Foundational Mathematics for Corequisite Courses: Succeeding at College Mathematics

But the longer answer to the question is that this moment of disruption is also the right time to try to make a change in how remedial mathematics (including corequisite courses) are taught and understood. I’ve spent over a decade working with students with weak mathematical backgrounds. I’ve watched how they think and how they attempt to learn mathematical material. I’ve come to the conclusion that I think there are better ways to help students become successful.

Of course, if we are talking about "success" then we need to define what that actually means. I will say that the traditional methods of remediation are pretty good at doing what they are designed to do. The primary emphasis of traditional remediation is to help students become proficient at specific algebraic manipulations. The entire structure of textbooks is based on that premise:

“Here is an example. Now try a problem exactly like that one. Now do that 50 more times (but only the odd numbered problems so you can check your answer).”

If all you want out of students is to get them to perform manipulations on command, this is a very good way of doing it.

But as I think about the students I encounter, I don’t think this is really that helpful. I question whether they actually remain proficient in those manipulations after the semester is over. After all, they’ve been through this before, perhaps even two or three times. Why is this time going to be different? I also question what they actually learned from the class, and I question the true value that the students get from courses like these. It all comes back down to a question that’s at the core of education: What do we really want students to learn?

If we only have one semester to teach students about mathematics, do we really want to teach them that he core of mathematics is doing algebraic manipulations? Is that what math is? Is that what we actually care about? I understand that some people would agree that it is. They think the goal is to get students to be proficient at these particular algebraic manipulations so that they can execute those manipulations in their other courses (college algebra/precalculus, physics, chemistry, statistics). I don’t think this is wrong, I just think it’s short-sighted.

I believe that the core skill that students should get is not mathematical manipulations, but the development of mathematical reasoning. The emphasis of the K-12 system is still currently heavily invested in mathematical manipulations. If you look at most Algebra 2 textbooks, you’ll find an incredibly broad range of topics that are covered. Most of these topics end up being manipulations piled on top of other manipulations.

When I look at students in remedial courses, I make two primary observations about them. The first is that students are often very confused about mathematics. They operate from a very rule-based perspective and often feel as though the bulk of their work is memorizing manipulations and memorizing when they need to use them. The second, which follows from the first, is that they completely lack confidence in their mathematical abilities. This is a learned helplessness from all the times they tried to memorize something and failed. They have had many years to develop the "not a math person" identity where they do not embody any level of mathematical confidence and show few signs of mathematical reasoning. Many students simply guess at whether they are doing the right thing, then sit back and wait to be told whether it’s right or wrong. And when you look at the educational system that they’ve come through, you can understand why this is.

My goal with this book is to change how students think about mathematics. I would love to have two or three semesters of math courses to really bring students to a place of thinking about mathematics at a college level. But there is only so much that can be done in one semester, especially when that semester is contextualized as a support for a college level math course. And that’s the reality. We simply need to do the best we can with the time that we have.

This book is written for all college students who are interested in improving their mathematical foundation. This includes students that are either destined for a liberal arts or statistics college math course, and students on the STEM trajectory headed towards calculus. I also think this has value to anyone who is teaching or will teach math at any level (especially K-12). There is a lot of systemic brokenness in the math education system, and I will be glad if these ideas help to fix even a smallest part of that.

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.

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 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.

The suggested course alignment below is based on having extra contact time with students for the corequisite content, which allows for the extra time up front to set the stage with the Main Branch of this book. (I don’t know how you would try to teach both the support material and the core content without extra contact time!) It’s possible to use this material without it, but students may not adopt the mindset and writing habits if those sections are not treated as the primary learning outcomes. After the Main Branch, very little in-class time is spent directly teaching the Foundations content. Students are expected to read the book, and only a small portion of the class time is dedicated to completing the worksheets and discussing the ideas directly.

Weeks	Foundations Textbook	Precalculus Content
1-2	Main Trunk (Core Algebra)
3-5	Branch 1 (Linear Equations/Coordinate Plane)	Introduction to Functions
6-10	Branch 2 (Fractions/Decimals)	Polynomials and Rational Functions
11-15	Branches 3-4 (Arithmetic/Applications)	Logarithmic Functions

Although the organization of this textbook is not built around other first semester courses (such as a quantitative literacy course or a "liberal arts" math course), the content of the course is perhaps even more valuable to those students and those classes. Those students are actually the ones I have had the most experience teaching. However, those courses tend to be quite varied in their structure, and so it would be impossible to align this text with that. While the branches are set up in a specific and intended order, it’s the ideas are sufficiently self-contained that they can be rearranged without much harm to the overall delivery of the material.

As with most things, this is a work in progress. If you find any errors in the textbook, please email me to let me know. Also, if you have suggestions for additional topics or alternative approaches, I’m always open to new ideas. The best way to contact me is through email (aaron.wong@nsc.edu).