I was in a position to provide some talking points for my company’s upper-level math textbooks. It was written in the aftermath of customer-induced pique regarding how we sell ourselves. While it’s focused on my company, the core idea of a problem-first approach extends beyond what we do in particular.
Nearly all existing math instructional materials follow the same pattern: Explain a concept once, focus on finding a formula or singular method, have students solve repetitive problems with small changes, and move on to the next topic. New approaches may emphasize doing this routine in groups, or using “real world data”, but the method itself hardly varies. It’s comfortable and can be made easy to implement, all at the cost of students only learning how to solve a handful of specific problems and failing to think independently. This approach also makes it difficult to differentiate learning within a class because the content has no room for increased depth.
From the very beginning, AoPS has crafted a problem-first approach to teaching math. We recognize math as a beautiful subject—we’re overjoyed when former students pursue careers in math specifically—yet also realize it’s an excellent field to focus on learning how to solve problems. We are developing materials for physics and computer science for this same reason: they’re full of interesting problems that require flexible thinking and application of a wide range of analytical tools.
To this end, you’ll notice the unique format of our textbooks. Nearly every section begins with a set of problems that motivate the topic of discussion. Instead of lecturing about a topic with a smattering of examples, we start with the example which motivates us to develop the relevant piece of mathematics. These problems are scaffolded to more advanced applications of a topic, and bring in previous topics to ensure students understand the connections between everything they’ve learned. This approach has several benefits.
First, it switches how students look at the world around them. Instead of being distraught if a problem they encounter doesn’t fit into one of the molds they were taught, they can look at a problem for what it is and suss out how they can break it down using their available tools.
Second, by repeatedly developing this problem-first philosophy, students will begin to gain confidence in approaching a problem where they have no tools that work. Instead, they understand that truly interesting problems require us to build what we need. Sometimes the first tool we build is enough; other times, we hope to refine that tool and understand it better. This mirrors the high-level thinking required for success in, for example, academic research.
Third, this approach offers a natural way to differentiate learners within a classroom. Students who show repeated success with a topic may be grouped together to focus on solving the problems from scratch, only using the rest of the text to confirm their findings or obtain the specific definitions and conventions for that topic. Other groups may require more scaffolding from their instructor, perhaps working through one or two problems as a group to understand the basic ideas, then flexing their analytical muscles on the later problems.
As the opening to our books say rather succinctly:
This book is probably very different from most of the math books that you have read before. We believe that the best way to learn mathematics is by solving problems. Lots and lots of problems. In fact, we believe that the best way to learn mathematics is to try to solve problems that you don’t know how to do. When you discover something on your own, you’ll understand it much better than if someone just tells it to you.