasian Journal of Gifted Education, 5(2), 7-11.

Lowrie, T. (1996b). Visual imagery in primary school mathematics. Unpublished doctoral dissertation, University of Newcastle, Australia.

Lowrie, T., & Kay, R. (2001). Relationship between visual and nonvisual solution methods and difficulty in elementary mathematics. The Journal of Educational Research, 94(4), 248-255.

Lowrie, T., & Hill, D. (1996). The development of a dynamic problem-solving model. Journal of Science and Mathematics Education in Southeast Asia, XIX(1), 1-11.

Mayer, R. E. (1992). Thinking, problem solving, cognition (2nd ed.). New York: W.H. Freeman.

Pirie, S., & Kieren, T. (1991). Folding back: Dynamics in the growth of mathematical understanding. In F. Fulvinghetti (Ed.), Proceedings of the Fifteenth PME Conference (Vol. 3, pp. 169-176). Assisi, Italy: International Group for the Psychology of Mathematics Education.

Presmeg, N. C. (1985). The role of visually mediated processes in high school mathematics: A classroom investigation. Unpublished doctoral dissertation, University of Cambridge, England.

Presmeg, N. C. (1986). Visualisation in high school mathematics. For the Learning of Mathematics, 6(3), 42-46.

Pylyshyn, Z. W. (1973). What the mind's eye tells the mind's brain: A critique of mental imagery. Psychological Bulletin, 80(1), 1-24.

Pylyshyn, Z. W. (1979). The rate of mental rotation of images: A test of a holistic analogue hypothesis. Memory and Cognition, 7(1), 19-28.

Pylyshyn, Z. W. (1981). The imagery debate: Analogue media versus tacit knowledge. Psychological Review, 88(1), 16-45.

Quinn, M. J. (1984). Visualization in learning mathematics. Unpublished doctoral thesis, Monash University, Melbourne, Australia.

Richardson, A. (1977). Verbalizer, visualizer, a cognitive style dimension. Journal of Mental Imagery, 1, 109-126.

Sheckels, M. P., & Eliot, J. (1983). Preference and solution patterns in mathematics performance. Perceptual and Motor Skills, 57, 811-816.

Suwarsono, S. (1982). Visual imagery in the mathematical thinking of seventh-grade students. Unpublished doctoral dissertation, Monash University, Melbourne, Australia.

[Figure 1 omitted]

[Figure 2 omitted]

[Figure 3 omitted]

[Figure 4 omitted]

[Figure 5 omitted]

[Figure 6 omitted]

Table 1

Mathematical Performance of the Students Involved in the Case Studies


Task or Assessment Activity          Jayne      Matthew     Robyn
Suwarsono (1982) Mathematics Test
Performance (/20)                      20          20        19
Visuality Score on Suwarsono
Instrument (0 - 20)                    0           10        20
                                    (highly    (balanced)  (highly
                                   nonvisual)              visual)
Australian Primary Mathematics
Competition score (/50) and
corresponding percentile rank          37          42        39
                                      (87)        (98)      (92)
Table 2

Approaches Used by the Three Students Solving the Swimming Pool Problem

QUESTION: An Olympic size swimming pool is 50 meters long and 20 meters
wide. If you assume that the average depth of the pool is 1.5 meters,
calculate how many liters of water would be required to fill an empty
pool.


Jayne's Response                      Matthew's Response
Jayne read through the                As Matthew began reading
problem with her eyes fixed to        the problem, a broad smile
the page for about 20 seconds.        covered his face, and he
She did not make any eye              stated, "This is going to be
contact with me until after           easy, can I use a calculator?"
she had picked up a pencil.           It appeared that he enjoyed
She wrote down the equation           the challenge of solving this
1 [cm.sup.3] = 1 mL. Glancing at me   particular problem (possibly
for approval, she then                enhanced by his love for
calculated 50 x 20 = 1000, and        swimming). He calculated (on
1000 x 1.5 = 1500. Her work           the calculator) 50 x 20 x 1.5
was not aligned in any                and wrote 1500 [m.sup.3] on his
particular order, but was             page. Underneath this he
spread across the page.               wrote 1[m.sup.3] - 1000 L. He then
Underneath her first equation         wrote 1500 and added three
(1 [cm.sup.3] = 1 mL) she wrote 1500  zeroes. I replied, "Well done,
[m.sup.3] = x. She then said to me    but don't forget the liters
"and now the hard part." It           sign." It took Matthew only
then took Jayne a consider-           about two minutes to solve
able amount of time (over 5           the entire problem.
minutes) to complete the task.
After she had finished she
asked, tentatively, "Does 1.5
million litres sound right?" In
the 5 minutes it took for her
to convert the [m.sup.3] into liters
she constantly used an eraser
when her untidy workings
distracted her too much.
Jayne demonstrated a strong
desire to solve the problem,
and was task persistent.


Jayne's Response                      Robyn's Response
Jayne read through the                Robyn read the question and
problem with her eyes fixed to        looked at me in disbelief. I
the page for about 20 seconds.        got the impression that she
She did not make any eye              felt the question was too
contact with me until after           difficult. For reassurance, I
she had picked up a pencil.           told her to take her time and
She wrote down the equation           think through the problem
1 [cm.sup.3] = 1 mL. Glancing at me   carefully. After a short time
for approval, she then                she drew a picture of a 3D
calculated 50 x 20 = 1000, and        representation of a swimming
1000 x 1.5 = 1500. Her work           pool with appropriate dimen-
was not aligned in any                sions on a piece of paper. She
particular order, but was             then calculated the volume of
spread across the page.               the pool by calculating 50 x 20
Underneath her first equation         and 1000 x 1.5 in a sequential
(1 [cm.sup.3] = 1 mL) she wrote 1500  order down her page. She
[m.sup.3] = x. She then said to me    highlighted the answer 1500
"and now the hard part." It           [m.sup.3] by underlining it twuce
then took Jayne a consider-           with her ruler. Underneath
able amount of time (over 5           this, she wrote 1500 L and
minutes) to complete the task.        again underlined this
After she had finished she            response twice.
asked, tentatively, "Does 1.5
million litres sound right?" In
the 5 minutes it took for her
to convert the [m.sup.3] into liters
she constantly used an eraser
when her untidy workings
distracted her too much.
Jayne demonstrated a strong
desire to solve the problem,
and was task persistent.

Jayne read through the problem with her eyes fixed to the page for about 20 seconds. She did not make any eye contact with me until after she had picked up a pencil. She wrote down the equation 1 [cm.sup.3] = 1 mL. Glancing at me for approval, she then calculated 50 x 20 = 1000, and 1000 x 1.5 = 1500. Her work was not aligned in any particular order, but was spread across the page. Underneath her first equation (1 [cm.sup.3] = 1 mL) she wrote 1500 [m.sup.3] = x. She then said to me "and now the hard part." It then took Jayne a considerable amount of time (over 5 minutes) to complete the task. After she had finished she asked, tentatively, "Does 1.5 million litres sound right?" In the 5 minutes it took for her to convert the [m.sup.3] into liters she constantly used an eraser when her untidy workings distracted her too much. Jayne demonstrated a strong desire to solve the problem, and was task persistent.

Appendix

The Mathematical Processing Instrument developed by Suwarsono (1982) was used to assess children's preference for solving mathematical problems in either a visual or nonvisual mode. The instrument consists of 30 mathematical problems (2 tests of 15 items), as well as a corresponding questionnaire containing written descriptions of different solution methods commonly used when working on each mathematical problem.

Solutions to each problem were categorized as either visual (where diagrams or drawings were used either on paper or in the mind) or nonvisual (solutions that are analytically based). To illustrate the use of the instrument an example of a mathematical problem and the relevant solution methods are presented. A nonvisual response would be indicated in Solution 1, and a visual response in Solutions 2 and 3.

A tourist traveled some of his journey by plane, and the rest by bus. The distance that he traveled by bus was half the distance he traveled by plane. Determine the length of his entire trip if the distance that he traveled by plane was 150km longer than the distance he traveled by bus.

Solution 1:

To solve this problem, I divided the journey into three equal sections, two sections being traveled by plane, one section by bus. The difference in the distance traveled by plane and that traveled by bus was one section. This was equal to 150km. Thus the length of the hole journey was 3 x 150km, or 450km.

(I did not draw or imagine any picture at all.)

Solution 2:

I solved this problem by drawing a diagram of the journey:

In the diagram it can be seen that the difference between the distance traveled by plane and that traveled by bus was one section. It was equal to 150km. In the diagram it is also clear that the length of the whole journey was 450km.

Solution 3:

I used the same method as for Solution 2, only I drew the diagram "in my head" (and not on paper).

I did not use any of the above methods.

I attempted the problem in this way: