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Different Ways to Conceptualize Math?
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."
This is similar to another question, however I found several interesting books but no comments toward learning a more efficient way to think."
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.
SJW: a person who perceives an injustice, and while correcting it, commits a greater injustice.
>>TV had taught me that math was difficult.
Go watch PBS you victim of TV
http://en.wikipedia.org/wiki/Jury_nullification
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.
I'll form my OWN solar system! With blackjack! And hookers!
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.
Microsoft cheerleader, blue flag waving, you got a problem with that?
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
watch this
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.
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 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".
... yeah I was suppose to be working but it's one in the morning, so what they hey ...
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
>>> "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
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 lateral thinking and practice.
A third aspect is pure math, were various "truths" are proven by starting with fundamental axioms and logically deriving the required result. This can help for a deeper understanding of a particular truth, such as the Pythagorean theorem.
The engineer will be most interested in the second aspect. The first aspect isn't used in practice due to computers. And frankly, engineers do not care about the correctness of the math in the same way that mathematicians care. So the third aspect won't help an engineer very much.
My advice is to solve a large and varied set of word problems, that range from basic algebra, to calculus, and then vectors. This is really how you develop a strong feeling for math and how to apply it to engineering. If you also solve a large number of mechanical problems that may have eventual application in solving the word problems, that would help as well.
Just realize that confusion is usually caused by a lack of familiarity. Solving a large number of problems is the first step towards clearing up the confusion. Books with a lot of examples are the best. Go straight to the examples, see how a particular problem is solved, and then apply the technique to problems listed in the excersises.
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.
The number you have dialed is imaginary, please rotate your phone 90 degrees and try again.
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
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 is not bad if you live Mathematica. For the rest of us, Maple is easier to use, has a better interface (tabs, advanced yet easy to use formatting, etc.) and is much more like the programming languages you're likely to know anyway.
That almost made me spit out my Diet Coke. Here at CU, we aren't allowed to use CAS programs on exams (or any calculator at all, for that matter), but on homework assignments they are absolutely essential for checking your work. In fact, we have three labs per course that absolutely require the use of a CAS system - the Applied Math department pushes Mathematica, but I use Maple.
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.
There's nothing wrong with a 12MHz M68000-based portable computer with a math-optimized keyboard, 240x160px screen, and 256k of memory. It's a fine device that works very, very well. You can compile for the calculator using GCC, there's a big standard library, and the built-in software is generally very, very good.
There's nothing wrong with RPN, but assuming that it is "natural" or "superior" is like saying that we should all use DVORAK keyboards. Use both, choose what you prefer. I'm an algebraic entry person myself, because I like the input to match the problems - it helps me visualize what I'm actually doing and helps eliminate error. Pushing numbers (or symbols) on to the stack is more abstract and, at least in my case, more prone to error. A few saved keystrokes don't mean that much to me.
No, no, no. I have had mathematics professors who were also EE professors - they tend to spend too much time focusing on specific applications and gloss over the fundamental mathematics. Real-world examples are great, but you need to understand the concepts first - and EE profs, in my experience, frequently do not understand that their students do not.
At CU, we have an entirely separate engineering math department (Applied Mathematics), with different courses and different textbooks. Our text is filled with engineering sample problems and our professors use them in class. But our Applied Mathematics professors do nothing but teach mathematics - they know their material (and how to teach it) very, very well because that's all they do.
At CU, we call engineering "pre-business". It's not for everyone. If you don't like it now, you won't like it as a career.
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.
"Some days you just can't get rid of a bomb."