I'm worried about math

Student Lounge College Life
1

<p>Hmm, if this is the wrong forum, please move it, mods. lol</p>

<hr>

<p>I'm a freshman who just completed linear algebra and multivariable calc this year. I want to take more math classes and do well. I'm not doing terrible right now, but I want to improve not just grade-wise but also in my understanding. Here is my situation, and I would really appreciate if you guys could help.</p>

<p>I found I could not keep up with the pace of things in class. Why? Maybe it's a lack of interest, a slow processing speed when it comes to the computations, or I'm just bad at learning conceptual subjects audibly. This didn't worry me too much, since I've always been like this and still did well in Calc BC in high school. And so I zoned out through most of my classes and did most of my learning through the book. I found that trying to read the book explicitly, meaning going line by line and stopping whenever I didn't fully understand a concept, got me nowhere. I could do it, but it took way too much time, and often I would hit mind-blocks that would be extraordinarily difficult to resolve. Often, I would end up more confused than informed. Such is the case when a misunderstanding of a new theorem undermines a correct understanding of a previous theorem (<em>sigh</em>). But I found the information was much more manageable if I absorbed the substance in a roundabout, implicit manner. </p>

<p>By this I mean that I would half-skim the material at a reasonable pace, gathering momentum as I proceeded and just bowling over any conceptual obstacles I encountered. I might end up doing this several times. But pure repetition was not enough. I also required time, where, I guess, the concepts could incubate in my head. I know this because I would fail most of our weekly quizzes, because they were so proximal, but come test time I would have a sufficient handle to score very high. After much experience, I realized my brain was doing most of the work subconsciously. Only slowly and implicitly could I get the gist of what was going on. This method produced test results (though I failed the quizzes and couldn't do the homework on time), but I feel like I haven't really learned anything. Of course I've learned 'something', and at times I feel like my brain has been stretched quite far. However, a nagging feeling tells me that I've just learned how to process new and complex algorithms, to tie vague ideas to a list of formulas, and to do shallow manipulations in very standardized ways. I feel like my experience has been more ad hoc than systematic.</p>

<p>To inject a little objectivity into my narrative, this situation makes perfect sense in light of an intelligence test I took at a psychologists' office a year ago. To compress a broad matter into a few sentences, I did poorly on the block design test but very well on the memory sub-portion of the Weschler test. The block design test is regarded as the most valid test of mathematical conceptual capabilities. Obviously, I'm weak here (very weak compared to the rest of my subscores). On the other hand, I have a particularly strong memory. Memory has an episodic (long-term) and a working (short-term) component, and both were very good. The low block test score suggests that I cannot conceptualize heavy mathematical ideas very well, but the high memory score suggests that the concepts I do learn will be retained effectively via episodic memory and manipulated efficiently via working memory.</p>

<p>Thanks for reading all the way down here. Now for some specific questions I have.... Given the information I have provided, do you think my self-evaluation is fair? If so, is there a point to continuing with math? I know that from now on, math will get much more abstract. From what I know, real analysis (building a deeper, firmer basis for everything we learned in calculus) seems like it would be really cool. But I might be beating my head against the wall if I can't fully digest the proof for double integrals. Given how easy and standardized my tests have been (at a top-20 school), is it even expected that we do more than just learn algorithms -- meaning, should I even be worried? Can most good multivariable students understand the proof for Gauss's theorem, for example? On one hand, I realize analysis would use less memorization. On the other hand, I find that I do better on conceptual work that starts from the ground up (I did much better in linear alg than multi calc). For someone like me, would a more theoretical course like analysis be worse or better? I feel like I'm missing a lot of the basic of mathematics because I've scraped by for so long on my memory alone (I was atrocious in this respect in high school). Is there some basic math theory book/website that I could get acquainted with, which could also provide me with some kind of indicator of whether I want to or even can progress further in math?</p>

2

<p>i just completed linear algebra as well, though i scored an A- , it was enough to see that i will not be happy as a math major.</p>

<p>do you enjoy the math, does being a math major conflict with any goals you have after college?</p>

<p>i am transfering to another university as a math major (applied before starting linear algebra), was accepted into math program. I curiously went to their engineering website…and guess what happened.</p>

<p>I will try to argue with advising at my new university to be able to change to engineering. If its not to late for you and your interested in changing…some schools don’t let you or make it difficult after too many credits! don’t carry a burden for too long, anyway thats my advice.</p>

<p>Btw i would describe math and me the same way you did, its alright and i can get by just fine but not enough to be excellent…i think in engineering hard work will shine through more.</p>

3

<p>Hey good luck with your new major and your new school. Engineering is a big no-no for me, the thought of E&M in high school still makes me shiver.</p>

<p>Anyways, I’m definitely not majoring in math. Maybe I’ll minor, but that’s not really important, neither is the course requirements or stuff like that. Whether I’m ‘interested’ in math itself… well it’s like exercise: a complete chore, but highly rewarding in the end. I just feel like it’s one of the few subjects worthwhile to study. I definitely excel more in the social sciences, but compared to math everything else just seems silly and fluffy. I’m sort of inexplicably drawn to it, though sometimes my frustration defies words. It’s a love-hate relationship, oh well, aren’t those the best sometimes?</p>

4

<p>If you want to go deeper into math, maybe the most important skill to learn is to be very aware of your own reasoning process. So you are stuck on this one line in a proof, or a definition does not make sense. What exactly is in your way? Is there a term that you don’t understand? Notation that does not make sense? Do you need an example to illustrate a concept? Does the proof use a fact that you do not remember? Does it seem to contradict a previous result? Which result? Can you write out all of the implicit assumptions you are making in your mind and see which one contradicts the assumptions in the text? Once you know exactly what you are stuck on, it is usually not too hard to correct that. But many students never make the step from “I am confused” to “I am confused by…”</p>

<p>Math is one of the subjects that you have to study actively: Ask yourself questions as you go through the text or your notes, find links between concepts. If you feel like you are not completely on top of the material yet, do more exercises than assigned. Pick exercises that you are unsure how to do, not exercises that you are comfortable with.</p>

5

<p>^ B@r!um is wise as always, good to see you around again!!</p>

6

<p>I’m probably in a similar predicament to you: I want to pursue a non-mathematical major, am quite interested in math, and probably have little aptitude for it (by which I mean insufficient acumen to contribute to the field, not relative to the general population). To use the example of Gauss’s Theorem, I was able to understand the proof in the book by going through it line by line. That is, I understood the logical connection between the various steps; consequently, I accepted the conclusion. But I still had no intuitive “feel” for the theorem; I understood it in a superficial, sequential way, but lacked a deeper knowledge of it. So if someone were to ask me, “Does a four-dimensional analogue to this theorem exist?” I would be utterly clueless. </p>

<p>I’m currently taking linear algebra and I do find it more straightforward and understandable in a conceptual way, but I still have a nagging feeling that I don’t fully appreciate its basis. In all likelihood, I would fail miserably if I were to attempt to discover a new concept by myself rather than by reading it in the book (for example, if someone told me to devise a method for QR factorization without looking at the procedure). This is partially due to the fact that I haven’t put virtually any effort into understanding the material (I’m a second-semester senior in high school, so sue me), but I’ve also reached the threshold of my ability to synthesize information from all the aspects of the course. Seeing as how you’ve already finished the course, do you know the rationale for some of its most basic premises? For example, why is matrix multiplication defined as it is and not in another way? Why doesn’t matrix division exist?</p>

<p>Overall, though, I think gaining a conceptual understanding of math can be one of the most rewarding things you can undertake. The more abstract math becomes, the more beautiful its results. Most of the people will probably be in the same boat as you (based on anecdotal evidence from my class; a majority wouldn’t be able to answer the above questions very well). Also, I think one of the most useful techniques you can employ is to constantly challenge your thematic understanding of a topic. If you do that, then you’ll probably be better off than most of your classmates who tend to stick to its mechanical, algorithmic facets. There’ll always be people who are much more naturally adept at math than you (and will go on to become exceptional block designers, as per psychological exams), but if your objective is just to obtain as broad an understanding as possible of mathematical theory, it’s definitely worth the struggle.</p>

7

<p>Linear Algebra is one of those subjects that make more sense after time, once you start using it in applications or build more theory onto it.</p>

<p>Just to answer a few of your questions:</p>

<p>

Because it allows you to express any linear transformation as a multiplication by a matrix, and the composition of two linear transformation is just multiplication by the product of the matrices of the individual transformations.
Symbolically: if f(x) = A<em>x and g(y) = B</em>y, where A and B are matrices, then f(g(y)) = A<em>B</em>y. That’s handy, isn’t it? Linear transformations are so useful that they form the basis for many applications and much of higher math. For example, the derivative of a function is a linear transformation. In fact, it is the best linear approximation of a function at a given point. And the chain rule is just a matrix multiplication in disguise! </p>

<p>

It does exist. Dividing by a matrix A means multiplying by the inverse of A. Of course you need A to be invertible! When you divide by a real number, you actually multiply by the inverse of that number as well. For example, 5/3 = 5 * (1/3), and 1/3 is the multiplicative inverse of 3. Why is it illegal to divide by 0? Because 0 does not have a multiplicative inverse. There is no number x such that 0 * x = 1.</p>

8

<p>my first real introduction to higher math was i.n. herstein’s “abstract algebra”
if you stare at it for a while and try some of the problems,
you’ll probably learn something
and you’ll get a good feel for what it’s like
i also recommend “introduction to analysis” by rosenlicht</p>

<p>and don’t be discouraged if some things aren’t immediately obvious,
everyone goes through that at the beginning,
and the more you look at it the better you’ll get
it usually takes a lot of playing with things to get new concepts</p>

<p>good luck and have fun :)</p>