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What college mathematics topics can be taught in such a way that causes students to physically move around?

Background: Recently, it was suggested to me by a faculty member (from another academic department) who visited my class that there are positive effects from having students physically move around during class. So, I wonder what mathematics topics are taught in such a way, and what are the students actually doing? Is this movement connected to the math they are learning, and do you find it beneficial?

While I could just ask my students to "stand up and stretch" halfway through class, I would not classify that as being directly tied to learning a math concept.

I am looking for things you have done (or would like to do) to this end.

Note: I would most benefit from examples at the community college level (arithmetic through calculus), but I'm sure this is practiced at every level of math.

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    $\begingroup$ jumping jacks, pushups, I hear even merely walking works. Put lectures on head sets for students, and send your students out for a walk why they listen. $\endgroup$ – amWhy Nov 23 '19 at 18:16
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    $\begingroup$ @Namaste If you are serious about this idea, consider turning it into an answer. $\endgroup$ – Nick C Nov 23 '19 at 18:20
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    $\begingroup$ Just a half-baked idea: Could you perform group theory with -- well, groups of people? Shuffle them around to show symmetry and stuff. $\endgroup$ – Peter - Reinstate Monica Nov 24 '19 at 22:19
  • $\begingroup$ @NickC Students can practice counting by counting the number of jumping jacks they perform, or the number of pushups. Since such counting invariably involves natural numbers, you could extend this to, e.g., elementary number theory: "If you just completed 15 push ups, what would be the number of pushups, mod 7?" You can get really creative in this way. $\endgroup$ – amWhy Dec 7 '19 at 23:03

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To start things off, some moments my students move around during class:

  • I have given students tape measures and had them determine how much they would spend at the paint store if they wanted to paint the walls and ceiling of the lecture room (while projecting two images on the overhead: one of a paint can label, showing the number of square feet per gallon; and another of an ad from the paint store, showing the cost per gallon). Many of my beginning algebra students do not have a good sense of size when various units are used (particularly metric), so actually measuring the room makes them directly confront a length like 18.3 meters (they must walk it).
  • I have drawn a chalk line outside, labeled with its length, and had students time each other walking the length of the line. Then they use their personal walking speed to determine the length of a much longer, unlabeled chalk line, as well as the area of a large concrete quad area. Again, most students don’t know things like how fast they walk, so they learn something about themselves. It becomes an easy story for later examples where we talk about speed: “Remember when you walked outside?”
  • I have given students tape measures and mirrors to have them find unknown heights (of trees or buildings) by using shadows or reflections (similar triangles). Students get to choose an object to indirectly measure, and they must work together to make sure things are set up properly. [e.g. If a mirror is used, is it sitting flat on the ground?]
  • I very often start class by having students write solutions to homework problems on the chalkboards around the room. This is mainly a way to get the writing process finished quickly, so we can discuss homework and I can give feedback on notation. Their physical movement is incidental, but students have given me positive feedback on the practice.
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    $\begingroup$ @chepner Note the community-colleges tag on this question. There, we teach everything from basic arithmetic through linear algebra, so there is a lot of room for folks here to tailor their answers. Units, rates of change, similar triangles -- all found in basic algebra (even when taught at a community college). $\endgroup$ – Nick C Nov 24 '19 at 14:50
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    $\begingroup$ I disagree with the notion that being taught at a community college makes something a college-mathematic topic. $\endgroup$ – chepner Nov 24 '19 at 16:03
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    $\begingroup$ @chepner Community colleges teach the math that the students in the community need. Many students who graduate high school do not know any algebra at all, so the community college teaches basic algebra. Many students who graduate high school do not know any differential equations at all, so the community college teaches an intro to differential equations. If any of these things make you cry, you should think seriously about why that is the case. $\endgroup$ – Chris Cunningham Nov 24 '19 at 17:34
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    $\begingroup$ I know exactly why that is the case: basic algebra should have been learned in high school as a prerequisite for college. $\endgroup$ – chepner Nov 24 '19 at 17:39
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    $\begingroup$ @Chronocidal No, chepner's position is that students who do not know certain things should not be admitted to college. This is off-topic at best and toxic at worst, so must be challenged. Community colleges teach calculus, differential equations, and linear algebra courses at the same level of rigor as the intro courses at universities (maybe you also consider these to be "high school level!"), but also teach arithmetic and basic algebra (and do not award college credit when doing so). Nothing about this should cause anyone to cry. $\endgroup$ – Chris Cunningham Nov 25 '19 at 17:47
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The Hungarian Quicksort Dance demonstrates a computer science algorithm with dance. It's pretty advanced, but the idea is to have your students physically act out algorithms.

Perhaps something similar can be done with a number line. Line your students up and have one walk down the line to demonstrate addition and subtraction, or take big steps for multiplication. Demonstrate complex numbers by having them move perpendicular to the line.

Geometry is particularly good, get out some chalk and string and physically work it out on the floor. Surveyor and navigation techniques serve well. For example, roll a hoop one revolution and measure the resulting distance to show how the radius, circumference, and pi relate. Or demonstrate its utility for navigation with basic triangulation to determine the distance to an object. Or demonstrate mathematical proofs with geometry, like why zero is so perilous.

Calculus is the mathematics of change and motion, and realizing this (sadly post university) gave me a new appreciation for the utility of calculus. Demonstrate calculus with some change and motion. I can't think of one off the top of my head, but you might find inspiration in A Tour of the Calculus, the flowery writing reviews complain about is an advantage here.

Numberphile occasionally does physical exercises to demonstrate geometry. For example, calculating Pi with Pies. You don't have to use pies, any set of equal size circles. Or higher dimensional geometry with darts. Or math-based card tricks.

If you're discussing graph theory, physically make the graph. For big demonstrations I've used Tinker Toys to build the graph and Post-It Notes to label them. For finer ones Post-It Notes on whiteboard (nodes) and drawing lines between with marker works.

The physicality of the famous Numberphile butcher paper helps with engagement. Rather than having students do exercises at their seats, you can distribute butcher paper and markers to groups of students to gather together and work out exercises together. In my own classes I've taped the paper to the wall so students have to physically get up, move around, stand, and talk to each other. Using colored markers makes it more enjoyable. And it allows me to less intrusively keep an eye on how each group is progressing.

Here are some folks to get inspiration from.

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    $\begingroup$ I've illustrated sorting algorithms by having students line up and execute, e.g., bubble sort. $\endgroup$ – Joseph O'Rourke Nov 24 '19 at 1:58
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    $\begingroup$ This is like the tenth time I've seen the Hungarian Quicksort Dance. Every time I expect that it will be multithreaded and every time my day is ruined because it isn't. $\endgroup$ – Matthew Daly Nov 24 '19 at 18:00
  • $\begingroup$ The Hungarian quicksort dance is fun to watch at 2x speed. Thanks! $\endgroup$ – Rusty Core Nov 27 '19 at 1:20
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A technique I use a lot is to use software to put students in random groups, then have them do active learning activities in those groups. These activities are sometimes the "conceptest" technique created by Mazur, or think-pair-share. My set of conceptest activities for freshman calc are here (click through to "active learning resources").

The conceptest technique involves showing a conceptual, multiple-choice question. Students think silently for a minute or two, then vote. (I have them hold up 1 finger for A, 2 for B, etc.) If there is a clear consensus for the right answer, we discuss it briefly and move on. If not, then students discuss it in groups. The theory is that the right answer is supposed to "win" in debate. Then we vote again.

Although this technique can be done without having students get up and move, I find that having them move around like this helps a lot. If I don't assign groups, then inevitably many students will not participate in discussion. If I try to assign permanent groups, then we get problems because of absences.

In this technique, often there will be situations in which, e.g., everyone in group 2 initially votes for C, while everyone in group 3 votes for A. In this situation, I swap some students between groups so that each group will have something to debate.

If I'm doing several of these questions, I usually also re-randomize the groups between questions. This is optional, but it helps, for example, with situations where one group is composed entirely of weak students. That group doesn't have to languish and keep failing.

The reasons for doing all this don't necessarily have anything to do with getting blood flowing or reducing boredom, but it certainly also has that effect, and students seem to appreciate that. I think it also creates a nice social atmosphere in the class, because everyone knows everyone else, and they all know they need to maintain good relations.

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  • $\begingroup$ @user1527: The forming of the groups, reassigning of students between groups, and forming of new groups has them walking around the classroom in order to gather with the group they're working with. $\endgroup$ – Ben Crowell Nov 26 '19 at 3:39
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To build intuition for the Cartesian plane, assign axes to the room (or to students, depending on how many you have and how they're arranged in the room). Then you can do things like:

  • Stand up if your $x$-coordinate is 0, 1, 2 etc.
  • Stand up if your $y$-coordinate is less than five.
  • Stand up if your $y$-coordinate is less than or equal to five.
  • Stand up if your $y$-coordinate equals your $x$-coordinate.
  • Everyone stand somewhere in the room so that your $y$-coordinate will be equal to twice your $x$-coordinate.
  • Everyone stand somewhere in the room so that your $y$-coordinate is equal to twice your $x$-coordinate, plus 2.

You can of course extend this to more complex equations, inequalities, loci etc. I find students often confuse $y=k$ and $x=k$ in terms of which produces a horizontal versus vertical line; this makes that quite intuitive.

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My favourite is addition and multiplication by walking. This is my go-to activity when I talk at schools.

$+2$ is two steps forward. Then we get to negative numbers, and $-3$ is three steps back. Multiplication scales and rotates- so multiplication by $-1$ turns you backwards (or rotates your "true north"), and $\times 2$ does something twice. Now we can do $(3 \times -1) + 4$.

Next, complex numbers. $+ i$ is a step to the left. $\times i$ is a $\pi/\,2$ rotation.

Very quickly, kindergarten children can do $(((3 \times i) + 4) \times i) - 3$. But they will experience it as a little walk around the classroom.

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One activity I've seen done at the high school level is a scavenger hunt. You come up with a group of questions with numeric answers, and then you print out a bunch of papers to hang around the room (or hallway or wherever). One paper has the answer to #$1$ and the question to #$2$, one has the answer to #$2$ and the question to #$3$, all the way around to one having the answer to #$N$ and the question to #$1$.

Here is a sample of two such cards from an equation-solving scavenger hunt where you'd solve the equation on the first card, see that the answer was $x=9$, and then search for the second card to find the next equation to solve.



The students are given a sheet of paper where they have to list the card they started and and where they went from there. The cards have a cycle to them, so students can start at any point in the cycle and still get the entire activity. Grading it is pretty much a breeze, because you're just checking to make sure that they visited all the cards in the correct order (although you could also obviously give them a longer sheet to fill out where they would have to show all their work). The activity also has a great separate formative assessment feel to it, because you can see the questions where there are larger crowds of students and understand that those are the topics you need to address more in the review.

This is the kind of activity that makes teacherspayteachers.com such a fabulous resource. For most math skills, someone has already gone through the effort of doing the layout for all the cards and it's worth a buck or two to save yourself the same effort. (Props to Robert Duncan for making the free hunt that the two above cards came from.) You might have to make your own calculus-based activities, but if you start with lower-level skills you can get a feel for the general format first.

Hope this helps! Students generally seem to enjoy getting up and seeing the room from different angles, and I don't think I've ever heard of a teacher who regretted it as an activity every now and then.

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  • $\begingroup$ Matthew, I reduced the image sizes so they fit side-by-side on a reasonably wide page. If it's not to your liking, roll back. $\endgroup$ – Joseph O'Rourke Nov 23 '19 at 14:08
  • $\begingroup$ @JosephO'Rourke That's brilliant, thank you! I've favorited this post to remind me how to keep SE from auto-sizing. $\endgroup$ – Matthew Daly Nov 23 '19 at 14:11
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    $\begingroup$ I've used something like this on a review day for calculus. There are hard derivative questions with multiple choice answers. Your choice leads you to another problem. The set has 20 signs, with 4 different circuits of 5 problems each. My colleague made it and shares with us. Students participate much more than they do for sit-down review activities. $\endgroup$ – Sue VanHattum Nov 24 '19 at 17:50
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    $\begingroup$ @MatthewDaly as an alternative, appending an m to the image url also makes them smaller: https://i.stack.imgur.com/7g27P.png -> https://i.stack.imgur.com/7g27Pm.png $\endgroup$ – JAD Nov 25 '19 at 12:28
  • $\begingroup$ Please don't do this. Among the students who can calculate the answers, those who are physically fitter will physically crowd round the locations first, and the bullies will try to stop their victims seeing the questions. $\endgroup$ – Rosie F Nov 26 '19 at 8:55
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How about drawing an ellipse, a parabola, and a hyperbola, using string and a straightedge:


         
         
         
          Snapshots from MathLapse: "pin-and-string conics."


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I took a small math class over the summer (about 11 people) where instead of lecturing the whole time (as is customary for many college math classes), the professor gave us some problems once in a while that we would do in groups. The act of forming groups, talking, and then explaining the solutions to the professor (at the group) I think allowed for a lot more interaction/movement than most math lectures. Also, if you are in the position where you do have time to have students come to the board to present solutions to groupwork/homework, that would be even better.

I think the professor's insistence in having us work together in class while the professor made her way around the room made such a strong impression on me that if/when I become a professor I will find some way to carve time to do those sorts of things into the curriculum.

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Take a look at Computer Science Unplugged. They have a well curated list of activities ready to use with students, quite a few of which include physical activity. You might need to adjust them a bit for older audience, but they are definitely worth the attention.

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For example when explaining the properties of a perpendicular bisector of a segment, you can take the students to a hall, place 2 sheets on the ground (that will be our segment) and then give students a measuring tape and ask them to find a spot that is equidistant from both extremities of the segment (the 2 sheets) and another student find another spot and si on... they will understand how every point on the perpendicular bisector is equidistant from the extremities of the segment, that it is a straight line, passes through the midpoint, ....

By a similar manner you can make them draw some conics and circles.

Another thing you can do is putting cardboards in the corners of your class, each with some questions on it (say 3). Divide the class into 4 groups (each group of 3 for our example) and a student from each group goes up to the carboard to solve one question and come back high-five another group member that will go up to solve a second question, ...

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Teaching the addition and subtraction of negative numbers by walking along a number line.

The idea of the exercise is that you draw out a number line on the floor, and have your students stand next to it on the zero position, facing the positive direction. You then have them calculate additions and subtractions by walking along the line; whenever they encounter a "-" symbol, they turn around to face the other way before walking and then turn around to face the positive direction again once they stop.

So, for instance, if you tell the students to work out "5+-3--2", they'd take five steps forward, turn around, take three steps forward, turn around to face the positive direction, then turn around twice and walk two steps forward, so that they're at the position on the number line corresponding to "4".

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  • $\begingroup$ @user1527 Well, the idea is to use it as an exercise to teach kids how the addition and subtraction of negative numbers works, in a way that they might find more intuitive. I think complicating things by adding in parentheses might obscure that - you'd have to perform one computation, then use the result of that to perform the next computation, I think. $\endgroup$ – nick012000 Nov 27 '19 at 5:55
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w.r.t. "Hungarian Quicksort Dance" - mentioned by @Schwern in his answer - Stephenson's book Anathem has a section where a group of physicists do the following at a sort of community "open house":

Three fraas and two suurs sang a five-part motet while twelve others milled around in front of them. Actually they weren't milling; it just looked that way from where we sat. Each one of them represented an upper or lower index in a theorical equation inolving certain tensors and a metric. As they moved to and fro, crossing over one another's paths and exchanging places wile traversing in front of the high table, they were acting out a calculation on the curvature of a four-dimensional manifold, involving various steps of symmetrization, antisymmetrization, and raising and lowering of indices. Seen from above by someone who didn't know any theorics, it would have looked like a country dance.

I've always wanted to see it.

(In his book the word "theorics" means "physics".)

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Graph traversals

Many students do a lot of graph doodling in discrete mathematics courses, at least at first before they master the theoretical techniques. Instead of having your students draw the graphs on paper, have them lay them out on the floor or ground. Having access to a gym or an outdoor area helps, but isn't truly necessary if you move desks and chairs out of the way. Students then walk along various paths to simulate various traversals. They can also drop "bread crumb" tokens along the way to track where they have been.

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As you say College-level: Simulate field biologists' jobs, for any statistics class (or for data input for mathematical modelling). Basically any physically spaced situation you can recreate, and choose a mathematical level appropriate for your group (how science-heavy is your course?; US college = university; UK college = 16--18y olds; etc).

From my background: a capture-mark-release-recapture methodology to estimate population properties (popsize, sexratio, individual characteristics, clusters/subgroups, ... ): Say you have 10 students, split them in two groups. Space a 10x10 nontransparent containers in the room, each representing a search area. They contain whatever you're trying to study --- marbles of 3-4colours and 2-3 sizes, or even just text. In the first phase, half the students get to each sample 10 random containers and mark the contents -- which is like ringing birds, or tagging any animal, -- followed by a second phase where the other half of students is in charge of randomizing the contents across boxes (should they stay in clusters? should they change independently or are some 'paired for life'? do some stay together? With 100 containers and 50 red balls, 30 small blue balls, 10 bigger blue balls, how many containers expected empty? the second group knows these numbers, the first is trying to estimate them; the second group can calculate the expected numbers the first group will find). Then third phase 1st group students sample another 10 boxes each (if the students capture 24 red first round, and 26 second round then you expect 12 are marked). So it's all confidence intervals and Student-tests etc.

Containers can be simple origami boxes from another lesson (harder to make them look uniform); or nontransparent paper cups; or ... .

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