How to know if you've mastered a topic?

PatternMan · 2014-03-26 14:53:50

Okay so with math when I go through a textbook by the time I finish it I can answer all the questions in there for a long time after. However a lot of the time they don't explain well enough for you to truly understand. Then when I look at more difficult questions and problems based on what I already know, I can't answer them.

For example after learning the laws of indices I see 2^0^3 and not be sure how to answer it. Or I will see a question like this in programming -(-(-(-2))) after doing algebra for a while. Are there any ways to learn these rules so you can apply them even to something unusual? Or even extrapolate?

bobbym · 2014-03-26 19:09:33

Hi;

Comes with experience, with doing problems. It takes time. Let's face it, math takes a lifetime, or perhaps a little longer...

Just remember to work from the top to the bottom on power towers.

ShivamS · 2014-03-27 08:11:51

For algebra/precalculus:
When you can solve contest/IMO problems based on it

For Calculus:
When you can solve Spivak/Apostle/Courant problems based on it

For real analysis:
When you can solve Rudin's problems based on it

PatternMan · 2014-03-27 10:48:10

ShivamS wrote:

For algebra/precalculus:
When you can solve contest/IMO problems based on it
For Calculus:
When you can solve Spivak/Apostle/Courant problems based on it
For real analysis:
When you can solve Rudin's problems based on it

This means I haven't really mastered anything in mathematics. Not even number theory lol.

Last edited by PatternMan (2014-03-27 10:49:43)

bobbym · 2014-03-27 11:11:59

Hi PatternMan;

You do not really master mathematics.

eigenguy · 2014-03-27 11:23:47

The first step is not to learn rules, but to learn why the rules hold. Some things are simply a matter of convention (such as working from the top down on power towers instead of the bottom up). But most mathematical rules have to be the way they are, such as -(-x) = x. When you understand why it has to be that way, then you have mastered the subject. And then you don't have to remember tons of rules, because you can usually figure out the rules again when you need them.

For -(-x) = x, this follows from two things:
1. the definition of the opposite: -x is the unique number which, when added to x, gives 0: x + (-x) = 0.
2. the commutivity of addition: if x + (-x) = 0, then (-x) + x = 0.
That is, x is the unique number which, when added to (-x), gives 0. Thus -(-x), has to be x.
(Okay - I fudged that by putting "unique" in the definition, which is really a property that needs to be proven of the opposite.)

Applying that to your example: -(-(-(-2))) = -(-(2)) = -(-2) = 2.

ShivamS · 2014-03-27 12:27:56

Bobbym, he probably meant mastering whatever he learnt. Not even Ramanujan had come close to mastering all of even number theory.

Patternman, rome wasn't created in a day. It will take time. Around this time 2 or 3 years ago, I couldn't even solve something like #25 on the AMC 10 even though I had learnt precalclus. It took me a lot of practice solving problems and experiencing productive failure to get me to a point where now I can solve IMO, Putnam etc problems with ease. Don't be put down if you can't solve most of the difficult problems. I did that and it harmed me a lot.

PatternMan · 2014-03-27 14:30:21

ShivamS wrote:

Bobbym, he probably meant mastering whatever he learnt. Not even Ramanujan had come close to mastering all of even number theory.
Patternman, rome wasn't created in a day. It will take time. Around this time 2 or 3 years ago, I couldn't even solve something like #25 on the AMC 10 even though I had learnt precalclus. It took me a lot of practice solving problems and experiencing productive failure to get me to a point where now I can solve IMO, Putnam etc problems with ease. Don't be put down if you can't solve most of the difficult problems. I did that and it harmed me a lot.

It's more worry and disillusionment than putting myself down. I had always found mathematics ridiculously easy in school. Even though there is evidence to suggest I may have some mathematical intuition, after looking at real maths problems, it will take a lot of study to even attempt to solve them. If I were to go through textbooks I could be done with the whole of school mathematics in 6 months but from what I'm seeing in AMC, STEP, Brilliant.com where they don't have the conventional easy problems, it will take way more time then I anticipated.

I will need to loop on myself, review and fully understand everything from the ground up. A lot of these questions rely on your ability to easily see that something mathematical can easily be transformed into something else using whatever method that you may have learnt. You really need to understand why things are the way they are. Oh I can apply this to that because of such and such. Most people wont be able to make those connections. I'll have to overlearn everything. overlearning one simple topic will takes me probably at least 4 times the amount of time as just learning it. Turn those 6 months into 2 years.

I may not be fully prepared in time for a mathematical degree if I want to do one. Oh well there are shortcuts but there's no way to avoid the long hard work to get results. Time to discipline myself to start doing mathematics 3+ hours a day lol. I just hope I have some talent or all this work may be in vain.

Last edited by PatternMan (2014-03-27 14:43:14)

bobbym · 2014-03-27 20:27:56

Hi PatternMan;

I had always found mathematics ridiculously easy in school.

The sites you are talking about test your problem solving capabilities using math. This takes a long time to get good at.

School is designed to teach you the math. Problem solving is a whole different thing. To simplify the concept, a person can be taught to play chess in about an hour. He can absorb all the knowledge he needs in about 40 hours. To apply that knowledge, to be good at the game takes the best several years, for the rest of us a few decades.

PatternMan · 2014-03-28 03:45:20

bobbym wrote:

Hi PatternMan;
I had always found mathematics ridiculously easy in school.
The sites you are talking about test your problem solving capabilities using math. This takes a long time to get good at.
School is designed to teach you the math. Problem solving is a whole different thing. To simplify the concept, a person can be taught to play chess in about an hour. He can absorb all the knowledge he needs in about 40 hours. To apply that knowledge, to be good at the game takes the best several years, for the rest of us a few decades.

I think I understand what you're saying. It's the same with this game Street Fighter I used to play. In fighting games you could learn certain combinations of keys to do different moves and combos. Some of these were very easy to learn. Towards the end commbos would become more difficult to learn because of how advanced they were. Even after learning a large portion of procedures you can use, it still didn't make you good. You would need to be almost fluent to pull them off in game.

Certain knowledge bases were only useful for certain situations. The advanced procedures were rarely ever used because of not being feasible. It would often be best to use the most simple techniques in creative ways rather than the advanced stuff. All of that would come with experience.
The people considered the great players would raise the bar by using some known procedures in ways nobody really has before. They would understand the mechanics in ways nobody had noticed and exploit it.

Outsiders watching couldn't see what was so interesting about a game. However any decent player could tell that the person assessed multiple different possible outcomes in a fraction of a second and reacted with the best procedure given their circumstances. The best players would figure out the mechanics of the game somehow. They would find out that this has priority over that, moves at this speed so if they react to x,y or z in 1/3 of a second then they can counter it.

It only took maybe 100 hours to get decent at the game but to compete at a high level it would take a lot longer because you would have to turn procedures and their best use in different situations into a reflex or instinct. It was like you could get into the top 60 % of players with only 200 hours learning and practice but for each 10% bracket higher you went the effort to get there doubled. To get from the top 40% to the top 30 you would need to put in 400 hours or something. Some people were naturally good at these things. I competed in the B league and everytime I got to A I was kicked out. I reached a brick wall where I couldn't compete unless I could relegate complex combinations, mechanics, and reaction times into instinct and focus on reading the opponents.

Getting good at mathematics will probably be similar I guess. Part of problem solving is knowing the problem, and knowing what tools are relevant to solving it. I got good at these games by learning the techniques, seeing how the greats used them, and experimenting myself. You can't really watcha mathematicians thought process and step by step solutions to a problem right? Anyhow this has given me some insight into the issue.

Last edited by PatternMan (2014-03-28 03:48:30)

bobbym · 2014-03-28 03:53:27

To get good at pool you have to hit a million balls. Since it takes about 10 seconds per shot that means about 1000 days or 3 years. Since you can only do it for about 8 hours a day productively that means it takes about 10 years of practice before you are world class.

It has taken me quite a bit longer in both math and pool so I am a bit dumb. Perhaps you will go faster. There is one thing for sure, if you do not try you will never get anywhere.

Do the math. Enjoy yourself. Let greatness come if it is going to. If it does not at least you had fun.

Math Is Fun Forum

#1 2014-03-26 14:53:50

How to know if you've mastered a topic?

#2 2014-03-26 19:09:33

Re: How to know if you've mastered a topic?

#3 2014-03-27 08:11:51

Re: How to know if you've mastered a topic?

#4 2014-03-27 10:48:10

Re: How to know if you've mastered a topic?

#5 2014-03-27 11:11:59

Re: How to know if you've mastered a topic?

#6 2014-03-27 11:23:47

Re: How to know if you've mastered a topic?

#7 2014-03-27 12:27:56

Re: How to know if you've mastered a topic?

#8 2014-03-27 14:30:21

Re: How to know if you've mastered a topic?

#9 2014-03-27 20:27:56

Re: How to know if you've mastered a topic?

#10 2014-03-28 03:45:20

Re: How to know if you've mastered a topic?

#11 2014-03-28 03:53:27

Re: How to know if you've mastered a topic?

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