“The times they are a-changin’” sang Bob Dylan, though I don’t think he was trying to communicate anything about math education with this song. It is true, however, that much has changed, and continues to change, when we look at the math education landscape, especially when we consider improvements in technology; our understanding of learning and the brain; and the increased need for people who can think critically, analytically, and creatively.
This list is hardly exhaustive, but here are six ways to engage students in math as the times continue a-changin’, along with some examples of how to execute each. As you read about each one, take note of the fact that these methods, like the discipline of math itself, do not exist in their own silos!
1. Ask students to engage with real things.
2. Make math interdisciplinary.
3. Provide opportunities for students to make their own discoveries.
4. Have students create their own problems.
5. Allow students to use what they know.
6. Emphasize process, not right or wrong answers.
1. Ask students to engage with real things. Concepts in math get increasingly abstract as one progresses through the discipline, and forcing students to solve problems that deal with meaningless quantities and concepts just compounds the issues. We’ve all seen the arrg! e-card that illustrates many students’ experience with math word problems:
This image is obviously an exaggeration, but we can reduce cognitive load for students by introducing questions that apply more directly to their lives. Look for ways to incorporate situations that students might encounter regularly and use numbers that are accessible. The best problems help students recognize the power and utility of math in ways they least expect it.
Here’s an example: A few years ago, as part of bringing a building up to code during a major renovation, we had to install a wheelchair lift in order to allow those with mobility impairments to move between the tiered-levels of a public gathering space; the lift covers a vertical distance of no more than two feet. Students balked at this new piece of equipment and questioned the decision to install the lift instead of a ramp. Therein lay a valuable problem for our students to solve: “What would it take to install a wheelchair-accessible ramp in this space?” By having students take measurements, research ADA guidelines for ramps, and perform the necessary calculations, students eventually recognized that a ramp simply would not have fit in the available space.
2. Make math interdisciplinary. Math should not exist in a silo. Find ways to tie in other subjects and allow students’ interests and strengths in other disciplines to spill into math class. Here are some examples:
a) Students who love language and words will enjoy considering the etymology of math vocabulary, much of which is not straightforward. For example, we all think of “quad” as meaning “four,” so what does “quadratic” have to do with “squared (power of two)”? Or why do we use the word “exponent” to refer to the small number that’s written smaller and above-and-to-the-right of another number? Or why do we use the word “radical” to refer to square roots?
b) Math possesses a fascinating history, and students should know these stories. Consider that the number zero was not always part of our number system. Or that math existed for centuries before the introduction of the Cartesian Plane. Math has also made its way into mainstream media with movies such as The Imitation Game and Hidden Figures, which highlights mathematicians’ prominent roles throughout important moments in history.
c) Have students read and write about math. One doesn’t have to look hard to find examples of math throughout literature. Some are more explicit or central to the story (such as the concept of probability in The Curious Incident of the Dog in the Nighttime) whereas others are tangential (population over time, costs associated with war, passage of time).
3. Provide opportunities for students to make their own discoveries. One of the things I find most fascinating about geometry (and much of math) is that someone, at some point in time (here’s the history theme again), had to discover the theorems that we study today. And yet, so much of math education involves telling students exactly what they need to figure out, thereby denying students the experience of discovery that makes this discipline so exciting in the first place. We need to take a few steps back and simply present students with information (numbers, data, a diagram, etc.) and simply ask, “What do you think might be true?” We need to put students in the driver’s seat and allow them to identify patterns, define their own rules, and then use what they have learned.
Early in my teaching career, I often redirected students when they “discovered” their own theorems in geometry, theorems that the textbook did not highlight. Usually, I argued, “Well, that is true and provable, but it’s not as useful and we won’t need it as often as the other theorems that the book outlines.” But then I realized that the logic is quite circular here: The theorems that the textbook authors deemed “important” were largely due to the fact that the textbook authors wrote problems that would necessitate said theorems. It finally struck me that my students’ theorems were equally valid and important if only we had a different set of problems.
This leads me to my next point…
Every year, one of my stated objectives is for my students to enhance their ability to ask good questions. Offering students the space to write their own problems definitely moves us closer to that goal, especially when we arrive at the end of the year and students have had a years’ worth of practice under their belts.
Moreover, when students create their own problems, they engage with the material on a deeper level. They have to look at existing problems to understand how the pieces fit together and they have to understand what types of information are required to ask (and solve) the question successfully.
Here’s an example: My students (even the strongest ones) are always fascinated by how quickly I can create quadratic trinomials that can be factored (especially if the quadratic coefficient is not 1). It’s not that hard! By asking students to produce ten expressions that can be factored, I really begin to see which students truly understand factoring.
5. Allow students to use what they know. This sounds obvious, but take a minute and ask yourself: how often do we stop students from using knowledge gained outside of our class? Have you ever said something along the lines of, “Well, we haven’t learned the Pythagorean Theorem yet, so let’s put that method on hold.” Or, because the textbook does not cover cover area of polygons until Chapter 11, do you refrain from asking any questions involving area, even though many students will have arrived at ninth-grade Geometry having done significant work with area in other classes?
If we prevent students from using what they already know, we put up the walls of the silo and teach students that the knowledge they learn in other places and bring to our classes does not matter. This is the exact opposite of what we are trying to accomplish.
This doesn’t mean, however, that we shouldn’t question or probe our students’ understanding of the knowledge they bring to our classes. Ask students to justify what they know, and explain when and why they would use one method versus another. Ask students to teach what they know to their classmates. These are great ways to empower students, place value on their knowledge and skills, and make them more critical consumers and users of knowledge.
6. Emphasize process, not right or wrong answers. I encounter many students who love math because “there’s always a right answer.” And there are many students who dislike math for the very same reason. Whether or not a “right answer exists,” there are ways in which we can shift the focus to the process of problem-solving, and away from the answer.
a) Ask questions to which you don’t know the answer. If I don’t know the answer, I can’t acknowledge whether a student’s solution is correct. The best I can do is to rely on the student’s reasoning and ability to communicate her thought process. This practice also forces me to think about how I would solve the problem, leading to opportunities for me to model a logical and sequential problem-solving process.
I love using Fermi-type problems for this very purpose, though keeping in line with bullet-point #1 above, ask questions about real things. Make the problem too abstract or removed from students’ daily lives, and we lose a valuable learning opportunity. So, don’t ask “What was the total revenue of pots and pans sold in the United States last year?” Instead, ask “If we were to fill your house with marshmallows, how much would all of those marshmallows weigh?”
b) Give students the answers! My students are often blown away when I hand out a test that already has all of the answers on it. To me, it’s the ultimate statement that I care more about the problem-solving process than I do about the answers. In this case, yes, there might be one right answer, but that does not necessarily mean that there is one right way to get the answer. I want to see their work.
c) Once students arrive at an answer, ask them to use a different method to solve the problem. This accomplishes several things: it challenges students to think about process, it offers students a way to check their own work, and it opens the door for a discussion about efficiency in one’s work as students compare the different methods.
Keeping students engaged is an important part of our roles as educators, especially in a discipline like math where students seem to arrive with strong emotions and attitudes toward the subject. This list is certainly not exhaustive! What other strategies do you use to engage students in math?