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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Homework and Grades in a Math Class

I am trying to reconcile two recent conversations, and I suspect that most of us have had similar discussions.

First, a student (after missing a passing grade by 5%):

I have worked hard, and did extra homework.  The instructor set up the class to fail.  Very unreasonable.

Second, a potential faculty member in response to a question about classroom assessment techniques:

I am really in favor of homework.  I don’t grade it, but it shows what students do not know.

The potential faculty member was reacting to the practice of some colleagues who assign significant points in a class for the completion of homework.  Their institution also suggested that they use the publisher’s homework system for the tests in online sections (My Math Lab, Connect Math, or Web Asssign).  The sad part of this candidates response was that they never mentioned any other assessment technique.

In the student’s situation, she was mostly responded out of frustration.  She is trying to overcome a false start in college a few years ago resulting in a very low GPA, and was dealing with a family health situation; a failing grade in our math class meant that she would not be able to continue in college.

The question is this:  What is the primary purpose of homework?  We know that students do not learn if they do not apply significant effort.  This leads many of my colleagues to give credit for completing homework, sometimes up to 10% of the course grade.  They either give all points regardless of ‘score’ on the homework, or they prorate the points based on the performance level.  The ‘all points’ approach tells students that the main goal of homework is to complete the problems regardless of learning; the prorated method tells students that mistakes in homework can cost them points.

Doing assignments (reading, studying, practice, checking answers, etc) is the critical learning activity in any mathematics class.

With good intentions, we award points for homework; however, that’s pretty much a no-win situation:  Awarding points will encourage students to get it done.  However, getting it done does not mean that students are learning anything.  As a sports metaphor, points for homework is a bit like telling a batter in baseball to swing at anything close to the plate: you need to swing to hit the ball.  The  problem is that swinging at anything means hitting the ball is mostly a coincidence — just like learning when points are awarded for homework.  I would prefer to not settle for accidental learning.

My own conclusion is that doing homework should not be connected directly to a grade in a math class.  Without a learning attitude, the homework will not help; with a learning attitude, the work will get done.  If I can build a learning attitude with my students, they are better prepared for success in any math class they take.

The current media treatment of this concept uses words like ‘grit’ and ‘perseverance’; these phrases reflect the infatuation with educational outsiders creating solutions for educational problems.   Two weeks ago, I sat through a day-long professional development session featuring a psychologist who tried to tell us how to flip our classrooms.  This was a practicing therapist with great expertise in generational issues, but with no particular understanding of learning in a classroom.  [He suggested awarding 25% of the course grade for completing the preparation for class, and another 25% for the assessment activities in class dealing with that preparation.]

The problem I have with this media approach is that grit and perseverance have strong cultural components reflecting the student’s history (especially familial). I much prefer to focus on a learning attitude; this concept is accessible to students, and we have some tools to  build a learning attitude.

A quick list of ingredients for a learning attitude in a mathematics classroom:

• All students involved in conversations about mathematics (I use directed small group work for this)
• Cold-calling on students (expecting every student to be engaged, understanding that “I don’t get it yet” is an acceptable answer)
• Encouraging discussion and reasonable disagreement in class (an initial step is to ask “is there a different way to do this?” … and then waiting 15 seconds or more)
• Quizzes at the start of class (I do a quiz in about half of the non-test days)
• Spending class time working with individual students on something they are struggling with (I do ‘test drives’ for students after doing examples, and worksheets with challenging problems)

This particular list is a practitioner’s list, not entirely founded in learning theory.  However, I can say that my reading of cognitive psychology leads me to believe that the critical necessary condition for learning is the brain engaged with significant and accessible material — this is reflected in most of the items above.

Homework is a poor assessment; homework points is a weak motivator for learning.  How do you build a learning attitude for your students?

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Avoiding Problems and Disasters in the Learning of Mathematics

In some implementations of reformed mathematics courses, there is a strong emphasis on particular ‘learning’ procedures in our effort to improve student outcomes.  For developmental mathematics, the learning procedures du jour are problem-based learning and discovery learning (both heavily influenced by constructivist viewpoints).  These methods have some basis in research and theory, but are easier to implement badly than well (as is true for most procedures).  The purpose of this post is to suggest some guidelines that can avoid issues with these methodologies.

First, I recommend people read a summary of such methods by Krischner, Sweller, and Clark called “Why Minimal Guidance During Instruction Does Not Work” available at http://www.cogtech.usc.edu/publications/kirschner_Sweller_Clark.pdf .  Here is their conclusion:

After a half-century of advocacy associated with instruction using minimal guidance, it appears that there is no body of
research supporting the technique. In so far as there is any evidence from controlled studies, it almost uniformly supports
direct, strong instructional guidance rather than constructivist-based minimal guidance during the instruction of novice to
intermediate learners. Even for students with considerable prior knowledge, strong guidance while learning is most often
found to be equally effective as unguided approaches. Not only is unguided instruction normally less effective; there is
also evidence that it may have negative results when students acquire misconceptions or incomplete or disorganized knowledge.  [pgs 83 and 84]

Second, here is my own summary of other research:

Lectures are a poor method of providing direct, strong instructional guidance.

Our problem, and confusion, stems from a reaction to the lecture methodologies.  Students tend to be passive and not engaged, so our reaction looks for methods that make activity visible.  However, visible activity may be worse than a passive student (that’s what the research summary above says).  We can not settle for what is easy to see; we must go beyond the ideology of ‘learner centered’ and focus on LEARNING.

How people learn is not that much of a mystery.  For people with a brain functioning in the normal ranges, here is the recipe:

• New information that does not fit existing knowledge
• Effort applied to reconcile this gap or conflict
• Access to information related to this gap
• Validation of the resulting new knowledge

Of course, this is overly simplified to be prescriptive for use in a classroom.  However, we need to keep our minds on these ingredients, not on visible activity.  When discovery learning fails, it is often due to a design where the focus is on the first two ingredients; the mythical lecture mode involves a focus on the third ingredient.

I received a message from somebody teaching in a Math Lit course, who was frustrated by the difficulties students were encountering.  To me, those difficulties originated from an almost exclusive focus on new information and effort; the message spoke to the missing information (step 3), though step 4 is an issue in some courses as well (practice and assessment).  We can’t expect students — at any level of mathematics — to spontaneously create good mathematics that is integrated in their brain.

Yesterday, I had a brief conversation with my Provost after a professional development session emphasizing the advantages of a flipped classroom (which is a different issue than those above).  When I told the Provost that I was not impressed with the presentation, the Provost responded with something like “That’s okay; we mostly wanted to get people moving away from lecture.”  Yes, we need to move away from lectures; most of my colleagues have already done that.  We also need to move away from the antithesis of lectures to models where students are active but not productive.

So, provide summaries and mathematical statements to your students.  I don’t care whether this occurs before or after they attempt problems, as long as you design the experience for learning.  And, provide instruction to your students; you are an expert, and they will tend to remain novices when they do not see how experts do the same work.  Keep a large emphasis on validation and assessment; however students learn, they tend to store information either partially correct or completely incorrect, and it is our job to provide a rich diet of feedback on this learning.  Remember that learning is never visible; what happens in a brain is hidden from most of us (who lack fMRI machines in our classrooms).  And, be sure that you don’t confuse activity with learning.

Learning is not easy; designing a math class for learning is not easy.  The ideas of learning are simple, but the application to a classroom is nuanced, requiring attention to all components of learning.  Any instructional design that emphasizes a subset of ingredients for learning is going to fail students.  In our rush away from ‘lecturing’, we need to avoid jumping off the cliff in to the lake labeled ‘active students’; this lake might look attractive, but being in the water does not mean there is learning about the water.

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