Contextual images in mathematics problem solving.

Abstract

Solving word-problems is a difficult task for many students. Understanding, the first step to solving a problem (Polya, 1945; Kintsch & Greeno, 1983), requires the activation of three schemata; the language, the contextual, and the mathematical. Students who are unable to construct a contextual understanding of a problem situation are limited in their understanding of the problem (Brown & Wheatley, 1997). This study investigated the effects of providing a dual system of contextual information, the words of the problem coupled with an image to activate the contextual schema, on the mathematical problem solving of 5th and 6th grade students. MANOVA indicated that the presentation of the contextual images had no effect on problem solving performance. An analysis of student work indicated that many students chose a correct strategy to solve a given problem. However, their application of the strategy (for example an algorithm) was incorrect or incomplete.

Background

Even though problem solving has long been a major focus of mathematics education (NCTM; 1989, 2000), students continue to score relatively low on tests of problem solving ability. An example of this is scores on the Stanford Nine (a test administered annually to students in my state) subscale "computation in context" (i.e. word problems) which are typically among the lowest of all the subtests administered. This study examined the effect of supplying a context setting image on students' ability to correctly solve word problems. The use of imagery to evoke a contextual setting for the problems presented was derived from Paivio's dual-coding theory of memory.

Solving word-problems, even the most routine ones typically found in Elementary grades textbooks, is a difficult task for many students. One of the first steps to solving any mathematics problem is "understanding the problem" (Polya, 1945; Kintsch & Greeno, 1983). Understanding requires the activation of three schemata: first, the contextual schema that relates to the situation of the problem, second, the language schema to understand what the problem is asking, and third, the mathematical schema that corresponds to the implied action of the problem. For example, a division word problem requires a student to read the words (their language schema), access their contextual schema for sharing, and then their schema (i.e., a mathematical division strategy) for resolving a "sharing" situation. The contextual image is crucial for doing mathematics and a student who fails to construct an adequate image in a problem solving situation is limited when asked to give meaning to the situation (Brown & Wheatley, 1997). In other words, schema activation for context setting allows a child better to understand and solve a presented problem.

Problem solving is a complex cognitive activity requiring students to see relationships in order to gain meaning. Thinking involves the manipulation of what is represented mentally. It involves forming and manipulating relationships between items of information and forming a representation of the problem in the mind (Marschark & Hunt, 1989). This was demonstrated by Bransford and Johnson (1972) who had subjects read a clear, simple passage that could not be understood without the benefit of an image. The illustration was necessary to provide a cognitive framework within which to understand. In other words, the illustration activated the subject's contextual schema. The pictures presented to the students in this study are also intended to activate their contextual schema.

Language

How children solve mathematics word problems provides an excellent domain to examine how knowledge is constructed and integrated. Unlike some other forms of discourse, many mathematics word problems contain clear-cat criteria. However, word problems, like all other texts, can share the ambiguity and fuzziness of natural language. This means that children must possess not only the formal mathematical knowledge, but they must also possess linguistic and situational knowledge. What makes word problems difficult and intriguing are not their formal properties, but the way a problem is represented linguistically (its lexicon) and the way formal mathematical relations map onto the problems. Word problems are ideal for this study because they allow us to understand how formal mathematical knowledge and linguistic knowledge are integrated.

Context

Word problems embedded into familiar situational contexts are much easier to solve than problems that must be solved without situational support. Context is an important element to understanding mathematics word problems. Computation isolated from context results in the manipulation of symbols without the benefit of understanding. It is a common practice in the elementary grades to use children's literature as a context setting device when presenting mathematical ideas or processes (Zambo & Cleland, 2001). Whitin (1992) states that using literature with mathematics helps children see mathematics as a "common human activity," useable in various contexts. Children's stories help to develop the contextual schema in which the problem situations are based. For example, the book The Doorbell Rang by Patricia Hutchins (1986) is often used to demonstrate how to divide a set of cookies. Using books or pictures with story problems is assumed to assist the child in breaking down the artificial dichotomy that sometimes exists between learning mathematics and living it. The connections between mathematics and other content areas, as well as between mathematics and the real-world, are an important part of children's developing mathematical understanding.

Although word problems supply context, the delivery is limited to verbal symbols. A mental image of a problem would entail a perceptual representation or ideational picture of a perceptual experience, remembered or imagined. The ability to imagine has been shown to improve students' memory (Levin & Divine-Hawkins, 1973) and inferential reasoning about written text (Borduin, Borduin, & Manley, 1994). A dual-coding view of memory suggests that information is best understood when it is encoded visually as well as with verbal symbols. This theory known as dual-coding (Paivio, 1971) states that when students are exposed to dual codes, two memory systems are activated with access links between them. Paivio's work on human memory found memory for pictorial material superior to memory for verbal information (pictures are easier remembered than paragraphs). The reciprocal connections between the two codes allow information in one code to activate information in the other. The connections develop from experience with objects and events and by hearing the language associated with them. The advantage of dual coding in mathematics is that the forming of multiple codes in one's mind creates a context to solve the problem and increases the likelihood that a math strategy will be later recalled. The argument, "two codes are better than one" (Paivio, 1971), has been widely used to explain the improvement in memory that results when a visual display (e.g., a contextual picture) and text are read simultaneously. Research on the brain empirically supports Paivio's dual-coding theory demonstrating considerable overlap between the neurological systems responsible for verbal perception and imagery formation (Farah, 1988). The educational implication of these findings is that visual displays and verbal information are better remembered when the two are instructionally combined.

Problem solving involves the activation of contextual schema, language, and mathematical schema stored in long-term memory. Elaboration is a cognitive process whereby information is extended or added to in order to make it more memorable. Higbee (1979) suggests that because our memories are highly visual the use of images is an important part of most mnemonic systems. Elaborations that involve forming associations between new material and what is already familiar result in better meaning. It is speculated that the pictures used in this study would help students to better elaborate on the material and more readily solve the problems. This study investigated the effects of providing a dual system of contextual information, the words of the problem and an image (a picture of the situation) that represents the contextual schema. For example, one division problem about sharing a number of cookies evenly between a given number of children was paired with an image of children with a plate of cookies. It was speculated that the image would provide support for the "sharing" schema and allow students to map their knowledge about division to the situation. The pictures, taken from children's books popular in the elementary schools, relate only to the general context of the problem. They did not directly represent the numeric information from the problem. Although others have studied the effects of imagery on problem solving performance, no one has conducted a focused study on the effects of contextual images on problem solving performance. Results from this study will provide information that could lead to new strategies to increasing problem- solving performance.

Methods

Test Booklets Four word problems, that aligned with the contexts presented in four images from children's books, were created. The problems targeted four mathematics topics: division, factors, multiplication, and fractions. The problems were reviewed by the two participating fifth grade teachers and the two participating sixth grade teachers. They all thought that the problems would be appropriate for their students and they estimated that about 50 percent of their students would be able to solve them correctly. Each test booklet contained four word problems, one on every second page. The blank page facing the word problem contained the contextual image, or not. One form of the test had a context setting image only for items 1 and 3. The other form had a context setting images only for problems 2 and 4.

The Word Problems Mrs. Smith just brought a tray of 72 cookies to Christie's house. Christie has 11 friends over. If Christie and her friends divide the cookies evenly among themselves, how many will each of them get? This problem was based on the book The Doorbell Rang (Hutchins, 1986). Twenty-four ants decided to march in equal rows. How many different ways can the ants line up? This problem was based on the book One Hundred Hungry Ants (Pinczes, 1993). In a folk tale, the animals are throwing spears. The lion can throw the spear 16 feet. The monkey wants to throw the spear 2.5 times as far. How far does the monkey have to throw the spear? This problem was based on the book Two Ways to Count to Ten (Dee, 1988). John, Judy and their father are going to share a pizza. The father takes half of the pizza and John and Judy equally share what is left. What fraction tells how much pizza each of the children will get? This problem was based on the book Give Me Half! (Murphy, 1996).

Administration Testing was completed in April. Booklets were distributed to whole classes of students randomly. Teachers instructed the students to read the problem, look at the picture, and then solve the problem. Students were allowed as much time as needed to complete the task. Questions to the teacher about the problems were answered with encouragement to "try your best".

Subjects Subjects (N=134) were all of the fifth (N=53) and sixth grade (N=81) students at a public elementary school in a southwestern metropolitan area.

Results

Items were scored as correct or incorrect. The total score was computed for each subject. Out of a possible total score of 4, fifth grade subjects' mean score was 1.3 (33 percent) and the sixth grade subjects' mean score was 1.73 (43 percent). A P-value (proportion of subjects answering correctly) was computed for each item. For the division problem, fifth and sixth grade Ps were .40 and .33 respectively; for the factors problem, .15 and .07; for the multiplication problem, .38 and .49; and for the fractions problem, .23 and .63. Multivariate Analysis of Variance (MANOVA), a statistical technique for determining if groups differ on more than one dependent variable, indicated no significant effect of form on the individual test items. In other words, the presentation, or not, of a contextual setting image had no effect. MANOVA indicated a significant effect for grade level, F (2, 129)=7.32, p< .0001. Follow-up Univariate analysis showed that the difference was due to scores on the fractions problem, F(1, 132)=24.405, p< .01.

An analysis of students' work revealed that in many cases students had chosen the correct algorithm or had drawn a correct diagram but arrived at an incorrect answer. The incorrect answers were the result of errors in computation, misreading a diagram, or creating an incomplete list. Because many subjects had chosen a method that could have led to the correct solution if correctly implemented, the items were scored again for appropriateness of method. For example, if a student set up the correct computation he or she was scored as correct for choice of method even though there was an error in computation. In addition, it was assumed that students who supplied the correct answer without showing any work had also employed a correct choice of method. It was inferred that if a student set up the problem correctly, then he or she understood that context and had selected an appropriate mathematics schema. The proportions of students selecting a correct method by grade level for each problem were: the division problem, fifth and sixth grade scores were .85 and .91 respectively; for the factors problem, .72 and .60; for the multiplication problem, .62 and .74; and for the fractions problem, .64 and .79. MANOVA found no significant effect for form of test or grade level.

Discussion

The most surprising outcome of this study, for both the researchers and the teachers involved, was the low performance of the students. Problem solving is difficult for many students, but these teachers felt that their students were good problem solvers and would do well. Teachers were surprised that students did so poorly on items that they thought would be relatively easy for the majority of their students. Review of the students' responses resulted in the conclusion that students made a wide variety of errors which ranged from misplacing a decimal in the multiplication problem, to drawing the correct model of the fractions problem and then misinterpreting it. As a result, the tests were scored again for correctness of the applied approach. Those scores showed that most of the students understood the problems, (they had the language schema), and were able to determine an approach that should have led them to the correct answer. However, computational errors and the misinterpretation of correctly drawn diagrams kept them from successfully solving the problems. Teachers who find their students making similar errors should help students learn to look at their drawings and think about how and why they made them. Correct drawings show that the students could reason about the problems, but drawings are of little use if students do not take time to look over and think about what they have drawn and what it means.

The previous finding also raises the issue of transfer, or the influence of previously learned math skills on problem solving. Students must be able to spontaneously and automatically transfer the skills they learn in their classrooms to mathematics word and real life problems. To assist in students' transfer abilities they should have extensive practice in the basic procedures of mathematics. This does not mean endless worksheets but plenty of opportunities to practice their math skills in games, real world problems, and puzzles. The grade level difference for correctly solving the problems was expected. It was not expected that the only significant difference would be on the fraction problem. It was anticipated that the sixth graders would do better on all four of the items. It is not clear why the sixth graders did not outscore the fifth grades on the other problems. It was also surprising that there was no significant difference between the fifth and sixth graders in choosing an appropriate method to the fraction problem even though there was a significant difference in arriving at the correct solution.

There was no effect of the images on problem solving. This is possibly due to the fact that the contexts of the problems were ordinary enough that the image added little to the lexical information, therefore, no effect. Most of the students seem to understand the context sufficiently to choose the correct mathematical schema for the problem. It was problems with the application of the mathematics processes that caused the errors. A future study using images with more difficult problems may prove interesting and help determine of the influence pictures have on problem solving. With more difficult problems students would have to rely more on images to set the problem in context and activate the proper schema. The students in this study appeared to have the language, the contextual, and the mathematical schemata necessary to engage in problem solving. However, they lacked in the needed accuracy and completeness of the computational procedures needed to solve the problems correctly. Students need to have both higher level problem solving skills and lower level computational skills to be successful problem solvers.

References

Bourduin, B. J., Borduin, C. M., & Manley, C. M. (1994). The use of imagery training to improve reading comprehension of second graders. Journal of Genetic Psychology, 155(1), 115-118.

Bransford, J. D., & Johnson, M. K. (1972). Contextual prerequisites for understanding: Some investigations of comprehension and recall. Journal of Verbal Learning and Verbal Behavior, 11(6), 717-726.

Brown, D. L. & Wheatley, G. H. (1997). A student's imaging in solving a nonroutine task. Teaching Children Mathematics, 4(2), 100-104.

Dee, R. (1988). Two ways to count to ten. New York: Henry Holt and Company

Farah, M. J. (1988). Is visual imagery really visual? Overlooked evidence from neuropsychology. Psychological Review, 95,307-317.

Higbee, K. L. (1979). Recent research on visual mnemonics. Historical roots and educational fruits. Review of Educational research, 49, 611-630.

Hutchins, P. (1986). The doorbell rang. New York: Mulberry:

Kintsch, W., & Greeno, J. G. (1983). Understanding and solving arithmetic problems. Psychological Review, 92, 109-129.

Levin, J. R., & Divine-Hawkins, P. (1973). Strategies in reading comprehension: Visual imagery as a psychological process. Report from the Research and Development Center for Cognitive Learning. University of Wisconsin: Madison.

Marschark, M., & Hunt, R. R. (1989). A reexamination of the role of imagery in learning and memory. Journal of Experimental Psychology: Learning, Memory, and Cognition, 15(4), 710-720.

Murphy, S. J. (1996). Give me half!. New York: Harper Collins.

National Council of Teachers of Mathematics. (1989). Curriculum and evaluation standards for school mathematics. Reston, VA: Author.

National Council of Teachers of Mathematics. (2000). Principles and Standards for School Mathematics. Reston, VA: Author.

Paivio, A. (1971). Imagery and verbal processes. New York: Holt, Rinehart, Winston.

Pinczes, E. J. (1993). One hundred hungry ants. Boston: Houghton Mifflin Company:.

Polya, G. (1945). How to solve it. Princeton: Princeton University Press.

Whitin, D. (1992). Exploring mathematics through children's literature. School Library Journal, 38(8) 24-28.

Zambo, R & Cleland, J. (2001, November). Teachers' beliefs and practices for elementary grade mathematics/language arts integration. Paper to be presented at the Annual Meeting of the School Science and Mathematics Association, Downers Grove, IL.

Ron Zambo, Arizona State University West 

Debby Zambo, Arizona State University West

Ron Zambo, Ph.D. is an Associate Professor of Mathematics Education interested in mathematics problem solving. Debby Zambo, Ph.D. is a Lecturer in Educational Psychology interested in the role of images in understanding.