Closing the Teach For America Blogging Gap
Oct 04 2012

Khan-A brilliant, caring man, who is sadly promoting problematic meanings for slope.

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.

6 Responses

  1. Cal

    Yeah. Um, about 5% of the population needs to care about this crap, and if they’re bright, they’ll get it when they need to. And “students who get As and Bs in math” is a useless metric. Control for SAT scores or don’t bother.

    Back in the real world, kids can’t figure out whether a slope is positive or negative. Fiddle while Rome burns.

  2. Ms. Math

    Yes, it is true that in our population a very small percentage of people understand math or science. It is true that many people who were going to be math or science majors change their minds when they get to college because Calculus and rates of change don’t go well for them.

    The drop out rate in Calculus is bad enough that people are doing studies on it. Even Obama is concerned that we must educate people in math and science to keep up. There are other countries where twice as many people have math and science degrees. We rank 20th in terms of the proportion of 24 year olds with math and science degrees. Check out the report from Congress where they express their concerns about it:

    I understand that grades in high school, and even a math degree don’t mean that someone understands the basics. The point of that comment was to establish that I didn’t find someone who was abnormally bad at math and hated it for my research. I research people with math degrees or who want to get them to be math teachers and still find them lacking in basic understandings that I believe they need to teach ideas in a way that will be useful in science and advanced math.

    Also, it may be true that because I’m good at math, slope more or less made sense for me. However, because of our education system the fundamental theorem of calculus never really made sense to me. I knew the history of it but always wondered why slope at a point and area were opposites. As part of my math ed program, I worked on a class where understanding the theorem was a major goal and it makes so much more sense now. The other day in Geophysics my professor was using the fundamental theorem and struggled to answer a question about a part of the fundamental theorem and admitted to not knowing why it worked. After class I explained to my professor why the theorem makes sense and how to answer this students question and he loved my version. He thought it had never seemed so clear before and how he would teach it my way in the future. Check out my article with Pat Thompson on “A Conceptual Approach to Calculus Made Possible by Technology” if you want to know what I’m talking about.
    Now we are talking about a very bright scientist who uses Calculus all the time in research, who never just figured out the real meaning of what we teach poorly in the major Calculus books. I don’t think that we automatically learn better ways of thinking just because we are bright.
    Thanks for your input.

  3. Andrew

    Is it really that complicated? Can’t you start with a rate of change of 0. Go to a constant rate of change and build to more general functions?

    As much as I think it’s nice to understand WHY differentiation and integration are inverse operations, one doesn’t need to understand this in order to make great use of it. This is, I think, what the current problem in math ed IS. It’s so focused on the understanding of the internals that it may neglect the notion of usefulness. A good metaphor is a car. Can I tell you how a car works? Kind of. Can I drive a car and make use of it and know when to take a car vs. taking a plane, etc? Yes. And that’s what matters. Specialists for cars are called mechanics — they understand the insides. Specialists for calculus are called mathematicians (or, hopefully, math teachers).

    There’s a whole range of knowledge with regard to cars. And a similar range in math. Not every person who learns the usefulness of the calculus need be a specialist. Many can (and do) use it without understanding the guts, and that’s not all bad.

    • Ms. Math

      A lot of students pass Calculus without knowing that a derivative is a rate of change. To me, that would make it hard to apply the derivative to solve a useful problem.

      I’m trying to figure out how math is useful to scientists by taking a Geophysics course and learning how mathematics is used to model gravity and the insides of planets.

      I think understanding the guts of Calculus is taking real analysis and proving the theorems and being very careful about the logical rigors of infinitesimals. I don’t think everyone needs to do this, but I think they should all know that derivatives are rates of change and what a rate of change is.

      • Andrew

        I’m with you. But I think when people talk about, say, understanding Calculus, they’re thinking Analysis, not the concept of rate of change. Similarly, one doesn’t have to understand WHY the long division algorithm works in order to get great use from it (assuming you have no calculator).

        You might find Comenetz’s Calculus book interesting, as it takes a decidedly non-geometric approach in order to emphasize the notion of rate of change.

  4. Ms. Math

    I don’t propose teaching WHY long division works. Just what a quotient means-a comparison of the relative size of two quantities.

    Or, the size of a group after the dividend is partitioned into equal pieces.

    These are the useful meanings for quotient that would allow a student to use a calculator or algorithm to solve a real problem. If they are not sure what division means beyond the amazingly common, “it always makes the answer smaller” they won’t be able to apply long division or a calculator. I don’t really care if students know why each step of the algorithm works as long as they know what the answer means. I am going to bet that people do interpret math ed to say that kids need to understand WHY the long division algorithm works, and I agree with you that those interpretations are a problem in our field. I think many have misinterpreted the ideas of constructivism and come up with crazy ideas about having students discover long division algorithms when all I would really like is that kids know what their answer means in terms of real quantities.

    I think it is great that you noticed that we had two different meanings for “understanding Calculus.” I think that you are right that many mathematicians think of analysis, without even having a concept that students don’t know a derivative is a rate of change.
    Thanks for book suggestion :)

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