By Ralph Pantozzi

As I open my math classes each year, motion is always on my mind. Emotion, motivation, and (physical) movement derive from the same Latin root, and each is essential to students’ learning—as is a little commotion!

First, I think about how curriculum and instruction will “move” students. What kinds of feelings, interests, and passions will this course inspire? How can this math class promote creativity and positive dispositions, rather than move students to tears (of the non-joyous kind)?

As I plan the syllabus, I think about motivation(s). Why are students taking the course? Is it designed to answer questions they have on their minds? Math can be a welcome escape from reality, but relevancy can also motivate students to engage. Similarly, does each topic serve to motivate a need to learn the next? Does my teaching motivate students to pose their own questions and be active participants in learning?

As I design lessons, I think about how physical motion is often absent from math class. Physical activity during a lesson has multiple benefits. Bodily motion helps keep the blood flowing to the brain, supporting mental activity. When students move about the room, ideas flow more easily too. The mathematics we learn often derives from or has connections to motion, from the beginnings of counting to group theory and beyond, and students deserve to experience that richness.

And what of commotion? All learning involves some perturbation of your previously held beliefs. A little actual excitement can be memorable! (Perhaps you’ve heard of Edward Burger asking students to figure out how they might remove their pants and put them back on with their ankles tied together.) An “out of the ordinary” experience has a good chance of sticking with you for the long term.

Here are some events that have moved my students to learn mathematics with joy, depth, and connection.

In calculus, the class moves—in and out—of some defined area, subject to their own individual whims. As you, the student, wanders in and out of the space, what do you notice? It seems like such a mess. But what do you wonder? Moving about motivates multiple questions about the number of humans who are in or out of the space at any moment. What is the rate of change of the number of people in or out of the space, and how does that impact the fluctuations of the number of people? What is the average number of people who were “in” over an interval of time? A logo in the middle of an athletic field, a sidewalk, an indoors space with multiple entrances/exits have served well for such an activity. A video recording of the event aids later investigation and reflection.

Recent questions from the AP Calculus program exams transport us in our imaginations: as we move around, we are fish in a lake, a crowd trying to get on an escalator, or bananas in a store being bought and restocked. At the same time, the context of motion also links us to real contemporary issues: immigration and other motions of people around the world, the rise of sea levels, incarceration rates, average hours spent on social media, the rate of spread of misinformation.

At another point in the year, students move along a line, taking a trip with “zero acceleration,” “positive acceleration,” and “negative acceleration.” For the latter, a commotion occurs as students move backwards with me 5 feet per second, then 4 feet, then 3, 2, 1, and 0, before finally moving forward in a similar fashion. We chant “positive acceleration, yes!” during the trip, working to link the physical experience with the abstract concept. What it means to “speed up” inevitably comes up as part of the discussion.

Throughout calculus, a course in the “language of change,” the conversation can be about things in our own experience that change. Watch the electricity meter at your house. Observe the time of sunrise and sunset. Pour water or sand into a leaky bucket. What do you notice as you run down the field towards the goal or speed towards the basket? Why are those cars seemingly moving so slowly when they are so far away? Watch the reflection of a light on a tile floor when you walk towards the light. Many other “field trips” are possible—try walking the harmonic series down the hallway—you can get anywhere, if you are patient enough!

In a probability lesson suitable for any age, get yourself in motion with a “random walk.” A flip of a coin will determine if you move one unit forward on a number line, or one unit back. Try doing this alone, as another person in the class flips for you. This “flip trip” is a weird experience. Not knowing where you’re headed or where you’ll end up is different from merely flipping the coin and recording the outcomes. You and your classmates will have many questions about your long-term chances and destination.

Get the whole class (or more than a few classes) to take a flip trip together. Everyone starts lined up shoulder to shoulder, each with their own set of coin flips. It’s best to record your coin flips in advance to speed the proceedings, but the commotion here is the dance of random motion that you will see around you. After everyone has made 6 steps, scattering to and fro, they have arrived at specific locations: -6, -4, -2, 0, 2, 4, or 6 units from their initial location. Then, they turn right, and march forward in individual columns to form a bell curve of the binomial distribution. It may be easier to see in this video:

Probability is an attempt to quantify uncertainty. Our human experience with probability is often personal: luck, as some have said, is “probability taken personally.” Experiments that involve tracking one’s own results within larger samples can help students look beyond the personal, and answer a question like “How far might a sample proportion be from the true population proportion?” Understanding statistical inference is part of investigating contexts from racial profiling to the occurrence of floods to the effectiveness of drug trials.

Math has a reputation for “cold logic,” topics devoid of emotion or opinion, and independence from time, place, and personality. There is another view, that the math we learn is very dependent on experience, community, and context. Over 25 years ago, I read a short report by Gary Davis, titled “What is the difference between remembering someone posting a letter and remembering the square root of 2?” He mused about the importance of grounding math learning in episodic memories of events, people and actions. Math can be “alive” in the mind, “about” something more than calculations, and full of imagery which might include actual or imagined physical actions. I remember where I was when I first read this.

Math class can be a place where episodic memories are made that complement the semantic memories of rules and procedures. Math education can place humans at the center—physically, emotionally, and cognitively. There are many “field trips” that allow your students to take journeys into their own lives and the lives of others. The feature that connects them all is the opportunity to feel recognized as humans and as mathematical thinkers. We can foster inclusion and belonging by honoring all the different places our students come from, and the range of emotions that are part of mathematics as practiced by humans.

Learners need the freedom to move about, fumble and stumble. That’s the direction I want to move in.

Ralph Pantozzi is a Presidential Awardee in 7-12 mathematics and has been a high school math teacher in New Jersey for 31 years. He works to spread innovative teaching practices through undergraduate math methods courses, NCTM, and the National Museum of Mathematics, and enjoys sharing mathematical joys with family, friends, and the public. He is currently the NCTM representative to the MAA’s Committee on the Teaching of Undergraduate Mathematics.