This article in Slate pointed me to this video clip of two math teachers, John Golden and Dave Coffey doing a Mystery Science Theater 3000 type commentary (but less glib) while watching a Kahn lecture on how to multiply signed numbers.  The two teachers bring up a number of points which I would like to emphasize and extend.

Choice of Examples

Kahn’s choice of -2 times -2 as the introduction to the topic is so wrong.  Besides the fact that 2 added to 2 is 4 and 2 multiplied by 2 is 4, any example where the multiplicands are the same is suspect.  People, i.e. students, are natural pattern detectors so an instructor needs to develop examples that show the pattern he/she wants to teach and needs to avoid irrelevant patterns.  Sometimes students see a pattern an instructor didn’t intend and hadn’t the imagination to think of.  Looking for patterns should be encouraged but a teacher shouldn’t introduce any that divert attention away from the lession.

Make Connections

Kahn Lectures seem episodic.  The lectures stand alone with few connections to other videos except necessary ones of algebraic sequence.  Normally good teachers, if lecturing, ease their students into the subject with examples from previous lectures.

Work Up to the Concept

Kahn first started with the most difficult idea, the multiplication of a negative number by a negative number.  It is better to work up to it, building connections as you go, first positive numbers by positive numbers, then positive numbers by negative numbers, then negative numbers by positive numbers and finally negative numbers by negative numbers and somewhere work in multiplying by zero.

Know Where You Are Going

There are more than one algorithm, more than one explanation and more than one notation for many mathematical concepts.  Math is very sequential, one skill or concept building on the previous one. The choice of which algorithm to teach will influence the ease of learning later concepts.  And details count.  And language also counts as Golden and Coffey point out.  A good teacher knows where the students will be going next and plans the current lesson accordingly.  By the way, the diversity of mathematical notations, models and algorithms can make changing teachers or texts or school districts or cultures very difficult for students.

Be Precise

Be precise in language.  Students hear and remember, if unconsciously, what a teacher says whatever that is.  Be precise in writing.  Write the steps carefully.  Despite my years of experience, I still have trouble with this since when I do my own mathematical thinking symbols or words seldom appear and I skip steps.

In Summary

I think we conflate being able to do mathematical processes with understanding mathematical processes.  Just because I can read and follow a series of steps in some text or other does not necessarily mean I have any useful understanding of the topic or can work a difficult problem or can teach the subject. I think Kahn’s math videos are popular because they are unfettered by the need to be a coherent whole.  If a math learner is required just to work a series of problems to prove his or her learning, then a bare-bones demonstration of the algorithm is all he/she needs to see and learn.  Math skill with algorithms is certainly a prerequisite for mathematical knowledge and necessary for science and engineering, but the ability to use math in novel situations can’t be had by watching Kahn videos.