Let’s Talk About Math Talk

Justifications and Questioning Strategies

I was starting to get worried that my focus on justifications for the year was getting away from me. Then I remembered a key element in nearly every day of my class — math talk (that is, productive conversations about math, including justifications). When I initially researched mathematical justifications, I read a number of articles about math talk, and I realized that this is the most common form of justifications in my classes. I am constantly asking students to explain their reasoning or investigate why their conclusion is true. Throughout the year, I have tried to elevate the quality of these justifications through my selection of problems and questioning techniques.

Selecting the Problems/Purpose

This year I have been attempting to focus on problems that allow students to make a variety of connections to the material. I wanted problems with a low floor but high ceiling that allow students to be creative and embrace a variety of methods. A fellow blogger and math educator Junaid Mubeen recently described this as the Goldilocks principle while describing a breakthrough in his doctoral research.

“Too easy and you lose any sense of challenge and engagement (worse still, you can induce complacency). Too difficult and you risk shattering a child’s confidence… But when the problem is pitched at just the right level — where it stretches the child’s knowledge, as well as their imagination — they will soar. It is the Goldilocks principle for maths educators.” — Junaid Mubeen

Once you have a worthy problem it is time to think about what you want students to understand and, better yet, justify. The goal is for students to move beyond the surface appearance of a problem to instead discuss the purpose of the problem, which is where the real beauty of justification lies. When I think about the justifications I want students discussing, the topics usually fall into one of two categories:

• Explaining Why — Why is this true? Why is it important? Why are you using that process?
• Making Connections — How does this connect to the bigger picture? How does this example relate to the target for today? How does the algebra relate to the graph? (I’ve actually dedicated an entire blog post to these connections here.)

Asking Questions

Now you have these great problems and justification topics, it’s time to facilitate the ever-important math talk! These discussions do not always come naturally, so I try to have questions ready to go. Questions are one of our most powerful tools as teachers. Students naturally want to stop when they have an answer. As GMWP teacher Mark Childs explained in his post, they receive their dopamine reward when they find an answer. Our job is to push them beyond the answer to the reasoning within. We want to focus them on the important aspects within the problem instead of funneling them to a correct answer. After reading Questioning Our Patterns of Questioning by Herbal-Eisemann and Breyfogle last year I have worked on phrasing my questions in an open yet focused manner that allows students to progress through the problem while still putting the direction of the process in their hands. I am by no means an expert at asking these great questions, but I have used them to get students talking about the connections and “why” element behind their work.

An even greater goal within the realm of math talk is to help students ask their own justification questions. Once in awhile I hear students ask questions like, “Why did you do that?” or “What does this have to do with the equation?” Although it is not yet the norm, I would love the hear students ask these types of questions without me. Not only do students feel more connected to the material when they are investigating their own question, but they are also attempting to think deeper and explore justifications without being prompted. There are many resources available on how to foster these questioning skills, but I haven’t tried any particular methods yet. There is always more to work on in the teaching world!

As I go into the last quarter of the school year, I can take a minute to enjoy the accomplishments I have made with justifications. Even though they are not always written, the power of justifying through math talk in my classroom has made a visible impact on the problem solving taking place in my room this year.