Category: Math curriculum in general

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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Why We Will Stop Doing Pathways in Mathematics

Currently, and for the past few years, “pathways” has been a big thing in community college mathematics education.  For students not needing calculus or similar courses, alternate paths have been established — with a focus on courses such as Statway™, Mathematical Literacy, and Foundations of Mathematical Reasoning.  The fact that all three of those courses are very similar in content is not an accident, and the fact that the three organizations involved collaborated is a key reason for their success.

The reasoning behind the creation of pathways is essentially “give them what they need, not what they don’t need”.  Students with a pre-calculus target are still placed into the old-fashioned developmental math courses, and students with other targets are placed into a ‘pathway’.  All students are generally required to meet some arithmetic criteria before starting at the Math Literacy level or beginning algebra.

My own work has certainly played a role in this creation of pathways.  However, that was not the intent of the efforts beginning this work.  Neither do pathways have a good prognosis for long-term survival.

Let’s go through some of the reasons why “pathways” are not a long-term strategy.

Reason 1: Pathways are a dis-service to “STEM” (calculus-bound) students!
The original design of the major pathways courses (Quantway™, Math Literacy and Foundations of Mathematical Reasoning) was based on identifying what all students needed in college-level mathematics — statistics, quantitative reasoning, AND pre-calculus.  These outcomes were then categorized in two clusters … those needed by ALL students became the core of the Math Literacy course, and those primarily needed by pre-calculus students became the core of Algebraic Literacy.  [Algebraic Literacy also includes some outcomes needed for technical programs.]

In effect, “pathways” is preventing STEM (calculus-bound) students from getting the learning they need for success.  We have accumulated data showing the the traditional developmental algebra courses do not add significant vale for these students when they take pre-calculus.  In addition, we also know that the traditional courses were not designed for this purpose — they were designed to replicate the 9th to 11th grade content of a 1970’s high school.

Pathways create a better experience for non-STEM students, at the price of harming (relatively) those bound for pre-calculus.

Reason 2: Curricular complexity costs too much
One of the extreme cases I have seen is a college with SIX different courses at the Math Literacy level.  Clearly, half of these are quite specialized for students in particular occupational programs.  However, half were general in nature — a Math Literacy course, and two basic algebra courses.

Curricular complexity raises the cost of support functions at an institution, advising in particular.  Few colleges can support this extra work in the long-term, even when the initial launch of those efforts is strongly supported by the then-current administration & governing board.  As time goes on, the focus on advising slips … mistakes are made … and a later administration will question why things are so complicated.

This curricular complexity also raises costs within the mathematics department.  More courses at the same level means more difficult scheduling, less predictable enrollments in each course, and a host of faculty coordination issues.  Unless an institution has excess resources not needed for other situations, the mathematics department will realize in a few years that they can not support the complex curriculum.

Reason 3: Pathways allow the continuation of arithmetic courses at colleges
The presence of arithmetic courses at a college involves several problems and costs; the fact that our profession has not accepted these are overwhelming rationales for discontinuing arithmetic courses is a failure with moral and economic dimensions.

First of all, these extra courses at the developmental level are primarily taken by students of poverty and minorities.  This is the moral dimension for us:  these are the students coming to college to get out of poverty, who are then required to take one or more courses prior to the course that is a prerequisite to their required course.  No possible benefit from learning arithmetic can justify this process; in fact, there is no evidence of any significant benefit for taking such arithmetic courses in college.

Secondly, arithmetic courses in a college create costs for the mathematics department. We often have a fairly discreet set of faculty (heavily adjunct), and these faculty are seldom qualified to teach a college mathematics course.  In many colleges, the arithmetic courses are administered in a separate department.  As faculty, we should want to design a curriculum that does not depend on a course at the arithmetic level.

Thirdly, the presence of arithmetic courses at a college will tend to perpetuate the outdated focus on procedures and answers.  This conflicts with the design of Math Literacy, and impedes development of basic reasoning needed even in a traditional basic algebra course.

Reason 4: External Forces Will Continue to Push Us To Change
So far, the evaluation of ‘pathways’ has focused exclusively on the impact for students taking Math Literacy (or companion course) as preparation for statistics or quantitative reasoning courses — specifically, students who enroll in stat or QR after passing Math Literacy.

Curricular complexity means that there will be a less successful experience for students needing pre-calculus … by definition, because those students need two courses (beginning algebra, intermediate algebra) compared to the one & done of Math Lit.  There are also operational causes for other ‘bad’ data to show up — students taking Math Literacy instead of the course they were supposed to take, for example.

In addition, we can predict that these change agents will critique our developmental math courses compared to modern standards (whether Common Core, or NCTM standards).  We are not ready for this critique, and have no response for the results that are bound to come from such a critique — that developmental mathematics operates as if the year is still 1975, ignorant of the fundamental changes in our students’ experiences in K-12 mathematics.


In a way, I am reminded of something I learned at a conference session on graph theory and traffic design.  Our intuition might say that it is better to have more options in street designs, where there are several north-south options and several east-west options.  The traffic design results were the opposite … that the best throughput for a traffic system is the fewest possible streets.

A pathways curricular design presumes the presence of at least two courses at the same level in a sequence.  This design is not particularly stable, as a system.  In the long term, I think the system will collapse down to one of the options.

We need to be prepared for the demise of pathways so that we can maintain the improvements from those efforts.  The danger is in assuming that both Math Literacy AND the old courses will ‘always’ be there.  Within a few years, one of them will be gone.  Which type of course do YOU want to survive?

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Intermediate Algebra … the Barrier Preventing Progress

The traditional math curriculum in colleges is significantly resistant to change and progress; I talked about some of the reasons for this condition in a recent post about the Common Core & the Common Vision related to the future of college mathematics (see  )  We carry some historical baggage which creates additional forces resisting efforts to make progress in the curriculum at the college level.

Our “Intermediate Algebra” course occupies a position of power.  First … it has long served as the only accepted demarcation between “college level” and courses which are not.  AMATYC recently approved a position statement to help clarify this demarcation (see )   Second … it has been used as the prerequisite to both college algebra and pre-calculus, which contradicts the origin of intermediate algebra as a copy of HS algebra II (which was never designed for this prerequisite role).

I’ve written previously about the need for Intermediate Algebra to be intentionally removed from the college curriculum; see

Intermediate Algebra must die … now!

Recently, we’ve had some email discussion in my state about the credential requirements for faculty … especially those teaching “intermediate algebra”.  Although we all want to provide students with quality faculty for every math course, we don’t agree on what this means.  Like most accrediting bodies, ours makes a distinction between developmental courses and general education courses; developmental courses require that faculty have a degree at least one level above what they teach … while general education courses require that faculty have 18 graduate credits in the field they are teaching.

Because of that credentialing difference, faculty teaching college mathematics courses tend to be functionally separate from those teaching developmental math courses (unless at a small institution).  A consequence of this faculty split is that the interface zone (intermediate algebra to college algebra in particular) is difficult to change in basic ways.  Faculty with a STEM focus are more concerned with their ‘upper level’ courses (calculus, linear algebra, etc), while those with a developmental focus are often more concerned with the beginning algebra level.

Intermediate algebra, just by its presence in our curriculum, is a barrier to making progress in modernizing our work.  If we were to remove Intermediate Algebra as a course, both levels of mathematics faculty would (by necessity) work together to create a more reasonable replacement.  If Intermediate Algebra had never existed, do you think we would create that same course now?  Obviously, no … we would do something much more reasonable.

Intermediate Algebra must die … now!

Efforts to ‘improve’ intermediate algebra typically involve micro-adjustments (different mix of skills).  Changes of this type have been tried over the past 30 years (or more) with almost no impact on any problem or outcome.  Our problems have become severe enough that no set of micro-changes will create a solution … we need macro-changes.

We need to remove the barrier — get rid of your intermediate algebra course (and mine!).  Replace it with a modern course like Algebraic Literacy ( if that makes sense to you.  Or, create a different solution for the problems.  Of course, part of the solution is to keep some of the students out of any course at the intermediate algebra level — developmental but preparing for college algebra.  Intermediate algebra is certainly not needed as preparation for statistics or quantitative reasoning at the college level.

Some of us are having a strong response to this proposal (of removing the intermediate algebra barrier).  If you live in a state that has a policy of ‘intermediate algebra for general education in college’, or your institution has such a policy, you are experiencing another reason why intermediate algebra is a barrier that must be removed.  Intermediate algebra is a copy (sometimes quite weak) of an old high school mathematics course in an era when the overwhelming majority of our students have experienced more advanced mathematics in their high school.  This was true before ‘the Common Core’, and is becoming more true as time goes on.

Intermediate Algebra must die … now!

We can create viable solutions, with modern courses about current mathematical needs, if we are just willing to toss this one course from our curriculum.  Intermediate Algebra must die, and die soon.  It is a barrier to progress that we … and our students … need urgently.  Don’t wait for a replacement to be ‘ready’ — the solution will be ready when we are committed to make a change.

Which of these is your choice?

  • Eliminate intermediate algebra at your institution effective Fall 2018
  • Eliminate intermediate algebra at your institution effective Fall 2019
  • Eliminate intermediate algebra at your institution effective Fall 2020
  • Ignore the intermediate algebra problem, and hope it goes away by itself.

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