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As both a PhD in math and a working programmer who started with machine code (not even assembly language) and now programs primarily in Python and C, I think the analogy you make is incorrect.

I don't think there is much insight to be gained about programming in Python from repeatedly simulating the program at the level of tracing stacks, etc. - on that we agree. But I don't agree that doing so is the equivalent of learning how to add, subtract, multiply and divide fractions, or graph quadratics and cubics, or solve linear equations.

I believe the insights gained from doing quite a lot of arithmetic by hand are genuine.

Having said all that, if people aren't going to go on to careers or jobs in hard sciences then I believe the concept of qualitative computation is more valuable than doing any kind of mental arithmetic. But if you want to do proper math, or use calculus for real, or understand Fourier Transforms, or manipulate sheaves over higher dimensional topological spaces, then the abstract insights gained from arithmetic are, in my opinion, more than are generally suspected.

That's not to say that calculators are bad. Enormous computing power has let me get insights that I otherwise wouldn't have had. I'm just saying that not learning and internalising the underlying arithmetic basics is like not learning how to carve basic joints in carpentry.

To do so is to deliberately limit your skill set. Can't be good.

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I believe the insights gained from doing quite a lot of arithmetic by hand are genuine.

But what are the insights gained from doing them by hand? I agree that learning arithmetic is important. But we spend a lot of time (or at least did) on things like times tables. I remember looking at a page of 30 long addition problems as a kid. I knew that the hardest ones were at the end, so I'd start there. After about three of them I realized -- every one of these is virtually the same thing. But I have 27 more to do.

I was lucky. I actually had a teacher in 3rd grade who let me work at my own pace and actually got math books not in the school curriculum (and looking back, she probably did this out of pocket). But I imagine there are a lot of bright kids who just gave up on math and school out of sheer boredom.

It seems that you did a lot of problems, and gained no insights. Perhaps you were never intended or destined to be a mathematician. This is no insult - I genuinely believe that different people think in different ways, and the balance is important. I work with engineers who do things I never could, and I do theoretical work in which they have no interest. I don't understand why everyone seems to want to be mathematicians. However, having good math is valuable.

But there are insights about differences of two squares, sums of cubes, divisibility tests, prime factorizations, smaller denominators imply larger numbers, greatest common divisors and lowest common multipliers, and many many more.

I find repeatedly that I show people small arithmetical tricks and they are intrigued and surprised. I then expand on the basic ideas and derive things like RSA and DHMW codes, or the fact that primes of the form 4k+1 are always the sum of two squares, or that for primes larger than 3, p^2-1 is divisible by 24.

And so on. People are often fascinated by these trinkets, and yet they are observations that for me arose from doing the arithmetic.

I don't deny that most math teaching is appalling, and that many bright kids give up out of sheer boredom, but without the basics they are equally ill-served. We need teachers who actually understand the math they are teaching, and not just regurgitating the curriculum they've been given.

That's why I spend around half my time going around talking about what math is really about, and how it can be interesting, useful, fun, and occasionally exciting.

Without a basic facility in arithmetic, so much of real math - as opposed to arithmetic - is denied. If every calculation you do requires that you reach for a calculator, or fire up a symbolic math package, you are slowed to a crawl.

It's like trying to programming without being able to type. The ideas can't flow when you are constantly held up by not having mastered an underlying skill.

And i suspect we are more in agreement than not, each colored by our own experiences. Mine were happy, full of discovery. Yours weren't. How can we make kids experience more discoveries if they won't actually play with the underlying basics?

It's like trying to programming without being able to type. The ideas can't flow when you are constantly held up by not having mastered an underlying skill.

I almost love this analogy. There's certainly an aspect of "menial mathematics" which is like typing in that it directly translates into fluency of thought. There's also an aspect of discovery, though, that's missing from the metaphor.

There is a large difference between inferring that some equality holds based on abstract principles and actually performing the evaluation and directly tracing out why that equality holds (even very non-generally). I liken it to statistical modeling sometimes: models allow you to talk about and comprehend data on a high level, but only by directly plotting all of the data at high resolution can you let your brain's natural pattern seeking tendencies reach out for further insight.

That's a common theme there: granularity versus generality.

Indeed. It was Knuth who said that mathematicians and computer scientists think in very different ways. With that I don't disagree.

How can we make kids experience more discoveries if they won't actually play with the underlying basics?

I'd argue that we should let them play with calculators and other tools. My discoveries came as I did programming. In 4th grade I wrote an arbitrary precision number package. I learned more about arithmetic doing that than all of the rote drills combined.

It's not the arithmetic you're trying to get insight into - it's the structures.

I have a feeling I'm never going to be able to convey what I'm trying to because you're not a mathematician, and I'm not a good enough writer. It's not just about becoming better at the arithmetic, it's about finding and creating structure, order, relationships and mappings.

But I'm going to stop now. It's clear that I'm just not expressing myself well enough to make the point, and I've spent far too long on it. I regret not being a good enough writer - and perhaps not a good enough mathematician - to explain in a way so as to make it clearer.