What are the traits and habits of someone who is good at math? If you were to ask this question to ten people you know, you would likely get a variety of responses. Someone who can do math in their head. Someone who thinks logically. Someone who checks their receipt at the store. Someone who enjoys playing chess and doing puzzles. Someone who doesn't pull out a calculator to figure out their tip.

None of these are bad answers in their own right. But for too long we have accepted that the only definition of what it means to be good at math is fast and efficient calculating. We are impressed by people who can get the right answer in math class, who are good at memorizing formulas, and who finish all the problems on the “mad minutes” we did in elementary school.

### The Problem

When these beliefs persist, students are excluded from opportunities for authentic mathematical sense making and we perpetuate, rather than disrupt, inequity.

There are two main problems when we define math competence in this way.

It’s not true to the mathematical discipline.

Students who don’t excel according to this narrow definition lack a sense of belonging in the math classroom and are made to assume their contributions are not valuable.

By broadening what it means to know and do mathematics, we are not only offering a more accurate view of the mathematical discipline, but we are increasing the participation and engagement of an entire group of students whose ideas were previously seen as informal, inferior, or even invalid. Why would a student engage in an activity when they have been made to believe they will not be good at it or don’t feel like they can even contribute? Why would a student push past confusion and persist at a challenging problem if they believe it will only solidify what they already know about their mathematical competence?

### Ideas for Moving Forward

It seems that the solution is to change the experience of math class for our students and provide a more honest picture of what it actually means to do math. But how do we do this? First, we have to redefine what it means to do math and who does math. Mathematical competence goes beyond fast and accurate calculations and the ability to execute algorithmic procedures. Furthermore, all students can, in the right environment, make sense of mathematics and make important contributions to the group. This is something we have to believe as teachers, and emphasize to our students. Researcher Ilana Horn at Vanderbilt University has written about these beliefs and messages about mathematical competence in depth in her book "Motivated" (I highly recommend this read!). Looking at the key contributions of mathematicians like Fermat, Andrew Wiles, and Poincaré (whose impact on the field of mathematics was not related to their fast and accurate calculations), she identified the following list of significant mathematical competencies:

Posing interesting questions

Making astute connections

Representing ideas clearly

Developing logical explanations

Working systematically and

Extending ideas

To these I would add:

Visualizing

Estimating

Determining the reasonableness of an answer

Keeping track of multiple parts of a problem

Understanding and building on someone else’s ideas

Justifying an answer and

Sharing incomplete ideas (taking risks)

When we promote this much broader list of what it means to do mathematics, we help foster our students’ mathematical identities and in turn increase their participation, allowing for greater access to deep learning. The goal is to alter and mix expectations of competence with activities that allow students to demonstrate a variety of “smarts”. On any given day we can’t always predict who’s going to excel at an activity and who’s going to struggle. These are always changing. As a result, we help foster group interdependence, because there’s no longer one student who’s always the strong student and one student who’s always the weak student. There is now a genuine need to explain one’s thinking and pay attention to a classmate’s thinking.

### Who is a "Doer of Mathematics?"

I heard the analogy of a “doer of mathematics” card a couple weeks ago and it has stuck with me. Here’s how I imagine the story would go. Every student holds on to cards that describe their identity. One of the cards says “doer of mathematics.” When a student struggles during class, he walks up to the teacher with his head hung low and returns the card. He believes that struggling negates his identity as a doer of mathematics. The teacher smiles and hands it back. “Quite the opposite. You’re a more honest owner of this card than ever before.”

What the teacher is trying to say is that wrestling with math, persevering even when something doesn’t immediately make sense, not knowing the answer--these are the marks of what it means to truly do mathematics.

So what about you? How do students earn the “doer of mathematics” card in your classroom? What makes them believe they have to return it to you?

### Next Steps to Try

Take inventory of your class. What kinds of mathematical competence are you valuing? How do you know?

Publicly highlight a student contribution that highlights one of the competencies from the lists above. Point out how that contribution was authentically mathematical.

Ask your students to describe what it means to be good at math either out loud or in writing. Use their answers to launch a discussion about what math is and who does math.

Excellent post, thank you! I shared with my teaching colleagues.

I absolutely love these articles. Thanks for sharing and I'm looking forward to future articles. They certainly make me re-think my actions and mindset in the classroom.