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Comment by lordnacho | Hacker News Reader

I think a major issue with math problems in school is that they're obvious.

By that I don't mean it's easy. But when you're grappling with some problem, whatever it is, eg find some angle or integrate some function, if you don't find the answer, someone will show you, and you'll think "OMG why didn't I think of that?"

And you won't have any excuses for why you didn't think of it. Because math is a bunch of little logical steps. If you'd followed them, you'd have gotten everything right.

Which is a good reason to feel stupid.

But don't worry. There are things that mathematicians, real ones with PhDs, will discover in the future. By taking a number of little logical steps that haven't been taken yet. They could have gone that way towards the next big theorem, but they haven't done it yet for whatever reason (eg there's a LOT of connections to be made).

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Reminds me of this quote:

"The view that machines cannot give rise to surprises is due, I believe, to a fallacy to which philosophers and mathematicians are particularly subject. This is the assumption that as soon as a fact is presented to a mind all consequences of that fact spring into the mind simultaneously with it. It is a very useful assumption under many circumstances, but one too easily forgets that it is false. A natural consequence of doing so is that one then assumes that there is no virtue in the mere working out of consequences from data and general principles."

Alan Turing

You might want to edit that to fix the name. It's "Turing", not "Turning".

Thanks. That's what I get for posting on my phone :)

> if you don't find the answer, someone will show you, and you'll think "OMG why didn't I think of that?"

Really? There are many times where I thought "How am I supposed to think of that?" In fact, that's probably what would happen most of the time.

The problem with high school math is that it has no context. It's just a bunch of seemingly arbitrary rules.

It's as if when learning programming, you would learn fragments of various languages here and there but never connect the various pieces into anything meaningful.

Lesson 1: boolean variables in Java

Lesson 2: If statements in python

Lesson 3: Three-part for-loop in C

Mid-term exam:

Q1: In the C language, how do you separate the components of a for loop? (correct answer: semicolons)

Q2: What can you put in the right hand side of the '=' symbol in a boolean variable declaration in Java? (correct answer: an expression that evaluates to either true or false).

Q3: You are supplied a function that can take any number and determine if it's a Crastomarian number. We have 20 foot ball players each wearing a T-shirt with its own number. The numbers go from 5 to 24. We want to find how many players have Crastomarian numbers. The program to find the correct answer will use a for loop. Write the header part of the for loop. (correct answer: `for(i = 5; i < 25; i++)`)

...

And this is supposed to be how you learn programming.

Also note: there's almost no way to know the answers without rote memorization. Of course you will never absorb these concept by osmosis because your day to day activity will not involve writing programs that do things. Instead, you're just memorizing the rules given to you and doing exercises that help you recall which rule to use for which situation.

If you haven't memorized the for syntax and written 25 different variations of it, you will not be able to know it.

And when I say that you write 25 different variations, I mean you write them as part of pointless exercises of course because these are not in the context of a program.

It's more like just repeating these lines until the pattern somehow hopefully sinks into your mind.

for(x = y; x < z; x++)

for(y = 10; y < j; j--, y++)

.. etc

My university education experience seems to contradict this notion. Many classes were heavily focused on Java to allow students to learn data structures and object-oriented programming, in general. Other classes, like Operating Systems Programming were focused on C. In the end, you learn that there is the right tool for the right job.

I think you misunderstood my post (or perhaps I wasn't able to articulate the point clearly).

My University computer science was also broken up as you describe. The introductory class was in Pascal, the next course was Java, and then each course used its own language.

That's not what I'm talking about at all.

I think the more pervasive problem in schools is a lack of mastery in basic foundational skills before moving on to more advanced topics.

Too many students lack the ability to perform basic algebra. I mean simple stuff like simplification, factoring, working with exponents and even fractions. Faced with a typical problem in calculus, of course they're going to flounder and make errors when each calculus problem involves perhaps a dozen simple algebraic manipulations.

These basic things have to be drilled and practiced A LOT before moving on to more advanced topics. There is no way around that. No amount of explanation and reasoning will help the students and instructors get around the fact that they just have to spend the time to develop fluency with the basics.

> No amount of explanation and reasoning will help the students and instructors get around the fact that they just have to spend the time to develop fluency with the basics.

Conversely, no amount of memorization and drilling will provide students with the intuition necessary to solve problems just beyond the grasp of naive symbolically-driven proof search algorithms.

In a typical non-proof-based calculus sequence, tricky u substitution and integration by parts problems are the two places where you really see "I was good at math in high school" students start to struggle because they lack a good conceptual foundation in trig and logarithms. Up until that point the students with trig identities etched into their skulls by practice and drilling do just fine. But from that point onward, they really start to struggle because they need a bit of geometric intuition and some practice with more effective problem solving techniques than "blindly apply equations until something works".

Really, both are needed -- drilling and practicing, as well as a solid (and preferably proof-backed) intuition about elementary properties of trig funcations and logarithms.

I agree, you need both. I am not optimistic about the state of mathematics education. So many things have to "go right" for a student to master mathematics.

The issue is that being able to see that little path of logical steps is a skill on it's own, and most students don't know even how to begin approaching a new mathematical problem.

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