Khan, Comfort, and the Doom of Mathematics

Perhaps you already knew this:

If students perceive instruction as clear, the result will be reinforcing existing knowledge (often not so good knowledge).

I recently ran into a reference to a fascinating item posted by Derek Muller, specifically about videos like the Khan academy; Dr Muller’s specific interest is science education (physics in particular), and you might find the presentation interesting http://www.youtube.com/watch?v=eVtCO84MDj8 (it’s just 8 minutes long).

In mathematics, even more than physics, students come to our classrooms with large amounts of prior experience with the material.  Of course, much of their existing knowledge is either incomplete or just plain wrong (whether they place into developmental math classes or not).  A ‘clear’ presentation means that the existing knowledge was not disturbed in any significant way.  Clear presentations make students even more confident in the validity of the knowledge they possess.  This is not learning.  Reinforcing wrong information is the doom of mathematics.

In Dr Muller’s study, two types of presentations were done.  The first were the ‘clear’ ones; students felt good about watching, but the result was absolutely no improvement in their learning.  The second type were ‘confusing’ ones, where the presentation deliberately stated common misunderstandings and explored them.  Students did not like watching these;  however, the result was significantly improved learning.

We see this in our classrooms.  This past Friday, a young man from my beginning algebra class came in to see me … he had left class in the middle, in a distracting way to other students.  Turns out that he left because he could not stand the confusion.  In talking to him, he believes he can do the algebra but he is getting very confused by the discussion in class about “why do that” and “here is another way to look at it”.  In fact, this student has a very low functioning level about algebra.  If he does not go through some confusion, his mathematical literacy will remain unchanged; that is to say … he won’t have any meaningful mathematical literacy.

Khan Academy videos are popular; I understand … I have watched some myself.  I consider them to be very clear and essentially useless for learning mathematics.  If a person already has good knowledge, they will not need them; if a person lacks some knowledge, they will not perceive what they lack from watching a video.  [Just like witness research in criminal justice, students perception is controlled by their understanding.]

The attraction of modules and NCAT-style redesign is often the clarity and focus.  Students do not generally see anything that might confuse them; the environment is artificially constrained to avoid as many confusing elements (inputs) as possible.  To the extent that students in these programs are ‘comfortable’ and the instruction ‘clear’, that is the extent to which existing knowledge is reinforced.  Learning can not happen if we primarily reinforce existing knowledge; confusion is an essential element in a learning environment.  [I sometimes tell my students that instead of being called a ‘teacher’ they should call me ‘confusion control expert’.]

I suspect some readers are thinking that “He has this wrong … I have data that shows that students do really learn.”  It’s true that I don’t have proof; I don’t even have my own research (though I would love to see some good cognitive research on these issues).  What I do know is that student performance on exams — especially procedural items — is a very poor measure of mathematical knowledge.  I suggest that you interview some average students that you think know their mathematics based on exam performance; have them explain why they did what they did … and have them explain the errors in another person’s work.  Based on what I have heard from students, I think that you will find that only the best students can show mathematical knowledge in an interview at a level equal to their exam performance; average students will struggle with the interview about their mathematics.

How do we avoid the doom of mathematics?  How do we prevent our classes from becoming reinforcers of existing knowledge?  I think we need to create environments for learning where every student faces some confusion on a regular basis … not overwhelming confusion, and not trivial confusion, but meaningful confusion about important mathematics.   Do we need an LCD to do that?  Must we move terms in an equation before we divide by the coefficient?  Is that distrubuting, or is that subtraction?   Confusion is where students bump into the areas of knowledge that need their attention.

Our students have a strong tendency to drive through our courses as fast as possible, without really dealing with mathematics.  They believe the myth that the experts always understand, that we are never confused.  We need to be comfortable in showing confusion to our students and model appropriate behavior to resolve it.  The appropriate response to confusion is figuring out where we went wrong … not running away for a comfortable explanation.  Confusion may call for some meta-cognitive efforts, or we may simply need to polish one particular mathematical idea.

Confusion is the fertile soil of learning.  Avoiding confusion creates a sterile environment without growth.  Comfort is fine, and we all need comfort; however, comfort never learned anything. 

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