Math Venns

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:

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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.

Graphs of Functions

mathvenn1

  • 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

Factoring Quadratics

mathvenn2

  • 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.

Continuous vs. Differentiable

mathvenn3

  • 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

Convergence

mathvenn4

  • 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
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Problem Posing SSDDs

Based on his book “How I Wish I’d Taught Maths”, Craig Barton started a new website about problems that are identical on the surface but have different underlying mathematical structures.  He calls them SSDDs (same surface, different deep structure).

His motivation for calling attention to SSDDs:

By always presenting students with a series of problems that are set in different contexts (i.e. have a different surface structure), but which are all from the same topic (i.e. have the same deep structure), we are robbing students of the opportunity to develop the ability to identify the problem’s deep structure and hence identify the strategy needed to solve the problem.

I love the idea, and the website has taken off in a short period of time.  Having explored it, I can’t help but ask myself: can the idea of presenting identical images be used to promote problem posing?

I’ve been gaining an increasing interest in problem posing over the past year.  Problem posing means shifting control over the problem-generating process from authority figures (me, textbooks, even MTBoS) to students.  The few times I’ve tried it, I’ve seen that it puts the math we’ve been doing in a sharper light for them.  Sometimes a dish tastes better when you’ve cooked it yourself.

Craig Burton’s website gave me the idea to give my students what is essentially a blank SSDD and ask them to create a different question for each image.  I tried creating some images on my own for Algebra 1, and here’s the result:

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I haven’t yet given this task to my students but I hope to do so in the near future.  My plan will be to pair them up so they can take advantage of one another’s strengths.  It wouldn’t be too shocking to discover that students were stronger at some topics over others.  I’ll also ask them to answer their own questions.  Maybe it’ll give them a sense of self-efficacy and the realization that they can answer a wide range of their own scholarly questions.  Now that I’m writing this, I realize it’s like Dan Meyer’s Act Ones, except there’s a greater emphasis on comparing and contrasting structures.

Maybe this problem posing version of SSDD can also serve as a type of formative assessment.  Ideally, it’ll root out misconceptions.  But more than that, it’ll show me what topics spring to mind and what topics they only remember through heavier prompting.  Too often assessment is about “do you know” vs. “do you not know”.  Having them craft their own questions might help me (and them) see what they’ve truly attended to.