Much more important than specific mathematical results are the habits of mind used by the people who create those results. … Although it is necessary to infuse courses and curricula with modern content, what is even more important is to give students the tools they will need in order to use, understand, and even make mathematics that does not yet exist. (Cuoco, Goldenberg, & Mark, 1996)
My research focuses on understanding secondary teachers’ mathematical habits of mind (MHoM). We define MHoM to be the specialized ways of approaching mathematical problems and thinking about mathematical concepts that resemble the ways employed by mathematicians. These habits are not about particular definitions, theorems, or algorithms that one might find in a textbook; instead, they’re about the thinking, mental habits, and research techniques that mathematicians employ to develop such definitions, theorems, or algorithms. Some examples of MHoM include:
- Discovering the underlying structure by experimenting and seeking regularity and/or coherence.
- Choosing a useful representation of a mathematical concept or object.
- Purposefully transforming and/or interpreting algebraic expressions—e.g., rewriting x2 – 6x + 10 as (x – 3)2 + 1 to reveal its minimum value.
The importance of MHoM for students and for teachers of mathematics, particularly at the secondary level, cannot be overstated. These habits foster the development and use of general purpose tools that make connections among various topics and techniques of secondary school mathematics; they can bring focus and coherence to teachers’ mathematical thinking and, in turn, to their work with students. In this sense, we envision MHoM as a critical component of mathematical knowledge for teaching at the secondary level.
I was a Principal Investigator for the NSF-funded Assessing Secondary Teachers’ Algebraic Habits of Mind, a research project that investigated the relationships between teachers’ own mathematical habits of mind, their use of such habits in teaching practice, and student achievement.