Theory of Everything … A Presentation on College Mathematics

Presentation done at the MichMATYC conference on October 13, 2018 at Kalamazoo Valley Community College … with a goal of understanding everything about college mathematics in the first two years.

Presentation:  Theory of Everything presentation Oct2018

References (handout): References Theory of Everything Oct2018

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45 Years of Dev Math

These are the materials from the November 11th presentation … history and future.

The presentation slides: Forty Five Years of Dev Math in 50 minutes web

The handout: 45 years of dev math in 50 minutes AMATYC 2017 S137

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Why We Have a Major Problem Now: Forty Five Years of Dev Math, Part V

This is the 5th post in a series on “forty five years’ of developmental mathematics, roughly coinciding with my professional experience.  The prior 4 posts took us from the early 1970’s to the late 1980’s, with the last post talking about ‘normalizing a bad curriculum’.  It’s time to move on to the early 1990s.

When the  NCTM released it’s standards in 1989, people teaching developmental mathematics could immediately see the implications for our work.  The conceptualization of the dev math curriculum was a one-to-one mapping to 9th to 11th grade mathematics (in ‘the old days’).  The NCTM calls for increased attention and decreased attention had a lot of appeal.

During this same time period, the graphing calculator became a reasonable tool for mathematics classrooms; both TI and Casio had good machines, with some design elements driven by what math teachers wanted.  [The HP calculators of the time were designed for engineer use, so they seldom had much traction in schools.]  These graphing calculators provided a tool that would help teachers implement the NCTM standards.

In the Spring of 1992, Ed Laughbaum had an opinion piece published in the AMATYC journal (called “The AMATYC Review” at that time), with the title A Time for Change in Remedial Mathematics.  One of Ed’s main points was:

To change the current pattern of instruction, I propose that teaching methods be changed to support implementation of the graphing calculator into the remedial sequence.

Ed’s article is primarily an agreement with Lynn Arthur Steen that most mathematics remediation is a failure.  You might notice that this is the same message being sent in the last 5 years by change agents such as Complete College America.  Twenty-five years ago, we were saying it.  What happened?

One thing that happened was that I wrote a response, which appeared in the AMATYC Review a year later, with the title “Time, Indeed, for a Change in Developmental Mathematics“.  http://files.eric.ed.gov/fulltext/ED373817.pdf   This was written just as I was about to become chair of the AMATYC Dev Math Committee (the first time, 1993 to 1997).  My response was a little too soft in terms of critiquing our work at the time; I regret that now.

This response was the first time I used the phrase “mathematical literacy”, written in the general sense (not course specific).  Sadly, one of the things I said was that graphing calculators should not be used in courses at the beginning algebra level.  My position on this changed over the subsequent five years, but my comments resulted in a number of AMATYC members thanking me … they felt supported in doing their traditional courses (which was not my intent).

My conclusion in that article had this:

The basic issue facing mathematics educators today is how to integrate the various forces attempting to drive our mathematics curriculum. The solution involves dialogue and consensus building. Institutions such as AMATYC provide a needed forum and structure for this work. As we work together, our theories and standards will converge, resulting in changes in our curriculum which will certainly integrate technology in many ways.

It’s clear that this change process did not occur … in spite of the NCTM standards resonating with our own interests as shown in the first AMATYC Standards (“Crossroads”).  What happened?

Our collective resistance to the graphing calculator is the primary reason that we did not make any progress when there was another opportunity.  Partially, this was due to the overwhelming resistance to calculators at the college math level (college algebra, pre-calculus) … and much of this still exists today.  The fact that students could not use numeric methods in the next course meant that our use of those methods in developmental mathematics was a possible risk to our students.

In some ways, the content of our courses became ‘locked in’ by 1990.  We resisted professional calls for numeric methods, we collectively ignored the NCTM standards; we even ignored most of our own AMATYC standards (which were being written during the early 1990s).  From 1995 to 2010, fewer natural opportunities for change would arise.  Our default support for an antiquated curriculum is exactly why dev math was an easy target for policy makers and change agents in 2012 … 20 years after the early 1990s.

We are facing a similar call for change today.  The Common Vision suggests that our courses emphasize numeric methods alongside symbolic ones, as well as suggesting that our teaching methods change.  This is the danger of ‘pathways’ … that only non-STEM students get a modern course with numeric & symbolic methods; STEM students are required to survive a series of courses overly focused on symbolic methods with little emphasis on reasoning, and far too little emphasis on connections between concepts.  “Right Answer” still is the goal in these courses, which is the wrong answer for students.

I am hopeful that we individually and collectively will respond today with “let us build better courses for ALL students”.   No student should be required to take a course known to be defective.  In particular, I am hoping that AMATYC will develop a project that links the Math Intensive committee with the New Life Project to work on revitalizing the courses which follow developmental mathematics.

If our profession fails to seize the current opportunity for creating our own modern curriculum, external change agents will control the primary playing field: the initial college level math course(s) such as college algebra, pre-calculus, and similar courses.  These courses suffer the same defects as the traditional developmental mathematics curriculum — antiquated topics delivered inefficiently and with harm to the overwhelming majority of college students who will never take a calculus course.  [Our calculus courses are just as antiquated and inefficient; external change agents just don’t care about calculus very much.  They should!]

We have a problem NOW (2017) because we did not have sufficient motivation to make systemic changes 25 years ago.  The profession let a few visionaries create boutique programs which were locally successful but totally isolated from the mainstream of our work.  Today’s boutique program is “Pathways”.  We need systemic change to create modern mathematics courses for ALL students.  Do we really think that non-STEM students deserve a modern course while STEM students  slog thru disfunctional artifacts clustered as pre-calculus & calculus courses?

It really is “Time for a Change” … not just in remedial mathematics, but in all college mathematics.

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Normalizing a Bad Curriculum: Forty Five Years of Dev Math, Part IV

This is another entry in a series of posts looking back at developmental mathematics history.  Previous posts dealt with origins, a golden age, and a missed opportunity … and now we look at the last half of the 1980s.

It might be difficult to believe that there was a time before people talked about standards.  The first great effort on standards came from NCTM in 1989 (“Curriculum and Evaluation Standards”, summarized at http://www.mathcurriculumcenter.org/PDFS/CCM/summaries/standards_summary.pdf   ) which was a follow up to “An Agenda for Action (http://www.nctm.org/Standards-and-Positions/More-NCTM-Standards/An-Agenda-for-Action-(1980s)/ ).  Whether these standards were even discussed at a college was more a coincidence of faculty connections than any organizational cooperation.

The period we are talking about preceded these initial standards.  However, collaborative activity across institutions and regions was increasing in the latter 1980s.  It is not a coincidence that my first AMATYC conference was in 1987 (“Going to Kansas City” theme song).  We, as a profession, were looking for stability and support.  The AMATYC Developmental Mathematics Committee (DMC) had several active subcommittees on issues such as “Student Learning Problems” and “Minimal Competencies”, as well as “Handheld Calculators”. I served as the editor of the DMC Newsletter for several years, a newsletter produced by printing stuff on a dot-matrix printer and physically cutting & pasting to make the pages of the newsletter.  Ah, for the good old days …

We entered this period having missed the great opportunity, which naturally led to the primary outcome of the time:

The existing pre-college and college curriculum was normalized and accepted as a “good thing”, or at least “the way it should be”.

Some of us knew that NCTM was working on their standards, though none of us were involved in any way (no community college faculty served on a team or as a writer).  In this period prior to the first AMATYC work on standards, we explicitly supported the grade-course structure (from K-12) which had been our inheritance. When a problem was identified (such as low pass rates), our response was to double-down … we created split courses for beginning algebra, and split courses for intermediate algebra; we often added a basic math course separate from a pre-algebra course.  This double-down trend resulted in horrific sequences for students.  We often went from our old sequence of 3 courses to a system where some students took 9 terms or semesters of developmental mathematics.  [These structures still exist, relatively intact, in some places … parts of California, for example.]

Another aspect of the ‘double-down’ response was an attempt to identify THE list of critical learning outcomes.  The DMC “MinComps” (minimal competency) subcommittee worked by snail mail and annual meetings to identify the arithmetic skills that all students should possess.   Although MinComps never achieved their goal of writing a position statement on this content, the group did have an impact on our courses and the textbooks used in those courses.

Never was our response to ask “What are the mathematical abilities which students need for college success and life success?”  The response was ‘what outcomes should be in this course?’.  There was a trend, especially during this time period, to have our textbooks converge to a common list of content topics and outcomes (very skill based).  Workbooks were very popular in this period, often consisting of ‘name topic, state property, show example, give practice’.  In some ways, the ‘programmed learning’ textbooks of a decade earlier were more supportive of student learning.

The content became the thing.  When students did not succeed, we looked to identify a student learning problem.  In some cases, we even tried to provide support to ‘overcome’ a student learning problem.  Our efforts were directed at improving course pass rates … at the expense, frequently, of the sequence pass rate.  Our friend ‘exponential attrition’ is very powerful …  a sequence of 5 courses will always be worse than a sequence of 3 courses, unless we can realize close a 50% improvement in course pass rate.  Going from 45% pass in all 3 courses to 67% pass rate in all 5 courses is not likely; if it has ever happened, you can be pretty sure that this improvement was temporary.

Since we were relatively ignorant of the sequence and attrition issues, we were pleased with longer sequences which reinforced the defective content we had inherited and then had normalized.

My younger colleagues will have a difficult time understanding the technological context for this work.  When we had the “hand held calculator” subcommittee in the DMC, we were not talking about graphing calculators — the work was focused on basic calculators, with a recognition that scientific calculators were available.  Our offices had computers (very slow) with no networking; I had a dial-up modem to connect to a nearby mainframe, but that was quite unusual.  We often hand-wrote our tests (and it’s a miracle that any student could pass such tests!).  Later in this period, we had the  initial efforts to provide students with access to computers — often done in a separate computer classroom, not related to any math course.  Homework, like our tests, were a hand written affair.

This technology did not cause any change in the content or delivery of instruction.  If anything, the status of the technology was part of the set of forces which led to the normalization of the defective content in college mathematics.  Our motto seemed to be “We don’t know if this stuff is really worth much, but at least we generally agree that it is what we should be doing because we are all doing roughly the same thing.”  Many math faculty today continue to look at curriculum primarily from this lens.

The trend in this period to normalize the defective content contributed to our response in the next period (the early 1990s) when the NCTM standards suggested that such content was, indeed, defective.  We had set up conditions which made us essentially immune to the  valid critiques.  That is where the next post will look at our history.

 
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