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:

- The paper clip is \(\dfrac{3}{4}\) of an inch long
- The child ate \(\dfrac{3}{4}\) of a brownie.
- The gas tank is \(\dfrac{3}{4}\) full
- \(\dfrac{3}{4}\) of the \(24\) students in the class brought lunch today
- \(3 \div 4= \dfrac{3}{4}\)
- \(\dfrac{3}{4}\) 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:

- 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
- 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.”)
- 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.

## 6 Responses

## Sheryl

Wow – I never knew what numerator and denominator meant. Very interesting. You mentioned sets in passing – how does that work with defining your unit? We did lots of chicken nugget and french fry math with the kids when they were young.

BTW in quilting, how you divide the fabric is really important – its generally sold by lengths – the fabric is usually 42 inches wide, so a quarter yard is 9 inches by 42 inches. But many quilters don’t want that “skinny quarter” and ask for a “fat quarter” instead – a piece that is 18 by 21 inches.

## dborkovitz

Thanks, Sheryl! Good question on the set as a unit. You can do something like draw 24 stick figures (or circles or something easier to draw) and then draw one big rectangle or other shape around them all and write “unit” or “1 class.” Of course, that’s tricky too because kids know that classes aren’t all the same size….

Interesting example with the quilting fabric — definitely another challenge w/an area model — to make sure that 1/2 doesn’t have to look like a semi-circle, for example, that it could be two differently shaped rectangles.

When to measure fabric in yards vs square yards would be another topic of discussion….

## Sue Jones

Well, even 3/4 on the number line is 3/4 of one. All those numbers are that many “of” spaces of that distance.

I have been wondering whether it would be better, with adults, to focus on sets at first because they’re more familiar with that (except for the folks needing the math for measuring).

## dborkovitz

Sue, yes 3/4 is 3/4 of 1, but then again, 2 or 3 or 17 are all just as relative to 1. If we think 2 is a number that doesn’t have to be “2 of something” then that’s no different than 3/4 being a number. Of course, 2 is an abstract concept, but it’s one that we start feeling is concrete before we start feeling that way about 3/4.

Try focusing on sets first and see how it goes! Mostly the students I work with on fractions are pre-service elementary teachers, so they need to be familiar with many models, but in another context, certainly could do something else.

Thanks for commenting!

## Gregory Taylor

Nicely done. The idea of a “unit” needing context is so true, and I found myself agreeing with I think all your points about Khan’s video.

Where things can really get interesting (speaking as a high school math teacher) is in line equations… you’ve got slope, written as a fraction (sometimes) but it’s representing a proportion between two things (a rise and a run) but the equation ALSO has the y-intercept written as a fraction (sometimes) representing a location between two values on a vertical number line. Like the best of both worlds at once – or the worst, if you only have an understanding of one of those representations.

## dborkovitz

Gregory, thanks for commenting. Nice example with slope — there was a whole big debate on the def’n of slope associated w/this controversy. With my college students, I find very few who have realized that if x increases by 1, then y must increases by the slope. I like to do some problems w/slope and no graph at all — e.g. if 37 deg celsius is normal, and that translates to 98.6 F, then if someone has a 39 degree fever in C what is their temp in F, without finding the equation of the whole line (where we’ve already used freezing/boiling points to establish slopes).