"I want to do calculus with you..."

[mjr]

My wife told me that my 9 year old said this to her the other day.

He's really into math, and we have a few calculus books lying around, including one that he likes called "The Cartoon Guide to Calculus".

So as best as I remember the story, he asked my wife for some paper and a pencil. She gets it for him, and he stands there looking at her...and she asks if he needs anything else, and he says, "I want to do calculus with you."

[Ted_The_IT_Guy]

Welp, that's not the direction I thought this thread was going...

*whistles innocently, walks away with hands clasped behind his back*

EDIT: And any future kid of mine who wants to do calculus with his dad will be SORELY dissapointed. I can barely solve for one variable. I'm praying my kids get my dad's, my brother's and my 1st cousin's talents for the math-y arts. Cousin is an astrophyisics due with JPL.

[Ironclad Alibi]

I have a minor in mathematics. The highest level of calculus I took involved non-linear differential equations. I remember being told that the only way to solve them was to make your best guess and see if it fits.

I haven't used a single bit of calculus since I graduated.

[mathnerd]

If you don't already have it, I strongly recommend "Calculus for Cats", by Amdahl and Loats. It's not a "how to" book, but a "why" book, and if he's a strong reader, likely very manageable even at his age.

[MoonCat]

It's not a "how to" book, but a "why" book,

Wait, you mean as in "why it's done this way"?

That was always one of the major reasons I had problems with math! I couldn't understand (and the teacher never explained) WHY we had to do things a certain way. I never took calculus, but it sure would have helped anyway for what I did take.

[mathnerd]

It's "why" as in "this is what is really going on here", not the "why" of the actual computations. Though, from there it could be quite easy to figure out the why of how the computations work, assuming your algebra is strong.

Now, if your algebra is not strong, then I highly recommend the book "Algebra Unplugged", by the same authors. Again, it's a "why" book, not a "how" book, but it makes understanding what's going on a whole lot easier, which makes learning the actual procedures a lot easier.

[mjr]

If you don't already have it, I strongly recommend "Calculus for Cats", by Amdahl and Loats. It's not a "how to" book, but a "why" book, and if he's a strong reader, likely very manageable even at his age.

Thanks!

I'll check that out. Might even use it for myself. I'm getting ready to take a Calculus II class myself.

[MoonCat]

"this is what is really going on here",

That would probably have helped. I could never get past that block, whenever I had to learn something beyond the four basic functions. The teachers would say "You do this, then this..." and I kept getting stuck on WHY. Weird, I know

I don't have a numbers brain. I have a words brain

[mathnerd]

See, this is the problem with how mathematics is taught. Math isn't "numbers", it's logic and problem solving. There's a bit of a spiral, as certain concepts do have to be learned before the "why" is really understandable, but as long as the spiral stays nice and orderly, conceptual learning should flow nicely. Unfortunately, with the push to teach the "why" lower level teachers often times teach a wrong "why", which winds up just confusing students even more. I could write a whole book on the topic. Actually, I've been working off and on for years on writing that book.

Mooncat, if you're interested just for your own knowledge, I do recommend "Algebra Unplugged". The last time I bought it (I've purchased it several times, as I keep loaning my copy out and not getting it back), it was around $15, so it's not terribly expensive.

[Kaylyn]

Mathnerd, I'm going to look into both of those books. They're available for Kindle. I'm trying to go back to school this fall for a biology degree and I'm going to have to jump back into algebra (plan to retake it regardless of my comfort level in the subject since it's been a while) and work up towards a calculus course. The furthest I got in high school (graduated in '97) was advanced math, which as far as I can tell is trig lite, so calculus is completely foreign to me. Luckily I like math, and problem solving in general, so I just need to brush up on my algebra to get prepared for school. So thanks for the recommendations!

[ComputerNecromancer]

Wait, you mean as in "why it's done this way"?

That was always one of the major reasons I had problems with math! I couldn't understand (and the teacher never explained) WHY we had to do things a certain way. I never took calculus, but it sure would have helped anyway for what I did take.

They are harder to find these days, but two books by the lat Isaac Asimov explain a *not* about math.

"Realm of Numbers" and "Realm of Algebra". That latter got me thru Algebra in High School.

I've had to tutor folks on math a few times and it always amazes me the things that no one ever taught them. Especially that multiplication is "just" repeated addition. That is, that "3 times 5" is shorthand for "3 added to itself 5 times" (Or "what do you get when you add 5 3s together?)

And the reverse, that division is "just" repeated subtraction. (ie 30 divided by 6 is "how many times can you subtract 6 from 30?")

Heck many don't realize that division and multiplication are "inverses" of each other. That is, that you can check a division by
multiplying the result by the divisor.

[MoonCat]

I have no problem with the four basic functions. It's anything above that that screws with my head. When I was in freshman year of high school my math class was called "Regents Math." I have no idea what the correct name would be for the functions being taught. All I know is that my teacher HAD to be an android. No human being could say the same words in the same exact tone of voice 13 times in a row.

It didn't matter how many times he explained something, because he always explained it the same way.

This class was where I developed the eternal question, "If X equals 2....then WHY DON'T YOU JUST USE 2??!" And where I developed the ironclad belief that letters DO NOT belong in mathematical equations.

[mathnerd]

Especially that multiplication is "just" repeated addition. That is, that "3 times 5" is shorthand for "3 added to itself 5 times" (Or "what do you get when you add 5 3s together?)

And the reverse, that division is "just" repeated subtraction. (ie 30 divided by 6 is "how many times can you subtract 6 from 30?")

Actually, multiplication is not repeated addition. It's sometimes a handy way to compute repeated addition of the positive whole numbers, but in no way is it actually repeated addition. This is a pet peeve of mine, as it's one of those things I was referring to earlier that in the rush to explain the "whys" of things, students are taught something completely wrong that just serves to confuse them in the end. If you try to define multiplication as repeated addition, it falls apart as soon as you introduce fractions or negative numbers, leading to confusion and the beginning of the mental block that far to many students develop regarding mathematics. Instead, it's a scalar operation. It stretches and shrinks things. That definition never fails, no matter what type of number you use, since that's it's actual definition. Likewise, exponentiation is not repeated multiplication. Again, it's a handy way to compute it with the natural numbers, but as soon as you add rational numbers or integers (and let's not even talk about complex numbers), it fails spectacularly.

If I ran the world teaching multiplication as repeated addition would be an offense punishable by flogging, as my experience as both a middle school teacher and a college professor leads me to believe that it's one of the primary reasons why students begin to fail at mathematics.

ETA: As long as I'm starting to get technical, division and subtraction don't actually exist. They're shorthand for something else entirely. In addition, what we call subtraction is actually addition of a negative number (so 5 - 2 is just shorthand for 5 + -2) and "division" is simply multiplication by a fraction (6 divided by 2 is actually 6 times 1/2).

[MoonCat]

Re: Mathnerd ---- everything you just said? "Numbers" brain.

See, it may be logic and all that, but you still have to have an inherent understanding of the way numbers work. Numbers are another way of describing something, and that concept is itself one that I'm just barely equipped to grasp (and one reason why I greatly admire physicists, astronomers and others in that type of field). To me, words are a whole world, they're like a sixth sense. I think numbers do that for some people. I admire that, but I can't get a handle on it.

I get the four basics, because that's a logic that makes sense to me. But the more advanced functions always seemed to be missing something whenever the teachers tried to explain them. Again, though, I think the right teacher makes a big difference. My freshman math teacher would literally say the same thing the same way, over and over. A classmate once explained something to me, in different words, and I got it. Unfortunately I couldn't expect her to keep doing that for everything.

I'm one of those people who was right when I said, "I'm never gonna use this!"

[Chromatix]

ETA: As long as I'm starting to get technical, division and subtraction don't actually exist. They're shorthand for something else entirely. In addition, what we call subtraction is actually addition of a negative number (so 5 - 2 is just shorthand for 5 + -2) and "division" is simply multiplication by a fraction (6 divided by 2 is actually 6 times 1/2).

I would disagree with you on that point. The two forms are equivalent in each case, and I often use that fact in my work (computers can multiply numbers much faster than dividing), but the concepts of negative numbers and fractions are themselves derived from the concepts of subtraction and division, not the other way around (so to perform a division with a multiply, I first have to compute the reciprocal of the division, which is just as expensive as dividing in the first place).

It's true that multiplication and division are both, fundamentally, scaling operations - but I don't see any better way to explain it, to a young child who's just got the hang of addition, than as the idea of adding things together this many times. And five-and-a-half eights is the same as five eights and half an eight added together (forty-four in total), and a decimal is just so many tenths or hundredths or thousandths, and by the way the easy way of getting a number multiplied or divided by ten is to move the decimal point, adding zeroes if you need to make it fit. Put in a few examples of how such calculations are actually useful, and it shouldn't be too hard to connect the dots.

But then, I'm not a teacher.

The quality of the teacher makes a huge difference to any student who is not precisely average; he can help the less gifted (or differently wired) to keep up, and at the same time allow those naturally talented to go ahead at their own pace. I was lucky enough to have an excellent teacher in secondary school, who kept me on each topic just long enough to be sure that I understood it properly - hour-long lessons were often condensed to a fraction of that time as a result, and I was taking (and passing!) 16-year-old's exams at 12. But in primary school I had only a generalist teacher, who had to cope with a full room of young children from Toxteth. I'll leave you to fill in the blanks.

I recently got hold of a very old book which tries to explain a specific application of advanced mathematics to a non-mathematician audience - namely that of celestial navigation at sea. It appears to assume the reader knows about fractions beforehand, since the first thing it does is explain decimal numbers in terms of them. Then it introduces logarithm tables, explains how to look numbers up in them - forward and backward - and how to perform multiplication and division using them; essentially:

- Look up the logarithms of the two numbers.
- Add the logarithms together (for multiplication) or subtract (for division).
- Look up the inverse logarithm of the result.

Before you know it, it's talking about spherical trigonometry (which is much more difficult than the plane trigonometry you probably learned - or at least encountered - at school), and describing intricate procedures for calculating all sorts of things about the relative motions of the heavens and yourself. But there's something extremely strange about the way it does so - no, two extremely strange things:

1) It uses no mathematical notation. Even greek letters are mostly reserved for the names of stars. Given the audience, this is forgivable, though it means instead trawling through a tedious bank of prose to work out what the procedure actually is. "Turn to Table XVIII and look up the log-secant of the observed altitude as calculated in Problem 42b, then..."

2) It never explains nor uses negative numbers. Instead, it describes subtle variations in the procedures to be followed, depending on whether the "names" of the declination, latitude, hour-angle and/or longitude happen to "agree" or "disagree" with each other. The "name" in turn means whether it is "north" or "south", "east" or "west". And this convention gives me a massive headache; it must have made navigation much harder in practice for the poor mariners who had to carry it out. The modern convention is to treat north as positive latitude and south as negative, likewise for east and west longitude; this allows a single procedure to compute any result.