As a professional mathematician, I say… who gives a shit whether your kids do or don’t remember whatever standard traditional algorithm for how to do long division with pen and paper, or decimal expansions of square roots, or what have you? We live in a world with calculators and computers. LHoD’s approach emphasizing understanding, while also presenting the algorithm in this context without making its unthinking memorization and application the goal, sounds just fine to me. Of course, I’m not a professional teacher of young children (but he is).
Oh, really? That’s not remotely my experience. And I’ve tried it out: I’ve asked adults to explain to me why the traditional algorithm for 2-digit multiplication works, and very, very few adults can do it, even if they can manage to do the algorithm correctly (in college, I designed a 2-week unit that would build students’ understanding of this algorithm, precisely because at the time I couldn’t possibly explain why it worked). Is your understanding of this fourth-grade algorithm good enough that you can give an explanation without having to think it through very carefully first?
So that’s why, if I taught fourth-grade, I’d use an array model to explore multiplication, dividing each side of the array into tens and ones and showing the four sections thereby created, giving kids a visual image of how 2-digit multiplication works.
As Indistinguishable says, nobody does calculations on paper anymore: that’s what your phone is for. What you need to know is how to set up the equation, and in order to do that accurately, you need to understand the underlying math.
But even then, I do teach algorithms. (I should’ve said before now that I don’t like absolutists on either side of the argument, I think they’re dumb). I just teach them as the end of the process, and I don’t let students use an algorithm they can’t explain. And I make it very clear to students that there are multiple ways to solve a problem.
It’s funny to me that you dismiss the sentence you bolded: what I was showing you there is that I teach kids to go beyond the algorithm. If neither of us has a calculator on us, and if you’re not Rain Man, I can probably do most problems using flexible strategies than you can using the traditional algorithm, because I’ve practiced finding and using shortcuts to solve problems, and I can do it because I can hold multiple numbers in my head at once, and because I know several go-to strategies. I’m no math genius; it’s just a skill like solving crossword puzzles.
If your kids need math tutors, there are several possible reasons why, but it just strikes me as intellectual laziness to blame it on the pedagogy itself.
Yes.
And if you forget your phone? Or your high school math teacher doesn’t let you use your calculator?
Yes, clearly the reason that the US is trailing behind the rest of the world in math education is because of my intellectual laziness.
In the state of Washington, parents and higher educators have begun a group/website called WheresTheMath.org to fight the horrible math education in the state. Maybe it’s better where you teach. The biggest problem seems to be the switch over to discovery-based math. This is from University of Washington’s professor of atmospheric science Cliff Mass, from Where’sTheMath.org:
There’s more on his excellent blog: Cliff Mass Weather Blog: How Good Are UW Students in Math? , including:
and from the math, science and engineering faculty:
Here’s another thread a little while back where this was discussed as well, starting with post #18.
A large part of the problem here is the assumption that “knowing math” means “knowing how to do long division”, or the like. There’s no connection whatsoever between them. Now, understanding why the long division algorithm works, that’s math. The student who has never seen the long division algorithm, but who, on seeing it, can figure out how it works, knows far more math than the student who can turn the crank to do long division, but has no clue why.
Most students won’t have any math classes at all until at least high school.
Then you get one from someone else. The “What if you’re stranded in the woods without access to technology and need to carry out lengthy arithmetic calculations but never managed to learn the traditional algorithms? What then, smart guy?” challenge isn’t very realistic. I’m sure you survive just fine resorting to the convenience of calculators for logarithms, sines, nth roots, and all the other such, and if you were honest with yourself, you’d recognize that everyone gets along just as fine doing the same for long division, nontrivial multiplications, and even additions of any burdensome length. The only point there is to teaching kids arithmetic algorithms in the modern age is in order to be able to explain how those algorithms work, as a launching point for discussions of more general mathematics. But if one’s goal is merely to have people possess the means to obtain the answers to numerical queries, well, that’s why we invented the calculator in the first fucking place. Show kids how to use one, and the ubiquity of their availability (standalone pocket calculators, pre-bundled OS calculators (Windows, Mac, Linux, what have you), Google, iPods, cellphones, …), and they’ll be fine. Really. Truly. Restaurants don’t care whether you use your phone or your pencil to determine the tip.
Any skill can artificially be made to appear useful by contriving to test for it specifically. But why shouldn’t your high school math teacher let you use a calculator for tedious arithmetic? A student who has never learnt to use a calculator is far more ignorant than one whose only sin is recognizing the convenience it offers.
It’s pretty hard to demonstrate understanding of the long division algorithm without being able to carry it out. The student who turns the crank and doesn’t understand why is far more likely to be holding a calculator than doing it on paper. (And again, good luck if you forget your calculator.) Obviously, though, lack of long division isn’t all of what’s holding students back in math; it’s just one small element.
???
Learning how to mechanically follow instructions is of course a basic life skill and has, in many contexts, gone by the name “math”, but it has very little to do with the work of mathematicians. Chronos’s point is that classes along these lines, which are for many students the only classes they take by the name “math” until high school, are no more teaching math than classes focused on the rules of staff notation without any actual singing or playing of instruments would be teaching music.
The point it that you have to make sure they can understand it without the calculator first.
I’ll have my daughter tell her algebra teacher this when she doesn’t allow her to use her calculator on the test.
I’m all for understanding. So is LHoD. And Chronos. Everyone here is pro-understanding. But let me ask you a question… do you understand the method your calculator uses to perform division? Do you even know what it is? It’s almost certainly not the long division you were taught in school, for example. Is that a problem? Does it matter? If a child with no particular interest in numerical analysis or algorithm design should happen to grow up to know what division is (“A divided by B is the number which you multiply by B to get A”), how to reason about its properties, and how to use a calculator to carry out decimal expansion of its results, but not be practiced or proficient at carrying out such decimal expansion by hand, what harm will they suffer?
Now to be able to see that this operation on integers generalizes to that of splitting polynomials into a quotient and a remainder modulo a given divisor, to realize how to mechanically produce canonical representations of such, and to appreciate that such an algorithm is the same thing as “long division”, well, that’s a fine level of understanding to reach (albeit it’s still the case that the second step of those three is far outweighed in significance by the others). But I reckon less than a hundredth of the people who are taught long division ever realized or were shown such a way of looking at it, and for those among that majority who can still remember the algorithm anyway, there’s not terribly much difference between the lives they’ll have with their unused and unnecessary ability to reckon with pen, paper, sliderule, and abacus, and the lives of their more forgetful peers…
I have no idea why you are nitpicking long division. The topic here is whether “new new math” is serving our students well. The evidence indicates that it’s turning out a generation of math illiterates. Apparently, demonstrating math concepts to kids and then throwing them a calculator isn’t working. In my district, they aren’t even being made to memorize math tables because they can just use a calculator if they need to know what 4x6 is.
Again, here is the blog by the UW professor where he describes the sad state of the math education of his incoming students.
The way I see it, calculators are fantastic tools for learning, not obstacles to it. What a wonderful era we live in, to have them available to us. As an anecdote from a slightly older level, my TI-89 essentially was my calculus teacher, and its manual my textbook, and I seem to have turned out alright. Having the ability to ask it questions of my choosing didn’t spoil my mind and stunt my mathematical growth; it opened doors while I was still young and curious enough to care to explore them. Calculators are the infinitely generous sherpas of an entire second universe, and by lessening the burden of making one’s way about this unfamiliar territory, they provide the sufficient leisure to appreciate its sights. We should be embracing these marvelous tools, not masochistically shunning them with the equivalent of grunts about hard work and character and paradoxical hills. That anyone, decades into the aftermath of the computer revolution, can still feel differently is frankly astonishing to me.
And if the cost should be that young children fail to memorize the product of 4 and 6 prior to such experience as would naturally set it in? Very well; I haven’t memorized the product of 44 and 66. If a student doesn’t realize how to turn 4 * 6 into 6 + 6 + 6 + 6, then I’ll be worried.
Perhaps the problem is that some children simply don’t understand math, and simply showing them cubes and strips doesn’t teach them anything at all, except how to memorise cubes and strips instead of an algorithm.
And the algorithm is much more useful in calculating your change than cubes and strips.
I wish more people understood how to turn 4 * 6 to 6 + 6 + 6 + 6. But most people won’t ever need to. They can still get the answer, but with a complete lack of understanding about the fundamental nature of multiplication. I think the answer is “that’s okay, for most people. After all, that’s why we have mathmaticians”.
Personally, as a product of the Singapore system, I say drill the algorithm and the formulae. Those interested in math will get into it at some higher level.
I actually learned my math by programming. Before I ever did algebra, I was typing x = x + 6 in my BASIC compiler. When I finally got to algebra, it took me a while before I adjusted to having x = 6 and learning that x = x + 6 was a nonsense statement in math.
Someone who can’t do everyday mathematical calculations with just a pencil and paper isn’t being educated properly.
What is the value of knowing by memory that 4 * 6 = 24 without even the most rudimentary understanding of the meaning of this statement? How can this possibly be of any use to anyone?
That having been said, I think, whatever the numerous failings of our standard mathematics education, that most people do understand the equivalence between 4 * 6 and 6 + 6 + 6 + 6, but perhaps I am naive.
(On another note, your having learnt math by programming is, I think, a fine example of what I am talking about as far as the pedagogical value of calculators/computers in mathematics.)
If I’m doing it in my head I tend to do it the opposite of how you describe it (starting with the 7 and 8, carrying the 1, etc).
I go left to right, starting at the most significant, (the 3 and 5 in this case), then moving on to the next, etc., adding any carries to the previous result (next highest decimal position) retroactively as needed. For whatever reason this works much better for me than going right to left, where I have difficulty keeping the intermediate results in my head for some reason.
I don’t think I was ever taught this way, it just naturally came to me. I doubt that it would have if I hadn’t had an understanding of decimal places, etc.
I’ve never heard of the “blocks and strips” stuff. Is that considered “new math”?
Put it this way: Do you know how to use the algorithm for finding logarithms by hand? I’m betting the answer is no. Does it bother you that you don’t, or do you just get a calculator every time you need to know a logarithm? Is the log algorithm really any less important than the long-division algorithm? Personally, I don’t know the algorithm for finding logs, either, but I understand logs well enough that, if I were ever stranded on a desert island without a calculator and needed to do logarithms, I could work out a method for doing them.
Quoth Indistinguishable:
Reminds me of the time in one of my high school math classes, when I asked my teacher “What’s the significance of the number 2.71828? My calculator appears to have a button for calculating logs to that base.”.
This jumped out at me while I read this thread.
I believe it has been fairly well established that humans group objects in no more than 5 units when they are visually breaking down a large set of objects.
That means that if a human sees 8 units, they see it as 5 + 3 (or 4 + 4 in some cases). This is done on a primitive level and may explain why some numerical methodologies are easier to grasp for a larger percentage of humans.
I really should get a cite for this…
Ah… there we go:
Corbetta, M., Shulman, G.L., Miezin, F.M., & Petersen, S.E. (1995). “Superior parietal cortex activation during spatial attention shifts and visual feature conjunction”. Science 270 (5237): 802–805. doi:10.1126/science.270.5237.802. PMID 7481770
And, I was wrong. Apparently the operation is called ‘subitizing’ and applies to natural groupings of 1 to 4 units. Not 5 as I said above.
Hey! I learned something new today!
I’m addressing current pedagogical theory (and also my own approach, which is somewhere between the extremes, as I think is the approach of most actual teachers, good or bad). Under current theory, your approach is absolutely fine: we encourage kids to come up with a method, or better yet methods, that make sense to them and that they can prove is true. The use of blocks and strips and such are just good tools for showing kids why it works. Very few kids understand why you’d want to carry a 1, but if you show them that what you’ve really done is put 7 and 8 blocks together to get 15, and that you can trade 10 of those blocks in for a 10-strip, which you then group with the other 10-strips, they get it.
I get alternately amused and annoyed by people whose experience of education is limited to their own time in grade school who think they know better than teachers what’s developmentally appropriate for most kids. Sure, if you’ve really put the time into studying child psychology and how mathematical schemata are modified, then by all means your opinion is valuable. But if all you’ve done is read the blog of a cranky atmospheric science professor who gives freshmen an inauthentic exam, well, that’s not exactly an informed opinion.
Heh–when I was showing these 10-frames to the kids, I actually told them about this research. Not because I expected them to remember it, but because it’s awesome, and I figure sprinkling addition facts with awesome science will make it more memorable. (However, I thought the cutoff was 6: I’ll correct my misunderstanding for the students on Tuesday. I also figure it’s a good lesson for them that when you make a mistake, you fix the mistake enthusiastically and without shame).
I wonder if the cutoff point is different for different individuals.
I ‘remembered’ 5 as the cutoff because I think I group things in fives.
Of course, the study was done using PET scans, not subjective assumptions, but I wonder if there was any variation found in a population.
You sound like a truly terrific teacher! Kudos.