Do you understand this first-grade child's homework?
boingboing.net
boingboing.net
However, this is a pretty good way of learning a general principle. Take the first example, "8+3". Over on the right you have boxes in groups of 10, and the first box has eight "counters" filled in. Now you add three by filling in another three boxes. This gives you ten full boxes, and another one left over. Thus 8+3 goes to 10+1.
ADDED IN EDIT: Now that I think of it, how many of you simply ignored the instructions to fill counters in the boxes to help you, and tried to see what was going on just by looking at the numbers?
Of the three that you are adding, two are used to make the eight up to ten, so you only have one left. In a sense you've moved 2 from the 3 into the 8, making the sum easier. Moving stuff around in an addition is very similar to moving stuff around (and then making a correction) in multiplication. This is how Art Benjamin squares two and three digit numbers faster than you can punch them into a calculator.
53 squared is 53 times 53. Move three from one into the other (remember the 3 you've moved around) so you get 50 times 56. Use the same trick but in a different way: move a factor of two from the 56 into the 50 so you get 100 times 28, or 2800. Now add on the square of the three you moved earlier, giving 2809.
ADDED IN EDIT: How this works can be demonstrated clearly with an appropriate diagram.
If you're multiplying, moving factors is the same sort of operation (both multiplications) so you don't need a correction. If you move by addition/subtraction it's a different operation, so you will need a correction.
However, all of this fits into a larger framework. Getting only a small part lets you do those bits, but as it fits together you end up with more than just the parts, you get a much larger framework.
This is actually, tangentially, related to the item I submitted 10 hours ago and which sank without a trace:
http://news.ycombinator.com/item?id=951250
The same trick is used twice in two different context to give the Infinite Ramsey Theorem: Every infinite graph contains an infinite complete subgraph or an infinite null graph.
Very similar.
I didn't ignore it - yet it didn't help. The instructions are ambiguous. How are you supposed to fill them in? What is the connection between the 8 in the first example and 8 counters? Are they the same by accident? Why should both results be the same (8+3 and 10+X) - I thought about adding 10 new counters, so that's 8+3=11, 10+8=18, 8+10=18 (although it doesn't make much sense either). Then thought about adding 3 to the lower box (not good). There are simply not enough information to "solve" this. Even if most people can guess what the right solution is, I don't think that's good enough for homework.
In general, homework is not intended to contain the entire lesson all over again. That means parents often don't know what to do, or how to help.
Further, lessons and homework are often not intended to be done by creative people without instruction. Creative people find unexpected ways to follow instructions the teachers (or lesson planners) thought were clear, obvious and unambiguous.
I see a lot of criticisms of lessons and homework like this, but I've also seen the frustration from teachers trying to create lessons that are EXCITING and ENGAGING and ENTERTAINING and THRILLING and WONDERFUL, because now they're not expected just to be good at teaching and experts in their subjects, but also entertainers.
This lesson/homework isn't perfect, but I think many of the criticisms aimed at it are misplaced and ignoring the context. This homework is being assessed as if it were a lesson, and that's unfair.
Finally, homework that my colleague's child is set is accompanied by a sheet explaining the lesson. My colleague hos no trouble helping with the homework, and constantly praises the variety of approaches and integrity of the work.
My answer to the original question is, "No, but I would if I'd seen it before." The children, presumably, have seen it before.
The proper way, I believe, is to just have the student learn the concept by rote. If you can't grasp conceptually why 8+3=11 then you should just memorize it.
After repeating enough times, eventually it will click and you'll understand it.
Additionally, there's no point in trying to teach the concepts of arithmetic. You'll be crippled trying to learn things like decimals, fractions, exponents or god forbid algebra or calculus if whenever you have to add you have to be "making ten" in your head. It needs to be memorized.
I believe this also holds true up to even much more complicated concepts than basic arithmetic.
For example, I find if I don't understand the proof of some theorem in one of my math classes, if I just copy it out of the book by rote, (translating the notation line by line to something I'm more comfortable with) then I sometimes will suddenly understand it.
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Addendum: I believe Kumon has a good method for teaching math. They put a very strong focus on a rock solid understanding of basic concepts. If things haven't changed, they give kids timed problem sets of basic problems to solve (memorize) with the expectation of 100% accuracy.
But it doesn't; at least not for everyone.
I had to quit maths at high school because it was just too frustrating memorizing apparently useless and meaningless formulas and definitions. My teacher wouldn't explain (she probably didn't even understand) what any of it was.
Later, at the university, the beauty of calculus suddenly made sense of everything, and I got a renewed interest in maths. I now hold a master's degree in algorithms and take university math in my spare time for fun.
I curse my high school math teacher's name to this day. I hope she retired early.
/joke: they should just start with category theory -- it will produce a fine new generation of HN readers ;-)
> Thus we have convoluted methods of teaching addition like this that in some way do make sense, but are mostly just another layer of confusion.
I think everyone conceptualizes and encodes basic arithmetic stuff in their own way. Some memorize it symbolically, some visualize, some think of number lines, graphs and so on.
Trying to "help" by providing tables, coins and all kind of "aides" just confuses some students who started to encode it differently.
I've always had a tetris thing going on in my head.
You should memorize it either way.
Imagine if every time you looked at some basic arithmetic, you had to make the choice, am I going to take the time to figure this out and lose my train of thought, or am I going to gloss it over and move on? Imagine having to gloss over 8+3 as "some number in the lowish tens." You might know how to add 8 and 3 conceptually, but to actually work it out would take some extra effort.
Thus, memorization. It's highly underrated.
I see 8+3 as a number in the lowish tens. If I want the exact result I quickly increment "9-10-11" in my head. Some special cases (0,1,9 and 5+5) are easier.
I guess ability in arithmetic isn't really correlated with ability in calculus, etc., since math has always been my forte.
I'm studying at a japanese university right now and I see the effect of raw memorization daily. It's not pretty. I don't think it's highly underrated.
> ... raw memorization daily. It's not pretty.
> I don't think it's highly underrated.
I was good at math, but my mother ran a Kumon center as her small business, and I had to do Kumon so that she could use me as an example. I felt sorry for those kids. In the end, I think the repetition method helped a lot of the students. But the style is so different from the American educational system that, from the perspective of someone who did fine in public school, the Kumon method just felt.. dirty. Like corporal punishment.
For those folks here who don't know, they're a Japanese juku (cram school) which expanded to America mostly to service the children of Japanese parents abroad. (A common fear is that a student who spends a few years in an American public school prior to returning to Japan will be academically crippled.)
I have only two things to say about mathematics education in Japan: it works very well for students in the academic track, and it isn't considered strange to have girls say they like it. So the next time you hear someone saying how we need to address gender balance, there is your answer: soul crushing drills!
I fully agree. My parents like to tell this story about how I sucked at math and one day, at Disney World, there was a ride with cars going by. The cars had numbers on them and somehow seeing that made addition click in my head. I spent the rest of the day annoying by finding some sum for every number I saw in the park.
Even if the child knew what they were supposed to do, they'd have the explain the assignment to the parent... which they might explain incorrectly. Just another level of confusion.
I can guess the intent of the exercise, but it would have been a lot easier if there had been a solved example.
It sounds a lot to me from this and other contributions you've made about math that you, like many others, have been damaged by some really bad teaching. Using these techniques properly gives the child/student powerful tools for analysing situations they haven't seen before. Teaching rote techniques doesn't - of itself - provide that power and flexibility.
Having said that, these "illustratory" techniques are often taught appallingly by well-meaning and gifted teachers who themselves have learned by rote what to do, but don't themselves have any real understanding of the math underneath. We need teachers who are gifted teachers, but who understand the underlying principles.
Unfortunately, those gifted in math are rarely gifted in teached, especially teaching children. Add to that the fact that they could earn more respect, reputation and money using the math, and what we have is a continuing disaster in math education.
I have no answers.
I've helped out in primary classes, and sometimes when I've done something like this kids have been so excited they've jumped up and run around, unable to contain themselves. They've suddenly seen why something works, not just been told to memorize it.
It's the range of ways of thinking about things that matters. Here's one that doesn't work for you. Fine. Find another. Use both. See how they're the same thing, but from different points of view.
Sorry, I'll go away and stop ranting now - I'm just getting angry and will say something I regret.
Another silly thing I read about is associating numbers with animals. So "1" is the crocodile and "2" is the elephant or whatever. I can only say - wtf? Just another example - numbers are numbers, not elephants...
I just think to understand maths is to REDUCE the number of concepts (that's the beauty of maths). A lot of teaching concepts seem to increase the number of concepts instead.
I am not a teacher, though.
A single worked example would save the student a lot of time.
Some educators like to say that students should be able to figure it all out for themselves. That students are better off not being taught. I guess it depends on the teacher - some teachers can add value.
I don't think the goal has to do with adding to ten....
e: I think I get it, but I don't know what you're supposed to do with the second grid.
e2: Oooohhh. The grids are there to help you get to your answer.
http://yudkowsky.net/rational/the-simple-truth
Especially how it initially presents the concept as magic. Actually, maybe not a good way of explaining numbers to children. But I like the story nonetheless.
Don't condemn or applaud it, it's just one of many ways the children will be taught to add. Some kids get it straight away and some don't.
It's just practice and practice is good.
This is what the child is supposed to do: Grid (Draw in counters for second number, then count total of both grids) -> Fill in both bottom blanks with counted total -> Ten plus what equals bottom blank?
This is what we're doing: Add numbers -> Why the hell is there only one grid? -> Then we post our child's homework onto the internet
They should've formatted it like 8+3=11=10+? Then it'd be clear we have 11=10+?, 1 equ. & 1 unknown, BAM! solved.
It that context, it's a plausibly useful exercise; it's analogous to how I (still) total and multiply numbers in my head: determine the similar problem that gets to either a nice round number, or a memorized result -- then adjust for the remainder.