Doing Mathematics (Part 1): A Thinking Classroom
How the "Building Thinking Classrooms" model helps students learn.
(The following is adapted from a talk I gave with Dr. Patrick Eggleton at ACMS.)
Welcome! This article is the first of a three part series I’m writing on Dr. Peter Liljedahl’s “Building Thinking Classrooms.” Here, I’m going to discuss the foundations of the framework and why it’s good for student learning. Next time, I’ll give an example of a thinking task that my collaborator Dr. Eggleton used in his class to flesh these ideas a bit more. In the last part of this series, I’ll discuss how this framework can help promote virtue in students.
What comes to mind when you think of “doing mathematics”?
I think about struggling through my undergraduate geometry class - standing in front of a chalkboard at an unreasonable hour with two of my classmates and working hard on unfamiliar proofs. Or sitting at a coffee shop with my study group, trying to decipher the problems our abstract algebra professor had assigned. These were some of my hardest classes, but the reward that came with finally solving a problem was satisfying in a way that’s hard to describe - it was as if I understood the universe a little bit better.
Many of us who have worked on upper-division maths are familiar with this feeling - “insightful beauty”, as Francis Su calls it - yet it is seriously lacking from our math classrooms. As harsh as it sounds, I rarely (if ever) “did” mathematics in any of my mathematics classrooms. As a teacher myself, I’m sympathetic to this - the demands of “the curriculum” can cause teachers to feel like there’s little room for this level of understanding in a classroom. We are often so concerned with making sure our students have the proper set of skills that we miss the forest for the trees.
What I hope to present to you today is the foundation of a framework that I have found incredibly useful in bridging this gap - the gap between practical skills and beautiful insight.
The book, "Building Thinking Classrooms in Mathematics" was published in October of 2020 and research for the "Thinking Classrooms" framework goes back to 2005. This research was centered on one key question: How can we get students thinking in mathematics classrooms? Researchers questioned everything: from how students are assessed to how the furniture is arranged - and the results that provided the most thinking were condensed into the fourteen practices contained in the book.
There are three main aspects that characterize a “thinking classroom”: what type of tasks we use, how we form collaborative groups, and where students work. Though these are only three of the fourteen practices, my hope is that discussing them will create a good picture of what this framework is trying to accomplish.
What types of tasks do we use?
In order to promote student thinking, we have to give thinking tasks. Educators often recoil at wording like this - how am I supposed to find tasks that both require creative problem solving and teach the objectives students need to learn? Fortunately, we can easily turn a normal set of problems into a thinking task simply by presenting it in a thoughtful manner. We want students to move from what they know to what they don’t know. A simple example of this is teaching factoring trinomials, for example:
Think back - how did you learn to solve problems like this? If you learned like I did, it was a list of rules “the two numbers in the solution add to the middle term and multiply to equal the last term” “if the middle term is positive and the constant is negative, the solution will have a plus and a minus” (not to mention the convoluted methods used to factor trinomials when the leading coefficient is not equal to 1).
If you didn’t follow that, you’re not alone. Being taught factoring in this way did not give me the deep understanding I discussed earlier. Although I learned the patterns of factoring, I didn’t have a feel for how they were developed - I was just fortunate to have a good enough memory to memorize the steps and pass the test. As someone who wants his students to understand the language the universe was written in, this is not acceptable!
How could we teach a concept like this in a way that helps students engage with the underlying principles? To start, we must anchor it in existing knowledge. For example, consider the reverse (the answer to a factoring problem): multiplying binomials. By the time factoring is introduced most students have been taught to do this. If they have a strong understanding of multiplying binomials, it is a short step to give them a trinomial and ask “what two binomials would make this trinomial?” All of the “rules”, then, find their proper place in patterns that are observed, not rigid structures that confine. Below is a sequence that could be given to students that would help them build knowledge themselves:
If you are teaching math, you are probably already giving your students “problem strings” like this. If we thoughtfully present these problems to students, we have a better chance of helping them build a secure foundation that won’t collapse when harder concepts are introduced.
(Of course, this is not the only way that we can teach students - I’ll give an example of a more conceptual “thinking task” in the next article. )
How do we form collaborative groups?
In order to create accountability for every student, we form visibly random groups.
The logic behind this is simple: when groups are random, students are more likely to take responsibility for their own learning, rather than depending on others to do the thinking for them. If we (as educators) intentionally choose groups for our students, they will think that there is a reason they were paired with so-and-so, essentially meaning we’ve assigned our task to the “strongest” student in each group. Similarly, if students choose the groups themselves, they will depend on their “smarter” friends to carry them through the problem.
Working in randomly assigned groups provides support for the students who are struggling, accountability for the ones who would otherwise slack off, and an opportunity to teach for students who already “get it.” This also provides relief for the instructor - turning a chaotic mass of thirty students into a (slightly) more manageable ten groups (three per group, the research found, is the gold standard).
(There are many other benefits to assigning group work in this way, but they have less to do with student learning and more to developing our student’s character. I’ll discuss this in my third article.)
Where do students work?
In order to optimize student engagement, we use vertical, non-permanent surfaces - a fancy way of saying “whiteboards.” There are three main advantages to doing this - student thinking becomes visible, mistakes are easy to erase, and students can learn from other groups.
It is not easy to make students’ thinking “visible”. The beauty of whiteboard work (especially work that’s done in groups) is that, at a glance, the teacher can see what all students in the room are doing. It is hard enough to see the work of just one student when they are working head down with pen-and-paper, much less the entire class. There have been many times when, after a lesson, I look over student work and see the same mistake repeated over and over again!
Part of what makes student thinking visible (not just their work) is the ease with which mistakes can be rectified on the whiteboard. Because students can erase their mistakes they are much more likely to try, which is why (researchers found) the “non-permanent” part is so important. When we see how students attempt to solve a problem, we can see what they think (regardless of if it’s right or wrong).
A third benefit of having upright surfaces is that students can see the work of their peers. This helps students not have to rely solely on the instructor for help - instead, other student work is a valuable (and accessible) resource for each group. Students are always concerned about copying (meaning: they’ll ask you if they can’t just copy, or they’ll point out other groups who did), but this is just one more opportunity for us as instructors to point out that explaining the answer is more important that arriving at it. Requiring a student to walk you through the problem is powerful, not only for their learning, but for the learning of everyone in their group.
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These three practices blend together in elegant ways to create a classroom environment where students are encouraged to think in a way that increases their learning and engagement. In the next article, I’ll expound on the application of these practices by looking at a specific example of classroom implementation.