Like abstraction and generalization which I described in my earlier posts here and here,visualization is central to the learning and understanding of mathematics. (Note that these processes are also natural human mental dispositions and so we can claim that doing mathematics is a natural human activity.)
Visualization used to be considered only for illustrating otherwise abstract ideas of mathematics but now visualization has become a key component of mathematical processes such as reasoning, problem solving, and even proving.
What is visualization?
Synthesizing the definitions of visualization offered by Zimmermann and Cunningham (1991, p. 3) and Hershkowitz (1989, Abraham Arcavi proposes the following definition:
Visualization is the ability, the process and the product of, creation, interpretation, use of and reflection upon pictures, images, diagrams, in our minds, on paper or with technological tools, with the purpose of depicting and communicating information, thinking about and developing previously unknown ideas and advancing understandings– Abraham Arcavi, ESM, 2003
What are examples of use of visualization in mathematics?
- For communicating information, the graph is perhaps the most recognizable of the visual representations of mathematics.
- For proving, visual proofs are already accepted as legitimate proofs. For example, here’s a visual proof of the Pythagorean Theorem. Click here for source of movie. [iframe 350 500]
- Of course, visuals can also be used to challenge students to reason and explain in words and symbols. For example teachers can show the visual in #2 then ask the students what the visual is telling them about the relationships between the areas of the three squares and about the sides of right triangles. Students should be asked to support their claim.
- Visualization tasks also trains students mind to ‘think outside the box’. Click here for an example of a problem solving tasks which can be solved by visualizing possible arrangements. Patterning activity like Counting Hexagons are great activities not only for generating formulas and algebraic expressions but trains the mind to look for relationships, an important component in algebraic thinking.
- Because what we see usually depends on what we know, visuals can also be used as context for assessing students knowledge of mathematics. Click here for an example on how to assess understanding by asking students to construct test items.