The annual NCTM conference is always one of the highlights of my year. Not only do I get to meet teachers in our community from across the country, but I get to take on the role of a learner as I attend sessions and hear from educators and researchers on topics they are passionate about. My time spent flying home from the conference is reserved for reflecting on what I learned. In this blog post I'll share my take-aways from three of my favorite sessions. Whether you were able to attend the conference or not, I hope this provides some inspiration for you as it did for me.

### 1. Secondary Students are Mathematicians! Taking Non-Canonical Student Ideas Seriously

Meghan Riling, Vanderbilt University

How we frame errors, mistakes, and misconceptions has a significant impact on what students learn about math and what math they learn.

"Every person, including students, brings creative potential to how mathematics could develop or change next."

Definitions:

Canonical idea: A mathematical definition, concept, or relationship that aligns with standard, professional mathematics. This is the kind of math that is represented in textbooks.

Non-canonical idea: A mathematical definition, concept, or relationship that a student offers in good faith but does not happen to align with standard, professional mathematics. It may or may not have mathematical validity.

In this session we looked at a few samples of students’ non-canonical ideas. The first was in response to a question that asked students what happens to the graph of a line when the “b value” of the equation y=mx+b changed. The student said that “it moves diagonally”. My favorite part of this was the prompt we were given as workshop participants. Write down 3 questions this makes you wonder about lines. The reason I loved this prompt was because of the invitation to not just correct or revise the student’s solution but to explore what the student may be noticing about lines that might actually deepen our own content knowledge. It is interesting to consider how the effect of changing the “b-value” can also be seen in the movement of the x-intercept, which could very reasonably be perceived as a diagonal movement of the line. Further, the student’s response prompted me to think about properties of parallel lines, and the fact that the distance between the two lines (as measured by the segment perpendicular to the line) stays constant. This perpendicular segment could represent the student’s conception of a diagonal movement.

##### Big take-away: Students’ non-canonical ideas can provide rich opportunities to deepen understanding of mathematical concepts for teachers and students alike. Ideas don’t have to be fully developed or even correct to be worth investigating.

### 2. Increasing Student Responsibility in a Thinking Classroom

Peter Liljedahl, author of "Building Thinking Classrooms"

Peter talked about how we can help students transition from collective knowing and doing (what students can do in groups) to individual knowing and doing. Of his 14 teaching practices, there are 4 practices that are specifically related to this topic: how we consolidate, how we take notes, how we give homework, and how we do formative assessment.

I most appreciated his take on the various types of “debriefs” that can follow a thinking task that students do in small groups. He explained that the type of task helps determine the most effective debriefing strategy. Tasks that are divergent (where each group is likely to use a different strategy) might best be discussed with a gallery walk, whereas tasks that are convergent (generating the same or very similar answers among groups) might best be consolidated with a set of new problems that allows students to notice some of the underlying structure of their work. Often in convergent tasks, student thinking isn’t really represented in their work, so while students are certainly making sense of each problem individually, they may not yet have noticed the connection between the problems, or at least not be able to articulate them well. Choosing three new problems similar but not identical to the ones they did in the task and asking students to first order them and then notice commonalities and differences, helps students attend to invariance (connections) and variance (nuance). Most importantly, he discussed the value of unpacking the structure of each problem without actually finding the answer to each problem. This helps communicate to students that process matters more than product, in a way that goes beyond just common classroom rhetoric.

##### Big take-away: The goal of a debrief is to help students understand the underlying structure of a problem, not necessarily its answer, and the style of debrief should be aligned to the task.

### 3. Top Five Reasons to Incorporate Rough Drafts and Revising in Mathematics Class

Amanda Jansen, author of "Rough Draft Math: Revising to Learn"

The concept of Rough Draft Thinking (RDT) is defined by Mandy in her 2020 book as what happens when “students share their unfinished, in-progress ideas and remain open to revising those ideas.” Revising is the process of making new connections and expanding what you think.

In this talk, Mandy connected RDT with the Rights of the Learner, a concept developed by Olga Torres and Crystal Kalinec-Craig, which essentially allows students to claim their right to make a mistake, revise their answer, and to share unfinished thinking without being judged.

While I had heard of Mandy’s work (and have her book on my shelf), I was not very well-versed with her framework before this session. I learned that RDT is not a specific classroom routine, but a layer of intentionality added on to what we already do. Normalizing the notion that our initial thinking is not our final thinking and reflecting on what caused us to change our mind is at the heart of RDT. She shared a simple protocol that we can use to encourage RDT which is to have students use the template below when working on a task.

Notice how the use of the phrase “next thinking” encourages students to think not just in terms of a first and final draft, but about many rounds of revisions. In the Rough Draft Math framework, all thinking is treated as a draft.

Mandy also suggests using the sentence frame “I used to think _______, but now I think ________” to help students feel comfortable and encouraged to continue updating and improving their ideas. Perhaps the reason I loved this talk so much is because I recently read Adam Grant’s brilliant book “Think Again.” Mandy Jensen’s session was a terrific companion piece for me, as she explores Grant’s same principles in the space of math education. As a result of Grant's book and this session, I have challenged myself to complete Jansen’s sentence frame on a weekly basis to encourage me to continue learning, growing, and revising my thinking.

##### Big take-away: Building a classroom culture where all thinking is treated as drafts, and all draft ideas are seen as having strengths that can be improved upon, is more imperative AND more practicable than I ever thought before.

A huge shout-out to all three presenters for challenging my thinking this past week with excellent sessions! So, what new ideas are YOU thinking about?

I'm thinking about trigonometry, having attended your session, Sarah. Your thoughtful consideration of the language we're using (distance, ratios, scaling) and the notion that we need to be building slowly from one view of trigonometry to the next, scaffolding intentionally, reminded me of how much I often fail to connect the dots for students before getting to "coordinate trig." Slowing that down with some pointed exploration is crucial, and I tend to zip past that too much, I think.

I did wonder (and never got to ask you in person) if in a trig class sequence you eventually share about the word origins of the names for the tangent and secant ratios, as well as for the complement-functions (co-functions). I've…