Math gets a bad rap. In popular culture, it’s the subject of droning teachers, mindless memorization, and endless sheaves of worksheets covered in problems (although to be fair, memorizing the times tables does continue to be valuable). It doesn’t have to be this way — and mathematicians certainly don’t think of their work this way. Math is about exploration, discovery, and understanding the underlying patterns in the universe — how could a mathematical playground possibly be boring?
While of course gifted students are not all gifted in mathematics, we can take advantage of the complexity of their thinking and the way they make connections to teach math in a way that approaches the way actual mathematicians think. Importantly, I create a classroom environment in which students work on problems that engage their minds and nurture creativity — a place where they can play with mathematical concepts and ideas. Creating a culture in which math is exciting and engaging is a matter of finding the right kinds of questions for them to tackle — and then getting out of their way.
The Pancake Challenge
I recently asked my math students to solve the following problem: they have four pancakes, each a different size, which are arranged in a stack as shown below. The goal is to rearrange them into a perfect pancake pyramid, also shown below. They are only allowed to move the pancakes by sliding a spatula underneath one of the pancakes and flipping over the pancakes above the spatula, thereby reversing their order in the stack. What is the minimum number of flips required to obtain the perfect pancake pyramid from the out-of-order stack below? Spoiler alert: if you would like to solve the problem yourself, please do so before reading on.
Students quickly got the hang of pancake flipping, and discovered that the minimum number of flips required for the stack above is three. There are myriad pancake flipping variations students can work on from here: for example, what are the three distinct orderings of four pancakes that require a minimum of four flips to complete? This question is more complex than the first, requiring a more systematic approach. For even further exploration, students can search for an algorithm that could solve any possible stack of n pancakes, which requires them to generalize their thinking from the previous two solutions. Or what if each pancake were burnt on one side? The new goal would be not just a perfect pyramid, but one in which all burnt sides are facing down. How would this constraint change their algorithm? If students are so inclined, they can attempt the ultimate pancake flipping problem: find a closed form for the minimum number of flips required to sort any stack of n pancakes. This is an open question in mathematics, meaning mathematicians themselves have yet to discover a solution.
What a mathematical playground looks like
This might seem like much ado about a hypothetical stack of pancakes, but what makes this family of problems special is the response it elicits from gifted students. The photo below shows my students solving the second problem (three distinct orderings of four pancakes that require a minimum of four flips). Without prompting, they began working together as a team to solve the challenge. They tried to find the solution systematically by checking all possible orderings of four pancakes, and realized that doing so together would be much more effective than if they worked individually. Spontaneous full-class collaboration is not an everyday occurrence in math classrooms across the country, but it is something we see our students do when the culture of the classroom encourages experimentation and collaboration.
For this problem, students organically created and chose specific roles. One was the coordinator of the problem-solving efforts; she delegated tasks to each member of the team and kept track of how many of the twenty-four possible orderings the team had tested. A second student discovered a way to find all the possible orderings; she also tested various orderings to find their minimum required flips. A third took it upon herself to ensure that each part of the solution the team found was indeed correct; she attempted to find more minimal flip patterns than her teammates had. A fourth kept track of solutions the team had found; she synthesized the work that each of her teammates was doing. The fifth and sixth students served as minimal flip pattern testers; they found minimal solutions for the various pancake orderings given to them. Working together in these roles, the students completed the challenge given to them in under twenty minutes. Upon being told they had indeed found all three orderings, they cheered unanimously and asked if there were other pancake problems they could solve together.
Principles of fierce engagement
Teachers at Grayson call this kind of intense focus “fierce engagement,” when students experience authentic and memorable learning. If students are indeed fiercely engaged, they have taken custody of the problem for themselves; the teacher chimes in only if students need to be reminded of constraints or details, but does not direct their efforts.
The pancake problem invites this kind of absorption, for a few key reasons:
- The problem is easily explained. It is accessible to students regardless of prior mathematical knowledge, and each student can understand what the problem is asking.
- The problem has meaningful constraints. Students are only allowed to flip the pancakes in one specific way, and they must end up with a perfect pyramid. These constraints narrow the problem’s scope just enough to help students feel a clear sense of direction about where a solution path might lie.
- The problem has numerous extensions of higher complexity that also satisfy the first two criteria. Once students are hooked by the initial problem, they become increasingly excited about the opportunity to solve more complex variations.
Problems that satisfy these criteria are special because they create a “mathematical playground” that allows students to explore, learn, and solve problems together. This particular playground creates fierce student engagement, rich and spontaneous teamwork, and genuine pride in their solutions.
When my students are given access to a mathematical playground, they become hungry for the intellectual challenge and sense of accomplishment they find within it. When they come into the classroom the next day — and the next day, and the next — asking for more pancake problems, they are reflecting the kind of genuine excitement about learning that every teacher hopes to create, no matter their discipline.
Mr. Derek Graves is a mathematics instructor for upper-level mathematics courses and electives. His experience includes developing math curriculum — both daily instruction and enrichment programming — for gifted Middle and Upper School students. He is particularly interested in helping support students’ ability to communicate their understanding of math effectively.