Options
Visualization in primary school mathematics : its processes and roles in mathematical problem solving
Loading...
Type
Thesis
Author
Ho, Siew Yin
Supervisor
Yeap, Ban Har
Abstract
In an information age where the ability to reason visually becomes increasingly important, the roles that visualization plays in the mathematical thinking and problem solving experiences of students too become more significant. Recognizing visualization as a powerful cognitive tool in problem solving, the Singapore Mathematics curriculum reemphasized the importance of visualization in the revised 2007 Primary Mathematics Syllabus. In order to provide insights into students’ use of visualization, and provide suggestions for the teaching and learning of visualization, this study attempted to document the processes and roles of visualization in mathematical problem solving.
The research design involved asking primary school students to solve verbal word problems having high degree of visuality and difficulty. The present study included four parts − Investigation 1, Investigation 2, Investigation 3 and Investigation 4. Investigation 1 was conducted to determine the grade level of students suitable for the study. Results of Investigation 1 showed that only eleven and twelve-year-olds were academically prepared to handle the type of word problems in the study, and hence resulted in the subsequent three investigations to involve only eleven and twelve-year-olds.
Investigation 2 investigated the problem-solving responses of 1167 students from five Singapore primary schools in a paper-and-pencil test. In this investigation, 3 types of solution methods were observed, namely drew picture or diagram, did not draw picture or diagram, and no written solution. In addition, eight categories of responses involving the use of a visual method (drawing a picture or a diagram) were observed. About 14% of the eleven-year-olds and around one third of the twelve-year-olds were successful in solving the problem. Majority of students who were unsuccessful solved the problem computationally.
Based on the findings of Investigation 2, 50 students were selected for Investigations 3 and 4 where they were asked in an interview setting to solve six verbal word problems which were variations of each other. In Investigation 3, two factors were observed to influence the use of a visual method: novelty of the problem and students’ perception of their classroom teacher’s problem-solving preference. Students preferred to use visual methods for novel problems while nonvisual methods were used for familiar problems. Some students used methods which they perceived as their classroom teacher’s problem-solving preferences.
Based on Investigation 4, five processes and seven roles of visualization in mathematical problem solving were observed and illustrated by vignettes. The five processes of visualization are: understanding the spatial relations of the elements in the problem, connecting to a previously solved problem, constructing a visual representation, using the visual representation to solve the problem and encoding the answer to the problem. The seven roles of visualization are: to understand the problem, to allow opportunities to work with a simpler version of the problem, to see connections to a related problem, to cater to individual learning styles, as a substitute for computation, as a tool to check the solution, and to transform a situation into mathematical forms.
This study found evidence of an intermediate shuttling phase between Pirie and Kieren’s (1991 & 1992) “fold back” and Lowrie and Kay’s (2001) “move forward” phases. In this intermediate phase, students who were not yet fluent in using visual and nonvisual methods “move forward” from visual representations to nonvisual representations, and also ‘fold back” to visual representations when their nonvisual methods do not work.
The research design involved asking primary school students to solve verbal word problems having high degree of visuality and difficulty. The present study included four parts − Investigation 1, Investigation 2, Investigation 3 and Investigation 4. Investigation 1 was conducted to determine the grade level of students suitable for the study. Results of Investigation 1 showed that only eleven and twelve-year-olds were academically prepared to handle the type of word problems in the study, and hence resulted in the subsequent three investigations to involve only eleven and twelve-year-olds.
Investigation 2 investigated the problem-solving responses of 1167 students from five Singapore primary schools in a paper-and-pencil test. In this investigation, 3 types of solution methods were observed, namely drew picture or diagram, did not draw picture or diagram, and no written solution. In addition, eight categories of responses involving the use of a visual method (drawing a picture or a diagram) were observed. About 14% of the eleven-year-olds and around one third of the twelve-year-olds were successful in solving the problem. Majority of students who were unsuccessful solved the problem computationally.
Based on the findings of Investigation 2, 50 students were selected for Investigations 3 and 4 where they were asked in an interview setting to solve six verbal word problems which were variations of each other. In Investigation 3, two factors were observed to influence the use of a visual method: novelty of the problem and students’ perception of their classroom teacher’s problem-solving preference. Students preferred to use visual methods for novel problems while nonvisual methods were used for familiar problems. Some students used methods which they perceived as their classroom teacher’s problem-solving preferences.
Based on Investigation 4, five processes and seven roles of visualization in mathematical problem solving were observed and illustrated by vignettes. The five processes of visualization are: understanding the spatial relations of the elements in the problem, connecting to a previously solved problem, constructing a visual representation, using the visual representation to solve the problem and encoding the answer to the problem. The seven roles of visualization are: to understand the problem, to allow opportunities to work with a simpler version of the problem, to see connections to a related problem, to cater to individual learning styles, as a substitute for computation, as a tool to check the solution, and to transform a situation into mathematical forms.
This study found evidence of an intermediate shuttling phase between Pirie and Kieren’s (1991 & 1992) “fold back” and Lowrie and Kay’s (2001) “move forward” phases. In this intermediate phase, students who were not yet fluent in using visual and nonvisual methods “move forward” from visual representations to nonvisual representations, and also ‘fold back” to visual representations when their nonvisual methods do not work.
Date Issued
2009
Call Number
QA63 Ho
Date Submitted
2009