None of Us Stink at Math: Elizabeth Green, Constructing Knowledge, and You

It’s not like clockwork.  However, a regular event is to have a high-profile article spur debate … and passion for … specific ‘teaching methods’.  The most recent one is an article by Elizabeth Green called “Why Do Americans Stink at Math?” (see ).  Well written, understandable … and wrong in all ways that matter.

First, almost all references to ‘math’ in these discussions is actually ‘procedural arithmetic’; yes, we uniformly “stink” at that.  I do not see that as a particular problem, since calculating results is no longer considered a human function but is a machine function.  Very little of ‘math’ is involved, and none of the important ideas.  We need to help writers for the layperson get this right, or risk future generations being doomed by the mythology.

More importantly, this article — like many (even in professional journals) — advocates the use of constructivist models for teaching mathematics.  The basic constructivist idea is fine … learning involves constructing knowledge; I use quite a bit of this in my classes, with good results.  However, the constructivist model flies in the face of cognitive psychology and decades of research; this model says that students ONLY learn when they construct knowledge by THEMSELVES.  (This is ‘radical constructivism; a moderate approach removes the only and says ‘best’.)  If you want to explore the details of how constructivism defies research and cognitive psychology, start with this summary:  Applications and Mis-Applications of Cognitive Psychology to Mathematics Education ( )  This is one of my favorite articles of all time; nothing seen since its writing would require a change.

The truth is that learning happens in a variety of ways, some in spite of instructional design.  As professionals, our job is to design instruction to produce the best quality learning for the most learning.  Theory — and research — tells us that this will involve a combination of direction and student struggle.  At no time have I seen research support the naive notion that novices can construct valid mathematics on their own OR with loosely guided activities.  Although constructivist classes appear to be positive learning environments, that facade does not survive closer examination.  Likewise, an “all telling” old-school lecture might have appeal for its clarity of message; this facade also fails when actual learning is examined.

No, we need to resist those telling us that there are simple answers — whether constructivist, Khan Academy, flipped, blended, co-requirsite, accelerated, modularized, or MOOC’d.  Solutions involve addressing root problems; we should be more concerned with professional development and engagement than with simple-looking answers.

No, we need to provide a clear message.  People in the United States are able to do significant mathematics with reasonable skill; procedural capabilities — arithmetic, algebraic, or other — are not generally present, and the question is “Does that present enough of  a problem for us to ‘solve’?”  Whether we are talking about ‘real-life’ or academic preparation, we need to focus on major needs of students; this will always result in a complex design, because there are no simple problems.  The appearance of simple problems is an illusion caused by multiple salient features being ignored.

All of this is our joint responsibility.  I look forward to seeing what YOU can do to help — in your neighborhood, your state, or nationally.

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Supporting All Math Instructors

Like other professions, mathematics educators in community colleges are not likely to be in attendance at national conferences (such as AMATYC 2014 ).  More of us should join AMATYC; I would like to think that membership is expected for all full-time math faculty in community colleges as well as those in universities with a focus on the first two years.

However, even in the best possible situation, only a small minority of us will be present at AMATYC conferences on a routine basis.  The question is:

How can we support all math instructors?

My view is that the critical component of an answer is the affiliates of AMATYC.  Each affiliate offers closer-to-home opportunities, with the resulting reduction in expenses.  Most affiliates have a low membership cost combined with a reasonable conference fee.  My affiliate (MichMATYC) is among the most economical: $5 annual membership, and conference registration is $35 to $40.

Part of the reason for this post is to highlight a specific activity that affiliates can undertake, in a mode that is accessible for most faculty in the state or region (full-time or adjunct).  Although the attention will shift to college level courses, right now developmental mathematics is in the ‘hot seat’.  The Michigan affiliate (MichMATYC) is hosting a state “Summit on Developmental Mathematics”, connected to our fall conference.  Here are some of the session topics for our Summit:

  • Pathways for general education mathematics
  • Acceleration models
  • Financial Aid issues
  • Implementing a New Life course like Mathematical Literacy (or Algebraic Literacy)
  • Comparing models (Dana Center NMP and AMATYC New Life)

Think about this … most states only have 20 to 30 faculty at the AMATYC conference in a given year.  At the affiliate conference, we can have 150 to 200 faculty.  This is still a minority of the math faculty in the affiliate region.  However, the proportion participating is approaching the level needed for sustained long-term improvement in the profession.

Of course, AMATYC also provides the wonderful webinars — which provide benefits without any travel expenses.  The participation in these webinars is not generally large (30 to 80, I think).  My guess is that faculty see them as a small part of their professional development needs.  Of course, one factor here (again) is AMATYC membership; participation in the webinars is limited to AMATYC members.  Another reason for membership to be expected of all full-time faculty.

The key point is that we need to include far more of our colleagues in all of our work, professional development in particular.  AMATYC membership is critical for full-time math faculty, and affiliate activity is our best chance of making a long-term difference by including a larger proportion of both full-time and adjunct faculty.

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Colorado Gets it Right with New Life

The New Life Project seeks to make basic improvements in math courses for college students — to provide them with modern courses, focusing on sound mathematical content, designed to serve the real needs of college students.  Although the New Life courses (Mathematical Literacy, Algebraic Literacy) can exist side-by-side with the traditional courses, my hope is that the new courses will replace the old courses.

Colorado has done that.  Effective this fall, the community colleges of Colorado are replacing their old developmental courses with a combination of Mathematical Literacy (Mat050) and Algebraic Literacy (Mat055).  The course titles vary from community college to community college, and colleges offer a co-requisite course for Algebraic Literacy (Mat025) which enables more students to begin with the second course.

For examples of the Colorado design, take a look at:

Pike’s Peak Community College

Community College of Denver

Arapahoe Community College

Colorado got it right.  Congratulations to them.  Their plan can be an inspiration to the rest of us.

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Quality Instruction and Class Design

Last year, my college created a new structure for departments and programs.  Instead of a chairperson for each department within the 3 academic divisions, we got associate deans and ‘faculty program chairs’.  The associate deans are the administrative players ‘in charge’ of two or three of our old departments.  In my case, math and science share an associate dean.  We have 7 faculty program chairs for the two departments; I am in the role of faculty program chair for developmental mathematics.  [Not much time provided in the workload, but the work is rewarding.]

Currently, I am focusing on one key idea for our program:

How do we create quality experiences for our students?

We want higher pass rates and completion (of course).  However, our students need classes that serve a real purpose.  Designing a course so that grades and scores are consistently higher than a student’s learning does not help students.  Some people talk about this under the umbrella of ‘grade inflation’, though our interest is in the striving for quality in instruction and class design.

So, here are some issues I have been thinking about:

  • Should any ‘points’ be awarded for completing homework?
  • Should points be awarded based on the level of performance during homework?
  • Does “dropping a low test” support or hinder a high quality class?
  • If a student does not come close to passing the final exam, should they get a passing grade if their other work creates a high enough ‘average’?
  • Is it okay if students with a 2.0 or 2.5 grade are not ready for the next math course?
  • Do high grades (3.5 and 4.0) uniformly mean that the student is ready for the next math course?

When courses are sequential, the preparation for the next math course is a critical purpose of a math class.  Assigning a passing grade, therefore, is a definite message to the student that they are ready to take the next class.  In practice, we know that this progression is seldom perfect — we usually provide some review in the next class, even though students ‘should’ know that material.  At this point, our efforts are dealing with the existing course outcomes, which tend to be more procedural than we would like; eventually, we will raise the reasoning expectations in our courses (with a corresponding reduction in procedural content).

Of special  interest to me are the issues related to homework.  Some faculty assign up to 25% of the course grade based on homework.  Like many places, we are heavy users of online homework systems (My Labs Plus as well as Connect Math).  When those systems work well for students, they support the learning process; most students are able to achieve a high ‘score’ on a homework assignment.  Should this level of achievement balance out a lower level on a test and/or final exam?  Take the scenario like this:

Derick completes all homework with a friend; with a lot of effort, his homework is consistently 90% and above.  All of Derick’s tests are between 61% and 68%, and he gets a 66% on the final exam.  The high homework average raises his course grade to 71%, and he receives a 2.0 (C) grade in the algebra class.

This scenario is a little extreme (it’s only possible with a high weight on homework … >15%).  What is fairly common is a situation where homework is 10% of the course grade and the student passes 2 of the 5 tests; one of of the 3 not-passed tests is ‘dropped’, and the student easily qualifies for a 2.0 (C) grade.  One of the cases I saw this past semester involved this type of student achieving a 52% on the final exam.

In our case,we already have a common department final exam for the primary courses (pre-algebra up to pre-calculus).  In the case of developmental courses, we have a policy that requires 25% of the course grade to be based on that final exam.  This design for the final exam is a good step towards the quality we are striving for.  We are realizing that we can not stop there.

Like most community colleges, our courses are taught by both full-time and adjunct faculty; the last figures I saw showed about 40% by full-time and 60% by adjunct.  Because adjuncts are not consistently engaged with our conversations, adjuncts tend to have more variations than full-time faculty.  We will be looking for ways to help our large group of adjuncts become better integrated within the program, even in the face of definite budgetary constraints.  Fortunately, many of my full-time colleagues are committed to helping these efforts to improve the quality of our program.

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