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.

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