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## Different Ways to Conceptualize Math?166

rook a asks: "I've always been an avid reader but my math skills were poor, and TV had taught me that math was difficult. I knew only the concepts of the basic operations. From seventh grade through high school, I did only what was needed to get by and so my math skills remained below par. Now, as a freshman pre-cal student, I am struggling. I believe that I have a flaw in the basic way I think about numbers. I can think logically, but it does not carry over to math. I read somewhere that Feynman gave a lecture on arithmetic but I could not find it. I believe that different people have different thought structures for the same ideas. Has there been any research or books on the difference between how a mathematician, or a Richard Feynman, thinks about math and the way that the average person thinks about math? Or, did any of you initially find math difficult in college but go on to higher maths? If so what changed for you?"
"I wanted to be an EE and want very much to be good at math but if my ability does not increase I will not be able to. I am willing to do anything to increase my skill. I hate rote and do not want to be merely 'good' at math, I want to speak it. If math is a mindset then it's one I want to be part of.

This is similar to another question, however I found several interesting books but no comments toward learning a more efficient way to think."
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## Different Ways to Conceptualize Math?

• #### Math is not difficult (Score:2, Interesting)

Math is unique in that there are many levels of abstraction, and you can't understand the higher levels without first acquiring a pretty good understanding of the lower levels. At each level, a certain amount of study and memorization is required, just as in any academic discipline.

However, the idea that one needs some special cognitive ability or conceptual skills is a complete myth. Once you have absorbed the concepts and vocabulary from one level, moving to the next level should require no more brain p

• #### Re: (Score:3, Insightful)

True to an extent, but organizing your brain so that you can call up the knowledge necessary to solve a particular problem is something that is very difficult for some people. This is mostly a problem on math tests, where not only do you need to know what to do, you need to be able to follow the steps quickly enough to complete the test on time. It's just something that some people are not naturally very good at.
• #### Different people learn differently. (Score:5, Interesting)

on Thursday October 05, 2006 @07:49PM (#16330113)
Sure there are levels of abstraction etc, but I think you got lost on the "cognitive ability" bit.

Many people don't "get it" with math because they are not cognitively wired to absorb stuff the way it is presented. Yet, if something is presented a bit differently they might then "get it" and be able to move on to the next step.

I was very fortunate to have an excellent math teacher. This teacher was able to teach kids who had previously not done well at math and get them scoring As. I think his secret was this: He used many different wasy to explain things to the kids. Some would get it immediately. Some would only get it when he explained things differently. Quite often he'd explain things in thee or four different ways. Now sometimes he'd be stumped and could not get an idea across.... So here's where he was different from other math teachers..... He'd get one of the kids that "got it" to sit and explain to the kid that didn't "get it", and he'd watch and take notes. Eventually someone would manage to get through. Better still, the teacher would then have a few more ways of explaining things to future classes.

• #### I as well... (Score:1)

I too had troubles with math. In fact, I failed my Algebra 1 class the first time I took it. But, the second time I took the class, I had a different teacher. Thanks to her, I gained a fondness for math (of which I had outright despised for years. Her main reason for getting me interested in math was for very similar reasons to yours. She let the students teach the other students whenever she was having trouble getting a concept across. Plus, she was only pro-active in our studies. She wouldn't just let us
• #### Re: (Score:2)

An interesting example is a game by Nintendo called electroplankton, it basically allows you to create music using diffrent systems. Peraps you simply need to find the metaphor that helps you understand math better.

This affects everyone, most people really need a graphical representation to understand say calculus.

Anyway find your own metaphor (pies, lines, etc) but that won't help with Calc.
• #### McMath jobs? (Score:2)

no more brain power than, say, learning to follow a recipe which is why a burger flippers should have a PhD!
• #### Re: (Score:2)

Burger flippers, AFAIK, don't do much from a recipe. I thought they just flipped burgers.
• #### Excuse me (Score:2)

That's the recipe!

• #### Re: (Score:3, Interesting)

It doesn't have to be difficult. I think the reason it is or isn't for most people is emotional, or psychological. I for one loved math as a student. It was the only subject where you were either right, or wrong. I could walk into an exam, write it, verify my answers, and be sure of how I did. The teacher couldn't slant, because if there was a mistake in the marking, it could be proven a mistake.

On the other hand, there's a friend of mine who hates math. He's no good at it, and he can't learn it because whe
• #### Re:Math is not difficult ... err, yes it is (Score:4, Insightful)

on Thursday October 05, 2006 @08:14PM (#16330385) Homepage Journal
Right, background: excelled at mathematics in primary school (up to age 11) but got bored as I'd finished (the concepts of) all the course texts and didn't like doing actual work. Was top set in secondary (up to 16) but never really shone until that final year. Did double maths A-level (maths and further maths) and went on to do Theoretical Physics and maths degree. Some of it came easily to me - complex numbers, fractal geometry, differential equations; some not so easy - quantum field theory, fluid dynamics.

I've never really considered that I could have a different approach to numbers that would make maths easier. Maths and Physics I loved at school as I have a very poor memory and could always go back to basic assumptions and build from there. Later on (eg fluid dynamics) I had to try and really on some rote learning as the stuff was too abstract for me grasp.

I don't really have a visual grasp of concepts - I've often tried to envision a four dimensional hypersphere or a fractional dimension without much success. When I turn my mind to dimensions folding in on themselves the images are often just (barely) 3D. But somehow I grasp many of these concepts ... I guess it's that step of going from "this is an electron, a solid minute particle orbitting an atomic centre" to "this is an electron a four dimensional probability wave".

>>> "moving to the next level should require no more brain power than, say, learning to follow a recipe in a cook book or installing a plumbing fixture"

Hmm, that's a very _now_ statement. I'm sure that if all you're trying to do is pass an exam that's true. If you're trying to understand and develop, indeed push the boundaries of, a concept then I don't think that's true. Have you ever just picked up a recipe book (for soufflé say) and just tried to follow the recipe. Sure you know what the words are and carry out the action, but you can just lack the knack to perform it well. It's a terrible analogy but I think as with musicality, a sportsmans eye for the ball, an artists abstraction of images to capture their essence, there's a mathematicians feel for the equations and their beauty or otherwise.

What was the question again ... yeah I was suppose to be working but it's one in the morning, so what they hey ...
• #### Problem in communication (Score:2)

There are several elements to this

1) Education is inherently a problem in communication
2) Some Data is NOT intuitive, and needs to be broken down into appropriate byte size pieces. Different humans have different byte sizes.

Add to this the concept of missing fundamentals. Sometimes the data you need is included in earlier study/work that was not completely of fully digested. You need to make sure you cover this.

Sometimes the data is new, and you will need to work it out with sufficient practice so tha
• #### Re: (Score:2)

Starting in 7th grade I was placed in a course track called Unified Modern Mathematics. We were studying set theory, probability and statistics. It was fun, but pretty abstract for a 12 year old. The next year it got more abstract and I bailed out for a standard Algebra class. In Algebra I scored 100% on each and every exam, up to and including the NY Regents exam. Algebra was pure common sense to me.

The next year was Geometry. I failed it and had to retake it. I have the spatial cognition of a rock. Using
• #### Some answers that worked for me (Score:5, Insightful)

<seebert42@gmail.com> on Thursday October 05, 2006 @07:30PM (#16329873) Homepage Journal
Numerical Methods. It's usually taught as an advanced, post calculus course for computer science majors. But it gives alternative methods for all sorts of things from trigonometry to calculus, and it does so in methods that can be programmed in Basic or even Assembly (you do know, don't you, that at a very basic level the most complex math any computer can do is 1 And 1 is 1, right? And that all the other math computers do is built up from simple AND gates?)

In addition to this, I also recommend Godel, Escher, and Bach: The Eternal Golden Braid for a totally different way to think about mathematics, philosophy, and religion.
• #### Re: (Score:3, Informative)

Sorry to troll, but it's a NAND gate, not an AND
• #### Re: (Score:2)

Some good backup for this and demonstrations of how to create any other logic gate from NANDs at http://en.wikipedia.org/wiki/NAND_logic [wikipedia.org]
• #### Re: (Score:1)

it's not trolling, he was making irrelevant statement in order to impress and you called him on an error which showed just how much he really knows.
• #### Re: (Score:2)

Yes.

But _you_ are trolling.

• #### Re: (Score:2)

No troll at all- I was dumbing down the information to the level I thought the article writer was at. A NAND is an AND with an inverter- and many early computers used AND instead of NAND logic. Many more modern ones use NOR gates instead.
• #### Memory, time and visualization (Score:3, Informative)

Have you already tried to check out which thinking methods fit with you?

For example, do you already have the habit of trying to find rational patterns, and enjoy visualizing them? If not, you could try that out and see if it fits with you. Visualization may be a two-edged weapon when it comes to math. Some people (including me) do it a lot and find it helpful. (But others handle math topics that may defy visualization, and claim that the visual-modeling habit ends up a hindrance.) Maybe you could find
• #### right... (Score:2, Insightful)

>>TV had taught me that math was difficult.

Go watch PBS you victim of TV
• #### Info on Richard Phillips Feynman (Score:2, Informative)

A summary of Richard Phillips Feynman [st-and.ac.uk]

Amazon search [amazon.com] for Richard Feynman

--
Mod +1 informative -5 Karma Slut
• #### age (Score:1)

learning a more efficient way to think is better, as any other learning, at a young age. my expience tells that not efficient, but rather practical way of thinking is more widespread in adults. I mean, if I need some specific things done, I know how, that's not always efficient though.
• #### Many (Score:2, Funny)

"One, Two, Many" works for me. Or is it "one too many"...or "one to many"???
• #### This book will change your life (Score:3, Interesting)

on Thursday October 05, 2006 @07:46PM (#16330073) Homepage
Look up "Mathematics: From the birth of numbers" by Jan Gullberg. It should do the trick. Incredible book.
• #### Re: (Score:2)

I concur. This book started my road to "Mathematics recovery".

I had scraped though HS with enough maths to get to do a BSc. Unfortunately, that weakness precluded me from doing well in the subjects I wanted to do. Now, over 2 decades later, I am doing a second attempt at a BSc (part-time) and am doing 3rd year maths. If someone told me five years ago I would be fronting up at a PDE course. I would have not thought it possible.

I am not saying that the book fixied me though

I still have the problem, li

• #### Some thoughts (Score:5, Informative)

by Anonymous Coward on Thursday October 05, 2006 @07:52PM (#16330149)
First of all, do you know your learning style? Auditory, visual, kinesthetic? Your writing suggests visual. Did you find geometry to be easy, or difficult? If the answer is easy, there's part of your answer - relate calculus and linear algebra to geometric problems. Hint: most EE math can be reasoned about algebraically (equations) or geometrically (pictures).

See if you can get your hands on a demo of Maple. There's a student version available, I don't know if it's crippled, but I know that it's a disgustingly great deal. It got me through EE school. Mathematica has better marketing, but I always found it to be a horrible program (at least, its syntax requires you not know anything about programming languages). Maple has some great somewhat-interactive graphing modes too. You can't / shouldn't use it for the math courses, but for EE courses, you'll need a really good math program to help you out.

Also see if you can get your hands on a HP48GX calculator. Real engineers use old-school HP calculators. Posers use TI. You'll thank me come EE exam time. I'm not convinced that the currently selling HP calculators fall in the "real calculator" camp, but they might be okay. You want RPN. Trust me, if you're an engineer, RPN is your friend. It also tends to keep people from swiping your precious calculator ;)

See if any of the professors in the EE department teach math classes; usually there'll be a few people who have a foot in each department. Make friends and see if they'll help you out during their office hours. In general, I have found that math professors can't teach math worth anything. Or at least not to engineers. It's just a different mindset / world view. And the result is that they're teaching math the way they think of math, and you're just going W-T-F?! The EE professors can teach it with an engineering spin, and they have the very distinct advantage of being able to map math problems to the real world EE problems you need that math to solve. The worst math professor phrase is "suppose you want to..." - well, suppose that I don't, ya damn hippie! EE profs can put the horse back in front of the cart and tell you WHY you NEED to do this or that math, and that insight alone makes it much easier to learn.

In general, I must emphasize that EE is a math intensive major, and it gets very very much uglier than basic calculus. If you truly aren't good at math and you aren't willing to put yourself through dramatic pain and sufferring to learn it anyway, change majors now. Really, seriously. If you're going to hit your limit and change majors, you're better off doing it while you're not as far along and don't have as much work to throw away. If you decide to stick with it, good for you, just understand that it's going to get *a*lot*worse*.
• #### Re: (Score:2)

Any 'math professor' that says 'suppose you want to...' to someone needs to turn in their math badge. A real mathematician knows this is all a game, and you are just playing with numbers. Anyone who thinks there is a practical reason for this stuff is an outsider. Math is to show how clever and interesting logic can be. That's the point. If they say that line, either they are reaching outside what they know, or they aren't math people at heart.

• #### Re: (Score:2)

RPN is your friend. It also tends to keep people from swiping your precious calculator.

Funny, but true. Flash (horror of horrors) might be a good application to demonstrate how replacing variables in equations can affect the graphical output and get some interest going in learners' minds. Maybe this already exists for all I know.
• #### Re: (Score:3, Insightful)

See if you can get your hands on a demo of Maple. There's a student version available, I don't know if it's crippled, but I know that it's a disgustingly great deal.

Absolutely. Maple is your friend. The student version is every bit as good as the full version (it's the same program), and it's \$100. Not bad for a CAS that does just about everything.

Mathematica has better marketing, but I always found it to be a horrible program (at least, its syntax requires you not know anything about programming languages)

• #### Re: (Score:2)

Most engineers I know use neither. Numerical computation can be better accomplished using purpose-driven software. Many EEs would be absolutely lost without Matlab, a SPICE simulator, and countless other software packages.

Actually, as an EE, I use all of it. I use Matlab for large simulations. I use spice for simple stuff and low frequency AC work, and I use "Virtual TI" for back of the envelope calculations.

The CAS inside the TI-89 and 92 is actually pretty powerful and quite useful.

In short, I us
• #### Re: (Score:2)

My thoughts exactly.

Determine HOW you think before you can understand how you learn. Many people are visually oriented (I am). Geometry, Trigonometry and Newtonian Physics are very easy for visual people because you can SEE how the math affects the outcome. You could possibly touch many of the problems you worked on. Algebra is almost completely conceptual and non-tangible. Calculus has some visual and physical aspects, but it typically is working with algebra within a geometric problem.

I'd expand on thi

• #### It's not math anymore. (Score:5, Insightful)

on Thursday October 05, 2006 @07:56PM (#16330189) Homepage Journal
It took me a long time to figure this out.

The math you learned in primary and secondary school, where it's numbers that have distinct values, is no longer really applicable. Don't try to "grasp" the concepts. It's not a small step like algebra was, it's a quantum leap. You are working with a fundamentally different question, which is the question of infinity. You need to learn new rules. Don't try to use the rules you learned with numbers; they don't apply. Your way of thinking needs to be fundamentally altered.

Where I always screwed up in learning higher mathematics was in trying to somehow relate it back to arithmetic. That doesn't work. If you keep trying to connect those two dots, you will be perpetually frustrated. Just learn it for what it is. It doesn't matter if you understand it any more than it mattered if you knew why 2 + 3 was 5 in elementary school. Trust me: you will be able to understand it later, once you know a certain critical mass of concepts, but you need to have enough dots before you can connect them into anything remotely like a picture.

This will take roughly your entire pre-calculus class and probably half of your first actual calculus class. You will be confused. It will not make sense. You will feel like you are learning nothing. The answers you give on exams will feel memorised and formulaic, almost like you are cheating.

But eventually, you will have that "Aha!" moment where you really do finally understand what a definite integral is. You just have to trust that the material you're learning is going to get you there, even if you don't know how.

Likewise, it's not really true that higher mathematics doesn't connect back to arithmetic. It just won't connect back for a really long time, and it's not productive to look that far ahead right now.
• #### Re:It's not math anymore. (Score:4, Insightful)

on Thursday October 05, 2006 @08:08PM (#16330315) Journal
Don't try to use the rules you learned with numbers
This is the worst advice ever.

Most of the time when you're doing EE you'll be working with equations in which the variables represent numbers. It's important to bear in mind, that every stage, that these aren't just meaningless symbols. An unknown variable, x, satisfies all the properties that all numbers do. For example xy=yx because 2*3=3*2 and 5*7=7*5, and (-1)*22=22*(-1) and so on. Sure, you can forget about this, and just use the rules of algebra to manipulate these symbols. But as long as you do this you'll have no insight and you'll be like a brute force chess playing machine that has to search out all possible sequences of moves. Keep in mind that these symbols are actually numbers, and all that's happening is that you're doing arithmetic, then you can let your intuition about numbers guide you, even if your equation doesn't even contain any numbers.

eventually, you will have that "Aha!" moment where you really do finally understand what a definite integral is
It's much easier to understand the concept of a definite integral than to memorize and use the rules for manipulating them. Properly explained, the idea is incredibly simple. And once you get the idea, many of the properties will be plainly obvious.

It just won't connect back for a really long time...
Your teachers must have been awful. And despite the fact that I have a PhD in math, you must have had way more stamina than me to learn all of this stuff without connecting it back to arithmetic until much later.
• #### Re: (Score:2)

Do you think it's possible that your brain is slightly different than most peoples', and you might have a natural 'knack' for math that most people don't have?

Conversely, let me ask this: have you encountered a subject in school that was so opaque, arbitrary and ridiculous that you thought that the people involved in it must be fooling a lot of people into thinking that this was a serious academic subject, instead of a bunch of hoakum? That they must just be making it up, because it really didn't make any
• #### Re: (Score:2)

have you encountered a subject in school that was so opaque, arbitrary and ridiculous that you thought that the people involved in it must be fooling a lot of people into thinking that this was a serious academic subject, instead of a bunch of hoakum? That they must just be making it up, because it really didn't make any sense at all?

For me, it would be math. For the first eight yeras in school (including elementry and high), the courses taught that "7-9 is impossible". Then suddenly, they introdced a new

• #### Re: (Score:2)

> For me, it would be math. For the first eight yeras in school (including elementry and high), the courses > taught that "7-9 is impossible". Stupid, stupid curricilum. I learned about numbers by having them on a numbered line; negative numbers were there, right in front of me, from (I think) the second or third week of school.
• #### Re: (Score:2)

Conversely, let me ask this: have you encountered a subject in school that was so opaque, arbitrary and ridiculous that you thought that the people involved in it must be fooling a lot of people into thinking that this was a serious academic subject, instead of a bunch of hoakum? That they must just be making it up, because it really didn't make any sense at all?

High school biology, where they told us "facts" about the structure of microscopic organisms without actually giving us any of the research suppo
• #### Re: (Score:1)

> Most of the time when you're doing EE you'll be
> working with equations in which the variables
> represent numbers.

That's true, but the question is about higher mathematics. Algebra I is not higher mathematics. About 80% of what I do for a living could be done by a four-year CS grad, but I'm far more interested in the 20% that can't... and so is my employer.

> Properly explained, the idea is incredibly simple.

Define "proper". Define "simple". Sure, a definite integral is the area under the curve
• #### No, it is simple (integration by blackboard) (Score:2)

Define "proper". Define "simple". Sure, a definite integral is the area under the curve defined by a function; that's proper. That's simple. And it's completely incomprehensible to the average student new to calculus.

No, integrals are incredibly simple to show to a student. You walk up to the board and draw a function. You then ask the class what an integral is. When they don't know (assuming noone has had AP or failed, and remembers) start shading the area under the curve.

Remember kiddies - math has m

• #### Re: (Score:2)

To a student being introduced to this for the first time, it's still incomprehensible. They'll understand it's the shaded area under the curve, sure. But that doesn't mean anything to them -- it hasn't been tied to something that they're already familiar with in a way that makes it seem /useful/ to want to know the area under a curve.

And not knowing that they'll promptly file it away in the brains in the "useless theoretical handwaving" category.
• #### Re: (Score:2)

So you're claiming that working out the amount of paint required to paint a wall whose height varies is such a bizarre and abstract concept that potential math canidates won't understand it?
• #### Re: (Score:2)

You can't do anything with a PhD in math... except more math

You're trying to set up a barrier between mathematics and the other subjects. I use mathematics every day in my working life. I develop graphics software for movie visual effects. Using mathematics I'm able to write code to solve problems in geometry, physics, optics, image processing and even areas like plant growth and crowd simulation. I have previously used my mathematics knowledge in game development and drug development (in computational c

• #### Re: (Score:2)

O cruel, needless misunderstanding! O stubborn, self-willed exile from the loving breast! Two gin-scented tears trickled down the sides of his nose. But it was all right, everything was all right, the struggle was finished. He had won the victory over himself. He loved Differential Geometry.
• #### Re: (Score:3, Interesting)

Wow, there are just so many ill-informed and probably very unwise pieces of advice here.

Let me make this clear to start with - I've worked teaching Maths to people struggling with it for many many years, starting with private tuition, through the Education Department and as a University Tutor/Lecturer. I've seen teaching of maths at all levels above primary. And I'm damn good at it, as I've only had a handfull out of hundreds of students over 20+ years who didnt show marked improvement, in their grades a
• #### Re: (Score:2)

Is it that precalculus was considered a high school course, or is it considered college level?

I happened to take precalculus back in 11th grade, if I'm not mistaken. I then went on to take calculus in 12th grade, and with a score of 3 on the AP test, I obtained college credit, despite the fact I'm currently taking an approximately equivalent course in college.

They need to consider increasing high school math classes by approximately 40% of the daily run-time, even if that simply means longer school hours. T
• #### I suck at math (Score:1)

There. I said it. I really, really, truly suck at math. I mean, i can add/subtract and multiply/divide. other than that, I'm horrible at it. I too would be interested if there was another method of learning math so that I could be better with it. Any "out there" suggestions would be welcome. Oh, I'm good with literature, art, history, etc, just so you know what my strengths are (which are commonly opposite that of those who are good with math.) Any help would be truly appreciated.
• #### You're psyching yourself out (Score:5, Insightful)

on Thursday October 05, 2006 @07:58PM (#16330215) Journal

The fact that you state "TV has taught me math is hard" and that you have a problem with "numbers" yet are good at logic leads me to believe the problem is in your mind. Mathematics really has very little to do with numbers. It's symbolic logic. Equations are just concise, precise statements. If you can do logic, then you can do math. The only time numbers comes in is at the very end (for engineering and science) when you plug numbers into the final result.

I'm not good with numbers and I have a poor memory but I have a Ph.D. in applied mathematics from one of the top institutes of science in the entire world. There's no magic to it and don't let popular culture tell you that mathematicans are somehow different from everyone else. Just take a deep breath and relax a bit.

I'm not going to recommend any books or tell you to meditate or anything else like that. You just need to have some faith in yourself or dig deeper to find out what the real problem is. When you say you're good at logic, what are you basing this on? Are you a whiz at logic puzzles or something? Most of math is logic, a little creativity, and a lot of hard work.

By the way, if you're struggling in a class, here's an idea to try. Go to some of the already-solved example problems in your textbook. Write down the problem on a piece of paper and close the book. Try to solve the problem. Write out all your thoughts, crazy ideas, questions, etc. Struggle with it for a good half an hour at least. Then open the book (assuming you didn't solve it) and look at how they solved it and see if your scribblings were even close. The act of trying to work through the problem will make your subsequent reading of the solution that much more meaningful.

GMD

• #### Think of it as a language... (Score:3, Insightful)

on Thursday October 05, 2006 @07:59PM (#16330219)
I like to think of math as a language for anything quantifiable. When people "talk math" they use these math terms because these terms precisely project their thoughts into words. I think the best way to understand math is to really contemplate everyday physical phenomena. Think about vector fields in your car when the A/C is blowing and trying to reach everyone in the car. Think about parabolas when something is thrown into the air. Hell, try to do your own experiment and figure out the parameters for it. You'll soon find that you'll be looking into a lot of things that change with time and hence, require derivatives. This should segue into your pre-calc learning.

For starters, I'd say look at the basic definition of a derivative: lim[h->0] (f(x+h)-f(x))/((x+h)-x) and compare it to finding the slope of a line: (y1-y2)/(x1-x2)=rise/run. A derivative is nothing more than finding the slope of two points on a curve as the two points get closer and closer together until they lie directly on top of each other (this gives you the slope of a line tangent to a point on the curve which is equivalent to the rate of change at that point on the curve). This is the only hard conceptual part about pre-calculus, really.

And a couple other notes on learning. Intelligence, imho, is just the ability to break things down into smaller and smaller parts or to divide concepts into many little parts. Any field you learn has two parts to it: concepts and vocabulary. When you come across something "hard," figure out what is stopping you: the concepts, or the vocab. If it's the concepts, have someone explain it to you in laymen's terms. If it's the vocab, look it up at mathworld.wolfram.com or of course, www.wikipedia.com.
• #### Re: (Score:1)

"Intelligence, imho, is just the ability to break things down into smaller and smaller parts or to divide concepts into many little parts." Smart people just have bigger sledgehammers.
• #### Re: (Score:2)

Most of what you say is excellent advice. I have to point out that the formula you present as the "definition of the derivative" is nothing of the kind. The derivative is simply lim[delta x -> 0] (delta y / delta x) where y=f(x). For real valued functions of real numbers that is the same thing as you write from a purely numerical point of view, but it's the shorter (and more general) definition that allows you to see what it is, how it works, and why we can interpret it as the slope of the curve, or t
• #### Re: (Score:2)

If it's the concepts, have someone explain it to you in laymen's terms.

And if that someone can't explain it to you effectively in that way they likely don't understand the subject matter fully enough themselves.
• #### Do problems (Score:2)

You say you hate rote learning. I hope this doesn't mean that you hate learning by doing problems, because that's (not only in my opinion) the best way to learn mathematics.

Try to find a textbook with a wide range of difficult in the problems. Start with problems that you think you should be able to do. If you have difficulties, don't hesitate to try easier ones first. If you feel confident with some problems, move up to more difficult ones (but be honest about that, try to get them 100% right, not just th

• #### Keith Devlin has looked at this issue. (Score:4, Interesting)

on Thursday October 05, 2006 @08:03PM (#16330265) Homepage Journal

Keith Devlin [stanford.edu] addresses your concerns. His recent book "Math Instinct" looks at the conundrum of mathematics being easier in practice than in theory.

I haven't read it but I have read his "Math Gene" book looking at innate abilities for mathematics.

TRUE FACTS FROM THE MATH INSTINCT When a dog runs along a beach and then jumps into the water to retrieve a ball thrown diagonally into a lake, it instinctively solves a problem that humans need calculus to solve. Lobsters have a built-in positioning system that is the equal of the hugely expensive and mathematically rich high-tech Global Positioning System (GPS) human travelers use today. Within a couple of days of being born, human babies know the numbers 1, 2, 3, and can distinguish between a correct addition or subtraction such as 1 + 2 = 3 and an incorrect one such as 3 - 1 = 1.
• #### Re: (Score:1)

When a dog runs along a beach and then jumps into the water to retrieve a ball thrown diagonally into a lake, it instinctively solves a problem that humans need calculus to solve.
This is such BS it's incredible that it could have appeared in print. You might as well say that a falling stone knows calculus because at any moment it knows how fast to go to keep moving on a parabola.
• #### Re: (Score:2)

You must have read it differently to me. He didn't say the dog solved the calculus, he did say it solved the optimisation problem. Here is a bit more information [mathinstinct.com]to help you.
• #### Re: (Score:1)

Maybe he meant "...that humans need calculus to describe"?

And the "falling stone" thing doesn't clarify anything - the stone has no capability of doing anything except following Newtonian mechanics. (Besides, for it to fall in a parabola, it'd have to be released someplace like here [theblowingrock.com]...)

• #### Re: (Score:2)

The dog certainly must solve the problem at some level, whether it's using calculus, geometry, optimization, or whatever. The dog moving to intercept the ball must make 'decisions' about when and where to move, how fast, etc. If the dog doesn't actively move, it would just slow down to a stop, following the laws of physics.

Unless you believe that there is another way of knowing where the ball is going to land, other than math, the dog's brain must be using *some* kind of math at *some* level, in order to
• #### Re: (Score:2)

Unless you believe that there is another way of knowing where the ball is going to land, other than math, the dog's brain must be using *some* kind of math at *some* level, in order to move its body to get to where the ball is.

Sure there's a way. Just from experience built from practice and observation. After a while, accurate pictures in the mind can be formed of what the ball is going to do and the dog can take the proper actions to get the ball. Just because all of this can be modeled using mathemat

• #### Re: (Score:2)

>>does not necessarily mean that the dog's brain is in some way doing mathematics.

Yeah it does. He's trying to intercept the moving ball. Balls can be thrown in any range of angles and speeds, and he's able to position himself at the right place at the right time.

I note the same thing when I'm approaching a light that I worry might be about to turn yellow. Based on my speed, my acceleration/braking potential, and the distance to the light, I actually feel an inflection point pass. On one side, I brake
• #### Re: (Score:2)

Sorry, the grandparent is right. Think of catching a ball (baseball, basketball - doesnt matter which). Most people learn to intuitivly allow for differences in elevation, wind speed, relative velocities of thrower and catcher, movement of the ball, spin on the ball, bouncing off uneven surfaces (in cricket at least) etc reasonably well. Much better than a Patriot Missile battery. The equations needed are pretty hairy compared to the gool old v=u+at stuff from High School. Yet almost all humans do it i
• #### Brains are different (Score:3, Interesting)

on Thursday October 05, 2006 @08:11PM (#16330337) Homepage Journal
Different people have different brains. Some people just can't do math after a certain level. A lot of stuck-up geeks will tell you it's just that you haven't learned the lower level math well enough -- that may not be true. They probably have a brain that is well-suited for doing math, and they think that everyone must be just like them, that math is easy, and anyone who says otherwise is lazy or doesn't care.

I consider myself to be a geek. I have always had a nerdy, intellectual personality. However, I had math difficulties since day one, starting with addition.

In high school, we had a geometry class. There were hardly any numbers in it, just images, compasses, and protractors. A lot of our assignments were proofs. I got an 'A' in the class. I remember one assignment in particular at the beginning of the class. There was a figure that was a bunch of triangles, and we just had to count how many different triangles we could find. Most kids got 12-15, but me and a few other kids who were good at art counted into the late 20s. There were actually 32 in the figure. The next year was Algebra II, and I got a C. :( My point in saying this is that my 'math' mind works visually. I had no problem doing geometric proofs as long as we were looking at figures and drawing. However, when it comes to reading 'number sentences' with abstract symbols, and solving equations, I'm sunk.

Another area of geekiness is reading and language. I taught myself to read before school started. I never had a problem with reading or writing assignments -- I typically did them the night before, skimming. That got me a magna cum laude degree in the honors program at Ohio state (in the honors program, you could only take classes that were designated 'honors' -- less than 30 students, taught by a professor, or a graduate level class. ) I took my math at a local community college and transferred in so as not to ruin my GPA ;) I have a BA in Anthropology and Religious Studies.

I'm pretty good with computers, but companies aren't very interested in a computer guy without a BS. I am doing alright with my LAMP job, but I will probably go back to school and get a masters in linguistics. I took a few classes and found it fascinating. I did really well with the grammar parts, such as diagramming sentences. From linguistics, I can use this as a launching pad into other areas that I am interested in, such as artificial intelligence or speech recognition. I couldn't get into those areas through CIS.

I guess my long winded point of all of this is just because you might not be good at certain types of math, doesn't mean that you aren't smart or aren't a true geek ;) You might see it worthwhile to try to get good at those maths, or, you might just find something that is more suitable to your natural abilities.
• #### Re: (Score:2)

"That got me a magna cum laude degree in the honors program at Ohio state.

Well, yeah, but it's not like that's all that hard anyway ;-D

rk, Miami University '94, who can't really talk because he dropped out of grad school (TWICE!) at Arizona State.

• #### Re: (Score:2)

Go Buckeyes!

honors program at Ohio state (in the honors program, you could only take classes that were designated 'honors'

That's not true -- you can and have to take regular courses as well. There's no Honors graduate-level engineering or Math courses. You just take 'em as an undergrad.

Cheers,
Matt (BS CSE 1998)
• #### Re: (Score:2)

That's true. The math and engineering departments are considered so rigorous that maintaining a high GPA in and of itself is considered 'honors' performance. However, in the 'soft' social sciences and Liberal Arts, we have special 'honors' classes that are tougher than the regular classes.
• #### If you want to improve... (Score:2, Informative)

Watch 'Algebra: In Simplest Terms' hosted by Sol Garfunkel, PhD.:
http://www.learner.org/resources/series66.html [learner.org]

26 half-hour videos covering all topics of Intermediate and College Algebra. The webcast videos are free (registration required), just click on the VoD symbol to watch them. If you use SDP ( http://sdp.ppona.com/ [ppona.com] ) you can download the ASF steams for repeat viewing. BTW... I got an A+ in my College Algebra class... It's absolutely critical that you fully understand advanced topics of Algebra before
• #### My advice to you... (Score:1, Insightful)

by Anonymous Coward
I have a degree in math. I was also an engineering major.

Math has many aspects to it. There is the mechanical aspect, like adding numbers, or long division. To learn the mechanics well, you need to simply solve a large number of problems. One aspect of calculus is mechanical. You apply mechanical rules to find derivatives, etc.

A second aspect is the application of the mechanical rules to solve more generally stated problems. Traditionally these problems are called "word" problems. These require some

Look for anything by William Poundstone - he goes into ideas about paradoxes, probability, game theory, AI, all sorts of things that should probably appeal and it'll encourage you to read more maths related works.
• #### My math (Score:2)

Math is just a series of simple rules. When put together in the correct order something larger is created, just like a computer program. I have a natural aptitude for math and always found the little rules to "help" more confusing than beneficial (x100 to get a percentage confuses me, x100% makes perfect sense, maybe I'm crazy.)

None of it is hard at the level of doing it, just forming the picture/design/structure of the solution might not be immediately straightforward.

I think the best way to approach learn
• #### Re: (Score:1)

I know people who started doing better in Calc when they threw away their silly picture filled BA Calculus book and used the "harder" one from the Math/Eng courses.

This reminds me of an interesting notion i don't think anyone here has really pointed out. One can think of there being two sides to math: there is the syntax and the semantics. Either can be (in some sense) a basis of reasoning. Some seem to agree with one more than the other: for example, the calc book with pictures attempts to explain c
• #### Well, here's one hurdle you need to overcome (Score:4, Informative)

on Thursday October 05, 2006 @08:40PM (#16330651)
> I hate rote

This insane allergy people have to simply memorizing some things gets in the way all the time. Just get over it. Despite what new-age bullshit you might be used to about how rote learning is 'just' memorizing lists of facts, it remains important to memorize those facts. Some things you just have to memorize, and math is full of them. What edges of the triangle a cosine relates to is an example. Once you start committing this stuff to memory things will start to fall into place. Worked for me. Got a degree in math and everything.
• #### Re: (Score:2)

For some things like grammer and spelling that often lack any true logic you have to just buckle down and memorize them. But for Math there is a fundamental reason why something is correct. I never bothered to memorize formulas and so for my classes. Instead I worked to understand and derive the formula. Once you understand what the formula means and the steps that lead up to it, math is easy.
• #### Question about Statistics then (Score:2)

Statistics I've found is more difficult to grasp. As I've come to understand the key terms and concepts and worked out some problems on paper sure, its become easier. I've picked up Statistics Demystified (Highly recommended) and its helped. I need to brush up before my next Stats class though (bear in mind doing more 'applied' and 'actual' research with it more in the Liberal Arts type and not in an actual Stats dept level) .... any suggestions (books, exercises, websites)?

IMHO, there are a number of 'open
• #### Re: (Score:1)

Larry Gonick's "A Cartoon Guide To Statistics" (Amazon [amazon.com]) is pretty good. It's mildly entertaining, clear, and doesn't slaughter the precise concepts as badly as many "popular" books do :) Good luck.
• #### Puzzle pieces versus steps (Score:3, Insightful)

on Thursday October 05, 2006 @09:04PM (#16330877)
I've been tutoring math from calculus to basic arithmetic for a number of years now. I also am drawing on my own experience when I first took an honors math analysis course. There is a radically different approach between how math (really arithmatic) is taught between high school and college.

High school typically chooses a rote approach - learn the steps required to complete the problem and regurgitate on request. Even some college courses are taught this way. You are given a collection of steps and are expected to remember the steps that are applicable for each problems. I have found, tutoring, that the best approach by far is to teach a collection of 'pieces' - a particular approach to a particular sub-problem - where my students also have to learn why it works. I then encourage each of my students to visualize any problem as a jigsaw puzzle where existing pieces are combined to find a solution for the problem at hand. (i.e. There exists a sequences of steps using known 'pieces' to solve the problem and the student is expected to eventually pick up an intuitive understanding of what kind of techniques to apply when facing a new kind of problem.)

I've experienced a great deal of success teaching with this technique and recommend it whole heartedly. Create a notebook listing every technique for solving a sub-problem you have been shown to date. Each technique should have a name, a set of conditions when it applies, and how to implement the technique. If you plan to remember the techniques for an exam, also include a description of why it works - preferrably worked out / thouroughly understood by you.

Obviously, this is what I have found to work - YMMV. But I have found that, as long as an individual is capable of viewing problems abstractly enough to grasp the approach, it has been an effective problem solving technique.

• #### Why is TV always to blame? (Score:3, Funny)

on Thursday October 05, 2006 @09:08PM (#16330949) Homepage Journal
"...and TV had taught me that math was difficult."

I thought that was Barbie's job.

• #### It is hard to "speak" Calculus (Score:2)

When arrived at my school a lot of (CS,PMATH) professors told me that they didn't believe that Calculus should be anyone's introduction to mathematics. Unfortunately, a lot of Calculus tends to be taught with "tricks" (this includes a fair number of proofs) that while are neat to look at as a math major provide very little benefit to most people.

One of the best things that happened to me was our first year classical algebra. It starts from the very basics of logic, surveys elementary number theory, modular
• #### IMO (Score:1)

First and foremost, you do NOT think logically. This you must accept as if you did, math wouldn't be so difficult for you.

Secondly, you must understand that no matter how good you are at math, you must spend hours and hours and ... and hours, and then some more time, studying math. Also, please note that studying math does NOT just include reading, and memorizing the definitions. It _mostly_ includes actually doing problems.

Also, reading a math text is different from reading any other non-science book i.
• #### practice (Score:2)

I pretty much know what you are going through. I left EE and went to physics because there was too much application of math and not so much understanding of it. To me it was rote plugging into matrixes, rote construction of taylor series, rote construction of algorithms. For some people it was fun, but I just did not have the ability to go through all the homework and understand the background.

There is no silver bullet. The most sensible thing is to practice all the problems in all the books. Learn t

• #### Miller's "Popular Mathematics" (Score:2, Informative)

See if you can find a copy of the 1942 book "Popular Mathematics" by Denning Miller. It goes from arithmetic to calculus, taking generally a more geometrical, physical, and historical approach than most math classes do these days.

I was pretty good in math, up until I hit differential equations; I bought this book just for curiosity, so I can't really say if it will help you. But it looks like copies can be found on eBay for just a few bucks [ebay.com], so I'd say it's worth the gamble.

• #### It's just like a game (Score:1)

That's what I learned in geometry.

Math is
- a set of propositions
- a set of ways to manipulate propositions

The game is to use the manipulations on the propositions to reach the answer.
• #### It has to do with passion. (Score:5, Insightful)

on Friday October 06, 2006 @12:11AM (#16332471)
All the people who have said that there is no difference in ability, and any arbitrary person can advance to any arbitrary level of mathematical ability are pretty unrealistic. I base this statement on my own abiding comfort and love of the subject, as well as five years tutoring and teaching it at levels varying from elementary school level to graduate school. That said, here are my own personal observations as to which people succeed in their math goals and which people fail.

First, what people said about practice is partway true. But HOW you practice is as important as how much. Many people think that if they do the same problem over and over and over, perhaps with minor variations, this will somehow improve their mathematics ability. Except for at a very base rote level, this is untrue. A far better challenge would be to INVENT problems like the ones you have been solving, and see if you can solve those. Frequently, the 'canned' problems you are given for most mathematics instruction below second year university level are designed to have 'neat' answers. This very quickly becomes a crutch for students, because they are so used to looking for the 'neat' answer that they are unable, or don't trust their ability (almost the same thing, in practice) to work a problem when it is unclean. In addition, when you start designing problems, you start to focus on the crucial idea of whether you are right or not. Having an answer handed to you is almost useless, because it short-circuits the other half of problem solving - how do you know whether you have a right answer? If you don't understand how to check your answers, you aren't qualified to be doing the problem! Right there, that suggests a different method of problem solving - trial and error. This is not to be scorned, but encouraged, because it means your brain is engaged again, and you are not just regurigitating the motions.

Second, most people who are good at math like it. What this means is that they are practicing far more often than people who don't like it, because they have some part of their mind on math problems throughout a day, or they find problems that have mathematical solutions. How do you get to like math if you don't? Tough question - I found that good teachers who enjoyed explaining how they got to an answer, what makes it fun or interesting, how it applies, or just how neat it is are better than the rote type. But at some level, you have to start figuring what you want to DO with your math - frequently, practicality and application focus the mind and make it easier to learn and enjoy it.

Third, don't let people who are better at it than you get you down. REAL math is messy. When solving a problem that has not been solved before, mathematicians go through all sorts of detours, false starts, unnecessary constructions... messy, messy, messy! But after thier adventures through the mathematical jungles, after they get the prize, they clean up the mess. They don't mention the false starts, the extra logic that really isn't needed, the play with ideas that turned out to be useless. They just show the clean, sparse, neat path. This is a modern fashion, and I think a bad one, because it removes the human element of play, adventure, and imperfect effort. Learning math is messy - you need to experiment, make mistakes, try to fix them, try different ideas, and PLAY with the stuff. They don't tell you this in the textbooks, at least not the modern ones (of course, there were flowery extremes on the other side - read Cardano for an illustration of 99% prose and 1% math! But he does tell you of his false starts, his dispair, his mistakes, and the joy of his ultimate triumph). AFTER you have made mistakes, tried alternatives, and played with other ways of solving a problem, then the 'standard' way of doing it makes much more sense, and you appreciate the WHY vs. the HOW. This is why, if you don't know how to check your answer for sure, you are not at the level where you should be attempting such a problem.

Feynman
• #### Re: (Score:2)

I think you mean CALCULUS by Michael Spivak, and I agree, it is a great calculus book.
• #### Re: (Score:2)

CALCULUS by Michael Spivak

Whoops! Thanks for the correction - it gave me an excuse to pull it off the shelf and thumb through it. It's STILL as good as I remembered - [cue nostalgic music while I wish I was 21 again...]

• #### Math books and other comments (Score:2)

Some math books do a poor job at explaining what they are trying to teach. Not showing the appropriate examples on what you're trying to do is one problem. Not following the proper format students are expected to follow when showing their work is another problem the books have.

They need to strengthen math skills while in k-12 schooling. If they can extend math class by 40% per day, that gives more time in which students can receive help and be given more clear instructions on how to solve the problems.

I bel
• #### 3 main alternatives (Score:2)

There was a Professor at the University of Minnesota in the '70's named Pedoe, who taught a class in Non-Euclidian Geometry over 3 quarters. The first quarter, the descriptions were oriented around numbers (for those who understood and liked numbers, arithmetic), the second quarter the descriptions were oriented around Algebra (for those people who liked general recipes and principles), and the third quarter the descriptions were oriented around actual visualizations as in graphs and geometric diagrams(for
• #### Ask a Secondary (High) School Maths Teacher (Score:3, Interesting)

on Friday October 06, 2006 @06:26AM (#16334237)
...like my missus. She's actually not that good at maths, but she understands how people think and learn about maths pretty well, as will any good maths teacher. There are hundreds of books on the subject, so find a Maths PGCE/Teaching course syllabus and look for the Recommended Reading section - that should give you some good grounding.

The important thing to understand about maths is that it isn't an intrinsic ability - our brains are not designed to deal with even counting, and certainly not with abstract mathematical concepts. We adapt various neural modules such as language, spatial perception etc by constantly using them in unique ways to consider mathematical concepts.

As an example, the notion of a "number-line" as something on which all natural numbers have a place is introduced at an early stage in teaching. This is later developed to deal with non-integers, and then extended backwards to develop an understanding of negative numbers (and how they're not "different" numbers, but a continuation of the line). Then at a higher level this is further developed to include imaginary numbers as a perpendicular axis to real numbers, and the notion of complex numbers is introduced. Through all of this, it is the spatial-perception module that is being used and thus adapted to deal with abstract space and its relationship to number.

One of the most important mathematical concepts to develop (though few high-school children do) is to stop thinking of numbers as abstract things in themselves, and see them more as names of matched sets of objects - four elephants can be "matched" to four marmosets on a one-to-one basis (unlikely and unproductive though that might be), so those menageries are in the set of all things that can be matched in this way, but they cannot be matched to any abstract "thing" called Four. Four is just the name of the set. This is a simple way of approaching the basis of Set Theory, which is irrelevant at high-school but vital at Uni. Admittedly, it might not be so useful for EE, but IANAEE.

One of the key areas you will need to master for EE, I suspect, is algebra. This is closely linked to the language centre of the brain, so you will find it easier to learn if you consider it as a language. Start with simple expressions and learn how to translate them either way, gain a familiarity with the most basic ones so they become second nature, and progressively move on by expanding your vocabulary and the complexity of expressions. When you face a challenge, slow down, break it down and try to translate it. Eventually you will become fluent and - more importantly - it will be like a second language in which you can converse without difficulty or any real conscious thought.

Interestingly, a lot of our perceptions and methods of thinking about mathematical issues are conceptually conflicting, and that is a barrier that is difficult to overcome. As an early example, moving from algebra to graphs to vectors & matrices is a serious stumbling block for many children - they can handle any concept individually and with practice they can translate one to the other, but until the mental connections are made they will find it difficult and obscure. Once those connections are made it is a rapid revelation, and they find their understanding and enjoyment of both topics is enhanced (as you might have guessed, this is precisely what my missus seeks as a reward for her hard work).

I mentioned algebra as a key player in EE. There are obviously other areas you will need to grapple with - trigonometry and graphs being obvious ones - and they will require different approaches, but if you find you have trouble with any of them then I strongly recommend you call in the professionals. Uni-level course books and materials tend to present the facts and concepts in a very clear way, but they do not tend to be very forgiving or understanding of those who have difficulties - if you don't get a concept, you will fall down later when you need to build upon it. The best thing you can do is enrol o

• #### Approach it like a language (Score:2)

Learning the practical skills of Calculus, particularly integration, is like learning a foreign language. The methods of integration are like the syntactic structures of language: they're ways of getting certain things done. Not the only way, just ways that work for some people.

Facility in a language comes when you can use its structures to think with, without thinking about them. When they become automatic, you've mastered that part of the language. That means while you can learn about a language by stu
• #### If Pre-Cal is bad... (Score:3, Insightful)

on Friday October 06, 2006 @07:55AM (#16334715) Journal
then you should quit now.
I wanted to be an EE and want very much to be good at math but if my ability does not increase I will not be able to. I am willing to do anything to increase my skill. I hate rote and do not want to be merely 'good' at math, I want to speak it. If math is a mindset then it's one I want to be part of.
You will never make an EE with bad pre-cal skills. You have yet to hit Calculus and are struggling already. Most every EE I know, and that was my degree so I know quite the few, were taking Calculus in high school. It will only get worse until Differential Equations, and if someone told you EE was not a lot of math, they lied to you.

Have you considered the option that maybe EE is not for you? I whole-heartedly suggest that you go and find a counselor or advisor and get their opinions, but I am pretty sure any one from your College of Engineering, will tell you that it probably is not a good idea to pursue EE (or any other engineering) if you are struggling with Pre-Calculus. I know I have completely skirted your question, but this is something you should really consider. If you are not good with Math, engineering is not for you and trying to learn math now is a bit late in the game.
• #### A lot of factors are involved (Score:2)

Do you think verbally or with imagery, or both? Do find yourself needing to move when learning something? How good are your reading skills?

If your reading skills suck, work on that first since they are so fundmental to everything.

If you use a lot of imagery, learn to draw pictures. Reading equations doesn't work so well for me until I can draw a model, even a simplified model, of the situation (what do n orthoganol vectors in n dimensions look like? Like a sea urchin).

Logic and verbal descriptions help me t
• #### Problem is with work. (Score:2)

From seventh grade through high school, I did only what was needed to get by and so my math skills remained below par ... I believe that I have a flaw in the basic way I think about numbers.

Seems to me the real problem is that you cruised for six years instead of engaging the material. Those years were there for you to develop a way of approaching math.

Yes, you need to go to your professor's office hours (and your TA's, if you have one.) But you also really, really need to find your university's academi

• #### Let go. (Score:2)

If you are struggling now, you wont make it, let go.

My story: I was always good at math, and so as a senior in high school I ended up in the highest math course offered. The teacher was great, I mean really great. Like state teacher of the year great. This guy Dr Corbin Smith, taught only the highest math and remedial math....nothing in between. Any way, we used to sit and discuss the mathematics of odd solids (without the calc) it was very fun. But I was failing his class, badly. I worked my butt off and

• #### Concept then application (Score:2)

There are, as you probably realize, many ways to approach learning. Sometimes it helps to get an explanation of what the concepts are, that is to say 'what is trying to be accomplished,' before delving in to find out the nitty-gritty How parts.

As one previous poster mentioned above (in probably different terms,) understanding a base point is usually sufficient for building up to the bigger thing and what you'll really remember is the path of logic that gets you there. What is to be developed in this case is
• #### Teach, of find a good 'personal' teacher (Score:2)

When I look at nearly all the stumbling blocks I've conquered - algebra, writing skills, engineering school topics - I got over each and every one only after some serious one on one time with a real person.

In a similar sense, read multiple books on tough topics. Don't just grunt through the explanation from the one 'official' course book or lecture. Find 10! Eventually, one of the sources will have an explanation that fits you...someone who got stuck on the same mental snag as you. THEN go back and read

The absent ones are always at fault.

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