Equity and Stand-Alone Remedial Math Courses

One of the key errors that co-requisite (mainstreaming) advocates make is the treatment of ‘developmental mathematics courses’ as a single concept.  We would not expect college students who place into arithmetic to have comparable outcomes to those who place into intermediate algebra.  However, most ‘research’ cited with damning results uses that approach.  We need to have a more sophisticated understanding of our work, especially with respect to equity (ethnicity in particular).

A local study by Elizabeth Mary Flow-Delwiche (2012) looked at a variety of issues in a particular community college over a 10 year period; the article is “Community College Developmental Mathematics: Is More Better?“, which you can see at http://mipar.umbc.edu/files/2015/01/Flow-Delwiche-Mathematics-2012.pdf   I want to look at two issues in particular.

The first issue is the basic distribution of original placement by ethnicity.  In this study, ‘minority’ means ‘black or hispanic’; although these ethnicity identities are not equivalent, the grouping makes enough sense to look at the results.  The study covers a 10 year period, using cohorts from an 8 year period; partway through the 8 year period, the cutoffs were raised for mathematics.

Here is the ‘original’ distribution of placement by ethnicity using the data in the study:
Distribution by level Flow-Delwiche 2012 Original









After the cutoff change, here is the distribution of placement:
Distribution by level Flow-Delwiche 2012 New HigherCutoffs









Clearly, the higher cutoffs did exactly what one would expect … lower initial placements in mathematics.  However, within this data is a very disturbing fact:

The modal placement for minorities is ‘3 levels below college’ (usually pre-algebra)

This ‘initial placement’ data appears to be difficult to obtain; I can’t share the data from my own college, because we do not have ‘3 levels below’ in our math courses.  However, the fact that minorities … black students in particular … place most commonly in the lowest dev math course is consistent with the summaries I have seen.

We know that a longer sequence of math courses always carries a higher risk, due to exponential attrition; see my post on that http://www.devmathrevival.net/?p=1685    Overall, the pass rates for minorities is less than the ‘average’ … which means that the exponential attrition risk is likely higher for minorities.

The response to this research is not ‘get rid of developmental mathematics’; the research, in fact, shows a consistent pattern of benefits for stand-alone remedial math courses.  This current study shows equivalent pass rates in college math courses, regardless of how low the original placement was (1-, 2-, or 3-levels below); in fact, the huge Achieve the Dream (ATD) data set shows the same thing.  See page 46 of the current research study.

The advocates of co-requisite (mainstreaming) focus on the fact that 20% or more of the students ‘referred’ to developmental mathematics never take any math AND the fact that only 10% to 15% of those who do ever pass a college math course.  The advocates suggest that a developmental math placement is a dis-motivator for students, and claim that placing them into college math will be a motivator.  Of all the research I’ve read, nothing backs this up — there are plenty of attitudinal measures, but not about placement; I suspect that if such studies existed, the advocates would be including this in their propaganda.

However, there is plenty of research to suggest that initial college courses … in any subject … create a higher risk for students; it’s not just mathematics.  So, the issue is not “all dev math is evil”; the issue is “can we shorten the path while still providing sufficient benefits for the students”.    This goes back to the good reasons to have stand-alone remedial math courses (see http://www.devmathrevival.net/?p=2461 ); although we often focus on just ‘getting ready for college math’, developmental mathematics plays a bigger role in preparing students.  The current reform efforts (such as the New Life Project with Math Literacy and Algebraic Literacy) provide guidance and models for a shorter dev math sequence.

Even if a course does not directly work on student skills and capabilities, modern developmental mathematics courses prepare students for a broad set of college courses (just like ‘reading’ and ‘writing’).  It’s not just math and science classes that need the preparation; the vast majority of academic disciplines are quantitatively focused in their modern work, though many introductory courses are still taught qualitatively … because the ‘students are not ready’.  Our colleagues in other disciplines should be up in arms over co-requisite remediation — because it is a direct threat to the success of their students.

Developmental mathematics is where dreams go to thrive; our job is to modernize our curriculum using a shorter sequence to give a powerful boost for all students … especially students of color.

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The Difference Between Mathematical Literacy & Algebraic Literacy

As more colleges implement Mathematical Literacy courses, we are running in to a point of confusion:  what is the difference between Algebraic Literacy and Mathematical Literacy?  The easy reference is problematic … comparing these courses to the traditional beginning & intermediate algebra courses; those traditional courses are at the ‘same level’ in a general way, but this fact does not help us deal with the details of new courses.

I’ve written previously on the comparison of the new courses to the old, especially Algebraic Literacy compared to the traditional course (http://www.devmathrevival.net/?p=2347 and http://www.devmathrevival.net/?p=2331 ).  However, I’ve not talked that much about the difference between these new courses that share a word in the title (“Literacy”).  That’s the goal of this post.

First, the course titles are not perfect … the word ‘literacy’ was meant to imply that the courses deal with pre-college material; ‘mathematical’ was meant to suggest that we did not start with algebra directly … while ‘algebraic’ was meant to suggest some directionality (headed towards STEM and STEM-like courses).  We have focused on the goals and outcomes documents for the new courses as a way to clarify what the courses are designed to deliver.

MLCS Goals and Outcomes Oct2013 cross referenced 2 by 2

Algebraic Literacy Goals and Outcomes Oct2013 cross referenced

Since these courses diverge from the traditional curriculum, these documents were not sufficient to clarify “what belongs in each course” for shared topics (especially algebra).

So, here is a side by side chart meant to provide some additional clarification.

Math Lit vs Algebraic Lit July2016















The intent is not to avoid any overlap between the courses, though there is less overlap than the traditional courses (in general).  As an example, many Math Lit courses introduce systems of linear equations; the solution methods are usually limited to numeric (graphing & intersect) and some substitution.  In an algebraic literacy course, the problems would be more diverse and so would the solution methods presented.

Another example is factoring polynomials.  The classic Math Literacy course might cover “GCF” factoring only (pardon the redundancy) … though that is not assumed.  The intent is that Math Literacy avoid most factoring beyond that which is a direct application of the distributive property; Algebraic Literacy picks up most of the factoring concepts necessary.  We note that most ‘needs’ to include factoring are contrived; a deep understanding of functions (the core goal of pre-calculus) does not depend upon all the typical methods presented in the albatross “Intermediate Algebra”.

A solid Mathematical Literacy course will involve some algebraic manipulation (limited in types as well as in complexity), and these procedures would be further enhanced Algebraic Literacy.  Therefore, the distinction between Math Lit & Algebraic Literacy can not be reduced to a particular ‘problem’ being present in one course but not the other.  We really want to keep the focus on the purposes of each course; see the ‘goals’ part of the course documents listed above.

If you have questions about the distinction between the two new courses, I would be glad to provide any information I have.

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What’s Wrong with That? Mainstreaming, Co-requisite Remediation

Some of us perceive “co-requisite remediation” as a risk to mathematics education because the model is based on a perceived lack of benefits from stand-alone remedial courses.  I believe that we also have additional reasons to be concerned.

I’ve written previously on academic research on the actual benefits of stand-alone remedial courses, including http://www.devmathrevival.net/?p=2541.  This does not mean that all developmental math courses provide benefits for all students; the data does mean that passing developmental math courses provides benefits in that later performance is greater than would be predicted for the abilities of the student at the onset.

The people and groups advocating co-requisite remediation (aka ‘mainstreaming’) do not use academic research to justify the approach.  Almost all data shared publicly in their efforts is demographic in nature and framed in a way that can only support what they want us to see, just like a graph with a scale chosen so that the results are skewed in the chosen direction.  I want to ignore this dishonesty issue, and move on to matters of substance.

Issue Number 1:  No Problem Being Solved
The advocates start with statements such as the following [TBR is Tennessee Board of Regents]:

Prior to 2014, more than 60 percent of TBR’s students system-wide began college needing remediation in math, reading and/or writing. In response, faculty across the system paid significant attention over the last decade to improving the effectiveness of developmental education. Schools implemented and developed nationally acclaimed models around modularization, computer-aided instruction and personalized learning support rather than traditional developmental instruction.
Despite all of this effort, while more students were completing their developmental work, credit-bearing classes were another matter. Overall, only 12.3 percent of the students who began in developmental instruction completed a credit-bearing mathematics class within an academic year, and 30.9 percent completed a credit-bearing writing class. Something had to change.

[See https://higheredtoday.org/2015/10/21/reimagining-remediation-in-tennessee/]

The implication is that the 12.3% rate is the problem being addressed.  This rate is the result of several factors: number of developmental math courses for each student; quality of those courses; quality of the faculty; institutional support for dev math; appropriateness of the math prerequisite of the college math course; quality of the college math course; advising about college math; quality of the college math faculty; institutional support for the college math course, etc.  The solution blames the ‘problem’ on the first set as a single factor, called ‘developmental math’.  As we know, an ill-defined ‘problem’ will not lead to scientific progress … progress will be luck and local coincidences.

Issue Number 2:  Ignoring Student Differences
By placing the blame on an ill-defined ‘developmental mathematics’, the advocates place all students in the same treatment.  Developmental mathematics courses can range from arithmetic to intermediate algebra; in our grade-level fixation, this corresponds to 4 to 6 different grade levels … can one treatment provide success for all of these students?

I’ve previously published this chart, which provides a more scientific method of matching the new treatment to students.

Matching students to remediation model






Issue Number 3:  Ignoring Course Issues at the College Level
What is an appropriate prerequisite to intro statistics?  How about liberal arts math?  A quantitative reasoning course?  College algebra?  The settings for most data cited by the advocates involve the same prerequisite for all of those — intermediate algebra.   We in AMATYC know that intermediate algebra is not an appropriate prerequisite for non-STEM math courses.

The advocates waste institutional and state resources by providing extra course support for courses which had inappropriate prerequisites … most of the improved throughput could have been achieved by simply correcting the faulty prerequisite on the non-STEM math courses.  However, the advocates don’t mind wasting these resources:  because many students would have passed the college math course WITHOUT support, their “results” are guaranteed to be better.

Issue Number 4:  Ignoring Course Issues at the Developmental Level
The advocates never seem to have raised the basic question:

What pre-college mathematics abilities can be justified as necessary and sufficient for success in various college math courses?

I have written repeatedly about the mis-match between traditional remedial math courses and the needs of college math courses (STEM and non-STEM).  Recall that the advocates treat ‘developmental math’ as a single concept and conclude that it does not help students.  Apparently, the advocates see no benefit in looking at fixing the basic problem … they would rather play the snake-oil-salesman role for co-requisite remediation.  There is no scientific method in their advocacy, so they will end up solving the wrong problem as well as creating some new problems.

If the advocates of co-requisite were really interested in solving the real problems, the advocates would also support basic reform in developmental mathematics such as Mathematical Literacy & Algebraic Literacy and similar efforts.  However, the advocates are generally just as dismissive of these professional efforts as they are the traditional courses.  If the advocates took a little time to investigate, they would discover that the reform courses generally use just-in-time remediation to minimize the number of pre-college math courses for every student.   I’ve never seen or heard an advocate voluntarily mention AMATYC New Life, Carnegie Pathways, or Dana Center New Mathways.

Issue Number 5:  Jeopardizing the STEM Path
Co-requisite remediation is almost always implemented only for non-STEM paths.  In some cases, a student who does not qualify for the first college-level STEM math course (pre-calculus, for example) is tracked OUT of the STEM programs.  This has led some people to comment that the recent movements are relegating community colleges to the trade-school and vocational roles, since relatively few students arrive ready for STEM math courses.

This relates to issue #2 (equating all students) … students arrive with gaps in abilities for a variety of reasons.  One of my students this semester is an immigrant from Cuba, who placed in to beginning algebra … with a goal of being a medical doctor.  I’ve no doubt that he will achieve his goal, and he is happy with the opportunity he’s had in our developmental math courses.  However, in the world envisioned by the advocates … this student would be told to select a different major.

Issue Number 6:  Ignoring Any Data that Does Not Support the Cause
In the scientific method, we make a hypotheses … we test it, and then we look for the results when others try ideas related to it (same, related or different).  The advocates never cite data that is not flattering to their cause.  I can’t cite much data like that either — because almost all of the data being collected is done at a low-level of sophistication and it therefore not published anywhere else.

Are you aware of any treatment for humans that ALWAYS works for any sample and any population?  That is what the advocates say: Co-requisite remediation is always better.  Do we think that it is reasonable for a model to always work, even when not implemented well?  We should be seeing some failure stories; analyzing failures teaches more than successes.  By only publicizing success stories, the advocates doom their own cause; they are more interested in selling their particular solution than they are in helping create long-term progress for our students.

I’m reminded of an interesting article by John Ioanidis  called “Why Most Published Research Findings Are False  http://robotics.cs.tamu.edu/RSS2015NegativeResults/pmed.0020124.pdf  .  Note that the advocates present data that is not research, so they have even more reason to be ‘false’ positives.


The advocates of co-requisite remediation (mainstreaming) include some within academia.  With the presence of so many issues and defects in their work, I am left with major concerns about the health of higher education:  How can we help our students achieve a quality education and upward mobility when such an ill-founded movement can control so much of our enterprise?

It’s time to push back; a prolonged conversation between doubters and advocates so that we can find real solutions to problems based on a deep analysis of root causes and other contributing factors.  A ‘quick fix’ is seldom either.  We need better solutions, starting with better math courses.

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Active Learning Methods in Developmental Mathematics

We’ve all had this … students who attempt to complete a math course based on memorization; such students often report frustration when asked to apply knowledge in any way that differs from what they ‘learned’.  As mathematicians, we see the process of that frustration as a key part of what mathematics is all about.

As teachers, we sometimes behave in the same memorizing way.  For example, we attend a workshop session on a particular active learning method or methods.  After some planning, we begin to use those methods in our classes and usually feel good about the experience.

Any teaching method will be more effective if the teacher understands the whole story — how the method works and WHY it works, with connections to other knowledge about the learning process.

If you look for data and research on active learning methods, the results look very good.  However, most of that data collection is done by experts using the methods.  An observational study was done using a random sample of college biology teachers, with an eye towards seeing this positive impact of active learning methods.  Some faculty used little ‘active learning’, some used a lot, and some were in-between.  The results?  Well, not so good for ‘active learning’.  See http://www.ncbi.nlm.nih.gov/pmc/articles/PMC3228657/?report=classic for the Andrews article “Active Learning Not Associated with Student Learning in a Random Sample of College Biology Courses”

It turns out that many faculty using active learning methods have memorized the ‘steps’ for the method but did not understand the method well enough to adjust it for their students … and also could not accurately monitor the process.  Putting students in pairs with a problem for ‘think-pair-share’ will not automatically produce better learning.  Understanding is important for us, as well.  The authors of the study listed above believed that was the critical factor for not seeing positive results.

I’ve been saying something for 20 years, and believe it is still true today:

Developmental mathematics … we are a desperate people!

Partially because we’ve been teaching the wrong courses, our work has not been successful over a long period of time and over a large range of locations.  That process results in us looking for something … almost anything … that might help in our classrooms.  In many ways, this is the same attitude that our students bring to our developmental math courses.  We see something — it might help, so we quickly try it out in our classes.  Learning a new teaching method is like any other learning: there is a process, and ‘knowing steps’ does not equal ‘learning’.

Here are some pointers on how to use new teaching methods effectively … ‘active learning’ or otherwise.

  1. Experience the method yourself repeatedly:  for example, use the think-pair-share process to learn something new.  Look for the how & why of the method, and develop an intuition for what it looks like when it works.
  2. Read and use multiple sources of information.  You are most likely hearing about a method from somebody who heard about it from somebody who heard about it … each of those stages involves filtering and distortion (just because it’s human communication).  Multiple sources will provide a more accurate picture of the method.
  3. Use the engineering principle: estimate the time it will take, then double the number and use the next larger size.  “10 minutes” becomes “20 hours”.  That’s a little extreme, but valuable as a guideline … nothing breaks a teaching method quicker than rushing it.  This applies to both your planning time, and to the operational time in the classroom for the method.
  4. Don’t be deceived by appearances and initial student reactions, which are often skewed (more positive) by ‘something new’.  Assess the results using multiple measures — direct observation, one-minute paper, survey, quiz, etc.
  5. Assume that your first use is a crude approximation requiring a number of adjustments based on analysis of results.  Proficiency is the result of lots of practice … and learning from that practice.
  6. Allow yourself to reject one method and switch to something else.  We all need to become effective teachers, but we don’t need to become the SAME teacher.  Use methods you can be enthusiastic about, since that helps students almost as much as the details of the method.
  7. Talk about your experiences with colleagues you trust.  You are learning something new, and it’s complicated … verbalizing helps your brain clarify the process and the results.  Ideally, you would form a ‘lesson study’ type group working on the teaching method.

I wrote those pointers with active learning methods in mind, but they apply to any method — including lecture (aka “direct instruction”).  Lecture, sometimes defined as continuous elaboration by the ‘teacher’, is a valuable tool for us; it’s just not adequate in general, and needs to be used intentionally.  My own classes tend to be several lectures of 5 to 10 minutes separated by some active learning method.  You might experience ‘experts’ who claim that we don’t remember what we hear; the irony is that the message the expert is delivering is one which they HEARD.   The critical thing is to have the learner’s brain engaged with the material using multiple teaching methods appropriate to the content; listening is an effective method for some things.

We each need to develop high levels of skills with a variety of teaching methods, because that is what experts do.  Limiting our methods, or using methods poorly, impedes our student’s learning … or even causes damage to their learning.  Just like our students, we need to have a growth mind set.

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