In Pursuit of Coherent and Formalizable Understanding: Reflections on David Tall’s Three Worlds Framework

Abstract

In “How Humans Learn to Think Mathematically”, David Tall endeavors to account for mathematical thinking and learning from childhood to mathematical research; he does this while maintaining an eye toward formalization, as one might expect of the editor of “Advanced Mathematical Thinking” (Tall, 1991) – a seminal work in proof-oriented undergraduate mathematics education. Indeed, “How Humans...” directly reflects Tall’s career in many ways, lending it an autobiographical flavor. I appreciated the fascinating insight into Professor Tall’s own thinking and learning about mathematical thinking and learning over his decades of work. The book serves as a synthesis of many of the major projects and ideas from Tall’s career and makes frequent reference to his work with various graduate students. As a result, many of the elements that comprise his overarching “three worlds” framework will be familiar to many mathematics educators. Like most mathematics education frameworks, the three worlds framework partitions mathematical thinking and doing in various ways, but most prominently into the embodied, symbolic, and formal worlds of mathematics. Tall provides a number of examples to show how students can encounter the same “concept” over the course of their schooling through the three worlds, generally in the embodied and symbolic worlds before formalizing. Unlike the authors of some frameworks, Tall tries to avoid privileging any one of these three ways of thinking about mathematics, because his ultimate goal is the coordination of various meanings into a unified whole called a crystalline concept. Unlike many texts, Tall’s presentation will be accessible to anyone with advanced mathematics training, although it reflects the sensibilities of someone who has engaged in pedagogical conversations with mathematicians, as evidenced by Byers’ (2014) complimentary review. Though Tall casts a vision for flexible and coherent understanding across the worlds, he acknowledges several barriers to understanding the unity of crystalline concepts. It was primarily these barriers that fostered the questions I was left pondering at the close of the book.