contributed by: Luke Wolcott
go back to category: Essences & Beholding


This activity was used in multivariable calculus, as a review exercise. Students worked in pairs or groups, with one person asking questions and the other answering, going "behind" the math concepts.

Class information

The Spring 2015 multivariable calculus class at Lawrence had 24 students. They were generally enthusiastic about mathematics, and planning to be math or science majors. These students tend to identify as good at math, and tend to find multivariable calculus material enjoyable and engaging.


This activity was done in the class periods before the midterm and the final exam, as an active way of reviewing material. We took 20-30 minutes on it.

Description of activity

I wrote the following list of questions on the board:
What’s behind that?
How do you know?
What’s an example?
Where does that come from?
What the definition of that?
How do you calculate that?
What’s really going on?
How should I think of that?

The students break into groups of two or three. One person is designated the question asker, the other one or two are answerers (or two askers and one answerer). They choose a mathematical idea — e.g. the gradient, continuity, Green’s Theorem — as a starting point.

By continually asking questions from this list, they trace the idea through their understanding. For example: the gradient.

Q: What’s behind that?
A: Partial derivatives.
Q: Where does that come from?
A: Partial derivatives show up when you have a function of many variables and want to take derivatives.
Q: A function of many variables — what’s an example?

Before we start, after I have explained the activity, the class brainstorms other questions that could be added to the list. Others that they came up with:

What is that symbol?
What can I do with that?
What's the essence of that?

They are encouraged to travel the mathematical landscape in an interesting way. Maybe to try to cycle back to where they started. To back up if they reach a boring place. Or to really go deep and get into “what’s really going on.” Students are also encouraged to use the exercise to generate a list of topics that they are confused about and need to review further.


This is a very different way of engaging the material than the students are used to, and it often takes some guidance. Sometimes I model it for the class. They enjoy the freedom, and seem to appreciate recognizing connections they were only barely aware of. Quite often they follow the ideas back to essences, and end up asking each other very philosophical questions, with glee.


This activity is inspired by one led by Dan Barbazet at a weekend workshop at the Omega Institute in August 2014. The idea of that exercise, in my understanding, was to trace back from specifics to essences, to go deep behind an idea. Honing in through this individual process and extracting a kernel of personal truth, this truth could serve as a signpost after once again zooming out.

Likewise the students are tracing through their personal frameworks of understanding, building some new bridges while recognizing and reenforcing ones that they’ve already built. The result may be a resonant essence of the original idea, or it may be an essential network of several ideas.


Students were engaged and seemed to enjoy pushing their knowledge in a new way. As mentioned in the comment section, they gleefully gravitated towards deep questions (e.g. How do you know?), in some cases deviating from mathematical understanding towards self-understanding.