It seems like after taking algebra, geometry, trigonometry etc., the next math class most students take is calculus. I’m currently taking a combinatorics class at the University of Minnesota, and almost no calculus is required to understand the material, which doesn’t seem too hard for highschool students. The only reason I can think of that calculus is seen as next step after algebra etc. is its connection to physics, but the teacher of my highschool physics class tells students to avoid using calculus even if they know it. So why is calculus so widely taught instead of some other branch of mathematics?
Because it’s so pervasive. Even statistics involves quite a bit of calculus, especially at degree level.
High school physics is generally taught using first-order approximations of physical laws - Newton’s laws of motion, and that sort of thing. In the real world, the situation is not quite so simple. Calculus is a tool to describe any relationship which is non-linear, or a function of more than one variable. Dealing with variables that change over time, or analyzing systems which have boundaries described by non-linear functions, is the strength of calculus.
You mention geometry - you know the formula for volume of a sphere (4/3 pi*r^3)? This result is arrived at by applying calculus to the problem - a simple one, but much more complicated volumes are analyzed with the same tools. Imagine a sphere, with a conical section subtracted from it, bisected by a plane at an angle to the conical axis. How do you find its volume? This is one of the many applications of calculus.
Because it’s far more useful than pretty much any other post-algebra branch, and the concepts are fundamental to most of what comes later. The reason your high school physics teacher says to avoid using the calculus is tied up in curriculum requirements. The class requires its students to know certain tools, so if you already know and use the proper tools and not the required ones, the class isn’t doing its job.
Yes, it’s stupid. Why do you think I don’t teach high school?
The idea is absolutely fundamental to handling dynamic problems. Breaking a process into small intervals seems to be the only way we know of so far to get a grip on such things.
And that is the reasonable way to get at anything. Begin with condition you want to exist and work backwards in small steps to where you are now is fundamental to the planning process.
I think it’s a good thing to be aware of even if you never use calculus again.
The real reason is its utility, but it’s also one of the few remaining topics that can be taught after your general high school education without a strong emphasis on theorems and proofs.
Calculus complements everything you’ve learned up to it very well. It is a natural extension of geometry and algebra. Also, trigonometric functions are dealt with (as well as polar graphing and other analytic geometry topics).
Calculus is extremely pervasive. At my high school, the two upper-level math courses are AP Statistics and AP Calculus (we have Calculus III also). I feel like Statistics is insufficient and very basic. Why is this? Because a good understanding of Statistics requires a whole lot of calculus.
My Calculus III class also jokes that we’ve learned more Physics in class than we ever had in Physics class. (It’s true-- we’ve done vector projections, center of mass/inertia with triple integrals, and I’ve learned line/path integrals, surface integrals, vector fields…etc.). In my opinion it’s not very effective to teach Physics and Economics without Calculus. It’s been extremely helpful to work out concepts like marginal cost and revenue using derivatives.
Simply put, just about every field that is somewhat analytical is founded on Calculus. Combinatorics and Discrete Mathematics might be more useful for a Computer Scientist, but a Physicist, Economist, Statistician, scientist…these people use calculus concepts on a daily basis.
I’m an electrical engineer with an advanced math degree, and I do use calculus regularly. As others have mentioned, it pops up as a basis for lots of mathematics. (E.g., you can teach discrete statistics without it. But the minute you introduce a continuous distribution, you have to use an integral).
But I believe that the relevance of number theory, combinatorics, binary logic/arithmetic and graph theory is much greater today than it was twenty years ago, and that curricula haven’t caught up yet (my teaching experience in the last few years is limited, so my opinions may be dated).
I agree that calculus is still very relevant, and should be widely taught. But there is no reason that other math that is more relevant to information-age jobs can’t be taught alongside it. As noted above, many of these subjects can be taught with little or no background.
[quelquechose] … why is calculus so widely taught instead of some other branch of mathematics?
Calculus is one of the great breakthroughs in the history of mathematics, and because it is so accessible to anyone with a grounding in basic math (at the high school level), it is taught. Why should you learn it? I don’t know where you are in your education nor your major, but if you go on to more advanced courses in physics, chemistry, economics, finance, statistics, computer science, engineering, and operations management, you will see the appearance of derivatives and integrals, and these are all part of the wonderful subject of calculus.
Calculus is fundamental to the study and genuine comprehension of physics (i.e. beyond plugging-and-chugging forumlas.) It also offers tools for dealing with non-discrete and time-varying phenomena that cannot be solved any other way. Even in combinatorics calculus sometimes rears its head (when dealing with variant or distributed quantities) and it is definitely vital for statistical analysis.
IMHO, (conceptual) calculus (and plane geometry, and basic statistics) should be taught at the elementary level. Once students have grasped the notion of whole numbers and the arithmatic operations performed upon them, distributions are a logical progression, and a very simplified (i.e. non-limit based approach) could easily be taught to any student of average ability, paving the way for acceptance of advanced mathematics, instead of this sudden introduction of algebra, geometry, and trigonometry in high school. We teach math in this country as if it is something to be worshiped and feared; if we taught reading comprehension and literature the same way we’d have students reading about Dick and Jane all the way up to eighth grade and then suddenly expect them to digest Shakespeare and Chaucer.
Stranger
No doubt about it, anyone who knows how add and to compute the area of a rectangle can grasp the principle behind the Riemann Integral.
What Stranger hit on and David furthered needs it’s own thread. The state of education in America is appalling.
For the OP, what others have said before me is basically the long and short of it. Integrals and rates of change even show up in froshie chemistry classes with mechanics and rates of reactions. Plus, given how crappy our education levels are, most kids can generally go from a pre-calc or an “advanced math” course into Calculus AB or regular calc with little problems. All that really needs to be understood is functions, variables, and basic trig. With the right teacher it’s, well, easy, especially compared to other sometimes more abstract post-algebra courses.
(At our school we go from pre-Algebra, Algebra, Geo, Alg. II, Trig, pre-Calc, Calc or Stat. Generally, this is followed, but some exceptions are made with classes that are bridges to or combined forms of the above. FWW)
Here’s an example of a real world application of calculus:
http://www.1728.com/minmax3.htm
Yes, it does come in handy.
OK, here’s a dumb question, since we’ve got all the math people together…
Can you suggest any online texts where I can re-learn math? Is it even possible to self-teach math?
I went all the way up to AP calculus in high school and did fairly well, but never took any more math after that. Now I am interested in queue theory due some problems that need solving at work. But looking over the web sites, there is some notation that I’d swear I have never seen before. I thought we’d seen all the symbols by the time we’d reached calculus, if not all the concepts.
http://mathworld.wolfram.com/ Although generally more advanced, this can help at times.
www.wikipedia.com A general catch-all refrence place even though the quality can be lacking at times.
www.google.com You can almost always find stuff on google.
Hmmm, I don’t know much about calculus, but as a carpenter/remodeler, I deal with this every day. I would simply figure the angles with my handy tools (squares, angle finders and such), measure them out, and figure it much more easily than using equations. I would then compare the cost/mile prices with the distance.
I deal with this a lot with for ex: ceramic tile in odd-shaped shower pans. (esoteric, I know) but I do admit I deal with feet and inches as opposed to miles.
Of course, maybe I am using applied Calculus with the help of tools.
Hell, to cut a pitched roof uses the Pythagorean Theorem, but I just use my framing square or look at the tables in the book that comes with every speed square.
I am not down on math, but I think it should be taught more from a practical real-world POV.
I’m speaking as someone who hasn’t directly used calculus in almost 20 years, but I still think it incredibly important. Calculus was the first math class I took that taught me to solve problems that weren’t just practical, but interesting, too - in other words, multivariates.
As a lawyer-cum-writer I have no daily need for it - yet I kinda miss it too.
NattoGuy writes:
> I thought we’d seen all the symbols by the time we’d reached calculus, if not all
> the concepts.
Oh, please. There are enormous amounts of mathematical concepts beyond calculus. There are also lots of mathematical symbols beyond it too.
Here is a good calculus site… http://www.karlscalculus.org/
If you took AP Calc, did you buy any study guide books for the AP exam at the time? I bought the Cliff notes edition to study for my test. It’s handy if you think you know most of it but need to be reminded of a few concepts. Also, did you save your notes? I kept mine to review for future classes. Personally, it’s too much for me to try any learn something like this off a screen, and I hate waste the paper printing it out.
Just in case you feel you can’t self-learn calculus, my high school had a free math tutoring program every week at the mall. Your local high school may have something similar. If such a thing exists, I bet they’d be happy to answer your questions. You may have to swallow your pride a little to be tutored by a high school student, but the event will probably bring in the kind of advanced math students you need help from. I seem to recall that they were open to whoever needed help, not just other students from my school. Yeah, it requires effort, but you’d be doing the job right. Right?