I've recently considered pursuing a degree in one of the math, science, or engineering fields. I have a degree in economics, but I have never been strong in math. Although I do enjoy math, but I never had that "natural" talent. When speaking about my plans everyone keeps brining up "are you good at math?" As if math can't be learned without some herculean effort.
What are your thoughts on this idea? I feel like it goes into the whole left brain vs right brain idea, which I have heard is erroneous.
I think that almost anyone can learn the math required to be an engineer (speaking as an engineer). It's just like any other skill that must be learned. The reason that I feel many people state that they are "bad at math" is they simply don't enjoy solving those types of problems.
If you don't enjoy doing something, you're certainly not going to spend much time learning that skill. Just like repairing cars, cooking, making music, or any number of other things, people tend to persue math if they enjoy it.
That's not to say that people don't have natural talent; they do. However, I think that most people have the ability to learn how to use the calculus, differential equations, and physics skills requried to be an engineer. If you like technology, tinkering, and figuring things out, I think it's a great career.
I agree, I studied math at the end of my degree, best decision ever.
Personally, i think people just suck at teaching math and it discourages people from ever trying.
Although I do enjoy math, but I never had that "natural" talent.
Have you observed it in others? Did you have math competitions at school?
Simply put like the other posters have said, if you enjoy it there's no reason you won't develop all the skills necessary. Practice has always been the secret to mastery in nearly any field- plus with all the resources splattered all over the internet to help you learn this kind of stuff, you have nothing but good luck on your side.
Natural "talent" in something like math is simply a bit of better intuition some people may have had while being involved in math that you haven't yet. But throw yourself into solving enough problems, and it'll all come to you naturally till everyone will start asking you how long you've had this "natural talent" in math.
I agree with the above poster who pointed out about enjoying learning and doing something as an important motivation. I don't know if i'm a good example since I had a bit of a natural talent for maths as a kid, but this might have equally well been due to early exposure from my dad for both me and my sister(he typically taught us what we were to later learn at school 1-2 years in advance). My real interest in the subject came later though as a motivation to understand physics, and this allowed me to enjoy learning it at a higher level.
That being said, mathematical economics sometimes almost bores me to tears. I don't think I would have liked maths as much if I had to learn it originally to do economics so I think I can sympathise with you on that. I wouldn't say a lack of a formal background should necessarily be a barrier either. One of the econ Phds at my current university who most impressed me with his mathematical abillity had a history degree as his bachelors.
With tenacity and enthusiasm you can be unstoppable.
"When the King is far the people are happy." Chinese proverb
For Alexander Zinoviev and the free market there is a shared delight:
"Where there are problems there is life."
In high school I hated economics. Now, I love economics. I was considering doing a math minor, but decided it would add too much time to my undergrad studies and I've already wasted too much time as an undergrad (super super senior status). It's a question of opening your mind and enjoying learning something new.
I agree, I studied math at the end of my degree, best decision ever. Personally, i think people just suck at teaching math and it discourages people from ever trying.
I agree with this. I learned more math from studying the history and philosophy of math than I did in any of my math classes.
In States a fresh law is looked upon as a remedy for evil. Instead of themselves altering what is bad, people begin by demanding a law to alter it. ... In short, a law everywhere and for everything!
~Peter Kropotkin
lol ditto, not that it is still anywhere near my radar of most thought about things. I think my aversion of all things math in school was that it wasbeing taught as a contextless abstraction.
Your better off learning music theory/ an insnturment, a second language, or logic.
"lol ditto, not that it is still anywhere near my radar of most thought about things. I think my aversion of all things math in school was that it wasbeing taught as a contextless abstraction.
Your better off learning music theory/ an insnturment, a second language, or logic. "
Interesting...i've played trumpet and guitar since i was 12, i speak french fluently, and love debate; makes sense.
I know for me, speaking as a college student majoring in Mechanical engineering, just finished my first year, that there is a vast difference in mathematical "skill/ability" among my classmates.
I was lucky and was able to take Calc I and II in high school, so I just finished Linear Algebra/Differential Equations this semester. But even at my school, people majoring in math- same degree as the math professors, are far and few between- my roommate is one of about 10 undergrads in the entire school of ~5000.
This does vary though as the actual math courses seem to be more involved of the why it is- although my LADE teacher just said it is what it is because it works for maintaining independence in the terms(by multiplying by ln(x) or x, based on the case. In my physics courses it seems more important that you can get from point A to the answer- the in between is up to you, especially as you can use a calculator there and in the math courses no calculator or formula sheets allowed- even with Laplace transforms.
This affects the students are there are some who seem to get the basic principles and fail to succeed in math, but succeed in the science side- I've been lucky/taught/self-taught just to power through the stuff.
im not a natural when it comes to math but find that solving something previously out of my reach to be greatly satisfying. my career in the field of laser and plasma tech i do a bit more math than im suited for. that being said... repetition works. repetition works. kahn academy on youtube helps.
Laotzu del Zinn: I agree with this. I learned more math from studying the history and philosophy of math than I did in any of my math classes. That is so fascinating! Which book did you like the most? Your experience agrees with this article: Final exams were coming, but I still could not afford the textbook for Botany 100...With nothing to cram from, I made my way to the library. But finding no botany texts in the stacks, I checked out the autobiography of Gregor Mendel, the “father” of genetics, whose field had been the major emphasis of Dr. Baker, my instructor. So while my classmates were poring over the graphs and study questions in the textbook, I lay on my bed, reading about Mendel’s childhood illnesses, his interest in plants, and his stream of consciousness musings. When I finally got to Mendel’s experiments growing green peas, I was pretty into it, having accompanied him on his “journey” since Day 1. “You received a perfect score on the genetics section,” said Dr. Baker. “That’s never happened.” I was no science genius. The D earned later in zoology confirmed that fact. But acing the test made me wonder: Were textbooks necessary? Could students succeed while saving money, using free resources from the library, or from life? Check out my video, Ron Paul vs Lincoln! And share my PowerPoint with your favorite neo-con | Post Points: 5
So while my classmates were poring over the graphs and study questions in the textbook, I lay on my bed, reading about Mendel’s childhood illnesses, his interest in plants, and his stream of consciousness musings.
When I finally got to Mendel’s experiments growing green peas, I was pretty into it, having accompanied him on his “journey” since Day 1.
“You received a perfect score on the genetics section,” said Dr. Baker. “That’s never happened.”
I was no science genius. The D earned later in zoology confirmed that fact.
But acing the test made me wonder: Were textbooks necessary? Could students succeed while saving money, using free resources from the library, or from life?
Most math instruction and math books are obfuscatory. I have no idea why except, perhaps, that the teachers and authors themselves were subjected to obfuscatory mathematical instruction and never really grasped the subjects themselves and so they are just aping whatever they know for sure isn't wrong.
My recommendation is that, in addition to your regular studies, you pick up a particular field of mathematics which has a strong foundation in intuition (say, Euclidean geometry or number theory or complex analysis, etc.) and then read old, out-of-print books on the subject form your university library, do the book problems and just generally tinker with the subject on your own until you feel you really have a grasp on it. Just don't ever let it go, keep tinkering. I play with integer sequences, for example. They're loads of fun and you can learn (maybe even discover) interesting things along the way. For example, I found an interesting fact about a certain kind of generalized Fibonacci sequences (turns out, it's already been discovered but it's a recent discovery!). The Fibonacci sequence is generated by beginning with 1, 1 and then adding the last two numbers to generate the next:
1, 1, 2, 3, 5, 8, 13, 21, ...
If you wanted, you could also start with 1, 1, 1 and add the first and third of the last three numbers, like so:
1, 1, 1, 2, 3, 4, 6, 9, 13, 19, ...
This is not a Fibonacci sequence but it looks a lot like one. I'll call it the Fraggle sequence. One of the amazing facts about the Fibonacci sequence is that you can take the ratio of any two numbers and plot them and they tend toward a limit:
1/1, 2/1, 3/2, 5/3, 8/5, 13/8, 21/13, ...
These fractions tend toward a number and the number to which they tend is... brace yourself... the Golden Mean!
1.61803
This number is often called phi (the Greek letter). One of the remarkable facts about phi is the following:
phin-1 + phin = phin+1
But this isn't even the best part. The formal definition of the Fibonacci sequence is the following:
an-1 + an = an+1
Do you see the formal similarity between the Fibonacci sequence definition and the relation between powers of phi?
If you take the ratios of the Fraggle sequence, you will find that these ratios also tend to a number, slightly smaller than phi. I'll call it frag. I haven't formally proved this but I worked out numerically that the following is true of frag (it has been proven by someone else):
fragn-2 + fragn = fragn+1
Now here's the coup de grace... look at the formal definition of the Fraggle sequence:
an-2 + an = an+1
And there are a whole class of numbers which conform to this pattern. In other words, for each sequence of the form:
an-k + an = an+1
... there is a real number x such that the following relation holds:
xn-k + xn = xn+1
This is a very remarkable fact to me since you are getting a real number from the ratios of an integer sequence with formal properties that mimic the formal definition of the original integer sequence! How cool is that!
Refs:
http://www.maths.surrey.ac.uk/hosted-sites/R.Knott/Fibonacci/phi.html
http://oeis.org/A000045
Clayton -
My practical suggestions:
1. Avoid using calculators as much as possible so you are forced to learn about prime numbers and calculating techniques
2. Use common sense and substitute tiny numbers to help figure out limits and derivatives; don't worry about logic or rigorousness. Accept Leibniz's infinitesimal numbers, e.g. (x+dx)^2, where dx is an infinitesimal number, is equal to x^2+2xdx+dx^2. Since the latter term dx^2 is infintely smaller than the former terms, it can be discarded, leaving x^2+2xdx. Thus, if x increases by dx to become x+dx, then x^2 increases by 2x dx; the derivative is then 2x. Infinitesimals caused endless pointless quibbling until Robinson's non-standard analysis put them on a firm footing anyway.
3. Learn Taylor series as they are extremely useful.
4. Learn the geometric series, 1/(1-x)=1+x+x^2+x^3+...
5. Accept Euler's divergent series, e.g.
1/(1-x)=1+x+x^2+x^3+...
Differentiating:
1/(1-x)^2=1+2x+3x^2+4x^3+...
Substituting x=-1 yields
1/4 = 1 - 2 + 3 - 4 + ...
Another example; it was either Goldbach or Euler (I can't remember) who computed the sum of the reciprocals of each number one less than a power
1/3+1/7+1/8+1/15+1/24+1/26+... = 1
and it can be proven easily using the divergent Harmonic series s=1+1/2+1/3+1/4+...
Mathematicians hate divergent series despite the best mathematicians ever (IMO) Euler using them like crazy. They are also used all the time in quantum field theory.
6. Learn the Zeta function / Bernoulli numbers and the Euler-Maclaurin formula as it's very useful for calculating
Math is not hard.
Everything else is ridiculously easy.
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"Even when leftists talk about discrimination and sexism, they're damn well talking about the results of the economic system" ~Neodoxy
When people ask whether you're good at math, 9 times out of 10 they really mean are you good at fractions, division and multiplication.
A lot of kids, and girls especially, totally lose interest in mathematics based on the way it's taught in elementary education. By 3rd or 4th grade, being "good at math" is more of an insult than a compliment.
Anyone can learn even the most complex, established mathematics subjects. Formulating advanced theories is quite a bit more difficult, and takes real talent. The most important qualities for any student of mathematics are patience and persistence.
My impression of the mathematics as it's taught and used in physics, engineering and especially economics is that it tends to be less rigorous in the proofs, and often skips over many steps in the process. But then my concentration was in mathematical theory where you were heavily penalized for skipping over such details. I can definitely see, however, how a student can get lost when their instructor doesn't cover the complete proof. It left me with the impression than these physicists, engineering and economics students were getting short changed.
Scroge - Honestly, get a book called How to Prove it,
Read it, then decide
I find it to be an issue of context and usage. If you learn a foreign language then never speak it, your skills dimish and/or vanish. Likewise, if you speak to someone with a bizarre dialect, you may not understand them despite speaking the same language.
I studied BioChemistry. You use a great deal of algebreic manipulations and a fair amount of Calculus within that subject. I did very well in all of those classes, but when placed into my required maths, I barely passed and even failed a few times. Considering that my major required maths all the way up to Calculus III, I struggled for quite some time.
I found this product line at www.mathtutordvd.com and it literally changed the entire game for me. For $130 USD I purchased lessons all the way from advanced college algebra to Calculus III. The few months of self study with these exams has even me a superior education to what would have taken me 6 semesters in a university setting to aquire. I simply CLEP'ed through everything. I tried highly regarded books, private tutors, everything. These DVDs were like magic. I couldn't more highly recommend them.
baxter: Mathematicians hate divergent series despite the best mathematicians ever (IMO) Euler using them like crazy. They are also used all the time in quantum field theory.
Good point. The greatest mathematicians of all time have been very comfortable using "non-standard" methods to arrive at conclusions that have taken years for the more "conservative" mathematicians to ground on the orthodox system (ZFC). There is this huge Aristotelean hang-up over working with "actual" infinities versus "potential" infinities which is precisely why Leibniz's infinitesimals are considered "un-rigorous" even though there is nothing un-rigorous about them, which is essentially what Robinson's non-standard analysis shows... if you rigorously define the concept of an infinitesimal and then manipulate it according to its algebraic rules, guess what, you get sensible answers that correspond to mainstream calculus with 1/10th the labor of taking the limit of everything and applying epsilon-delta arguments. There is a field of mathematics (large cardinal theory) that takes actual infinities as proper objects of discussion and manipulates them like any other mathematical object.
Another big hang up is on the concept of incompleteness. Mathematicians are still running around nearly a century after Godel proved the limits of formal systems acting like all we need is the "right set of axioms" and everything will just become clear. The reality is that no set of axioms will exhaust mathematical truth since there are always true mathematical statements which are not provable from a given set of axioms. Contemporary mathematician Gregory Chaitin has significantly extended Godel's results to show that almost all true mathematical statements are not provable from a given set of axioms. A whole new field of mathematics (algorithmic information theory) has grown up around these results. But despite these well-established results, mainstream mathematicians (the analysis/group theory/topology types) continue to act as if the goal of mathematics is the search for the perfect set of axioms and, once these are found, mathematics will be "finished", that is, it will be solved like the game of checkers. Godel, Turing, Chaitin and others have conclusively proved this is not true.
@baxter: If you're not already familiar with them, take a look at p-adic numbers... divergent series are the norm there but the metric works out such that what would be divergent series in a standard metric converge... for example, 2^1+2^2+2^3+...+2^n = -1 (sum as n->oo) in 2-adic. I'm not finding enough online resources for p-adics so I'm going to have to buy a book. My interest in p-adics comes from my profession as a computer engineer... it turns out that truncated 2-adic numbers are precisely how we represent integers in the machine (also known as "2's complement" in computer lingo) and there is something so compellingly elegant about these numbers that I just have to understand more about them.
Thanks for the responses. Clayton and others, do you have high level formal math training, or do you study it more as a hobby?
I should clarify about not being good at math. I can add, subtract, multiply, divide, etc in my head very quickly. It is with the rules of algebra that I struggle. I managed to make it through college level algebra, precalculus, trig, and "business calculus." Even if I do not pursue another degree, I would like to be able to look at complex equations and understand them and not be intimidated by them. I would like to do this with equations for both math and physics. For example, I would like to be able to understand the equations which explain M-theory.
Once again, I would suggest reading How to Prove it, by Velleman. It lays a great foundation for learning linear algebra, number systems through logic.
If i had read that book before Linear Algebra, i would have gotten an A instead of a C+(which was still higher than most in the class)
It might help you a little in calc 2, calc 3 for sure.
Myself, I have a bachelor of science in mathematics and in computer science, and I'm now getting my master's in actuarial science. I started my undergrad in pre-pharm and, when taking a Calc I class, discovered that math was pretty fun.
Linear algebra is one of those courses they use to weed out students. Of course it depends on who's teaching it.
I have a minor in mathematics. I personally recommend that you take up some kind of programming language. My personal favorite is Perl (you can download and run ActivePerl on your local machine if you have Windows, it's standard on most Linux systems) but you might consider Ruby as a more elegant language. A programming language will teach you to think about variables, which is what the "x" in an algebraic equation like "x^2+3x-7=0" really is. Once you can fully wrap your mind all the way around the idea of a variable (the programming will help you with that), then you will find it much less intimidating to look at equations filled with variables. The next step will be learning to differentiate between the common kinds of variables and the common conventions for differentiating between them... i.e. x and y are frequently used for real-numbers where as i, n and k are frequently used for natural numbers, especially as indices into summations, etc.
Message me if you are interested in programming, I can provide any assistance you might need.
I find tensor calculus and the calculus of variations to be very useful tools for solving field problems in engineering, but I think that most engineers just stick with plane old calculus and differential equations in their day-to-day work.
These days, there are handy computer programs that do a lot of the 'behind-the-scenes' math for engineers when solving more complex problems.
Anyhow, you shouldn't be scared off by math. Again, it's simply a skill that must be learned. Folks like Euler and Newton have already figured out the hard bits of it, so for most people it's simply a matter of learning how to use the tools that they gave us.
Also, per the above post about learning programming: do it. Computer programming is a great tool for solving tough problems. Also, there are great programs out there that are quite useful for high-level mathematics problems. Look into Maxima/wxMaxima and Octave/qtOctave. They're free and similar to Maple and Matlab respectively. In engineering, I find Octave to be the most useful to me (Matlab is even better, but pricy). Maxima, however, does symbolic math, which can be quite handy for mathematicians.
"Linear algebra is one of those courses they use to weed out students. Of course it depends on who's teaching it."
I agree, its kind of a turning point for a lot of people. In my class especially; we had a teacher who didnt help you at all after class, and he was such a stickler for "writing correct mathamatics,"
You could have the right answer but if the logical for wasnt correct, or if you put "there exists" in stead of "For all." You lost all the points.
>The greatest mathematicians of all time have been very comfortable using "non-standard" methods
Yes, Euler and Ramanujan used divergent series a lot. The Euler Archive http://www.math.dartmouth.edu/~euler/index.html is a great place to learn math and you'll see plenty of divergent series there. And of course infintesimals/fluxions were of course non-standard when Leibniz and Newton toyed with them to develop calculus. Read "Where does Mathematics Come From?" for an eye-opening experience on the thought processes that people use to do mathematics. It's based cognitive neuroscience (maybe pseudoscience?). It makes math feel a lot more like a human invention or a priori logical structure of the mind rather than some kind of Platonic math universe.
>the orthodox system (ZFC)
No such thing! A lot of mathematicians reject the Axiom of Choice. A lot of mathematicians disparage the concept of proof. On the huge disputes raging in the history of mathematics, ioncluding unearthed defects like Godel's discovery, read "Mathematics: The Loss of Certainty".
>@baxter: If you're not already familiar with them, take a look at p-adic numbers
I'm aware of them but I'm too steeped in Euler's style (simply writing and summing divergent series) to see the merit in them. Maybe I'm missing something, but they just seem like divergent series dressed up to look less objectionable but very awkward.
>it turns out that truncated 2-adic numbers are precisely how we represent integers in the machine
Yes, -1 in 2's complement is the infinite sequence ...11111 = 1 + 2 + 4 + 8 + 16 + ... = -1
Anything seems difficult when you are not good at it. Anything seems easy when you are good at it.
@baxter: Yeah, the p-adic literature is pretty impenetrable. I like p-adic numbers because they actually form a number system and you can do addition, subtraction, etc. on them just like you would decimal or binary numbers. In addition, they are, at least in some sense, more "natural" in that they include negative as well as positive numbers where standard decimal or binary requires the use of an external "minus sign" to indicate the positivity or negativity of a number.
IIRC p-adic numbers have some ambiguity. Here's an example
Since -1/3 = 1/(1-4) =1+4+16+64+...=....101010101xb
Then 2/3 = ...101010101xb + 1 = ...101010110xb
But 2/3 can also be represented as 0.1010101...xb
Some people have a higher aptitude for math than others, but I've noticed that anyone can learn any level of math as long as they (1) are very careful with definitions and (2) don't get intimidated no matter how many mistakes they make. People that hate math usually have a problem with one of those two.
Why anarchy fails
AJ, can you elaborate on (1)? As I think more about this topic it seems apparent that individuals wrap math in a sheet of mysticism. I don't know if it is because these individuals are mathematically ignorant and/or that some math is so complex and specialized that it intimidates some and seems unknowable. But given that math is reason and logic, it should be knowable by all.
@Scrooge: I think there are two different sorts of math. The first sort of math is more constructive in nature, it is math that you can easily build a machine to perform. All forms of numerical calculation fall into this category.
The second sort of math is less constructive, it's more definitional and I think this kind of math is more contextual to the human brain, specifically language. When you speak of a "sphere" for example, an image pops into your mind, maybe something like a super-thin, see-through basketball. From this image, you are able to "visualize" the rules of the geometry of spheres. Then, these rules which you began with (which you could visualize in your head) can be generalized to other dimensions which you may not be able to visualize... a two-dimensional sphere (circle), one-dimensional sphere (line segment), zero-dimensional sphere (point), four-dimensional sphere (hard to visualize) and so on. The rules of the geometry of spheres were not chosen because "these are the only possible rules" but, rather, they were chosen because they were the best formalization of the mental picture that we share when we speak of a "sphere in the abstract." I believe this mental sphere is a by-product of the human brain. An alien race intelligent enough to engage in mathematics and geometry would not necessarily visualize the world in the same way we do and may not choose to begin with the same rules of geometry and may derive a different, conflicting set of rules regarding the behavior of geometric spheres.
Where I think people run into trouble is in connecting the formalism (rules of symbol manipulation) with the base visualization. You might learn the rules of symbol manipulation but then it's "empty symbol manipulation" until you connect it to the visualization. But then, you might understand how mathematical objects that you can visualize interact while having difficulty translating this to the rules of symbol manipulation and thereby lose the ability to extend into what cannot be visualized.
Clayton, to your last paragraph, what level of mathematics is required to understand this? Perhaps my math knowledge is so poor that I am not understanding your point. I think I understand the gist of your post, and I see math used to explain visuals, but I don't understand math well enough to "get" those equations with regard to visualization. As I mentioned earlier, I have no idea how math explains M-theory or anything like that and I'm not sure what level of math is needed to do so.
Yes, it absolutely makes sense. I forget intermediate rules of algebra and get intimidated by big equations in calculus, trig functions, etc.
@Scrooge: Well, I think everyone is constantly using the "visualization" aspect whenever they speak to one another, especially about physical objects. Consider the surface of a table. Geometrically, it is a "bounded plane". Now, imagine that surface in your mind's eye, without the table, just the surface. Now, remove the bounds of the surface and allow it to extend indefinitely. This is a geometric plane or just "plane". Now, envision two such planes which are not parallel... they must eventually "intersect" or cross through each other. Now, envision the "joint" where there two planes intersect... what could you describe it as? We are now in a position to state a geometric fact about planes: the intersection of two planes is a line.
For a long time, arithmetic and geometry were separate disciplines. Arithmetic was an artisan skill of the merchant class and geometry was mathematics. But by the time of Rene Descartes, the foundations had been laid for the arithmetization of geometry. The key concept is imagining a line as if it were a ruler with regularly spaced numbers ordered from 0 in the middle to infinity and negative infinity in either direction. This is called "the number line." By arranging two such lines together, you can speak of "functions" and their "graphs" which are a visual, quasi-geometric representation or summary of the behavior of the function. A function is simply an equation into which you can plug one or more numbers and get one (or more) numbers out. The discipline of the study of the geometry of algebraic functions is called analytic geometry.
It turns out that an ordinary geometric plane can be thought of as the graph of a function. Imagine a 3-dimensional space with three number lines all intersecting at their zeros and each at 90 degree angles to one another (also called "axes"). Each axis is normally labeled x, y and z. There are actually three planes already defined in this space just from what we've described... the first plane is defined by the function x = 0, the second by y = 0 and the third by z = 0. Now, we can make an arithmetic statement of the geometric fact described above... the intersection of the plane x=0 and the plane y=0 is the line (x+y=0). That may not have made sense to you but my point is simply that geometric facts which you can visualize have a corresponding arithmetization and one of the first tricks you need to learn is how to switch between them.
This video is a not-half-bad introduction to analytic geometry. There are other videos linked which continue explaining more advanced concepts.
Good luck.
Yes you can be good at math. It is the same thing as having a high IQ. But you can be good at math and fail the class, or not be that good at math and stil max out your grade. Where being good at math shines through is math competitions where it is a must. We had those and the results were similar year in year out. There was a correlation to grades, but not really. Eg there were honor students who consistently failed to achieve a noteable score etc.
Scrooge McDuck: AJ, can you elaborate on (1)? As I think more about this topic it seems apparent that individuals wrap math in a sheet of mysticism. I don't know if it is because these individuals are mathematically ignorant and/or that some math is so complex and specialized that it intimidates some and seems unknowable. But given that math is reason and logic, it should be knowable by all.
As far as advice about definitions in math, there's not much to say except simply to examine definitions very carefully. Like if something seems wrong, go back to the definitions and make sure you have them straight. If you don't understand a term in a definition (or theorem), look up its definition.
Then just always assume that you misinterpreted things when you get the wrong answer, not that you can't do it. (2) I've seen what you might call "not the sharpest tools in the shed" who excel at math simply because they are excessively confident that it is all very easy, and if they mess up they just assume they misread something, never that it's beyond them. Since it is just logic - the same logic people use for doing everything in everyday life - it shouldn't be hard as long as the terminology and symbols don't throw you off.