"You will never hear in any Khan Academy video 'memorize this formula.' " - Sal KhanWell, that should be easy to check. And he's wrong.

- "What we are going to do in this video is to give you just a ton of more practice and start you on your memorization of the multiplication tables." (Multiplication Tables, 0:13)
- "If you don't get it, maybe you just want to memorize this." (Squeeze Theorem, 0:37)
- "And it's good to memorize this." (Basic Ratios, 0:41)
- "I think this is a good time to just memorize the sides of a 30, 60, 90 triangle, because that's something that one needs to know in life. It's surprisingly useful, especially once you start taking standardized tests or do trigonometry." (Circle Area, 1:59)
- "You really just need to memorize that the hypotenuse is twice the shortest side." (Same video, 2:46)
- "You also should just need to memorize PV is equal to NRT." (Thermodynamics iv, 9:47)
- "All you have to really memorize is this." (Harmonic Motion, 6:52)
- "You can just memorize this... and you might want to just memorize it." (Trig Identities, 1:21 and 5:02)
- "You should memorize this." (Law of Cosines, 8:46)
- "So you should, to some degree, memorize these." (Quotient Rule, 3:38)
- "And now I'm going to give you one of the few things in math that's probably a good idea to memorize." (Quadratic Formula, 5:15)
- "And it's useful to memorize to some degree what this means." (Binomial Expansion, 1:33)
- "It's something good to memorize," (Conservation of Energy, 2:53)
- "This is something that you really should just memorize." (Trig Identities - Secant, 8:19)
- "It's probably not a bad idea to memorize some form of this formula." (Projectile Motion, 4:55)
- "It doesn't hurt to memorize this." (Radians and Degrees, 5:06)
- "I'd like to help you memorize this." (Normal Distribution, 7:57)
- "You probably won't be using this in your everyday life five or ten years from now, so it's OK if you don't memorize it, but temporarily, put this in your medium-term memory." (Subspace, 2:31)
- "In future videos I will give you a little bit more intuition for why this works, or I'll actually show you how this came about. But for now, it's almost better just to memorize the steps." (Inverse Matrices, 6:24)
- "And then this formula...just memorize. This "P e^rt' will make a lot of sense to you, and you will have a permanent neuron for it the rest of your life." (Compound Interest, 7:51)

There are a lot, lot, lot more times that Sal Khan asks students to memorize stuff. The way to get access to this is through the Khan Academy interactive transcripts. So just use your favorite search engine to look for the word "memorize" through khanacademy.org, and then click on the "Interactive Transcript" button on the video page. Then use your browser to search through the page for the word "memorize."

Anyway, happy searching.

Is your issue with the fact that he asks us to memorize within his lessons, or that he misrepresents the degree to which he asks us to?

ReplyDeleteI get the latter and agree that Sal has some work to do in the communication department. As for the former, I think all of us have asked students to memorize something from time to time. I'm not ready to crucify him for that.

And, in his defense, he never actually uses the quoted phrase, "memorize this formula."

ReplyDelete@Anyonymous:

ReplyDelete"You could immediately -- if you memorize this formula although you should know where it comes from, you could immediately say..."

@Marshall

What's an example of something that you'd ask your students to memorize? I'm having trouble thinking of something that I've asked kids to memorize.

There are lots of things that I want me kids to commit to memory. But I think that "memorizing" something is often the least effective way of committing it to memory. Much more effective, I think, is to put a student in a variety of situations where the knowledge of some formula proves useful.

And doesn't repeated calls for memorizing betray a poor understanding of how learning works? If you don't understand something, how on earth will memorizing help you? (See the quote from the Squeeze Theorem, "If you don't get this, maybe you should just memorize it.") We've all dealt with students with just bizarre mistakes that come from memorizing.

Also, we might disagree here, but I don't think that Sal asks kids to memorize stuff "from time to time." I think that it's pretty often, given the evidence that I pulled out.

And, look, plenty of teachers ask their kids to memorize stuff. And lots of teachers present math as purely procedural knowledge. But Sal is claiming that Khan Academy isn't like those teachers.

On what planet?

No I get it that Sal's claims and Sal's product don't quite match up.

ReplyDeleteThe best example of something kids need to commit to memory is the quadratic formula. Sure, you could derive it every time if you wanted to I suppose, or stick to completing the square, but life after factoring is so much easier if they memorize it. And I think that's a completely defensible position because the memorization of that formula gives the kids another tool to use.

Some of my colleagues like the song (yuck) but I prefer to make them write it 50 times if they don't find a way to remember it on their own. No, not fun, but necessary. That's my opinion, anyway.

My kids last year preferred using completing the square to memorizing the quadratic formula. It made sense to them, and the quadratic formula didn't. I'm not convinced that the formula is any easier than completing the square, either in terms of memory or in terms of frequency of errors.

DeleteNot sure I completely understand your argument. Are you saying there are no cases memorization is advantageous? No cases it's necessary? Few cases? And are you arguing that Sal is over-emphasizing it? Or do you find just that fact that he encourages at all is an issue? I definitely agree with you, and as Marshall put it, that his claim and product don't "match up", as he obviously tells people to memorize stuff. Just trying to understand the nuances of your argument clearer.

DeleteFor example, how do you feel about Sal asking students to memorize the multiplication table? To me, some things that require "fluency" should be memorized. Which I guess brings me to another question... are you treating "memorize" as a process? It sounds like you distinguish between "memorize" and "commit to memory", which I wouldn't assume most other people would treat differently.

Also, to clarify a couple things...

Delete1. My reply is to MBP in case it's not clear.

2. I'm Frank Lee, in case people immediately associate "Frank" and educational blogs with Frank Noschese, the better-known Frank in these areas. ;)

Ha! Welcome, Frank.

DeleteFirst, as you've both mentioned, Sal contradicted himself. Oops. But who cares?

It matters because Sal, rightly, thinks that memorizing formulas is a bad idea for math students. He's right in that interview -- you should (almost?) never ask students to memorize formulas. And the evidence shows that he asks kids to memorize formulas all the time. Not even super rarely -- he does it all over the place, and for a variety of reasons. I think that Sal asks students to memorize stuff too often. That's my first claim.

My second claim is about asking kids to memorize stuff at all. And here I am making a distinction between "memorize" and "commit to memory." Memorizing is an activity. When you ask kids to

memorizea formula, you're asking them to spend time intentionally and directly committing a formula to memory. There are different ways of trying to do this -- some like flashcards, some like saying things out loud, others like mnemonics. Having somethingcommitted to memory, on the other hand, is a state. There are lots of things that I have committed to memory that I never had to memorize. For example, my name, my cousins' names, the price of grapes at the local bodega, etc.Memorizing isn't always the best way to commit something to memory. For example, we want kids to commit the procedure for solving linear equations. We could have kids memorize an algorithm for solving linear equations. But that would be a bad idea -- as Khan and pretty much everybody knows. Instead we give kids problems in lots of different contexts that forces them to develop and commit to memory procedures of their own that allow them to easily solve these problems.

So when is memorizing the best way to commit something to memory? I'll throw out these as a few first-shot criteria:

1) When the thing that we want kids to remember isn't long and complicated. Humans aren't good at directly memorizing long and complicated things.

2) When the thing that we want kids to remember can't be better remembered through other techniques, such as problem-solving, rederiving, explaining, etc.

I suppose what I'm really getting at is a prediction, a testable hypothesis. Multiplication tables should be committed to memory. Kids should know them off the top of their heads. But memorizing, I think, will not be consolidated into long-term memory in an efficient way. I'm predicting that retrieval through application in novel contexts is more efficient.

Worth noting 20/3000 (number of calls to memorize/total number of videos) puts the lie to "all the time".

ReplyDeleteWhat ratio would you want before we could make a claim for "all the time"? I'm not looking for precision or an exact number here, just a ball-park figure.

DeleteIs the point here just to pick our phrases more carefully? Sal's "never" definitely wasn't a never. When MBP said "all the time", I don't think anyone would interpret that literally as *all the time*, but as he mentioned somewhere above, Sal asks students to memorizes things too often.

DeleteYes, it's 20/3000, but how many of the 3000 are videos that actually involve math and formulas in which Sal would even have the opportunity to suggest memorization as part of problem solving? That certainly doesn't bring it down to near 20, but it does bring it down some.

Regardless, I think the evidence shows (and the point being?) that Sal suggests memorization significantly more often than he claims (but for his sake, let's not put a percentage on that).

You are employing the same broad-stroke mischaracterizations that sparked the writing of this blog post. I hope the irony of this is not lost on you. Stones and glass houses and all of that.

DeleteWhat I'm trying to say is the conversation is unproductive if we focus on literal interpretations of the authors' words in this case (especially since the common usage of "all the time" is more debatable than the common usage of "never"). If we go with the intent of their messages, I believe the point still stands that Sal encourages memorization significantly more often than he thinks he does. Do you disagree?

DeleteHow about this:

DeleteForget "all the time." My bad. I don't know what counts as "all the time", and so it's entirely possible that I was misusing the phrase. Subjectively, it seemed like an OK characterization but I can't say that I'm sure.

Here's all that I care about us agreeing on: There are about 90 videos where he asks students to memorize something. I just went through my search above and counted. If you think that I'm wrong, please check my work and do your own count. (By my count there are 42 times when Khan tells kids not to memorize a formula.)

From going through the search, it seems to me as if Khan asks kids to memorize things that he takes to be basic: multiplication tables, slope, physical formulas, elemental properties. He also asks kids to memorize things whose derivation is complex: various theorems in Calculus, some advanced stuff in Linear Algebra. In other words, the two options for Sal are derivation and memorization, but for teachers there is (at least) one more option: pattern observation. That's something that Sal hasn't figured out how to do in videos, but is a way of helping kids learn stuff without memorization, though also without the full sophistication of a proof.

Also, thanks everybody for pushing me on this post. I'll hopefully put together a follow-up out of these comments.

First, I don't think there really is such a thing as "pure memorization" for those of us without a photographic memory. We have to develop understandings and make connections in order to remember or commit to memory.

ReplyDeleteI found memorizing proofs (the majority of the book) to be quite helpful when I was taking more advanced classes in college. I couldn't remember it all without good understanding. I also think it allowed my subconcious to further develop my understanding.

I suspect that high school students would also benefit if they were able to memorize a large amount of well connected content. Of course, most high school content is not very well connected.

Right. We're talking about memorizing exponent rules instead of seeing negative exponents on the same continuum as positive exponents. We're talking about memorizing a way to calculate slope instead of seeing it as a rate. We're talking about memorizing a multiplication table instead of learning calculation shortcuts through the properties of numbers.

DeleteMemorization sometimes comes along with good learning, but sometimes doesn't. And isn't that the point? That memorization is an unreliable technique to recommend to students?

This is a very important conversation, and I'm glad you posted this!

ReplyDeleteThe idea of memorizing something purely on what it appears to be on the outside (just the formula part: the letters nd operations involved in an equation, for example) versus "committing it to memory" based on the nuances and all of the aspects that are hidden upon initially seeing the formula,like how it's used in situation A, B, and C or what specific contexts it would useful in, is so important in mathematics education. It's the difference between a student memorizing something for a short bit during the time period they use it or UNDERSTANDING the concept and comprehending the *whys*, perhaps for a lifetime, because of the numerous mental connections that are involved in committing a concept to memory.

I, for one, believe that it is our job as math educators to challenge students to commit ideas to memory without boring rote memorization that often becomes useless in the long run. Through a variety of problems, collaborative demonstrations, and other activities, a math classroom should be one in which students can explore the beauty of math through the concepts. Because, hey, what's the point of learning math if it's just an exercise in memorization? The only thing that'd teach is students to be drones of tomorrow.

I really love the message you're trying to get at- that even the seemingly most basic ideas DO NOT have to be memorized if teachers can appropriately guide students to viewing the reasons and whys behind the formula. Instead of being a simple, 1-step "just memorize this for tomorrow" process, "committing something to memory" takes any learner through multiple scenarios and levels of understanding before they finally *get* it conceptually. This happens to me all the time- I think I understand a seemingly basic concept to its extent but then realize that there is more to the idea than meets the eye that I have yet to realize. That's when I sit back and think how cool math is... these moments are what we need in math education to make it interesting and fun, and this conversation has encouraged me to continually strive toward a "no memorization" goal. Thank you, MBP!

(I hope that I understood your main point after reading your clarifications... if I misconstrued it, please feel free to correct me.)

The painfully consistent reality is that Sal Khan keeps claiming that he really wants people to understand math... and provides what he says are innovative and amazing tools for doing this... except that even when he doesn't use the words "memorize this," he does not ground his explanations of math in the understanding he claims is important. He describes procedures and provides opportunities to practice, practice, practice the procedures. Practicing a procedure helps you memorize the procedure. Students generally reach their own conclusions as to why the procedure works -- and unless they've got other support the conclusions tend to be wrong.

ReplyDeleteSal Khan, however claims that the "best way" to understand math is to do a whole bunch of problems -- and then demonstrates the procedures, not the conceptual background, for getting the right answer to problems. (He doesn't say why he believes this.)

Now, there's rather a whole lot of evidence -- tediously gathered as well as anecdotal -- that says that practicing procedures actually *doesn't* help students understand the math. ( http://www.resourceroom.net/math/ida_math_spring2011_booth.asp and http://www.carnegiefoundation.org/carnegie-perspectives/what-were-learning/what-community-college-developmental-mathematics-students-understand-about-math are two examples of rather many).

It's a bit counterintuitive, too -- when does it make sense to actively teach the thing you claim is less important?

I work with so many students who have been told that they need to practice procedures and they assume that math is fun for that select group of math people... and the rest of us have to memorize procedures and survive courses so they can forget those silly rules and move on with life.

I see so much of the Khan Academy that reinforces that and that's why I question the claims that he's all about "understanding."