# Schools How does high school maths compare to university maths?

1. Dec 5, 2012

### Synchronised

I am in my final year of high school, I am doing the highest level of maths available but I don't find it difficult at all, so I was wondering whether or not I should cover some university level mathematics before doing bachelor of science (Advanced) next year.

For this BSc I am allowed to do two majors (1 major replaces electives) over 3 years, the available majors are physics, chemistry, maths, nanotech, medical science, computer science, biology erc... so I am thinking about doing physics and maths.

The topics I will cover by the end of high school (October 2013) are curve sketching, complex numbers, polynomials, advanced calculus, mechanics, conics, series and induction, inequalities, circle geo, combinations and permutations, binomial theorem, probability, trig, parametric equations and more... so which topics should I study during the 2013 December holidays before uni starts? I am thinking differential equations because we don't study those in HS, what else? By the way the university website doesn't say what topics are covered in first semester for physics and maths so I have no idea what to study?

2. Dec 5, 2012

### micromass

University mathematics is vastly different from high school mathematics. Unless you went to a really good high school, your current high school experience in mathematics is probably memorizing formulas and rules and plug-n-chug exercises. This will end in university. Mathematics there is much more proof based. Instead of being able to calculate things, you will be expected to know why and how something works.

That you did well in high school is not a guarantee that you will be good in university math. The opposite is also true: I know people who were very bad in high school math, but who excelled in university.

If I were you, I would absolutely try to learn basic logics and proofs. Being able to prove things is so important in mathematics.

You can learn proofs either from a proof book or by working through an actual math course. If you want a proof book, then I don't think there is a better book than https://www.amazon.com/How-Prove-It...id=1354719872&sr=8-1&keywords=Velleman+proofs.
If you want to do some actual mathematics, then you might want to think of going through https://www.amazon.com/Calculus-4th...ie=UTF8&qid=1354719931&sr=1-1&keywords=Spivak (this is a highly nontrivial book!!!) or you might want to consider this gem: https://www.amazon.com/Abels-Theore...&qid=1354719968&sr=1-2&keywords=Abels+theorem

The trick to writing proofs is to make as much proofs as you can, and then to present the proofs you have written to somebody else (for example, on our homework forum). Then you should ask that other person to give as much criticism of the proof as he can. In the beginning, he will probably rip your entire proof to shreds. But you will see that you will get gradually better in time. This is really the only way of learning proofs.

Last edited by a moderator: May 6, 2017
3. Dec 5, 2012

### Synchronised

Should I start by learning famous proofs like Wallis Product, Taylor Series for e, Gregory/Leibniz Series, Sine Integral and the Riemann-Lebesgue Lemma, Basel Problem, Ceva's theorem, Gama function, The fundamental theorem of algebra, riemann zeta function, Irrationality of various real numbers etc...? After that I will buy one of the books you suggested to learn more proofs.
What other proofs can I learn? and at university do we get asked to just prove something or are we given step by step instructions?
Thank you a lot micromass, this kind of opened my eyes, I now realize that the level of maths I am studying is just the basics even though it is labeled advanced mathematics at high school level and 90% of the students struggle with it.

4. Dec 5, 2012

### micromass

You seem to have the wrong idea. The goal is not that you should learn famous proofs. The goal should be that you could invent and make a proof yourself. Reading proofs is good, but it won't help you as much as doing proofs yourself. It's a bit like watching others swim: you might learn from seeing their technique, but it won't teach you to swim.

So no, you shouldn't "learn proofs", you should learn how to prove!! That's a big difference. At university, they will often just give a statement and ask you to prove it yourself. In the beginning, they might give some hints or instructions, but that won't last very long.

You mention all these famous theorems. But you won't really see the significance of them until you know how to prove things yourself. If you read mathematics, then your focus should be on "how can I use this theory to prove things myself later?" and "Does this proof contain important ideas that might be helpful?" So you should always read mathematics with the goal of proving things yourself later on. Doing exercises is an essential part of this.

5. Dec 5, 2012

### Synchronised

So what I should concentrate on now is doing exercises where I can prove things myself. Do the textbooks you mentioned have exercises?
In my high school textbook there are a lot of prove questions and I do them all but they are very simple, for example, if n is a positive integer, prove that $$\sqrt[n]{n!}\geq \sqrt{n}$$. If they were to ask us to prove something more difficult like prove pi is irrational or prove the Basel problem they would step us through the question making it much easier.
I will learn the famous proofs I mentioned earlier but as you said I will try to learn the methods used in the prove and how I can use them to prove other problems but where can I find other problems?

Last edited: Dec 5, 2012
6. Dec 5, 2012

### Klungo

My way of looking at a introductory pure math course at the University level is much like building something completely from scratch.

You are given axioms (laws) that "govern" how the objects they describe are related.
From these axioms, you construct results or theorems by combining them. And from there, more and more abstract theorems and objects are manufactured until you construct an entire subfield of mathematics.

But it's a bit more complicated than that. Different math subfields collaborate for instance.

7. Dec 5, 2012

### micromass

All the textbooks I mentioned have exercises. In fact, I suggested the textbooks especially because they have good exercises.

8. Dec 5, 2012

### pasmith

You may want to read this post by Tim Gowers, which although expressly aimed at Cambridge students is in fact generally applicable.

9. Dec 5, 2012

### Synchronised

I will buy the Daniel J. Velleman's book since it is the cheapest :) thank you again!

10. Dec 5, 2012

### Alex Wiseman

I'm just about to finish first year Linear Algebra tomorrow morning (that's my final), and all I can say is it truly does not compare to high-school maths at all. High-school was based on, "here's an equation, here's some value, solve for x", university level maths are more like, "here are some values, what can I do with them and why am I doing it? How can I take these values and apply them to things we encounter on a daily basis?".

As an example, in high-school I learned how to solve simple matrices, in university I've learned how these matrices essentially make up the way Googletm ranks pages in a search. Or how the 3x3 lights out game is mathematically defined.

There's not a whole lot you can do to prepare for university maths, seeing as it is like starting from scratch all over again. But as micromass suggested, learning to prove things on your own could be of benefit to you.