A number line doesn't help some students decompose numbers. The key insight in this example is noticing that 0.35 is "3 tenths and 5 hundredths", in addition to being "35 hundredths". The former is a decomposition. Number line approaches more commonly rely on students being able to think of 0.4 as 0.40, which some interpret as an arbitrary rule, rather than as related to facts about equivalent fractions.
The particular approach offered by the Rational Number Project in the article I linked to in the original post focuses kids on that decomposition from the very beginning using area models and colors to highlight multiple ways of seeing (e.g.) 0.35
On the Rational Number Project matter, you'll notice that I didn't recommend the entire body of work, but rather one article relevant to the matter at hand. It's worth a read—totally practical and applicable, and it reinforces Mr. Khan's intuitive sense that decomposing decimal numbers is important.
I do look forward to future iterations of the decimals stuff. The teaching and learning of fractions and decimals is tremendously complicated stuff. Problems in this area are arguably at the root of many, many people's struggles in mathematics, which is why I think getting it right matters so very much.
I am so glad to hear that your child finds Khan Academy videos stimulating and thought-provoking. I really do think that's wonderful and I am glad this resource exists for you and your child. I mean this in all sincerity.
At the same time, the popular press rhetoric around Mr. Khan is about his revolutionary teaching. The medium (free availability of instructional videos) gets confused with the message (hastily produced rehashing of low-quality traditional content). It is the medium you celebrate in your note here—that the videos are freely available and better than (a) nothing, or (b) very expensive alternatives.
In my letter, I was asking Mr. Khan to pay closer attention to the message. Mr. Kamens and others here have indicated that this is coming. I suppose I shall take their word on that for now.
Quickly about your 10-year-old? I suggest any of Martin Gardener's books, and "On Numbers" by Isaac Asimov as inexpensive ways to stimulate your child's mathematical mind. "On Numbers" may be tough to find—it has been out of print for years—but it will be worth the effort.
Thanks for the thorough response here. You wrote, "Our recent push to improve our math content hasn't yet touched the decimals exercises but it will soon." But Mr. Khan posted a video of a new decimal exercise just last week, which is what prompted the post. I get that everything can't be perfect right away. Somebody at KA has been working on decimals instruction in the past year, so it was frustrating to see that this instruction was not substantively changed. I look forward to the more robust content to come.
In this particular case, the exercise is one that we've had since November 2011. This week we made a push to have videos created whenever an exercise doesn't have an accompanying explanation, which is why that video was created specifically for exercise.
Reviewing content of the exercises is a separate and important task which is ongoing – as you point out, the exercise itself has a few problems and we'll be sure to revisit its content in the coming months. Thanks for your useful criticism.
(And sorry for not responding yet directly on your blog or to you on Twitter – it was on my to-do list but as I opened Hacker News this morning, I felt the need to comment here, as response speed is key on a site like this.)
As I was looking through our content, I found we have a new exercise on comparing decimals! It was written just 10 days ago (and hasn't yet been added to the main site):
With questions like 0.3 __ 0.03 and 0.02 __ 0.1, I think you'll find that the exercise touches on exactly the misconceptions that you've seen. In addition, the hints use area models to explain the comparison, just as you suggest in your post.
Minor bug report, hints 3 and 5 (where the values are displayed as hundreths) overlap with the previous hints (where they are shown as combinations of tenths and hundredsths) for me on Firefox/Ubuntu, but look fine on Chromium/Ubuntu.
Hi. Original poster here. I have amended the line in the post to make clear that the "useless" claim was not intended to apply to all of Khan's videos; only to the decimal ones.
As for motives, I would invite you to read deeper on the blog to get a sense of what motivates my work. See especially the "Talking Math with Your Kids" series, where you will find the following sentiment expressed in multiple ways:
"But I don’t want Griffin and Tabitha‘s mathematical educations to depend on better telling. I want them to explore and to wonder. I want them to commit to their ideas and see what the consequences of those ideas are, and to revise their thinking when their present ideas are not good enough to explain what’s going on in the world.
And what I want for my own children is no different from what I want for my students, and no different from what I want for all children.
There is a place for good, mathematically correct explanations. I want kids to experience those when they’re the right move.
More importantly, I want them to learn to think for themselves."
> As for motives, I would invite you to read deeper on the blog to get a sense of what motivates my work. See especially the "Talking Math with Your Kids" series, where you will find the following sentiment expressed in multiple ways:
Do you have any videos of your own we can watch, so as to get a better handle on the way you think these things should be made?
As one of those who worked on one of the Washington Post pieces, I would invite you to come by my blog, <a href="http://christopherdanielson.wordpress.com>Overthinking My Teaching</a>, where I think you will find my motives to quite pure-better understanding of student learning of mathematics.
In fact it was very important to me that the piece I co-wrote <strong>not</strong> be a smear piece. I can't speak for the comments I did not write. But a careful read of the post itself should reveal a claim backed up by evidence. The claim is that Sal Khan is lacking some important knowledge for teaching. That's not a smear. I would gladly engage in a discussion that pointed to evidence that he does know some important things about how people learn.
I looked at your blog. From my perspective, you folks are down in the weeds. Khan isn't using the optimal technique for teaching decimals? Who cares? The problem isn't that kids aren't learning optimal methods.
The problem with education in this country is that we funnel most all of our public education dollars into a system and related organizations that are only tangentially motivated to educate our children. Watch how the public school systems and teachers unions react to vouchers and charter schools and you'll see how much they care about educating kids vs protecting their turf and power base.
I've noted that those within the education system or closely aligned with it absolutely hated "Waiting for Superman". They nitpick it and try to blow up minor faults with it in a similar way to how you're going after Khan. To those of us outside of that power structure, that documentary showed how the establishment is utterly opposed to actually solving problems in education in this country if those solutions occur outside of that establishment.
I view your articles vs Khan to be in a similar vein. What Khan has done is he's added a solution to our society for our education problems that operates outside of the current public education power structure. Rather than build on what he's done and improve on it through working with Khan or doing something better -- you folks have taken the route of trying to publicly take him down in the media.
"Khan Academy should be ideally placed..." I would say 'yes' and 'no' to this. Yes if the questions ask, and no if they don't. And that's the point of the piece. Khan won't pick up these misconceptions because he doesn't ask. He doesn't ask about decimals with different numbers of decimal places; he doesn't ask about the meaning of the equal sign. Maybe it's because he doesn't know that he needs to ask these things. That is a kind of knowledge he didn't need in his old profession and that he does in his new one. Maybe as KA grows, he will improve on this point and he will bring in people with this knowledge. But given Khan's dismissive ness of education research in the Harvard EdCast interview, I am skeptical.
The particular approach offered by the Rational Number Project in the article I linked to in the original post focuses kids on that decomposition from the very beginning using area models and colors to highlight multiple ways of seeing (e.g.) 0.35