by Mark Illingworth

At its highest level, math is a creative endeavor. Do you give your students opportunities to develop their creativity? I’m not talking about students creating collages or doing improvisational dance to represent the triangle proportionality theorems. I’m talking about students doing original thinking. Here is what that can look like.

A simple example would be in how you present proofs. Do you always tell students what they should prove? Or do you give them the chance to tell you what they think might be provable? For instance, when you get to quadrilaterals, do you tell them the properties of a rhombus and then ask them to prove those properties? If so, you might instead start by asking them to work in groups to list the properties that might be true. Then they can prove (or disprove) those properties. Adding this one step to your proof process elevates the activity to the top of Bloom’s Taxonomy. (If any of you are using the Trapeze Education Honors Geometry program, this is what the Quadrilateral Gang project is all about.)

Let’s go to the other extreme. My final project in my Honors Geometry course was to have students find things worth analyzing and then to conduct original investigations and present their findings. I didn’t want them to leave my course without having had opportunities to do original thinking in math and to find the problems worth studying on their own. In other words, I wanted them to feel confident in forging new paths.

The students started getting ideas for their investigations during the winter. Ben wanted to trace rays through a glass lens modeled by two congruent arcs. Katrina wanted to create a spreadsheet that would track the path of a cue ball hit from a given coordinate at a given angle. Vinnie wanted to calculate the probability of a randomly hit baseball resulting in a base hit. You can see a list of project ideas here.

Although the level of sophistication of the final projects varied, all students were—given this opportunity—able to generate that they could find opportunities to create original math models to analyze or solve problems. Nobody had to spell out the steps for them or give them all the data.

I didn’t start with the projects of course. Students coming into my class were not used to thinking independently and creatively in math. Notice that we didn’t start the project in September. I first needed to give them smaller experiences that would model the exploration process. You can of course keep feeding them a diet of problems in which you tell them exactly what to find and then provide them with all the pertinent data. With simple changes, you can pose problems that require more creative thought. This can be as simple as providing a diagram like this one (Insert visual “visualthinking.jpg”.) Before you pose a problem, ask students to explore relationships within the diagrams.

You can change any problem by showing a diagram without any data and then asking them what relationships they might be able to investigate.

You can create bigger investigations like The Squirrel Problem by describing or showing a scenario and then asking what the mathematical relationships might be. For example, the letters on a road are purposely elongated so that they appear normal in height to the approaching drivers. Instead of creating a problem with all the data needed, show them a picture of elongated letters and ask them to talk about what math might be involved. Students need to be spoon fed very little. You can find a lot of this type of problem at Dan Meyer’s blog. In fact, you can learn a lot about problem solving from Dan. Go to this “Three-Act Math” link.

I’m just trying to get you started by making an important shift. Strengthening students’ confidence in thinking creatively and independently is behind several components of my Honors Geometry course, but you don’t need me to get started. Just give the way you pose problems and approach proofs some careful thought, and you can give them a lot more by giving them less… if that makes sense.

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Creativity is the highest level of Bloom's Taxonomy. It's also one of the goals of 21st Century Learning.

Do you always tell students what to prove, or do you also ask them what they think could be proved?

Do your students have opportunities to do independent investigations and forge new paths.

Provide open-ended investigations that don't come with all the data and directions needed. You'll be surprised by what students can do.