Calculus I & II (BC, roughly) - can it be done in 8 weeks?

[brum]

Calculus I & II (BC, roughly) --- can it be done in 8 weeks?

I was recently accepted to the http://www.ssp.harvard.edu , where I will take either two 4-unit courses, or one 8-unit course over the 8 week period during the summer. I am now at the stage of choosing a course (or 2, if I do two 4-unit courses).

I am considering taking the http://www.ssp.harvard.edu/2004/academics/courses/math.jsp#s-1ab -- 8 credits, meets every day M-F 10am-12pm (not sure what the "Required sections M-F 2-3 pm" are... anyone know what those are?)

Then, instead of taking AP Calculus next year (my senior year in high school), I would take a math class (or two) at a nearby college, probably University of Wisconsin-Milwaukee. I would probably take multivariable calculus or something. (What other 3rd semester math courses are there that I could do after Calculus I & II? Number theory? that's something I'll have to look into, as well).


However, when running this idea by the head of the math department at my school (the idea of taking calc at Harvard Summer Program and then a math class at UWM), he was unsure about me taking calculus in just 8 weeks.

He felt that I would not learn calculus very well if I learned it all in just 8 weeks. He thinks that you have to have time for it to "sink in" to truly understand and master calculus. My question is: from your experience, would taking calculus (the equivalent of Calculus BC, more or less) in just 8 weeks at the Harvard Summer program seem doable? Worth it? Or should I take something like an Astronomy and Statistics class, instead, and take AP Calculus at my high school senior year (where I would learn calculus over a school year's time, not 8 weeks)?

Let me remind you that this will be the ONLY class I will be taking, and the teachers will probably be very good. Also, I love math and am pretty darn good at it. Right now, in my junior year, I am taking a Pre-Calculus course (and acing it), so that will help to prepare me for it.

I would really appreciate any comments or advice.

Basically, my questions are:
1) what are these -- "Required sections M-F 2-3 pm"
2) what are the 3rd semester math courses (ie, after 2 semesters of calculus)? multivariable calculus, number theory, ... ?

and, most importantly
3) Do you think this Calculus course (plus then something like multivariable calculus at UWM during first semester senior year) is a good idea for a kid interested in math and physics? Would I grasp calculus well enough to move on to, say, multivariable calculus?

Thanks!

 

[cookiemonster]

None of us are qualified to tell you if you are capable of learning it in 8 weeks. However, it is very possible to learn calculus in 8 weeks. Many institutions in the US have "fast-track" calculus programs that either get students up to speed in the summer before their freshman year or get them ahead in the summer before their freshman year. If you are having no difficulty with pre-calculus and find that the concepts come easily, then you'll probably be comfortable in these courses. I suppose the best question to ask yourself is "In your current pre-calculus class, do you think you need the extra time and review they devote to each subject, or do you think you could speed up significantly?"

MF 10am-12pm and MF 2pm-3pm required sections means that you're going to be in class 10-12 and 2-3 all week. The "required sections" are usually recitation sessions taught by a TA that is essentially practicing the concepts you learned in the lecture (10am-12pm). It looks like they expect you to be in class in the morning, give you some time for lunch, then come back for recitation.

Following Calculus I and II is Calculus III, which is multi-variable and vector calculus. After that it branches out by topic. Some topics include Ordinary Differential Equations, Set Theory, Linear Algebra, Introductory Number Theory, and Introductory Statistics.

There are some other things to consider in your decision. For instance, will your high school allow you to take college classes? Will your high school schedule conflict with your college schedule next year? Do you have the means to go to the college probably two or three times (and maybe even five times) a week to attend class? Can your family handle the tuition paid to the college to allow you to take the class?

All in all, it certainly will not hurt you to get ahead in math if you're planning on pursuing math and physics. However, you do not have to take calculus. You could take the calculus class your high school offers and pursue some physics classes or different math classes (of the classes I listed above, only DEQs requires calculus as a pre-requisite) at the university. Whatever you do, make sure you carefully assess your options and plan out all of the details that must be handled for your path.

cookiemonster

 

[NateTG]

There's a whole bunch of stuff that's unrelated or after calculus.

That said, you should have little trouble getting all of the calculus under your belt in eight weeks. The question is whether you can handle the lecture time. If you consider that most college students see 3 hours of lecture per week, you'll be getting the equivalent exposure to about 27 weeks of class. The problem with summer school in general is that it's intense and time consuming, you may be expected to put in more than 40 hours of schoolwork every week.

That said, calculus AB-BC is conceptually relatively simple. If you have a good algebra background, you should be fine.

Here are some topics that you might see after calculus, even if many of them do not require it:

Modern Algeba
Differential Geometry
Differential Equations
Linear Algebra
Number Theory
Topology
Analysis
Statistics
Numerical Methods

 

[brum]

thanks for the info!

Originally posted by cookiemonster
There are some other things to consider in your decision. For instance, will your high school allow you to take college classes? Will your high school schedule conflict with your college schedule next year? Do you have the means to go to the college probably two or three times (and maybe even five times) a week to attend class? Can your family handle the tuition paid to the college to allow you to take the class?

cookiemonster

Yes, my high school has a "youth options" program where students can take classes at UWM or some other nearby college if that course is not offered at the high school.
And the school district pays for the tuition, as well.


If I don't take calculus this summer, I would probably take Astronomy (Fundamentals of Contemporary Astronomy: Stars, Galaxies, and the Universe) and Statistics (Introduction to Quantitative Methods).

There aren't any physics courses available that I would take (they only have physics 1-2, which I'm taking right now in high school, and Laboratory Electronics: Analog and Digital Circuit Design, which doesn't really interest me).

And the other math courses (multivariable calculus, linear algebra and differential equations, Discovering Numbers and Functions: Rigor with Vigor, Discrete Mathematics with Computer Science Applications, and The Mathematics of Symmetry) all seem to require calculus or knowledge of Java (which I'll get next year in my AP Computer Science class).

Except for the one that's called "Discovering Numbers and Functions: Rigor with Vigor". That one doesn't seem to require calculus, just a "thorough command of high school algebra."

Could someone tell me more about what this class is? Difficulty? Will it come in handy to have taken?

(Here is that class's http://www.ssp.harvard.edu/2004/academics/courses/math.jsp#s-101 )

 

[matt grime]

Looking at the schedule/syllabus for that course you link to, I think most of us mathematicians, who deal with students who find the idea of a 'proof' baffling when confronted with it for the first time, would say you should take it: the sooner you deal with rigour the better.
Even though though the course has got a bloody stupid name.

There are several things in maths that are often (mistakenly) presumed of students: that they know about 'directions' of implication; will understand what iff (sic) means; know when a proof is finished; can indicate when they believe a proof is finished; that they can define 'well-defined' (the least well defined term in mathematics as one Fields Medallist put it).

If you can get these out of the way and if they can actually teach you to be rigorous (usually one just learns through experience and repetition) then it will be well worth taking if you've any intention of doing anything mathematical.

 

[NateTG]

I understand the the US use of calc and pre-calc is a lot more general than in the UK, but really the whole point of Discrete maths is that it isn't calculus, so it seems surprising to me that it requires calc, when a course that sets out to deal with functions and limits etc doesn't require calculus. Can you explain to this increasingly bewildered Englishman what high-school calc is then? (And I've taught 200 level university calc in the US so you'd think I'd know...) Or does discrete fall under the banner of needing to know java?

Java (as you are probably aware) is a programming language, and has apparently insinuated itself into the CS curricula of US unversities. "Discrete Mathematics with Computer Science Applications" is likely to require some programming.

In the US, there really isn't any 'high school calculus'. Calculus is an advanced placement topic. But in response to your comment , the notion of limit is certainly more general that most of the notions that appear in calculus, so it seems much more sensible to talk about limits before calculus instead of the reverse.

As I recall, high school calculus essentially covers various integrals and derivatives without a whole lot of theory. It's primarily designed to give students sufficient background to handle kinematics and other phyisics courses, in a fashion similar to lower division College course.

That said, the 'rigor with vigor' class is likely to be a college-level pre-calc course and I would advise asking the prof about who the target audience is.

 

[matt grime]

Yes, I rapidly realized that the 'with computer' part would mean java, hence I deleted that part (in case anyone wonders where that quote comes from)

 

[curiousbystander]

I can't offer any advice, just experiences to consider.

When I taught a 4 week Summer Session course (on Discrete Math ironically with no Java) it was like forcing the students to drink water from a firehose. There were one or two who were able to absorb the material quickly enough, but the majority struggled the entire month. I'm really not certain how much they gained from the experience. There really is something to be said for soak time.

On the other hand when I attended a two week MSRI session on mathematics and computer graphics it was one of the best (and most intense) learning experiences I've ever had. So the moral of my story is it can go both ways...

Oh... and I agree with Matt, the sooner you are exposed to proof techniques the better.

 

[GRQC]

Originally posted by matt grime

If you can get these out of the way and if they can actually teach you to be rigorous (usually one just learns through experience and repetition) then it will be well worth taking if you've any intention of doing anything mathematical.

My opinion from having taught a few college-level calculus courses is that most high schools do an exceptionally poor job at teaching calculus! I was amazed at how many of my students claimed to "know everything" we were going to cover, yet their knowledge of integration techniques and derivations of same was horribly limited (if not incorrect). I personally think that high schools would do much better in teaching more algebra-based subjects and easing off calc. Intro complex analysis or a decent geometry sequence would serve those students much better, so that they don't go in with the sense that they're "reviewing" the material when they reach college.

In terms of intensity level (for calc), gnerally the "rigor" is obtained from analysis courses, and not calculus (although I suppose many schools will blur the distinction). Formal proof structuring and methodology will usually not be covered in the "standard" calc-I/II/III stream (single diff, int. and series, and multivariate).

In that sense, I am of the opinion that calculus can be learned in two weeks or so, but the formal analysis part (from which you would gain a truly deep understanding) will take some time.

To be specific, I would qualify analysis courses as those which would use, for example, Spivak's intro book (Calculus, the big gray one). An introductory analysis course would include epsilon-delta proofs, formal definitions of limits using same, a more indepth look at series and convergence using sets, and so on... An advanced analysis course would cover calculus on manifolds, Heine-Borel thm, boundedness and compactedness, integration on chains, etc...

 

[PrudensOptimus]

Originally posted by brum
I was recently accepted to the http://www.ssp.harvard.edu , where I will take either two 4-unit courses, or one 8-unit course over the 8 week period during the summer. I am now at the stage of choosing a course (or 2, if I do two 4-unit courses).

I am considering taking the http://www.ssp.harvard.edu/2004/academics/courses/math.jsp#s-1ab -- 8 credits, meets every day M-F 10am-12pm (not sure what the "Required sections M-F 2-3 pm" are... anyone know what those are?)

Then, instead of taking AP Calculus next year (my senior year in high school), I would take a math class (or two) at a nearby college, probably University of Wisconsin-Milwaukee. I would probably take multivariable calculus or something. (What other 3rd semester math courses are there that I could do after Calculus I & II? Number theory? that's something I'll have to look into, as well).


However, when running this idea by the head of the math department at my school (the idea of taking calc at Harvard Summer Program and then a math class at UWM), he was unsure about me taking calculus in just 8 weeks.

He felt that I would not learn calculus very well if I learned it all in just 8 weeks. He thinks that you have to have time for it to "sink in" to truly understand and master calculus. My question is: from your experience, would taking calculus (the equivalent of Calculus BC, more or less) in just 8 weeks at the Harvard Summer program seem doable? Worth it? Or should I take something like an Astronomy and Statistics class, instead, and take AP Calculus at my high school senior year (where I would learn calculus over a school year's time, not 8 weeks)?

Let me remind you that this will be the ONLY class I will be taking, and the teachers will probably be very good. Also, I love math and am pretty darn good at it. Right now, in my junior year, I am taking a Pre-Calculus course (and acing it), so that will help to prepare me for it.

I would really appreciate any comments or advice.

Basically, my questions are:
1) what are these -- "Required sections M-F 2-3 pm"
2) what are the 3rd semester math courses (ie, after 2 semesters of calculus)? multivariable calculus, number theory, ... ?

and, most importantly
3) Do you think this Calculus course (plus then something like multivariable calculus at UWM during first semester senior year) is a good idea for a kid interested in math and physics? Would I grasp calculus well enough to move on to, say, multivariable calculus?

Thanks!

Don't take it, you'll probably end up getting an insatisfied grade and get out of the course know nothing. Time is too short. You'll forget by the end of the summer lol.

And Java, no. Don't touch Java.

Physics. Maybe. Calculus won't help you much in physics at that level. Trig may help you a little.

 

[brum]

I am now thinking of taking two 4-unit courses

1) http://www.ssp.harvard.edu/2004/academics/courses/astr.jsp#s-35 (Fundamentals of Contemporary Astronomy: Stars, Galaxies, and the Universe)

2) http://www.ssp.harvard.edu/2004/academics/courses/stat.jsp#s-100 (Introduction to Quantitative Methods)

...instead of the Calculus I & II course.

How does that sound to you guys?

I like it because it's less rigorous, which will be a bit more fun for the summer. Now, I will probably take AP Calculus at my high school.

 

[PrudensOptimus]

Originally posted by brum
I am now thinking of taking two 4-unit courses

1) http://www.ssp.harvard.edu/2004/academics/courses/astr.jsp#s-35 (Fundamentals of Contemporary Astronomy: Stars, Galaxies, and the Universe)

2) http://www.ssp.harvard.edu/2004/academics/courses/stat.jsp#s-100 (Introduction to Quantitative Methods)

...instead of the Calculus I & II course.

How does that sound to you guys?

I like it because it's less rigorous, which will be a bit more fun for the summer. Now, I will probably take AP Calculus at my high school.

Don't take AP Calculus in high school. Save it for later, when your brain works better.

Stat is better course for summer than Calc.

Astronomy yes, take that.

You got the perfect 2 there. Have fun.

Thread closed.

 

[cookiemonster]

Er, exposure to calculus never hurt anybody. Why shouldn't he take calculus in high school?

cookiemonster