Section 3 The Goal of the Book
So we return to the driving question: What do we really want students to learn? The answer that this book gives is that the goal is to get students to become better and more confident mathematical thinkers. They've spent enough time doing the lather-rinse-repeat of pure manipulations. They need to develop a different intellectual foundation. The good news is that we're not working from scratch. Because although students may struggle with certain types of algebraic manipulations, it's rare that all the students need to relearn all of them. Those past experiences are our starting point.
So instead of treating students as if they've never seen these manipulations before (which is how the majority of remedial coursework approaches topics), we're going to work with students as people who have experience but have not carefully reflected on those experiences. A lot of the manipulations that we ask of them are already somewhere in their heads, and we're simply working with them to connect those neurons to other neurons and strengthen that signal.
This begins with the very first section. The emphasis is on mathematical communication. This is already a significant divergence from most other approaches, where the goal is to get the right answer. The majority of students don't even have a basic framework for understanding mathematical communication. We lay out simple but clear expectations for how to begin to organize their mathematical writing, and that foundation is the tool that we use to help students reorganize the information in their heads. Once we can get them to write their work in an organized manner, it becomes easier to begin to isolate specific struggles that students are having, and they can even begin to start to recognize them for themselves.
Every section that follows treats students as adults who already have mathematical experiences. We do not treat the students as kids who need to have to be told exactly what to do all the time. In fact, we emphasize throughout the book that mathematics is not about following rules, but about being able to think through situations and understanding what they're doing and why they're doing it. You will find that some of the "Try It" problems do not have an example that models the exact thing they need to do. Those examples often become a crutch for students as they realize that they don't actually need to think for themselves, and simply have to hunt down the right example to show them exactly what to do. You will see this philosophy expressed in the worksheets as well, as many of them will touch topics that were not directly discussed in the text.
We have done this specifically because we want students to learn to think for themselves. The goal is not just that students will learn the idea, but that they will begin to develop metacognitive strategies for how they approach new mathematical ideas. This is a much broader foundation that they are more likely to carry forward with them into their future classes. If you just teach the manipulations, then it's going to be hit-or-miss whether they will remember those manipulations when they need them in the future. But if you give them the tools to think about the mathematics effectively (and the confidence to trust their thinking), they will be far more able to reconstruct the ideas behind the manipulations if they've forgotten them. It simply puts them in a much better position for long-term success.