Now here's something I hope you'll really like

High achievers are often high achievers in the U.S. system because they are procedurally fast. Often these students have not learned to think deeply about ideas, explain their work, or see mathematics from different perspectives because they have never been asked to do so. When they work in groups with different thinkers they are helped, both by going deeper and by having the opportunity to explain work, which deepens their understanding. [Jo Boaler, Mathematical Mindsets, pg 138]

In my quest to find meaningful assessments, presentations are presently high on my list. In my second year integrated mathematics course, presentations are a prominent component of the course.

Students learn in all sorts of ways. Once upon a time, I learned that there are three modes of learning: visual, auditory, and kinesthetic. That is, one can learn by watching, by listening, and by doing. Some students, I was told, will be strong in one mode and weaker in others. It was also explained that the best learning happens when all three modes are utilized, but that kinesthetic learning was generally considered the best, so should be very prominent.

This model of learning seems very simplistic and I believe is outdated. Notably, it does not mention learning by explaining. Perhaps one might argue explaining to be a form of kinesthetic learning, but in my mind these are two separate activities.

In my math class, kinesthetic learning happens when a group of students tries to make sense of a possible pattern by drawing a model graph. It continues as the group constructs a table of values or a model equation. And it happens while the group attempts to use its models to make predictions. Significant and meaningful learning will certainly take place during this process. The learning will undoubtedly be richer than had the students merely watched a teacher working at the board.

But this sort of learning is starkly different from learning through explanation. As Dr. Boaler’s quote notes, explaining work deepens one’s understanding. How can it not? When a student attempts to explain an idea to another, she must struggle to find the words to convey her understanding. She must think through the idea in a way that is much different than when she just wants to understand it for herself.

This is my primary motivation behind having students present to the class. It is one thing for a student to use mathematics to figure out how long it will take for a garden hose to fill a water tank or how many stars there are in the universe. It is altogether different for a student to attempt to explain his work to others, to verbally justify his results and describe the evidence and reasons. It is very common in my classroom for a student to struggle to explain his reasoning even when I know he understands very well how to do the problem.

It is this struggle that is so important. This is no time to invoke the lifeguard metaphor. A student struggling to find the words to explain herself is not drowning. She is no way about to die. There is no need for the teacher to dive in and save her. No, she is holding the map, she has a compass, she is making her way through the woods and she has all the tools she needs to find the lake. She does not need saving. She needs a chance to find her way without a teacher’s help.

Three to four times a semester, I ask my students to prepare a 5-10 minute lesson in which they must try to teach their classmates. Typically, I provide them with a problem that will be the focus of the lesson. The students are tasked with showing a full solution to the problem, but they must also find creative ways to improve the understanding of even the students who find the problem easy. Their teaching techniques might include extending the problem, or looking at common mistakes, or linking the problem to other ideas from the class.

A discussion is always a part of this presentation. It generally happens at the end, but sometimes it happens along the way. Sometimes, the discussions are started purposefully by the student teacher. Other times, a student in the audience brings up an interesting point. Otherwise, I will pose a question or make a conjecture for the teacher and students to consider. Because the student teacher has prepared in depth for the presentation, I know I have at least one student who has thought deeply about the topic to help grow the discussion. The discussions we have after these presentations are often some of the best of the year. As the student teacher wrestles with answering, I swear you can hear the synapses firing.

Certainly, one difficulty in presentations is the anxiety many feel towards public speaking. Add to that a dose of math anxiety and it can potentially be a very frightening experience. Thus, I work intentionally to help students become more comfortable. I provide time in class for preparation. I assign the lessons a week in advance so that they have ample time to get help should they want it. We discuss what effective presentations look like and what it means to be a helpful audience member. And generally, we work to create a classroom atmosphere where all voices are respected, where mistakes are celebrated and not shamed, where it is expected to express confusion so that all can help.

In my experience, these presentations have succeeded in deepening the learning of the presenter. Students gain confidence and comment favorably about them.