Kindergarten Calculus

This project was inspired by Jerome Bruner and Richard Feynman. It turns out to relate to Squeak Etoys on the OLPC XO by Alan Kay.

Note that this is not meant as the basis for a textbook for children to read and study. It describes visual activities where it is not necessary or even desirable to describe in words what you are doing. The point is for the children to see it and be able to do the same. For little children, and even sometimes for adults, explanations often get in the way of understanding or even remembering what they see.

Some of the ideas described here will work in kindergarten with no trouble. Others will require further testing and development, with the assistance of children of various ages. It may be that some should be delayed to a later grade. We shall see.

Differential Calculus

• To find the slope of a curve.

Place a straightedge tangent to any convex curved object. (Cylinders of various shapes are the best for this demonstration.) The straight line shows the direction of the curve. The steepness of the line is the slope of the curve. If it is angled, up on one side and down on the other, the curve is also rising on one side and falling on the other. The slope of the tangent tells us how fast the curve is rising at that point.

We can demonstrate how to measure the steepness of a curve by taking a right triangle with a base of one unit of length. As the hypotenuse tilts, so the length of the upright side the triangle changes. This length is called the slope of the line.

At some age, we can graph the steepness of the tangent at various points, and sketch in a curve that interpolates between them. This is an approximation to the derivative of the original curve.

• To find the maxima and minima of a curve.

At the top or bottom of the curve, the tangent is level. (Also at inflection points.)

Integral Calculus

• To find the area under the curve of a function.

Find the side of a square of paper of a convenient weight, such as one gram. Cut out several such squares for the children to use as units of area.

The problem statement means that the curve is continuous and single-valued. We don't bother the children with the details of this definition at this point, or even mention its existence. For our first demonstration, we use a curve that is everywhere positive. Draw the curve, draw vertical lines at its endpoints, and draw the X axis, making a closed figure. Cut out the figure and weigh it.

The integral of a segment of a curve is the sum of integrals of any pieces we may cut it into along vertical lines. Note and graph the weight of the integral. Then cut off strips of some simple fraction of the width of the unit square. Graph the weight of the remaining paper at the appropriate distance to the left of the previous point. Continue cutting, weighing, and graphing. Interpolate the graph. The resulting curve is an indefinite integral of the curve we started with.

Try integrating a straight line. What does your integral curve look like? Can you make such a curve using household objects? (Hint: flashlight).

• To demonstrate that differentiation (finding how fast a curve is rising at every point) is the inverse of integration (finding the area under any segment of a curve). (The Fundamental Theorem of Calculus)

Cut a very thin slice from one edge of an integral cutout. The difference between the integrals with and without that strip tells us approximately how fast the integral is increasing at that point. That rate is determined by the height of the curve, which is therefore equal to the slope of the indefinite integral at that point.

So the rate of change of the integral, which is the derivative of the integral, is the curve you started with. Similarly, taking a derivative and integrating gives you a curve of the same shape as the original, but everywhere a certain distance above or below it. Try it.

Thus we have covered the fundamental ideas of the calculus, with no symbols, limits, proofs, calculations, or other apparatus. We can leave all of that until later, and in the meantime apply these fundamental ideas to any subject where they are appropriate, such as falling objects, the shape of seashells, or population dynamics.

Future Grades

I don't know at what age the concept of negative length begins to make sense. It should be pretty early, using thermometers as examples. Negative area might be a little harder. There should be versions of the full mathematical concepts used in calculus that we can bring to kindergarten, even if we have to leave the beginnings of written math and calculation to first grade, and algebra for some years after that.