Comment by Atiscant | Hacker News Reader

Comment by Atiscant | Hacker News Reader

The single most important thing IMHO is to get the basics right

For context, I have a PhD in theoretical computer science and a master in mathematics. I've been teaching mathematics for several years ranging from high school to university.

The biggest problem for students, especially at the university level, is that they don't have the basics down. If you are not fluid in fractions and manipulating equations, you can't focus on the important part of learning induction proofs. This is just one example, but the return on investment on getting the basics down is huge.

For each new subject, you need to take the prerequisites serious. If you can't read a new definition and focus only on the new part, you should go back. The is of course very time consuming if you are missing a lot of prerequisites but if you want to get good, this is the way.

A well planned program will do this for you to some degree but the reason most people struggle is that they don't take prerequisites seriously or simple haven't learned them probably. You should be able to solve exercises in each prerequisite subject to a fairly good degree before tackling the next.

For anybody starting or doing university level mathematics I suggest your start refreshing pre-uni. Make sure you have fractions, linear and quadratic equations, exponentiation and logorithms and basic properties of functions are really down. If you have time, it can be very beneficial for pure mathematics to start out with some formal logic and natural deduction to really understand the way mathematicians reason in proofs and definitions.

Finally, ask all the stupid questions. Nobody, except yourself maybe, care if you misunderstood a basic point and got something wrong because of it. If "you just don't get" something, there is a high chance you missed something. If you don't understand the answer, same thing applies. Mathematics should be, if done properly, straight forward.

What online coursework do you recommend for getting the basics / fundamentals down? Is Khan Academy sufficient for this?

It is decent for getting the concepts, but the number of practice questions (about 5 per mini-concept) are far too few to reach the kind of mastery GP is talking about. You probably need to do an order of magnitude more questions to get to the point where you stop having to think when doing these things.

Khan Academy is not bad if you like their style of presentation, but as another comment noted, the number of exercises is too few. Unfortunately, I have yet to find a good recourse with focus on exercises, so I tend to create them myself for courses I teach. If you (or any other for that matter) are interested, I can upload a work sheet somewhere.

There properly are tons of sites providing work sheets, but the focus then tends to be on K-12 education and they are often to simple IMO.

For university the importing thing is often manipulating say fractions rather then "calculating" with fractions.

Tangentially, would there be any interest in a service providing this? Pay say 2$ or something and get 50 exercises in the basics with an eye towards university with sample solution? That space is properly swamped but maybe there is a niche? I often think that there is room for textbooks of this short but then again, that market is also swamped with big publishers.

I am interested in this. Do you have a website / would you mind posting your contact info?

I think there is absolutely a niche: students (or graduates) who have some experience with undergraduate-level math courses but that need to improve their fundamentals. Most US universities give credit for "Advanced Placement" courses taken in high school, even though these courses are rarely taught to university standards. I skipped the intro calculus sequence at my school, and since the proof-based math courses are usually fairly self-contained, I never got a chance to address the gaps in my fundamentals. I've now found myself in this somewhat awkward position where I can e.g. explain to you what sequential compactness means but can't solve for the roots of a third-degree polynomial.

It doesn't make sense to read a high-school level textbook, and books on competition math are either too hard or the focus is more on exposing you to hard problems rather than teaching fundamentals. There needs to be something which reviews fundamentals in a rigorous way.

I've added an e-mail to my profile.

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