Towards a Theory of Teaching and Learning to Think Mathematically at University

Alon Pinto

The purpose of this study is to set the foundations for a long-term research program on the teaching and learning of mathematics at the post-secondary level. My overarching goal is to develop an empirically based theory of how math teaching at the university can shape the mathematical ways of thinking and doing (e.g. beliefs, habits of mind, practices) that students develop.There is no doubt that there is more to mathematics than the content that is taught and discussed explicitly in math classrooms. Thus the lessons students learn about mathematics extend far beyond the scope of explicit mathematical knowledge. Ways of thinking about and doing mathematics that may be beneficial for the students are often left implicit in the curriculum, and may not be considered the teachers’ responsibility, if at all teachable. There is extensive literature describing how students pick up mathematical habits, practices and beliefs through instruction regardless of what is taught and how, and that these mathematical ways of thinking and doing are often unproductive and undesirable. This problem is even more apparent at university, due in part to the widening gap between school mathematics and mathematics as practiced by mathematicians. Students at university are required to pick up the norms, language and ways of thinking and doing of the mathematics community, which are often tacit and rarely discussed explicitly.I propose a study of the mathematics that is left implicit by various university instructors teaching the same concepts, theorems and proofs. The goals of this study will be: (1) To describe how different instructors adapt a common curriculum and how these adaptations affect the mathematics addressed in their lessons, particularly mathematical ways of thinking and doing; and (2) To propose explanations as to why the instructors implement the curriculum and address the mathematics the way they do. In order to achieve these objectives, I will attend lessons, examine the mathematics that is addressed explicitly and implicitly in the lessons and discuss with the instructors selected episodes from their lessons. The data will be collected and analyzed in iterative cycles in order to uncover the instructors’ beliefs and goals and their impact on the mathematics in their lessons. The analysis will be guided by the Resources-Orientations-Goals (ROG) theory on decision making processes develop by Alan Schoenfeld and his Teacher Model Group at the University of California at Berkeley.This study takes a novel approach of examining practices not only in light of how mathematics is taught, but also what mathematics is addressed and why. This will provide important new insights both regarding the factors that shape teaching practices and the mathematics that is left implicit at university math lessons. The potential impact of this study extends beyond undergraduate math teaching. Improving the math education of math undergraduates, the teachers of tomorrow, by leading them to a better understanding and appreciation of mathematics, would significantly impact math teaching at all levels.

About Alon Pinto

Alon Pinto is a Postdoctoral Fellow in the Science Teaching Department at the Weizmann Institute of Science. He earned his Ph.D. in Mathematics from the Hebrew University of Jerusalem in the field of geometric group theory. Pinto’s research focuses on mathematics teaching and learning at the post-secondary level. His work aims to uncover how university instructors’ beliefs and knowledge shape their instructional practices and consequently the mathematical ways of thinking and doing that their students develop. He is currently studying the mathematics left implicit by various instructors teaching in parallel the same curriculum for a single course. In his NAEd/Spencer project, Pinto intends to extend this work as a step towards an empirically based theory of how math teaching at the university can lead math undergraduates – the math teachers of tomorrow – to a better understanding and appreciation of mathematics.