by Ho Siew Yin
Seeing is believing, or so the saying goes. We depend on our sight for many things in life - using a map to find our way, using a picture to aid recognition, or using diagrams to better describe what our words fail to communicate. In the math classroom, sometimes the solution to a problem is right before our eyes.
Visualization in Math Problem Solving
Inasmuch as the ability to solve problems is at the heart of mathematics, visualization is at the heart of mathematical problem solving.
Visualization is the ability to see and understand a problem situation. Visualizing a situation or an object involves “mentally manipulating various alternatives for solving a problem related to a situation or object without benefit of concrete manipulatives” (MOE, 2001, p. 51).
Visualization can be a powerful cognitive tool in problem solving. In the revised Primary Mathematics syllabus (MOE, 2007), it is highlighted as an important skill “essential in the learning and application of mathematics” (p. 13).
This ability to reason visually is increasingly important in the information age. Thus, the role that visualization plays in students’ mathematical thinking and problem-solving experiences has become more significant.
The Research Study
A recent study by Dr Ho Siew Yin attempted to provide insights into the use of visualization in mathematical problem solving among primary school students.
She asked 50 Primary 5 and Primary 6 students to solve word problems with a high degree of visuality and difficulty. Here is an example of such a word problem:
A man plants seedlings along a straight path.
He plants a seedling every 4 cm along a path.
The length of the path is 60 cm.
How many seedlings, at most, can he plant?
This problem is typical of what students would face in the PSLE exam. The students were asked to solve six such problems in an interview setting. Siew Yin documented five processes and seven roles of visualization in their problem solving.
Processes of Visualization
Siew Yin noted that the students went through the following processes when solving the math problems:
- Understanding the spatial relations of the elements in the problem
- Connecting to a previously solved problem
- Constructing a visual representation (in the mind, on paper, or through the use of technological tools)
- Using the visual representation to solve the problem
- Encoding the answer to the problem
As part of the problem-solving process, the students would construct visual representations, often in the form of diagrams drawn on paper.
However, Siew Yin also noted that a number of students in the study were creating visual representations that were not helpful to them. Diezmann (2000) describes three types of unusable diagrams:
- where the diagram is too small to represent all the relevant information in the problem;
- where the diagram is too untidy for the problem-solver to see the elements of the problem clearly; and
- where there is insufficient space around the diagram to extend it.
Roles of visualization
Visualization plays different functions or roles as students use it to solve problems. Siew Yin identified seven roles:
- To understand the problem
By representing the problem visually, students can understand how the elements in the problem relate to each other.
- To simplify the problem
Visualization allows students to identify a simpler version of the problem, solving the problem and then formalizing the understanding of the given problem and identifying a method that works for all such problems.
- To see connections to a related problem
This involves relating the given problem to previous problem-solving experiences.
- To cater to individual learning styles
Each student has his or her own preference when it comes to the use of visual representations when solving problems.
- As a substitute for computation
The answer to the problem can be obtained directly from the visual representation itself, without the need for computation.
- As a tool to check the solution
The visual representation may be used to check for the reasonableness of the answer obtained.
- To transform the problem into a mathematical form
Mathematical forms may be obtained from the visual representation to solve the problem.
Developing Visualization Skills
To help students develop visualization skills, classroom teachers and designers of curriculum materials should first be mindful of the factors that influence students’ choice of problem-solving method, and of the processes and roles that visualization plays in mathematical problem solving.
Siew Yin also recommends that teachers increase students’ awareness of the three types of unusable diagrams by illustrating the disadvantages of using such diagrams during problem solving.
If visualization is at the heart of mathematical problem solving, then it is vital that both teachers and students see the role of visualization clearly and use it to help them in their problem-solving process.
Diezmann, C. M. (2000). The difficulties students experience in generating diagrams for novel problems. In T. Nakahara & M. Koyama (Eds.), Proceedings of the 24th Conference of the International Group for the Psychology of Mathematics Education (Vol. 2, pp. 241-248). Hiroshima, Japan: PME.
Ho, S. Y. (2009). Visualization in primary school mathematics: Its roles and processes in mathematical problem solving. Unpublished doctoral dissertation, National Institute of Education, Nanyang Technological University, Singapore.
Ministry of Education, Curriculum Planning and Development Division. (2001). Mathematics Syllabus. Singapore: Author.
Ministry of Education, Curriculum Planning and Development Division. (2007). Mathematics Syllabus Primary. Retrieved from the Singapore Ministry of Education website: http://www.moe.edu.sg/education/syllabuses/sciences/files/maths-primary-2007.pdf
Presmeg, N. C. (1986). Visualization in high school mathematics. For the Learning of Mathematics, 6(3), 42-46.
Bishop, A. J. (1989). Review of research on visualization in mathematics education. Focus on Learning Problems in Mathematics, 11(1), 7-16.
Lowrie, T., & Kay, R. (2001). Relationship between visual and nonvisual solution methods and difficulty in elementary mathematics. Journal of Educational Research, 94(4), 248-255.
Presmeg, N. C. (2008). Spatial abilities research as a foundation for visualization in teaching and learning mathematics. In N. C. Presmeg & P. C. Clarkson (Eds.), Critical issues in mathematics education: Major contributions of Alan Bishop (pp. 83-95). New York: Springer.
About the author
This article was contributed by Ho Siew Yin, a Lecturer with the Mathematics and Mathematics Education Academic Group at the National Institute of Education, Singapore. The article is based on her doctoral research.