CBE … Competency Based Education in Collegiate Mathematics

Recently, I wrote about “Benny” in a post related to Individual Personalized Instruction (IPI).  We don’t hear about IPI like we once did, though we do hear about the online homework systems that implement an individual study plan or ‘pie’.  Instead of IPI, we are hearing about “CBE” — Competency Based Education (or Learning); take a look at this note on the US Department of Education site http://www.ed.gov/oii-news/competency-based-learning-or-personalized-learning

That particular piece is directed towards a K-12 audience; we are hearing very similar things for the college situations.  The Department (Education) sent accreditors a Dear Colleague Letter (GEN-14-23) this past December, as academia responds to the call to move away from “seat time” as the standard for documenting progress towards degrees and certification.  A former Provost at my college predicted that colleges will no longer issue grades by 2016, because we would be using CBE and portfolios (said this about 10 years ago); clearly, that has not happened … but we should not assume that the status quo is ‘safe’.

In my experience, most faculty have a strong opinion on the use of CBE … some favoring it, probably more opposing it.  As implemented at most institutions in mathematics, I think CBE is a disservice to faculty and students.  However, this is more about the learning objectives and assessments used, rather than CBE itself.

We need to understand that the world outside academia has real suspicions about the learning in our classes.  The doubts are based on the sometimes vague outcomes declared for our courses, and the perceptions are especially skewed about mathematics.  We tend to base grades on a combination of effort (attendance, completing homework, etc) along with tests written by classroom teachers (often perceived to be picky or focused on one type of problem).

One of the projects I did this past year was a study of pre-calculus courses at different institutions in my state, which lacks a controlling or governing body for colleges.  To understand the variation in courses, I wanted to look at the learning outcomes.  This effort did not last long … because most of the institutions treated learning outcomes as corporate ‘secret recipes’.  Other states do have transparency on learning outcomes — when all institutions are required to use the same ones.

This relates to the political and policy interest in CBE:

CBE will improve education by making outcomes explicit, and ensuring that assessment is aligned with those outcomes.

Sometimes, I think those outside of academia believe that we (inside) prefer to have ill-defined outcomes so that we can hide what we are doing.  We are facing pressure to change this, from a variety of sources.  Mathematics in the first two years can improve our reputation … while helping our students … if we respond in a positive manner to these pressures.

So, here is the basic problem:

Most mathematics courses are defined by the topics included, and learning outcomes focus on manipulating the objects within those topics.  The use of CBE tends to result in finely-grained assessments of those procedures.
Understanding, reasoning, and application of ideas are usually not included in the CBE implementation.

Compare these two learning outcomes (whether used in CBE or not):

• Given an appropriate function with polynomial terms, the student will derive a formula for the inverse function.
• Given an appropriate function with polynomial terms,  the student will explain how to find the inverse function, will find the inverse function, and will then verify that the inverse function meets the definition.

Showing competence on the first outcome deals with a low level learning process; the second rises to higher levels … and reflects the type of emphasis I am hearing from faculty across the country.

I do not see “CBE” as a problem.  The problem is our learning outcomes for mathematics courses, which are focused on behaviors of limited value in mathematics.  A related problem is that mathematics faculty need more professional development on assessment ideas, so that we can improve the quality of our assessments.  Without changing our learning outcomes, the use of a methodology like CBE will wrap a system around some bad stuff — which can make the result look better, without improving the value to students.

We need to answer the question “What does learning mathematics mean in THIS course?”  for every course we teach.  Assessments (whether CBE or not) follow from the learning outcomes we write as an answer to that question.

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Walking the STEM Path I: Take Time to Smell the Functions

As we engage in a conversation and discussion about “pre-calculus” (or ‘college algebra’ to some), I am thinking of our curricular goals and how we emphasize what is apparently important.  When the two align, we have potential for success; when our goals differ from what we emphasize, non-success is guaranteed.

Our work in pre-calculus deals primarily with functions (of all kinds).  That makes sense.  However, take the case of ‘inverse function’; whether we are talking about a specific relationship (exponents and logarithms) or the general concept, the idea is important on the STEM path.  The emphasis for most of our courses is on the following:

1. Replace y with x (once), and x with y (all times).
2. Solve for y
3. This is the inverse function, called f^-1

We often feel good about this when combined with the identification of one-to-one functions.  Once we practice finding the inverse, we sometimes explore what the inverse does … sometimes, we present this in terms of composite functions.

This procedural emphasis on ‘finding the inverse’ hides the purpose:  All inverse functions are a matter of undoing.  Algebra starts with inverse operations to solve equations of limited types, where we almost always emphasize the WHY.  In pre-calculus, we take a remedial approach:

• The ‘why’ is too difficult, and we wait until calculus to deal with it.
• Correct answers are an accepted proxy for understanding mathematics.

The procedural approach submerges and prevents understanding; transfer of learning will not occur in most cases.  We can do better: Inverse functions can be approached from the ‘undoing’ perspective, in two senses:  We undo the operations in the function in the appropriate order, and the output for f, when substituted into f^-1 results in the original input.  [We should really create a more reasonable notation for inverse functions.]

Another example is ‘end behavior’ of rational functions.  Our typical approach is:

• If the leading term of the numerator is a higher degree than the leading term of the denominator, the function approaches positive or negative infinity as indicated by the coefficient of the numerator’s leading term.
• If the leading term of the numerator is a lower degree than the leading term of the denominator, the function approaches zero.
• If the leading terms have equal degrees, the function approaches the value of the quotient of the coefficients of those two terms.

Some textbooks do base this end-behavior topic on a discussion of limits (a good idea).  Seldom do we approach end-behavior with an understanding base, which might go something like this:

• End behavior analysis has nothing to do with reducing a fraction.
• Terms never ‘reduce’; factors do.
• End behavior is based on analyzing the terms with the greatest influence on the values of the numerator and denominator.

Our complaint in calculus is that students do not know algebra; however, many pre-algebra topics are approached in a way that avoids dealing with those algebraic struggles — like ‘when does a fraction reduce’.

The pre-calculus experience must involve deep work with functions, combined with a focus on fundamental algebraic ideas.  Procedures can help students become efficient; when presented without that deeper understanding of functions and basic algebra, we create our own potholes and ditches in calculus.

Unless your calculus students never struggle with function ideas, your pre-calculus course deserves a critical analysis — does the course provide a good sense (feeling, smell, vision, etc) for functions and covariation?  Unless your calculus students never make algebraic faux pas, your pre-calculus course deserves a critical analysis — does an emphasis on procedures avoid dealing with basic algebraic ideas?

 
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Equity and Participation; our Response

This was one of those news reports that really got my attention as an educator.  There are many, of course, which should get my attention as a person and citizen; this report talked right to the ‘teacher’ in my brain.

The role of AP courses has been debated; in the mathematics community, we have some concern about how much benefit actually accrues to the majority of students in AP calculus.  However, as long as AP courses are offered … our goal needs to be equity: no group of students should be participating or not-participating at significantly different rates.

My college serves a blended district — a small-to-medium city (Lansing) and a surrounding area made up of suburbs and rural communities.  AP courses are offered in both the city and suburbs.  Here is a quick breakdown for two of the local districts for student population and AP population.

The school data came from a tool at marketplace.org; see http://www.marketplace.org/topics/education/learning-curve/spending-100-million-break-down-ap-class-barriers.  The city population data came from the 2010 Census.

One way to look at this data:  The Lansing high schools are about 73% ‘minority’; the AP classes in Lansing are about 60% minority.  Another view: the AP participation rate for white students is 50% higher than their proportion of the population would indicate, while the black student participation is 25% lower.

We might conclude that this discrepancy is a Lansing school problem; that is not the case.  The same pattern is present in Holt … just not quite as extreme, due to the smaller minority student population.

In case you are wondering, the national figures are:

We’ve known that minority students are over-represented in developmental math courses in college.  This recent data suggests that the equity problem extends through the whole range of abilities.  I respect the difficult work that our K-12 colleagues are doing, often without support or respect; this equity problem is not about the teachers … it’s about society and us.

We can, and must, do better.  How will we respond?  I think we recognize signs of a problem; what actions can be taken?

 
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Math Paths Workshop for Michigan Community Colleges

When:  June 17 – 18, 2015

Where: North Central Michigan College (Petoskey, MI)

Here is the flyer: MI workshop flyer_final_2015_04_14

To register, use this link: http://www.cvent.com/d/trqzrd