As an aside on the full-length mirror problem: If you decide to draw stick figures to try to figure it out, it helps conceptually if your stick figure is Marge Simpson.
I gave up hunting for factors of 54931 after nothing up to 19 worked and I’d have to try every prime up 233 to brute-force it. Have we improved on Eratosthenes?
But I question whether someone could really understand how to carry out some kind of calculation unless they had actually done so.
Judging from personal experience, when I’m learning something new, very often what I find effective is to work through a problem or two where I “follow mechanical rules slowly, patiently, carefully, with the rules in front of them.” The process of doing this helps me to understand the procedure and what’s really going on, and is an important step towards doing the thing more fluently and with greater understanding of what’s really going on.
And I think you’re making a mistake if you’re saying that understanding does or should necessarily come prior to doing.
There are faster prime number sieves, but to generate a short list of small primes up to 233 there is not much point in using anything more complicated than the Sieve of Eratosthenes or the Sieve of Sundaram. A better question is, how many digits in a number does it take before you are better off trying something other than dividing by small primes? Even so, you might want to start with trial division first, just in case your number has a small factor. Maybe some mental factorizing prodigies are willing to share some trade secrets?
FWIW 54931 factors almost immediately using this algorithm starting with k=2. I used a piece of paper, though.
It’s not clear to me that everyone actually needs a lot of lessons on factorization; I’m not inclined to start from “Well, of course we have to have SOME class on X, because we always have, so if not the dumb stuff that was in there, then what?”.
But, I guess I would note that sometimes bigger values are products of smaller values (people should understand multiplication, conceptually, what it means most of all, how it relates to repeated addition, to area, how it distributes over addition, how the order of the factors in a product doesn’t matter, why it’s possible to divide by nonzero values, and so on. And even how they could calculate the digits of long complicated multiplications if they so desired, though I don’t care about drilling them on actually doing lots of long complicated calculations quickly or memorizing any particular algorithm as opposed to any other one). And show how sometimes this breaking down of large values as a product of smaller values is useful for thinking about them, and for thinking about GCDs and LCMs and stuff, sure.
I might even talk about uniqueness of prime factorizations, though in my own childhood math education, almost nothing on primes ever came up in the standard curriculum, and I’ve never yet met people whose standard elementary math courses bother soundly explaining why prime factorization is unique (or, for that matter, why every divisor divides the GCD, or why any fraction can be brought into its form with lowest possible denominator by cancelling common factors between numerator and denominator, or any of these related facts, much less how to efficiently compute GCDs as linear combinations). So clearly it’s possible to get by in life without understanding these things fully, but still, uniqueness of prime factorization is a basic fact which I’m happy exposing everyone to, fine by me.
On my own, though if I had ever taken an AP Calc class, I’m under the impression this sort of thing is covered implicitly, if not explicitly.
As I said, I’m not motivated in my discussions by “How will they do things the traditionally structured curriculum makes them try to do?”, when my whole point is to reconsider the traditionally structured curriculum entirely. I will say, I do not feel experience efficiently factoring integers by hand is particularly vital to understanding trigonometry or calculus. For what it’s worth, my personal calculus education was built off playing around with a TI-89 which could do all kinds of things for me I’d never care to do by hand.
How do you feel about the New Math topics (symbolic logic, non-decimal bases, matrices, and the rest) that actually made it onto the curriculum?
Most of these didn’t make it onto the elementary school curriculum I experienced, but, I’d say non-decimal bases are potentially good for helping better understand what’s going on with decimal bases, which in turn are of such ubiquity in our world that everyone does need understanding them jammed into them early, like it or not. So I see reasonable value here, sure.
As for matrices, from having asked around about this before, my general experience is that lots of students are taught things like matrix multiplication as an arbitrary symbol-shuffling ritual devoid of clear purpose or understanding why anyone would care to define the operation this way, just “that’s the rules”. (And nearly so with determinant calculations as well…). That’s no good. That’s pointless. I’d love to see everyone understand matrix algebra better, but I also acknowledge that mostly, they get along fine without any need for it. I wouldn’t make anything like this particularly mandatory for people to spend time on.
As for symbolic logic, what kinds of things do you have in mind? Like Venn diagrams and truth tables? Doing what with them?
In case anyone is curious to check them out, some old threads here where many of the same or related arguments and discussions as we’re having now were previously carried out:
I remember straightforward exercises like having to prove whether two statements are logically equivalent, whether one implies another, simple Boolean algebra with finite sets, things like that. No advanced non-elementary applications, obviously. I don’t remember anyone being particularly mystified or wondering what it all meant, not that anyone polled the class.
I don’t have a strong opinion about it, but that seems alright to me as a topic. I’m all for conceptual understanding of basic ubiquitous things, and what’s more ubiquitous, in some sense, than basic logic? But like everything else I wouldn’t want to turn it into an algorithmic execution drill-fest (“Here are 50 pairs of propositional formulae; determine whether they are or are not equivalent, by truth-table evaluation”).
I might have quibbles with specific problems or presentations of them (if anyone’s getting chided over Peirce’s Law phrased using ordinary language “implies” or something, I might feel weird about it), but the general idea of just covering such logic topics seems fine to me.
I was certainly exposed to matrices in high school, but we never did anything useful with them until college linear algebra (where we still didn’t do much with them, because that course was split between linalg and diffeq, but at least we did something). In high school, we learned how to define matrix multiplication, and we learned two different methods to use matrices to solve systems of equations, one of which was just the same method we’d been using anyway but made harder to keep track of, and the other of which was far less efficient. In retrospect, I don’t think that I ever saw matrices put to good use until I saw them presented as special cases of tensors (and it just so happens that the sorts of tensors which are most often useful are the ones which can be represented as matrices).
Out of curiosity, what were the two methods?
Of course, for systems of affine equations, one can always solve by simply applying intro algebra, one by one taking an equation as a definition of one variable in terms of the others, thus reducing the number of variables and equations by one, till reaching a contradiction or only tautologies. A matrix can help keep track of this process, but isn’t particularly necessary; it’s the same thing whether you write it with matrix notation or not. I suspect this is what Chronos is getting at with “one of which was just the same method we’d been using anyway but made harder to keep track of”.
(And when I mentioned “And nearly so with determinant calculations as well”, I did have in mind that some apparently somewhat widespread early exposures to matrices for this purpose do manage to impart that the relevant determinant is nonzero if and only if the system is guaranteed a unique solution simply by virtue of the matrix. Which is good, so far as it goes. But almost no students learn why this is true, and indeed they appear to be trained to only think of the determinant as fundamentally defined by Laplace expansion as an arbitrary magic formula, destroying all intuition for it. But this, then, is the way all math classes end up running: trained to mechanically run calculational algorithms, not to actually understand the concepts at hand.)
Some choice excerpts from “A Mathematician’s Lament” by Paul Lockhart, which is a very good read on the topic:
“But if your math teacher gives you the impression, either expressly or by default, that mathematics is about formulas and definitions and memorizing algorithms, who will set you straight? The cultural problem is a self-perpetuating monster: students learn about math from their teachers, and teachers learn about it from their teachers, so this lack of understanding and appreciation for mathematics in our culture replicates itself indefinitely. Worse, the perpetuation of this ‘pseudo-mathematics’, this emphasis on the accurate yet mindless manipulation of symbols, creates its own culture and its own set of values. Those who have become adept at it derive a great deal of self-esteem from their success. The last thing they want to hear is that math is really about raw creativity and aesthetic sensitivity. Many a graduate student has come to grief when they discover, after a decade of being told they were ‘good at math’, that in fact they have no real mathematical talent and are just very good at following directions. Math is not about following directions, it’s about making new directions.”
“LOWER SCHOOL MATH. The indoctrination begins. Students learn that mathematics is not something you do, but something that is done to you. Emphasis is placed on sitting still, filling out worksheets, and following directions. Children are expected to master a complex set of algorithms for manipulating Hindi symbols, unrelated to any real desire or curiosity on their part, and regarded only a few centuries ago as too difficult for the average adult. Multiplication tables are stressed, as are parents, teachers, and the kids themselves.”
“SIMPLICIO: But don’t we need third graders to be able to do arithmetic?
SALVIATI: Why? You want to train them to calculate 427 plus 389? It’s just not a question that very many eight-year-olds are asking. For that matter, most adults don’t fully understand decimal place-value arithmetic, and you expect third graders to have a clear conception? Or do you not care if they understand it? It is simply too early for that kind of technical training. Of course it can be done, but I think it ultimately does more harm than good. Much better to wait until their own natural curiosity about numbers kicks in.
SIMPLICIO: Then what should we do with young children in math class?
SALVIATI: Play games! Teach them Chess and Go, Hex and Backgammon, Sprouts and Nim, whatever. Make up a game. Do puzzles. Expose them to situations where deductive reasoning is necessary. Don’t worry about notation and technique, help them to become active and creative mathematical thinkers.
SIMPLICIO: It seems like we’d be taking an awful risk. What if we de-emphasize arithmetic so much that our students end up not being able to add and subtract?
SALVIATI: I think the far greater risk is that of creating schools devoid of creative expression of any kind, where the function of the students is to memorize dates, formulas, and vocabulary lists, and then regurgitate them on standardized tests—‘Preparing tomorrow’s workforce today!’
SIMPLICIO: But surely there is some body of mathematical facts of which an educated person should be cognizant.
SALVIATI: Yes, the most important of which is that mathematics is an art form done by human beings for pleasure! Alright, yes, it would be nice if people knew a few basic things about numbers and shapes, for instance. But this will never come from rote memorization, drills, lectures, and exercises. You learn things by doing them and you remember what matters to you. We have millions of adults wandering around with ‘negative b plus or minus the square root of b squared minus 4ac all over 2a’ in their heads, and absolutely no idea whatsoever what it means.”
Cramer’s Rule is commonly taught in high school.
But you are right that properly motivated linear algebra is typically introduced only at the university level, and perhaps only for mathematics students (otherwise, say, the physics department starts to complain how their students are being taught all this useless abstract theory rather than how to solve this or that particular partial differential equation). That is really the fault of the high-school teacher for not introducing things properly; obviously no one is going to remember some random complicated algorithm without the necessary “number sense”, in this case for vector spaces.
OK, so we have decades of experience with which to compare the Old Math, New Math, New New Math, Core Mathematics, etc (substitute other classical subjects instead of mathematics if you want). Any definitive conclusions?
My contentions in this thread, for what it’s worth, aren’t about Old Math vs New Math vs Common Core Math or whatever. Math classes in all these eras have been shot through with a desire for children to become organic calculating machines, as far as I can tell, just with different approaches for how to achieve that. (And, yes, some amount of differing exposure to other more worthwhile knowledge as well, but still an overriding focus on these calculational drills as major assessed goal).
My contentions are about “What are even our goals here with these math classes? Are we even achieving our nominal goals ultimately, and are these goals actually useful? Perhaps we should have different goals than training children to be organic calculators”, that sort of thing.
From the students I’ve taught and from my general conversations with adults, I wager (I mean, I don’t know, I just… wager) that I could not find one person in fifty off the street who had learnt Cramer’s rule in high school, remembered it today, and could still apply it.
And why should they be taught Cramer’s rule (in universal compulsory classes if they have no interest in it, I mean? Everyone should be taught anything if they have some interest in it, of course). To solve systems of affine equations without a computer which will happily do it for them? They already can solve these systems by plug-and-chug intro algebra, as noted before. They gain neither a new ability nor new conceptual understanding.
(And Cramer’s rule breaks down if the system has multiple solutions, sending them scurrying back to some other method they could just as well have employed in the first place. And Cramer’s rule is highly inefficient for solving systems of equations regardless! Even in a computer-less world, no serious real-world approach to training human calculators to solve systems of equations would be via Cramer’s rule.)
So… what is the purpose? Seems to me it functions as mere ritual and inertia.
The only way I remember being taught to solve (linear) equations with matrices involved those same manipulations where you multiply the equations through by a constant and add or subtract them to isolate one variable. The matrix version just ended with a matrix like (line 1) [10] (line 2) [01] equal to a matrix that gave you the answers.
It always seemed entirely pointless to do it with matrices, to the point that we did it for one lesson and then were told we could use them if we wanted to after that, but we didn’t have to. And pretty much no one did.
What other ways are there to use matrices to solve equations? We were definitely taught them as just some weird set of rules with no real purpose. I always assumed it was just to introduce us to the idea that some forms of math exist where operations are different, where not all inputs can be multiplied, and AB !== BA. Just more of an “expand your horizons” idea, similar to learning other bases.
I’m guessing that’s the one Chronos referred to as “one of which was just the same method we’d been using anyway but made harder to keep track of”, and that Chronos’s “the other of which was far less efficient” was, as DPRK notes, Cramer’s rule. You could think of these as two different ways to compute a matrix’s inverse (when it exists) and multiply by it, as well, if you like.