Visual Thinking in Mathematics
After last's week's recommended text, I was hesitant to overview another "heavy" resource. But, this new text seems like a good followup to David Tall's opus.
Marcus Giaquinto, University College in London, focuses on one element in mathematical thinking: the visual. His text is an epistemological study, which means that it is a study of the origin, nature, methods, and limits of visual thinking within a mathematical context.
With those philosophyladen words, I know that perhaps most of you have clicked to another page already, but for those that remain, hang in there! His book is Visual Thinking in Mathematics: An Epistemological Study (2007), which I just discovered.
First, Giaquinto defines visual thinking as "visual imagination or perception of diagrams and symbol arrays, and mental operations on them"...all within a mathematical context. Second, his big question: Is visual thinking at most a psychological aid that helps interpret what else is "gathered" or does it function as a "means of discovery, understanding, and even proof"?
I suggest that all reflective mathematics teacher (and perhaps a great many mathematical learners) have asked that question regarding both the value and the educational use of the visual. He separates his argument into three broad content areas:
 Geometry (Chapters 25)
 Airthmetic (Chapters 68)
 Other mathematical areas such as analysis and algebra (Chapters 812)
Without giving too much away, I can state Giaquinto's conclusion: visual thinking in mathematics is rarely just a superfluous aid; it usually has epistemological value, often as a means of discovery." As part of his argument, he shows how learners "discern abstract general truths by means of specific images, how synthetic a priori knowledge is possible, and how visual means can help us grasp abstract structures."
Again, this book is not an easy read, as it enters the realms of philsophy (e.g. Plato, Kant, Mill) and the history of mathematics (e.g. Pythagoras, Bolzano, Peano) to support its investigations of common notions of mathematics. I found it interesting and thoughtprovoking, amidst its analysis of mathematical thinkers as they confront and use visual images. For those who take the effort to preuse the text, I think you will find the effort well worth it.
