Constructive Criticism of Khan Academy’s 1st Video on Fractions

The video below is my entry into the MTT2 contest, where people are making commentary videos on videos from the Khan Academy (KA). This contest began when John Golden and David Coffey posted their “Mystery Teacher Theater 2000” video, where they provided running commentary on a KA video on arithmetic with integers. Their video was quite sarcastic, and many people criticized it for not being constructive. I aimed for constructive in my video, but I think the effect of John and David’s video was also quite constructive, as it’s opened up more space for dialogue (some productive, some less so) than a less provocative video (like mine) is likely to do.

(Note: The audio was fine on the computer I used to make the video, but I tried watching on another computer and had to turn the volume up or down depending on who was talking; apologies if the volume is uneven for you.)

My goals were more informative than entertaining, so I don’t expect to win the contest, but I found it worthwhile to push myself to articulate the problems in the video I chose, “Numerator and Denominator of Fractions,” and to suggest alternative ways of presenting the material. After a while, however, there was a limit to how many hours I was willing to put into commenting on a two and a half minute video, so I declared my work “good enough.” If my goal were to create a video to replace the original, then I would need to do more work (and I don’t think video is the best medium for what I’d be trying to do).

I chose the first video in the KA sequence on fractions because I have found fractions to be a topic that is deceptively difficult to teach. I very rarely get to do it right the first time; instead I work with college students who often have a bag of half remembered, poorly understood algorithms and tricks, with attitudes and beliefs that include deep shame about not understanding such a “basic” topic, thoughts that they are stupid and cannot do math and/or that math is boring and doesn’t make sense and is just something to get over with. I’ve spent a lot of time listening to my students, learning from colleagues, and trying to “unpack” my own knowledge of fractions to understand why things that are obvious to me are so difficult for others.

My conclusion: students have trouble with fractions because they’re hard.

Fractions have many different meanings, an infinite number of ways of representing the same number (\(\dfrac{1}{2}=\dfrac{2}{4}=\dfrac{93}{186}\)), and at their core are about the relationship between two quantities (see my post Trigonometry Yoga for more on this issue).

To illustrate a few points that didn’t fit into the video, consider the fraction \(\dfrac{3}{4}\), which is the focus of the KA video. Here are several contexts where we might encounter this fraction:

1. The paper clip is \(\dfrac{3}{4}\) of an inch long
2. The child ate \(\dfrac{3}{4}\) of a brownie.
3. The gas tank is \(\dfrac{3}{4}\) full
4. \(\dfrac{3}{4}\) of the \(24\) students in the class brought lunch today
5. \(3 \div 4= \dfrac{3}{4}\)
6. \(\dfrac{3}{4}\) is a number that’s halfway between \(\dfrac{1}{2}\) and \(1\)

It’s not hard for me to see the similarities between these contexts, and I’m sure it’s also not hard for Sal Khan to see them. But let’s take a look at a few of the differences:

1. 1. In some of the contexts (paper clip, eating brownies, division) it makes perfect sense to have a fraction greater than \(1\); in others (gas tank, fraction of the class who brought lunch) it makes no sense

1. 1. Some of the contexts (the last two) are abstract; to understand them we must see \(\dfrac{3}{4}\) as a number. With only the more concrete contexts, it’s not clear that \(\dfrac{3}{4}\) is a number on the number line (I sometimes lead debates in my classes on whether \(\dfrac{1}{2}\) is number; many students say that you can’t have half “by itself;” it has to be “half of something.”)

1. The paper clip is best modeled as length, the brownie as area, the gas tank as volume (or perhaps as length with a gauge), and the class with a set. Area models are often over represented in school mathematics (and pies and pizzas are especially over represented), and students of a more concrete bent often conclude that an area model is integral to the meaning of a fraction (e.g. if there’s no pizza, there’s no fraction) and have trouble connecting models that look so different.

As I said in the video, the subject of fractions is one of the key places in the curriculum where some kids give up on math (timed multiplication tests and starting to use variables in algebra are some others). Other kids, with or without the help of teachers, push through and make sense of the many meanings of fractions, until they’ve internalized the concept so well that the connections seem obvious. Some kids remain in between.

When the kids grow up and some find themselves teaching fractions to the next generation, it’s clear that to be effective, the ones who don’t understand fractions and have disengaged from mathematics have to reengage and improve their understanding. What may be less clear, however, is that those who already understand fractions also have a lot of work to do before they are ready to effectively help novices make sense of this very tricky subject.

Constructive feedback is welcome, please feel free to comment below.