Should an undergrad accept that some things don't make sense, or study the foundation of mathematics to resolve this?

Question

I'm a second year math student. And I've the following problem.

When I prepare myself for an exam, I can distinguish two phases. First I'm mainly interested in whatever is necessary to pass the exam. This means that I do not always read the theory too intensely, and I'm more focused on finding the "tricks" that I need to solve the problems.

Second phase: After some time, I feel like I've found most of the tricks, and at this point, I usually spend more time in understanding the theory. Just because, I know I will pass the exam, and I think it is fun to understand the theory better.

However, sometimes when I really try to understand the theory, I feel like understanding the foundation of something, doesn't make me feel I understand it better, it makes me only feel like I never really understood it. And this kind of frustrates me, and I'm not sure if it is even smart to try to understand a concept in that way as an undergraduate student.

For example, I'm studying complex analysis. I skipped the formal definition of complex numbers a little bit, and I just began making calculation and showing if mappings are conformal/holomorphic etc. Now I'm back trying to understand the construction of the complex numbers as pairs of real numbers etc. My brain is getting more and more confused the more I think about it.

My brains goes like: "If you define $\mathbb{C}$ as pairs of real numbers with complex addition and complex multiplication in the usual way. Okay that makes sense, we clearly don't have $\mathbb{R}$ as a subset of $\mathbb{C}$, but we can construct a bijection $f:\mathbb{R} \to \{(x,y):y=0\}:x\mapsto (x,0)$ that preserves every single property you want real numbers to have. So in that sense there is no difference between $\mathbb{R}$ and $\{(x,y):y=0\}$, and if we would define the real numbers as $\{(x,y):y=0\}$, then we have $\mathbb{R} \subset \mathbb{C}$. "Euphoric" Two seconds later, but wait... now I basicly define $\mathbb{R}$ as $\mathbb{R} \times 0$.... this doesn't make sense.... "Frustration" (just to give an example, I wouldn't post the whole list of things about theory of complex analysis that confuses me)

Sometimes I wonder if it there is any use of trying to understand mathematical concepts in this way as an undergraduate student. Should I just accept that it doesn't totally make sense and go on with solving problems?

Answer 1

Mathematics is really about relations between things. Therefore, while constructions are good and useful, you should never take them very seriously.

Construction agnosticism

Consider the real and complex numbers. Given $\mathbb R$, you can construct a ring which is isomorphic to $\mathbb C$ by taking pairs of real numbers and defining addition and multiplication in the usual way. Note here that I say that you can define a ring isomorphic to $\mathbb C$. We can ask the following question:

• Is the ring we have defined actually the ring of complex numbers $\mathbb C$ itself?

But you shouldn't ask yourself that question. (Just because you can ask a question, doesn't mean you should.) It won't do you any harm; it's just not useful to.

What really matters? That there is a ring which has all the properties that we want from the complex numbers — such a ring exists. Then we can say: let $\mathbb C$ be such a ring, and then study the maps $f: \mathbb C \to \mathbb C$.

When we say "the" complex number field, the word "the" is a red herring; we just want to talk about some ring which works that way. Similarly, we don't really care about "the" real numbers. If $\mathbb C$ is "the" complex number field, then it contains a principal ideal domain $Z$ coinciding with the abelian group generated by the multiplicative identity; a quotient field $Q$ induced by $Z$; and a subfield $F$ which is the analytic completion of $Q$. We can then call these "the" real numbers. Just as we don't care if "the" complex numbers consist of ordered pairs of objects from some ring $S$, we don't care if "the" real numbers are the ring $S$ or some set of ordered pairs $(s,0)$ for $s \in S$ and $0$ the additive identity of $S$. It just doesn't matter — we just care about the proper subfield $F \subseteq \mathbb C$ which has all of the same properties as the real numbers, so we may as well adopt the convention that this subfield is the field of real numbers.

This applies to the integers as well. The von Neumann construction of the natural numbers makes $3 = \{ \varnothing, \{\varnothing\}, \{\varnothing,\{\varnothing\}\}\} $. Does this mean that $3$ "really is" a set which e.g. has the empty set as a member? Not really, because these ideas are totally irrelevant to what we care about the number $3$. We could consider any other "construction" of the natural numbers, in which case $3$ might not be a set at all (for instance, if we consider a set theory in which the natural numbers are atoms), in which case it is not only irrelevant to consider the maps $f:3\to3$, but these would not even be defined. All we care about is that $3$ is part of a collection of objects $\mathbb N$ which forms a monoid with some specific properties. The "true identity" of $3$ is beside the point.

Mathematical "interfaces" (in place of "foundations")

If this ambiguity bothers you, you can think of it axiomatically as follows: treat each of the sets we care about — such as $\mathbb N$, $\mathbb Z$, $\mathbb Q$, $\mathbb R$, and $\mathbb C$ — as underspecified objects, where we specify all of the properties about them that we could care about, and only those properties.

• von Neumann's construction of ordinals describes a certain countable well-ordered monoid: we don't define $\mathbb N$ to be that monoid, but merely say that it is isomorphic to it, leaving further details to be filled in later.

• From equivalence classes of ordered pairs of elements of $\mathbb N$, you can define an ordered ring $Z$, which contains a monoid $M \cong \mathbb N$ and which is the closure of $M$ under differences. Now, $\mathbb N$ can never be a subset of this set of equivalence classes; but it can be a subset of some other set. We never pretended to characterize precisely what object $\mathbb N$ is, so who is to say that $\mathbb N$ is not itself contained in a ring which is isomorphic to $Z$? Nobody, that's who; without loss of generality we may define $\mathbb Z$ to be a ring isomorphic to $Z$, and declare as a refinement of the earlier specification that in fact $\mathbb N$ is contained in $\mathbb Z$.

• We may similarly declare that $\mathbb Z \subseteq \mathbb Q$, where $\mathbb Q$ is isomorphic to the ring of equivalence classes of ordered pairs over $\mathbb Z$ in the usual way. We also declare that $\mathbb Q \subseteq \mathbb R$, where $\mathbb R$ is isomorphic to the field of Dedekind cuts over $\mathbb Q$, or (equivalently) to the field of equivalence classes of Cauchy sequences, or any of the typical constructions of the real numbers. The set $\mathbb R$ isn't defined to be any particular one of these constructions, because (a) any of these constructions is as good as the others, and (b) we don't really care about any of the details lying underneath any of the constructions, so long as the properties we care about hold for each.

You should think of these refinements as axioms which we add during the process of doing mathematics. A definition is in the first place is only an axiom: one which defines a constant, such as defining ∅ by asserting ∀x:¬(x∈∅). These mathematical underspecifications — mathematical interfaces — are also axioms: having proven that a certain sort of monoid exists satisfying the Peano axioms, we assert that $\mathbb N$ is such a monoid, saying nothing more until it suits us to; and similarly declaring that $\mathbb C$ is a sort of number field of a kind which we've proven exists, and which happens to contain the field $\mathbb R$ which we mentioned previously without quite defining it completely.

Fundamentally, this approach to mathematics is not really all that different from what we usually do: it merely substitutes complete descriptions (what Bertrand Russell would call simply "a description") for objects, with partial descriptions. But pragmatically, in the real world as in mathematics, partial descriptions tend to be all that we care about (and in the real world, they are all that we ever have access to). Embracing this allows you to focus on what really matters.

If you are a mathematical "realist", in which the real numbers has an identity separate from our descriptions of it and has some fixed location in the mathematical firmament, this sort of wishy-washiness as to the "exact identity" of these objects may bother you. After all, if you imagine the possible identities of the objects $\mathbb N$, $\mathbb Z$, $\mathbb R$, etc. as you subsume them into more and more complicated objects, it would seem that the set of objects with which a set such as $\mathbb N$ could be identified recedes to infinity as our mathematical framework grows more elaborate. To this I can only say, "so much the worse for realism". If you want the freedom to construct objects and only concern yourself with the relationships that matter, in the end it is better to abandon this preoccupation with the precise identity of a mathematical object, and engage in mathematics as the creative, descriptive, and above all incomplete and ongoing endeavor that it is.

Answer 2

It really depends on what kind of things you are talking about.

You certainly shouldn't accept theorems as true just because someone tells you they hold - especially not at an undergraduate level. Any good math curriculum should start with the very basics (say, natural numbers), and work its way up from there, proving every theorem that is encountered as the curriculum progresses.

What you seem to be struggling with, however, doesn't seeem to be a particular theorem, but rather notation. There, my advice is to just not take notation too seriously.

Take $\mathbb{R}$ for example. People will often say "... the set of real numbers $\mathbb{R}$...". That makes it sound as if there was exactly one set of real numbers, and that set, and nothing else, went by the name of $\mathbb{R}$. The problem with this is - it isn't true. There are different ways to construct the real numbers (as dedekind cuts, as sequences of rationals, as infinite decimals, ...), and strictly speaking all of these constructions yield a different set. After all, the set of all possible sequences of rationals certainy isn't the same as the set of all infinite decimals, which in turn isn't the same as all dedekind cuts of the rationals. But you can define an ordering and the basic arithmetic operations on all of these sets, and then find bijections between these sets which are compatible with these operations and the ordering. Thus, for all intents and purposed, it doesn't matter which of these sets of real numbers you pick, and hence $\mathbb{R}$ really stands for any of those, instead of for one particular one.

What you have discovered that there's another way to arrive at $\mathbb{R}$ - you can start with $\mathbb{C}$, and view $\mathbb{R}$ as a subset. Now, it's true that, strictly peaking, $\mathbb{R}$ cannot be that subset if you also want $\mathbb{C}$ to be the set $\mathbb{R}^2$. But if you view $\mathbb{R}$ as just something that stands for any representation of the real numbers, that problem goes away. $\mathbb{C} = \mathbb{R}^2$ then just tells you that one way to view $\mathbb{C}$ is as the set of pairs of real numbers, and $\mathbb{R} \subset \mathbb{C}$ tells you that one way to view $\mathbb{R}$ is as the set of complex numbers with imaginary part zero.

So again - don't take notation too seriously. It's just a tool used to communicate ideas, just as any language is. And just as any other language, it's sometimes ambiguous, and sometimes a bit confusing. The trick is to try to see beneath that.

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Since you seem to be dealing with complex analysis currently, here's a list of things (off the top of my head, surely incomplete, and in no particular order) that I think are worthwhile to think about and try to understand

• What does $f$ being holomophic mean geometrically (compare to the situation in $\mathbb{R}^2$)
• Why $z^n$ is integrable, except for $n=-1$. Watch how that single fact shapes the whole subject (c.f. laurent series)
• Exponential function and logarithms. Especially the fact that $x \to e^x$ is not injective over $\mathbb{C}$.
• What does exponentation mean over $\mathbb{C}$. $z^n$ for $n \in \mathbb{Z}$ is clear enough, but what about $z^q$ for $q \in \mathbb{Q}$?

Answer 3

Aaaaauaauaghaguaghaguaghaguahguaghaughaguahguahguahgh!

Sorry.

I don't mean this as an insult, so please don't take this the wrong way, but your entire approach is precisely backwards. To be fair, just from the way you express yourself, it sounds like you do actually have a pretty decent understanding of what you're studying, and I'm thrilled to hear that you think it's "fun" to think about the underlying concepts. (Yes! It is fun! That's why we do it! I mean, yeah, it's also useful for science and stuff, but pure math is art, not just a means to an end.) So while you may have exaggerated your inclination to just "gloss over" the core of your studies rather than plumbing their depths, I'm still going to put forth my argument for why I disagree with the approach you described.

As a student, even though you're paying for a degree, your goal should be to understand mathematics--and therefore your attempt to understand the underlying theory should never be a last-minute "if I have time" activity. This is true regardless of whether you're an undergraduate or not. It seems both dangerous and unfulfilling to assume that the "tricks" are they key to "how mathematics is done," while true understanding is only in the purview of "advanced" mathematicians (whether that means graduate students, post-docs, professors, or whatever).

Instead of remembering "tricks" and procedures to solve particular problems, try to figure out why each trick works, then try to re-derive the trick yourself. If you really understand the core concepts, then in theory you should be able to take a test without having memorized anything; you can simply re-derive every problem-solving method you need. (In fact, when I was an undergraduate in mathematics I used to tell people that I'd chosen that major because I hated memorization, and memorization was generally useless for the way I took tests.) In practice, you'll probably find that the "tricks" are actually easier to remember once you understand them in this way, and if you "mostly" remember a trick but don't quite remember some details (for instance, the order of a couple terms, or whether an operation is an addition or a subtraction), you can quickly figure out the missing parts and then proceed with confidence.

Regarding your particular question about the real and complex numbers considered as formal sets, I hope you've found your discussion in the comments with Malice Vidrine enlightening; I don't know that I have much to add. (Though personally I do consider $\mathbb{R}$ to be a subset of $\mathbb{C}$, which should add a bit of weight to Vidrine's statement that the structure of the thing matters more than the formal definition.)

Answer 4

You (presuambly) have a prior notion that allows you to interpret the elements of $\mathbb{R}$ as expressing the abstract concept of a "real number".

What this construction is saying is that we also want to interpret the elements of $\mathbb{R} \times \{ 0 \}$ as expressing the abstract concept of "a real number" (but only in the context of interpreting elements of $\mathbb{R} \times \mathbb{R}$ as expressing the abstract concept of "a complex number" in the way you describe). And we want these two interpretations to be related in that we want to interpret $x$ and $(x,0)$ as expressing the same thing.

Different expressions for the same thing is a familiar concept: e.g. $1$, $2-1$, and $3/3$.

The problem comes when you become too invested in the idea that some particular representation is what something "really is" -- e.g. when you simultaneously hold the notions that the abstract concept of "a real number" really is the same thing as "an element of $\mathbb{R}$" and "when I'm talking about elements of $\mathbb{C}$, I'm talking about what they really are", then it becomes very difficult to digest the idea that an element of $\mathbb{C}$ can be a real number.

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Sometimes, you see other approaches used to allow you to work around this dissonance rather than trying to work through it, which have their advantages and disadvantages: e.g.

• There is an injection map $i : \mathbb{R} \to \mathbb{C}$, but we usually won't write it explicitly. e.g. when I say "$x + \bar{x}$ is a real number", I really mean "there is a real number $r$ such that $i(r) = x + \bar{x}$".

• That definition of $\mathbb{R}$ you learned in analysis was just a temporary definition of the real numbers that we used so that we could define $\mathbb{C}$ properly. Now that we have our definition of $\mathbb{C}$, we will start using the "correct" definition of the real numbers which is the corresponding subset of $\mathbb{C}$

• We could define $\mathbb{C} = \mathbb{R} \times \mathbb{R}$ this way, but we really want it to contain $\mathbb{R}$. So what we'll actually do is define $\mathbb{C} = (\mathbb{R} \times (\mathbb{R} \setminus \{ 0 \})) \cup \mathbb{R}$.

Answer 5

My strategy is to keep a journal - write down everything that doesn't make sense right now, study what you have to to get a good grade so you can move on, and keep moving forward. Of course, also constantly ask professors, friends and older students if they have any insights, but don't get too stuck on one idea because you'll always be able to come back to it later!

Answer 6

I too believe that an exam mostly needs knowledge of so called 'tricks' instead of a complete understanding of the material. To understand some subject completely takes much more time, and is probably not reachable by an undergraduate student, since solving one question probably only brings more new questions. However, digging further into the material than your course does is certainly possible (and I think very enjoyable).

In your example: There are many ways to define the complex numbers, but probably the most common is that it is an extension of $\mathbb{R}$ which contains all the roots of all polynomials with coefficients in $\mathbb{R}$. It is called the algebraic closure of $\mathbb{R}$ and in algebra it can be formally defined as $\mathbb{R}[X]/(X^2+1)$. Of course this is not really understandable if you haven't had any group and ring/field theory, but these subjects are almost certainly handled in undergraduate courses (at my university in the second semester of the second year).

So just keep digging deeper step by step and you will learn more and more, even as an undergraduate! Also just ask people when your stuck!;)

Answer 7

I recognize certain things in your story here. When I was in High School, there were a lot of things in Math that did not really make sense to me, but I knew the rules and formulas (didn't care how to derive them) and I was pretty ok in math. Then in my undergrad, a lot of math topics from High school start to make sense because I learned more math to view High school from a different angle and I did a lot of high school tutoring. I was also getting more mature (at home they still disagree to this day...) This all continued into grad studies where a lot of undergrad topics started to become clear why it worked the way I learned them. Now I am in the teaching field and still take grad classes myself off and on and believe me, by the passage of time and experience, things do clear up. I don't know how old you are (don't need to reveal either), but I can guarantee you that if you continue to use math higher up or go into teaching, you will have your questions answered! Right now ,pass those exams by knowing the rules and get back to the material at a later stage of life. It certainly will become clear.

Answer 8

If you want to do research in mathematics or in other subjects with a heavy mathematical content, you should definitely try to really understand what you are doing starting at the undergraduate level.

Note that I said try, in some cases you will manage to, in some other you will not and, curiously, in some cases you will think you really have understood but you will have not.

In order to better understand mathematics you need experience and so called mathematical maturity. But you build very little of those if you just learn tricks and apply formulas, the more you try to understand things the more you will eventually understand things.

Don't get discouraged if you feel that, despite your best efforts, you still cannot understand something, it is normal. In many cases, things will become clearer later - not for free, but because all the thinking you have done will give its fruits.

Don't go overboard, you should not risk to compromise your marks. If you get stuck, keep thinking for a while but at some point move on, there will surely be some things which you cannot make sense for the moment. Sometimes it is not because the topic itself is very difficult, but it requires a level of abstraction, or mathematical maturity, or confidence with the material which you do not have.

In a line: as you are an undergraduate student, don't get stuck for too long, but by all means keep asking yourself questions and try to understand what you are doing.

Answer 9

I can't speak to your exact questions about the reals and complex numbers, but I can definitely relate to your desire to understand the fundamentals rather than taking things on faith. In fact, this was one of my primary stumbling blocks to enjoying math, and it really wasn't until my very unusual calculus teacher taught calculus very rigorously from the ground-up (with deltas and epsilon proofs, which I understand is not how it's usually taught), and encouraged us never to move on until we thoroughly understood something, that I finally began to really fall in love with math.

I think it's insane and pointless to move on without understanding the fundamentals. To do otherwise is just to turn us into number-crunchers, which, to be honest, it seems like all school is really trying to do. "Actually learn the material? Pfft! Just get an A on the test!" (Though, of course, truly learning the material can very much help with acing the test.)

For example, I got an A in Algebra II without understanding logs and matrices one bit! Okay, so I had a general sense that logs were inverse exponents, but I had no clue as to why they were invented, what they were good for, and how, really, they worked. It wasn't until literally just a couple weeks ago when I finally decided to wrestle the question to the ground. I went alllll the way back to John Napier and even read his and Briggs' original papers until I got it. And now, finally, I understand. It's wonderful!! When I just crunched the numbers without understanding, I felt an uncomfortable sort of cognitive dissonance, but after understanding something, I'm like, well yes, of course.

That's my two cents. I know a friend of mine who's very advanced in math had the opposite view--that sometimes you just have to take something on faith and move on--but in my opinion, that takes all the fun out of it. I'd say that's good to do really only when necessary--when you're coming up to a deadline or a test, or perhaps when you've gone down a side tangent that's actually not all that necessary to your primary focus (for instance, Napier logs, which aren't even used anymore).

Otherwise, I heartily encourage you to understand the fundamentals as much as you can, even when you're not "ready" for them. Personally, I think the way math is parsed out and incremented over the years is unnecessary and ridiculous. (And I'm not the only one.) I think algebra and calculus, for instance, should be taught MUCH earlier than they are. I say go as far as you can, and when you hit that wall of incomprehensibility, okay, stop then. But I think you'll be surprised by how far you can go -- and how much going that far will help you. :)

Answer 10

I guess you have problem with formal language of math which may hide the beauty of the idea behind the theorem.

Suppose you have an set with elements apple,orrange and lemon.If I say that I can see these set as subset of $Z_6$,does it make sense?

Actually,answer is yes since with defining suitable operetion,I can see apple,orrange and lemon as $\{0,2,4 \}$ which is subset of $Z_6$.So,$R$ can be seen 'same' with $0\times R$ or $15\times R$ .

Thus,My suggestion is

$*$ try to understand the idea behind the theorems written in formal language

$*$ Avoid to make rigid defination of objects (you can see everything as an element of set,an elemet as function,a function as a morphism ....a letter as number,a number as a letter...)

$*$Do not just try to understand the theorem,also try to understand the defined concept.( why is it defined in that way.)

$*$Be patient to see big picture,do not expect yourself to grasp every consept deeply.

When you grasp the idea behind theorems and concepts,you can feel yourself in 'safe'.

Answer 11

I think a natural part of learning mathematics is growing comfortable with the fact that there can be (and usually are) many different ways to describe a single idea. I think this fact of life is essential for our sanity actually, if for example I had to start talking about Dedekind cuts whenever I use real numbers in even a basic calculation I would switch jobs.

At some point the "ordered pair" approach to complex numbers may make more sense to you, after you have some more context to back up that definition and some experience in moving between different representations in other contexts as well.

So I would say: Don't worry too much if something like this doesn't make perfect sense immediately. If it is a math topic that you intend to apply often, then experience will naturally improve your understanding. It takes a very rare type of individual who will genuinely improve their understanding by trying to work "from the ground up." Remember that historically mathematicians did not work this way. It was only in hindsight that we have come up with the definitions which are so efficient in wordcounts but perhaps not in pedagogy.

To gain more understanding, just keep doing math and don't stop. Read good math books (especially the introduction sections by respected mathematicians), ask plenty of questions when something doesn't make sense. But most of all just don't stop doing math.

Answer 12

This learning problem is a human trait, and occurs in almost all subjects. It is common in the science, technology and engineering subjects in the same way. Do not be worried by an open and inquisitive mind.

It is worth getting to the core of the understanding as from there you can get to any of the derived results in one step, rather than taking 2 pi steps to walk around the periphery of rote learning (the usual exam revision problem).

There are two techniques I've used over the years.

One is to skim read text books 'back to front' so that you know what the good bits and conclusions will be, and you already know you won't fully understand the end sections yet. So you don't worry. And you know where a technique will lead.

The other technique is to go back and read the introduction to the good text books (that's the Preface and Chapter 1), because that's were they hide those little nuggets of 'obvious' information and understanding that we all struggle with when researching new areas of interest (you should think of your course as guided research).

As an Engineer I have to drop into all sorts of obtuse maths to get an understanding of how some calculation is completed (e.g. currently Radar Cross Section calculations led to Fast Multipole Methods and then onto Gegenbauer series and glazed eyes, and then some comprehension & enlightenment...).

Even Astrologers have to build up confidence in what they understand, having take the axiom of choice one step further than most ;-)

Once you have conquered complex numbers have a look at quaternions (or are they bi-complex?) to see more roots of -1 if you want a little fun.

Answer 13

In mathematics, proof part is the only main thing to learn. If every immediate logical implications are written down, then math would become the most easiest and smallest subject in a university course. Theorems are democratic in the sense that every one is entitled to know it. They should be presented in the actual length. To test the ingenuity of a student, there is a separate section of Exercises or Problems. Textbook authors should not intermingle Theorems with Exercises. How many people will believe that the actual size of a proof without any missing steps of Implicit Function Theorem takes nearly twenty pages. On the other hand, no book on Analysis allows more than three pages for it. Authors and publishers supply various excuses. Such a bad situation does not prevail in chemistry or biology or history, etc. All hue and cry for being stuck in math has this one as the main reason. Fortunately, it is not an insurmountable difficulty. First of all, authors should discover a way for how to write every step in a proof while consuming the least amount of space on paper? Authors should avoid the temptation of writing the phrase "clearly", because there is nothing in math that is clear. One such author is Rajnikant Sinha. Check his book "Calculus of tensors and differential forms" in this connection.

Answer 14

I think you have to be practical and make a balance. Obviously the more you understand bases, the easier it is to remember techniques. All that said, things would crawl to a halt if you had to build every basis before learning a technique. Should we teach 5 year olds to add or should we teach them some theory about intervals?

It is a practical, human trait to learn iteratively. To learn simple models first. It actually helps you to understand relativity to have learned Newtonian mechanics first. And there is a lot of power in math techniques also. A lot of use. And a lot of explanation of natural science.

So my PRACTICAL advice is to NAIL the techniques, tricks, etc. A+ level. And then with extra time (or later with harder courses) come back and prove some theorems.

Even for a research mathematician, life is not just inspecting other people's proofs. Manipulative skill and puzzle-solving ability will help you to discover new math. And it will make it easier to read papers.

Answer 15

From my experience, future job prospects will value an applicant for both high undergraduate scores and depth of knowledge. But, if you want to excel in an environment where your prospective employer wants you to do problems that can be solved by knowing the 'tricks', be prepared for that inquisitive mind to be wasted. If this is how you study in your undergrad, this is what you will be prepared for in the start of your career.

Educators have a responsibility to educate, but not all of them are preparing you for YOUR dream position (and not all of them are good at their jobs). Start preparing yourself for YOUR dream position by studying material in a way that will land you that job. Bottom line is: professors will not walk you to your definition of success, that is up to you. The OP's question was proposed in a binary fashion, but I believe the answer is not: Do what is necessary to excel in the scholastic arena because GPA will be important AND learn theory, foundation, accessory information, etc. while you do it. The most ideal candidate will have all of these things. Find your own balance with the time that you have. GPA will help you get the interview, but depth of knowledge, an ability to self-educate to gain that depth of knowledge, and an inquisitive mentality will gain you tons as an employee. That mentality is learned NOW, not after you graduate. The requirements of success in the class room (GPA) should be considered a minimum, and if you are inspired to understand the depth of the material, do so in your own time, and it will be a great investment to your future.

I was fortunate during my undergraduate education to be taught by a professor who was very interested in the science of education. His requirements where very similar to the process outlined by Philip Oakley's response: he would require us to take a high level quiz over the material before he started lecturing on it, which would require us to have already been exposed to the knowledge before even starting to teach it. It was a tremendous approach that created many more logical connections to materials than other approaches had, and it helped bridge the gap between concepts.

As a final point: read Euclid's Elements. There after, I think you will appreciate why professors must encapsulate material, it would be impossible to grow some of those concepts from axiom up with the time frame allotted. And most people just don't care to know. If you do, however, learn what you can outside of the constraints the classroom puts on the educational process, and become great because of it!