Mathematics and Visual Representation

Being a textbook editor for mathematics, I find that more and more diagrams and visual display of mathematics concepts can be found in mathematics textbooks in Hong Kong while the length of paragraphs is shortened. It seems that textbook editors implicitly agree with Tufte (1983) that visuals can communicate complex ideas with clarity, precision, efficiency and convey the most knowledge in the shortest time in the small space. From my experience, because of the diversity of learner differences in Chinese and English languages, simple and clear annotated visual representation of information can help explain mathematics concepts.

For example, when the method of completing the square is introduced, some students do not understand why the process of completing the square works even after teachers have explained the algebraic proof (e.g. http://students.ou.edu/H/Layla.Hayavi-1/Episode%202.html). Teachers can try to help students visualize the algebraic proof by providing the corresponding geometry proof.

From: http://www-groups.dcs.st-and.ac.uk/~history/Biographies/Al-Khwarizmi.html

An interactive geometric proof can be found at http://illuminations.nctm.org/ActivityDetail.aspx?ID=132. The visualization can help students see the unseen and enhance better understanding (Arcavi, 2003). Teachers can also use this example to let students know that in mathematics, one can sometimes find more than one method to solve a question.

Another example is the visual representation for sum to infinity of geometric series. See the following two GP sum. With these visual representations of proofs without words, students should be able to 'see' what the sum to infinity of geometric series on the left hand side of the equality sign represents.

From: http://www.mathland.idv.tw/


In these examples, diagrams group together all the information necessary for explaining certain mathematics concepts so that students would not be distracted by unnecessary information during learning (Larkin & Simon, 1987). However, how to prepare a good visual representation is really another issue for study.

References:

Arcavi, A. (2003). The role of visual representation in the learning of mathematics. Educational Studies in Mathematics, 52(2003), 215-241.

Lakin, J.H. & Simon H.A. (1987). Why a diagram is (sometimes) worth ten thousand words. Cognitive Science, 11(1987), 65-99.

Tufte, E. R. (1983). The visual display of quantitative information. Cheshire, Connecticut: Graphics Press.