Are entry level mathematics really the hardest?

[Zill1]

Ive often heard from people that the lower-level maths (calc 1-2, intro to linear algebra etc etc) are difficult because they "weed" people out of the hard science programs... and the way things have been going with me I kind of am starting to hope its true.

I am a BSC joint major in mathematics and economics. After some serious struggling with introductory calculus and linear algebra I got past those and finished all of the requirements for the economics part of my degree (econometrics, theory I-III in micro/macro, probability, statistics, financial econometrics, and a few social sciences.)

So now I'm getting ready to take on the math side, which consists of multivariable calculus, elementary analysis, algebraic structure, applied linear algebra, discrete mathematics, ODE, and a few electives which will probably be financial mathematics and another abstract algebra.

But to be honest I am stressing so hard as of right now just because I'm looking back at my calculus/linear algebra marks... I barely scraped by those courses. Cs and Ds to be honest, and I had to retake calc 2. I feel very worried about divulging into higher level math (even having taken probability/stats/econometrics). Has anyone else struggled at the lower level but ended up "blooming" when push came to shove? Maybe I'm surrounded by too many math prodigy kids but it really worried me when they talk about 4.0 gpas from the get go.

 

[Number Nine]

You should keep in mind that higher level "real" math course are very different (qualitatively different) than anything you've done so far. A course in, say, abstract algebra requires very different skills than a course in calculus. Personally, I'm extremely comfortable "upper level" math courses compared to first year calculus, in which I was borderline incompetent.

 

[Zill1]

You should keep in mind that higher level "real" math course are very different (qualitatively different) than anything you've done so far. A course in, say, abstract algebra requires very different skills than a course in calculus. Personally, I'm extremely comfortable "upper level" math courses compared to first year calculus, in which I was borderline incompetent.

I'm really excited/nervous about learning abstract algebra... the only slight exposure I have had to it is the cusp of vector spaces (which I know isn't even really anywhere close). But I think I understand... leaving the optimization/plug n solve type problems for abstract proof/definition based work?

On the other hand, specifically what is stressing me and making me question myself is number theory and analysis... the horror stories I have heard from some students makes me cringe.

 

[DeadOriginal]

"Engineering" calculus as weeder courses? lol.

Sorry. I am not trying to be mean. Computational calculus classes is called engineering calculus by professors at my school. To me, there is no such thing as geniuses. Those kids who seem like they are, are not. Otherwise they wouldn't be there. The secret to doing these problems is to do a lot of them, over and over and over and over and over and over and over again. Do them to the point where when it comes time to study for the final, you go back to the problems and can remember the answers from doing them the previous times. If that doesn't work I'll stand in a corner and recite The Art of War by Sun Tze one thousand times. That is my guarantee to you.

I am skeptical of geniuses. 99.999999999% of us either work hard or don't which separates us into "genius" and stupid as seen by others.

 

[20Tauri]

As far as I can tell, the way my college does it is by weeding people out in linear algebra, sort of leveling off in difficulty during the sophomore-junior elective math classes, and then making things very difficult again in advanced classes like algebra and analysis.

I haven't taken any of the advanced sequences yet, but I did better in my mid-level electives than I did in calculus or linear algebra. Some of them were easier than linear and some were not, but I think there's also an element of learning how to learn math. Once you figure out how you study best, you do better.

Honestly, I would be nervous if I were in your position. Algebra and analysis are not likely to be easy. That said, many people have had to retake courses and gone on to be successful math majors.

 

[nickadams]

"Engineering" calculus as weeder courses? lol.

Sorry. I am not trying to be mean. Computational calculus classes is called engineering calculus by professors at my school. To me, there is no such thing as geniuses. Those kids who seem like they are, are not. Otherwise they wouldn't be there. The secret to doing these problems is to do a lot of them, over and over and over and over and over and over and over again. Do them to the point where when it comes time to study for the final, you go back to the problems and can remember the answers from doing them the previous times. If that doesn't work I'll stand in a corner and recite The Art of War by Sun Tze one thousand times. That is my guarantee to you.

I am skeptical of geniuses. 99.999999999% of us either work hard or don't which separates us into "genius" and stupid as seen by others.

you wouldn't happen to be the poster formerly known as edin dzeko would you?

 

[DeadOriginal]

you wouldn't happen to be the poster formerly known as edin dzeko would you?

Should I be worried that I am being related to that person?
I'm insulted that I am not myself and am instead "edin dzeko".

 

[bpatrick]

My vote would have to be: those intro courses are the easiest courses you'll take ... they're 100 level (freshman) for a reason. There may be an element of difficulty when it comes to the speed of the courses being set on "college" rather than on "high school" that makes them more difficult when you look at new college students adaptation to college in general.

Conceptually, it's pretty basic stuff that, in some cases, is offered in US high school curriculum anyway ... at least my high school offered AP Calculus-BC in 12th grade, analytical geometry and trig in 11th grade, and our three year algebra sequence from 8-10th grade had us (at the end) doing elementary matrix stuff (Gauss - Jordan + determinants).

I wouldn't worry about it too much though, since you're doing a double major, and most of your math classes sound like applied stuff, you'll encounter much of the same level of difficulty / thinking that you've seen thus far:

"multivariable calculus, applied linear algebra, discrete mathematics, ODE, and a few electives which will probably be financial mathematics" ... these are all what DeadOriginal described as "engineering classes". I would consider them to still be "lower level" courses as they are very algorithmic, mostly focused on applications for when you get out into the workplace. These courses are offered as either "lower level" or "sophomore level" courses at the universities I've attended.

The two classes: elementary analysis and algebraic structures are the only courses that would qualify as "upper level". I have seen these offered as "sophomore level" or "junior level" where I have attended. They will probably seem more abstract to you, and given that you will only need to take one of each, you will probably find very little applicability to what you learn during them ... maybe you'll get some good things out of learning about groups in algebra. Most likely the rest will just seem more like a peek into the history and development of how we discovered and laid the foundations of the calculus you've learned so far.

Since they are intro courses, they will probably not give you any trouble. Keep in mind to devote plenty of time to them. Read the topics from the book for the next lecture before it happens so you're absorbing stuff rather than mindlessly taking notes during class, read it again after the lecture, and memorize as many of the theorems and definitions as you can, especially if these are your first exposure to proof based mathematics. I like to analogize it to drawing via connecting the dots. You see a proof and know you need to draw a house (my analogy for getting from A to Z in the proof). You have to set up the dots (B, C, D, ... X, Y) in enough places to make sure that when you connect them via steps in your proof that your final product will end up looking like the house and not something else. The way you know what the dots are / where they are placed is by knowing all the theorems and definitions so you can just use them as necessary to make your drawing manifest itself.

Anyway, I think you'll be fine for pretty much the rest of your degree. Most of it is merely a continuation of the calc and algebra you've learned already ... just more variables / equations and more solution methods to learn and applications to use the math tools on.

Good luck ... ooh, and if I can give one more piece of advice: you may want to take Discrete Math before you take algebra/analysis. Most discrete classes give you an intro to mathematical induction during some chapters on analyzing computer algorithms, which could be helpful for when you see induction used as a prominent proof method during analysis. Many DM classes expose you to stuff like permutations/relations/graphs/trees, which may help when you get to algebra ... so in some cases may serve as a bit of a bridge between all the "engineering math" you've taken and the two intro courses in "pure math" you'll need to take.

 

[Zill1]

My vote would have to be: those intro courses are the easiest courses you'll take ... they're 100 level (freshman) for a reason. There may be an element of difficulty when it comes to the speed of the courses being set on "college" rather than on "high school" that makes them more difficult when you look at new college students adaptation to college in general.

Conceptually, it's pretty basic stuff that, in some cases, is offered in US high school curriculum anyway ... at least my high school offered AP Calculus-BC in 12th grade, analytical geometry and trig in 11th grade, and our three year algebra sequence from 8-10th grade had us (at the end) doing elementary matrix stuff (Gauss - Jordan + determinants).

I wouldn't worry about it too much though, since you're doing a double major, and most of your math classes sound like applied stuff, you'll encounter much of the same level of difficulty / thinking that you've seen thus far:

"multivariable calculus, applied linear algebra, discrete mathematics, ODE, and a few electives which will probably be financial mathematics" ... these are all what DeadOriginal described as "engineering classes". I would consider them to still be "lower level" courses as they are very algorithmic, mostly focused on applications for when you get out into the workplace. These courses are offered as either "lower level" or "sophomore level" courses at the universities I've attended.

The two classes: elementary analysis and algebraic structures are the only courses that would qualify as "upper level". I have seen these offered as "sophomore level" or "junior level" where I have attended. They will probably seem more abstract to you, and given that you will only need to take one of each, you will probably find very little applicability to what you learn during them ... maybe you'll get some good things out of learning about groups in algebra. Most likely the rest will just seem more like a peek into the history and development of how we discovered and laid the foundations of the calculus you've learned so far.

Since they are intro courses, they will probably not give you any trouble. Keep in mind to devote plenty of time to them. Read the topics from the book for the next lecture before it happens so you're absorbing stuff rather than mindlessly taking notes during class, read it again after the lecture, and memorize as many of the theorems and definitions as you can, especially if these are your first exposure to proof based mathematics. I like to analogize it to drawing via connecting the dots. You see a proof and know you need to draw a house (my analogy for getting from A to Z in the proof). You have to set up the dots (B, C, D, ... X, Y) in enough places to make sure that when you connect them via steps in your proof that your final product will end up looking like the house and not something else. The way you know what the dots are / where they are placed is by knowing all the theorems and definitions so you can just use them as necessary to make your drawing manifest itself.

Anyway, I think you'll be fine for pretty much the rest of your degree. Most of it is merely a continuation of the calc and algebra you've learned already ... just more variables / equations and more solution methods to learn and applications to use the math tools on.

Good luck ... ooh, and if I can give one more piece of advice: you may want to take Discrete Math before you take algebra/analysis. Most discrete classes give you an intro to mathematical induction during some chapters on analyzing computer algorithms, which could be helpful for when you see induction used as a prominent proof method during analysis. Many DM classes expose you to stuff like permutations/relations/graphs/trees, which may help when you get to algebra ... so in some cases may serve as a bit of a bridge between all the "engineering math" you've taken and the two intro courses in "pure math" you'll need to take.

Thanks a lot for this post. It really does solidify the idea I had that really even what I perceive as looking daunting or difficult isn't as bad as it seems... I always had this notion looking at the stuff that I figured it couldn't be THAT hard since I'm barely divulging into each subject. And luckily applied math is all that I am actually interested in; the workplace is my goal, not academia. As much as I wish I had the capacity and interest to do graduate level hard science like many of you here, I just don't think I could ever have it in me, haha.

And re: your last point. For some reason the analysis course is a prerequisite for the Discrete Mathematics at my school unfortunately.

 

[Jackx]

No,
But after a class is finished I always look back and forgot all the times I was ripping my hair out. So then I say it was easy.

 

[nickadams]

As much as I wish I had the capacity to do graduate level hard science like many of you here, I just don't think I could ever have it in me, haha.

this is exactly how i feel

 

[chiro]

As much as I wish I had the capacity and interest to do graduate level hard science like many of you here, I just don't think I could ever have it in me, haha.

Don't be a prisoner of your mind.

I am going to go out on a limb and say that all of mathematics has intuition and a motivation behind it that can be grasped in the spoken or written word using the vocabulary of the layman.

Always keep this in mind that we, human beings, wrote these symbols down and we have translated through time, refinement of ideas, and effort these symbols that express not only a symbol but a refinement of an idea. The motivation may indeed have a large component that is mathematical, but I gaurantee you that there is always a suitable analogy that can be used to convey this same idea.

 

[SophusLies]

"Engineering" calculus as weeder courses? lol.

Actually, I got much lower grades and struggled more in my first year calculus classes than my upper level ones. I don't think this is the case for most but it was for me. I believe things make more sense as the classes get higher up because the ideas are more precise.

 

[rolledoats]

"Engineering" calculus as weeder courses? lol.

Yes. Weeder (aka gatekeeper) courses are more about the gradelines than the material. Few schools have a surplus of math majors but many have a surplus of would-be engineers, so "engineering calculus" is used to separate the weak from the strong.

 

[20Tauri]

I believe things make more sense as the classes get higher up because the ideas are more precise.

I'd believe that. Intro calculus always felt a little shaky, like we didn't actually know what we were doing when we took limits and things like that.

 

[DeadOriginal]

Yes. Weeder (aka gatekeeper) courses are more about the gradelines than the material. Few schools have a surplus of math majors but many have a surplus of would-be engineers, so "engineering calculus" is used to separate the weak from the strong.

I fear for my safety if engineers find that difficult.

 

[nickadams]

I fear for my safety if engineers find that difficult.

you are so smart and cool

 

[Number Nine]

I fear for my safety if engineers find that difficult.

Engineers don't take engineering calculus; first year students do.

 

[rolledoats]

I fear for my safety if engineers find that difficult.

Classic gatekeeper courses use a fixed bell curve to assign grades. The bottom x% fail.

 

[DrummingAtom]

you are so smart and cool

LOL, I second that.

 

[QuarkCharmer]

Classic gatekeeper courses use a fixed bell curve to assign grades. The bottom x% fail.

Pretty much this.

At my university E&M 1 is that course. I have seen a test with 32 points out of 100 that scored a B. Typically the bottom 50-60% of the course fails or gets a C (which means it cannot be used as a pre-requisite, thus requiring they retake it). I'm taking it soon, hopefully with a class full of dummies.

Also Calculus II (Integral Calculus) seems to be a weeder course here to a lesser extent. When I finished Calc 2 only about 15 students remained. The course started with about 35 and a few people in the course were taking it a second time. I don't get it, that course was awesome.

 

[Bearded Man]

When really low grades (like <40%) are considered passing, are students even learning anything?

 

[Mholnic-]

I'm a little worried about this myself. The only Calc II class I could schedule in is rumored to be... well. Difficult. By that I mean the professor is described as "brilliant", but his tests are "handwritten mind games". My brother took this prof for College Algebra with 35 other students and dropped it just before the withdrawal deadline. At that time there were 5 students left. Another class, Finite Math, started with 14 and had 3 by midterms. I have this professor for both Calc II and Linear Algebra in the fall.

I'll admit to being a wee bit scared. xD

 

[chiro]

I'm a little worried about this myself. The only Calc II class I could schedule in is rumored to be... well. Difficult. By that I mean the professor is described as "brilliant", but his tests are "handwritten mind games". My brother took this prof for College Algebra with 35 other students and dropped it just before the withdrawal deadline. At that time there were 5 students left. Another class, Finite Math, started with 14 and had 3 by midterms. I have this professor for both Calc II and Linear Algebra in the fall.

I'll admit to being a wee bit scared. xD

Don't fear the reaper dude! (*insert bill and ted guitar lick here*)