Teacher as Confusion Manager — The Key for Student Learning

Early in my career, I focused on being clear — as close to perfectly clear as I could manage.  Class time was easy for students to follow.  Eventually, I realized that my students were not learning very much and decided that I was part of the problem.

Since then, I have seen my role as “confusion manager”.  In planning for what to do in class, I would look for a sequence of activities or problems that were likely to lead to some confusion.  In a basic way, confusion is the brain’s assessment that there is a gap between existing knowledge and needed knowledge.  Without confusion, learning new material is limited for most people.

There is a recent article in The Chronicle  called “Confuse Students to Help Them Learn” by Steve Kolowich  (http://chronicle.com/article/Confuse-Students-to-Help-Them/148385/?cid=cc&utm_source=cc&utm_medium=en)    The initial part of the article covers the experience of another teacher noticing this ‘confusion to help’ property.  Later, the article brings in some experts in psychology.  Certainly, the points in this article are consistent with what I know as a teacher and as a reader of cognitive psychology.

So … here is what I think is so important about the concept of “confusion to help learning”:  There is a great pressure to utilize digital resources, such as short videos.  These videos are almost always created with an emphasis on BEING CLEAR and avoiding confusion.  To check this out, just do a search for videos on your favorite math topics and watch.

Of course, it is possible to create short videos made with productive confusion in mind.  The article listed above suggests that it is not possible; however, I believe that I could create a video that would confuse my students just enough to be helpful.  The problem is that it is much easier to avoid confusion entirely OR to create a lot of confusion.  Productive confusion videos would take the type of planing that goes in to commercial production or movies.  Who has the resources to invest 20 hours in order to create a 5 minute video?

I think students who watch clear videos (or clear lectures) see the experience as validating their existing knowledge — even when their knowledge conflicts with what they just experienced.  A little additional information might survive through short-term memorization, or limited amounts of disconnected long-term information.  If all we are after is recall of facts or replication of simple procedures, clear videos might be sufficient.

As mathematicians, however, we are more interested in goals of reasoning … of application … and connecting knowledge.  This type of learning needs deliberate effort by students, and that means that we need to create and manage confusion.  Confusion, like anxiety, is a natural state for humans; too little of either leads to less learning just as too much of either impedes learning.  Our professional judgment and connection to every student is needed to provide productive confusion.

Have you confused any students yet today?  I hope so!

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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 http://www.nytimes.com/2014/07/27/magazine/why-do-americans-stink-at-math.html?ref=magazine&_r=1# ).  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 (http://act-r.psy.cmu.edu/papers/misapplied.html )  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 https://amatyc.site-ym.com/?page=2014ConfHome ).  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 http://www.ppcc.edu/app/catalog/current/mat-055-algebraic-literacy.htm

Community College of Denver http://www.ccd.edu/ccd.nsf/html/WEBB87UAA8-CEA+New+Math+Classes

Arapahoe Community College  http://www.arapahoe.edu/departments-and-programs/a-z-programs/mathematics/syllabi-mathematics-department

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

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