Is Math Cumulative & is Memorization Important?

[pandaexpress]

What's up everyone!

I'm a junior in HS right now and have been trying to improve my math skills after reading a bunch of stuff on jobs and how math and science skills are very important for getting a good job in our economy here in the U.S.A. I've been sort of "ok" my entire life at math. I've gotten mostly A's and B's, but was never that straight A math whiz. It's also been my experience that I've had to work for my math grades and they usually didn't come easy (except maybe arithmetic! :tongue:).

Anyhow, I'm wondering if higher math that you take in college will constantly build on all math taken in high school? I'm going to be taking Calculus my senior year here and I've found that literally every single year it takes me about two weeks to remember my math skills after taking the summer off. I'm not a person who can just solve a problem or even remember order of operations that well without practice. So it's like a constant practice issue for me. I'm wondering how you guys who are good at math do things? Is there just a lot of memorization that is required? Does math always end up building on previous math and become a cumulative skill set? Or do you ever take math where you start fresh again?

Sorry if my question is kind of general, but hopefully you guys understand me. Feel free to ask me to clarify things if you want me to. I just find that I lose my math skills when I don't use them and wondering how "good" students take on math and do so well.

 

[ActionPotential]

The first thing that I would suggest is to try to think about why you are performing certain algebraic or trigonometric operations as opposed to trying to mindlessly memorize. There is obviously an element of memorization involved in mathematics, however, from my own experience, it is much easier to remember things through understanding and repetition. Repetition is very important. Mathematics is not a spectator sport.

As for Calculus, you will want to feel confident in algebra and trigonometry. Make sure you understand functions and rational expressions (you will see a lot of them). It is also helpful to have a good intuitive understanding of the unit circle (this makes things like finding the zeros (min/max) of some trig function much easier), trig identities (you will become comfortable looking for possible trig substitutions to make things simpler), and trig formulas.

In addition to all these things, you must relax and not stress too much. Mathematics is only easier when you are enjoying what you are doing and feel like you are understanding the larger picture (and not just mindless computations). Also, it is common for people to struggle in math at different points in their life and then one day it clicks and they take giant leaps forward (I can speak from experience).

Keep working, keep thinking, keep practicing and stay positive. Also, make sure you use all of the resources at physicsforums as plenty of people are more than happy to explain something you don't understand.

Peace out brotha! Have a wonderful day dude.

 

[SteamKing]

All of you previous knowledge, experience, and skill will be important when you take your college level courses. If you don't know the basics of arithmetic and algebra, you will continue to have problems not only in all your math courses, but any course which has math in it.

While memorization is derided in some quarters, knowing basic facts like the multiplication tables or how to carry out an algorithm like long division is always useful. If you have to stop and derive everything you do from first principles all the time, you'll never be able to progress.

Memorizing everything is not required as long as you can remember where to find the information you need. That's why it's important to have handy good reference works or to know how to search for information online or in a library.

 

[jbunniii]

Anyhow, I'm wondering if higher math that you take in college will constantly build on all math taken in high school?

Generally, yes it will. You won't necessarily use every concept you learned in high school every day, but certainly algebra is ubiquitous and calculus and trigonometry also show up in higher mathematics as well as in other fields such as physics, statistics, engineering, etc.

I'm going to be taking Calculus my senior year here and I've found that literally every single year it takes me about two weeks to remember my math skills after taking the summer off. I'm not a person who can just solve a problem or even remember order of operations that well without practice. So it's like a constant practice issue for me. I'm wondering how you guys who are good at math do things?

Yes, practice is the key at any level and any subfield of mathematics. You can't really learn without doing lots of problems.

Is there just a lot of memorization that is required?

Not really. Some things are useful to memorize, such as the quadratic formula, trig identities, and the pythagorean theorem. But it is much more important to learn when and how to apply these things than it is to memorize them. The ones that you use most often you will probably end up memorizing through repetition anyway.

Does math always end up building on previous math and become a cumulative skill set? Or do you ever take math where you start fresh again?

Some fields of math are more dependent on what you have previously learned than others. For example, real analysis requires pretty much all high school math, whereas abstract algebra does not use much calculus or trigonometry. But you never really start fresh. Some concepts will be used in almost every field of mathematics: sets, functions, arithmetic, etc.

I just find that I lose my math skills when I don't use them and wondering how "good" students take on math and do so well.

Well, there is certainly such a thing as natural mathematical talent - see child prodigies such as Terence Tao. So I am sure some people are able to pick up where they left off the previous year without forgetting much. But probably most people will forget some of it. It's never a bad idea to spend some time as the new school year approaches, and review what you did in the previous year. Maybe look over your old exams and see what you have forgotten, so you will know what you need to focus on.

 

[Axel Harper]

It seems the others have covered your question well, but I want to speak more on memorization in math. I used to have the idea that math was memorizing a bunch of formulas and knowing when to use them, but of course, you'll run into a dead-end if you get a problem that doesn't fit any of your formulas. The large majority of math involves understanding the underlying processes and how to apply them - basically why we use the formulas and methods we do, why they work, and (in higher math) how they were derived. As someone else said, it definitely helps to memorize commonly used formulas and algorithms to save time. A really good book to help you better understand this is Algebra by Israeli Gelfand. This book covers high school algebra very concisely; in each section, you are presented with the material and a couple of very straight-forward examples, much like you would see in any textbook. The real fun is in the more difficult problems that seem completely unrelated to what you just learned. It's possible to solve these problems with only the material presented, but finding the solution requires a thorough understanding (and some creative thinking) in how the math actually works.

 

[Yellowflash]

Most people above have expressed opinions which I agree with. I definitely believe that there is some, but very little, memorization involved in mathematics. Generally, if you can understand the material, you will get used to the formulas and will not need memorization - or you can re-derive them (generally you will just get used to it).

 

[ZombieFeynman]

My view is that remembering (though not necessarily by rote) things is a crucially necessary but insufficient condition to success in science or mathematics.

 

[Lavabug]

It seems the others have covered your question well, but I want to speak more on memorization in math. I used to have the idea that math was memorizing a bunch of formulas and knowing when to use them, but of course, you'll run into a dead-end if you get a problem that doesn't fit any of your formulas. The large majority of math involves understanding the underlying processes and how to apply them - basically why we use the formulas and methods we do, why they work, and (in higher math) how they were derived. As someone else said, it definitely helps to memorize commonly used formulas and algorithms to save time. A really good book to help you better understand this is Algebra by Israeli Gelfand. This book covers high school algebra very concisely; in each section, you are presented with the material and a couple of very straight-forward examples, much like you would see in any textbook. The real fun is in the more difficult problems that seem completely unrelated to what you just learned. It's possible to solve these problems with only the material presented, but finding the solution requires a thorough understanding (and some creative thinking) in how the math actually works.

This is all fine and well. But we shouldn't completely downplay the importance of some memorization in a pure math curriculum. All the peers I knew in math spent a good portion of their time repeating proof after proof exercise to drill them in for their exams. So did I, in my early math classes that were of the Apostol/Spivak flavor.

Much like in physics, I don't think one starts an exercise from bare-bones first principles unless that is what was asked of you, you have to have seen everything at least once.

 

[micromass]

My view is that remembering (though not necessarily by rote) things is a crucially necessary but insufficient condition to success in science or mathematics.

Quite right!

While math is a very beautiful structure that you could possibly build up from scratch every time, nobody actually has the time for this. For example, if I have a differential equation to solve and have to show some integral converges, I can of course build up the entire integration theory from scratch. But that won't get me very far if I need to do it every time. It's better to have memorized the properties (and of course to understand why they're true!) and then just apply those properties to show that the integral converges.

A lot of people always say there is little memorization involved in mathematics, but I really disagree. You can't do advanced mathematics like functional analysis or differential geometry without remembering all the stuff that came before. In my students, I see that the person who still remembers a lot of the previous material has a real and definite advantage over the rest.

Sure, it's not like in medicine where you need to memorize hundreds of bones and muscles. The memorization in mathematics is "easier" because of all the innerconnections and visualizations. But there is still a lot of memorization involved.

 

[jtbell]

Sure, it's not like in medicine where you need to memorize hundreds of bones and muscles. The memorization in mathematics is "easier" because of all the innerconnections and visualizations.

Hey, those bones are connected, aren't they?

http://kids.niehs.nih.gov/games/songs/childrens/bonesmp3.htm

 

[Borek]

Hey, those bones are connected, aren't they?

http://kids.niehs.nih.gov/games/songs/childrens/bonesmp3.htm

My favorite version: https://www.youtube.com/watch?v=X4aa-nn3IxI&feature=player_detailpage#t=60

 

[homeomorphic]

While math is a very beautiful structure that you could possibly build up from scratch every time, nobody actually has the time for this. For example, if I have a differential equation to solve and have to show some integral converges, I can of course build up the entire integration theory from scratch. But that won't get me very far if I need to do it every time. It's better to have memorized the properties (and of course to understand why they're true!) and then just apply those properties to show that the integral converges.

Actually, people as famous as Atiyah and Feynman have said things to the effect that they would just re-derive things, rather than remember them. Atiyah said you don't have to have a great memory to be a mathematician because you can just derive what you forget. But then, you still have to remember how to derive it, of course. Feynman said he remembered the results, but would re-derive the explanations behind them, if needed. Personally, I like to go more towards the approach of trying to remember a lot of stuff--particularly, I like to remember intuitive reasons why things are true.

A lot of people always say there is little memorization involved in mathematics, but I really disagree.

I think this is partly the semantics of whether you want to call it memorization or understanding. I think they mean you don't memorize by rote (at least, not very much).

Sure, it's not like in medicine where you need to memorize hundreds of bones and muscles.

I'd rather learn anatomy like an artist than a doctor (though I'm not entirely sure what strategies they actually use--I would guess, at least a handful of them are artists as well and take advantage of it). Of course, a doctor needs much more detailed knowledge, but I suspect learning it like a medical illustrator would be the way to go. Somehow, that seems a little closer to learning math.

 

[Borek]

OTOH:

a field is a nonzero commutative ring that contains a multiplicative inverse for every nonzero element, or equivalently a ring whose nonzero elements form an abelian group under multiplication.

Plenty of definitions that you have to strictly memorize before you can understand what it says.

 

[homeomorphic]

Plenty of definitions that you have to strictly memorize before you can understand what it says.

Well, that's why I'm not a big fan of straight axiomatic, theorem-definition-proof or Bourbaki-style approaches. For the example of a field, it's better to keep in mind examples of real or complex numbers, so that you at least have a hook for the more general concept. Even with examples in mind, I'm really not a big fan of pulling the definition of a field out of a hat, as it were. For something a little more my style, you could look at a book like Ian Stewart's about Galois theory, where fields are just subfields of the complex numbers and then you abstract it from there later on.

If only more people would be more questioning of the poor motivation they are often presented with, mathematics would not be in such a pedagogical mess. And things would be generally easier to remember and make more sense, and less rote memorization would be necessary.

That being said, it's not the end of the world if you sometimes temporarily have to memorize a few things by rote. It can occasionally be a useful crutch, perhaps, though I don't derive any satisfaction from it (and the whole reason I do math is to get satisfaction).

 

[AlephZero]

Actually, people as famous as Atiyah and Feynman have said things to the effect that they would just re-derive things, rather than remember them.

That's fine, except for one thing: most people aren't as smart as Atiyah and Feynman. You need a strategy that works for you, not for somebody else.

 

[Vanadium 50]

I think you need to think "remember" instead of "memorize".

 

[mpresic]

Interesting that you say it takes you two weeks to come up to speed after a summer off.
I was always good at math. But let me start this way.
I once took Karate, if I quit for 3-4 months would I need (at least) two weeks to regain my form.
I once started guitar, if I quit for 3-4 months, could I get as good as I ever was in two weeks.
I play competitive chess. Am I as good when I do not play in tournaments for 3-4 month. Does it take at least two weeks to regain my playing strength.
I can give more examples from soccer, step aerobics, jogging, etc.

You put the answer yourself. You go out of practice.

Many "good" mathematicians and science professionals even as "kids" studied math all year round, although not always in a structured classroom way. For example growing up, along with some friends our parents took us to the library once in a while to borrow some books. (Usually some were math books)

If you can, get a slide rule and learn how to use it. Computers can do the calculations faster, and you might never use it in real life but who knows, you might even find it cool. I think math education all over would improve if there were slide rules sold in drug stores the way they were in the 1960's.

Maybe some of your teachers could give advise over the summer on how not to lose your edge.

If you are worried that it is hard to find the time, do not worry. As you get better and better, you will do more, and it may even take more time, but by then it won't seem as long because you will start to enjoy it. (like the theory of relativity (lol)).
Suppose you work 2 hours a day on math during the school year, and you are two weeks behind. This is 14 hours of math. Sounds like a lot. Now Suppose a 7 week summer. That's 2 hours a week of math. Average 30 minutes a day is plenty. Of course, you will be even better off with more.

 

[pandaexpress]

All of you previous knowledge, experience, and skill will be important when you take your college level courses. If you don't know the basics of arithmetic and algebra, you will continue to have problems not only in all your math courses, but any course which has math in it.

While memorization is derided in some quarters, knowing basic facts like the multiplication tables or how to carry out an algorithm like long division is always useful. If you have to stop and derive everything you do from first principles all the time, you'll never be able to progress.

Memorizing everything is not required as long as you can remember where to find the information you need. That's why it's important to have handy good reference works or to know how to search for information online or in a library.

Thanks everyone for the responses! Slow catching up.

I was actually going to ask what you meant by "first principles" here, because I've seen that phrase used in some threads before?

My memory is OK in math, but as I said it's really bad if I don't do anything all summer. I'm just amazed at people who seem to just literally remember all the various formulas and rules seemingly without having studied it in months.

Is that just natural talent? Is it rare?

Are there ever instances where even college students or math majors/engineers, etc. have to like pull out a formula or rule sheet to know how to solve a problem? Or is it all just imprinted in these guys' brains?

 

[micromass]

Thanks everyone for the responses! Slow catching up.

I was actually going to ask what you meant by "first principles" here, because I've seen that phrase used in some threads before?

Depends on the context. The true first principles in math are the ZFC axioms of set theory. Almost all of math can be derived from those axioms.

But it could also mean the field axioms of real numbers. It is safe to say that almost all of high school math can be deduced from these.

My memory is OK in math, but as I said it's really bad if I don't do anything all summer. I'm just amazed at people who seem to just literally remember all the various formulas and rules seemingly without having studied it in months.

Is that just natural talent? Is it rare?

Are there ever instances where even college students or math majors/engineers, etc. have to like pull out a formula or rule sheet to know how to solve a problem? Or is it all just imprinted in these guys' brains?

It's part natural talent and practice. Natural talent because some people remember and memorize easier than others. Some people have to see a math formula once and immediately know it.

Practice is more important than natural talent. The more you use a formula, the better you will remember it. I am ashamed to say that I really can't remember the multivariable chain rule anymore because I haven't used it in years. However, if I were to need it, I would just do a lot of exercises, and I'll know it again.

So "do mathematicians need formula sheets"? I'd say yes, not everybody can remember all formulas. Some formulas will be known very well (these will be the formulas you use all the time), other formulas you just know that they exist but you don't know the specifics.

 

[pandaexpress]

Depends on the context. The true first principles in math are the ZFC axioms of set theory. Almost all of math can be derived from those axioms.

But it could also mean the field axioms of real numbers. It is safe to say that almost all of high school math can be deduced from these.



It's part natural talent and practice. Natural talent because some people remember and memorize easier than others. Some people have to see a math formula once and immediately know it.

Practice is more important than natural talent. The more you use a formula, the better you will remember it. I am ashamed to say that I really can't remember the multivariable chain rule anymore because I haven't used it in years. However, if I were to need it, I would just do a lot of exercises, and I'll know it again.

So "do mathematicians need formula sheets"? I'd say yes, not everybody can remember all formulas. Some formulas will be known very well (these will be the formulas you use all the time), other formulas you just know that they exist but you don't know the specifics.

You probably didn't realize I have no idea what you are talking about. Hahaha. :yuck: I have no idea what a ZFC or an axiom is. But I'm sure I'll find out one day. :thumbs:

That's at least slightly comforting to know you guys have to pull a formula sheet every now and then. The only thing for me is that I don't use math everyday. I do use English, so my vocab is constantly in use and accumulating. But other than arithmetic, I'd never use stuff like a quadratic equation. That's why I feel I lose my skills every summer. I'll try to practice a bit this summer to see just how much difference it would make.

 

[micromass]

You probably didn't realize I have no idea what you are talking about. Hahaha. :yuck: I have no idea what a ZFC or an axiom is. But I'm sure I'll find out one day. :thumbs:

An axiom is a statement which is taken to be true and which does not need to be proven. All of math is then derived from that axiom(s). Which axioms we accept depends a lot on context and depends a lot on observations in nature.

For example, one of Euclid's axioms was "through every two distinct points, there is a unique line".
Another axiom (and part of the field axioms) is "For every two real numbers $x$ and $y$ holds $x+y=y+x$.

Both of the previous statements are evidently true if we look around us. It's something we want to be true! Other statements such as the quadratic formula can then be proven by only making use of the axioms.

In the past, the Greeks thought that axioms were true statements about the real world. Nowadays, we don't think this anymore. In fact, the ancient axioms of geometry that Euclid worked with are false in our universe! Nowadays, we think of axioms of setting the rules of some universe which might not be ours. In this sense, mathematics can describe many possible and different universes.

It's a lot like writing a fantasy novel. When you start your novel, you make certain assumptions about your world. For example, "there is a magical ring which can make people invisible and which is actually property of a dark wizard named Sauron". This can be seen as an axiom. The rest of the story must then satisfy this axiom.

So there is a huge amount of creativity involved in mathematics. You have the awesome liberty of choosing which statements you accept to be true and which you do not accept to be true. If you wanted $1+1=0$ to be true, then you can!

The difference between fantasy and math however, is that in fantasy books, you need to keep on inventing new stuff. In math, this is not so. Once you stated the basic rules for the mathematical world, the world starts living on its own. So while the world itself is invented, once that invention stops, you can actually visit the world you just created and look around at its magnificent properties.

That's at least slightly comforting to know you guys have to pull a formula sheet every now and then. The only thing for me is that I don't use math everyday. I do use English, so my vocab is constantly in use and accumulating. But other than arithmetic, I'd never use stuff like a quadratic equation. That's why I feel I lose my skills every summer. I'll try to practice a bit this summer to see just how much difference it would make.

Yes, if you don't do math everyday, then you are bound to forget a lot of it. This is true for most people. Some people will forget after an hour, other people might remember stuff for months.

This might be interesting to you:

 

[homeomorphic]

I'm just amazed at people who seem to just literally remember all the various formulas and rules seemingly without having studied it in months.

I can remember Maxwell's equations off the top of my head, even though it was 10 years ago that I learned them. There are several reasons I remember them. Firstly, I understand their meaning, not just the symbols, which is very visual. Visualization has been proven to enhance recall. Secondly, when I was learning them, I used to think about their meaning every day, so repetition and deep understanding. And I think of them from time to time and on a few occasions I have done something with them since then. Also, Maxwell's equations are not just any equations. They have a nice sort of symmetry to them and are fairly simple once you understand them deeply. They also are connected to various other pieces of knowledge that I have, like special relativity, electrical circuits, multi-variable calculus and so on. Things are more easily remember as part of a big web of knowledge than as isolated facts.

Here's a link I like to give to demonstrate that there are powerful secrets hidden in everyone's mind that they may not be aware of:

http://www.ted.com/talks/joshua_foer_feats_of_memory_anyone_can_do

I'm not sure of how useful this sort of memory stuff is in math, though because in math, you have to have understanding rather than just memory.

 

[pandaexpress]

An axiom is a statement which is taken to be true and which does not need to be proven. All of math is then derived from that axiom(s). Which axioms we accept depends a lot on context and depends a lot on observations in nature.

For example, one of Euclid's axioms was "through every two distinct points, there is a unique line".
Another axiom (and part of the field axioms) is "For every two real numbers $x$ and $y$ holds $x+y=y+x$.

Both of the previous statements are evidently true if we look around us. It's something we want to be true! Other statements such as the quadratic formula can then be proven by only making use of the axioms.

Hey - I appreciate your effort man! To be honest, this might currently be a bit over my head. I think I get the gist of what you might be saying, but of course it's hard to tell. You're saying that there are truths of math that other truths can be based on?

What I didn't get was what you meant by saying you can prove the quadratic formula with axioms. We did learn the quadratic formula in my classes thus far, but nothing about axioms.

In the past, the Greeks thought that axioms were true statements about the real world. Nowadays, we don't think this anymore. In fact, the ancient axioms of geometry that Euclid worked with are false in our universe! Nowadays, we think of axioms of setting the rules of some universe which might not be ours. In this sense, mathematics can describe many possible and different universes.

It's a lot like writing a fantasy novel. When you start your novel, you make certain assumptions about your world. For example, "there is a magical ring which can make people invisible and which is actually property of a dark wizard named Sauron". This can be seen as an axiom. The rest of the story must then satisfy this axiom.

So there is a huge amount of creativity involved in mathematics. You have the awesome liberty of choosing which statements you accept to be true and which you do not accept to be true. If you wanted $1+1=0$ to be true, then you can!

The difference between fantasy and math however, is that in fantasy books, you need to keep on inventing new stuff. In math, this is not so. Once you stated the basic rules for the mathematical world, the world starts living on its own. So while the world itself is invented, once that invention stops, you can actually visit the world you just created and look around at its magnificent properties.

OK, this is actually very interesting to me! I'm a HUGE LOTR fan!

Having said that, I honestly don't think I understand what you mean. You were saying that there are axioms that are true and that we base other math on earlier, but then you say later here that these don't have to be true? ...and that we can invent our own truths like 1 + 1 = 0 Did I hear that correctly? You need to come talk to my math teacher! hahaha

Thanks for the memorization info. though. Very intriguing!

 

[Borek]

Having said that, I honestly don't think I understand what you mean. You were saying that there are axioms that are true and that we base other math on earlier, but then you say later here that these don't have to be true? ...and that we can invent our own truths like 1 + 1 = 0 Did I hear that correctly? You need to come talk to my math teacher! hahaha

You can think about it as a "parallel maths" - that is, you can create one "math" using one set of axioms (including 1+1=2) and another "math" using different set of axioms (including 1+1=0).

ATM you are learning one of these "maths".

(My wording is attrocious here, as these "maths" are all part of the math; perhaps someone will be able to use better - or even correct - name).

 

[Jorriss]

Having said that, I honestly don't think I understand what you mean. You were saying that there are axioms that are true and that we base other math on earlier, but then you say later here that these don't have to be true? ...and that we can invent our own truths like 1 + 1 = 0 Did I hear that correctly? You need to come talk to my math teacher! hahaha

You use this type of math quite often. Think about addition on a clock. On a clock 6 + 7 is not 13, it is 1. And 6 + 12 does not equal 18, it is 6 (assuming you use a 12 hour system).

 

[micromass]

Hey - I appreciate your effort man! To be honest, this might currently be a bit over my head. I think I get the gist of what you might be saying, but of course it's hard to tell. You're saying that there are truths of math that other truths can be based on?

What I didn't get was what you meant by saying you can prove the quadratic formula with axioms. We did learn the quadratic formula in my classes thus far, but nothing about axioms.




OK, this is actually very interesting to me! I'm a HUGE LOTR fan!

Having said that, I honestly don't think I understand what you mean. You were saying that there are axioms that are true and that we base other math on earlier, but then you say later here that these don't have to be true? ...and that we can invent our own truths like 1 + 1 = 0 Did I hear that correctly? You need to come talk to my math teacher! hahaha

Thanks for the memorization info. though. Very intriguing!

Yes, you heard that correctly. Axioms are true because we take them to be true. So in the rest of what we do, the axioms are true. However, they might not be true in real life!

Since we both like Lord of the Rings, let me give another example. For example, we might ask ourselves if Tom Bombadil could keep the ring safe and away from Sauron. An answer like "none of this is real" clearly is lame and insufficient here. Indeed, for the sake of argument, we have assumed that everything in lord of the rings is real.

In the same way, when we propose $6+6=0$, we assume for the sake of argument that this is true, and then we see what else we can deduce from this.

Of course, if the axioms aren't applicable to real life, then none of the deductions will be either. However, if the axioms happen to be applicable to real life, then so will the consequences. Clearly, something like the Pythagorean theorem of the quadratic theorem is something applicable to real life. This is because we use it in physics and engineering all the time and it yields good results.

Something like $6+6=0$ is not applicable to real life because it's clearly not true. Or is it? It is definitely not the usual arithmetic we work with in our daily life. But this is an arithmetic that we use in reading the clock. Indeed, if it is now 6 hours, then 6 hours later it will be 0 hours. And if it is now 7 hours, then 11 hours later it will be 6 hours. So 7+11=6. So something like 6+6=0 isn't nonsense at all, it is actually useful. It's just that its uses are clearly different from what we typically use arithmetic for (like counting money).

Then again, there are some weird versions of arithmetic which aren't useful in real life at all. Still, they make up satisfactory theories which are mathematically acceptable.

So math really gives a many different theories (all based on different axioms). But only one (or a few) will be applicable to real life. In the same sense that if I give you three books, namely "The Lord of the rings", "The Furies of Calderon" and "World History of ancient times". All three are fascinating to read, but only one will truly be about our world. The rest are about worlds which are internally consistent, but are not real.

Which math theory is real (and which axioms are real) is something we must find out through physical experimentation and common sense (both of which can be deceiving).

 

[jbunniii]

You were saying that there are axioms that are true and that we base other math on earlier, but then you say later here that these don't have to be true? ...and that we can invent our own truths like 1 + 1 = 0

Indeed, this is how your computer adds binary digits!

0 + 0 = 0
0 + 1 = 1
1 + 0 = 1
1 + 1 = 0

This is also known as the "exclusive or" operation. Of course when multiple binary digits are used to represent a number, the computer will "carry the 1" if there is enough room to do so, resulting in 1 + 1 = 10 (the binary representation for 2), instead of 0.

 

[homeomorphic]

We did learn the quadratic formula in my classes thus far, but nothing about axioms.

Probably there was SOMETHING about axioms, but you didn't know it. You may have heard of something like the distributive property, associativity, commutativity, additive inverses (-5 is the additive inverse of 5), and multiplicative inverses (1/4 is the multiplicative inverse of 4). Those are examples of axioms, although you can actually deduce them from more basic axioms and definitions. Axioms are basically just starting assumptions. So, you can start with assumptions like associativity and call those axioms, or maybe you think there should be even more basic principles that those are based on, so you start with simpler assumptions and see if you can prove associativity and all that from those simpler axioms. In an elementary algebra class, it's not really foundational math, so you you'd start with things like associativity and so on.

The proof of the quadratic formula involves a calculation something like this:

x^2+2bx
= x^2+2bx + b^2 - b^2
= (x + b)^2 -b^2

Which relies on that sort of stuff, plus some idea of the number 2 and how it behaves. You also end up taking a square root a couple steps later, the existence of which would be a bit annoying to prove (not to mention it would normally require another somewhat complicated axiom) and out of the scope of your algebra class, so you definitely wouldn't derive the whole thing carefully from scratch at that level. But that's what's involved in it.

There's a more sophisticated way of looking at it, due to Lagrange, which, in essence, says you can try to solve polynomial equations by seeking symmetry. I find that a little more poetic than the calculation above ("completing the square"), but it's also quite a bit deeper and harder to actually understand.

 

[MidgetDwarf]

Read math book and become pro active having a paper/pen ready. It is rather sad when I see classmates with books they have used for a full 3 months with no writing.

Question the material being read. Think of mathematics as reading Sir Bacon, " 4 idols." You have to ask question and write in the margins to understand.

 

[QuantumCurt]

It really is quite cumulative. Generally speaking, you're never going to stop using algebra in a math class. People that struggle in calculus are often struggling with the relevant algebra more than they're struggling with the calculus itself.

Some people are of the opinion that math has little or nothing to do with memorization. I can appreciate this sentiment. I've always found that understanding the process of arriving at a conclusion is a much better method of doing math than simply jumping to the conclusion. Clearly though, memorization is a very large part of math. It would be silly for a math, physics, or other math-heavy major to not take the time to memorize things like the multiplication tables, addition/multiplication algorithms, basic factoring formulas, the quadratic formula, and tricks for finding integrals or derivatives more quickly. Simple things like the shortcuts for factoring expressions like $x^2-a^2$ can and should be memorized. It's silly to work through something like that formally when you KNOW that the factors will always be of the form $(x-a)(x+a)$. The same can be said of differential equations. My physics professor last semester gave me some valuable advice - "The best way to solve a differential equation is to skip the work and just write down the solution." This of course is a reference to the many different common types of differential equations that have standard solutions into which one can simply plug their constant values or otherwise manipulate as needed. My differential equations professor last semester didn't seem to be of the same opinion. He liked to see the work. ;)

edit - My physics professor did stress (after that fact) that it's important to see where these solutions come from and not just take them for granted. One should work through the differential equations for circuits, oscillations, waves, etc. when one first encounters them. After that though? You already did your work the first time you solved it. Now those solutions are part of your toolkit.