Mathematics thrives on the ability to imagine—but can that actually be taught in the classroom? Christof Weber says yes, and he wrote a book called Mathematical Imagining to show us how it works.
In this excerpt, you are invited into Christof’s classroom as he guides his students through an exercise called Mathematical Imagining. Read on to discover what this original routine looks like and how it can help secondary students engage with mathematical content in a very personal way, and allow you to get a glimpse into their thinking as well.
My high school students stream into class late in the afternoon, many coming from gym. They drop their bags next to their chairs and put their math materials on their tables. I switch off the lights and ask students to prepare for an exercise in Mathematical Imagining. There is a perceptible change in energy—students are familiar with this weekly routine, and they push their books, notes, and pencils out of their way and take a few focused breaths. I invite students to sit comfortably and close their eyes. Some students sit up, hands relaxed in their laps, with their eyes closed. Others choose to fold their arms on their tables and drop their heads into the spaces they’ve made, to further block out light and distractions. They rest their foreheads on their forearms and listen, expectantly. I pick up my written exercise and begin reading it out loud, slowly and clearly.
Imagine you are walking on a grassy field. You see three long strips of cloth lying in front of you in the shape of a large triangle. Stand in the middle of one of the strips of cloth with your nose and toes pointing into the triangle in front of you and stretch your arms out sideways to the left and right: they form a line above one strip—that is, parallel to and above one side of the triangle. . . .
I watch as students imagine. They are relaxed, concentrating, each one creating a personal field and triangle. I notice that, when I told them to imagine stretching their arms out sideways, a few students made barely perceptible movements in their shoulders or fluttered their fingertips a tiny bit, their physical bodies echoing what their imaginary bodies are doing inside their minds. I continue.
Now begin moving sideways along the strip of cloth, keeping your nose and toes pointing into the triangle as you sidestep in the direction of your right arm, placing one foot next to the other, until you reach a corner where two strips of cloth meet. . . .
I wonder whether students are moving sideways as I intended when I wrote the exercise or whether some students have mentally turned their bodies and walked toward the first vertex. I remind myself to explore this question in our discussions later.
Your right arm now extends beyond the figure, and your left arm is above the side of the triangle that you have just sidestepped along. Rotate slowly about your body’s axis with your arms firmly outstretched so that your left arm begins to point into the figure. Keep rotating your body until your left arm has swept out the corner and arrived above the next strip of cloth, and then stop rotating.
I am curious how students are experiencing this moment, this chance to consider an interior angle of a triangle as a dynamic rotation, rather than a static measurement. They’ll have two more chances to think about these turns.
Begin sidestepping along the triangle’s side, this time in the direction of your left arm, placing one foot next to the other, until you reach the second corner. Your left arm now extends beyond the triangle, and your right arm is above the side of the triangle that you have just walked along. Rotate again slowly so that your right arm initially points into the triangle, sweeps out the corner, and arrives parallel to and above the next side. . . .
I take a brief moment to conjure up my own imagined triangle—the one I created when I planned this exercise and can now access in an instant. Taking this moment to recollect helps me connect to the mathematics and also helps me ensure students have sufficient time to imagine their triangles in detail. After this pause, I bring them back to the start.
Continue sidestepping and rotating like this until you arrive back where you started from. . . .
I am delighted to hear some audible gasps as students return to the start. I ask my first questions.
How are you standing now?
What has happened?
I give students time to contemplate these questions individually, quietly. Asking students, “How are you standing now?” encourages them to compare which way they were facing when they started and which way they are facing when they ended up. I expect some students are mentally going around the triangle again, either deepening or modifying their original journey. Students who may have turned in different directions at the vertices, rather than making all their turns in the same direction, have their first chance to adjust their image and move around the triangle again. The question “What has happened?” encourages students to think about their mental journey around the triangle as a whole, perhaps even from a new, bird’s-eye perspective.
When I sense the class is ready, I ask the question that I always ask at the close of this routine:
What did you imagine during this exercise in Mathematical Imagining?
I wait, again, for students to consider this question, before inviting to them to “come back to class.” Students open their eyes, eager to discuss their imaginings with their peers. I invite students to talk about their imagining in pairs or threes. I walk around, listening as students talk with one another about which way they were facing at various points along their routes. To help their classmates understand their points of view, some students spontaneously make quick sketches or use gestures.
Students are eager to share and are curious about one another’s imaginings, so they listen intently. I see a few students close their eyes and revise their triangles in response to what they’ve heard from their classmates.
After a few minutes, I ask students a new question.
What is the sum of the interior angles of a triangle?
Students now have a chance to reason together about why they have made a halfturn from beginning to end. Why a half-turn, exactly? And is it exactly a half-turn? Will it always be a half-turn, regardless of the triangle? These questions help lay the mathematical foundation for the interior angles theorem.
When conversation naturally begins to trail off, I ask students to take out their journals. Sometimes, I close our imaginings without journaling, but in this case, I decide to make time for students to capture their imagining before it starts to fade because their imaginings will be foundational to our upcoming work. For example, we’ll eventually want to consider what happens when shuffling around a flat polygon with four vertices? Five? n? That discussion will be much richer if students can readily access their personal, mental triangles and modify them by adding sides and vertices. Therefore, I ask students to take a few minutes to respond to these prompts in writing before moving on to the rest of my lesson:
Record the mental images—both pictures and actions—that you imagined during this exercise.
Which of these images and actions were useful to you? Which got in your way? What revisions did you make?
Later, I’ll look closely at their writing and sketches. I might select a few examples to share and have students discuss tomorrow. Or, I might choose to have all students leave their open journals on their tables and then walk around to read and comment on other students’ work. Or, I might read them solely to inform my planning for upcoming lessons. Either way, this Mathematical Imagining gives my students a chance to create mathematics for themselves, and it helps me to see what is endlessly fascinating but not so easy to see: how my students think mathematically.
About the Author
Christof Weber is an associate professor in the School of Education at the University of Applied Sciences Northwestern Switzerland. He taught mathematics for twenty-five years at a “Gymnasium” in Switzerland, a post-compulsory public high school preparing students for tertiary education.
