Calculus Without Tears
The Idea That Will Revolutionize Mathematics Education - Teach Computational Calculus First
Modern math and physics began when Newton discovered gravity and the laws of motion, and developed calculus to solve the equation Force = Mass*Acceleration, F=M*A, to determine the orbits of heavenly bodies
Physics, the study and analysis of the physical world around us, is the motivation for studying math. The most important physical laws are written as differential equations. Differential equations are the language of physics and engineering. And yet, differential equations are not part of the math curriculum. Why not? Because there is no comprehensive theory, and even innocuous looking differential equations, like F=M*A, can be very difficult or impossible to solve.
Without differential equations, the student never learns to formulate and solve basic problems in physics and engineering. As a result, the entire math program, including algebra, geometry, trigonometry, as well as calculus, which is taught without differential equations, is unmotivated.
There are two types of calculus, analytical calculus, and computational calculus. Only analytical calculus is difficult. Computational calculus is very easy. Really! And, I don't mean 'easy' like algebra is easy (algebra isn't easy) or like geometry is easy (geometry isn't easy). I mean easy like the equation distance equals velocity times time is easy - this equation IS computational calculus. Computational calculus uses this equation for everything, including computing solutions to differential equations.
Computational calculus was not useful before the advent of computers because it requires a lot of computations to get accurate results. Now, with the advent of computers, it is the dominant form of calculus used in the scientific world. Computational calculus is overlooked in calculus texts and is not taught, because it was not useful until fifty years ago, and, it's too easy! The educational system has not caught up with the math that is being used in science and engineering.
For more info on computational versus analytical calculus, click here
Calculus Without Tears begins with computational calculus. There are many advantages to this approach:
This new approach to teaching calculus will revolutionize the entire math and physics 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
From the review of Vol 1 on EclecticHomeSchool.org
From the review of Vol 2 on EclecticHomeSchool.org
From the discussion of CWT on HomeSchoolMath.BlogSpot.com
From the review of CWT on RainbowResource.com
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.
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.