As a teacher of mathematics who focuses on inquiry-based instruction, I often get asked how I go about developing my students’ conceptual understanding. Early in my teaching career, I became very good at teaching procedures through direct instruction.
As my teaching methods have evolved, especially since embedding the math practices in every lesson, I’ve pushed myself to use inquiry-based tasks and hands-on manipulatives with my students as a way to help them develop their conceptual understanding. Yet there are still topics (such as solving inequalities with a negative coefficient) where I feel unprepared to teach in this way.
I recently got a sneak peak of Leah Alcala’s new video, Concept First, Notation Last. While Leah is probably best known on Teaching Channel for her My Favorite No and Highlighting Mistakes strategy videos, this new example of practice is a chance for us to see how Leah helps her students develop conceptual understanding in math. After watching, I felt as if some of my own questions on the topic had been answered.
Here are my 5 takeaways, which are now reminders to myself and tips for you:
Leah Alcala has the utmost belief that her students will be able to access and attempt the task at hand. In the video, you’ll hear her make the statement, “Everyone likes to learn.” It’s clear that her beliefs move students forward. Take inventory of yourself. Do you believe that ALL students in your classroom can access mathematical ideas without being told what to do and how to do it? To continue to deepen your beliefs, I recommend reading Jo Boaler’s research on how all students are capable of developing mathematical ideas.
2. Sense Making
When my teaching started to evolve into one that focused more on conceptual understanding rather than rote memorization, I started asking one question of all my students: “How/Why does your solution make sense to you?” I found that by developing justification as a norm, I could understand exactly what students believed, why they believed it, and respond with more effective questions. Moreover, in order to have those conversations, the task they were responding to had to be a rich, inquiry-based task. That meant I was no longer teaching tricks or math shortcuts. Students had to be able to construct viable arguments and critique the reasoning of others. In the video, you’ll see Leah posing carefully developed tasks alongside constant questioning, as opposed to teaching mathematical tricks. This allows her students to use sense making to come to a conclusion on their own.
From the outset of this lesson, Leah keeps the numbers easy to manipulate while the mathematics remains demanding. If the numbers become too difficult at the outset, students can get overwhelmed and will often shut down. As Leah mentions, easier numbers help give an access point to all students. Over time, you can increase to more challenging numbers while maintaining challenging thinking as well.
Conceptual ideas are not built in one day or even two. They are developed after repeated exposure to a particular mathematical idea in various contexts. Students have to struggle and resolve that struggle to internalize a concept. This process takes time. Leah mentions that the one day her class is being filmed will, by itself, not be sufficient for complete learning to occur. In my classroom, this means a learning objective may remain consistent for a week or two as we are developing the idea.
5. Multiple Representations
Leah allows her students to engage with the mathematical idea of solving inequalities through graphs, lists, and/or mathematical notation. When developing conceptual understanding, it’s imperative to give students freedom of choice in how they might potentially respond. Narrowing to one representation too often causes students to try to find the one correct path towards a solution, rather than thinking expansively and for themselves.
Since I’ve begun to focus on the development of conceptual understanding, I’ve run into some challenges. Students and parents alike want me to teach short cuts and algorithms. Yet, when I hear students talking and thinking mathematically, I’m certain that this struggle will prepare them to be risk-takers, not only in my classroom but in their daily lives, too. My hope is that this will lead to self confidence and an internal belief that when faced with any problem, they have ideas worth exploring.
I encourage you to watch this video and think about how developing conceptual understanding is both doable and essential in your classroom. I would love to read your responses in the comments section below on how you were able to take these ideas and apply them with your students!
Crystal Morey is a middle school math teacher in Enumclaw, WA. She has served on the county’s Common Core transition team, and is a teacher leader on a statewide project with Washington State’s Office of the Superintendent of Public Instruction and the Mathematics Education Collaborative. Connect with Crystal on Twitter: @TheMathDancer.