Teaching calculus to people who only learned algebra 2.

[AlfredPyo]

So, I want to teach calculus to my math club.
I know that differentiation and integration is easy. But how do I teach the CONCEPT of DERIVATIVES and INTEGRATION to people who only learned algebra 2?

Algebra 2 people because that's the bare minimum of mathematics you need to learn calculus, with the exception of trigonometry.

Also, how do I teach them the concept of dy, dx, dy/dx, d/dx, d^n/dx^n?
I don't want them to be confused. I have to treat them like little kids.

 

[Simon Bridge]

Start with geometry - can they find the areas of triangles and rectangles? Trapezia?
Do they know about functions?

Move them onto calculating areas of arbitrary polygons by breaking them up into triangles.

Then move onto smooth shapes - approximate by triangles drawn from a center ... you'll probably find they will automatically start using polar coordinates without realizing. Work out what happens to the equation of the area as the number of small radial triangles gets bigger.

You can introduce limits here as a form of notation.

Try not to do it by "rule" - which is how most of us got taught.

I stumbled upon this approach when I was looking for a way to motivate students to learn trignonemtry ... decided to say that every shape is made out of triangles, so learning all about triangles means less work, lead to questions about smooth shapes, and from there a couple of the brighter students invented a basic calculus without realizing.

 

[AlfredPyo]

Well, integration is finding the area under the curve, SA of one revolution, volume under a surface, and etc.
But I didn't learn integration by breaking up areas into triangles - we broke them up into rectangles using sigma notation.

But how should I teach that to them, and also the concept of derivatives.

 

[Vanadium 50]

I think you might want to back up a bit. Why do you think people will want to join an organization where the president "treats them like little kids". You have to solve that problem first.

 

[AlfredPyo]

Treat them like little kids as in teaching something. I can't move on to topics that might be confusing to them, so I have to be very gentle about the topics I teach at the beginning.

Later on I'll teach them differential equations. That's what I'm trying to do.

 

[IGU]

So, I want to teach calculus to my math club.

You might find it interesting to learn and then teach them Visual Calculus. This is a recent interest of Tom Apostol. Pretty cool stuff. You'll want some geometry though.

 

[lurflurf]

Algebra 2 is not a meaningful term. What you call algebra 2 might be algebra 1 to some people and algebra 3 to others. To learn calculus (traditional presentation) you need some algebra to not become lost. The exact amount of algebra is less important. This is even more true if you are presenting some concepts from calculus without attempting comprehensive coverage. If so omit a few algebra details and slow down at times, it will cause no problems. What specifically do you worry about them not knowing? The concepts of calculus are pretty simple. The difficulties are in the details. You might say that the only purpose of algebra 3 is to give students another chance to learn algebra 2 and to finish a few things that were left out.

 

[Yellowflash]

"treat them like kids'?

 

[Yellowflash]

@yellowflash I've already explained that you little

I think a bigger problem for you will be your attitude. Teaching this stuff requires patience. If you explain integration by parts and someone does not understand it, you will have to restate it multiple times.

What is algebra 2 for you? Does it cover functions and graphs?

 

[Simon Bridge]

I'm inclined to go along with the others: your approach needs work - isn't this a maths club?
The members are already interested in maths so you shouldn't have to be giving lessons at all.

But you are talking like it's a forum for you to expound on topics of your choosing.
Maybe the format of the club is that members take turns giving lessons?
You shouldn't be trying to control things - you'll just lose members.

The approach I outlined is just an outline - you have to develop it should you choose to use it.
Nobody is just going to hand over a set of lesson plans.

The way you learned calculus was the non-fun approach that I call "by rule". Your teachers just said "this is how you do this thing called integration" with no reason behind it: it's just the rules of an abstract game. This is a fast way to learn the mechanics of calculus, and the application can follow so it's fine for serious maths students aiming to use maths as part of their future job but terrible for people doing maths as a hobby.

Note: the lecture-by-rule approach is among the least effective ways of teaching that actually get used.
It is only used at all because it covers the material quickly with little effort on the part of the teacher.

If your club is actually a course in maths then fine - get a textbook and follow it.
But if you really mean that it should be a club for enthusiasts, let go the control - slow down, be patient - start with the applications and develop the maths from there.

Good luck.

 

[AlfredPyo]

Ok. I'll go with your approach on integration.

On the maths club, I'm from America, not Britain.

 

[AlfredPyo]

Back to the topic.
You've explained how to explain integration.

But how do you suppose teaching the concept of the derivative to someone?

 

[Physics_UG]

Back to the topic.
You've explained how to explain integration.

But how do you suppose teaching the concept of the derivative to someone?

The concept of a derivative is quite easy to explain. First teach the idea behind a limit. Then, assuming your audience knows something about calculating the slope of a line you can immediately write down the definition of a derivative in terms of the limit.

Remember, you need to first explain what a limit is, since calculus is the study of limits. Derivatives and Integrals are applications of limits.

 

[Simon Bridge]

Ok. I'll go with your approach on integration.

On the maths club, I'm from America, not Britain.

So noted.
America = USA?

I am not in Britain either, or the UK for that matter, and I try not to assume things about where people are writing from, just like I'm trying to to assume too much about what your math club is all about.
You still have not told us that.

Back to the topic.
You've explained how to explain integration.

How to motivate the topic - sure.
The approach outlined makes no attempt to explain anything.

But how do you suppose teaching the concept of the derivative to someone?

The principle is the same - what do you use derivatives for?
Start with why they would want to know and work backwards.

Once students/members are familiar with finding the areas of smooth shapes - a process that involves making the triangles very thin ... so the length of one side gets very small ... motivates a need for the concept of a limit, which is a key concept for derivatives as well as for integration.

You can describe a smooth shape with a function so you have the option of the area of a shape being the area under a graph if you like - you will have to guide members to make this jump though - if that feels natural from how the members are going.

From a function you can find the slope of the tangent to the function for example.
Areas of smooth shapes from triangles requires being able to find the tangent to a shape - so there may be a way in there too.

You could also introduce derivatives as an anti-integration if that seems natural. It depends on the members - and different people will find different things intreguing.

You need to use the members enthusiasm to work out where to go and some of them will have snatches of information here and there from other sources, so they will make connections and you can use that.

You are mostly just providing the notation and terminology as they need it, and listening to their learning process. Listening and waiting are big parts of this approach.

To figure out how this works, you will need to work through this path yourself, thinking about the different ways that people are going to try things out and how to nudge the more way-out ones back in line. You will probably find yourself using a mix of different approaches and that's perfectly fine: you have to be comfortable too.

It's harder work than the usual approach ... focus on the bits your members find fun.
Assuming of course that math club is supposed to be fun...

good luck.

 

[mathwonk]

my first take on your question, was: wow! what a privilege to have calculus students who have actually learned algebra 2! the main problem a calculus teacher had at my uni was how to teach it to students who have not learned algebra at all.