The "Aha" Series: Annika's Pythagorean Theorem Story

This week, Annika Voynow, our Assistant Head of School, shares her favorite story illustrating a moment of joyful discovery when an idea clicks for a student, the mental light bulb switches on, and the “aha!” epiphany brings new life to learning.

ahacolorAnnika’s Pythagorean Theorem Story

One of the things I love most about Montessori is that in giving children space to explore, we give them the opportunity to find their own interests and passions. In my experience, they find ways to amaze us everyday and do more and work harder than we would every think to ask them to.

When I was teaching in an upper elementary classroom, I was introducing the Pythagorean theorem to a small group of sixth graders. The Montessori materials allows children to see a concrete manifestation of the Pythagorean which states: The area of the square built upon the hypotenuse of a right triangle is equal to the sum of the areas of the squares upon the remaining sides or, as we learned in high school math, a2 + b2 = c2.

I noticed as we were working, a fourth grader peeking at our lesson from a near by table. He watched the whole lesson and then, when all the older children had gone off to do other work, came over to me. “Can you show me that work?” he asked. I hesitated for a moment, as he was only nine and I wasn’t sure he would understand it. “Of course,” I said, “I’ll show you and you can work with the materials.” I showed him the lesson, we chatted briefly about how the theorem might be used, and then as soon as I was done, he returned the material to the shelf and went on to another piece of work.

A few weeks later, my nine-year-old student approached me with his math notebook in his hand. “Can I show you my work?” he asked. “I would love to see your work,” I told him. He opened his notebook to reveal page after page of drawings, each with a math equation utilizing the Pythagorean theorem at the bottom. He had drawn sailboats, barns, flagpoles, tents, pyramids, and staircases. He began to tell me the story for each drawing with great excitement, having imagined each scenario when he might be able to use the Pythagorean. When he was finished with his stories, he asked, “I like these kind of puzzles, are there more like this?” Needless to say, he and I spent many more days exploring all aspects of geometry. Because of his interest, geometry took off in my class that year in a way that I had never seen before.

[This post originally appeared in Rhythm & News, the Inly School newsletter, on February 13, 2009.]


  1. Hi there!

    I love the image at the top of the blog of the bulb with AHA!

    I am a writer (and a part-time education dayjob:)

    I was wondering if it would be possible to use the image for a short story collection about “Aha!” moments?
    I’d of course include credit and could also include a contact if you’d like…

    Please let me know your thoughts….

    Again, image, brilliant,

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