A while ago I discovered an interesting web site, Berkeley Science Books, that publishes a set of very comprehensive Ebooks called "Calculus Without Tears." Author Will Flannery has a pretty detailed explanation on the home page of his web-site of why he thinks Calculus can be taught in elementary school. His view is that Calculus in college is bogged down with lots of theory; if you change the focus of Calculus to application first and theory later, and if you teach the fundamentals of Calculus that don't require algebra, trigonometry, or geometry (except for the formula for the area of a rectangle) then you can teach Calculus to 4th graders. Flannery sees the motivation for all of mathematics, beyond basic arithmetic, to be physics, and the building basics - derivatives, integrals, and differential equations, which are fundamental to physics and to Calculus - can be taught to those with no mathematical sophistication.
Flannery questions the wisdom of the Math and science curriculum teaching algebra, geometry, and trigonometry before teaching the physics that drives the need for these other branches of mathematics. To be honest, part of me agrees with Flannery and part of me doesn't. I've always enjoyed pure and recreational Math. I absolutely love Math for the sake of doing Math. I love the logic, the creativity, the problem solving, the beauty, the joy, and the elegance of mathematics. But, I get that I'm not typical. Many people find Math to be too abstract and don't see the value of manipulating abstractions. For those people I can see the value of learning Math in a very concrete fashion. I can see the value in approaching Math from the desire to understand how our physical world works, starting with basic formulas for force and distance, and proceeding from there. I believe that someone with an engineering mindset or teachers who want to approach Math from the very concrete will really appreciate Flannery's books. I'm not an educator so I can't speak to what works best in the classroom. I would suspect that a combination of concrete and abstract might work best but I'm not sure in what combination or sequence.
I now provide a review of "Calculus Without Tears." My review is more about Flannery's approach to the subject matter, and not so much about the content, which you can read about on his web-site. You should know that I received review copies of the books from the author. Beyond that I received no payment of any kind and no conditions were placed on my review.
"Calculus Without Tears" comes in four volumes. I've seen the first three:
- Volume 1: Constant Velocity Motion, is 154 pages long and contains 13 lessons
- Volume 2: Newton's Apple, is 166 pages long and contains 14 lessons
- Volume 3: Nature's Favorite Functions, is 198 pages long and contains 18 lessons
Cost and ordering information is in the buy page.
The books are all in a workbook form. There are lessons with explanations of concepts and tons of problems per lesson. I really like this approach to learning. The exercises are so well structured that they guide the learner through the material one step at a time and thus build confidence. Thus, they are perfect for homeschool students or those who enjoy a self-paced approach to learning. I have to say, though, that while I believe that a motivated 4th grader could learn the material from these books, I'm not sure that a student with a 4th grade reading level would understand all of the explanatory material. Much of the material is easy to understand but I think some of it would require help from an adult teacher or tutor. An adult learner would sail through the explanatory material and benefit as much as a child from the numerous examples to work out.
True to Flannery's claim, very little mathematical background is required to learn the material. Volume 1, for example starts with calculating the position of a runner (running at constant velocity) against time. The student is given many opportunities to compute values of the distance function. First, values are entered into a table, later they are graphed. Derivatives are introduced as are integrals, all in a very natural progression from calculating values of simple linear functions, to learning how to plot these points, to learning about properties of lines, including their slopes and the areas underneath them.
Flannery employs a very clever technique to letting students know if they got the answers to the problems right. Many sets of problems have a "checksum", which is just a number that represents the total of the numbers of the answers of some or all of the problems in the set. So, a student computes the answer to the problems in the set, adds up those numbers, and compares it to the checksum. If his total matches the checksum then there's an excellent chance that he got them all right. If not, then one or more answers are incorrect and the student can check his work and change one or more answers until his total matches the checksum. While I like this approach I can also see some students becoming frustrated as there is no answer key and no solutions to the problems.
I appreciate Flannery's approach as an engineer to teaching this material. He has very clearly thought through all of the concepts he needs to convey to get from A to B and he has created numerous progressive exercises to cover that path. I'm quite impressed.
As a comparison of Flannery's approach to teaching Calculus to traditional methods, I pulled out my copy of Thomas/Finney Calculus and Analytic Geometry, which I used as a text during my senior year in high school when I took BC Calculus. Chapter 1, The Rate of Change of a Function, must have been meant as a review. It very quickly covered directions and quadrants, increments (deltas), slopes of lines, angles of inclinations, equations of parallel and perpendicular lines, point-slope equations, and general linear equations. And, that's just in the first 12 pages. Later in Chapter 1 is a discussion of limits, including limits involving infinity and those involving trigonometric functions. In my judgment, the Thomas/Finney approach requires a fair amount of sophistication and, as Flannery states, a solid understanding of algebra and trigonometry. Plus, Thomas/Finney is not relating their content to the physical world, i.e. their approach does not make the connection between Calculus and its motivation the way "Calculus Without Tears" does.
So, would I recommend the books? Yes. I really like the idea of demystifying Calculus, of having it be a natural part of early mathematics education. I don't like that Calculus is seen as something that requires enough sophistication that it can only be learned in an honors Calculus class in one's senior year in high school or in college. However, I don't see Calculus as fitting into a box where you either learn it early on or you learn it later. There's a place for college-level Calculus, and there's a place for getting grounded in the relationship between Calculus and the basic physics that motivates it. So, I would recommend that students get a hold of at least Flannery's first book and work through it to get grounded in the basics. If they resonate with Flannery's teaching approach then they can purchase additional volumes. Once students are grounded in the basics ala Flannery, when Calculus comes up again later in their academic career they'll have a perspective on what's motivating the material that will serve them regardless of how it's taught in high school or college.