Can Calculus be made pain-free? Should it?

I know from nothing on the subject. If it's just a flat formula type thing, let them do it the hard way. Builds character! If they will understand and apply the reasoning, why not?

I hope my lack of knowledge didn't trap me into a silly answer. It happens sometimes.

roger your character building is as [the integral of (e^i*pi) -1] compared with your gentle understanding and shall forever be differentiated

It is more philosophical than math. I knew A Straight A philosophy student who would accept Calculus on merely the concept. Remember philospher Berkeley rejected Calculus as well. His reasoning was that the infinitesimal quantity could never disappear no matter how small. and Integration is similar in the you get the area under the curve if the slice gets smaller and smaller till the tangent matches the curve.

I think if you can devise a methodical approach that benefits students, by all means teach it to them. Understanding your approach would probably teach them plenty.

Ooops, I meant the philosophy student rejected Calculus.

Ive found that I had to teach calc to students who were changing from a BA to a BS discipline compact. Ive had to get past the " elementary philosophical" concepts , like "functions" and "limits" and get right into examples of what calculus is really about (motion, dimension, change etc) . Ive had pretty good success with kids whove been calc ignorant than with kids whove been exposed to the HS AP calc programs where the applications were not examined so we can understand how we use the calc .

Its like teaching a pilot to fly an airliner after hes spent time only flying crop dusters. Its often easier to start from scratch.

YEH calc can be made easier and more intuitive. I basically taught calc to myself cause I missed almost all of my 10 grade due to an acute illness. I had my dad get me some books from the engineering dept of the railroad and he let me talk to one of the system design engineers of the Reading Company. I learned loits about calc in boiler design and pressure analyses and things like "catenary curves"
Sometimes the educational system gets in the way of the education process.

Which reminds me - the best book about Excel I personally ever came across (most software guides seem to be written by people who only know software, but no applications, and no English) is called "What Every Engineer Should Know About Excel", by J.P. Holman, published by Taylor and Francis.

Opening it at random I find an exercise starting with Planck's blackbody radiation formula (given) with (given) constants and asking for numerical integration of the formula over specific intervals for specific temperatures - it's so much easier to do calculus if you know what you're looking for!

Im actually going to have to get that because, as you know, most of our models are huge algebraic functions run ad nauseum. I remember the beginning days of Basic "A" when you would just write several equations of things like crystal structure or melting temps etc and just string them together. VERY VERY klutzy, but everyone who worked in the field could follow the math. Now with all these app languages, if it werent for calculus, Id have no idea what the hell the computer was telling em or how to plop down some formulae.

I recall getting one of my grad students (a mineralogy student) to do some derivations in partial diff equations and this guy never made it that far. We starte with the applications (del operators) , did expansions as word games, and he caught on in a few days of relatively painless problem solving. Im a big fan of jumping right in to apps and problem solving rather than stringing together crap to build one up to diff equations or complex identities. Its no big deal to learn a specific calculus application and then spread out to back design a program from there . Thats sort of the way I did it in 10th grade and kept it up in college.

Okay ... thank you for green-lighting me, Ralph.

Warning: seasoned Calculus teachers are the target audience of this post; new students are not.

Mostly what I had in mind was a methodical approach that would help with those trickier, more tedious, indefinite integrals. Usually I'm able to work along in the standard textbook and explain everything else to them, including most of the concepts like functions and limits, in the standard way that those concepts are taught. But I have found the integration techniques are often poorly explained.

In the book I just picked up from the Salvation Army store, James Stewart's Calculus (4 ed.), he talks about 4 principal techniques for finding anti-derivatives:

(1) Immediate recognition of what certain basic functions integrate to.

For example:

cos x --> sin x + c
x^2 --> (x^3) / 3 + c
e^x --> e^x + c

(2) Simple substitution.
(3) Integration by parts.
(4) Fancy trigonometric substitution.

Section 5.5 through Chapter 8 of the book covers the topic of "how to integrate expressions."

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It's more or less exactly the same book I used in high school. I loved it then, but now I realise that while the order in which he taught things worked well for me, it might not work best for everyone. First he introduces the integrals of a few basic functions, and then he covers (2), (3), and (4) in that order. While he's teaching (2) - (4), he continues to introduce new basic forms that you can integrate. So he's throwing a ton of different tricks at you all at once, instead of presenting integration as a systematic process.

Here is the new approach I was thinking of using with some students:

[Okay, this approach does assume they are already comfortable with the whole "function" concept, which isn't really very difficult, as I'll explain later. Any way, my explanation in real life probably wouldn't be so heavy on function notation]

(Phase 1) Introduce all the basic functions first, before you even teach substitution. If this bores them, spice it up with some real-world applications. By the time they get to substitution they should know all the basic forms like the backs of their hands.

(In Stewart's book, there are only 16 "basic" functions to learn, so learning them all at once isn't as hard as it sounds. And it's possible to separate some of these 16 functions into different groups, to make them all easier to remember. For example, 10 involve trig, whereas 6 do not.)

After they have been through Phase 1, they should instantly recognise all these 16 basic forms instantly, whenever they see them.

So they'll know...

cos x --> sin x + c
x^2 --> (x^3) / 3 + c
e^x --> e^x + c

...on sight.

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(Phase 2) When they get to simple substitution, teach it to them in the following, very formal way:

Okay, kids, let's look at these new integrals. These things we want to integrate are not basic functions, but they all have a lot in common in terms of how they look. Each of them is the product of two different functions [I.e. in every case we have one "glob of numbers, symbols and x's" times some function which I'm going to call "f(x)." So we've always got "some glob of stuff" times f(x).]. Now that "glob" will be a composition of two more functions. [So that thing I called a glob will just be one function inside of another. Let's write this glob as "g(h(x))" ... because we can.] When you see this kind of thing, kids, "substitution" may be in play.

So at this point, we have something inside the integral that looks like g(h(x))*f(x) dx. Now ask yourself a couple questions: (i) Are g(x) and f(x) "basic functions" that we would know how to integrate? (ii) Is f(x) the derivative of h(x)? If you've answered "yes" to both questions, then substitute the new variable "u" for "h(x)", and your integrand transforms to "g(u) du."

For example, when we want to integrate [sin (x)]^3 * cos (x) dx, we recognise that it is a "glob" times the function cos (x). That "glob" is the function g(x) = x^3, composed with the function h(x) = sin (x). They should identify the glob [sin (x)]^3 as a composition, because visually it looks like [ ... ]^3, with "sin(x)" stuck inside it, and that's how compositions usually look. The student notes the x^3 and cos (x) are basic functions that we know how to integrate; notes that cos (x) is the derivative of sin (x); then substitutes "u" for sin(x) and "du" for cos (x) dx. They end up with u^3 du, and they know how to integrate that.

(This is what they are taught do anyway, except that I'm making it more formal, giving names to more things, and making them conscious of every step.)

(Phase 2a) At this point in the course, take some time to revisit some of the basic forms, since some of those 16 of those could be figured from other basic forms out using the substitution method. It will reinforce the knowledge from Phase 1 they already have.

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(Phase 3 & 4) I haven't figured out the details yet on how to incorporate integration by parts and trig substitution into this formal thought process that I may end up teaching to more process-oriented kids. But I don't think doing so will be too hard. It isn't too hard for me to look at an expression and tell whether it's something that you can use integration by parts on, so the formal thought process I give to my students for analysing something and ascertaining whether it's something you can use integration by parts on won't be a long one.

(Phase 5) Finally, I'll have to develop strategies for combining all the different techniques. There are also times when you have to tinker with the integrand a bit before it turns into something you recognise. I'd have to think up some formal way they could do that.

===

In summary, here's what I'm aiming for... I'm trying to take my own approach to solving these integrals, which began as a collection of unconscious gut feelings that told me what to do next while looking at a particular problem, and which became more reliable as I did problem after problem; and then turn it into some kind of conscious, formal, deliberate process that the vast majority of people -- who don't rely on gut feelings as much as James Stewart and I do when it comes to Math -- can remember and use.

Stewart doesn't give students a formal procedure for how to integrate. He shows them a few examples of how to apply the substitution rules and expects readers to figure out the procedure on their own. Then later he gives students a mess of integrals and expects them to intuitively "know" which integration tricks to use on which ones. (I'm not singling him out at all for criticism; that is how they all teach it.) I want to give students a formal, analytical way of deciding which trick works for which integral.

I'm trying to play the part of that Indiana Jones character kicking sand on the invisible bridge in The Last Crusade. You need to make things more systematic, concrete and familiar for some people. Not everyone can blindly lunge correctly in the direction of the right answer.

Well, you went in a completely different direction than I did in this thread.

But my approach was only intended for very procedural types. And, of course, when it's a question of teaching what the things actually mean, I'm far less formal.

I usually use concepts like motion (position, velocity, acceleration) when explaining the concepts of first and second derivatives to them.

One trick I've found very useful in describing what a derivative is, is to talk about instantaneous velocity, because that's something very tangible to anyone who has stuck his hand out of a car window. If we use t for time and s for position, then average velocity is over a shot period is:

v = [s(t + Δt) - s(t)] / Δt

Shrinking Δt, the time between your measurements of position, gives better and better estimates of instantaneous velocity, which can be understood in terms of how hard the air is hitting your hand. So theory, the value that v approaches as Δt approaches zero is your instantaneous velocity.

All other rates of change can be understand by analogy once you know how s, v and a are related.

Interesting comments here from Oylok and Farmerman.

I too did best at learning math in various specific Engineering courses in which I was aided by intuition about concrete physical processes. A course in elasticity of materials gave me as much or more understanding of the mechanics of Tensor Analysis than I got from the Math department -- same goes with the standard Vector Calculus everyone gets in Engineering courses. Certainly the common application of relating location, velocity, and acceleration in linear motion is a near universal teaching aid.

From a teacher's perspective I suspect the basic difficulty in calculus springs more from the algebra involved than it does from the basic theorems of calculus involved,; i.e. limits, the mean value theorem, derivatives & anti derivatives. I have long believed that a firmer foundation in algebra was the solution to most of the difficulties students encounter incourses in calculus and differential equations.

Ultimately algebra involves messing around with mathematically equivalent expressions until you recognize the answer. The problem of finding an algebraic expression that is a perfect differential of another function (i.e. integration) certainly illustrates that observation. I think the methodology Oylok outlined above is a good one.

Algebra is really just an elegant and compact shorthand, and, as with any new language, it requires practice and fluency to see equivalances and patterns. I believe that too is an important point to emphasize to students.

Seems that were going in the same direction even though were on different routes.
Ground water is loaded with "drumhead" and vector analyses

Geophysics , is series , spetcral analyses, Fourier, and del ops.
Geochem and petrology is all diffusion,Thermo and Variograms (teeny bits of calc ) mostly stats.

Same goes with fluid mechanics. Stokes Theorem, analytic functions and even Helmholtz' vorticity theorem (Curl or del cross V) were everywhere. (I am even learning to talk to geologists about groundwater & transport modelling. They usually tolerated me badly though.)

I do remember feeling that I had become the victim of a massive fraud the day I realized that a finite Fourier transform was merely an Hermitian Matrix. Suddenly two disjoint paths (Generalized Harmonic Analysis, and Linear algebra) revealed themselves as the same thing.

You are always welcome at our klaverns of Ki.

There is no eaasy road to Mathematics of any branch, as it is the language of the universe

I have been rather busy, or I would have replied to this sooner...



Thanks, George Ob1, you've almost inspired me to type up the entire lesson and present it here. Unfortunately, this BBCode really isn't all that handy for Math.



Yes, that is a good point. Farmerman actually mentioned teaching advanced Math as a word game at some point in the thread, too. I've posted some much easier word games below that could allow students to become more enterprising in jumping from expression to expression. They could learn these and play them long before they began basic Algebra, but the games would help kids become comfortable at an early age with the type of intuition involved in Trig, Calculus and mathematical proofs.

Game 1:
"Shooting Blanks"

Procedure for Tutors to Follow:

(1) In your head, come up with a chain of words, so that each two consecutive words make up some common expression. For example, I could start with "Halloween" and get "Halloween candy" from that, then move to "candy cane", and then "sugar cane." So I'd have linked the first word, "Halloween", with the last, "sugar", through a series of associations.

(2) Write out the word associations in a vertical list, giving each association it's own line. However, the tutor should represent all words except the first and last as blanks. So you'd give the student a work-sheet that looked like this:

Halloween _____________(1)
_____________(1) _____________(2)
sugar _____________(2)

The kid would then fill in the blanks to complete the problem; putting "candy" where the 1's were, and "cane" where the 2's were.

Better Procedure for Teachers to Follow:

Replace "Step 1" with the following: ask fourth graders to come up with their own word associations for you to use on them. Make Jack's word associations into a puzzle for Jill, and vice versa. That way, all the associations will be ones that the fourth graders are comfortable and familiar with.

Relevance:

In Trig, I might tell someone that 0 < θ < 90 deg., and sin(θ) = 0.8, and then tell the kid to prove that cot(θ) = 0.75. The information you've provided needs to spark a couple of "word associations" between trig functions: (i) cot(θ) = cos(θ) / sin(θ), and (ii) [sin(θ)]^2 + [cos(θ)]^2 = 1. You can then plug a number into (ii) to get that cos(θ) = 0.6; then plug the result into (i) to get cot(θ) = 0.75. But the hard part is in recognising how Trig's words are associated.

Game 2:
"Metamorphosis"

People have probably seen this one. You're given two words, "Word A" and "Word B". Start with Word A, then, replacing one letter at a time and passing through a series of other valid words, transform A into B.

Example: "Change SHOE to LACE."
SHOE --> SHOT --> SOOT --> SORT --> SORE --> PORE --> PARE --> PACE --> LACE

Relevance:

This is a lot like integration by substitution when you think about it...

There might also be a lesson here for Calc-A teachers:

Do a lot of simple numerical integrations to find areas until the students are sick of it; then teach integration in terms of Riemann sums once you've motivated them to learn the subject.

I didn't even bother paying attention to the lessons on Riemann sums, since I knew you could bypass that whole stupid process just by guessing the anti-derivative. I only started caring about how summations were related to integrals when I got to University Physics courses. (Perhaps that's why I only scored 4/5 on the AP Calc exam.)

Do you know a poster here named Boomerang? She's a world-class photographer but if you as much as mumble anything along the lines of "e to the x du/dx, e to the x dx" in her presence you're going to lose her attention and never catch it again, so - see if you can help her with a concept she's developing:
http://able2know.org/topic/172617-1#post-4624003