Oh dear.

No.

No.

No, no, no.

No.

I know we went over this. I **know** we did.

Seriously, Mr. Khan. Gimme a ring next time, OK? I’ll talk you through it. I promise.

I’m easy to find.

(“Thanks” to Frank Noschese for alerting me.)

The mathematics I encounter in classrooms

Oh dear.

No.

No.

No, no, no.

No.

I know we went over this. I **know** we did.

Seriously, Mr. Khan. Gimme a ring next time, OK? I’ll talk you through it. I promise.

I’m easy to find.

(“Thanks” to Frank Noschese for alerting me.)

This isn’t available on the Khan Academy site yet; just YouTube so far. But it responds to my original critique—that nowhere does Khan Academy help students to compare decimals with different numbers of places.

I initially observed that the my feedback was incorporated in an awfully literal fashion. Frank Noschese came to Mr. Khan’s defense:

@Trianglemancsd C’mon, man. Obviously he’s taken your pretty straight forward feedback and very clearly used it in his video. He’s trying.

— Frank Noschese (@fnoschese) August 14, 2013

Maybe Mr. Khan and I can have an extended conversation in New York in September? (Although I am suspicious that he may be telecommuting to that thing!)

Ben Alpert from Khan Academy responded to my open letter to Sal Khan and provided a link to an improved set of decimal exercises.

They are better.

You end up having to compare two-digit decimals to one-digit decimals.

My major objections now boil down to pedagogy, on which point I understand that I will make no progress with Khan Academy, so I won’t make the effort. I’ll leave that to Frank Noschese.

If you accept that people learn mathematics by doing lots of multiple choice exercises, then all I have left are technical details.

They are these:

- In the U.S., money is a good enough model to get students through two-digit decimals. It is not uncommon for children to be able to reason about two-digit decimals, but not generalize to three- and four-digit decimals.
- These are randomly generated, I assume. And the probability of getting two-digit decimal>one-digit decimal seems artificially low. As I ran through a bunch of these, I began to build a model in which I could (1) treat comparisons with same number of decimal places as whole numbers, and (2) claim that the one-digit decimal is larger. So I ran an experiment. Twenty exercises using my model. I got 90%. (See video of a repeat of this experiment; I don’t know that I did 20 this time, but I did a bunch and only got one wrong)
- Related to this, there is no need to click through the hints. None of the decimals came out equally (i.e. no 0.1 v. 0.10). So when I got a wrong answer, I just chose the other inequality. Pattern matching and process of elimination allowed me to avoid instruction of any kind and to get an A.

See, here’s the thing. Teaching requires a mix of knowledge and assumptions on which to base decisions. When everything is pre-programmed, deeper knowledge is required in order to create meaningful instruction, not more analytic data.

Carnegie-Mellon is working on a deep model for diagnosing student misconceptions with decimals [pdf] (and presumably many other domains). Again there’s the pedagogy thing, but I am impressed with the effort to build a solid theoretical foundation for their work. Here is a sample of a taxonomy of decimal misconceptions they have developed.

Without that deep knowledge base, all that’s left are assumptions. Which is fine, as long as the assumptions are not flawed.

Dear Mr. Khan,

A year ago, I expressed my concerns on the Washington Post’s blog that your decimal place value videos and exercises failed to incorporate very basic knowledge about how people learn place value.

I wrote that your decimal comparison videos were problematic because they only addressed decimal numbers with the same number of decimal places, and that a very basic, robust finding in rational number learning research is that students do not struggle with these comparisons—because students can treat them like whole numbers and get correct answers. Instead, students struggle with comparisons where the decimals have *different *numbers of decimal places because here, the whole number place value rules do not apply.

Together with my co-author, I wrote,

A student who thinks that 0.435 > 0.76 is offered nothing in the way of correction on Khan Academy. In fact, one of the top questions on the page for this video (as of July 18, 2012) is “So is .02009 greater than .0207?” This is exactly the sort of question that a competent teacher of arithmetic needs to anticipate and to answer. Khan fails to pose it.

In short, these decimal videos and their accompanying exercises are useless.

You must have read our piece, as it came out at the same time as Karim Kai Ani’s critique of your treatment of slope, which you responded to directly in writing and video.

But you do not seem to have taken the critique seriously. Consider the following video, which you posted yesterday.

Notice how 1000 is the same size as in that exercise?

Mr. Khan, that matters. It matters very much.

The hard thing about learning decimal place value isn’t learning the *names* of the places. It is learning the relationships among these places. That is of , for instance, and that 10 make . And that these two relationships are themselves related.

When we fail to emphasize these ideas in instruction, we get the following results. (The following is a short excerpt from a longer video that is part of IMAP at San Diego State University, on a CD-ROM published by Pearson.)

These two girls (earlier in the video) correctly identified that 1.8=1.80 because you can *add a zero *to 1.8. But then, if it’s “You Cannot Add a Zero Day” they decide that 1.80 is larger because it’s “1 and **80** hundredths” while 1.8 is only “1 and **8** tenths”.

These two students have learned all the rules that you seek to teach them, and they do not understand decimals at all. What we see in these two girls’ thinking is precisely the problem you set out to solve with Khan Academy. But you aren’t solving the problem, Mr. Khan. You are perpetuating it.

The problem is that students learn names for places and rules for operating without thinking about the values and the relationships among these values that our place value system represents.

In the imagery of the exercise in your new video these students would be imagining 80 hundredths boxes for 1.80 and 8 tenths boxes for 1.8. The exercise builds a mental model for students that feeds their misconceptions.

Mr. Khan, you have a team of teacher advisors. If none of them can identify these gaps for you, you need to ask for help from the larger community (and then to reexamine your hiring practices).

You might consider starting with Twitter. Like this:

Dear Twitter: What is one common mistake or misconception made with decimals? #mathchat#MTBoS

— Christopher (@Trianglemancsd) August 1, 2013

You have many more followers than I do, so you should be able to generate in a few minutes several dozen times what I got back in a couple of hours. You might get responses such as the following…

@davidwees @Trianglemancsd “0.05 is not equivalent to 0.050.”

— ATLT Games (@ATLTGames) August 1, 2013

@Trianglemancsd 0.5,0.36,0.004, etc are decimals. They forget about 3.8, 6.2 etc. Or 2 is not a decimal b/c no decimal point.

— Jamie Rykse (@jrykse) August 1, 2013

@Trianglemancsd 0.05 is larger than 0.5 “because it has more zeros in it.”

— David Wees (@davidwees) August 1, 2013

@Trianglemancsd @mathtans When making comparisons, Ss think the more digits in the decimal number, the larger the number.

— Math Minds (@MathMinds) August 1, 2013

@emwdx @Trianglemancsd blindly use algorithms wo thought/estimation. So 2.1*5.25 can end up 110.25 or other such variation.

— Julie (@jreulbach) August 1, 2013

@Trianglemancsd Mistake: 0.3*0.3=0.9

— Evan Weinberg (@emwdx) August 1, 2013

@Trianglemancsd Appending zeros at the end creates a larger quantity.

— Chris Robinson (@absvalteaching) August 1, 2013

@Trianglemancsd Assuming universality of decimal system units? 4.25 hours isn’t 4 hrs, 25 minutes.

— Gregory Taylor (@mathtans) August 1, 2013

Do you see that none of these has to do with the instructional purpose of your video or exercise? None of these has to do with *naming* the decimal places. They all have to do with understanding the relationships that decimals are intended to notate.

What your work presents as being the whole mathematical story (naming decimal places) is just the tip of the iceberg.

You could hire experts, Mr. Khan—on an ad hoc or long-term basis—to advise you in these matters, if you don’t trust Twitter to provide good guidance.

Or you could educate yourself (as we require of all licensed teachers) on what is known about how people learn mathematics. I’m not talking about reading everybody’s blogs, or years of professional teaching journals. You don’t have time for that.

I’m talking about reading a few reports of robust research. You should start with *Children’s Mathematics* and *Extending Children’s Mathematics*. These are highly readable accounts of how children develop early ideas about whole numbers and operations (in the case of the former title), and about fractions and operations (in the case of the latter).

Then you could move to some of the work of the Rational Number Project. Now, they have many, many years of research that is challenging to wrap one’s mind around. Their work is overwhelming. Because we are talking about decimals, I’ll recommend one article in particular: “Models for Initial Decimal Ideas“. (Behind paywall, but someone at Khan Academy is an NCTM member, right? Right? If not, shoot me a note. We’ll get a copy to you.)

If you read that article, you’ll see that you are on to something at about the 10-second mark of your new video.

We could say this is one-hundred and twenty-three thousandths, or we could say it is one tenth, two hundredths and three thousandths.

That right there? Gold.

That’s the important bit. That is where you need to expand your instructional videos and your exercises. Good, long-running research projects will show you how.

I am not for hire, Mr. Khan. I am not lobbying for a job here. I am advocating for you to do what’s right, which is to use your visibility, your reputation and your capital investments to produce and promote informed instruction. As is often noted, you get millions of hits every day on Khan Academy. I want those students to get something better than they’re getting right now.

But I will make the same offer to you that I do to everyone I communicate with on Twitter and on this blog, which is this: Let me know how I can be helpful. You can do that through the About/Contact page on this blog or through Twitter.

Sincerely,

Christopher Danielson

NOTE (1): This letter has been edited a couple of times for clarity since originally being posted.

NOTE (2): We seem to have gotten ourselves stuck in an endless feedback loop. Comments are now closed (as of August 7, 2013). There are a few threads below that are interesting enough to follow up on, and I’ll do so. In the meantime, if you want to continue your conversations elsewhere, you can link back here; pingbacks remain open.