Category: Math curriculum in general

What Does the Future Look Like? College Mathematics that Works!

We live in a transition time, for college mathematics.  Developmental Mathematics is shifting away from the traditional curriculum, with an over-use of “prerequisite remediation” in the short term.  At the same time, both of the primary professional organizations for our work (MAA and AMATYC) have been calling for basic shifts in both what we teach and how we teach within the ‘standard’ college level mathematics courses.  What does our eventual ‘target’ look like?  Can we anticipate where we will end up?

In a basic way, the answer to the last question is ‘yes’, due to the fact that all of the forces shaping the future are known at the present time.  We don’t know precisely which forces will have a larger influence, and that is fundamental since the forces are not operating in the same direction.  Imagine yourself in an n-dimensional force field where you can see the vectors around you.  Although the wind varies over time, some types of vectors dominate your environment.

These vectors around us originate from power sources.  Professional standards (MAA, AMATYC, etc) send out vectors in the direction of higher levels of reasoning, modern content, more diverse content, and more sophisticated instructional methodologies.  The K-12 educational system, the Common Core in particular, send out vectors in very similar directions.  Policy influencers, higher education provosts and chancellors, and state legislators send out vectors representing forces in different directions from those in the prior lists.

In the short term, this latter set of forces will dominate … because some of the individuals involved have sufficient decision making power that they can impose a set of practices on portions of our work.  However, these practices will not survive long term except to the extent that they support the prevailing set of forces around us.  As the people in authority change faces, the practices will tend to revert … either to the pre-existing conditions (bad) or to a condition making progress in the direction of the prevailing forces.

Here is a description, a picture, of where we will be in 10 to 15 years.

  • Remediation will be smaller than in the past, but still normally discrete (not combined with college courses as in co-requisite models).  Arithmetic will be ‘taught’ but never as a separate course and never will be a barrier to a college education.  Content will focus on the primary domains of basic mathematical reasoning — algebra, geometry, trigonometry, statistics, and modeling.  No more than two remedial courses will ever be required of students, regardless of their ‘starting condition’.
  • “College Algebra” will not be used as a course title.  Similar courses for non-STEM majors will have titles such as “Functions and Modeling in a Modern World.  The content of this course, never used as a prerequisite to standard calculus, will be from the same domains as remedial mathematics — algebra, geometry, trigonometry, statistics, and modeling.
  • “Pre-calculus” courses will be replaced by a one-semester “Intro to Math Analysis” course which focuses on the primary issue for success in calculus: reasoning with flexibility supported by procedural understanding.  This course will have a very strategic focus in terms of objects and skills involved, with a shorter topic list than prior courses … taught in a way which results in a true readiness for calculus.
  • “Calculus” courses will be re-structured to focus on a combination of symbolic and numeric work.  The first semester of the two-semester sequence will include derivatives and integration for basic forms, as well as an introduction to scientific modeling using matrices such as those encountered in the client disciplines; this eliminates the need for our client disciplines to teach basic quantitative methods, and provides modern content to serve those disciplines.  The second (and final) semester calculus course focuses on multi-variable processes combined with a more complete approach to scientific modeling — appropriate for students who may eventually conduct their own research in a client discipline
  • “Liberal Arts Math” and “Quantitative Reasoning”  will have merged in to a new QR course at most institutions.  At some institutions, these courses are replaced by the “Functions and Modeling” course (which is fundamentally a QR course).  Where QR exists as a separate course, the ‘practical’ content will be de-emphasized relative to today’s courses, with an increase in symbolic mathematics. The primary distinction between QR and Functions and Modeling is that QR does not include as much trigonometry.
  • “Intro Statistics” will exist with similar content to the best of today’s courses.  The primary change will be a relative decrease in the number of students taking a Stat course to meet a degree requirement, as program planners realize that their mathematical needs are more diverse than statistics … and that requiring statistics should not be based on just a desire to avoid college algebra (which does not exist in this ‘now’).
  • Students will become inspired to consider a major in mathematical sciences by the diverse quality content along with the effective methods used within the courses.  Instead of a focus on weeding out students not ‘worthy’ of majoring in mathematics, we will focus on including all students on the mathematical road to maximize the distance covered.

I see an exiting future, once we get past the relatively short-term impacts of changes imposed from outside.  In the long-term, nothing can stop us from achieving a desired goal … except for our own doubts and lack of clarity.

My hope is that you see something in this image of the future to get excited about, something that plays the role of a beautiful sunrise in the forest.  If you can SEE where you want to go, you can get there … and it is a lot easier to survive temporary struggles along the way.

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The Selfishness of the Corequisite Model

One of the major ‘things’ right now is “corequisite remediation”, referring specifically to the practice of placing (most or all) students in a college mathematics course with a requirement that certain students take a support class.  Over time, we will discover that this has sufficient promise to justify further exploration and use.  The problem is … the practice is very selfish on our part in many of the common implementations.

Most data on this practice comes from two college math courses — introductory statistics and liberal arts math.  For most offerings in those courses, few prior skills are needed for success; in both cases, the most common need is for expressing fractions as percents and some proportional reasoning.  Algebraic reasoning is seldom needed.  In both cases, the legacy prerequisite (usually intermediate algebra) was an artifact more related to establishing transfer than to course needs.  Few co-requisites structures have been done in college algebra nor in quantitative reasoning with an algebraic emphasis

My contention is that using co-requisite methods for a non-STEM math course amounts to a selfish decision on our part.  We place a higher priority on improving our ‘measures’ of completing that one math course … at the expense of preparing students for other quantitative needs in college.  This is especially an issue for our friends in science, who often depend on a variety of algebraic concepts in their courses (as they should).  The co-requisite model focuses almost totally on “What do they need for THIS terminal math class?” (which is a small set); ignored is the larger set “What do students need for college quantitative work?”.

Now, it is true that the traditional developmental mathematics courses do not deliver on that larger set — at least, not in an efficient manner or with good results.  Replacing developmental math courses with the co-requisite model (as is being suggested) is placing students at risk … just so a math course can ‘look better’.  Our response should be:  “How can we make fundamental improvements in the course content and design so that developmental mathematics works for almost all students across their college program?”

Our reason for existence in developmental mathematics is the whole student for their whole college experience.  We can achieve short term ‘better results’ for ourselves with co-requisite remediation.  This comes at the expense of leveling the playing field, equity, and student success in general.  Can we be that selfish?

I realize that I am attributing a personal motivation to a practice in the profession.  I’m okay with that … most of us are in this profession because of personal motivations.  I think large numbers of ‘us’ have a deep commitment to equity, as well as upward mobility.   Co-requisite remediation creates a system focused on the short-term ‘results’, often involving minority-heavy support classes, with few long-term benefits (if any).

Our responsibilities involve much more than “one math course”.  Let’s do our job.  Instead of taking the easy ‘out’ of co-requisite remediation, we should replace the traditional sequence of developmental math courses with a very short structure to serve all college students and all of their needs … Mathematical Literacy and Algebraic Literacy (or similar courses).

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Transitioning Learners to Calculus in Community Colleges (TLC3)

You might have heard of the MAA project “National Study of College Calculus”  (see ).  That work was very broad, as it studied calculus in all 3 settings (high school, community colleges, and universities).

A recent effort is focused on community colleges  with the title “Transitioning Learners to Calculus in Community Colleges”   (info at  )  Take a look at their web site!

One component of their research is an extensive survey being completed by administrators of mathematics at associate degree granting public community colleges, including the collection of outcomes data.  A focus is on “under represented minorities” (URM), which relates closely to a number of recent posts here (on equity in college mathematics).

I am expecting that the TLC3 data will show that very few community colleges are successful in getting significant numbers of “URM” students through calculus II (the target of this project).  The ‘outliers’, especially community colleges succeeding with numbers proportional to the local population of URM, will provide us with some ideas about what needs to change.

Further, I think the recent emphasis on ‘pathways’ has actually decreased our effectiveness at getting URM students through calculus; the primary assumption behind this (based on available data) is that minorities tend to come from under-performing K-12 systems which then results in larger portions placed in developmental mathematics.  The focus on pathways and ‘completion’ then results in more URM students being tracked into statistics or quantitative reasoning (QR) pathways — which do not prepare them for the calculus path.  [Note that the basic “New Life” curricular vision does not ‘track’ students; Math Literacy is part of the ‘STEM’ path. See ]

Some readers will respond with this thought:

Don’t you realize that the vast majority of students never intend to study calculus?

Of course I understand that; something like 80% of our remedial math students never even intend to take pre-calculus.  Nobody seems to worry about the implication of these trends.

Students are choosing (with encouragement from colleges) programs with lower probabilities of upward mobility.

The most common ‘major’ at my college is “general associates” degree.  Some of these students will transfer in order to work on a bachelor degree; most will not.  Most of the other common majors are health careers (a bit better choice) and a mix of business along with human services.  Upward mobility works when students get the education required for occupations with (1) predicted needs and (2) reasonable income levels.  Take a look at lists of jobs (such as the US News list at )  I do not expect 100% of our students to select a program requiring calculus, nor even 50%; I think the current rate (<20%) is artificially low … 30% to 40% would better reflect the occupational needs and opportunities.

Our colleges will not be successful in supporting our communities until URM students select programs for these jobs and then complete the programs (where URM students select and complete at the same rates as ‘majority’ students).  Quite a few of these ‘hot jobs’ require some calculus.  [Though I note that many of these programs are oriented towards the biological sciences, not the engineering that often drives the traditional calculus curriculum.]

I hope the TLC3 project produces some useful results; in other words, I hope that we pay attention to their results and take responsibility for correcting the inequities that may be highlighted.  We need to work with our colleges so that all societal groups select and achieve equally lofty academic goals.

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Segregation in College Mathematics: Corequisites! Pathways?

So, this post will mostly apply to those of us located in urban colleges (more than rural).  The idea is to remind of the societal costs of “separate and not equal”.

As a general statement, urban public schools have more challenges than suburban schools (and more than rural schools).  The role of poverty in this situation appears substantial, and the burden of this poverty tends to fall on ‘minority’ students more than ‘majority’.  In this post, I’m focusing on two categories — black/African American and white/Caucasian.

If you track the proportion of each course that is black, you are likely to get a chart like this one.  Note that “0” represents a college-level math course (most commonly ‘college algebra’ … more on that later).











This comes from a college where black students represent about 10% of the population; the college does not have a “-3 course” (pre-algebra).  The pattern in course enrollment is a similar pattern to the ‘placement levels’ of each group … the mean placement level for black students is about -1.4 compared to -0.6 for white students.  If all students are in a sequence (‘path’) that produces an equal chance of succeeding to all college mathematics, there is ‘equality’ (given the unequal starting points).

However, two current trends break that ‘equality’ and produce a system of separate and unequal.  In many co-requisite models, students who do not place into college mathematics are given only the option to take a non-STEM math course (statistics or quantitative reasoning aka ‘QR’).  In general, colleges using a co-requisite model find that their ‘support sections’ (ones taken by non-placing students) are predominantly minority.  I know some colleges have tried to use co-requisite models in college algebra (though more often ‘intermediate algebra’); these results are seldom published, and I think this is due to the much lower ‘results’ than statistics or QR.  The result of this type of system is an unequal result for minority students — they are discouraged (or even prevented) from pursuing a STEM or high-tech program.  A new segregation is being sold to colleges, in the name of ‘better results’; more on that later!

Some ‘pathways’ implementations also produce this same unequal pattern.  Those placing ‘lowest’ and ‘struggling students’ are strongly encouraged to take a stat or QR pathway program; some of these programs actually do allow students to select a STEM or high-tech program, but many do not.  The most common model is a side-by-side design … Math Literacy (or similar course) as an option to beginning algebra, where the Math Literacy course only leads to stat or QR.  In the K-12 world, this is called “Tracking”.  Pathways often create a segregated condition, due to the impacts of the lower-performing K-12 schools.

One argument is that the co-requisite models (and pathways) at least get students to complete a college math course, most commonly stat or QR.  The question remains … so WHAT?  There is an assumption that this stat/QR approach results in more students getting a degree (likely to be true).  But … what good is the degree?  Are there actually jobs for that program?

Obviously, the answer to that last question is ‘in some cases’.  In some regions, nursing requires either statistics or QR for their associate degrees, and the employment outlook is often good.  However, these health careers programs can be ‘selective admission’.  My experience has been that students accepted in to a nursing program tend to be ‘whiter’ than the college population in general … which likely goes back to the urban school system problems.  As a practical matter, I don’t think that a focus on stat or QR, in either co-requisite or pathways, results in ‘equal’.  We are creating separate in a deliberate strategy, without ensuring that they are equal.  [Of course, it’s also reasonable to say that we should avoid “separate” in the first place.]

Now, I’m not saying that co-requisite and pathways have no place in college mathematics.  The concern deals with the ‘scaling up’ that is often sought with them, as well as the target population.  Co-requisite remediation can be quite effective at the boundary … students who “just miss” qualifying for their college course (stat, QR, or college algebra); this system can be used to partially offset the negative impacts of lower-performing K-12 schools.  Pathways keep our focus on getting out of the way as much as possible … get them to their college course quickly; however, all pathways should preserve student options.  Any pathway that blocks student options is very likely to result in ‘unequal’ conditions.

Both of these efforts (co-requisites, pathways) remind me of the segregation caused by ‘school of choice’.  Do we really want to institutionalize segregation in these new ways?

I think the better response is to modernize the entire mathematics curriculum in colleges.  Start by replacing arithmetic and basic algebra courses with Math Literacy with an intentional design to provide students options at the next level.  Replace intermediate algebra with Algebraic Literacy with its intentional design to prepare students for modern college mathematics courses.  Replace college algebra with a course likewise designed to actually prepare students for calculus.  Reduce the calculus curriculum to fewer courses while incorporating more numeric methods (see “Common Vision”).

We do not need to create separate conditions for students, not nearly as much as we need to modernize our curriculum.

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