# Teaching Calculus for the First Time

## Main Question or Discussion Point

Hi,

I am a graduate student at the University of Michigan and I have been tutoring a high school student in math and physics for about a year now. As he will be taking Calc AB next fall, he asked me to teach him calculus this summer. I am excited to do this, but am not sure the best way to approach it.

When I was in high school, I did a little independent study using the book Calculus Made Easy by Thompson and Gardner. I remember this to be beneficial; however, returning to the book now, there is very little treatment given to limits, a topic my student has yet to cover in pre-calc. I'm thinking the best place to start a mini-course in calculus is with the concept of limits, but am not sure of this.

Does anyone have any advice on the best way to introduce calculus to a student?

Thanks,
Eric

I found "Single Variable Calculus - Early Transcendentals" by James Stewart helpful. A lot of attention is paid to limits and will cover everything he will see in high school Calculus.

bartle and sherbert !

I suggest a custom set of lecture notes, design to appeal to the students interests and curiosities so that they feel motivated to learn.

I've always thought that it would be neat to introduce calculus by asking your students to derive the formula for the area of a circle.

Gradually, they'd realize that a circle is just a polygon with an infinite number of sides, and that you can break up a polygon into triangles. Eventually they'd have a formula of n sides, and realize that when n = infinity they have the area for a circle.

I'm only a student, not a teacher, so I can't say how well that'd work

My calculus teacher introduced derivatives/integrals to us by showing us cars, and then telling us to make graphs representing the acceleration, velocity, and distance.

mathwonk
Homework Helper
well you yourself said you were benefited by thompson's non limit approach. why would your student be different?

in my opinion limits are much too hard and are actually inappropriate for a first encounter with calculus. just start teaching him and play it by ear. you will see yourself what works and what does not.

in fact limits have almost nothing to do with calculus of polynomials, but only with calculus of functions that do not even have taylor series.

if you want to introduce limits i also like doing the are of a circle as a limit of areas of triangles, with vertices at the center. this is essentially archimedes' point of view. i.e. he regarded a circle as a triangle with base equal to the circumference and vertex t the center.

an analogous view deals with the sphere as a pyramid with vertex at the center and surface area as base. this also explains their volume formulas in relation to their circumference or surface area formulas.

Actually, I took AP Calculus A/B and B/C this year, and I think one of the best ways to tutor calculus (and one of the best ways to review it when preparing for the AP exam or even final exams later on) is to simply get an AP review book (Princeton has a great version). You can then go over the lessons with him straight through (as well as the practice problems at the end of each section). I think this would be effective because each section is short, which is advantageous since you are trying to give him a taste of calculus, plus though this book may not get into all of the fun parts of calculus, like the word problems, it does touch on all of the fundamental aspects. If you teach as you go and have him explain back to you the concepts and have him ask questions, I think the tutoring sessions have the potential to be quite effective.

PS--it is great to start with limits then move on to derivatives and the short-cut, then addition/multiplication/division rules, et cetera.

PPS--I've also heard that Calculus for Dummies is good at giving a brief but logical explanations.

I would get James Stewart's Calculus: Early Transcendentals. The first five chapters generally cover everything that is covered in a Calculus I course.

My general approach to tutoring calculus is to stay away from "useless memorization," and to show the student that a given problem can be solved in many different ways.

Can't go wrong with Apostol's Calculus text

What I find most troubling in the teaching of calculus (in One dim) is to call attention on the fact the ubiquitous delta x:
a) does not go to zero in the denominator, one divides by deltax before letting deltax tend to zero
b) if a problems calls for the solution of a differential or an integral equation by numerical methods deltax can not go to zero

Hi !

I have thought a lot about the best way to learn and to teach calculus.
Students need strong algebra and trig skills to do well in calculus

Arithmetic - > Algebra - > Calculus - > Advanced Math

Understanding the idea of a limit is essential to understanding the process
of differentiation

functions - > limits - > derivative

My advice is to concentrate on the basics and forget advanced stuff. There are
too many people trying to run when they can't even crawl. I include myself in this.

What is the point of trying to do hard advanced stuff when you can barely differentiate
and integrate ?

I would forget about things like

Infinite Series
Vector Analysis
Differential Equations

until you have given the student a hard test in elementary differentiation and integration
problems so you know they are ready for it. Otherwise what's the point ?

John

>I am a graduate student at the University of Michigan and I have been tutoring a high >school student in math and physics for about a year now. As he will be taking Calc AB next >fall, he asked me to teach him calculus this summer. I am excited to do this, but am not >sure the best way to approach it.
>
>When I was in high school, I did a little independent study using the book Calculus Made >Easy by Thompson and Gardner. I remember this to be beneficial; however, returning to >the book now, there is very little treatment given to limits, a topic my student has yet to >cover in pre-calc. I'm thinking the best place to start a mini-course in calculus is with the >concept of limits, but am not sure of this.
>
>Does anyone have any advice on the best way to introduce calculus to a student?
>
>Thanks,
>Eric