To solve a “Math Venn”, one must generate examples for each of the regions in a Venn diagram (or explain why no example exists for that region), where each region is labeled with a math concept or constraint. I found them through Craig Barton, whose website contains Math Venns created by himself and others. Here’s a screenshot from the website:

I like the idea of asking students to generate their own examples. The NCTM calls examples created by students “learner-generated examples” (LGE), stating:

“When students generate their own examples, they behave more like mathematicians, drawing on connections and taking ownership of the concepts. As a result, generating examples can be motivating for students at all levels. Students benefit by becoming better problem solvers and developing a rich array of example types. Further, when students generate their own examples, they reveal information about their thinking that is not readily available otherwise.”

In the current climate of math pedagogy, students are being exposed to an increasing number of problems with multiple solution paths. Still, it’s likely (and students probably know it) that their solution is probably shared by someone else. LGEs have the benefit of increasing the likelihood that what a student comes up with is completely unique. It’s *their* example which they’ve created.

I see Math Venns as a great structure to promote LGEs. They provide just the right amount of constraint but also a good deal of intellectual space to explore. I wonder if students need a healthy bit of domain specific knowledge before they can truly benefit from engaging with Math Venns; I’d be hesitant to use a Math Venn while introducing a topic.

Here are a few Math Venns I made, along with some discussion about the topics they cover and ways of raising the ceiling.

- Potential Areas of Discussion: This Math Venn might raise discussions about the definition of a “function”, domain and range, piecewise functions, equations of functions, asymptotes, and end behavior
- Extensions: If students have already learned them, this could be made more challenging by prohibiting piecewise functions or functions that pass through the origin

- Potential Areas of Discussion: This one could raise discussions about factoring quadratics and sums and products of even and odd numbers. It also brings up the question “what counts as factoring?”. For example, does (x – 2.5)(x + 6) count?
- Extensions: Change “b is even” to “b is prime”. I haven’t thought much about it but I’m hoping it yields interesting results.

- Potential Areas of Discussion: This one is about the definitions of continuous and differentiable and which implies which (and which doesn’t). H is particularly interesting because of nowhere differentiable functions.
- Extensions: I can’t think of an extension here but if you do let me know: @ m3lvinp3ralta

- Potential Areas of Discussion: This one raises common misconceptions such as students believing that the sum of 1 + 1/2 + 1/3 + … converges just because 1, 1/2, 1/3, … converges. It also raises the relationship between convergence and boundedness
- Extensions: I can’t think of an extension here but if you do let me know: @ m3lvinp3ralta