Tuesday, June 10, 2008

learning Math

This post is an answer to a question one of my readers asked using the Skribit suggestion widget.

We all have to study math at some point at our lives. Even if we fail to find a use for it in the future, we still have to learn it at school. At this post I want to present a few general tips for learning different aspects of math.

Photo by foundphotoslj

The math taught at school can be divided into two main subjects - algebra and geometry. In university the difference between the two becomes less obvious, yet most of math can be put under one of these subjects. I am using the terms algebra and geometry in a rather general way in this post - all involving drawing is geometry and the rest is algebra.

There is no "royal road in math" - all who study math face some difficulties. However there are methods that can be used to improve your result.
Math requires two qualities - good memory and the ability to use the formulas and ideas learned effectively. Not all of us have these qualities, but there are ways to acquire them.

1. The first thing you need to memorize is the multiplication table. Remembering it well helps a lot (at least thats what my experience taught me). The easiest way to learn it is the following: Firstly memorize all the squares - preferably up to 13 or 14. Then memorize all the multiplications by 5 - it should be very simple to do. Finally memorize all the multiplications by two and three. Now you can multiply any two numbers. All you need to do for this is to use the closets pair whose multiplication you remember and then add to it. For example, to multiply 6*7 you would do the following:


I also would recommend to memorize 3*15, 3*7, 4*7 - for some reason these three are often encountered.

2. A simple example to a formula we need to memorize in school is the sum of geometric series. The geometric series is a series in which every number is the previous number multiplied by some number q. For example:


These are the first 7 numbers of a geometric progression that starts with one, for q=2. The formula for the sum of such a progression is:

A1 is the first number in the series, and n is the number of numbers in the progression. Obviously the formula doesn't work for q=1. To prove that this formula is correct you just need to multiply - I am not going to prove it here. It is very easy to remember the formula itself - but I personally found it difficult to remember that it is n and not n+1. To remember it you just need to do a simple calculation:

3. The example shows how to remember the derivatives of inverse functions. One way to do it is to remember a short formula, but it is not a nice one to remember, although it is possible to use the method in the second example for this. However there is a more user-friendly way. Lets for example find the derivative of arcsin. It goes the following way:







And this is exactly the derivative.

4. It happens sometimes that you need to memorize a lot of theorems that are relatively simple (even obvious) but are hard to remember, because there are so many of them. The approach here is to remember the general idea of a theorem. Lets look on this theorem:

Theorem: If two polynomials p(x) and q(x) of degree n-1 and n difference numbers .... are such that p()=q() for all i between 1 and n, than p(x)=q(x).
The theorem is relatively simple - perhaps even too simple to be called an theorem. But the exact form of the theorem might prove hard to remember, especially when it is just one out of 40 you need to remember. However it is simple to remember the general idea of the theorem - it basically states a condition for equality of two polynomials. If you remember this idea, it is easy to derive the rest. The only thing you need to remember is that polynomials of degree n-1 form an n dimensional space - so you need n numbers and the rest follows easily (for those who didn't study Linear Algebra this should be meaningless) . Another way is to remember the Wan-Der-Monde matrix.

5. Sometimes you just fail to remember if there is a theorem that says what you need to solve the problem. In this case it is sometimes useful to see if the claim you need seems obvious and general enough. If so it is safe (to some degree) to assume that there is a theorem and then formulate it. This might work, but it is better not to use this during a test - unless you can prove the theorem during the test itself.

6. For geometry, an analytical approach is often useful. This is, it is a good idea to use tools of analytical geometry and vectors to check if you remember a theorem correctly. The example I thought of is rather trivial, however it shows the point. We all know the Pythagoras theorem - for a right triangle with sides x,y,z the following equality holds:


Now lets write the same equality using vectors. For any triangle:


Lets suppose that v=(x,0) u=(0,y) and w stand for the third side. It follows than that w=(x,y). Lets calculate w^2:


And this is exactly what we wanted to show.

In these examples you saw the following methods:
1. Memorize some parts, and then use simple operations to calculate the rest.
2. Use simple example, for which you can easily calculate the result, in order to remember the more general case.
3. Learn how to get to the result by hand. The last method is especially useful for physics - you often have a lot of formulas that can be easily derived by hand.
4. Go from general idea to concrete theorems using logic.
5. Make an educated guess.
6. Use different methods and approaches to derive your conclusion, and to check it.

Also. don't forget the old tried method of writing the formulas/theorems on paper until you start seeing them in your dreams. It is a lot of hard work, but it works.

Using what you learned:
This is difficult to do. The only true method that works is to sit down and solve problems. It doesn't help much if you know all the formulas/theorems perfectly - you still need to practice to be able to solve a problem. One of my lectures in the University told the class once that while he knows the course material very well, he doubt that he would be able to solve out homework. He didn't practice solving problems for a long time, all he did was proving theorems. I doubt that he was serious, but this is something to consider.
What helps is positive thinking, and being really stubborn. Perhaps it will take an hour (or more) to solve a problem - you will solve it in the end, if you will try hard enough. And never feel that you wasted your time. Just today I fought for about an hour with a really simple question - for some reason I decided to try to solve it in a way in which it was impossible to solve. Finally I tried a different approach and I solved the problem in half a minute. I don't care that it took so much time - it was spent trying different approaches, which will hopefully contribute to better understanding of this particular subject.

Final thought:
Math is portrayed as a hard subject. There are also people who want to change this mindset and who claim that "math is your friend". In my opinion, both approaches are wrong.
I believe that math is interesting and fun. Sometimes it presents us with difficulties that are too great for us, but even by doing so it lets us try and attempt to solve this problem. However, it is also certainly not friendly to all people - it is a "friend" to only those who already like it. If you will decide that you hate math, don't expect it to be fun for you. Look at this little story from Clientcopia, for example.

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