Freshman Student Worried About Order of Courses

[currently]

How did you find PF?: Various searches, finally decided to sign up. Happy to be here.

Hi, I am a freshman student who has taken too many sophomore courses and am slightly worried. I will probably continue my habit of taking courses that are far beyond my capability because I think it'll give me more free time when I am a junior/senior.

Am I correct in assuming that?

 

[fresh_42]

Hello and !

Why do you think so? If they are beyond your capability, then they are probably in vain and you waste time. You cannot study general relativity if you never heard about curvilinear coordinates or manifolds. You cannot study manifolds if you haven't basic knowledge of differentiation and topology.

But this is only a general comment, as you haven't given us details.

 

[currently]

Hello and !

Why do you think so? If they are beyond your capability, then they are probably in vain and you waste time. You cannot study general relativity if you never heard about curvilinear coordinates or manifolds. You cannot study manifolds if you haven't basic knowledge of differentiation and topology.

But this is only a general comment, as you haven't given us details.

Thanks for the warm welcome.

I signed up for Differential Equations and Vector Calculus in my first year of university. It hasn't been great, if I must be honest. Among other things of course.

I thought I was giving myself a head start but it doesn't feel that way anymore!

 

[Dr. Courtney]

The content in the usual pacing of courses in a Physics major is pretty fast relative to most students' ability to master it. Very, very few students could do as well with the content flowing any faster.

 

[currently]

The content in the usual pacing of courses in a Physics major is pretty fast relative to most students' ability to master it. Very, very few students could do as well with the content flowing any faster.

Oh, I'm not a Physics major. I'm an engineering major, so I'm sure the pacing is still, even if not as fast.

Thanks for the concern though. If it means anything, I took Differential Equations before I even started learning linear algebra and now I have homework questions I fail to even comprehend...

 

[Orodruin]

You cannot study general relativity if you never heard about curvilinear coordinates or manifolds.

I agree with the geist of your post, but not this particular part. Most GR courses contain a crash course in differential geometry or take a physics-first approach where the mathematical foundation is suppressed for practical application (not saying I would prefer that approach, just that it is there and seems to be working for the purpose). In neither case do you neef to have learned about manifolds before you start the course (although in the first you are typically taught about manifolds before you start learning actual GR).

 

[symbolipoint]

How did you find PF?: Various searches, finally decided to sign up. Happy to be here.

Hi, I am a freshman student who has taken too many sophomore courses and am slightly worried. I will probably continue my habit of taking courses that are far beyond my capability because I think it'll give me more free time when I am a junior/senior.

Am I correct in assuming that?

You may have taken enough courses, hopefully enough variety that you know what you want to study (degree objective), and you need to pick a program and go through the necessary courses in the correct sequence. Since you told us you chose Engineering, you should take advice as needed from the department.

 

[Vanadium 50]

will probably continue my habit of taking courses that are far beyond my capability because I think it'll give me more free time when I am a junior/senior.

If you don't have a good foundation, you will have lots of free time in a year or two.
Lots of free time.
Lots and lots of free time.

You really want to get a solid foundation rather than rushing through.

 

[Zexuo]

Thanks for the warm welcome.

I signed up for Differential Equations and Vector Calculus in my first year of university. It hasn't been great, if I must be honest. Among other things of course.

I thought I was giving myself a head start but it doesn't feel that way anymore!

I would expect AP Calculus to suffice to prepare you for differential equations and vector calculus in your freshman year. If you hadn't done AP Calculus I'd consider it a mistake for you to sign up for those classes and for the university to allow you to do so. If you have gone through AP Calculus and have trouble with the courses keep an introductory calculus textbook handy to go over the basis material for the problem at hand.

 

[Nugatory]

I will probably continue my habit of taking courses that are far beyond my capability...

As long as you’re able to keep up... the idea is to stretch yourself, not break yourself. You’ll get much more out of a hard course that you survive than an easy one that you float through, but you will get zero or negative return from one that you can’t handle.

Get to know your professors and TAs, and above all else be engaged with your academic advisor.

 

[DEvens]

Hi, I am a freshman student who has taken too many sophomore courses and am slightly worried. I will probably continue my habit of taking courses that are far beyond my capability because I think it'll give me more free time when I am a junior/senior.

Am I correct in assuming that?

The most probable way that gives you free time is to cause you to flunk out in first year.

The next most probable thing is that you have to re-take some classes, probably the classes at the level you skipped over.

Unless you are the sort of person who can skim through a text in two hours and understand it completely, I'd say take the courses in the order suggested in the course calendar.

Maybe you want to be an auto-mechanic. And maybe you think you don't need to learn to drive a car. It's conceivable you are correct, but it's not the way to bet.

Heh. I knew a guy like that in my undergrad. Got a cumulative average on all his courses in undergrad of about 98%. Was always finished his homework by 8PM. Now I occasionally see mentions of him in various popular publications with phrases like "the great" in front of his name. Sigh.

 

[vela]

I signed up for Differential Equations and Vector Calculus in my first year of university. It hasn't been great, if I must be honest. Among other things of course.

Is it safe to assume you had already taken one year of calculus before getting to university? If so, taking these courses as a freshman would have been a natural choice to make. If you hadn't already finished a year of calculus, how were you able to enroll in these courses without satisfying the prerequisites?

 

[mathwonk]

It is impossible for me to make reliable comments because course difficulty varies so much from place to place, and even from professor to professor. E.g. when I was a math professor at a state university, I would not have recommended anyone even with good grades in high school AP calculus courses to take my vector calculus course, much less diff eq. That's because high school AP courses are by definition high school level and not university level and do not even compare favorably to my basic non honors freshman calculus. The right course to take would have been an honors level first year one variable calculus course.

But things changed slowly, because parents lobbied our university to give college credit for high school AP courses even thought they did not cover as much material nor as deeply as we did in college. But they wanted to save money, even if the education they got in return was inferior. To keep those parents happy, we lowered the level of our college courses down to that of high school AP and generally dumbed down the whole program to match. In that scenario, you can successfully take a non honors college level second year course with just high school AP prep.

But some professors, like me, are dinosaurs, and do not agree this dumbing down is a good idea, so you may get in one of those old fashioned genuine university level courses and be lost.

To know just what courses you should be in, you need to seek advice from a competent professor/advisor at your school. In an ideal world, you should take all basic courtses at uni, at an honors level if you have good background, and should always take linear algebra as early as feasible, since it underlies both vector calculus and differential equations.

I.e. before you can understand vector calculus you need to understand vectors. and before you can understand even linear differential operators, the easiest kind of differential operators on functions, you need to understand basic linear operators on vectors.

Basically, I suggest you stop shooting yourself in the foot by taking precisely those courses you are unprepared for. The better the quality of your school, the more harmful that will be.

good luck.

 

[currently]

I would expect AP Calculus to suffice to prepare you for differential equations and vector calculus in your freshman year. If you hadn't done AP Calculus I'd consider it a mistake for you to sign up for those classes and for the university to allow you to do so. If you have gone through AP Calculus and have trouble with the courses keep an introductory calculus textbook handy to go over the basis material for the problem at hand.

Is it safe to assume you had already taken one year of calculus before getting to university? If so, taking these courses as a freshman would have been a natural choice to make. If you hadn't already finished a year of calculus, how were you able to enroll in these courses without satisfying the prerequisites?

Yes, I did take AP Calculus and I did fine in it. Difference is, my current university does not allow usage of calculators like my high school AP courses did. It's harder than I expected. Old textbooks definitely still help though

 

[currently]

It is impossible for me to make reliable comments because course difficulty varies so much from place to place, and even from professor to professor. E.g. when I was a math professor at a state university, I would not have recommended anyone even with good grades in high school AP calculus courses to take my vector calculus course, much less diff eq. That's because high school AP courses are by definition high school level and not university level and do not even compare favorably to my basic non honors freshman calculus. The right course to take would have been an honors level first year one variable calculus course.

But things changed slowly, because parents lobbied our university to give college credit for high school AP courses even thought they did not cover as much material nor as deeply as we did in college. But they wanted to save money, even if the education they got in return was inferior. To keep those parents happy, we lowered the level of our college courses down to that of high school AP and generally dumbed down the whole program to match. In that scenario, you can successfully take a non honors college level second year course with just high school AP prep.

But some professors, like me, are dinosaurs, and do not agree this dumbing down is a good idea, so you may get in one of those old fashioned genuine university level courses and be lost.

To know just what courses you should be in, you need to seek advice from a competent professor/advisor at your school. In an ideal world, you should take all basic courtses at uni, at an honors level if you have good background, and should always take linear algebra as early as feasible, since it underlies both vector calculus and differential equations.

I.e. before you can understand vector calculus you need to understand vectors. and before you can understand even linear differential operators, the easiest kind of differential operators on functions, you need to understand basic linear operators on vectors.

Basically, I suggest you stop shooting yourself in the foot by taking precisely those courses you are unprepared for. The better the quality of your school, the more harmful that will be.

good luck.

Thanks for the elaborate answer. I enjoyed reading about your perspective, I don't usually hear anyone speak so honestly of AP course credit!

I only took AP courses because, well, they were the norm in my high school. I eventually went to a school which didn't accept AP course as credit; definitely a high-ranking school in my country. (My classmates mostly went to the US and that's why they all took AP courses.) I'm glad for the knowledge they offered though I don't really think it was quite enough to prepare me for differential equation in first year >.>

 

[mathwonk]

One year I taught a course in my son's high school for 5 or 6 of the best math students who had already completed the AP courses, from a vector calculus book by Marsden and Tromba, a book used at Berkeley, University of California. It also covered linear algebra and I included differential forms. The students from this course did survive their second year math courses at schools including Harvard and Yale, and some took PhD's in math and physics afterwards, at least one becoming a professional mathematician. ( I was never allowed to teach that course again at the high school because it was only suitable for a few students.)

Oh yes, in the following summer I taught a course for some of these same students and a few others from the appendix to Spivak's Calculus, on the definition and properties of the real numbers, including least upper bounds, thus giving them an introduction to rigorous concepts of analysis. To be precise, we did not follow Spivak's construction using Dedekind cuts, but rather his problem outline treating "high school students' real numbers", problem 2 chapter 28, using infinite decimals, but employing the key idea of least upper bounds.

In addition to those mentioned above, some of these students went on, apparently successfully, to Chicago and Duke.

 

[vela]

Yes, I did take AP Calculus and I did fine in it. Difference is, my current university does not allow usage of calculators like my high school AP courses did. It's harder than I expected. Old textbooks definitely still help though

I have to admit, I'm wondering how useful a calculator can actually be in calculus, much less differential equations and vector calculus.

 

[mathwonk]

my friend Dave Penney, co-author of a successful calculus book, advocated use of numerical calculators, and purposely made the exercises in his (first edition) calculus book so hard numerically that one required a calculator just to work out the lengthy numbers and decimals. I suspect there are programmable calculators today that can do easy, say linear first order constant coefficient, differential equations, and add vectors, and solve linear systems of equations, as well. In fact a robust enough one could probably host mathematica, and I guess with an internet connection a calculator or iphone can access mathematica online. way beyond my experience or preference, but likely exists out there today.

 

[currently]

I have to admit, I'm wondering how useful a calculator can actually be in calculus, much less differential equations and vector calculus.

It's not that useful in differential equation or in vector calc.

It's a problem because I calculated integrals and summations with a graphing calculator throughout high school and I don't think I have the best fundamentals.

 

[Zexuo]

It's not that useful in differential equation or in vector calc.

It's a problem because I calculated integrals and summations with a graphing calculator throughout high school and I don't think I have the best fundamentals.

That'll do it. Reviewing the chapter on integration techniques (radical and trig/hyp substitution, partial fraction decomposition, etc) in your calculus textbook and doing at least half of the exercises could remedy the problem.