The essence of our problems are what Jo Boaler and others have called “low-floor, high-ceiling” problems. This means that our problems are accessible and engaging for everyone. Specialized knowledge of math is not required to begin tackling them, so that those who are not confident and experienced with thinking mathematically can still experience success. Further, our problems also have rich and deep connections in higher mathematics, so that those who are comfortable with mathematical formalisms and abstractions can pursue such strands, as well.
What is important, we think, is that such problems offer opportunities for our attendees to engage in explaining, modeling, demonstrating, drawing, and showing their mathematical thinking. Problems need not be solved mechanically, using algebraic language or symbols–and, in fact, employing different sorts of mathematical tools besides algebra can yield key insights.
Here are two sample problems from recent PAMTC workshops:
THE BROWNIE PROBLEM. At the end of the school term, you bake a tray of brownies to share among your two 7th grade classes. While the brownies are cooling on the counter, some mischievous scamp sneaks into the kitchen, cuts out a rectangular piece, and steals it. The tray of brownies must be split evenly between the two classes, and you have only moments to do so. Is it possible to divide the brownie tray evenly, using a single cut from your knife? Why or why not?
THE KEY-CAP PROBLEM. Rushing to get inside your house, you drop your key ring on the ground. What is the smallest number of colored key caps that you need to distinguish all of your otherwise indistinguishable keys?
Notice that these two problems are deceptively simple, but that they also inspire questions, such as:
- Does the knife-cut need to be “straight?” (What does “straight” mean, exactly?)
- Can you measure the brownie tray with a ruler and do some calculations?
- Should we agree that all the keys are physically indistinguishable?
- Is the key ring marked in some way?
- How many keys are on the ring? Does this even matter?
We call this question-asking “interrogating the problem,” and it is an essential routine in our workshops and, generally, a key component of mathematical thinking. We hope that you are intrigued by such problems and will want to join us to discuss approaches to solving them and explore other problems like them!