David Tall's Opus
You may not have heard of David Tall (University of Warwick), a dedicated researcher in the area of learning mathematics. I first met him about 25 years ago at a conference in Sophia, Bulgaria of all places. In conversations over numerous dinners, David always impressed me with his knowledge, insight, and approach to things mathematical...plus he has an extremely generous humor.
David Tall recently published a resource text that is a collective overview of his 30 years of research on mathematical thinking and learning. The book How Humans Learn to Think Mathematically: Exploring the Three Worlds of Mathematics (2013) is an opus of wealth and perspectives for mathematics teachers.
Two initial comments. First, throughout the book, David Tall shows the influence of his mentor Richard Skemp, who produced many great resource texts for teachers, as well as the fundamental notions of "instrumental understanding" and "relational understanding." And second, David's text is unusal in that it attends to a framework of mathematical thinking that extends from birth through adulthood.
Originally proposed in one of his articles in 2004, David Tall's three worlds of mathematical thinking are:
 Conceptual Embodiment: Builds on perceptions and actions developing mental images that are verbalized in increasingly sophisticated ways
 Operational Symbolism: Starts with physical actions that evolve into mathematical procedures, which some even transcend to even operate on the procedures themselves
 Axiomatic Formalism: Generates formal knowledge akin to axiomatic systems based in set theoretic definitions, whose properties and relationships are deduced and verified by mathematical proof
To share some flavor of the book's content, consider these comments from near the end of the text: "The evolutionary framework of the three worlds of mathematics is based on the fundamentals of human thinkinghow we make connections and build knowledge structures that grow in sophistication. As this happens, new ideas that prove to be useful are strengthened and old ideas that are superseded are lost." (. 418). I like that perspective!
The book is not an easy read...you have to be committed to the task of trying to understand both your own understanding and your own mathematical thinking. In return, it creates many reflective moments, and I predict even some Ahas!
