Using Geometry To Make Kids Curious

I don't think of geometry as just another subject, but as a way of thinking and living life.

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Curiosity is the root of all fun

"This is so cool" was the expression on the face of my students. I have been teaching kids geometry for more than a year now, and this happened again last Sunday during my class. While teaching, my goal is to make my students curious, as if they were in an escape room. Curiosity leads to a desire for exploring and understanding the mysteries of the world. Curiosity leads to a desire to achieve mastery over things. It results in students wanting to learn, rather than students being unwillingly forced into learning. Curiosity leads them to more happiness that comes from the ability to connect the dots.

But the journey to curiosity starts from contrived, and then takes a couple of pit stops at clever and cool , before getting to its final destination. Let me explain what actually happened in my classroom this Sunday. The topic of the class was the principle of reflection in geometry. I wanted to show my students that nature favors efficiency, and there is efficiency in symmetry. These are profound concepts that go far beyond the reflection principle and are covered in topics like the symmetry group.

Contrived

We started with a simple but arguably contrived problem:

Joe stands at point A. He needs to get to point B, but before doing so, he must touch the wall (any point on it). What is the shortest path he can take to get from point A to point B (and touching the wall in between)?

Clearly, he can pick from any number of points on the wall. What he is required to do, is to pick a point that will minimize the distance he traverses. It is obvious, for example, that going from A --->1 --> B or A--> 4--> B is not efficient. At this point, the kids felt that the problem was contrived but still interesting. They wondered how they could solve it.

Clever

After a while, I showed them how reflection could make this simple to solve. The solution is to take the reflection of point B about the wall (we'll call that point B'), and draw the line AB'. The point where this line intersects the wall (C in the diagram below) is the point that minimizes the distance traversed by Joe. It was easy enough to prove with elementary geometry. The kids thought this was extremely clever.

Cool

The next step though was to show that the point of intersection has a unique property. This property is that the angle of incidence is equal to the angle of reflection at this point (notice the two angles, x, being equal). This is the principle of reflection. It holds true for all physical entities bouncing off walls or barriers. Light reflecting from a mirror, billiards balls bouncing off the table walls, etc. It seems like things in nature bounce around in ways that minimize the distance they have to traverse. Nature favors efficiency. You can use the reflection principle to find out which direction to hit a billiards ball for the best possible results. That was cool.

Clearly, nature favors efficiency. But what does this have to do with symmetry? If you were to now simply frame the problem slightly differently, other cool things start to emerge. For example, you can extend this to prove that an equilateral triangle is a triangle with the smallest perimeter of all the triangles with the same area. Similarly, the surface with the minimum area that encloses a given volume is a sphere, which is the most symmetrical shape in the 3D space. Soon it becomes evident that there is efficiency in symmetry. Suddenly, they started to notice that nature is filled with symmetry and experienced the joy of connecting the dots.

Curioser & Curioser

The experience of learning geometry should be like going to an escape room where kids look forward to the hour of exploration and, in fact, wish they had more time to explore. They attack open-ended problems with gusto and by the end of it are intellectually exhausted, but can't wait to come back. Curiosity is the ultimate motivator and geometry is replete with intrigue, elegance, insights, and drama required to make kids curiouser and curiouser. It just needs a lot of work on the part of the teacher to create that experience in the classroom.