«Recent curriculum reforms in some Australian states (e.g. Department of Education Tasmanian, 2002) have highlighted the importance of thinking, ...»
Talking and Writing About the Problem Solving Process
Kim Beswick Tracey Muir
University of Tasmania University of Tasmania
This paper reports one aspect of a larger study which looked at strategies used by Grade six
students to solve six non-routine mathematical problems. One focus of the study was the
relative effectiveness of students’ written and verbal communication in revealing their thinking during the problem solving process. The results suggest that students may benefit from instruction on communicating their thinking in writing and that emphasising writing as a guide to students’ thinking may disadvantage lower ability students.
Recent curriculum reforms in some Australian states (e.g. Department of Education Tasmanian, 2002) have highlighted the importance of thinking, communicating and instilling deep understanding in our students and have seen a re-emphasis on mathematical problem solving as an important mechanism for enhancing these skills. Problem solving is, of course, not a new idea in mathematics education. Over half of a century ago, the importance of problem solving was recognised (Brownell, 1942, cited in Suydam, 1980) and its importance was emphasised strongly throughout the 1980’s (Suydam, 1980). Polya (1949) and others (e.g. Branca, 1980; Schoenfeld, 2002) maintain that problem solving is the goal of mathematics learning while the National Council of Teachers of Mathematics (2000) go further saying that problem solving “is not only a goal of learning mathematics but also a major means of doing so” (NCTM, 2000, p.4).
Similarly, communication is and always has been an important part of mathematical problem solving (Willoughby, 1990) and a key to gaining an insight into students’ understanding. This study examined both students’ written and verbal communications, in response to a number of mathematical problems, in terms of what they revealed about students’ problem solving processes.
Written and Verbal Communication in Problem Solving Flewelling and Higgenson (2003) explain how communication relates to Polya’s (1957) four-step problem-solving model. Students can talk about the problem to understand it better and to clarify the method. They can listen to the other people’s ideas, draw pictures, use manipulatives, write words and symbols to represent the steps involved in solving the problem, share results of computer or calculator operations, and decide upon the best way to describe and explain how the solution was reached (Flewelling & Higgenson, 2003). Furthermore, Mousley (1999) found that many of the indicators that teachers use to gauge the extent of children’s understanding relied on students’ ability to communicate. They included such things as an ability to articulate understanding, to respond to probing questions, to explain concepts in their own language, and the ability to teach the relevant concept to others (Mousley, 1999).
The benefits of writing about the executive processes of problem solving are well documented (Williams, 2003; Pugalee, 2001; Azzolino, 1990; Clarke, Clarke & Lovitt, 1990). However, some students express a dislike for writing explanations and justifications in mathematics and according to Bicknell (1999), many students find writing explanations a difficult and tedious exercise. In contrast with this even quite young children are able verbally to articulate their thinking (Franke & Carey, 1997). Kaur and Blane (1994) found that it was often difficult to diagnose the difficulties experienced by students in solving problems by examining their written solutions alone. The limitations of relying on written recordings of solutions to gauge students’ understanding were clearly illustrated by Clements and Ellerton (1985). Their study aimed to determine whether or not students who gave correct answers to pen and paper tests had a comprehensive understanding of the mathematical concepts and relationships which the tests were designed to measure. They found that about one quarter of children’s responses could be classified as (a) correct answers given by students who did not have a sound understanding of the mathematical knowledge, skills, concepts and principles behind the question, or (b) incorrect answers given by students who had partial or full understanding.
According to Huniker and Laughlin (1996), students become aware of what they really know and what more they need to know as they talk about their experiences and test their new ideas with words. With regard to interviews aimed at revealing students’ thinking, Stoyanova (2000) emphasized the importance of framing ‘good’ questions in order for teachers to understand their students’ problem solving processes, and indicated that a dialogue needed to occur between teacher and student in order to elicit student responses.
Ginsburg (1987) also advocated the interview as an effective means to enable students to communicate their mathematical thinking and Norbury (2003) recommended its use particularly with younger children.
Despite the fact that problem solving is a complex process and that children’s problem solving or thinking ability cannot simply be assessed through pen and paper means, the difficulty of conducting one-to-one interviews in classroom contexts means that this remains the norm.
The Study The study examined the use of problem solving strategies by Grade six students and their ability to communicate their thinking both verbally and in writing. The specific research question addressed in this paper is: Does reliance on written recordings of the problem solving process differentially disadvantage students according to their mathematical ability?
Subjects Four grade six students were selected from each of five primary schools, making a total of twenty students (12 girls and eight boys). Grade six students were targeted because they could be expected to have a sufficient level of literacy in order to be able to record their solutions in writing. Teachers in each school were asked to nominate the four students who would participate in the study. Each teacher nominated with respect to mathematical ability, one ‘above average’, two ‘average’ and one ‘below average’ student, maintaining a gender balance if possible. This nomination was based on the professional judgment of the individual teachers, but in most cases was also supported by the students’ Year Five Numeracy Testing results which rated their overall numeracy in relation to other students in the state. The total sample was thus made up of five ‘above average’, ten ‘average’ and five ‘below average’ students with respect to mathematical ability.
The Problems Six mathematical problems were chosen based on the researcher’s classroom teaching experience with regard to their appropriateness in terms of the degree of challenge offered to the selected age group, and their potential to be answered using a variety of strategies both across the problems and for any given problem. The problems are listed below.
1. Jenny is making towers of cubes using red, blue, yellow and green. How many different towers can she make by changing the order of the colours?
2. Jim got into a lift. He went down five floors, up six floors and down seven floors.
He was then on the second floor. At what floor did he get on?
3. Susan worked at an apple orchard. When she was sorting the apples for sale, she noticed that two out of every seven apples had worm holes. If there were 70 apples in the basket, how many could be expected to be ‘good apples?
4. Some children were playing with some rabbits in a yard. I tried to count them and found that there were 30 legs and 11 heads. How many children and how many rabbits were in the yard?
5. At a meeting of the Good Friend’s Society, everyone begins by shaking hands with each other once. If there were ten people at the meeting, how many handshakes were there?
6. I made some triangles using matchsticks. I used three matches to make one triangle, five to make two triangles and seven to make three triangles. If I continued in this way, how many matches would I need to make 12 triangles?
Procedure Students were interviewed using a semi-structured approach (Burns, 2000). A copy of each problem was presented and read aloud to the student by the interviewer. The student was then asked to solve the problem and to make a written recording of any working that they used in this process. Potentially helpful concrete materials such as cubes, counters and matchsticks were available to be used as the students chose and in relation to any of the problems. Following the completion of each problem, each student was asked to verbally explain what they had done, referring to their written solution as they wished, while the interviewer tape recorded their responses, made observational notes and asked prompting questions when and if clarification was required.
The rating scales shown in Table 1 were used to separately classify each of the students’ written and verbal responses in terms of the clarity with which they communicated the student’s thinking, and the correctness of the solutions obtained. These scales were also used by Adibnia and Putt (1998) in their study of problem solving by year six students. Judgments of the clarity of the communication were made independently of an assessment of the correctness of their solutions. That is, very clear communications of incorrect solutions, as exemplified by Belinda’s verbal response to Problem Two, were
rated ‘2’. Belinda said:
I added up 5, 6 and 7 and got 18, except that I thought that can’t be right because I don’t think there’s an 18th floor … then I wrote down a different sort of pattern and I wrote the first D (for down) and I wrote 5 and then 6, then 7, then … I did 57 how many 6’s … and I just wrote all the lines to 57 and counted 6 and put a line around them and then I got 9 remainder 3 so I just added 9 and I put 3 with it and got 12. So he got on at the 12th floor.
More than ninety percent of the verbal explanations provided by students at each of the ability levels were at least partially clear in their communication of the students’ thinking.
Table 4 Ratings of Solutions in Terms of Correctness Rating Above average Average Below average No. % No. % No. % Total numbers 30 60 30 Only five percent or less of the solutions produced by average and above average students rated ‘0’, while more than one third of the solutions produced by below average students received this rating. From Tables 2, 3 and 4 it is clear that more able students are more likely than other students to produce clear and appropriate written records of their problem solving activity, more likely to be able to clearly verbalise their thinking, and more likely to obtain correct answers.
The numbers of ‘0’ ratings for written recordings obtained at each ability level is related to the numbers of students in each ability category who obtained ‘0’ for the correctness of their solutions. Although the correctness or otherwise of the solution was not taken into account in rating the written or verbal responses it is clear that the students’ ability to engage with and attempt a problem would influence what they were able to convey either in writing or verbally.
Verbal communications rated more highly than written responses for all ability levels.
This is consistent with the other findings (e.g. Kaur and Blane, 1994) and also unsurprising given that the students communicated verbally, with their written recordings available to remind them of the processes they undertook and to assist with structuring their responses.
Nevertheless it is noteworthy that just one student at each ability level produced a verbal response that rated ‘0’. That is, in all other cases (97.5%), regardless of the ability of the student, the verbal communication provided an at least partially clear explanation of the student’s thinking. This is important given that written responses alone would have provided potentially useful insights into the students’ thinking (i.e. ratings of ‘1’ or ‘2’) in just 65% of cases with the overwhelming majority of this difference accounted for by average, and particularly below average, students. The following examples are illustrative of the differences between the written and verbal communications of students considered to be of below average mathematics ability.
In response to Problem Six, the triangle problem, Jack wrote only the incorrect answer, 23, which was rated ‘0’. However, when asked to verbalise his thinking he revealed that his reasoning in relation to the problem was sound and that his incorrect answer was
simply the result of a computational error. He said:
I just started um, with, how it said, how it’s got on the sheet there and just kept adding it on, like, um, just added the top bit and then added the bottom bit to it and it worked out to 10 triangles first and then I got another um, 4 toothpicks and just added them on and made triangles and kept going until 23.
In response to Problem Five, the handshake problem, Peter wrote an incorrect answer, 41, at the top of the page and at the bottom of the page he wrote the numbers nine down to one vertically. There was no indication of the purpose of these numbers but nevertheless they were relevant to an appropriate solution of the problem and hence his written communication was rated ‘1’. Peter’s verbal explanation rated ‘2’ and revealed that he did understand what the problem was asking and had chosen an appropriate solution strategy.
I got ten matchsticks and I got one out and he would shake the other matchsticks’ hands, every matchstick’s hand once, so that’s nine times and one of the others would shake the other matchsticks’ hands eight times and then another one would shake them seven times and another one would shake other hands and so forth and so forth.
When asked what he then did with those numbers, he responded that he added them up and got forty one. He had correctly identified that the numbers from one to nine needed adding, he simply made an error when calculating their total. Interestingly, Peter’s use of matchsticks in this problem is one of just 14 of a possible 120 instances in which concrete materials were used. Of these two were by above average students, four by average students, and eight by below average students.