Feeling the Angle Sum of a Triangle
An Introduction to Embodied Learning in Mathematics
By Hortensia Soto
Learning and the Senses
Growing up, my mom used to say “usen sus cinco sentidos” (use your five senses) when she wanted us kids to use reason or to behave in a logical manner. This phrase gained new meaning for me when I discovered embodied learning, which is a philosophy that claims that learning occurs when the meaning of what is learned is grounded in body movement (action), perception, and sensation. As such, those of us who follow this philosophy of learning create learning environments where learners engage in seeing, hearing, and feeling — three of the five senses. Let’s see what this looks like.
Walking the Sides and Angles of a Triangle
In elementary or middle school, students may discover that the sum of the interior angles of a Euclidean triangle is 180 degrees by drawing a triangle, measuring each angle, and summing up their angles to obtain something that is approximately 180 degrees. The students might also draw a triangle, color the angles different colors, tear the angles, and line them up to illustrate that the angles create a straight line, which has a measure of 180 degrees as shown in Figure 1, above.
Both of these activities extend beyond hearing the teacher say or seeing the teacher write “the sum of the interior angles of a triangle is 180 degrees.” Another way to help students discover this fact is by having them walk the sides of the triangle while traversing the interior angles. This activity gets students to move, make mistakes, learn from them, and begin to feel the mathematics.
The activity begins by creating a large triangle on the floor using any materials that are handy, such as painter’s tape. A volunteer, whom we will refer to as the walker, then stands on a vertex of the triangle and walks on a side of the triangle. The walker then rotates by the amount of the first interior angle, then walks the second side, and so forth until they get back to the first vertex. The goal of this activity is to have the students recognize that they will end up facing the opposite direction from where they started. Thus, they will have rotated 180 degrees, as illustrated in Figure 2 with the red and black arrows at vertex A.
Distinguishing Between Interior and Exterior Angles
This may sound like an easy task, but it is not at all intuitive and in fact, it can seem unnatural for learners. Learners of all ages often end up facing the same direction they started in, indicating that they rotated a full 360 degrees instead of only 180 degrees. In this section, I describe what happened when I implemented the activity virtually as part of an AIM Math Circle. The short video clip below illustrates a participant traversing the exterior angles instead of the interior angles, which occurs quite frequently because rotating by the measure of the interior angle at vertex B, requires that one walk backwards along segment BC.
The fact that the walker ends up facing the same direction can result in cognitive dissonance for everyone in the room, not just the walker. This is an opportunity for a rich discussion about what happened. For example, at this Math Circle, one participant tried to explain by saying “so you turned and each time you turned an angle,” as she gestured a turn with her hand each time that she uttered the words “turned” and “angle” as shown in the video:
A bit befuddled, she then went on to explain that the sum of the angles is 180 degrees, so that she expected the walker to be facing the other direction.
A second participant then conveyed that the walker traversed the exterior angles instead of the interior angles as she gestured the exterior and interior angles:
This realization is a great learning opportunity because in traversing the exterior angles instead of the interior angles, students also discover that the sum of the exterior angles of a triangle is 360 degrees. At this Math Circle, the walker stated, “I actually can feel the math. I can tell I am doing that [walking the exterior angle].”
Another participant also illustrated that if one extends one’s arms as one walks and goes “to the tip of the vertex, then you rotate” — as she rotated her whole body with arms extended — “and you see you are sweeping the exterior angle so you can do this action,” as she re-gestured from left to right (see Figure 3).
After the walker realized that she had walked the exterior angles, she re-walked the triangle and the interior angles as the activity was designed and illustrated on Figure 2. Besides feeling the math, she also conveyed that she enjoyed walking the exterior angles because she felt like she was dancing. She remarked, “I felt like I was making a dance turn.”
While this activity is engaging, it tends to mostly engage the walker. Other students can also participate at their desk by drawing a large triangle on a piece of paper and pushing a small toy, such as a droid, along the sides and angle of their triangle as shown in Figure 5.
Notice that the droid in the last image is facing the opposite direction compared to the first image. Wanting the participant to also feel the rotation in her hand, I asked her to push the droid without taking her hand off the droid. This allowed her to see that her hand also ended up facing the opposite direction. She remarked that this made her “feel like a contortionist a little bit.” The short video clip illustrates this movement.
Assimilating New Movements
Certainly, a skier does not do a 180 degree turn in this way. Turns such as those are made up of a continuous rotation following the same path of an arc. This might be why the walker expressed feeling like a dancer when she traversed the exterior angles.
That rotating 180 degrees to traverse the interior angles of a triangle could make one feel awkward or like a contortionist sounds about right: This is a movement where one experiences rotating 180 degrees in a new way. Similarly, we might also say that learning itself is moving in a new way.
About the Author
Hortensia Soto is a Professor of Mathematics at Colorado State University. She is currently the Associate Secretary of the Mathematical Association of America (MAA) and the upcoming President-Elect of the MAA. Much of her teaching and research embraces embodied learning. Learn more about Embodied Mathematics Imagination and Cognition and follow her on Twitter @HortensiaSoto3.
