Open-Ended Math

Over the last several years, I have spent significant time trying to understand and implement the best methods for teaching mathematics–especially to students who think they don’t like mathematics. Several ideas have risen to the top, but the very best is this:

 Students should be given the opportunity to engage with real problems that require significant effort, but are within their ability to solve.

Put another way, students should engage in meaningful struggle to learn real mathematics.

This ideal is difficult to reach for several reasons. Students do not always have the tenacity to keep going when a question feels difficult. Teachers are busy, and don’t always have time to come up with appropriate questions. It’s easier just to assign the practice sets found in the curriculum. Questions are very difficult to judge—some that appear easy are beyond the reach of students, and others that look complex are easier than they appear.

Nevertheless, it is worthwhile to present challenging problems to students and to give them the opportunity to engage in meaningful struggle as they learn mathematics.

I will not attempt here to answer all the challenges to this approach. Rather, I will point out several resources I have found helpful as I try to come up with challenging but solvable questions for my students.

This site offers interesting problems at many difficulty levels. It’s a subscription-based service, but offers weekly challenge problems for free. These weekly problems often require solvers to think laterally rather than try to solve the problem directly. Check out their principles and note that this is another way of saying that students ought to engage in meaningful struggle to learn real mathematics.

Art of Mathematics

This project of the National Science Foundation offers inquiry-based “textbooks” (free pdf downloads) on a wide range of subjects from Knot Theory to Calculus. Each book has a “Note to the Explorer” that whets the student’s appetite for the subject at hand. Each book starts with the basics and works up to some very challenging problems.

Art of Problem Solving

This is a curriculum aimed at high-achieving students. No simple questions here—many of the practice problems come from national and international math competitions. Students are challenged to think outside their normal constraints as they work to solve the problems. The elementary counterpart, Beast Academy, offers the same learning philosophy to younger children.

Phillips Exeter Academy

PEA is an expensive private school in Exeter, NH, with an approach to education that is based on collaborative problem-solving. They’ve put their mathematics materials online for free. These are written and edited by PEA faculty, all of whom are highly skilled at inquiry-based learning. The curriculum is full of interesting questions. Unfortunately (or fortunately, perhaps), it does not offer solutions to these questions. With no answer key to consult, it forces students to collaborate with each other (and the teacher) to decide whether their proposed solutions make sense.

There are many, many other sources for open-ended questions. When I need to challenge my students with questions of a type they don’t normally see, I cast about on these kinds of sites until I find some that interest me. I have found that these have a good chance of interesting my students as well.  With so many free (or nearly free) resources available, I have no excuse to be a dry and boring math teacher.


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