High School Calculus: Differentiation vs. Integration

[The_Z_Factor]

Today I asked an upper-classman friend of mine what she was doing currently in pre-calculus. She told me she's doing differentiation, I woulda guessed that. However, she has a friend in calculus AP and I asked her what they did in AP, if she knew, since I'm planning on taking AP calculus my senior year, and she said they're doing differentiation as well. Its almost the end of the semester there's only 4 weeks left so I guess my high school doesn't get into integral calculus. So I was wondering, once I get into college and start taking calculus in college, will it jump right into integration or continue differentiation and then get into integration? If the first is the case, should I study integral calculus and try to be ready for integrating?

 

[Invictious]

You probably will get into integral calculus, but for a brief moment only, aroun a week or two, not even scratching the surface.

Once you get into college, you will be doing differentiation again, but at greater depth with other various techniques.

 

[mathwonk]

be sure to learn algebra and geometry. calculus in high school is a BS idea intended to prove high schools are high quality, but not to help students.

 

[The_Z_Factor]

be sure to learn algebra and geometry. calculus in high school is a BS idea intended to prove high schools are high quality, but not to help students.

I read in a calculus book about how teachers in high school don't know how to teach calculus and can't give an idea or show how brilliant and beautiful true calculus can be which is why high school calculus courses usually are so poor because students arent interested or complain that its so hard. I study calculus because I like the subject and I like studying alone although it can make it harder for me personally but I think studying the subject alone let's me have a pure perspective of calculus, and not a teachers. I'd rather have my own view on something that I love than a teacher who doesn't even want to teach it.

Which brings up the fact that at my high school I believe the calculus teacher (we only have one calculus teacher I think) is a sports coach

Perhaps itd be better if I learned what I need to know by myself that way my mind won't be 'manipulated' by a teachers view of the subject

 

[sennyk]

The students are normally the problem.

I taught calculus at an inner city high school for 2 years. I can tell you that I only had one student that was ready for calculus. He was from China. The students couldn't find the equation of a line given two points; they couldn't graph y = x^2, without their calculator. Forget about finding the zeros of a polynomial. They were super super super super weak. Their precal teacher wrote notes on the board and gave them a worksheet everyday. Don't even get me started about their trig. I don't expect trig geniuses, but they couldn't tell me what cos(0) was without their calculators. (rant, rant, rant, etc...)

 

[colby2152]

My high school calculus teacher was one of the best I ever had. Not one of my college professors could even touch his ability to relate to students.

 

[HallsofIvy]

Today I asked an upper-classman friend of mine what she was doing currently in pre-calculus. She told me she's doing differentiation, I woulda guessed that.

I would not have guessed that! "Differentiation" is definitely part of calculus, not precalculus. Precalculus is generally a review of algebra and an introduction to limits.

However, she has a friend in calculus AP and I asked her what they did in AP, if she knew, since I'm planning on taking AP calculus my senior year, and she said they're doing differentiation as well. Its almost the end of the semester there's only 4 weeks left so I guess my high school doesn't get into integral calculus. So I was wondering, once I get into college and start taking calculus in college, will it jump right into integration or continue differentiation and then get into integration? If the first is the case, should I study integral calculus and try to be ready for integrating?

I would expect that "precalculus" might involve some elementary differentiation formulas near the end of the course. Calculus itself would introduce differentiation early, involve more theory, more complex differentiation problems, and more applications of differentiation. What the calculus course you take in college covers will depend on which calculus course it is. A calculus I course has to assume that at least some of its students will never have seen calculus before and perhaps not limits. If you take AP calculus in High School, then it is possible that you can skip Calculus I (that's the whole point of "Advanced" Placement, isn't it?) and go directly into Calculus II which will assume knowledge of differentiation and basic integration and be mostly methods of integrating more complicated functions.


Z_factor, I find it very difficult to believe that a Calculus book would about talk about "how teachers in high school don't know how to teach calculus". Whether it is true or not, such a statement would have no place in a textbook, and the author would be alienating a large portion of his market! I doubt any publisher would allow that.

 

[SiddharthM]

generally the calculus in one variable is taught over one year. Calculus 1 the first semester, you typically learn what is covered in AB calculus except at a higher level. Calculus 2 the second semester the 2nd sem except at a higher level.

The AP exams are pretty weak, and generally you are missing out on a lot if you skip out on your college calc 2 class.

 

[maphco]

I'm taking calculus at my high school right now and I'm finding it pretty easy, we've been doing limits and differentiating this first semester and have been doing optimization problems for the past couple weeks. I've heard from my calculus teacher, and other teachers in the school, that taking calculus in high school is a great benefit, but when you go to university you should retake the calculus I course because the learning environment drastically changes. This would allow you to basically review the course in a university setting, making it easier for you to adapt to the new way of learning.

 

[watanake]

I am taking Calculus I at university. We started with derivatives and ended with Integration of inverse trig functions.

 

[k3N70n]

be sure to learn algebra and geometry. calculus in high school is a BS idea intended to prove high schools are high quality, but not to help students.

quoted for truth

 

[martial]

be sure to learn algebra and geometry. calculus in high school is a BS idea intended to prove high schools are high quality, but not to help students.

It very much depends on the quality of the teacher. Starting when I was fourteen I was taught by a Harvard grad who was really gung-ho and almost ruthlessly demanding about calculus, lin al, and (elementary) analysis. Absolutely loved it and feel I learned and retained tons.

 

[mathwonk]

I am of course speaking in general terms, about the 95% of all post AP students I get in college, who think calculus is a list of rules to memorize, and who have never mastered algebra, much less geometry. You would be amazed how many students get into calc and do not know how to use similar triangles or pythagoras theorem to set up the problems before calculus can be used on them. I almost never get a student who knows how to factor even x^3-a^3, or the binomial theorem, and without that how can one even do the derivative of X^3?

Of course one can memorize that it is 3X^2 but that is not sufficient for college. students here think i am joking, but a good mastery of algebra is very rare, and essential for success in college calculus. and proofs in geometry are less and less common, in the US of course. I know many of you come from other countries which respect high school math more than we do here.

and there is a big difference between knowing how to relate to students and knowing how calculus works. both are nice of course but i myself prefer a teacher who knows the subject.

 

[mathwonk]

but perhaps my advice goes misunderstood. although i do believe calculus is really much too advanced for most high schoolers, it is possible to learn calculus well, and to learn algebra and geometry badly.

my real concern is not with the substance but the spirit of AP courses. They are not abiout why, but merely what. So go ahead and take any course you find interesting, and that ahs a good teacher, but keep asking why, keep trying to understand. do not be satisfied with formulas you do not know the origin and justification of.

and if you really want to know if you understand geometry, read euclid and archimedes. and of course you will find a lot of wonderful number theory also in euclid that is now taught in college at the upper levels, like gcd's and the infinitude of primes, and irrationality of sqrt(2).

these topics would be much better in high school than calculus.

high school math should consist of algebra geometry, and linear algebra and matrices. maybe some limits in the context of areas of circles and volumes of spheres as done by euclid and archimedes. this is already more sophisticated than most high school AP courses.

teaching derivatives without understanding limits is not really depth. it is a fraud practiced on high school students to make people, including themselves, believe they are studying advanced material.

notice that many high schools have actually removed geometry with proofs from the curriculum as too difficult, but they offer calculus. how absurd is that?

 

[mathwonk]

one problem with my advice is that if you do not take ap courses some admissions officials may think you are not advanced and not challenging yourself, and may not admit you to the best schools. they do not know what they are talking about of course but you cannot afford to be misunderstood by them. after entering college knowing the subject matters more, but when vying for entrance appearing to do so is key. this is frustrating to me but true.

so take whatever the prevailing "wisdom" says to take at the highest level, but try to insure that you actually learn something by questioning the teacher at every stage as to why the material is presented as it is. if they cannot answer then you may suspect you have an unqualified teacher.

still you must be diplomatic and learn from them whatever they have to offer, without insulting them.

 

[mathwonk]

here is a simple little test of whether you are learning differentiation: do you realize that if you define a moving volume function for a solid, as the portion of volume lying below a moving plane, moving perpendicularly to some line, that the derivative of this moving volume function is the area of the leading face of the moving volume? i.e. the area of the plane region cut from the solid by the moving plane?

if you do not know this you do not know even as much as archimedes about area and volume.

 

[martial]

if you do not know this you do not know even as much as archimedes about area and volume.

In all fairness, Archimedes was one of the greatest mathematicians in the history of the world along with Newton and Gauss, so what's apparent to him cannot be expected to be apparent to any arbitrary high school or university student ;-).

Overall an excellent series of posts elaborating on what you meant. Provided that a student has a good grasp of algebra, geometry, and trigonometry (I admit I'm a bit weak in geometry myself, but I certainly know how to do the examples from your students!), I think learning some calculus could only help them because it's far more interesting than memorizing double-angle formulae or the equations for conic sections or something (not that you won't need those) and because the earlier they learn it, within reason, the better it will stick (e.g., it won't be one of those things where a couple years later they're trying to figure out which is which with regard to integrals and derivatives).

As to your point about "memorizing" calculus, I totally agree. For example, I can't remember the last time I used the ring or washer method for finding the volume of a 3D object, but I could figure it out in less than a minute because I know calculus reasonably well and I know exactly what the idea is.

Also I recommend T.W. Körner's Analysis text, which ought to resonate well with students at the university level; it certainly did with me. It's actually an enjoyable book to sit down and read, with occasional pauses to figure out some mathematics.

 

[mathwonk]

i think your remark about archimedes is off the mark. It is correct that no student could be expected to discover what archimedes did on his own, but he definitely should be aware of it 2,000 years after it has been explained and has grown and evolved into the foundations of the subject.

my comment in fact amounts merely to the reason behind what is now called "volumes by slicing" or volumes by the "disc or washer method".

my point is that even the most basic insights which entered the subject at its inception are being omitted in todays shallow treatments of the subject.

the problem is that students merely memorize that they should integrate pi y^2 instead of understanding why this is true. since you are able to reconstruct this method, apparently you do understand it.

 

[Count Iblis]

Why don't we teach maths in primary school? Most 9 year olds are capable of enough abstract reasoning to be able to learn algebra from high school books. At an earlier age it is more difficult, but I do think it is possible to teach algebra to 5 year olds using computer games.

If you learn things at a very young age then certain specialized brain areas, like e.g. the part that is used to recognize faces, are still available to be programmed. On NGC channel it was shown that Polgar who learned to play chess at age 3 by her father, uses the brain part that is used to recognize faces when she looks at the pattern of pieces on the chess board.

So, if we teach maths to children when they have just left kindergarten, they'll become much better at maths. Children at age 9 or 10 could start to learn calculus. By age 15 they would have mastered what most Ph.Ds know today (but then in most subjects, not just maths, of course).

I think that language education is the best developed of all subjects that are taught. language also involves abstract concepts but that doesn't stop us from teaching it to babies. If we were to teach English the way we teach maths then you would read your first book at university.

 

[mathwonk]

well i taught eulers formula to third graders, and one if my students is now an engineer. you teach whatever you know. we need to encourage teachers to learn more, and people who know more to teach.i brought in colored polyhedra to the class and had kids count faces edges and vertices. there was a mentally retarded kid who was being mainstreamed and even he enjoyed the models.

the other end of the spectrum was the 8 year old girl who guessed eulers formula. (V-E+F = 2)

 

[mathwonk]

Count, your suggestions are apparently those of someone who has never had children, as most of them do not care to be force fed mathematics at an early age.

the obstacle is not their intellectual capacity, but their desire to have fun and to be like other children.

they also need to be allowed to play, and eventually choose their own specialization area.

maybe you were joking, but some of us who have been parents may have been tempted at one point to think the same thing, since in the abstract it may make sense, but only there.

 

[Count Iblis]

Mathwonk, yes, but then if every child is treated in the same way then they won't complain about being forced to study maths while everyone lese can play. And I don't think we need to cut back on free time at all. Currently in primary school children are taught almost nothing about science. They are taught to do simple arithmetic. The same type of sums over and over again from age 6 to 12. No wonder they don't like it.

 

[sadhu]

i like to study everything at home so i don't know how well my teacher is doing at school