<p>Does it depend on the school? Is it more than calculus? I'm a high school senior, and I take Applied Calculus at a comment college. The class is very easy, because it is based on the book. I am learning new techniques and calculus problems that can be applied but I think that this school math's department is very easy on students. When I go to a 4 -year school, will math be based on books and memorizing concepts?
I still don't know if I want to major in math. </p>
<p>At the college level, math tends to become more about rigorous proofs and very abstract concepts (think analysis, modern algebra etc), as opposed to being about computation and applications like calculus. Books will be important, but you will have to synthesize your own knowledge too, with less memorization involved.</p>
<p>Math is about far more than calculus. Calculus is an important part of mathematics, but it is but one tiny part of it. </p>
<p>Upper level mathematics becomes more abstract and focused on rigorous proofs. Introductory algebra and calculus are all basically courses about different methods of “solving for x.” Once you get into the upper level, math takes on an entirely new form and often barely resembles that math that you currently know. </p>
<p>As already mentioned, people who major in math do so because they want to get into more theoretical and esoteric stuff than calculus. People who are just interested in calculus would be more likely to study engineering, I think.</p>
<p>It gets VERY hard. Avoid like the plague.</p>
<p>Just kidding. Really, the math you learned in high school is nothing like the math you will see when you get to college as a math major. You probably have done a few proofs in geometry and trigonometry. But these are nothing like the proofs you will do in higher math.</p>
<p>Take linear algebra or abstract algebra, which are typically sophomore and junior level courses in college. In linear algebra, you learn how to perform algebra with vectors. I don’t necessarily mean vectors from precalculus. Linear algebra is the class where you learn that almost anything can be thought of as a vector: matrices, power series, the real numbers, etc… Thus, you start abstracting the whole, “solve for x” mindset and you begin to really see what it’s like to do algebra on something other than the real numbers. You’ll also learn that your teachers from junior high and high school basically dumbed down the material to make it presentable to people who’ve never seen math before. Does that mean it’s hard? No! I just mean that a lot of your teachers never used the “real” notation for math and its concepts.</p>
<p>Abstract algebra is where you study what math would be like if we moved to other dimensions and spaces, and what it means to actually do math in spaces that don’t necessarily make any sense whatsoever unless you consult the rule sheet laid out. In other words, how well can you follow directions, especially directions that go against everything you know and believe in? For example, how much of math actually holds if we live in a space where 2 + 2 = 5 and 0 × 4 = -1? The answer might actually surprise you.</p>
<p>Proofs will really take a toll on you if you really don’t understand what it means to actually prove. Proving will require you to memorize special axioms and theorems and apply them to come to a conclusion the problem tells you to. Sometimes there’s only one way and sometimes there area multiple ways. Many students have difficulty with even knowing where to start, and if they manage to start, they never really know if they’ve “proved” the problem actually.</p>
<p>I took linear algebra last spring. It was complete and total culture shock since I took a theory based class. There were easy parts for me such as showing something is a vector space. But proofs otherwise were just terrible because I didn’t really understand the theorems all that well. Looking back, I understand them now, but man I thought I was going to fail at the time.</p>
<p>The point is, try to do some independent study from Larson’s Elementary Linear Algebra. If you’re familiar with matrice, start in chapter 4 and try many of the proofs. That’ll give you an idea of what it’s like to be a math major. And when I say “familiar with matrices,” I don’t mean putting in A^(-1) B into your TI graphing calculator to solve a system. You absolutely CANNOT do that anymore once you hit linear algebra or higher classes. You almost guaranteed will not succeed.</p>
<p>@cameraphone Thanks. It’s ironic how these things that you told me aren’t really exposed to us. It is interesting that math is beyond the scope and that you have to reason it, sounds cool but challenging, almost like Newton finding out a new theory and he has to prove it. And my community college doesn’t even go up to those courses, Calculus 3 is the sec on or third to the last course, I think. </p>
<p>I can tell you that the math my son is studying as an applied math major is way beyond me, and I was an engineering major! I look at the stuff he’s doing and just shake my head. I don’t know how he does it. He likes it, though!</p>
<p>@MaineLonghorn LOL, hoe he does it? He has a deep interest for it so he would handle it. And I thought that engineer majors would take similar courses as a math major or even more advances courses like Physics. </p>
<p>I wouldn’t really say that it’s “ironic.” Mathematics is cumulative. People gain their mathematical maturity through algebra and calculus. Problem solving skills that students develop in these subjects are still absolutely crucial to doing higher level mathematics, even if the techniques used in those courses aren’t exactly comparable. High school students are typically nowhere near mathematically mature enough to be able to really ‘get’ things like abstract algebra (which is nothing like the algebra you know), analysis, topology etc. </p>
<p>Community colleges rarely offer anything beyond the introductory calculus sequence. Many of them also offer introductory courses in both differential equations and linear algebra, but that’s less common. That’s the nature of what a community college is though. They offer courses that are comparable to freshman/sophomore courses at a university. The upper level courses are the types of courses one would take during their junior and senior years and are beyond the scope of a community college, and certainly well beyond the scope of high school. There are a handful of private high schools out there that will go through a whole calculus sequence and into some upper level mathematics, but those are exceedingly rare. </p>
<p>Some of the concepts can get pretty difficult but I don’t feel like math is as rigorous as some of the engineering or science majors because it doesn’t have labs. My school has multiple math concentrations which include: Actuarial Sciencies, Statistics, Computational, Applied, Pure, and Teaching. I will be graduating with around 3-4 proof courses. If I was doing the pure mathematics option, it would be a bit more. I pretty comfortably dominated math courses from elementary through high school but I am challenged in college. Most math majors finish calculus their first year so doing well in calculus far from guarantees you will be good at math but it is true that many of the upper division math classes have concepts that require knowledge of calculus to do. Usually in that case, the calculus is easy because the course is not intended to judge how well you know calculus (you demonstrated that when you took calculus). I suggest you spend a lot of time looking into employment opportunities and careers in mathematics to see if it would interest you. Just being good at mathematics is very far from enough. You need to be passionate about it and enjoy it.</p>
<p>@SadHippo @comfortablycurt Are all of these abstract courses useful or applied? I do enjoy math, but I haven’t really looked deep into what I want to use for in the future, thus I might minor in math and major in Interior Design. Competition is very extent and so many people have started challenging themselves at young age and continued their advances, thus far superior than me, an average person who thinks he’s good at math. LOL, I’m not disparaging myself. Just wanting to get opinions from math majors or college students. </p>
<p>Pure mathematics courses aren’t necessarily ‘applied’ as such. Pure math often exists solely for its own sake. It may not have any usefulness in the “real world,” and serves the sole purpose of being mathematics. That said, many forms of pure, abstract math has proven very useful after the fact. The Lorentz Transformation existed before Einstein formed the Theory of Relativity, but it became crucial to the functionality of his theory. Knot theory was an obscure branch of pure mathematics that later proved crucial to understanding the helical structures of DNA. </p>
<p>SadHippo makes a good point. Many upper level courses will still use calculus. Differential equations is essentially calculus, but the calculus is often taken for granted in a Diff EQ course. I’m in differential equations right now, and we generally skip the more basic calculus in the in-class examples. There’s a lot of integration involved and it can sometimes get pretty time consuming. The professor likes to say “and then a miracle occurs, and the function integrates to this…” It’s assumed that you know how to integrate a function in a plethora of ways. It’s assumed that you know how to do all of the basic algebra involved in manipulating an equation. Some of the problems in differential equations can get very lengthy, and it’s not because of the process of solving the actual differential equation. It’s because of the algebra more than anything. My professor even says that a better name for this course would be “a ton of algebra, with a little bit of differential equations.” </p>
<p>If you’re interesting in majoring in pure mathematics, it has to be due to the fact that you love mathematics purely for its own sake, rather than it’s usefulness or application. Many areas of mathematics are obviously very applicable to the real world, but a lot of it has no connection to the real world, and could never even exist in any real way. I’m a physics major, so the areas of mathematics that appeal most to me are the ones that are most relevant to physics. I love pure math too, but the majority of my math courses (I’m doing a math minor) are going to be focused on more applied aspects of math. I’m doing a couple of purely theoretical math courses as well though, plus a course focused entirely on proof writing. </p>
<p>I don’t know much about pure mathematics or usefulness of proofs. It is not my specialization and I don’t like it so I would not be the right person to give advice on it. </p>
<p><em>proceeds to give advice</em></p>
<p>I can say that of the proof courses I have taken, I do not recall any scenario in which a professor has mentioned which industry or where in the real world that this stuff is used. I would guess that of the math concentrations, it would be less marketable to have a pure mathematics degree than a different type of mathematics degree.</p>
<p>I would say that after calculus, the most closely related math course for me was Differential Equations because a greater percentage of material in that class required knowledge of calculus than any of the other 12ish math courses I’ve taken since. The calculus sequence at my school has the option of doing it in three semesters or two semesters. Both covered the exact same material but the two semester version is a lot faster and intended for the people good at math. The first calculus course was called “Differential and Integral Calculus” this course was extremely important for many math courses to come. Many future math courses expect you to know how to do derivatives and integrals as well as what they actually mean conceptually. The last course in my two semester calculus sequence was called “Sequences, Series, and Multivariable Calculus.” This course was not nearly as useful or important as the differential and integral calculus. It is extremely important to understand what a series is and what a sequence is but the problems we were given about them in class seemed unimportant. Taylor Series and portions of the multivariable calculus were the only things important from that class that were necessary for me to be prepared for future math classes.</p>
<p>If you want to know the biggest difference between high school and college math, it’s the pace. Professors teach stuff a lot faster, assume you know a lot more, and work out far less examples.</p>
<p>Edit: I forgot to answer your question about memorizing. I have taken around 40 hours of mathematics now and I remember there was a lot of memorizing in Differential Equations and Vector Calculus. I felt like a large percentage of subjects in both of those you were able to either memorize the formula or memorize the steps needed to solve a problem of each type without needing to understand why they work in order to get an A. I guess that’s not bad if it was just those two courses which made up 8 credit hours of the 40 I’ve taken so far.</p>
<p>Sequences and series can have a lot of utility in the real world. In physics, we use Taylor Series and Power Series Expansions quite often to approximate solutions. There are alternative ways of doing it that are sometimes simpler, but it often ends up being the case that a Taylor Series will get us close enough to the real answer. </p>
<p>Pure math tends to have fewer paths or career options than a more applied math degree. Pure math is a route to becoming a mathematics professor or a researcher. While it’s true that the logical thinking and problem solving that is taught in a pure math degree is beneficial in many areas, it doesn’t offer the versatile problem solving skill set that would accompany a physics degree in most instances. A math degree is more narrow in scope. This is why physics majors tend to have a lot more possibilities for career paths than many math majors. This obviously isn’t true in all instances though. </p>
<p>I hated math in high school. I definitely wanted to do a degree in journalism. Interests change. I wanted to do physics cuz I read a book by Richard Feynman.</p>
<p>I recently graduated with a B.S. in Mathematics, Applied Mathematics/Statistics Option. I think it really depends on your work ethic and somewhat on how smart you are. Obviously, the smarter you are, the less time you have to spend memorizing, figuring things out, or pouring over your lecture notes. </p>
<p>I think the math at community college was one huge joke. The courses were easy unless you had a teacher that was really harsh and unforgiving. There is an intellectual jump from community college to university for most people. The gap varies. I was the kind of student who didn’t really know what he wanted to do with his life. I thought I had it all figured out that I was going to be a physicist and that was that. Well, I ended up switching my major from physics to mathematics after my first semester. Go figure. It’s not that I didn’t like physics either. When I was transferring, I was either going to be a physics major at my girlfriend’s school or a double major in physics and mathematics at the school I ended up graduating from. Well, the school I ended up graduating from decided to institute a policy that you couldn’t do a double major unless you could finish both within the same degree time, which was not feasible for me. So, I stuck with my single major. I got to tell you, I was pretty dorky as a 19 year old. My favorite mathematician at the time was Evariste Galois. He definitely inspired me to want to do mathematics. I think I read a book called Men of Mathematics or something like that, I don’t remember. But I was very inspired to pursue the major. I was a very lazy student who never worked hard in any of his classes during community college. I did not know how to study.</p>
<p>Well, university was a real wake up call because I ended up getting bad grades my first quarter of university. I guess I had to study? Well, turns out, I needed to study, but I didn’t need to study very much for my Applied Mathematics courses. Although, partial differential equations is the exception. I had always liked this concept of “proving” things. I absolutely loved proving identities in trigonometry. I also loved it when my calculus professors would step away from the engineering plug and chug teaching and give proofs to anyone who dared try it. I always did them. </p>
<p>Well, I ended up taking an intro to proofs class. I liked the homework a lot. What really kept me on my game was the fact that she’d pick on people who she thought were bright students and while we were learning something we’d have to tell her what the next steps were and what to put in the proof. Or she’d make you go up to the board and finish a proof or present a proof you had done in the homework. She’d also get you into groups to solve proofs together or think of geometric proofs in odd ways. I wasn’t interested by a good portion of it, but I was ahead of the average student. I’m no genius by any means. </p>
<p>After that class, I had to take a ton of proof classes to finish out my degree. I think when I took Real Analysis I really struggled in the beginning, but I rocked the midterm and final exams with a good couple days of studying. I think what really kept me on topic were the weekly quizzes, so that’d force me to look over my notes for an hour or so. Abstract Algebra came very naturally to me. I’d get a text from a friend of mine who was fishing for answers to a homework assignment and they’d be with 9 other people who have no idea how to figure out the problem…then I’d work on it, figure it out, and then tell them what I thought. I mean, my advice to you is to put down what makes sense to you and not just follow what someone else is doing. I never put down what someone else thought when it made absolutely no sense to me. I always did the extra credit too at the end of tests, which I almost always got 100% right, which I think comes from liking proofs and being motivated to think hard about something. If you like doing that, awesome. I think your brain has to work in a certain way for everything to gel. A lot of people just never get it.</p>
<p>Anyways, on that note, people did really bad in these proof classes. At least 75% of the people were failing for every proof class I took, each with different professors (except the intro one). Curves for classes would vary by professor. I did pretty good, but it helps to have an interest in the subject. A lot of people just wanted to become teachers, which was not really satisfying to me. It was stressful at times to take tests, but they were usually not that bad. My friends I studied with later on always failed them, which was funny. Sometimes I felt the professors graded me more harshly than them. I’d explain as much as I could to them, but it’s not like you can hold someone’s hand throughout the test. They have to be able to think on their feet and come up with proofs on the spot. I am the kind of person who thrives on high pressure situations. Unfortunately, towards the end I became severely anti-math major. I think this influenced my two main study partners because one ended up switching into a business major and the other is still finishing his degree since he only had one more elective to finish, but he’s going to be doing a master’s in computer science. He’ll be working in the software industry as well. I did not like what I was doing anymore, but I was only a few weeks shy of graduating. I think I was just tired of doing what I felt were a waste of time for me (abstract algebra proofs). I don’t think liking proofs will take you very far unless you’re going to become a professor. I’ve since then started working in software because I didn’t really gain any marketable skills (they took me on with no previous job experience other than an internship I did as a trader, which I hated but was good at) and am coming back in the fall to do a master’s in aerospace engineering. </p>
<p>I would say that there are some things you cannot get around memorizing because knowing the exact definition of something can be useful. However, you truly need to understand the concepts to be able to apply things to a given situation. If you’re a logical and creative person, mathematics can be fun. A lot of people freaked out over memorizing 70 some definitions for the test, any of which could be used in a proof we had to do. Honestly, I didn’t just memorize, I knew the very important ones in my head because I nailed the concept down forwards and backwards. I didn’t waste a lot of time memorizing unless a professor wanted something verbatim spit back to them. </p>
<p>Lack of examples suck but you learn to look in other places where you might find some. </p>
<p>What I enjoyed most were my upper-divison differential equations classes as well as mathematical physics. Numerical methods was pretty good too. I absolutely hated my statistics classes (I took a good portion and I was at least top 3 in each class). One of my friends, who I consider incredibly bright, was doing his degree in Mechanical Engineering and working on several things I thought were interesting. I liked how close knit the engineering community became and I was included in it as a math major. I think it is important to realize interests change. I definitely remember not wanting to become an engineer at 18. I don’t regret my math degree, but I know what direction I want to take my career in now. I don’t like that I didn’t get a double major in physics & math, but I think I’ll get an adequate amount of physics in a master’s degree in engineering to satisfy me.</p>
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<p>Is the intellectual jump more due to the difference between lower division courses and proof-oriented upper division courses?</p>
<p>@CalDud thanks bro, reasoning and proving is a nice way to have and it does mean something to me because I don’t just like memorizing formulas but I have always thought that knowing the roots of theories or formulas are important because you never what math problem or test will intake much more than memorization. Like, at my community college, my professors go by the book, I mean Literally by the book, like the Pearson edition Textbook is where they get their content and HW questions, and the exams are derive from questions that we did on the Textbook. I’m a HS senior I will take advantage of this but at the end I do see the importance of understand the depth of mathematics.
@SadHippo thanks,for your accounts of your experience, and good luck. </p>
<p>I don’t feel that the math at community college is necessarily a joke. I think that’s a drastically generalized statement. People often claim that community college math/science classes are subpar. I don’t see it personally. Of course this is going to vary from school to school. </p>
<p>My math professors at community college have all been top notch. They’re passionate about the subject, know it like the back of their hand, and do an amazing job of developing the theory and then applying it. There are just as many math classes at universities that are a joke. How many threads have been posted on here from university students who are complaining that their math professors are horrible, don’t care about teaching, are more worried about their research, teach by the book etc? A lot. </p>
<p>There is obviously an intellectual jump in math after transferring. You’re moving from the lower level courses into the upper level courses. By their very nature, there is an intellectual jump involved. University students experience that same jump when moving from their sophomore to junior years.</p>
<p>I wasn’t speaking for all community colleges, just the one I attended. My first semester I took pre-calculus, which was one of the hardest classes I ever took after not having taken math in two years. I think there were only 15 students in the class by the end. Actually, maybe even half that. However, after that there was a steep drop off. My Calculus I professor would miss weeks at a time. He was in the process of trying to get married and spending time with his bride-to-be in India. No substitute. The homework was on the honor system, so he assumed that everyone would grade their own assignments fairly without question. There were quizzes, but they were very watered down problems. The final was a cakewalk and should’ve been for anyone since it tested the basics… I think he got his master’s in electrical engineering at Purdue and did engineering physics during undergrad. I don’t know why I remember that. I thought this would really mess me up once I got to Calculus II. My Calculus II teacher was okay. She was from Greece and did her master’s at UCLA. The class was less of a joke because of the way she made it hard on everyone because it was a summer class, but it was still relatively easy. She had us show her our transcripts to prove we did not get less than an A or B. She said that anyone with a C would fail her class and a B would barely make it. I did the homework the day before the tests. A friend of mine never did it at all and sometimes got higher marks than I did. A huge percentage of the class got high grades. She’d give us two quizzes and then a test each week consisting of 35-45 questions. But the material was all plug and chug for the most part and I hate that. Where’s the creativity? I understand though as I think that community colleges are OK for people preparing to become engineers. The homework she assigned was enough for me without really looking over the material too much. My Calculus III class I don’t remember ever studying for beyond being in class or ever doing the homework, so that was one big joke to me. I don’t know where he got his training from, but he made many mistakes during the lecture. He kept a tally of it too. He’d stop mid lecture because he couldn’t figure out how to finish a proof that he wanted to show us or solve an integral and let us go home. Or someone would ask him a question and he would not know how to answer it. Differential Equations (sophomore level) I never studied for except the day before the final. I ditched a lot. Same kind of deal as Calculus III but different teacher. Linear Algebra actually was quite tough, but that’s because my teacher was preparing us for a four year university, in my opinion. I think there were 45 of us when we started and only 10 or so by the time it ended. That class was proofs big time. She was a very good teacher, but she was a brutal grader. I think 2 of us went to Cal Poly Pomona, 1 MIT, the rest UC Berkeley or UCLA. I think I was the only math major out of that group though, everyone else wanted to do Mechanical Engineering. She made us do hundreds of problems each week from the textbook, gave us her sheets with her own problems, and gave us practice tests which were more difficult, and then the tests you had to be extremely quick with computation (probably 45 problems or so minimum, quite a few long computations) so you could crank out the proofs. </p>
<p>When I say university was challenging for me at first, it was probably due to a number of things. Not that the material was necessarily harder at first, it was a huge adjustment for me to move 3 hours away from home and actually have to spend more than a few hours of my time on homework when I didn’t want to. I’d frequently procrastinate from doing it and do it all in one night before they were due. I had not developed proper studying techniques. It was also harder for me because I was losing interest in mathematics fast. I was thinking heavily of becoming a lawyer and I was taking applied math classes that I really did not care for (operations research I and II, never again), so much that I could barely sit through a lecture of it. I think I missed about 75% of the lectures my first two quarters of my math classes because they were just so boring and uninteresting. I still had that habit of doing my homework the day before or even the day of. Or doing math projects the night before that should’ve taken more than 4 people to split up the work. I was stressed all the time and unconfident in my skills as a math major since for some reason I thought you simply just have to be good at it and barely study. Somehow I managed. Still, I considered what I had learned in community college insufficient since it seemed like at the time everyone was so much further along that they knew things from being there originally whereas I had to fill in the gaps of my knowledge. It also kind of made me feel afraid of the upper-divison classes (in proofs) because many of the people that I considered smarter than me that were in the classes I was taking had failed those classes.</p>
<p>It wasn’t until I hit the proofs classes at university did things become easier for me. They were more challenging, but I was better at them and I was finally taking all math classes instead of having a mix of whatever other classes that were history, psychology, or whatever. I actually cared about what I was learning about and would read the material for my classes. I was paying more attention in my applied math classes that I actually thought were fun and blowing off classes that I thought weren’t (basically for three statistics classes I probably showed up the day of the midterm and the day of the final for each one (i did top 3 most times), although I had taken more statistics classes that I showed up to every day but studied only the day before). I think I showed up twice to the lectures of Real Analysis (there were 2 sections) sometimes and went to most of the study sessions my professor held before exams. That class I was super into. I think I studied for Complex Analysis twice the whole quarter (I took it after Real Analysis I). Real Analysis and Abstract Algebra I studied quite a bit because they were interesting and challenging. I spent quite a bit of my time studying my senior year versus barely any my junior year. It was a wake up call, though. Most of the time I was thinking “What am I going to do with this degree?”</p>
<p>The physics classes at my community college were in worse shape. I only went to class for the labs for my first physics class and we didn’t even have a final because the teacher knew almost everyone was failing, even though it wasn’t all that hard. I specifically avoided taking him again for E&M because I knew I wouldn’t learn anything from him again. I took someone who was a better teacher, but he was not reasonable. He came from my alma mater and had gone to graduate school for a PhD in Physics at UCLA after doing Mathematics for undergrad. I question some of his teaching style, though. He’d play video games while he’d pick a random problem in the book for us to work on in groups. He’d distract you during lecture and go off on wild tangents. His tests were hard, but there was such a fat curve that you could get away without really knowing anything. My quantum mechanics professor confused the crap out of me and everyone else. No one had any idea what he was talking about for the most part and he had written the textbook himself. I had to teach myself quantum physics alongside what I was taking in university because I was severely lacking the prerequisites for that class (astrophysics), even though I had it. Somehow I managed. He kept saying “When you all get to UCLA, this class counts for 5 classes! But retake them again if you like! I took all of them.” Of course he did, he went there from the beginning. </p>