I have a story that is too long here about my struggles with math in high school, and while I'm maybe not the model CS student for a variety of related reasons, I think that I turned out mostly fine...

None of the symbolic manipulation really made sense to me until I started learning programming (and, much later, mathematical foundations) and how to define these operations. The answer to "Why isn't sqrt(x+y) = sqrt(x) + sqrt(y)" shouldn't be "It's not in the table of 'laws' we're allowed to use". Sure, you can (easily) provide examples where that purported identity is false, but trying to understand why was really hard for me, and I wouldn't accept it until I understood why.

/ramble

> The answer to "Why isn't sqrt(x+y) = sqrt(x) + sqrt(y)" shouldn't be "It's not in the table of 'laws' we're allowed to use".

This is the most rigorous and correct answer I can imagine! It's not true because we have not proven it to be true or learnt it from someone who has.

But it takes a lot of "playing" with math to understand the value in thinking that way. Similarly to how some people start out programming with LOGO or BASIC or bastardized javascript, people who start out in math need some extra help in order be put in the right mental place before they can think about it "correctly."

By the same line of reasoning it would be not false until we have proven it false.

Yes, we should consider it neither true nor false, and not use the rule (or its negation) to prove whatever it is we actually want to prove.

> Sure, you can (easily) provide examples where that purported identity is false, but trying to understand why was really hard for me

What do you mean by understanding why?

You can look at a graph of sqrt(x) and it's immediately obvious that sqrt(1+3) is much larger compared to sqrt(1) than sqrt(10+3) is compared to sqrt(10), but the graph is just an aggregation of an infinite number of counterexamples. The answer to "why isn't something true?" really is just a counterexample.

What "why" were you looking for beyond "that isn't a property that sqrt(x) has"?

It's just that you could literally substitute 1 for both x and y and it would be immediately obvious that the equations cannot hold.

That's not really relevant. You only need to check if the equations hold if you don't understand the math involved. You've already lost at that point.

Checking is the smart thing to do then, but ideally what is important is the understanding rather than just the diligence to make sure a guess isn't trivially shown incorrect.

I strongly disagree. Brains are fallible; a quick check at the end lets you unit test your understanding. I find the ability to identify one's own mistakes and correct them to be a sign of an excellent student.

I agree with you in general, but not when only talking about the mathematical understanding.

If you want to add fractions by individually adding the numerators and denominators and the only reason you don't is one of these checks, you have some kind of fundamental lack of understanding about fractions. It's great that you are wise enough to verify, but that's irrelevant to your mathematical understanding.

Misunderstanding also exists, though, at many stages of the game. There's a pair of famous books - counterexamples in topology and another on analysis - dedicated entirely to counterexamples which dispel some common mis understandings.

And these misunderstanding are not limited to undergrads; often they stand in the way of progress at the edge off research, simply because we don't actually know what's possible. Research level math is often guided by folklore: important conjectures and shadows of what might be. And when the folklore is wrong, we go down the wrong track until someone finds a counterexample...

>That's not really relevant. You only need to check if the equations hold if you don't understand the math involved. You've already lost at that point.

You have no idea how much understanding I gained by doing such 'quick checks', sometimes even during exams. Just a little sanity check to be sure you're still on track.

Besides, inserting 1 for both x and y is an actual mathematical proof that the equation does not hold! It's always a fantastic thing to strengthen your understanding with formal proofs that you came up with by yourself.

I've always done those quick checks myself because otherwise I'm prone to making errors. They give me a quick answer of "no, you can't do this" but I never found they actually improved my understanding of anything. It just told me I can't do the thing I wanted to do and it was time to move to the next idea. It's a valid proof, but not one that really ever helped me get a grasp on what's happening.

Mostly it just means I'm in trouble and need to be really careful and do more checks because I'm working with things I don't understand fully.

For example, I've never needed to do that sort of check with adding fractions. I have enough understanding of what those numbers represent and how they work together that I've just never thought I might want to add numerators and denominators separately.

I regularely teach math for university students, and get favourable feedback.

When somebody asks why isn't 1/(x+y) = 1/x + 1/y, then I ask why it should be. Just because something looks intuitive or pleasing does not mean it's true. While this is only a meta-level answer for the question at hand, I consider it to be more important to learn than the special case. The students need to learn basic scientific methodology, for then they wouldn't ask such question, as one of the basic pillars of science, scepticism, easily empowers the student to answer the question on their own. Many haven't learned the scientific method in school, since in school pupils are conditioned to say what the teachers want to hear, not what is true.

When I'm convinced that the former point has been made clear, I proceed to the next step of how to approach a question like this. First, what is the statement supposed to mean? What are x and y allowed to be? Naturals? Integers? Reals? Complex Numbers? Matrices? Polynomials? All, of course, restricted to those where the expressions are defined, since worrying about the truth of an undefined statement is not sound. The context usually defines the possible range of the variables, but one needs to learn to identify that, which is akin to type interference.

Then, maybe someone simply forgot whether that equation was one of the proved ones, or not. The overwhelming majority of wrong, practically ocurring mathematical statements have small counterexamples. That's why I recommend plugging in arbitrary values for the variables, which seem interesting or make the evaluation easy. Whereas intuition and beauty don't imply truth, they remarkably successfully guide us to where to look first. To this specific problem, x = 1 and y = 1 seem like a good idea. We immediately see that they form a counterexample inside all the mentioned possible ranges for the variables. Now, we now the statement is in general wrong, and we can move on.

Depending on whether the time allows, I would show that for real x and y we have

1/(x+y) = 1/x + 1/y

<=> (multiplied with xy(x+y)

xy = y(x+y) + x(x+y)

<=> (use of law of distribution and substraction of xy)

0 = y^2 + xy + x^2

<=> (use of well-known p-q-method)

y in {-x/2-sqrt(x^2/4-x^2), -x/2+sqrt(x^2/4-x^2)}

However, the expression under the root, x^2/4-x^2, simplifies to -3/4*x^2, and is for real, nonzero x always negative, so there is no real solution in y. We just showed that the statement is not just in general wrong, we proved that it is always false, when the variables are only allowed to be real, and that there do exist true cases for complex variables.

It's a bit funny that I explain the things in that order while I despise philosophy. Learning the scientific method is not philosophy, or — to state it better — it's as much philosophy as learning mathematics as a physicist is.

Now, I don't really know anymore why exactly I wrote this down here. Probably it was just the pedagog in me wanting to make clear that questions like these do not indicate lack of mathematical talent, but lack of scientific education.

Do you know a way to teach science and math in a more interactive way than blog posts or youtube videos on the internet? I would love to do that.

Similarly, very little of the trig and pre-calculus stuff that I had to learn in math class made any sense to me, until I either had to use it for drawing graphics with QBasic in my programming class, or for a plethora of things in my shop classes.

I don't think that's as bad as it looks.

Part of the point of inventing concise notation for mathematics is so that you can "let the formalism carry you through"; a great deal of the time you can write down the next line of a proof by looking at the previous line or two and simplifying something without thinking too hard about what it means.

One of the things that the students are practising is performing a longish series of operations while thinking about the important stuff and letting the simple stuff take care of itself.

So I don't think it should be too concerning if a student starts to make mistakes like this: what's gone wrong may not be "the student doesn't understand what sin means", but "the student has gone astray in converting their understanding to an internalised rule of formal manipulation".

(It's a bit like if a younger pupil mistakenly gets it stuck in their head that 6x7 is 49: it doesn't mean they've fundamentally misunderstood what multiplication is.)

My algebra teacher in college did the first mistake. I still remember it after 21 years:

1. She was reducing some formula, and got it down to cos(a - 0), which is obviously cos(a). The problem is, her cheat sheet had the solution as sin(a). 2. She "expanded" to cos(a - 0) = cos(a) - cos(0) 3. We know cos(0), so this leads us to cos(a) - 1 4. This is where she got stuck. After thinking for a long time she wrote on the board: "(from trigonometry) cos(a) - 1 = sin(a)" 5. Final answer: sin(a)

I went up to the board and explained step 2 was wrong and how she should've done it, and I explained that step 4 was wronger, because that would mean that for some angles cos/sin would be outside the [-1, 1] range.

She said, "let's try your way then, young man", and doing the expansion on cos(a - 0) got it to cos(a). She then proceed to say I was wrong, because the answer had to be sin(a).

I still thank her for teaching me to question authority.

Mathematics is usually taught as rote. Many, even at higher levels of academia, never realize that mathematics is a language. Truth be told, I'm not sure most are cut out to fully grasp much beyond rote.

I hated math. I hated it, with a passion. I was figuring out what area of right triangles was, when the teacher spoke from behind me. He said, "You know, you're just squaring the triangles and dividing that in half."

I realize how trivial and silly that is. But, it was that step that made me realize that the notation was telling me something. I realized that it was a language, one which I didn't understand and still hated.

I'd eventually get my Ph.D. in Applied Mathematics, all because I learned you could square a triangle, find the area of that, and divide it in half. I'm not sure how many folks can point to the exact moment their life changed? I can, and I am glad.

I don't know the solution, because math really can be hard. Not many folks want to learn that much. I am reluctant to say this, but maybe they don't really need to?

I've graded freshman and sophomore calculus homeworks where students do things like sin(x)/x = sin/1 = 1.

That's an exact example.

What? That's clearly wrong. The answer is "sin", right? :)

I remember in first year of University (Math/Physics) someone saying that arctan(x+y) = (arctan(x) + arctan(y))/(1-arctan(x)arctan(y)).

(Basically he thought that the expansion of arctan(x+y) was just like the expansion of tan(x+y))

On the other hand, sin(x-y)sin(x+y) = sin(x)^2 - sin(y)^2.

It's really a travesty how much of the twelve or thirteen years of compulsory schooling we waste, and how little return is generated from it