Encouraging Alternative Approaches in Math Class

As a middle and high school student, I liked mathematics because, as I saw it, there was one clear path to the correct solution. I thought that if I followed the prescribed steps correctly, I would come to the correct conclusion. As my knowledge and experience with mathematics deepened, I realized I was mistaken in my assumption of “one clear path.” Now I see that the beauty and joy of mathematics come from the huge variety of approaches that a group of creative minds can produce.
We have a variety of opportunities for creativity in the mathematics curriculum. As we try to generalize theorems from specific examples, a student sometimes phrases their understanding in a way that is, on its face, completely different from the phrasing a textbook or I would have used, but captures an excellent summary of the mathematics involved.  

Often the best mathematics happens in the classroom when I’m not a part of the conversation. When they discuss their approaches to a problem with each other, students are sometimes surprised by the path a classmate takes. I am delighted by the creative approaches I get to see, and I love the opportunity to add to my mathematical understanding by thinking through a student’s path I had not considered.

In Upper School Mathematics courses, we emphasize multiple modes of understanding: numerical, analytical, and visual. Often the visual exploration leads to creative outcomes. A visual pattern made up of emojis becomes a quadratic function. A construction made with a compass and straightedge generalizes to a theorem. Those outcomes become more vivid in a student’s mind because she can conjure that visualization to keep the details available to her. As she progresses to calculus, she can see the solid formed by a function rotating around an axis because she is practiced in visualizing mathematics.

The visual components of mathematics lead to entertainment. In a mini-course called “The Art of Mathematics,” students have the opportunity to play with mathematical tools and explore outcomes they don’t necessarily see in the standard curriculum. Chords in a circle, either drawn with colored pencils or stretched across a frame with thread, teach about modular arithmetic. Compass and straightedge constructions lead to tessellations that appear in Islamic art. Sculptures made with laser-cut plywood or wire hangers show how faces and edges of platonic solids meet.

Students apply the creativity of using multiple modes in other courses: a well-designed graph along with a table of data in science reveals a trend and tells a story, or a verbal description along with an image in art history helps tease out a theme. For students learning mathematics, the ability to develop their knowledge in multiple modes allows them to see connections between concepts, make the abstract concrete, and imagine alternate scenarios. Their creative approach to problem-solving expands over time, each student sharing their unique thought processes and unexpected paths along the way.