# Mathematical Habits of Mind

The widespread utility and effectiveness of mathematics come not just from mastering specific skills, topics, and techniques, but more importantly, from developing the ways of thinking—the habits of mind—used to create the results. —Al Cuoco, Paul Goldenberg, and June Mark, Habits of Mind: An Organizing Principle for Mathematics Curriculum

In our work, we define mathematical habits of mind (MHoM) to be the specialized ways of approaching mathematical problems and thinking about mathematical concepts that resemble the ways employed by mathematicians. Understanding the MHoM and general purpose tools underlying the various topics and techniques of secondary mathematics content can bring parsimony and coherence to teachers’ mathematical thinking and to their work with students. In this sense, we envision MHoM as a critical component of mathematical knowledge for teaching, i.e., the knowledge necessary to carry out the work of teaching mathematics.

At this time, we have focused our research on algebraic habits of mind, and we have also narrowed to three categories of mathematical habits that we think are particularly important to secondary teachers:

• Using mathematical structure,
• Seeking mathematical structure, and
• Using mathematical language clearly

Using mathematical structure. This habit entails taking advantage of the underlying structure of a given situation to facilitate problem solving.

For example, consider the task of finding the maximum value of the function f(x) = 11 – (3x – 4)2. Since (3x – 4)2 represents the square of some number, it is ≥ 0. Thus in f(x) = 11 – (3x – 4)2, we are subtracting a non-negative number from 11. To maximize f(x), we need (3x – 4)2 = 0 so the maximum value is 11. Such an approach illustrates how one interprets and makes use of the underlying algebraic structure of the given expression for f(x).

Alternatively, consider the task of simplifying the expression 3(992 – 1) + 8(992 – 1) – 11(992 – 1). By viewing (992 – 1) as a common term, the expression becomes 3♧ + 8♧ – 11♧, which equals 0♧ or 0. This approach, which treats (992 – 1) as a single object ♧, is called “chunking,” which entails intentionally avoiding certain details to hide the complexity of an algebraic expression.

Both examples above involve algebraic expressions with a particular structure (or form) that were well-suited for the task at hand. Recognizing and utilizing that structure (as opposed to further manipulating the expressions) is at the heart of this mathematical habit.

Seeking mathematical structure. This habit is about the search for useful structures that are not immediately apparent. It typically involves experimental processes such as writing expressions in equivalent forms, working with smaller/special cases, seeking regularity in repeated calculations, and generalizing from concrete examples.

For example, consider the task: Use the fact that 1764 ⨉ 1765 = 3113460 to find 1762 ⨉ 1767. The original product has the form x(x + 1) = x2 + x and the new product has the form (x – 2)(x + 3) = x2 + x – 6. Thus the latter is 6 less than the former, so that 1762 ⨉ 1767 = 3113454. This approach unravels the relationship between these products by representing them symbolically and comparing their forms. Another approach involves the use of smaller examples—e.g., start with 8 ⨉ 9 and 6 ⨉ 11 and notice that they differ by 6. This approach, however, does not explain the relationship between the products.

Remark: The habits of using and seeking structure are closely related, but we distinguish them as follows. In the task of computing 1762 ⨉ 1767, for example, the underlying structure of the expression is not readily apparent. Some work is needed to uncover this structure, which makes the example more illustrative of the seeking mathematical structure habit.

Using mathematical language clearly. This habit entails communicating clearly with others. Features of this habit may include (but are not limited to) understanding and appropriately using mathematical terminologies and definitions, using mathematical notation correctly, and giving clear descriptions of the steps in a process.

For example, a student asks about the question: Determine if r = –2 is a solution to 6r + 2 = 12 + r. The teacher asks the class what “solution” means, to which students respond with phrases such as “when it works” and “the answer.” This imprecise language does not help unravel the problem to understand what they are being asked to do. Thus, the teacher encourages them to be more specific. Eventually, they arrive at, “something that makes the equation true.” The class can now use this definition to substitute r = –2 into the equation and check if it makes it true.

Alternatively, suppose a student writes 82 + 8 = 90 ÷ 3 = 30 – 5 = 25. This illustrates an incorrect use of the equal sign, since the expressions 82 + 8, 90 ÷ 3, and 30 – 5 are not equal. The teacher corrects the student, calling it a “run-on sentence in math.”