1. An exploration of students’ problem solving behaviors Presenter: Chun-Yi Lee Advisor: Ming-Puu Chen Muir, T., Beswick, K., & Williamson, J. (2008). I am not very good at solving problems: An exploration of students’ problem solving behaviors. Journal of Mathematical Behavior, doi: 10.1016/j.jmathb.2008.04.003

2. Introduction Research questions  What behaviours do a selected number of grade 6 students engage in when solving non- routine problems?  To what extent are these behaviours consistently displayed by each individual student?

3. Motivation International studies have included documentation of problem solving approaches involving large numbers of students (e.g., Cai, 2003), describing the success or otherwise of the implementation of problem solving approaches to mathematics teaching (e.g., Thomas, 2006) and contrasting cross-cultural approaches to problem solving (e.g., Cai, 2004). Detailed studies on individuals’ problem solving behaviour are rare.

4. Literature Review The problem solving process Problem solving can be defined as attempting to find a solution in situations where the series of actions which need to be performed to reach that solution are not immediately known (Cooper, 1986). Problem solving requires a variety of skills including interpreting information, planning and working methodically, checking results and trying alternative strategies.

5. Literature Review The problem solving process A successful problem solver must incorporate a range of heuristic approaches when solving problems (Schoenfeld, 1980). This heuristic framework was used as a basis for analysing the data in the study discussed in this paper.  understand the problem, devise a plan, carry out the plan and look back and examine the solution obtained The following strategies have been identified as the most widely used in solving problems: guess and check, draw a picture, make a list, make a table, work backwards, look for a pattern and the use of logical reasoning, solve a simpler problem and write an equation (Charles, Mason, & Garner, 1985; O’Daffer, 1985).

6. Literature Review Conceptual understanding Conceptual understanding (relational understanding)  one knows “both what to do and why” Instrumental understanding (procedural understanding)  a tendency to apply a procedure without being able to explain why or how the procedure works

7. Literature Review Problem solving behaviors According to Lester and Kroll (1993) problem solving performance is influenced by five factors:  knowledge acquisition and utilization,  control,  beliefs,  affects and  socio-cultural contexts

8. Literature Review Expert and novice problem solvers According to Bransford et al. (1999) there are a number of characteristics which distinguish expert problem solvers from novice problem solvers.  knowledge acquisition and utilization,  control,  beliefs,  affects

9. Methodology Participants Four grade 6 students were selected from each of five primary schools, comprising 20 students in all. Schools that varied in their socio-economic status, size and location were selected in an attempt to address reliability and validity issues that may be raised in relation to a small sample size (Patton, 1990). Grade 6 students were targeted because they could be expected to have a reasonable level of literacy in order to understand the questions and record their answers in a written way. Teachers in each school were asked to nominate the four students who would participate in the study, based on their mathematical ability. Each teacher nominated one ‘above average’, two ‘average’ and one ‘below average’ student in relation to their mathematics/numeracy ability. This nomination was based on the professional judgment of the individual teachers and consistent with the students’ Year 5 Statewide Testing Numeracy results.

10. Methodology Tasks

11. Methodology Procedure Students were interviewed using a semi-structured approach (Burns, 2000). A copy of each problem was presented and read aloud to the child by the interviewer and each respondent was then asked to respond to the question and make a written recording of any working that they used in this process. Following the completion of each problem, each respondent was asked to verbally explain what they had done, while the interviewer tape recorded their responses, made observational notes and asked prompting questions when and if clarification was required.

12. Methodology Data analysis

14. Results and Discussion

15. Conclusions and Implications The continuum provides a model for examining students’ problem solving behaviours and also describes what a successful or sophisticated problem solver does. The below average students in the study consistently used the numbers in the problem and tried to apply operations to them. A teaching strategy to avoid this trend may simply be to provide these students with problems where the numbers are written in words, rather than numerals.

16. Conclusions and Implications Some of the students’ problem solving behaviour could be attributed to the way in which they have been taught. Telling students strategies or rules for solving problems in advance is not an effective way of improving their problem solving skills, particularly if the strategy is supplied along with a collection of problems which all use that one strategy (Willoughby,1990). Teaching practices that encourage students to produce their own lists of strategies, to think about their own and other people’s strategies and apply them accordingly to a variety of problems (Willoughby, 1990) would possibly promote more sophisticated problem solving behaviour.

17. Conclusions and Implications Why did so many of these students display poor problem solving behaviours? Is it because problem solving has been treated superficially, as claimed by Schoenfeld (1992), or because many problems presented to students in mathematics classes are not authentic and do not require students to engage in higher order thinking processes (Willoughby, 1990)?

18. Conclusions and Implications Through analyzing the errors made by these students, it appears that they would benefit from some heuristic instruction, provided that this instruction “illustrates how the procedures work, gives ample opportunity for discussion, practice and reflection and supports and encourages the learner’s efforts” (Kilpatrick, 1978, cited in Suydam, 1980, p. 45).  The provision of a variety of problems  Worked examples and instructional examples  Problem posing  Write problems