Are Math Courses Worthwhile?

Colleges usually require students to take math courses, whether for an associates degree or a bachelor degree.  The common approach is to identify a level on the “developmental math to calculus” ladder that seems like the best fit.  I believe that this approach is bound to failure, partially because these math courses are normally not worth taking … from the students’ point of view.

Much of the current conversation focuses on developmental mathematics, and whether those courses are worth the investment of institutional resources.  This approach hides the assumption that the gateway college courses are worthwhile for the institution and its students.

During a meeting about general education last week, a comment was made that we all know what college algebra is; to be fair, the speaker meant college algebra as opposed to intermediate algebra.  It’s true that math faculty can tell when they see a college algebra course — because it matches a generic description of college algebra.  The suggestion was also made that a college algebra course includes more demanding problem solving than intermediate algebra.

A student perspective on courses is naturally simpler than ours, but perhaps we need to attend to that perspective to solve our deep-rooted curricular problems.  A course is worthwhile for students when one or more of these conditions is met:

• The content of the course is naturally appealing to a curious mind.
• The abilities developed in the course enable success in other courses (easily seen as such).
• The process of learning in the course is stimulating and/or rewarding (innately).

I’m not describing students who think a course is worthwhile because it was easy, nor those who see primarily value in the social relationships.  I am thinking of students who are looking for an academic reason for taking a course.

A course such as college algebra is doomed to fail all student criteria, at least for most students.  It seems like we, as mathematicians, want students to take these courses so that we can spot the unusual students for whom such an artificial set of content appeals to students via the third condition (the learning is innately stimulating or rewarding).  We seem to take pride in the tidy logic and coherence of the traditional content, forgetting that students might need something different for their needs.

In other disciplines, a gateway course is often seen as an attractor for students — show students how wonderful the discipline is so that they want to see more.  Sociology and french inspire students, sometimes, because wise faculty design such courses to be potentially inspiring to a broad cross-section of students.  When was the last time your college algebra course inspired somebody who was not already STEM-bound?

We would like to have more math and STEM students, but we put courses in front of students that have a strong track record of discouraging student interest in our discipline.  Whether we call it college algebra or pre-calculus, a central goal of the course should be student inspiration.  I do not think our typical courses serve students sufficiently well to be worthwhile.

My concern also applies to other courses besides college algebra or pre-calculus.  We often use statistics as an easier path for students, a sort of “we can’t win anyway, so let’s make it easier” approach.  It’s time for us to re-build our gateway math courses so that they are appropriate introductions into the science called mathematics.  The emerging models of developmental mathematics — AMATYC New Life, Dana Center Mathways, Carnegie Foundations Pathways — can form a foundation for these new gateway math courses.

Our gateway math courses are not usually worthwhile for students, but they can be … and we should make them be worthwhile.

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Acceleration in Developmental Mathematics

Whether it is the recent Achieving the Dream conference, reports from Complete College America, or even education blogs, “acceleration” is a hot topic; acceleration is deemed ‘a good thing’.  Is it?

In physical contexts, acceleration is the second derivative of a position function … the rate of change of the rate of change.  To model acceleration, a valid position function has to be established (along with technical requirements such as continuity).  We can estimate acceleration outside of these limitations by use of numeric methods; however, numeric methods can not yield results appropriate for comparisons of different conditions in a scientific manner.

In education, what is ‘acceleration’?  The applied definition is something like “progression to college mathematics done significantly faster” (usually compared to a tedious sequence of traditional developmental mathematics).  Assessment of acceleration models in education is most often done by anecdote — we now get n% of students through their college course in two semesters, and previously we only got t% through.  This corresponds to the numeric methods for physical situations; are the results valid?

Before we can interpret results from acceleration efforts, we need to have a valid model for position.  We do not currently.  The traditional mathematics curriculum in the first two years is primarily a historical artifact continued though social inertia.  We (mathematicians) have not established what mathematics is required and how this mathematics should be ‘packaged’ into steps; without these steps (courses), we lack valid measurements of progress — which is the heart of the acceleration work.

The AMATYC New Life model (Developmental Mathematics Committee) and the Dana Center (University of Texas – Austin) “New Mathways” provide a consistent message about a package (sequence) that offers a scheme to measure valid progress.  Here is a segment of the New Life vision of the curriculum:

With all of the attention on developmental mathematics, there is a tendency to neglect the critical courses which follow:  pre-calculus needs to be a proven preparation for calculus, college algebra needs to be a proven preparation for other STEM courses, and general education mathematics needs to be proven preparation for other quantitative needs.

Until we tackle this large problem area, acceleration may (or may not) be a waste of effort — getting students to the on-ramp faster does not help if the highway is going in the wrong direction (or if the highway is full of unneeded hazards).  Acceleration efforts make the statement that the college mathematics is both a reliable and a valid position (goal); this statement is questionable.    In this way, I see acceleration as sharing a risk with modularization — they both will tend to entrench existing curricular structures at a time when we need to re-build the structure.

Acceleration is done through a variety of methods, and someday we can determine which method is valid for our position function (curriculum).  Until we get closer to that goal, I would not invest significant resources in acceleration.

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Towards Effective Remediation: Culture of Learning

Two events are fairly common in my classes: (1) A student says “just tell us” when we are doing group work, and (2) A student says “I don’t get that at all” when I am doing a (mini) lecture.  I suspect that these are also common in other classes, and wonder what meaning we each see in these statements.

Both statements deal with confusion and frustration.  One occurs when a student is struggling to find the idea in their own work, and the other occurs when a student struggles to understand the idea in my work.  Both are normal, and both are part of a learning process.

A culture of learning would be shown by an acceptance of these frustrations, combined with a determination to learn in spite of (or because of) that frustration.  Learning is rewarding just for its own sake as we see how ideas connect and build on each other.  A focus on comfort defeats a learning attitude.  Perhaps a focus on the learner raises the same risk.

We tend to see the phrase “student centered” as a positive goal usually implying a process whereby students find ideas about mathematics.  For some of us, this means that we seek to minimize frustration and/or confusion.  I think a better goal is to manage the frustration and confusion to maximize learning and build a culture of learning.  I want my students to see learning mathematics as a set of goals which are attainable given effort and attitude.

We can also see ‘student centered’ as an idea leading to a focus on context and applications, perhaps to the extent that we only cover mathematics that can be applied to problems of interest to students.  As much as I am enthusiastic about applications (I teach a course 100% ‘applications’) I think it is a mistake to construct a curriculum around problems that students can understand and care about — these must be included, but a culture of learning means that we look at extending beyond the immediately practical to the larger ideas and even the artistic beauty of the subject.

In every course, I seek to present some beautiful and useless mathematics.

I know that few of my students achieve this culture of learning, even though my goal is to get them so motivated to learn that nothing will stop them from learning more mathematics.  I know that most of my students will stop taking mathematics as soon as that becomes an option, even though my goal is to inspire them to take at least one more math course than they are required to take.

Students seldom achieve more than our goals and expectations, so I have this culture of learning as a goal in my classes.  Rather than a limited range of ‘student centered’ ideas, I am looking at the largest possible picture of what that means — including how we deal with frustrations and confusions.  Learning, as in life, mostly is determined by how we deal with such problems; learning, as in life, is damaged by attempts to avoid confusion and frustration.

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Towards Effective Remediation: Quantity and Pacing

I’ve been having conversations about arithmetic and similar topics (at the Achieving the Dream conference, and online at MATHEDCC).  In some cases, the conversation was the result of telling people about the New Life model (see https://www.devmathrevival.net/?p=1401).  In other cases, we had been discussing other issues.

So, I have been thinking about related issues.  As a result, I have some ideas of how to frame a conversation about developmental mathematics that might help us make progress.  To start with, we often describe developmental mathematics by using names of courses or by listing topics with implied outcomes.  In our New Life work, we actually started by examining the types of mathematics that students need to succeed, and dealt later with course names and lists of topics.

Our curricular designs are based on our assumptions and goals, which are often unstated.  One of the most problematic assumptions is:

It is effective to deal with 8 general topics, with 10 to 15 outcomes within each, in one course.

There are two fundamental problems with this.  First, the courses we design cover so much ‘material’ that we prevent the learners’ brains from dealing with the associations and connections that are part of learning; this results in most students focusing on just remembering what to do, rather than making sense of it.  Second, the design is based on the absence of prior learning (good and bad) in the learner; this is obviously not true in almost all cases.  Time is needed for us and students to identify where there are conflicts between prior learning and current need, and time is needed to deal with these conflicts.  The result is that we add another layer of ‘learning’, one that is weaker than prior learning; students after our courses are notorious for returning to wrong methods and ideas after our course is done … because we do not provide a method of correction.

We need to slow down; learning is much more complex than having a list of 80 to 120 outcomes.  Since we need to go slower, we must be strategic about what areas to focus on .. trying to do it all (or even most of it) means that we are willing to accomplish little of significance.

This strategic work should be based on our judgments as mathematicians about which mathematical ideas are most important in particular cases.  Do we want work with fractions, or do we want work with proportional reasoning?  If we want both, what are we willing to give up … percents or linear models (as examples)?

We need to do some critical thinking about our goals and purposes, and apply our problem solving skills so that our courses are effective learning experiences for our students.

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