Clearly they are identities, but I’m not clear on how they’re useful in practice. Can you give an example?
Powers &8^]
Actually, I think it’s because addition is much easier to grasp, and easier to figure out with fingers. Kids just pick it up; they don’t have to be drilled on it – or at least, not drilled on it as much. Addition tends to be taught by repetition, rather than by memorization of a table.
They’re also about two years younger when they learn addition, so by the time they reach 4th grade, they remember learning the times tables but the addition tables are ancient history.
Powers &8^]
OK, but understand that a large part of this is that I have the values of 10-0 through 10-9 so internalized that I’m not even consciously aware of it being a step.
7 + 8 = 10 + 8 - 3 = 10 + 5, or, alternatively, since addition is commutative:
7 + 8 = 10 + 7 - 2 = 10 + 5
Similarly, to subtract 5 from 13, I subtract 3 from 5 and get 2, which means the answer is 8.
I’m not saying that this is a good way to do it; in fact, I know perfectly well that it isn’t. My point is only that, if I worked out that kind of thing on my own at the age of seven or eight (I think I was visualizing the overrun on a linear [addition and subtraction] slide rule when I hit upon it), it’s hopeless for me (with no children, and no nieces or nephews living within 300 miles) to try to understand the mental processes of normal children. And, as I said earlier, as far as reading goes, I passed so quickly from
Look, look.
Oh, oh, oh!
Oh, oh, oh! Look, look!
(the lapidary constructions are still seared into my brain) to
The year 1866 was signalised by a remarkable incident, a mysterious and puzzling phenomenon, which doubtless no one has yet forgotten. Not to mention rumours which agitated the maritime population and excited the public mind, even in the interior of continents, seafaring men were particularly excited. Merchants, common sailors, captains of vessels, skippers, both of Europe and America, naval officers of all countries, and the Governments of several States on the two continents, were deeply interested in the matter.
that I cannot remember the transition. So I can see the educational problem, but I have no idea what the solution is, beyond the obvious observations that different children have different needs, but that the final needs of the fully developed adult mind are to master the whole-word and “new math” skills, no matter how you get there.
There were definitely addition and subtraction tables in my arithmetic texts of the 50s. I think you’re probably right – many of us simply forget them.
The opening of 20,000 Leagues Under the Sea?
AFAIAC, a major part of the solution is allowing different pathways to the answer. I had a kid in my class last year, drove me crazy with his constant chatter to friends, couldn’t shut him up with a roll of duct tape, but he did the same sort of numerical decomposition you’re describing. He’d break numbers apart and add them back together in ways that I found totally crazy, and when I asked for his proof of his method, he’d rattle it off so quickly that I could barely follow it (much less could his classmates follow it)–but it worked for him, and it allowed him to solve problems quickly, so I slathered on the praise and made it clear that he was more than welcome to solve problems that way.
Today I showed the kids some 10-frames–2x5 grids with dots in some of the spaces. The idea was for them to ascertain the number of dots very quickly. When I showed 7 dots, someone saw 4 dots above 3 dots: awesome. Someone saw a column of 5 (which they automatically knew, since it filled one complete column of the grid) plus 2. Keen.
Then one girl explained her method: she saw a column of 5, plus a column of 5 missing three, resulting in 5 + (5-3)=7. Kind of similar to what you did, JWK.
I told her frankly that I never would have solved it that way, it sounded like way too much work for my brain, but that that was totally beside the point: the point is that her method allowed her to solve the problem quickly and accurately, and if it worked for her, by all means she should keep using it.
That’s part of constructivism that I really like. I tell kids the following bedrock principles:
- Math makes sense. If you don’t understand why something works, study it some more until you understand why it works.
- There are a lot of ways to find the answer. As long as your way is valid and you can prove that it works, go crazy using whatever method you like.
- Mathematicians are lazy. Look for the simplest way to solve the problem, especially when we’re working on easy problems, because later on we’ll be working on harder problems where, if you haven’t found a simple method, you’ll spend all day working on it.
These are pretty constructivist principles, although I’ve rephrased them.
Just to go back to a question from before, why do you use the label “constructivism” for that pedagogical ideology?
And THIS is why my kids need math tutoring.
I realize you guys are back on the main topic, and not the hijack I was responding to, but a private message indicated that I did not indicate my age, and that that might be important. At least, I assume it must have been about this thread.
I am 25–so my story happened 15-20 years ago. (I left the Montessori school after 4th grade. Man, was 5th and 6th grade a piece of cake after that.)
I can actually talk about what method I was taught math: Iti’s similar to what LHoD described. One lesson in particular that I remember was using beads. There were individual beads for the one’s place, groups of 10 on a wire for the tens place, a square made of 100 of those for the hundreds place, and a cube made of 100s for the thousands place (although both the squares and cubes were quickly replaced with wooden blocks with the beads stamped on them). We learned carrying by realizing that we were trading in 10 beads/10 sets of 10/etc. for the appropriate larger amount. Only after we learned this for a while were we introduced to being able to do the same thing on paper–i.e. the algorithm.
Now I was quite abnormal in this system, as I eventually maxed out the math curriculum there, reaching the ability to do square roots of 6 digit numbers on paper (using a method I don’t believe I’ve ever seen anywhere else). But my point is that I knew no one by fourth grade who was below the public school level–although there were some who regressed when put in a less rigorous environment.
I don’t know what exact method you used, but would I be right in guessing that it’s an iterative method? That is, you start with some approximation to the square root, and then do something with that that gives you a better approximation, and repeat?
Apparently what I learned in the sixties and early seventies was the so-called “new math”. Speaking strictly from my own experience (for whatever a single anecdote is worth) I found it vital to gaining my computer science degree, especially the stuff about number bases.
Granted, if it hadn’t been taught in K through 12 it would have been taught in college, but that would have meant less time learning other things and it wouldn’t have been as intuitive.
The whole point of “new math” was to advance science and technical achievement in the U.S. and I think it worked at least in my case.
To me it seems intuitive that understanding is at least as important, if not more so, than rote memorization.
By the way, the “x = 5” stuff is algebra, which I don’t think was ever considered to be new math.
That would be the Newton-Raphson method – start with a quick approximation G, divide G into X, get the average of G and the quotient and use it as the new G. Lightning fast – I don’t know of any computer that doesn’t do it that way.
But because of all the divisions, working it on paper is best done by the method (similar to long division) that I remember being taught in school.
I actually know of a method that only requires a single division for the entire operation, though at the expense of requiring more variables. Basically, you store all of the approximations as rationals (i.e., a separate variable for the numerator and denominator), and use the mediant rather than the average to get the next approximation. Then, after you’ve got a sufficiently-large number in your denominator, you do a single division to get the final answer.
We had something called the School Mathematics Project, which had us delving deeply into matrices, and something called Critical Path Analysis which I’ve never had any occasion to use since. I can’t now recall how to multiply two matrices together, and don’t seem to miss it.
It seems to me that students are often initially taught the operation of matrix multiplication with almost no motivation of the concept, which is rather a pointless exercise except as a test of the ability to memorize and follow an algorithm (which really oughtn’t be in doubt after overly much time has already been spent on algorithms for manual arithmetic in elementary school). If I had my druthers, students would be exposed to the ideas of vector spaces and linear maps between them first, and only after familiarity with these concepts was initially established would the mechanics of representing linear maps in coordinates with matrices be introduced, matrix multiplication then being clearly understandable and even obvious as the operation of composition of maps.
For that matter, I’d also have students become a lot more familiar with vector spaces and linear maps before they ever took their first derivative; in my new new math curriculum, basic abstract linear algebra would precede calculus (it astounds me that students (like those I am currently teaching) are expected to grasp multivariable calculus without the concept of a linear map).
(The above, of course, is only relevant to those students who are expected to learn such things; obviously, many make it through life blissfully ignorant of both matrices and calculus)
What do you mean?
I’m not entirely sure I understand what you’re asking–are you familiar with constructivism?
Ah, I am not. I am familiar with constructivism, you see, which was causing confusion.
Ah, okay–and I wasn’t familiar with that definition of the word. No wonder you were asking me what the hell I was talking about! 
Anyway, constructivism in education usually refers to practices in which students create their own knowledge instead of having it told to them. The teacher’s job is to engineer experiences in which students can construct the knowledge. If I want them to understand the commutative property of addition, I might provide them with a stack of 7 cubes and a stack of 3 cubes and ask them to tell me how many cubes I have altogether; I’ll then reverse the position of the stacks and ask them to tell me how many cubes there are, without re-counting them. If they tell me I have 10, I ask them to explain how they know that’s true. Then I’ll write an equation for each example (7+3=10 and 3+7=10), and I’ll ask them to generalize: if 17+41=58, can they tell me what 41+17 equals? How do they know?
The idea is that, by constructing knowledge themselves instead of having it explained to them, their understanding will be both deeper and more permanent. I think it’s a great idea, but as I said before, it’s one approach of many to use in teaching mathematics. Folks who adhere too closely to any one theory are, in my opinion, foolish.
The teachers have spent all the class time demonstrating regrouping with blocks and strips and how 495 + 342 = 342 + 500 - 5, that no one ever bothers emphasizing teaching the algorithm so that they can just add the damned numbers. Do you know for a fact if the next grade’s teachers teach the algorhithm, or do they just assume that you’ve taught it? My kids were barely taught adding columns of numbers, never taught how to multiply two 2-digit numbers on paper, and never taught how to do long division, because all that was ever emphasized was the blocks and the strips. My district’s parents are frustrated and the elementary school learning specialist is frustrated and the high school math teachers are dumbfounded at how little math the students are able to actually do; the local colleges have to teach remedial math, and one of the biggest growth industries here is math tutoring.
Math is one subject where oftentimes the full understanding sinks in after the algorithms are mastered, through the process of doing the algorithm. And at least they’ll be able to perform the calculation in the meantime. Do all those successful math students in Singapore learning Singapore Math not tell us anything? Why is it that when students get math tutoring help, this style of math (Singapore-type basics) is what they are taught to get them up to speed? Probably because they were taught everything BUT the algorithms. In looking back on my own math education, there was some amount of blocks/regrouping demos, but then it was all algorithm. There was not a single student who was unsuccessful in math at the end of each school year.
I just really hope that your students are learning the algorithms, and that someone makes sure it actually happens.