Section 4 The Structure of the Book
One of the features of this book to help students accomplish the goal is that only part of the book is intended to be "taught" in the classroom. Self-efficacy comes with the practice of being self-efficacious. This was an intentional decision based on the use of this material as part of a corequisite math course. As much as we can want to aim for deeper learning outcomes, we still need to confront the practical reality of our students' struggles with mathematics.
The book is broken into five sections:
The Main Trunk: Core algebra (Sections 1-11)
Branch 1: Linear equations and the coordinate plane (Sections 12-16)
Branch 2: Fractions and decimals (Sections 17-23)
Branch 3: A review of arithmetic (Sections 24-32)
Branch 4: A few applications (Sections 33-36)
The main trunk is the set of core algebra that we think are absolutely critical before launching into any college level mathematics. We spend the first two weeks of the semester covering these sections. They are meant to be reminders of things students have already learned, not a complete reteaching of the content from scratch.
The most important section is the first one, where we introduce the idea of mathematical communication. In that section, we provide students a framework that we expect them to follow. The idea is to slow them down and get them to think about the algebraic manipulations, and to otherwise disrupt the bad habits they've developed. Without this shift, it's significantly more difficult for students to make the necessary changes.
Beyond the main trunk, the students are expected to work on the sections more or less on their own. We do have times of in-class activities that will include demonstrating that they've worked through these sections, but we generally do not intend for our instructors to directly teach out of the book after completing the main trunk. The textbook is written in a conversational tone that students should be able to read and understand.
Rather than being a few big, complex ideas, it's really more a collection of a lot of little ideas to help math "make sense" to students. The instructors are expected to be available to help students if they get stuck on topics, but we are generally confident in the students' ability to mostly work things out on their own.
The emphasis of the branches is not about getting students to execute technical manipulations. It's about getting them to start to think and interact differently with mathematical concepts. Remember that the goal is to have increased mathematical thinking and mathematical confidence, not growing their catalog of algebraic manipulations. So while we do also review manipulations (especially see Section 17.5), the context and presentation lend themselves more towards thinking accurately about these mathematical ideas and not treating them as a series of rule-based manipulations.
This perspective is the opposite of how many math textbooks are written. Most of the time, there is a core thread of content and the branches point outward to bigger ideas and more distant horizons. For this book, the branches point inward to the main trunk. Rather than using the main trunk as a launching point for new ideas, it is the home base that we return to over and over again. And this is how we achieve the depth of mathematical thinking.
Simply put, for a support course like this we would rather that students make the mental effort to connect the dots of ideas that already exist in their heads instead of asking them to pursue new topics.