In a lot of teaching (particularly math teaching) it seems that when a student doesn't understand the concept, the immediate conclusion is that there must be something wrong with the way it's being explained. 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.
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.
> 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.
/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 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.
I've always had a tetris thing going on in my head.
If you can't grasp conceptually why 8+3=11 then you should just memorize it.
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.
I find it amusing that you say "imagine it", since it's like this for me. People think differently :)
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.
Raw memorization is dreadfully damaging. Equally, having to add 5 and 8 by counting up is a complete block to other tasks. What's needed is a range of techniques and skills, each assisting the others, and each helping students to get that insight that stops math being a collection of tricks and unmotivated rules, and starts it being a coherent set of principles, facts, tools and techniques.
I think they should start with memorization, and let each individual develop they own heuristics and shortcuts. When those abstract shortcuts are enforced and don't match with the ones a person is already using, it can get confusing.
I very much agree. Unfortunately, memorization is out of vogue in the teaching community. You don't get a PhD in teaching for saying what we already do works so you have to come up with newfangled ways of teaching that, sadly, often miss the mark. They are then, of course, quickly copied by all the textbook companies.
> You don't get a PhD in teaching for saying what we already do works
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.
You mentioned Kumon, and I felt a cold chill go up my spine from a long lost, forgotten childhood trauma. <nod/>
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!
After repeating enough times, eventually it will click and you'll understand it.
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.
Firstly, I agree with other contributors that memorization is under-rated, and agree that lots of memorization gives a basis of "instant knowledge" for when you are working on more difficult problems.
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:
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.
> 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?
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.
As has been said elsewhere, most likely examples were done in class, and this is extra work to act as spaced repetition. Without that lesson it might not be obvious to some. To me, it was, but only after I'd continued filling in boxes in the grid.
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.
We can guess they want different groups of numbers that lead to the same sum (e.g. 7+4=12; 10+2=12) but this lesson clearly has context outside of the paper itself - that is, the teacher explained what to do, and these are exercises based around that. Without that context, we're just guessing what the goal is.
That's a good point, but that doesn't make this assignment any less poorly crafted. It is common for parents to help their children with homework, and I would expect that to be taken into consideration when creating a worksheet that is less straight forward.
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 don't like this kind of "illustratory" maths. In my opinion it will only distract the brain and make maths look more complicated than it is. My neighbour's kid once had to draw squares on paper to do multiplication. If I had been forced to draw 100 squares just to multiply 10*10, I would probably have refused to do any more maths for the rest of my life.
You're being up-modded for that comment, so people obviously agree with you. I think it's important to realise that not everyone does, because not everyone thinks the same way.
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.
Sure, if it works for others, I have nothing against them using it. But don't force it on kids and make them hate maths.
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.
EDIT: This got down-modded, and I'm not sure why. You may disagree with it, but I think it adds value to the discussion. I've touched it lightly to make it slightly more generic, but I still stand by what it says. Teaching is hard, math is hard, someone who can do both can make more money elsewhere. If you disagree I'd be delighted to hear your case and thoughts.
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.
To play the role of devil's advocate, I can see what this assignment is trying to do. It's trying to teach that numbers (particularly discrete ones) are representations of a quantity, not just abstract symbols that follow some arbitrary set of rules. I wouldn't be suprised if the kids who successfully learned this way would have an advantage over the rote memorization kids if you were to stick them in a Discrete class 12 years later. As soon as I saw the grids, my mind was screaming "pigeonhole principle!" But that might just be because I have a combinatorics exam in 6 hours.
In modern educational parlance, this is "present, practice, produce". But it's missing the bloody present part. And the production.
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.
It's funny, I just realized the way I do certain simple arithmetic is by actually counting in my head. 8 + 3 is 8... 9, 10, 11. But something like 6 + 5 I just have memorized. Maybe cause my brain has decided it takes too long to go 6...7, 8, 9, 10, 11. Or that I'd forget where I was without using my fingers or something. And now that I think about it, for 8 + 7 I use a third technique: I double 8, and then subtract one. Even though I'm pretty sure I have that one memorized correctly, I still seem to always do that check in my head. Ok, I think it's bedtime. Probably shouldn't admit any of this in a public forum. :)
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.
I don't get the strong reactions to this homework. As far as I understand the assignment, it pretty much describes how I do arithmetic. It seems like a natural way of working to me, since nobody taught me to do it this way (if my memory doesn't fail me).
just change the arrows into equal signs and it looks pretty clear, though i'm not sure if a first-grade child is more familiar with that notation. anyway this homework is part of a larger lesson in which almost identical exercises were solved ad nauseam. unclear textbooks isn't really one of the problems with education at any level. also: flagged.
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
Agreed. The visual placement of the grid is a large part of the problem. You'd be able to discern the flow (and logic) of the assignment far better if the grid were placed in between the two sums, for example.
Before giving this out, several identical questions would have been worked out in front of the class. So it's not a "figure this out from the worksheet" matter; it's a "recognize it's like the problems we did earlier".
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.
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.
--
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.