The Shortest Distance to Authentic Learning

The moment I walk into Erin Hansen’s middle school geometry class, a flood of math memories comes rushing back to me.  I think about Pythagoras and proofs, oblique angles and octagons.  I remember memorizing rules and postulates, and wondering how any of this would ever connect with the real world. I’ll be honest: all of my math memories are not great.

So I’m definitely not expecting it when I hear Erin tell the class “Today, we’re going to build kaleidoscopes.”

A polygon as seen in the ‘kaleidoscope’

The ‘kaleidoscope,’ as Erin demonstrates, is composed of a hinged mirror—two small glass squares taped together at one shared edge—and an orange sticky note.

Erin asks her students to look at a series of questions in their textbooks that they will eventually need to answer. “But first,” she says, making her expectations clear, “you should experiment and explore. Place the sticky note underneath and try out different angles with your mirrors to see what happens.” Erin knows that learning this new geometric content on angles and polygons (i.e., multi-sided shapes) requires her students to take this essential first step.

One student from each of the classroom’s four worktables gathers the necessary materials and distributes them.  Then, the students commence their work: really, to play, in an intellectual and developmentally appropriate way. These simple kaleidoscopes, combined with each student’s imagination, produce an array of geometric figures—from triangles, to hexagons, and heptagons (that’s a figure with seven sides, I learned).

8th graders Dani L. and Henry A. seek an up-close perspective

By encouraging learning through experimentation and play, Erin taps into her students’ natural curiosity—which drives discovery, and results in learning.  This research with a simple kaleidoscope will provide each student with valuable reference points when studying future geometric content and concepts.

Now it’s time for the class to answer the questions in the textbook, and the answers come quickly, and easily.

A handsome hexagon

“What’s the relationship between the size of the angle and number of sides it produced?” Erin asks.  The students hypothesize that the smaller the angle made with the mirror, the larger the number of sides—illustrating an inverse relationship.  Erin suggests students address another relationship, this one between the angle of the mirror and the number of sides.  Because the students could generalize from their experiments, they were able to infer that the number of sides multiplied by the mirror’s angle would always be 360—the number of degrees in a circle.

At Wildwood, classes like Erin’s use the textbook to support student learning, not drive it.

8th grader, Jacob R., stages a surreal pose with his kaleidoscope

For these middle school mathematicians, the possible extensions for learning are many.  Erin’s students can use this lesson as a foundation to learn any number of geometric concepts that the course requires—determining unknown sides and angles, understanding the Pythagorean Theorem (a2 + b2 = c2), or completing geometric proofs.

And, this lesson is a reminder that sometimes, in order to understand the sublime simplicity of geometry—you just need to play.

~ By Steve Barrett, Director of Outreach, Teaching and Learning

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