The "Aha" Series: Thinking More Deeply in Middle School

This week, we asked Derek Stolp, our Middle School math teacher, to share a favorite story that illustrates a moment of joyful discovery when an idea really clicks and the “aha!” epiphany brings new life to learning.


Derek’s Fractal Story

Every day in school, children are working at the boundaries of their knowledge. And, every day, they experience “Aha” moments—many small ones, a few that are more profound and, not every day, ones that seem momentous.

One child who was anxious about learning mathematics at the beginning of the year discovered that, with perseverance, she could sometimes solve even the most challenging problems; and, though these were optional, she pushed herself to tackle them, expressing a quiet sense of delight when successful. Another, who every day has struggled with fractions, found that she understood areas and perimeters, and she eagerly volunteered to show her solutions to the class on the board. A young man who has struggled to make sense of mathematical concepts has discovered that, when he focuses in class, asks questions when confused, and completes his work in an organized way, he’s able to overcome his difficulties and master the ideas.

On the other hand, for those students who seem to grasp mathematical principles effortlessly, the “Aha” moment may take the form of a disorienting perplexity. While studying a fractal shape (the Koch Snowflake), two students discovered that it was a figure with a finite area but with an infinite boundary. “How can this be?” remarked one. “It doesn’t make sense,” said the other. This counterintuitive result forced them to think more deeply about notions of “infinity” because one infinite sum may converge to a finite number while another may grow without bound. This was an “Aha” moment that, rather than answering a question, suggested a whole new set of unanticipated questions, ones that they will revisit for years to come.

[This post originally appeared in Rhythm & News, the Inly School newsletter, on May 15, 2009.]

In addition to teaching math at Inly Middle School, Derek is the author of a middle school algebra curriculum that he has posted on his website,

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