## Mathematical habits of mind

## Math encounters — mind-bending paradoxes & the possibility

reflect symbolically: to use algebra to solve problems more easily and with greater trust in one’s response, as well as to better communicate solutions, gain a better understanding of problems, and investigate the possibility of multiple solutions.

demonstrate: to wish for a statement to be proven to you or by you; to participate in a discussion aimed at clarifying an argument; to create a deductive proof; to create an argument using indirect reasoning or a counter-example, search for plausibility: To check the reasonableness of any assertion in a problem or its proposed solution on a regular basis, regardless of whether it appears valid or false on first impression; to be particularly suspicious of findings that seem contradictory or implausible, whether the source is a peer, instructor, evening news, book, newspaper, internet, or other; and to look at special and limiting cases to see if a formula or a method works; and to look at special and limiting cases to see if a formula or

## Teaching habits that promote productive struggle in math

Every university professor would be ecstatic if their students came to their mathematics classes with the ability to construct viable arguments and criticize others’ reasoning; if their tendency was to do so.

However, how do students acquire these mathematical skills? During a student’s 13 years of mathematics classes in grades K-12, they learn from their teachers and engage in mathematics with their peers, laying the foundation. The eight Mathematical Practice Standards, which are part of the Common Core State Standards for Mathematics, have increased the awareness and value of positive mathematical habits of mind in K-12 education. It is no longer a bonus, but rather an expectation. Are instructors, on the other hand, prepared to assist their students in developing the practices before they become second nature? Do teachers themselves have useful mathematical habits of mind?

From studies, we know quite a bit about the mental behaviors of pre-service teachers (Karen King: Because I love mathematics, Mathfest 2012, Madison). Pre-service teachers with a mathematics degree, for example, have a proclivity to first state laws (Floden & Maniketti, 2005). They aren’t in the habit of searching for meaning, which is a critical mathematical mental habit. We can think of habits as learned patterns that we have done so many times that we no longer have to think about them. They are initially selected with consideration, but they ultimately become automatic.

### Lms popular lecture series 2012, attempting to model the

Mathematics’ widespread usefulness and effectiveness stem not only from mastering specific skills, subjects, and techniques, but also from cultivating the ways of thinking—the habits of mind—that allow the results to be produced. — Habits of Mind: An Guiding Concept for Mathematics Education, by Al Cuoco, Paul Goldenberg, and June Mark

Mathematical habits of mind (MHoM) are specialized ways of addressing mathematical problems and thinking about mathematical principles that mimic the methods used by mathematicians, according to our research. Teachers will add parsimony and coherence to their mathematical thought and work with students by understanding the MHoM and general purpose tools that underpin the different topics and techniques of secondary mathematics material. In this sense, we see MHoM as an essential part of mathematical knowledge for teaching, i.e., the knowledge needed to do the work of teaching mathematics.

We’ve narrowed our emphasis on algebraic habits of mind at this time, and we’ve narrowed it down to three categories of mathematical habits that we believe are especially important to secondary teachers:

### Computational thinking as habits of mind for mathematical

Paul Goldenberg, June Mark, and their colleagues examine how our students think about mathematics in Making Sense of Algebra. They look at five “Habits of Mind” that reflect on how proficient students think rather than just the effects of mathematical thought.

The ability to solve new and unexpected problems necessitates mastery not only of the outcomes of mathematical thought (the well-known facts and procedures), but also of the methods by which mathematically proficient people think. This is particularly true now that our economy is becoming increasingly reliant on fields that involve math.

In reality, as technology, economics, suppliers, legislation, and other factors shift, even conventional businesses—not only high-tech start-ups—frequently face brand-new problems to solve, problems for which no process, formula, or technique has yet been devised. When the real world throws a dilemma at us, it never asks what chapter we just finished learning!

The intrepid readiness to solve issues with just the information one has or can find and without a pre-learned solution method—the I-can-puzzle-it-out disposition—is an effective starting point. Other mental habits that enable one to define problems (and solutions) precisely, to subdivide and explore problems by presenting new and related problems, and to “play” (concretely or with thought experiments) to gain experience and insights from which some regularity or structure can be extracted are also needed for mathematical proficiency. There are also other mental habits to remember, such as the willingness to seek and express underlying framework that could connect new problems to previously solved problems, the capacity to select approaches strategically and flexibly, and so on.