Khan has some fantastic ideas about distributing education using technology and is good at mathematics.
However, when I watch his videos on topics such as slope and proportions, I can see that they are likely to promote the same misconceptions that are widely documented in math education literature. I don’t take offense to the ideas of clearly and directly telling students a meaning you hope they will develop for a mathematical concept. However, I believe that what students often learn from Khan is not identical to the mathematics he has in his head. This is sometimes clear from the students’ comments because they ask questions that make it clear that they didn’t understand what he said and were just trying to figure out where to move the symbols and numbers on their homework to get an answer.
For example, I believe Khan understands that if a slope is 7.3, this means that a change in y is 7.3 times as large as an associated change in x. I believe that he knows this change in x could be of any size. However, I frequently interview college students and high school math teachers who believe that if the slope is 7.3 that you go up 7.3 in the y direction for every time you go over 1 in the x direction. These meanings may seem identical for people who understand slope well, but often math teachers and students are unable to predict any of the points in between the ones they get from the up and over process. I often talk to high school math teachers and university students who cannot distinguish between a rate of change and a change in y values because they are used to the slope being measured for a unit change in x. There is also research evidence that high school math teachers do not always know why you divide in the slope formula. They think of the fraction bar as separating the number that tells you “rise” and the number that tells you “run.” To clarify, you divide because division gives you a measure of the relative size of a change in x and an associated change in y.
I’m currently taking science and statistics courses and focusing on the meanings for rate of change that I use in these classes. In my geophysics course it is essential that slope doesn’t not mean rise over run because most of our computations are in three-dimensional polar coordinate systems. A change in independent variable is not associated with an “rise” motion on the graph. When faced with polar coordinates, university students still assume that slope is horizontal rise over vertical run because of what they learned in school.
In statistics slope is essential for interpreting the correlation coefficients in linear regression. To understand what the situation is modeling we need to know that this coefficient gives a relative size of changes in two variables.
Calculus students I’ve researched often think of rate of change as “the change in y’s per 1 unit change in x.” They primarily focus on the change in y’s as the rate of change and actually don’t know how to determine what constant rates of change look like if the changes in x are of different sizes. For example, I gave the student measurements for the weight of a puppy after 1, 2 and 4 weeks and he said he couldn’t determine if it the puppy had the same average rate of change in each interval because the intervals were different sizes. In the definition of the derivative it is essential to consider arbitrarily small changes in x and compare the relative size of changes in y. This would require a multiplicative meaning for division and fractions.
Khan typically assumes that when he says “divide” that high school students will understand what that means with respect to the quantities involved. However, in my research on Calculus students I find that even people who have received all A’s and B’s in secondary mathematics can’t explain what a quotient means. For example A is A/B times as large as B, would be a statement that they couldn’t reproduce. Others can’t explain that A/B tells you the size of a group if you divide A objects into B groups. There are multiple meanings for fractions and division and often the ones they have are almost primarily related to the formulas for computing quotients. These meanings for fractions and division are helpful in understanding slope, but most students I talk to have very weak foundational meanings. If I was making videos to teach slope, I would take into account my learner’s current knowledge as well as the meanings that I knew would be useful in future courses.
I know that some math educators do not like Khan’s work because he is “telling” students what to do. I actually don’t mind that aspect at all, but I wish that the meanings he promoted for the critical idea of slope were consistent with meanings that would be useful in statistics, calculus and science. It would make me very happy if Khan and math educators could work together to make videos that take advantage of math education research findings on how students learn math. Both groups have a lot to learn from each other. Khan’s immediate impact on students is much greater than most math educators, and most math educators would have better ideas about what students might be taking away from his videos. Even our best states and affluent schools are still doing poorly in math compared to countries in Asia. Our entire tradition of math instruction is deeply flawed, and unless Khan provides videos that break free from traditional methods of teaching math, he will continue to reinforce the current culture where kids can find answers to certain problems but don’t know what the computations mean.
This is a reply to an article Khan wrote-the topics are only slightly related. I’m hoping that Khan will understand what I wrote and work on improving his videos in conjunction with translating and distributing them.