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Comment by bitwize | Hacker News Reader

Unless you are a math prodigy, drills are paramount to understanding the concepts, especially before about high school. Almost no one needs deep understanding why three and four make seven, or why long division works. But if they cannot call upon these elementary tools on demand, in an instant, they will be flummoxed in later math classes. Hence, drilling until they've nailed it.

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But the problem is that we spend ~9 years on those drills which is more time than a typical engineering student spend on algebra and above. How come we can teach so much higher math in such a short amount of time? It is not possible from scratch, the reason we can do it is because some bored students played around with the drills instead of just doing them and thus learned a lot of related concepts which are cached in on in later years.

The losers are those who just spent years learning how to do the drills, getting good grades but didn't learn anything worthwhile at all. It would be good if we could catch those and correct their behavior before they had wasted 9 years on it, but as is we don't notice that most students fail to learn anything before we test them on more interesting problems in high-school.

Note: I am not saying that we shouldn't teach arithmetic, on the contrary I'd argue that arithmetic is one of the most important things to learn in math since everything else is built on top of that intuition. We just have to realize that we practice arithmetic to build intuition and not to become a human calculator.

Because it takes about 9 years for a kid with an underdeveloped brain to learn the skills and the self-discipline necessary to stick with a problem and call upon lower level math skills on demand.

And no, they don't need to understand the rationale behind things like carrying and borrowing. They just need to do it. Understanding can come later. It is much easier to drill known good practices to mastery than it is to explain why they work and teach them to mastery. These are kids whose parents are still in the "because I said so" stage of parenting.

Intuition comes from mastery. I can factor polynomials in my head; I'd be flummoxed at that task if I didn't know my times tables by heart. And I wouldn't know those if I didn't drill.

Drilling is paramount to early math education. Nothing else that's been tried works.

Let me put it this way: who is more likely to get a programming job, a bricoleur with an incorrigible need to understand HTTP, networking, etc. from first principles before he can use them, or a certified professional who has chosen a framework and practiced building apps using that framework and related best practices, while knowing nothing about how the framework actually works?

The world needs surprisingly few bricoleurs and surprisingly many people who can get shit done.

> How come we can teach so much higher math in such a short amount of time?

1. Because, higher ed students are a subset of school students who are on average better than and more interested in solving math problems.

2. Because, the mental micro-skills needed for higher math are many and diverse and learning one during school doesn't make you better at any other. You have to learn all of them, which is why initial training takes longer. Once you have most of the micro-skills trained, you can use them to cover new material faster. Micro-skills include fast mental arithmetic, application of logical tautologies (If A=>B, then NOT B=> NOT A), visualization skills, the hand-eye coordination to draw and recognize complex symbols. There are also macro-life-skills like being able to sit and listen to a teacher for an hour without getting bored.

An engineering student can learn more advanced math in a shorter time because a) they're older, b) they have mastered the elementary stuff already.

Also let's be honest, how many engineers actually learn the more advanced math. Most of them cram it into their brain for the test and then promptly forget it.

Surely the understanding is the main purpose though. The value of being able to do long division is pretty minimal; the value of understanding how division actually works and the fact that you can break down one hard problem into multiple easier sub problem is surely more important.

I've seen this helping my son with addition: he knew to carry the one, but no one had explained what that actually meant. It was just a magic series of steps that gave the right answer, with no connection to related issues of place value etc that they then teach separately.

No one needs to know long division of numbers, not anymore. Source: am mathematician who didn't learn long division until I had to do it with polynomials at university (at which point it was easy due to accumulated knowledge) and that was 20 years ago. IMHO long division of numbers has been thoroughly obsolete for decades, and shouldn't be taught except as a history lesson, like slide rules or log tables.

I don't disagree with your general point, about drilling essential skills, but I'm pretty sure long division isn't one.

I teach computer science, and you need to know long division in order to understand the % operator.

This is actually an issue I run up against because my 15 and 16 year olds ask to use a calculator for things like 23 - 7 ... And then I want them to do crazy math like 23 % 7.

You need to understand the idea of division, but not the method of long division (I understood % years before long division). Kids learn about remainders from division at primary school, and that's enough to understand %. The idea of remainders, and multiplication tables, make 23 % 7 easy by stepping through the 7 times table.

I think multiplication tables are still pretty important, but a lot of my first and second year university students are patchy on them. It doesn't bother me too much when someone learning about modulo arithmetic needs a calculator to figure out that 7 goes into 23 3 and a bit times. Then they can do 7 * 3 and figure out there's 2 left over.

Needing a calculator for 23 - 7 is concerning at 16 though!

I have taught university engineering freshmen in physics courses who routinely used a calculator to simplify fractions as simple as 4/2 or a group of four students who put down 1/3+1/2 = 2/5 and could not identify the mistake when it was pointed out to them. This is what happens when you stop the math drills in school.

Requiring math students to know the concepts and not be proficient at the mechanics is like telling a basketball player they just need to know that if they spin the ball when shooting, it is more likely to go in. That doesn't help them become better players. Shooting a spun ball 10,000 times is what does.

I didn't say anything about stopping drilling in general, just that I think long division is at this point merely a historical curiosity.

I totally agree that the number sense from drilling fundamentals at school is essential for higher maths. Addition, subtraction, multiplication, fractions, powers, etc. are all very important, as are intuitions about them that only come from practice. Fractions are something many first year university students are incredibly weak at - I've seen the same things you have.

> No one needs to know long division of numbers, not anymore.

I agree completely.

It made me sick to my stomach to be told this is the formula and here is how you must calculate it, but we won't tell you why it works even if you ask.

It is army culture. Here is your shovel, now dig.

It's also the only thing that produces proficiency at math.

This exact theory made me utterly sick of math at a very early age. The response of the school was to assign increasingly more time to math bookwork, which I had completely stalled on. The more I was forced to do it, the more it made me sick, and the slower I progressed. For me at least, having a real application for knowledge is key.

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