Three Strategies to Help Improve Students’ Conception of Math – Part 1: Do Creative Math

(Note: Readers of Necessary Conditions may recognize these themes from Chapter 2. I feel like revisiting these since there have been advances in resources, technology, and scholarship around these three themes.)

(Another note: I know I said, I hadn’t planned on doing my typical Summer Mini-series, rather, I applied my allotted blogging time to Algebra Warm Ups for Geometry Teachers. But I just couldn’t help myself. This’ll be relatively short though, I promise. Consider it a mini Mini-series.)

Most of us are now a month or two from the end of school. A few of us are a mere week or two before its resumption. This seems like a good time to do some reflective work and consider: how can we give students a good and true mathematical experience? Students often have a very limited sense of the discipline of math. Here’s a word cloud generated from a Mathematical Mindset and Attitude Survey responding to the question, “Free write: In your best estimation, what do mathematicians do?”

Yes, the most common response to “what do mathematicians do” is far away “math.” Maybe it’s just me, but I’d like to get a few more verbs in there. If I’m envisioning what I’d like the word cloud to say, I’d want to see words like “draw,” “conjecture,” “guess,” “create,” “play,” “build” etc.

But how to we engender such views about math? In this and two successive posts, I’ll discuss three “types” of math that may engender a view of mathematics that is truer in nature to the work of mathematicians, while also opening up the discipline to students who previously felt unwelcome to it.

Strategy #1: Do creative math.

In research literature, mathematical creativity focuses on open ended problems, model-building, multiple-solution problems and problem-posing (Bicer, 2021). I’d extend the definition to activities that encourage (and even utilize) creativity around art and design. Just a little bit of drawing goes a long way to encourage mathematical creativity. One of my favorite is an activity I share often, Quarter the Cross.

Typically I have students do this twice. Then I “force” them to do it a third time to really stretch their thinking. By that point I have dozens of crosses I can compare. My go to prompt is “what’s the same? What’s different?“

If you’re stumped on how to generate mathematical creativity in a lesson, here’s a template I follow (not always, but sometimes).

Step 1. Have students draw or create something, possibly on grid paper (but not necessarily). It should be new and unique for each student.
Step 2. Have students apply mathematical content to it.

For example, I designed a lesson in which 5th grade students create a mosaic using PolyPad (specifically a 100’s number tile, split up into individual boxes).

Students then calculate the “value” of their mosaic according to a pricing sheet like so.

It’s not too dissimilar to Fawn’s “Hotel Snap” activity. In that instance, they are building a hotel using snap cubes and ascribing value to the number of “corner units” and “window units” etc. In that activity students are trying to develop a hotel that’s worth the most. It’s more involved than my description, so be sure to check out Fawn’s original post.

Another example that follows the create-a-thing-and-do-math-on-it template (CATADMOIT template) is an activity on parallel and perpendicular slopes. The lesson goes like this.

Step 1. Draw some lines of a graph. Make some of them perpendicular. Make some of them parallel. Color in the spaces with colored pencils. It should like kinda like a very math-y stained glass window when you’re done.

Step 2. Let’s look at all those parallel lines. What do you notice about their slopes?

Step 3. Let’s look at all those perpendicular lines. What do you notice about their slopes?

Step 4. Nice. Now let’s capture that information and hang up your beautiful drawings on the wall.

This is clearly oversimplified – but only by a little. You’ll want to facilitate the discussion and little more heavily. This is something where the 5 Practices could help.

Here are some more of my favorite creative mathematical tasks

Graphing Stories
Generalizing Patterns: Table Tiles (MARS)
Where Can We Visit? (NRICH)
Describing Designs (NCTM) – need to be a NCTM member
Map Distance and Midpoint (Pam)
Sprinklers (Geoff)

If you have a favorite task that utilizes creativity, toss ’em in the comments and I’ll add it to the body of the post!

In the next post, I’ll discuss another type of math that can help improve students’ conception of math: useful math.

Bicer, A. (2021). A Systematic Literature Review: Discipline-Specific and General Instructional Practices Fostering the Mathematical Creativity of Students. International Journal of Education in Mathematics, Science and Technology, 9(2), 252–281. https://doi.org/10.46328/ijemst.1254