A guest post I recently published critiquing the Khan Academy received a great deal of response, including an e-mail from Salman Khan, founder of the academy. Now here’s the next part of the debate: A response from the critic.
For those who may not know, the Khan Academy is essentially a library of more than 3,300 videos on subjects including math, physics, and history that are intended to allow students to learn at their own pace. Questions were raised about the quality of some of the math videos in the guest post mentioned above as well as in others elsewhere on the Web.
Below is Khan’s e-mail to me, and following that is a response to Khan from the author of the original post, Karim Kai Ani, a former middle school teacher and math coach who started a company called Mathalicious. I’ve also posted some responses to the debate that I received from readers.
Here’s Khan’s e-mail:
We here at the Khan Academy appreciate a public discourse on education and really encourage as much feedback as possible. We believe that we are in the early days of what we are and feedback will only make that better. I agree with you that no organization should be upheld as a magic bullet for education woes. We have never said that we are a cure-all and think we have a lot to do just to fulfill our potential as a valuable tool for students and teachers. Unfortunately, some of the headlines on articles are more grandiose, but we have no say in this.
In your previous post, you talk about the value of experiential learning versus lecture-based. We agree 100% with you; that is what KA is about too--allowing classrooms to be more interactive and experiential. See this video: http://www.khanacademy.org/talks-and-interviews/v/ideal-math-and-science-class-time.
We are also running project based summer camps.
With that said, there have been some major errors on your blog. In particular, Karim’s corrections are very incorrect (I encourage you to seek out an impartial math professor).
Slope actually is defined as change in y over change in x (or rise over run).
Karim’s definition is actually incorrect. Slope is not just a rate between two variables. It is how the variable plotted on the vertical axis changes with respect to the variable plotted on the horizontal axis (or “rise over run”). For example, if price were on the x-axis and memory on the y-axis, then Karim’s “how the price of an iPod changes as you buy more memory” would not be slope (it would be the inverse).
And, yes, slope is often unit-less (especially when measuring the slope of say the surface of a mountain which is where the whole idea comes from — you are dividing a distance by a distance so the units cancel).
I walk through this in a video at: http://www.youtube.com/watch?v=TNaQJjLAhkI
I also think you might misunderstand our “business model.” Unlike Mathalicious which is for-profit (and if it does well, Karim will become very wealthy), Khan Academy is a 501c3 not-for-profit. I take a salary from it that is approved by the board, but I do not own it (no one does). It can never IPO or be sold. It is a public charity.
Let me know if you’d like to chat further.
And here is Karim Kai Ani’s response (in bold italics are quotes from Khan’s email):
“Karim’s definition is actually incorrect. Slope is not just a rate between two variables. It is how the variable plotted on the vertical axis changes with respect to the variable plotted on the horizontal axis (or ‘rise over run’).”
Sal’s right: slope is not just a “rate between two variables.” But that’s not what I wrote. What I wrote was, “slope is a rate that describes how two variables change in relation to one another.”
Effective teaching requires the precise use of language. Khan’s original definition was that slope is simply “rise over run” which, while procedurally helpful, does not accurately convey what slope actually means. That said, his amended definition — how y changes with respect to x — is better. It’s not perfect — perhaps it would be more accurate to say, how the dependent variable changes when the independent variable increases by one [unit] — but it is an improvement.
So why not just use that definition in the first place?
The mark of effective teachers is that they’re constantly looking for the best way to communicate mathematical concepts to students. And while they may not always find it — indeed, while there may not even be a best way — it’s the pursuit that characterizes them. Yet this commitment to quality seems absent from Sal Khan’s approach to teaching which is characterized by his own admission, “I don’t know what I’m going to say half the time.”
For a college student preparing for the GMAT or even a high school student who needs to review for a test, this may be fine. But that’s not why we’re talking about Khan Academy. We’re talking about it as a wholesale revolution in public education. Across the country cash-strapped school districts are looking to incorporate the free Khan Academy as a core instructional resource, which is not something it was ever designed to be. Sal Khan started the service to help his cousins with their math homework, and there’s a real danger in extending it beyond its natural limits.
“If price were on the x-axis and memory on the y-axis, then Karim’s “how the price of an iPod changes as you buy more memory” would not be slope (it would be the inverse).”
The 8GB iPod Touch costs $199. The 32GB model costs $299. Once again, Sal’s right: if price were on the x-axis and if memory were on the y-axis, then slope would be inverted and written in terms of GB/$.
But they weren’t. Once again, Sal is arguing against a point I didn’t make.
Still, it is true that there are two ways to think about the relationship between the size of an iPod Touch and its cost, and therefore two ways to describe the slope.
The version on the left describes how the price of the iPod changes as the memory changes: price as a function of memory. Here slope is measured in dollars per gigabyte: $4.17/GB.
The version on the right describes how the size of the iPod changes as the price changes: size as a function of price. Here the slope is measured in gigabytes per dollar: 0.24GB/$.
So which version is correct? They’re both correct, and the one we use depends on which relationship we want to discuss.
What matters more, though, is that students are able to have a discussion in the first place: Is Apple really charging a constant amount per additional gigabyte, and is pricing linear? If the iPod doesn’t come in every possible size, should we even be drawing a line? This requires a teacher who is intentional about creating learning experiences that are meaningful. Absent this, as in Khan’s original video, the best students could do would be to describe slope as 4.17 or 0.24. 0.24, what?
Effective teaching requires more than Sal’s “two minutes of research on Google.” By definition this will yield only the most popular explanations, but not necessarily the best ones. Again, though, this isn’t a knock on Sal Khan. People have been teaching slope in a decontextualized, “rise over run” way for years…which is exactly why we need to teach it differently. If it didn’t work before, recording it into YouTube won’t change that.
“Slope is often unit-less (especially when measuring the slope of say the surface of a mountain which is where the whole idea comes from — you are dividing a distance by a distance so the units cancel).”
This is actually a very interesting question, and one we teachers were debating at Twitter Math Camp: Can you describe slope without units?
On one hand, as highlighted above, units are certainly a valuable pedagogical device. $4.17/GB is much more concrete than the unit-less 4.17, particularly for middle school students whose brains are yet to fully develop the capacity for abstract thought.
On the other hand, does this mean that units are absolutely necessary? Take Sal’s example of climbing a mountain. Let’s say that for every two feet we move in the horizontal direction (x), we climb three feet in the vertical direction (y). In this case the slope is 1.5 vertical feet per horizontal foot.
According to Sal, the units are the same (feet) and would therefore “cancel,” leaving a slope of 1.5. But is this correct: is a vertical foot really the same as a horizontal foot, or do they in fact describe two very different things?
There’s no simple answer. And that’s okay. Because the goal of the debate wasn’t to resolve some existential question about the nature of feet, but rather to figure out the best way to introduce slope to students. Personally (and brain development aside), I think slope does require units; after all, if units don’t exist, what exactly are we counting?
Of course, others may disagree, and indeed other teachers did disagree. Does slope require units? Some said yes. Others said no. What Sal said, though, was, “you’re wrong.” This seemingly defensive attitude is antithetical to effective teaching.
The real issue isn’t whether slope requires units or not. It’s whether we expect our teachers to engage in conversations like these, and whether we appreciate the value in their doing so.
“I also think you might misunderstand our ‘business model’. Unlike Mathalicious which is for-profit (and if it does well, Karim will become very wealthy), Khan Academy is a 501c3 not-for-profit. I take a salary from it that is approved by the board, but I do not own it (no one does). It can never IPO or be sold. It is a public charity.”
Given that my original post made no mention of Khan Academy’s business model, I’m not entirely sure where his comment is coming from. Nonetheless, I can appreciate why Sal (and others) may be suspicious of my motives.
As highlighted in the blog comments, many people believe that I’m pushing back against Khan Academy because it’s somehow “bad for business.” In reality, Mathalicious and Khan Academy aren’t competitors. Khan Academy creates videos to help students learn basic skills. We create lessons that help teachers apply those skills to real-world situations. Khan talks about a “flipped classroom:” we’re the other side of the coin! Thus, an effective Khan Academy would actually be good for Mathalicious, insofar as it would free teachers to spend more time on the types of projects and applications we create.
The problem, though, is that Khan Academy’s do-this-then-do-this style of instruction does little to foster true mastery, and students can’t apply what they don’t understand. I don’t say this as an entrepreneur; I say it as a teacher.
That said, Sal’s right: Mathalicious is a for-profit. Yet this is a legal/tax designation, not a motive. In fact, the reason I started Mathalicious -- the reason I’ve worked on it for the past three years, burned through my teacher savings, sold my camera equipment on eBay and even turned to Kickstarter — is to support classroom teachers with engaging and effective content.
Still, Mathalicious has expenses like everyone else and does charge. We have to. But we do it in a way that tries to honor the people we serve: instead of charging what the “market will bear,” we allow teachers to pay what they can. Of course, it would be even better if we didn’t have to charge teachers at all, but we don’t have the luxury of the more than $16 million that Khan Academy has received from the likes of Bill Gates (Microsoft), Eric Schmidt (Google) and Reed Hastings (Netflix).
Even if I were some educational monopolist – even if Mathalicious were a multibillion dollar company that charged $7,000/teacher – it wouldn’t change the substance of the argument: that “I’m not sure what I’m going to say” is incompatible with good teaching and does not constitute a revolution in education. That’s the conversation we need to be having. Everything else is sleight of hand.
People may interpret my criticism as a sign that I want Khan Academy to disappear. I don’t. I believe it has the potential to be a useful tool for students and teachers. For this to happen, though, Khan first has to acknowledge that his instruction isn’t perfect, and be willing to engage the community of educators who can help make it better.
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