These lessons are a revolutionary approach to learning calculus. The fundmental concepts are introduced in Calculus Without Tears Vol 1 - Constant Velocity Motion, and there is no algebra, no trigonometry, and no geometry (beyond the formula for the area of a rectangle). Why? - because they are not needed. Calculus is the mathematics of change, and change is represented by functions. The basic operations in calculus are differentiation (in plain English - calculating velocity) and integration (calculating distance), and solving differential equations. If the functions and equations are easy, there is no need for any algebra or trigonometry at all. The approach in these lessons is to learn calculus using a simple physical model, a runner running at a constant velocity, and the easy functions that describe it. Once the fundamentals are understood using the simple model, more complex models and functions can be studied (in Vols 2 and 3).

The Beginning of the Scientific and Technological Revolutions

Modern math and science started with a single differential equation. In the 17th century Isaac Newton discovered gravity, and wrote the equation Force = Mass * Acceleration (F=MA) that explains the motion of the moon and the planets, solving a problem that had baffled the best minds for thousands of years. This equation launched the scientific and technological revolutions that have transformed our world. It is the basis for physics, and is used every time motion is analyzed, from the calculation of the trajectory for the Apollo spacecraft, to the design of the rotor in your electric toothbrush. Differential equations are the key that unlocked the mystery of the physical universe.

Motivating the Math and Science Curriculum

The motivation for all mathematics beyond arithmetic is understanding the world around us, i.e., physics, and physics starts with differential equations (see previous paragraph). Yet in secondary school we are teaching complex algebra, geometry, and trigonometry before teaching the physics necessary to motivate their study. The current math curriculum is upside down - putting it right side up, that is by teaching calculus and differential equations early (instead of last or not at all), will make it possible to study problems from physics and electronics (differential equations are the basis of circuit analysis) that will motivate the entire math and science curriculum.

How Computers Have Changed Everything - By Making It Easy to Solve Differential Equations Numerically Using the Formula distance=velocity*time

Differential equations are, in plain English, velocity equations, and velocity equations are easy to understand, even if you don't have any math training at all. For example F=MA, the equation that is the basis for mechanics, is easy to understand: you push (with force F) an object (with mass M) and it accelerates (with acceleration A). Note: velocity is the rate of change of distance, acceleration is the rate of change of velocity, so F=MA is a 'second order' differential equation. The differential equations governing electrical components and circuits are just as easy to understand.

However, while easy to understand, differential equations were notoriously difficult to solve. But, that was before computers.

Before computers, calculus was mainly 'analytical', that is, problems were solved by analysis, invention, and algebra. Most calculus problems are difficult to solve analytically, and many are impossible to solve. With computers there is another way to solve calculus problems, that is, numerically. It is easy to solve every calculus problem numerically using the formula distance = velocity * time ! An example: while working at Honeywell I was responsible for estimating the accuracy of rocket trajectories used to launch satellites into orbit. Differential equations characterize the performance of the many system components, but the idea of combining and solving these differential equations analytically is not even imaginable. However, it's easy to solve the system numerically. Take the time of the flight, divide it into 1 millisecond intervals, and project the system through each millisecond using the formula distance=velocity*time. This is how it is actually done. See the Airplane Simulator page linked to the left for more details on the method.

So, differential equations are the basis of modern science, they were always easy to understand, and now they are easy to solve. It is time to include them in the math/science curriculum.

Calculus is Easy and Intuitive......Really

These lessons were written to teach calculus to a student starting in the 4th grade. The formal prerequisite is decimal arithmetic, that is, adding, subtracting, multiplying, and dividing easy decimal numbers. Surprisingly, the fundamentals of calculus are easy and intuitive. Here is a shocker: differentiation is a generalization of the formula velocity = distance / time, and integration is a generalization of the formula distance = velocity * time! (In Vol. 1 we don't generalize, we stick with distance = velocity * time). The presentation is rigorous in essence but not weighted down by technical details. The goal is for the student to understand calculus and differential equations the way someone who works with them every day understands them, with a good intuitive grasp of the fundamental concepts.

The lesson sheets are modeled after the ExcelMath lesson sheets used to teach the elementary school math curriculum. Each lesson consists of a brief presentation of some aspect of the subject being studied, followed by numerous easy exercises. Math is a subject that must be learned by doing, and the exercises reinforce the subject of the lesson as well as review and integrate the material from previous lessons. The books are designed for self-study and homeschooling in that the pace is very slow and nothing is ommited. Most of the exercises have multiple parts and the answers are added and the 'checksum' is printed in the upper right corner of the exercise so that you can check your work.

Each lesson is on a double-sided 8.5x14 page, there are 74 lessons in Volume 1.

Read the Reviews of CWT

From the review on HomeSchoolMath.net
"A TOTALLY different way to learn calculus - you can do it even before high school."

From the review of Vol 1 on EclecticHomeSchool.org
"Even a mathophobe like myself was able to work through the book without trouble, and my third-grade student, who is working on a fourth-grade level in our math program, has made it part-way through the lessons without any trouble. I wanted my husband's opinion on whether this was really calculus, for it seemed way too easy."

From the review of Vol 2 on EclecticHomeSchool.org
"In any event, yes, your elementary-school age student (or older; you don't have to limit this to elementary grades) can learn concepts that their counterparts in high school and college physics and math classes are struggling with, and the practical applications make the study relevant and not just an abstract juggling of numbers."

From the discussion of CWT on HomeSchoolMath.BlogSpot.com
Distance = velocity * time, or Calculus Without Tears. You can learn calculus concepts starting from the formula distance = velocity * time. Yes, that's true. That's what the book Calculus Without Tears is all about. It starts from the simple situation of a runner running with constant speed (velocity), and goes very step-by-step into actual calculus concepts, such as derivative, area under curve (integration), and differential equations.

From the review of CWT on RainbowResource.com
As I said before, I really loved the study of mathematics - especially calculus. But I never realized its use and richness until perusing these books. Mine was a theoretical study. This book gives you all application. This is an answer to 'When are we ever going to have to use this?' - a great answer. This 'study of change' could well transform your unmotivated mathematician into a rocket scientist (or an engineer, physicist, astronaut....)

About the Author

I have an engineering PhD from Berkeley, and have worked many years in the aerospace industry on a variety of projects, including Star Wars(!), the Galileo Space Probe, the Mars Observer, Space Station Freedom, the Centaur rocket, the F-117, and GPS. I also have taught high school math. I currently (2004) have a daughter in the 4th grade. Hobby: playing the saxophone.

Contact Me for More Information

Questions? Comments? What do you agree with, disagree with? Send me an email, I'll be happy to hear from you.

Calculus Without Tears, the Movie (beta)

A series of short videos covering the material in CWT Vol. 1 - Constant Velocity Motion.

The Mathematics of Motion
Graphs
How Fast Does it Go? ( Differentiation )
The Area Under a Graph ( Integration )
The Fundamental Theorem of Calculus ( Distance = Velocity*Time ! )
The First Differential Equation ( F = M*A )

Beyond CWT - FREEMAT, and the CWT Mantra

I have included a page on the web site on FREEMAT ..... every reader of CWT should download FREEMAT and start using it. This program is infinitely superior to a programmable calculator, and it's free.

The CWT mantra is that physics should motivate the study of mathematics in secondary school. Well, what physics? CWT has examples from mechanics and circuit theory, but, what about the rest of physics? The pillars of classical (not quantum) physics are Newton's mechanics, Maxwell's theory of electromagnetism, and Einstein's relativity. The mathematical language of these subjects is differential equations, and, here is a little surprise: beyond that the math required is modest, calculus as covered in CWT expanded to include partial differentiation, and vectors. The Wave Equation, Airplane Simulator, Planetary Motion, Maxwell's Equations, Relativity and GR (general relativity) pages have been added to the website to show you how it's done.

Calculus in the 4th 5th 6th Grade .... Really ???

There is a detailed synopsis of CWT Vol. 1 on the page linked to the left. Vol. 1 covers the basic operations of calculus applied to a very simple example, that of constant velocity motion. All there is to know about constant velocity motion is contained in the formula distance = velocity * time, and its alternate forms velocity = distance / time, and time = distance / velocity. In CWT Vol. 1 new terminology and concepts are presented, but all the calculations are based on these formulas. Thus, if a student can solve easy problems using these formulas, he/she is ready to start studying calculus. For example: A runner runs with a velocity of 3 yards/second for 5 seconds. How far does the runner travel?

CWT Vol. 1 presents the concepts and terminology of calculus in a simple context where the calculations are easy. Here is the important point, this wasn't a trick, the concepts and formulas in Vol. 1 really are the basis of calculus and generalize directly to more complicated examples, as is shown in Vols. 2 (7th, 8th, and 9th grades) and 3 (10th 11th and 12th grades). Calculus Without Tears is the only calculus book that takes this approach.

The Importance of Differential Equations

Differential equations are the connection between calculus and the real world, 'where the rubber meets the road'. Many of the laws of physics are written as differential equations; examples are on the physics pages linked to the left. CWT starts with easy differential equations in Vol. 1 . If there is no force acting on an object, F = 0 and Newton's equation become 0 = M*A. Since the mass of the object is not 0, it must be the case that A = 0. If an object's acceleration is always 0, what can we say about its velocity? If you guessed that the object's velocity is constant, you have just solved the differential equation.

In the standard curriculum, a student encounters differential equations sometime midway through college!

Here is another shocker: it's easy to solve any differential equation numerically using our favorite formula distance = velocity * time, and this is the method engineers use most often to solve differential equations (see for example the airplane simulator page linked to the left.) Thus, to understand the fundamental principles of physics it is necessary to know what differential equations are, but you don't need to know a lot of high powered math to solve them.

Funny but True - If Calculus Is So Easy, Why Has It Been Such a Mystery?

Calculus has always been taught 'theory first', that is, before a student studies calculus, he/she spends years studying abstract and difficult mathematics including geometry, algebra, and trigonometry. Then the study of calculus is encumbered with the notion of mathematical proof, and the student is required to mathematically prove the simplest facts about calculus before using calculus to solve problems.

We could take the same approach to teaching arithmetic. We could start with a series of courses on symbolic logic. Then, as our arithmetic textbook we could use Whitehead and Russell's Principia Mathematica, an important work that proves the basic properties of arithmetic. Never mind that it is two thousand pages long and comes in three volumes. The definition of number is on page 234, and the proof that '1+1=2' is on page 362 (see the proof at http://www.idt.mdh.se/~icc/1+1=2.htm ). Using this approach, multiplication would be taught midway through college! Fortunately, it's not the way arithmetic is taught. Unfortunately, it is the way calculus has been taught. CWT teaches calculus the way arithmetic is taught, by starting with the basic operations applied to easy examples. The result is that the student has a good intuitive grasp of calculus, something that often eludes students in college calculus classes.