Representations and Proportional Reasoning in a Problem Solving Context. (c2015)

Abstract

Research and experience show that a successful problem-solving teaching and learning model needs to include a construction of visual schematic representations that show spatial relationships between the different components of the problem. Such a model is particularly important when teaching proportions as students find difficulty identifying direct and inverse proportionality situations. This study aims to investigate whether or not using representations such as tables, graphs, diagrams, pictures and number lines extensively and functionally when teaching proportions to grade-eight students improves the development of students’ proportional reasoning and problem solving abilities. Thirty-four students participated in the study. They were taught “Proportions” and “Proportional Reasoning” by the same teacher using two differently designed instructional units. The participants were divided into a control group of 17 students learning proportions using a standard textbook plan and the usual teaching approach, and an experimental group of the same number following a plan with greater emphasis on multiple representations as tools for thinking and problem solving. Data were collected and analyzed using a mixed method design consisting of qualitative and quantitative methods: the qualitative method was used to detect the representations employed by the teacher and students through class observations, examine the representations used in both the textbook and the curriculum, and analyze clinical interviews conducted throughout the post-test with 24 selected students. On the other hand, an experimental design was implemented to study the students’ success level in solving problems using a pre-test and a post-test. Results of the qualitative analysis of the tests show that students in the experimental group have developed better understanding of proportionality than those in the control group. Moreover, students of the experimental group were able to use multiple strategies to approach the problems. Likewise, results of the quantitative analysis of the tests point to a significant statistical difference between the mean scores of high-achievers as well as average-achievers of the two groups, regarding the use of representations as a problem solving strategy, validating the practices implemented to teach the experimental group.