What you need in math is lots of unique problems to solve, not repetition of the same equation with different constants.
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.
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.
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.
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.
I don't disagree with your general point, about drilling essential skills, but I'm pretty sure long division isn't one.
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.
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!
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 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.
I agree completely.
It is army culture. Here is your shovel, now dig.
Learning to follow the specific formula by rote is a good thing.
For example you really should know your multiplication table up to 10 basically by heart. And the algorithms for doing basic arithmetic on large numbers needs to be practiced until you don't have to think about them. As should the algorithms for simple algebraic manipulation.
If you try to move on without having those and similar skills under control you will never get to the understanding part of mathematics since you'll spend all your time dealing with the mechanics of arithmetic.
What you need in math is lots of unique problems to solve, not repetition of the same equation with different constants.
Repetition of the same problem with different constants is how you learn to spot patterns and develop techniques you can build upon. Like if you find that f(x)=7, f(3x)=21, f(5x)=35, f(9x)=63 then you might spot a pattern that you can explore and that will make it really easy to solve the rest of the problems.
Whereas once I was older (as in like 20) I actually got the hang of them from real application, and then realised my struggles with memorisation were easily solved by finding simpler combinations to do so.
Obviously, there is also such a thing as too much repetition. But understanding of how it works alone and being able to prove the formula is not enough for ability to solve problems.
Do I agree that this is a completely fine strategy for getting good grades? Sure. But you're gonna have a lot of catching up to do if you want to apply this knowledge to anything that isn't a text book task that was made up specifically for you without any context.
When I went to college and was ambitious in math class that required calculations, it sucked and I was loosing. Until someone recommended me book full of exercises and I started doing them including boring ones.
That is why I say you need both. Knowing that derivation is related to speed or that integral is area under the curve does not help to solve the integral at hand.
But for solving an actual problem, I would say that understanding is way more important. Because in actual problems you tend to spend most of your time transferring whatever information you have into solvable equations, and much less time (comparatively) solving those equations.
So maybe my problem is more that tests aren't a great representation of real life problems, and less that homework doesn't prepare you well for tests.
In homework we had equations and series that took hours till you figured it. On test, we had equations and series that took over 20 minutes to solve when you was smart and good. No because there were many of them, but because it was hard to figure out what to do and because it took multiple experiments to find it.
It is like people blaming school for not teaching everything. There has to be some line because there is not enough time and money to teach people everything.
Bad side is that a lot of people think that everything they learn at school is enough. No one is going to teach you how to live your life. No one is going to hold your hand while applying complex math stuff because people who can do that have better things to do and they have limited time.