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 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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Open Educational Resources in Developmental Mathematics

Tired of your students paying over $100 for a textbook?  Frustrated by the cost of getting access to the online homework system that goes with the book?  It’s no news that “OER” is a general movement in colleges & universities.  At some institutions, including mine, there is a direct push for faculty to consider Open Educational Resources (OER) in an effort to save students money.

One concern with OER is that the materials available are almost always (perhaps always) designed for traditional courses, so that OER is a force opposing change in our curriculum.  For example, the “Open Stax” information ( information includes this:

“All textbooks meet standard scope and sequence requirements, making them seamlessly adaptable into existing courses.”  [in “About Our Textbooks”]

I’ve not seen any OER materials for Mathematical Literacy, nor for a modern pre-calculus course.  It is easy to understand why OER is traditional in orientation … the resources are judged by how many uses are tracked, and that can be done most easily by fitting materials to old courses.  What might not be as easily seen is the fact that OER is missing an element in the publishing business — the developmental editor, where ‘developmental’ refers to the creation of new textbooks for changing or new markets.  People say that OER is driven by users (faculty); that is not entirely true … I think OER is more driven by carrying on tradition in the name of saving students some money.

Of course, I know that some individual authors deliberately go someplace new.  For example,   see Schremmer’s work at where you’ll find nothing traditional.  The problem with this approach is that the materials … as interesting and high quality as they might be … remain on the fringes of the profession.  Perhaps the long-term benefit of these textbooks from the underground is more in the maturation of the profession, more than the particular materials themselves.

Within my college, “OER” even includes generic resources like the Khan Academy and Purple Math.  This inclusion is a bit humorous … math teachers have a strong tradition of connecting their students to such resources, but it has little to do with textbooks.  These free resources also represent a force which encourages us to maintain a traditional curriculum.

If our profession were content with the status quo, then there are few reasons to avoid using OER materials — you might get a traditional ‘book’ complete with online homework system for a lot less than a typical commercial textbook at full-retail price.  Faculty can even modify some of the material to represent their own well-founded (and not well-founded) views … like “never mention PEMDAS, because it’s an awful approach to mathematics”.

Almost everybody teaching developmental mathematics as part of their full-time load has been in contact with representatives of the main commercial publishers.  The publishers are sophisticated, in general, and know that they need to “do something” to keep our business in the face of the OER push, not to mention the presence of Amazon in the used-book market.  I’ve had conversations with field reps & editors from the big 3 (Pearson, McGraw Hill, Cengage); you can get a deal from the companies, though the discount rate appears to be inversely proportional to the company’s current market share.

With these discount deals, you can get a commercial textbook (as an e-book) with the online homework system for $50 to $80 per student.  The question might be:

Is the small savings ($10 to $40 less for OER with homework, compared to commercial book) a significant factor for students?

I think that the difference in cost between OER and commercial materials is relatively small now, and will tend to stay small.

So, I return to a prior statement, paraphrased:

OER materials tend to perpetuate the traditional math curriculum in colleges.

If you find OER materials that you are happy with, you might be able to save your current students some money (depending on how well your department can negotiate with publishers).  However, using OER will generally take you out of the process of basic improvements to your curriculum.  In my view, we should avoid using OER in both developmental mathematics and college mathematics so that we can maintain our focus on improving the curriculum first.

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Regional Accreditation and the Problems in Developmental Mathematics

This post is directed at my colleagues in community colleges and similar institutions … and the bodies that conduct our accreditation processes.  My conjecture is that the accreditation process contributes to the problems we have in developmental mathematics, and that this situation deserves corrective action on the part of the regional accreditation bodies.

The regional accreditation bodies use criteria for faculty credentials; in the case of the HLC, the specific wording is:

Faculty teaching general education courses, or other non-occupational courses, hold a master’s degree or higher in the discipline or subfield. If a faculty member holds a master’s degree or higher in a discipline or subfield other than that in which he or she is teaching, that faculty member should have completed a minimum of 18 graduate credit hours in the discipline or subfield in which they teach.

In all cases that I am aware of, remedial courses are not included in the ‘other non-occupational courses’ category.  The result is the common practice:

Anybody holding a bachelor’s degree, in any field, is qualified to teach developmental mathematics.

Within this common practice, a significant portion of faculty teaching developmental mathematics were original credentialed for high school teaching … usually in mathematics, but not always.  Teaching high school mathematics is a worthy profession, often undertaken by dedicated individuals who are either not-appreciated or blatantly disrespected.  However, the context for teaching developmental mathematics is fundamentally different from teaching high school mathematics.

Among those fundamental differences is the fact that developmental mathematics at an institution is directly connected to college-level math courses.  The developmental algebra courses are expected to prepare students for specific college-algebra or pre-calculus courses, with an expectation of content mastery and retention … those elements have a much lower priority in the high school setting.

Another critical difference between the high school and developmental math contexts is that the developmental math faculty need to interact positively with faculty teaching the college level courses.  Since so many of the developmental mathematics faculty have less qualifications, this presents a cultural and social problem:

How can faculty of college-level mathematics have professional respect for faculty of developmental math courses with ‘lower’ qualifications?

A typical developmental math course has a focus on procedural skills and passing, while the college-level math courses tend to emphasize application and theory … sometimes with a much lower emphasis on passing.  In many colleges, this difference in emphasis leads to either a de facto or official separation of developmental math from college math.

The biggest single problem we have in developmental mathematics is the emphasis on a long sequence of courses — 3 or 4 courses below college level.  The inertia for this structure is based, in large part, on the parallel to grade levels in K-12 work … arithmetic (K-6), pre-algebra (7-8), beginning algebra (9) and intermediate algebra (10 or 11).  I have found that many faculty in developmental mathematics have a difficult time letting go of this grade-level focus (courses in K-12).

The fact that the accreditation process ‘ignores’ developmental math teaching qualifications is the problem I think needs to be addressed.  Should faculty teaching developmental mathematics have the same credential requirement as college-level math faculty?  There are strong arguments for this approach.  Should faculty teaching developmental mathematics have credential requirements beyond that of K-12 math teachers?  In my view, definitely yes.

At this point in time, it is not realistic to hold developmental math faculty to the same credential requirement as college level math — we just don’t have enough people qualified at that level.  However, I think we can develop some reasonable standard which approaches that goal.  Perhaps  ‘masters in math education, or a minimum of 9 graduate credits in mathematics’ could be used as an alternative (in addition to the ‘regular’ credential for general education).  The professional organizations, primarily AMATYC, could develop such a criteria in collaboration with the accrediting bodies.

My purpose is more about pointing out the problem and need to develop a solution, rather than advocate a particular criteria.  Achieving a solution could be measured practically:

Can all mathematics faculty in a community college, regardless of normal teaching assignments, understand and contribute to all curricular discussions involving any math course at the institution?

Until we see this result, students will continue to experience a developmental math program that tends to be too long and overly connected to the K-12 ‘grade level’ structure.


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Data on Co-requisite Statistics (‘mainstreaming’)

Should students who appear to need beginning algebra be placed directly in a college statistics course?  For some people, this is no longer a question — they have concluded that the answer is an unqualified ‘yes’.  A recent research study appears to provide evidence; however, the study measured properties outside of what they intended and does not answer a basic question.

So, the study is “Should Students Assessed as Needing Remedial Mathematics Take College-Level Quantitative Courses Instead? A Randomized Controlled Trial” by Logue et al.  You can read they report at

The design is reasonably good.  About 2000 students who had been placed into beginning algebra at a CUNY community college were invited to participate in the experiment.  Of those who agreed (about 900), participants were randomly assigned in to one of 3 treatments:

  1. Elementary Algebra regular    39% passed
  2. Elementary Algebra with weekly workshops   45% passed
  3. College Statistics with weekly workshops    56% passed

At these colleges, the typical pass rate for elementary algebra was 37% while statistics had a normal pass rate of 69%.

The first question about this study should be … Why is the normal pass rate in elementary algebra so appallingly low?  I suspect that the CUNY community colleges are not isolated in having such a low pass rate, but that does not change the fact that the rate is unacceptable.

The second question about the study should be … Would we expect a strong connection between completing remediation (or not) with performance in elementary statistics?   The authors of this study make the following statement:

it has been proposed that students can pass college-level statistics more easily than remedial algebra because the former is less abstract and ses everyday examples

In other words, statistics is not abstract … not mathematics at the college level.  The fact that statistics focuses on ‘real world’ data is not the problem; the fact that the study of statistics does not involve properties and relationships within a mathematical system IS a problem.  I’ve written on that previously (see “Plus Four: The Role of Statistics in Mathematics Education at

The study uses ‘mainstreaming’ in their descriptions of the statistics sections in their experiment; I find that an interesting and perhaps better phrase than ‘co-requisite’.  It’s unlikely that the policy makers will move to a different phrase.

The authors of this study conclude that many students who place into elementary algebra could take college-level math (represented by statistics in their study) with additional support.  The problem is that they never dealt with the connection question:  How much algebra does a student need to know in order to succeed in basic statistics?  The analysis I am aware of is “not much”; in the Statway (™) program, most of the remediation is in the domains of numeracy and proportional reasoning … very limited algebra.

This is the basic problem posed in all of the ‘research’ on co-requisite remediation:  students are placed into low-algebra courses (statistics, liberal arts math), and … when they generally succeed .. the proclamation is the ‘co-requisite remediation works!’.  That’s not what is happening at all.  Mostly what the research is ‘proving’ is that those particular college ‘math’ courses had an inappropriate prerequisite of algebra (beginning or intermediate).

Part of our responsibility is to explain to non-math experts what the relationships are between various math courses, using language and concepts that they can understand while preserving fidelity with our own work.  We need to make sure that policy makers understand that it is not an issue of us ‘not wanting to change’ … the issue is that we have a different understanding of the problem and potential solutions.  In many colleges, the math department is already ahead of where the policy makers want us to ‘go’.

I encourage you to read this study thoroughly;  Because it using a ‘control’ and ‘random assignment’ design, this study is likely to become a star for policy makers.  We need to understand the study and provide a better interpretation.

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