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Exploring Strategies Used to Solve a Non-Routine Problem by Chilean Students; an Example of “Sharing
Chocolates”
Farzaneh Saadati, Mayra Cerda
To cite this version:
Farzaneh Saadati, Mayra Cerda. Exploring Strategies Used to Solve a Non-Routine Problem by Chilean Students; an Example of “Sharing Chocolates”. Eleventh Congress of the European Society for Research in Mathematics Education, Utrecht University, Feb 2019, Utrecht, Netherlands. �hal- 02435323�
Exploring Strategies Used to Solve a Non-Routine Problem by Chilean Students; an Example of “Sharing Chocolates”
Farzaneh Saadati and Mayra Cerda
Universidad de Chile, Chile; farzaneh.saadati@gmail.com, mayra.cerda@ciae.uchile.cl Keywords: Mathematical thinking, problem solving, representation strategy.
Introduction
In Chile, mathematical problem solving has been incorporated in the school curriculum. However, it has been shown that problem solving activities are practically absent in the classrooms (Felmer &
Perdomo-Díaz, 2016). Furthermore, despite an emphasis on problem solving in the new curriculum for grades 1-8 (MINEDUC, 2012), teachers appear reluctant or unable to incorporate rich problem solving practices in their classrooms. Cai and Nei (2007) suggested that students’ performance in problem solving is affected by the cultural context, teachers’ beliefs and their practices. In another study, we found a mismatch between Chilean mathematics teachers’ beliefs and practices (Saadati, Cerda, Giaconi, Reyes, & Felmer, 2018). Teachers have sort of reformed beliefs about problem solving, however in classrooms, problem solving is predominantly seen as a set of concrete techniques, and teachers focus on the repeated practice of procedures. These conditions and circumstances can obviously have an impact on students’ problem-solving ability and on their mathematical thinking. In this study, we are going to explore Chilean students’ representation strategies to solve the problem on sharing chocolates:
3 boys share 2 bars of chocolate equally and 8 girls share 6 bars of chocolate equally. Who gets more chocolate, the boys or the girls? Explain or show how you found your answer.
This problem is a process-constrained problem categorized as a non-routine problem for Chilean elementary students since they are not familiar with this type of problem. To solve this problem, they might fail to consider all of the information presented in the problem, or “compute first and think later” (Hegarty, Mayer, & Monk, 1995). Students’ representation might be directed towards using division because of the presence of the term “share” and the numbers. It can be represented as 3/2 or 2/3. We refer to this form of representation as direct-translation. The students could also try to construct a representation of the situation being described in the problem (Hegarty et al., 1995).
The representation constructed based on this approach is usually a more meaningful and concrete way that involves the construction of a mental model. We call it a meaning-making representation strategy which can become a basis for construction of a solution plan. In this study, we explore the dominant and most accurate representation strategies and the rate of success among students.
Methodology and Results
The students were asked to solve three problems presented by Cai (2000). Here we discuss the problem “sharing chocolates”. Participants were 143 students from several public schools; 72 girls, and 71 boys, included 52 sixth graders, 52 seventh graders, and 39 eighth graders.
There were 74 students who had a plan to solve the problem by following the three-step process. In contrast, 52 students did not devise any plan to solve the problem. These students skipped the first
two steps and tried to answer only the third part of the problem which was the sense-making question. The rest of the students (17) did not try to solve the problem and left it blank. The results revealed that 46 out of 74 (32% of students who followed the three-step process) were able to finish it, and 28 students stopped before reaching the third part or sense-making question. Among the students who followed the process, 17 students out of 74 showed a direct-translation representation – correctly or incorrectly – to solve the problem. The rest of the students used a meaning-making strategy. By using the meaning-making strategy, 45 students pictorially represented their solution (drawing only or mixed with a verbal explanation), the other 12 students just verbally explained their solutions. Finally, only 9 students could solve the problem completely without any mistakes; 3 students used a direct-translation and 6 got the answer by using the meaning-making representation strategy.
Discussion
The results of the Chilean grade 6th to 8th students revealed that using the meaning-making representation was more prevalent among them. The students’ performance was comparable with U.S. students in a similar study done by Cai (2000). However, there was a large rate of failures among these students and just a few of them (about 6%) could solve the problem successfully. We believe, the mismatch between teachers’ beliefs and practice can explain the gap between their tendency of using meaning-making strategies and their failure. The teachers’ reformed beliefs may lead them to put more value on meaning-making representation strategies, while their traditional teaching approach explain their students’ lack of skills and inexperience with non-routine problems.
In fact, the failure highlights the necessity of a shift of instruction towards more student-centered practices in order to give students more space to develop their mathematical thinking while working with those non-routine problems. Therefore, we suggest changing school practices and organizing effective and related teachers’ professional development programs.
Acknowledgment
The research is financially supported by CONICYT/Fondecyt Postdoctoral Project 3170673. PIA- CONICYT Basal Funds for Centers of Excellence Project FB0003 are gratefully acknowledged.
References
Cai, J. (2000). Mathematical thinking involved in US and Chinese students' solving of process- constrained and process-open problems. MTL, 2(4), 309–340.
Cai, J., & Nie, B. (2007). Problem solving in Chinese mathematics education: Research and practice. ZDM, 39(5-6), 459–473.
Felmer, P., & Perdomo-Díaz, J. (2016). Novice Chilean secondary mathematics teachers as problem solvers. In P. Felmer, E. Pehkonen, & J. Kilpatrick (Eds), Posing and solving mathematical problems (pp. 287–308). Cham: Springer International Publishing.
Hegarty, M., Mayer, R. E., & Monk, C. A. (1995). Comprehension of arithmetic word problems: A comparison of successful and unsuccessful problem solvers. Journal of educational psychology, 87(1), 18–32.
Saadati, F., Cerda, G., Giaconi, V., Reyes, C., & Felmer, P. (2019). Modeling Chilean Mathematics Teachers’ Instructional Beliefs on Problem Solving Practices. International Journal of Science and Mathematics Education, 7(5), 1009–1029.
