Explain this 'new math'

Factual Questions
121

Let me also clarify pre-emptively that it is important that students understand that a calculator is not doing anything they could not, with enough time, work out themselves. I just don’t care that they train to actually do the tedious working out with any efficiency. I think (optimistically?) this is the way most people understand the computation of square roots.

That is, students ought understand that the reason 3 * 6 = 18 is not fundamentally because “If I enter ‘3 * 6’ into the calculator, it prints ‘18’” (nor because “In the table you found so important for me to memorize, the ‘3 * 6’ entry is ‘18’”), but because if you count out a rectangle of dots which went “1, 2, 3” on one side, and then “1, 2, 3, 4, 5, 6” on the other, the total number of dots would come out to “1, 2, 3, 4, 5, 6, 7, 8, 9, ten” and then another “1, 2, 3, 4, 5, 6, 7, 8”.

This being in just the same way that, though we do not teach manual computation of square roots, nonetheless, we expect students to understand that the reason sqrt(16) = 4 is not because “If I enter ‘sqrt(16)’ into the calculator, it prints ‘4’”, but because if you keep adjusting the side lengths of a square of dots up and down till it has 16 dots in total, you will find the corresponding side-length to be 4 dots long. The calculator is just doing quickly something you could do tediously by hand.

The calculation has a meaning; it’s not just a ritual. In my lack of vitriol for calculators, I am sometimes accused of losing sight of this principle, but it is in fact the one that matters most dearly to me. (Even armed with a memorized algorithm, mind you, there is the danger that calculation is divorced from its meaning and treated as just a rote ritual; how many times I’ve seen this at all different levels of the math curriculum…)

122

Why in the world is arithmetic not part of mathematics?

123

Arithmetic isn’t part of mathematics? Maybe things have changed since I was in grad school. But the integers, last time I checked, were still a ring under the normal operations of addition and multiplication, and the rationals and the reals were still fields under the same operations. And the integers, rationals, and reals are where most people do arithmetic.

Sounds like math to me.

124

Probably Chronos should explain his own comment, but he may have just meant that “arithmetic” and “mathematics” are not the same thing. (Arithmetic is not math in the sense that Set A is not equal to Set B, not in the sense that Set A and Set B are disjoint.) This is a claim that no mathematician would find remotely controversial, but there are too many laymen who equate “math” with nothing more than calculation. And being good at calculating is no more the essence of being a mathematician than being good at typing or spelling is the essence of being a writer.

The relationship between arithmetic and mathematics is something that has been discussed before in past threads (this one, for example).

125

I should have said that what is taught in most schools under the label of “arithmetic” is not math. Show of hands: How many people went through any proofs in your elementary-school classes?

126

Depending on just how formal a proof you want. Can you even “prove” any of the Five Fundamental Laws of Arithmetic (those being axioms and all)? One of my elementary school teachers did (4th grade IIRC).

He showed us, by means of a diagram, why multiplication is commutative.

ETA: It has come to my attention that first semester college Calculus classes don’t include any proofs any more, at least in some arrangements of the curriculum. Those are delayed until some later time. Is calculus (at the first semester level) no longer math? (When I took Calc I, the text included proofs. We learned the epsilon-delta definition of a limit and how to do ε-δ proofs for some really simple cases. Does that make it math?)

127

I can imagine what these “Five Fundamental Laws of Arithmetic” you keep referring to are, but it’s not at all a standard term (Google brings up less than one page of hits, including this very discussion…).

Being shown why multiplication is commutative via diagram is a great proof. For what it’s worth, one of my formative memories is asking my mother in 3rd grade why the order of multiplication didn’t matter, and her having me visualize it via a rectangle. That it had not been made clear enough to me by the curriculum which had taught me multiplication in the first place does not speak well to what was emphasized in those lessons… But, thankfully, this particular insight is an easy thing to teach better, and I’m sure many schools do.

128

Responding to part of the same post once more:

One thing I like to stress is that proof is always relative; that is, relative to what you want to take as the rules of your proof-game (whether formal or informal). If you want to take it as a primitive rule of your game that you can assert one of these “Five Fundamental Laws” (which is often what we would mean by calling them axioms), then there’s no nontrivial work involved in proving them; the game lets you simply assert them. If you want to take the rules of your game to be something different, then there may be more work involved in proving these, the different work depending on precisely what those different rules of your proof-game are. But there’s no such thing as proof simpliciter; only proof relative to some particular notion of what counts as proof.

At the level of elementary school mathematics first being introduced to children, there’s no point being terribly formal about any of this; the relevant notion of proof is just “Reasoning that humans are inclined to consider conclusive regarding their intuitive notions of counting and measuring and so on”.

I’m guessing your Five Fundamental Laws are that addition is commutative and associative, multiplication is commutative and associative, and multiplication distributes over addition (the laws of a commutative semiring [taking associativity to include the 0-ary case of identity elements]). You could give reasonable “proofs”, in the sense of the above paragraph, of all of these as concerns intuitive counting or measuring with corresponding definitions of arithmetic and multiplication, and in this sense, you might not consider any of these laws to be truly primitive axioms in that context. (You could also imagine having introduced these laws as abstract axioms formally defining counting, measuring, arithmetic, etc., in the first place, but stripping them of intuitive meaning in that way is probably not the best approach for introducing children to the concepts of arithmetic)

129

Continuing to say things, because I want to:

Chronos and I are generally on the same page on the issues of this thread, but I have some wording quibbles:

I dislike the idea, which frightens students in later math classes, that “proof” is some alien concept one is first exposed to in high school or college geometry or discrete math courses or such things. Proof is just sound reasoning for a conclusion, and all human beings have been reasoning in various ways, to various standards, their entire sentient lives, inside and outside of mathematics. So I would not want to say of students that they had not seen any proofs in elementary school, since this is implausible except on a very restrictive formal account of proof. Rather, what I would lament is that proof (i.e., reasoning), while not entirely absent, is far from the focus in elementary math classes, with rote algorithm execution being emphasized much more strongly.

130

A better word than “alien” above might have been “unfamiliar” or “esoteric”.

131

As I noted earlier (see Post #71), I thought the teaching of “New Math” was too abstract too soon. (The example I gave there was at the Algebra-I level, with its legalistic introduction to negative numbers.) I’ve long been under the impression that “New Math” (and all the subsequent generations of Newer Math) erred in pushing the abstract-first presentation down to earlier and earlier grade levels. Seasoned mathematicians, of course, already understand their stuff at the abstract levels, and have it all worked out from the ground up from axioms, definitions, and logical development, and it’s all clear to us that way because we’re used to it.

Where I thought New Math failed was in being designed by mathematicians who felt it was just natural to teach that abstract development from the get-go. These were mathematicians, not grade-school-level teachers, and seemed to have no concept of how to teach math to children.

(BTW: Yes, the Five Fundamental Laws of Arithmetic are just the five basic rules you think I mean. That is what I remember them being called when I learned them. I don’t remember what grade that was, but 4th at the latest, and probably not earlier.)

132

I don’t, of course, expect the same level of rigor in proofs from a third-grader, a freshman taking geometry, and a math PhD student. But what I do expect in something I can call a proof is an answer to the question “but does that always work?”. A diagram like the one Indistinguishable’s mother used to illustrate the commutivity of multiplication can work for this, if it’s generalized enough. That is, it won’t work if the rectangle is specifically a 3x5 rectangle, because that just proves that 35 = 53, but it will work if the sides of the rectangle are unspecified.

Just providing examples doesn’t prove anything, because that leaves us with things like reducing fractions by canceling digits. What’s 16/64? Just cancel the 6s. What’s 19/95? Just cancel the 9s. There we go, two perfectly good examples! And yet this method almost never works.

133

I agree in principle, but I’d like to know from an actual childhood development specialist at what age children develop certain levels of abductive and inductive reasoning.

134

I opined, above, that I thought “New Math” was written by mathematicians, for mathematicians; and that they got too abstract too soon, without enough “concrete” development mixed in along with.

In other words, what you’re asking for here is exactly what I thought they didn’t do. (And maybe still don’t?)

Indistinguishable’s proof with the 3x5 rectangle and the 5x3 rectangle is exactly what I saw too (in 4th grade, IIRC). It’s “proof enough” for me to note that it would work as well for rectangles of any other dimensions. My 7th grade teacher showed us something similar to “prove” the Distributive Rule.

These diagrammatic proofs have their limits, but they work well enough to establish good plausibility, especially at the grade-school levels. This is better than those legalistic-sounding definitions that are so common in math texts (like the example I gave about negative numbers). Those rectangle diagrams have one limit in that they might really only work for integer numbers: Show a rectangular array of 3 rows of 5 blocks, equal to an array of 5 rows of 3 blocks. Now, how does that generalize to work for fractions too? And forget about generalizing that to work with negative numbers. But I still like techniques like that. The particular “proof” I saw for the Distributive Rule had even more limiting limitations. (If you’re thinking of a rectangle-diagram for the Distributive Rule, it probably isn’t the one I was shown.)

135

A picture is worth a thousand words. For example, (p+y)^3 = p^3 + 3p^2y + 3py^2 + y^3 is nice and everything, you can work it out and show it to be true, But I take just one look at this image (or better yet, a physical model) and I instantly grok it.

136

That picture is the exact same proof of the formula as anyone would write out “symbolically”, only rendered in the particular way we think of as “visual”. It’s nice, because your brain has special hardware acceleration for thinking in terms of this sort of rendering. But move on to (p + y)[sup]4[/sup] = p[sup]4[/sup] + 4p[sup]3[/sup]y + 6p[sup]2[/sup]y[sup]2[/sup] + 4py[sup]3[/sup] + y[sup]4[/sup], which works in exactly the same way, and your brain will probably no longer be so happy with visualization.

(Alas, we are particular beings living in a contingent universe, saddled with non-necessary physical facts…)

137

(To be clear, this is me lamenting the way our minds work, not me criticizing standingwave for presenting that example of visual proof)

138

Nah, we’re fine. Yeah, the physical models obviously have their limitations, e.g. they’re limited to three dimensions. But it gives this engineer a warm fuzzy down in the cockles of his heart just to see it expressed like that. And it provides a little added confidence when you extrapolate it to higher orders.

139

:slight_smile:

140

To which Jack (who is only just now learning subtraction) replies “What the heck is that? How do you subtract three-digit numbers-- We only know how to do it for one and sometimes two digits. What makes you think that’s going to work? It looks awfully complicated to me”.

And to which I reply “You’re an EE, and you can’t identify a simple error using a simple method and explain it to the user? How do you expect to ever get employment in your field if you’re that incompetent?”.

Then again, I’ve been involved in educating many of the future generation of EEs, and I’m not too surprised to find one who either doesn’t know how to count to six, or doesn’t know the difference between ones and tens. Maybe this guy can actually compete after all.