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 2: One Course, or “APL Design”

In the early days of personal computing, it was clear that digital storage was very limited; initial on-board memory was often measured in kilobytes (great by those standards in the 1970s).  The computer speed was decent for that time; as a result, programming languages faced issues and constraints.

As a mathematician, the most beautiful programming language was “APL” … the acronym for the obvious name “A Programming Language”.  You say you’ve never seen this  language?  Well, take a look at the stuff over at http://en.wikipedia.org/wiki/APL_%28programming_language%29 .

APL used an applied mathematics approach to programming.  Need a matrix invert operation?  One symbol did that.  Need a row operation?  One symbol.  Each symbol in APL was a wonderful contraction of a big idea, just like mathematics.  Of course, you needed a special keyboard to use APL.  Small price to pay.

Here is the theme song for the person who ran the local training for APL back in the day:

If your program does not fit on one line, you have not thought about it enough!

In other words, if you have not analyzed the problem intelligently and with insight, your program becomes multi-line and shows that you have more work to do.  Of course, programming has gone in a totally different direction, where we worry about ‘time’ more than lines of code.

In the STEM path, we are talking about connecting developmental-level mathematics with Calculus I. Think about this path as a problem to solve.  If we can not write this program for one semester, we have not thought about it enough.

Over the years, we have developed several ‘solutions’ for this path. Some involve a two course sequence of ‘college algebra’ and trigonometry.  Others involve ‘college algebra’ then pre-calculus.  Some have 3 courses — college algebra, trig, and pre-calculus.  Some institutions have a one-semester option (often called ‘pre-calculus’ or ‘college algebra and trig’).  A few other combinations exist.

We often allow content inflation in these courses by focusing on procedures rather than capabilities.  A well-prepared student can either figure out a needed procedure, or look it up once.  On the other hand, a student who has experienced the “100 most important tricks before calculus I” will not be able to figure out much, and will lose most of these tricks quickly.

What are the capabilities needed for calculus I?  We have a very good starting point for that conversation.  Take a look at the MAA Calculus Concepts Readiness test (http://www.maa.org/publications/periodicals/maa-focus/maa-updates-its-test-for-calculus-readiness).  The first item on that web page shows this problem:

Suppose you have a ladder leaning against a wall. Now suppose that you adjust the slant of the ladder so that it reaches exactly twice as high on the wall.  The slope of the ladder [now] is:  a. Less than twice what it was   b. Exactly twice what it was …

A student knowing how to handle that problem is likely to be better prepared than a student who can correctly evaluate a difference quotient for some arbitrary function.

If your pre-calculus path has more than one course between developmental and calculus I, you have not thought about the problem enough.

This “one semester … if not, finish solving the problem so it is” approach has been a recent trend at the developmental level.  Many of us are replacing 3 (or 4) procedural courses with 2 courses which provide both skills and reasoning.

We need national leadership from MAA and AMATYC on these issues; those organizations are ready.  We need many of us involved with an effort to upgrade and reform the STEM path.  Are YOU ready?

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Toward a Modern View of Mathematics

We face many opportunities in the coming years, in our professions of mathematics and mathematics education.  Will we seize the opportunities, or merely survive with the least efforts that avoid the largest problems?

As professionals, we know that mathematics is a collection of sciences dealing with quantities, shapes and relationships.  We have allowed one of these sciences — calculus — to dominate the mathematical experience of our students, and often only have students study other mathematical sciences after a mastery of calculus (even when there is not conceptual connection).

Now, I realize (as we all must) that calculus deserves a prominent location in undergraduate mathematics.  Not only are the concepts and methods of calculus used in a variety of fields, but the study of calculus allows students to experience some of the greatest achievements in science (and see the beauty as well).  I would like more students to learn calculus.

However, we lack balance in our curriculum.  The vast majority of undergraduate mathematics courses are part of the path to calculus, where the content is (loosely) based on what is needed to learn calculus.  The fact that this path is not effective and needs a new design is a related but separate conversation.

Many recent conversations have amounted to “calculus/calculus-path OR statistics”, with the refrain “people can actually use statistics”.  I question the accuracy of that statement in many ways, but more importantly — are there no other areas of mathematics that have a modern practicality?  Do we really believe that life begins after calculus … that study of other areas must be delayed?

Graph theory is ‘hot’; much of our modern technology is related to this work.  Is there a reason not to include a basic understanding of graph theory in undergraduate mathematics?  The work of graph theory seems accessible.  How about basic number theory and ideas of cryptography?  Discrete mathematical ideas? Matrices and numeric method?

Forum 5 of the Conference Board of Mathematical Sciences (October 2014) focused on mathematics in the first two years of college, with a prime motivation coming from the book “Mathematical Sciences in 2025” http://www.nap.edu/catalog/15269/the-mathematical-sciences-in-2025   As people talked about the vitality of mathematics, my question was (and still is):

Do we integrate any of these topics or concepts into basic college mathematics, or do those courses continue as single-minded diversions into mathematics that nobody cares about?

Many of you have a deeper understanding of the mathematics described in the “2025” book.  What I recognize is that our students are (in general) prevented from seeing any topics or concepts related to current mathematical research until after the first two years.  Perhaps we can not avoid that condition; however, I think we can include multiple mathematical sciences within the basic mathematics courses our students take.

I hoe that mathematical diversity is coming to a math course near you.

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QR Courses: Resources to Build Good Quantitative Reasoning Courses

We had a workshop this winter on Quantitative Reasoning courses (QR) in Michigan.  The information shared at that workshop is now available on the MichMATYC web site.  Here:  http://michmatyc.org/QRCourses.html

[This workshop was sponsored by MichMATYC with operational support from the Michigan Center for Student Success.]

Take a look … information from several colleges is included, and some math path maps are available on the page.

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