If you know me, you know that I approach math with trepidation, if not outright fear and loathing. Thus, when I first became acquainted with Calculus Without Tears, Volume 1 I was, safe to say, extremely skeptical.
Yes, I took Calculus in college. (Three times.) No, it wasn't pretty.
I was very pleasantly surprised, working through Volume 1 of this series, to find that Calculus was not only doable, but it was understandable. Even my math-phobic fourth grader (at the time) could do the math, having mastered the multiplication table before tackling this introduction to Calculus, and it was quite a confidence-booster! ("You mean I'm doing college math? Cool!") Our only complaint was that Volume 1 was a little dull, with most of the problems involving a runner. Even though my husband is a runner, and some of the children are following in his footsteps, there's just so much you can do with track-and-field.
However, I'm glad to say that as Volume 2 of the series starts out, the runner is nowhere to be found. This might have something to do with the fact that we're now looking at the law of gravity and falling bodies. Frankly, I shudder at the implications, but no, we're not studying what happens to that runner who stumbles and falls. We're talking apples and grapefruit, and for a change of pace a little later on in the book, we have a race between the Greek hero Achilles and a hopeful tortoise. But wait (as we like to say around here), there's more!
Now, if you're math-phobic like I am, you don't want to turn to a random page in this book, to see such frightening phrases as "Can you write a differential equation for the position function of a falling apple?" and "area under a sum of functions" and "integral of a sum of functions is the sum of integrals" and other such incomprehensible stuff that makes my husband smile in fond recollection of college math.
No, you want to start with the first page (or maybe even work your way through Volume 1 first, for a kind-and-gentle introduction that draws you into the water without you noticing your feet getting wet). As a matter of fact, the author does recommend that the student be thoroughly familiar with the concepts in Volume 1 of the series before proceeding to Volume 2. I second the motion. Volume 2 moves rapidly into math jargon that might have frightened us away if we hadn't just finished Volume 1. But then, remember, both myself and my fifth-grader are math-phobic. I wish my husband would take over math in our home education, but he just doesn't have the time. He did, however, approve of the author's approach to "higher math." (By the way, I like the way this guy thinks. On the website order page, he even provides a Genuine Calculus Without Tears Crying Towel. Rather than putting me off, it made me laugh. Laughter is a rarity, a thing to be cherished, when dealing with math around here.)
The author uses the method of linear approximation to solve differential equations (note how the term flows easily from my lips - I've done the math, you see!) with a simple concept: distance = velocity * time. This involves multiplication, an elementary-age skill. No need to wait until you've gotten through trigonometry and advanced topics in math; you don't have to have an engineer's training, needed to solve differential equations exactly, to gain an understanding of the concept through approximation!
As you and your student work your way through Volume 2 you'll start out using Newton's Law of Gravity and his Second Law of Motion to determine a differential equation for a piece of falling fruit. Next you'll graph such equations, a method engineers use to solve real-life problems! (That was one trouble I had with higher math; it never seemed to mean anything! My concrete-thinking fifth grader and I are very thankful for the concrete examples in Calculus Without Tears!) Next we get into derivatives... uh-oh, it's starting to sound mathematical, or algebraic, or something problematical, but bear with me. The water's fine. Really, it is. Come on in!
Next, we use the Fundamental Theorem of Calculus for linear velocity function, with the reassuring reminder that we've already done this for constant velocity function in Volume 1. Yes, we got through Volume 1. We can do this!
Finally, we solve differential equations to determine the trajectories for falling objects on Earth, Mars and the moon, to analyze changing current and voltage in an electrical circuit (and this was fascinating, after we spent several months last year creating circuits and drawing circuit diagrams), and finally we calculate the orbital velocity of the moon, which fits right in with our study of astronomy this year.
Calculus Without Tears is designed so that a student can work through the lessons without a teacher, though in our case I worked the lessons a day ahead of my student, as insurance. I found the lessons, taken in order from the beginning, to be fairly straightforward, but I wouldn't want to have to jump in somewhere in the middle of the book, if my student ran into a problem and wanted my help. There is no answer key, but the student can self-correct using the checksum function built into the lessons. (That is, you add up the answers within a block of questions and if your sum matches the number in the checksum box, you got everything right. If not, you need to take another look at your work.)
Each lesson consists of introductory material followed by exercises, and is designed to be completed in 30 minutes or less. The author recommends doing a lesson daily, and I've found that helps us not to be overwhelmed by some of the equations and scary-looking symbols. (Scary to me and my fifth-grader, that is. My husband glances over the book and shrugs.)
An appendix gives instructions on how to use a freely available "clone" of MATLAB, the standard programming language used by engineers, as a Super Calculator. There is also a google.com discussion group for those who are using Calculus Without Tears, where you can get answers when your student stumps you (if you're not working through the book alongside your student, or even if you are), and the author has included his e-mail address in case you're really stuck, though I have not had to avail myself of this lifeline, having a husband who actually enjoys math and seeks out math challenges for fun and diversion. Imagine that.
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.
I won't call it "painless" (Math? Painless?), but Calculus Without Tears is, as my husband likes to say, "Eminently doable."