Calculus in the Fourth Grade


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.

For example, Flannery 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 explains “Well, we do compute integrals. We also cover the fundamental theorem of calculus. We even solve differential equations, which I claim are the sine qua non of calculus.
How do we do it? We restrict ourselves to studying constant velocity motion. Everything there is to know about constant velocity motion is in the formula distance = velocity*time (d=v*t). I did make sure my own 4th grader was good with this formula. That’s all the prior math you need.

We only integrate the velocity function. The area under a constant function is given by the formula for the area of a rectangle, the base is the time interval, the height is the constant velocity, the integral is v*t. I use the integral symbol for fun, but the calculation is v*t.

How about the FTOC, which I can phrase as follows: the area under a velocity curve equals distance traveled. As above, the area under the curve is v*t, so the FTOC is v*t = d . We already knew it !
DE’s? Here’s a DE, in words: a runner’s velocity is 5. The runner’s starting position is 3. Solve the DE (p(t) = 3 + 5*t)
So, you have to watch out for the language, there are big words in the book, differentiate, integrate, differential equation …. but the concepts are simple and the calculations always are d=v*t, or v=d/t. Here’s the dictionary:
Differentiate – calculate velocity
Integrate – find the area under a curve
Differential equation – velocity equation
It might be a good idea to eliminate the big words altogether.
Here’s the thing – I claim this approach really presents the fundamental ideas of calculus the way they should be presented, regardless of the age of the student. This simple minded approach is the way I understood it when I used it as an engineer. The rest is elaboration!”


r f f