Mathematics education is hard, in part because the skills it takes to understand mathematics and to teach mathematics are actually quite different. In addition, once grasped, many mathematical ideas switch in an instant from impossible to easy. It can then be hard to put oneself in the position of someone who does not understand. To be honest in my own teaching I know well the frustration of not being able to see how a student who has all the pieces of understanding cannot yet grasp the whole. I also know as a student and as a teacher the joy of breaking through and having things become clear.
One of the biggest barriers in mathematics is that one often has to unpick a previous understanding in order to go further. This is true both of individual learning and the progress of the subject as a whole. I feel that these issues, combined with the desire to assess and make learning visible have caused some deep issues in the way we teach mathematics. The canonical work on this topic is, of course, Lockhart’s lament. In many ways I hoped that the disruption that technology is bringing to established models of education might change this. I believe that effective mathematical education is of great importance to the future will we live in.
Today, I felt cold fear through my veins as a long building realisation crystallised. There is a serious danger that it is already making things worse. The depression really kicked in as I found myself changing my opinion of Sal Khan dramatically. My old opinion, was that he was of great value in making information of reasonable, and often good, quality available. Though perhaps this could only ever replace part of the role of a teacher. An interaction between Khan and Wired blogger/physics educator Rhett Allen, made me change that. Let me set the scene. A couple of math teachers made a video talking about some of the issues in one of Khan’s videos. To Khan’s credit this lead immediately to changes to the treatment of that topic. Dan Meyer and Justin Reich responded by suggesting a competition to find other issues in Khan’s videos. The spirit was to help, by providing free peer review, improve the quality of material within what has become a standard resource, rather than to criticise Khan’s work.
Rhett Allain’s response was quite simple, and did not seem to even deal with the subtlties of pedagogy. Rather it looked at the use of vectors, and pointed out some, perhaps slightly subtle, factual errors in Khan’s treatment. Perhaps because of Rhett’s profile in Wired this video received a personal rebuttal. In many ways what Khan says there is correct. The way he presents the idea of vector certainly makes things easier for the particular problem he is working on. Yet, I also feel he has completely missed the point of Rhett’s criticism. This is one of the situations where making somethings easier can perhaps introduce issues that will actually make things harder later. On the other hand Rhett and I might be wrong and, in any case Khan has every right to defend himself. In particular I do not want this to become a personal attack. He has personally done amazing work. My fear comes instead from the authority that he has gained from this. You can see it in the comments that people made in response to the “Correction Correction”.
Khan’s magnanimity in responding to what was clearly aimed at discrediting a valuable online tool is admirable.
This highly reeks of the criticisms targeted towards Wikipedia from supporters of the traditional encyclopedia. Well, look what happened to Britannica – their physical editions have been discontinued. Those who cannot embrace change are destined to become obsolete.
FATALITY! Rhett Allain, go to amazon.com and buy a rope and a stool! Sal, you are the best!
These are, of course, comments on the internet and should not be taken too seriously, and there are other defending Allen. I believe they do demonstrate something, however. A more worrying example (to me) came from a previous discussion the Khan Academy and education on the edgryders site. There Alberto Cottica, who is a serious and subtle intellect, with experience around education and communication defended Khan saying:
So my crushing fear on this is that people might not be able to recognise the best mathematics teaching. Thus enabling an even smaller number of “master teachers” to dominate, perhaps including Khan. Even worse it might be that the system cannot even choose the best candidates for the role of “master teacher”. My hope instead is that we can move to a new model, summed up by another edgeryder, James Wallbank in what I would love to make a mantra:
Everyone needs to be taught to learn, and to learn to teach.