Foundational Mathematics for Corequisite Courses: Succeeding at College Mathematics

Section 1 Underlying Philosophy

Why write this book? That's a question that I needed to have a clear answer to before I began the process. The short answer is that I felt like I had little choice. Here is our reality:

• Nevada is historically among the lowest-ranked states for K-12 math education in the nation.

• The Nevada System of Higher Education passed a policy that completely eliminates any standalone math remediation for students, forcing us to adopt a full corequisite model.

• We want students to be successful in our college level math courses while maintaining our standards.

• The materials that currently exist are insufficient for the task.

And so by the necessity of the situation, we needed to come up with something to help our students.

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.

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.