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Comment by spinchange | Hacker News Reader

Can anyone weigh in on their differing arguments regarding the definition of slope? I am not a maths expert by any means, but am genuinely curious who is correct here. Or is this a matter of one being technically correct versus the other being correct in practice?

(Cross posted from the original thread because I genuinely would like an opinion more informed than my own and the totally unhelpful "conclusion" provided by Valerie Strauss.)

*Edit, I missed that he (Khan) posted a video defending his definition http://www.youtube.com/watch?v=TNaQJjLAhkI

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I'm not a math expert either, but rise over run sounds like a perfectly good definition of slope to me, at least for the purposes of an introduction to high school algebra. It is certainly how I was taught about slope, and I went on to graduate from MIT, so whatever nuance this definition allegedly didn't catch did me no harm.

Once you get to calculus, things get a little more complicated, as the slope at a given point is defined using derivatives, but I'm pretty sure that ends up being the same as the rise over the run of a tangent line.

In any case, anyone who would make such a lame criticism, should just STFU. If that's the best criticism they can come up with, they surely can't have much of value to say. Also, when teaching something like algebra, it's more important to make the material approachable and comprehensible, rather than define everything to a level of rigor that would make Russell and Whitehead happy.

>It is certainly how I was taught about slope, and I went on to graduate from MIT, so whatever nuance this definition allegedly didn't catch did me no harm.

In that case, this debate really isn't about you. You were probably an exceptional student, who saw the connections between mathematical concepts easily, regardless of instruction. You probably found yourself predicting the next concept a teacher would introduce, because it just "makes sense." Not all students are that way. Most are not.

The two people involved here are fighting over two different ideas. Sal is being pedantic, but is right, slope is defined as ∆x/∆y. What the other guy was saying is that slope represents rate of change, which is a much more important concept to early algebra, and the underpinning of why you actually care about slope in physics and calculus. You probably made the connection effortlessly. I assure you, many students do not.

I teach high school mathematics to both honors and special needs students, and it's important to keep in mind that the instruction is very different between the two populations.

"slope is defined as ∆x/∆y"

Do you mean ∆y/∆x?

Yup. Caught the typo when I wrote the post, even, but I apparently retyped it the same way.

I'm still not sure I understand the criticism. I'm sure that Kahn must eventually get to rate of change in his algebra course. The criticism is then supposed to be that Kahn didn't motivate his students on why they should care about slope soon enough?

If so, that's a completely different criticism, however, from the criticism that Khan is putatively making an alarmingly dense stream of gross factual errors.

I think that we can all agree that Kahn is not the best possible teacher that exists in the world for each given subject. Is that a decent argument against what he has done? Hardly! That would be letting the perfect be the enemy of the good.

Considering that so many people learn from the Kahn Academy these days, an argument can certainly be made that Kahn's lectures should all eventually be replaced with lectures by the actual best teacher in the world for that given topic. For all we know, this is already in the works.

> slope represents rate of change

You may as well say it represents a tangent. (Pun not entirely intended.)

You could say that, but it would be wrong. A tangent is an equation of the form y = ax + b at point P, which just happens to have (well, by definition) a value for a that equals the rate of change at P (of the original equation).

Usually when you talk about slope we're assuming "slope of a line". In that case, Khan's definition is fine. If you're trying to really rigorously define "slope", which is fairly silly since you should just start using the term "gradient" which is more rigorously defined, then I could see this going two ways:

1. You could argue that if you're talking about slope, you mean slope of a line. If you want to talk about the generalized notion of slope of a line, you should use terms like gradient, derivative, etc. If Khan had taken this stance, I would have been fine with his defense.

2. You go with the fully rigorous definition of slope as basically being a synonym for gradient. This is what mathworld actually does. This is where Khan's reputation gets really knocked, in my opinion. He quotes mathworld (http://mathworld.wolfram.com/Slope.html) but only selectively. What he failed to mention are these key points:

- The very first sentence defines slope this way: "A quantity which gives the inclination of a curve or line with respect to another curve or line." - The sentence he did quote begins with "For a line in the xy plane..."

For the vast majority of his students, including those other points when they're first learning about slopes would do more harm than good. He shouldn't be judged on his rigor, but on how effectively he enables learning and understanding. They wouldn't know what to do with gradients.

Sure, but I'm not talking about how effective his teaching is. I'm simply addressing the technical definition of slope.

I am perfectly fine with Khan teaching it as "rise/run". I thought criticizing this was silly on Karim's part. But then Khan was the one who came back to argue that his definition was correct and Karim's was wrong. I think Khan's wrong here and shouldn't have even engaged in such a trivial dispute...

I'm not a mathematician, but I think that the initial response by Karim saying that Khan was incorrect about the definition is really nitpicky and petty. Karim is saying that "Rise over run" is how you calculate it, but he claims it's not the definition. Even if it wasn't the precise definition, for most people, rise over run is good enough for a basic understanding.

Khan saying that Karim was wrong, is wrong and actually made him look worse in my eyes. In the video he said "Slope can present rate of change." No, slope is the rate of change. Him saying Karim was wrong about the price vs gigabyte being the inverse if you switch the axes is ridiculous. By definition, you always say the first variable vs the second variable, where the first variable is the y axis and the second is the x. Khan should have left it at that instead of trying to twist things around to make Karim look wrong, it was a poor and transparent attempt at being vindictive.

In math, is not accurate to say the "definition" and the "calculation" are one in the same? Sometimes we require ambiguous non-math language to start understanding a concept but ultimately, once understanding has happened, the calculation and the definition are identical.

Definition and technique to calculate are not the same thing. Consider integer division, which is defined by an equation, but calculated by an incremental long division algorithm.

Most definition use a phrase like "such that", which says nothing at all about calculating.

"Slope" implies that the denominator is a distance. In the context of a graph on paper or screen, it is clear how rise and run are mapped to distances. It's a totally sensible definition: slope only makes sense given a choice of axes.

"Rate" implies that the denominator is a timespan. This does not make sense, as there are many slopes which are not rates and which are not presented as rates.

It's kind of a silly thing to worry about, but Khan's answer is unequivocally better.

Khan’s definition is fine. Karim Kai Ani is being foolish by insisting that “slope is a rate that describes how two variables change in relation to one another.” No, that’s a rate of change. A “slope” is a commonly used rate-of-change in which there’s a reasonably clear “rise over run” relationship implied between the variables of interest.

But when that relationship isn’t so clearly implied, sensible people don’t try to describe it by calling it a “slope.” Instead, they say what relationship they really care about: Is it the instantaneous rate of change at a specific point? Or at all points along a curve? (Or surface?) Or is it the average rate of change over some interval? Or the weighted average rate of change over some interval of varying density? Or the weighted average rate of change over constant-width intervals centered at certain (or all) points along a curve? Or is it really that they care about—?

You get the drift: If you care about rates of change, you’ll use the term “slope” to describe them only when the context implies a clear “rise over run” relationship between the variables of interest. Kahn seems to get this; Karim Kai Ani, not so much.

This point is an interesting crux of disagreement about methods, not just nitpicky. Technically, slope is the rate of change of the tangent to a function, and the disagreement is about how to communicate that to students.

Approach 1, Mathalicious: "Unless you give students the right information from the first, even if it is a bit more abstract, they won't be prepared later on."

Approach 2, Kahn: "The best way to prepare students is to keep things simple. Later you can give them refinements about tangents and derivatives and such."

This is an interesting point, and I'd be pretty surprised if there isn't already a wealth of knowledge surrounding it in educational research. (Neither Kahn nor Ani appeals to such research in the articles.)

I think this debate does sadly just come down to semantics, and I tend to side with Kahn on that merit. Once I got past a certain point in my math education (first half of intro of calculus 1) "slope" wasn't used anymore. We moved to derivatives and gradients. In my education we've always used slope as a purposefully simple/geometric term for 1 dimensional lines plotted in two dimensional x-y planes.

Agreed, "slope" is a rather useless concept beyond algebra. When you're plotting, say, a position function, the slope of the function at any given point is of limited meaningfulness--the velocity, on the other hand, much more so! Likewise, the slope of the velocity curve at any given point is of little import, but the acceleration is very meaningful.

Of course, the slope of the position curve is the velocity; what I'm suggesting is that referring to it as slope detracts from understanding, while referring to it as velocity enhances understanding. Consequently, getting all wrapped around the axle over the canonical definition of slope is a shining example of majoring in the minors.

Sal is right but I understand the critique... which I might clarify as the importance of understanding the larger concept that slope is a comparison of the rate of change of one variable to another in specific order.

To take the example of price of an iPod vs. more memory. The slope is inverse depending on which variable you choose to plot on which axis. If you just watched Sal's one video, you would not understand why that is... your only context is 'rise over run'. If given sample data points and asked for a slope, you might not be able to figure out which way to plot it based on the question.

To be fair to Sal though, this was just a single video subtitled "Figuring out the slope of a line" and so Sal may have covered the larger concept and its importance in a different video.

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