What are Lower-Division Pathways and why are they important?


Greetings from the moderator of this category! Here’s a quick post to get us started on this topic:

Outside of TPSE Math, lower-division pathways are often called simply “mathematics pathways.” They refer to a course or sequence of courses that meet the program requirements for a degree or credential. Almost every undergraduate program requires some mathematics (hence the common term “gateway”). For most students, this still means College Algebra or Calculus, and whatever developmental courses are needed to get them there.

Although many colleges and universities have improved the math pathways they offer, most are still stuck with one or more of these problems:

  1. Too many students fail gateway math (often 2-3 times);
  2. developmental sequences are too long, and many students drop out before getting to college-level math;
  3. required math is often not relevant to a student’s chosen major (e.g., 90% of students are in majors that don’t require Calculus, yet College Algebra is basically pre-Calculus!);
  4. required math often doesn’t transfer to other institutions - or if the credits transfer, the course often isn’t allowed to apply to the student’s major program (and 40% of students transfer at least once).

Many institutions and a few systems and states have done a lot to redesign math pathways and align them with students’ majors. The most common type of redesign includes quantitative reasoning for humanities and fine arts majors, statistics for social science majors, and College Algebra and Calculus for STEM majors. The most successful redesigns also:

  • include accelerated developmental sequences, so students can get through college math within a year;
  • deal with issues of transfer, placement, advising, pedagogy, etc.
  • align with or feed complementary efforts on student success

Many are finding that students pass college-level math at twice the rate in half the time and that students at 2-year institutions transfer or get a degree at far higher rates.

TPSE Math and other organizations are collaborating on a major initiative to help institutions, systems, and states to redesign mathematics pathways in ways that will lead to increased student success. This collaboration is called Advancing Mathematics Pathways for Student Success (AMPSS).

Next steps for this thread:
Did I capture the main points related to lower-division pathways? I want this thread to be a helpful introduction to the topic.

Next steps for this category (Lower-Division Pathways):
Many of the issues and points I raise in this introduction should be their own topics! I invite you to create a new topic on any one of them, or on new or cross-cutting topics. I will get the process started by starting a thread (“Topic”) on the AMPSS partnership, and possibly some other topics.

I look forward to the discussion!
—David May



I would add one more common problem to your list of 4 (which I also agree with!) There are many students who are away from schooling for years, or even decades. By that point, they no longer know how to use prerequisite knowledge for the mathematics courses they are put into. For example, I have many students going back to school after raising their kids. They take College Algebra and are completely lost and overwhelmed.


Very good point, athompson! These “nontraditional” students are a growing group, and as you say may have different needs than those who are coming to college straight out of high school. In fact, a new report from the Community College Research Center presents the idea that “virtually all entering students need help developing skills and habits to thrive in college,” whether those students are considered “college-ready” or not.

The solution is not only to design math pathways that work for students of all kinds, but to ensure that placement and advising practices are put in place that help make sure the right students get the right kinds of support on the right pathways.

Other student groups deserve particular attention as well, such as first-generation college students, students from low-income families, immigrants, and students from minority groups that are underrepresented in college. The problems I list, and the problem you brought up that I didn’t list, often impact these students disproportionately, making math pathways redesign an issue of equity and justice. If someone wants to open a new thread on that topic, go for it!

Thanks again for your comment, athompson7.



No, college algebra is distinct from pre-calculus at many institutions. From the MAA’s CRAFTY report Partner Discipline Recommendations for Introductory College Mathematics and the Implications for College Algebra :

"Nationally, there is no general agreement on the content of college algebra. Some institutions teach college algebra as a terminal mathematics course while other institutions view college algebra as part of the pre-calculus track. In one of the participating institutions a long list of topics for inclusion in college algebra was mandated by the state. In other institutions a similar mandate came from the departments."



Thanks for the clarification. I suppose I overgeneralized, and you’re correct: the CRAFTY report does indeed show that the content of College Algebra varies from place to place.

The point I was trying to make about College Algebra is essentially the same as the CRAFTY report’s point: the course generally doesn’t meet most students’ needs. CRAFTY was making the case for a careful overhaul of College Algebra, and I was promoting the design of other pathways, but I think both are needed.

Even if College Algebra isn’t everywhere the same as Pre-Calculus, in general I think there is significant overlap in the content of the two courses - see Burdman, p.10 (PDF), for example, or the MAA’s Common Vision report which refers to College Algebra as “a curriculum designed to prepare students for calculus” (p13). Yet, as your quote from the CRAFTY report implies, most institutions don’t intend for these students to go on to take Calculus, and most don’t. Hence the mismatch I was getting at.

Deciding what to do about math pathways is highly context dependent: each institution, system, and state will have its own policies and structures, and each student or group of students will have their own different needs. The purpose of looking at general trends and for general strategies is so that reformers at each different site need not reinvent the wheel completely from scratch.

Thanks again for clarifying and wading into this complex issue with me!



I do not find it necessarily damning that there is significant overlap in precalculus and college algebra. I would be surprised if there weren’t significant overlap in the content of a math pathways course and any precalculus course that emphasizes conceptual understanding over algebraic manipulation skills.

CRAFTY’s documents (and AMATYC’s The Right Stuff) promoted college algebra courses that would be relevant to students not necessarily needing calculus.

I’m saddened when useful innovative programs go unacknowledged or are even denigrated because another program of the same or similar name is flawed.

For another example, I know a math professor who consistently disparages math pathways, describing them as “a total disaster for our most marginalized students”–evidently reaching this conclusion because of disliking a program designated as a math pathway and imposed on this person’s institution.


I’ve always argued that the right terminal math class for non-STEM students is Finite Mathematics. You hit combinatorics, probability, linear systems, Markov chains, linear programming, etc., These are great ideas and more importantly the students learn how to think (!).

At BYU several years ago, I spend a year convincing the business school to dump business calculus (which is an abomination) and switch to finite mathematics instead. I think the results have been great.

Another benefit of finite math is that it can be taught in larger classrooms and exams can reasonably be made multiple choice. These efficiencies allow for scalability at larger universities. Many of the proposed pathways (like a math modeling approach) just aren’t practical in a large university where graduate students, lecturers, and adjuncts are often tapped to lead the instruction. Like it or not that’s a reality that needs to be considered in these discussions.



Some of the most compelling work I’ve seen is colleges coming together to provide a more coherent framework for first year students across the board. I’m seeing a growing number of colleges think about math pathways for all students. What that requires is that a team from the math department meets with every program/discipline to take a hard look at the math knowledge and skills students need to master the math content in intermediate and upper lever courses. This puts faculty in exactly the role they should play. Defining the learning objectives of math pathways based on what students need to excel in the field of study.

I agree that some of the great work that AMATYC, MAA, and ASA have done has languished on bookshelves. That could be said for much of the great work the NSF has funded. These innovations need a systems approach do spread the knowledge of what works. We know from research on the spread of research findings in medicine that it takes between 15 and 20 years for a new finding to work its way into day to day practice. My greatest hope for TPSE Math is that we can shorten that time frame for our underserved students.


Has anyone used AMATYC’s “The Right Stuff” modules and, if so, what are your thoughts?


Jeff, I agree that finite math is a great topic for all students, particularly the “learning how to think” reason you emphasized. I’m sure many departments could be convinced to require finite math (or a course that has a lot of finite math) for their majors. I do think it’s important to find out what faculty in each discipline think their students need, however, which can sometimes be surprising. I’ve seen business colleges where some majors need calculus and others need statistics and yet others need other math.

Many are also taking the important step of redesigning pedagogy and course structures, such as by using “flipped classrooms” and interactive engagement strategies, that often have a lot of evidence behind them. I know changes like that have worked for very large classes that are usually lecture-based; I expect some finite math classes are among them. But I do understand that sometimes you have to start with getting faculty to talk about what math students need before they’ll take on the conversation of how best to teach it. Sometimes.


That’s why I said it took a year to talk them into abandoning business calculus. Many felt strongly that they needed calculus, but none of them could cite an example where a manager would use calculus out in the wild, and yet it’s easy to come up with countless examples where they would use Finite Mathematics.

So just as time and pressure moves mountains, it too moves deeply entrenched faculty.

I avoid discussions on pedagogy. It can become a religion, and I prefer to remain agnostic on the issue. I would suggest that, as mathematicians, we avoid those discussions and stick to content. It’s where we are the authorities and where we have the high ground.