- Bruner: "Any subject can be taught effectively in some intellectually honest form to any child at any stage of development."
- Gattegno: "Living a life is changing time into experience."
- Minsky: "You do not understand something unless you understand it in more than one way."
- Feynman: "We do not understand a topic [in physics] unless we can prepare a freshman lecture on it." Others say that you don't understand something unless you can explain it to your grandmother, or a child.
Note that this is not meant as the basis for a conventional textbook for children to read and study. It describes visual activities where it is not necessary, and may not even be 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 also sometimes for adults, explanations can 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.
There are two essential questions in calculus: to determine the direction of a curve, and to determine the area inside a curve. Several other fundamental ideas follow from these two discoveries of Isaac Newton, including most of classical physics and several branches of mathematics. Later physics requires a few more fundamental ideas, such as complex numbers, linear algebra, non-Euclidean geometries, higher-dimensional geometries, and more. Each can be made visual and tactile, but those are topics for another day and for other pages.
For the first question, the direction of a curve, take any curved object and put a Cuisenaire rod or other straightedge up to it. (Convex cylinders of various cross-sections are the best for this demonstration.) The straight line of the rod is the tangent to the curve at the point where they touch, and shows the direction of the curve at that one point. Physics gives us this tangent visually, with no notation and no calculation.
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. Then we can demonstrate that the top and bottom of the curve (maximum and minimum of a function) are points where the tangent is level. These are the two fundamental ideas of differential calculus. The rest is detail, for anyone who has the fundamental understanding.
Later on, 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 function of the original curve.
Secondly, we want to demonstrate integrals. One bit of preparation is needed. Take a standard grade of paper, draw a square on it, cut it out and weigh it on a sensitive balance. Then calculate the size of a square that will weigh one gram. Cut out some one-gram squares. (Weigh at least one to check your work.)
Now, on the same grade of paper draw the standard x and y axes and any curve that is always positive. Draw vertical lines at the ends of the curve to make a closed figure. 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. Cut out the figure and weigh it on a sensitive balance. The weight in grams is the area in your standard squares. This gives an accurate value for the definite integral.
Now take that same cutout and fill it with Cuisenaire rods aligned vertically. In the same way, we can weigh the rods that approximate an area, and weigh the smallest rod, and get an approximate area for our figure. A few more steps with narrower and narrower strips (easier to do on a computer than physically) will bring us to the idea of the Riemann integral.
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, or the width of a Cuisenaire rod. 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. Now we can demonstrate constants of integration and the idea behind the Fundamental Theorem of Calculus, that the derivative of the integral is the original function, because the derivative at each point of the integral is approximately the area of the last strip. 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.
Try integrating a straight line. What does your integral curve look like? Can you make such a curve using household objects? (Hint: flashlight, candle shining through a circular hole...). Where else do you see this figure in the world? Hints: Look for it upside down. Water, balls...
Again, we have used no notation, done no calculations, and proved no theorems, and yet we have successfully demonstrated all of the core ideas of calculus and begun on Galilean mechanics (uniform gravity). Then we can consider introducing bits of deductive and analytic geometry, precalculus, including limits, trig functions, and exponentials, at whatever age works for children, with a similar progression from the visual and tactile to the formal and numeric.
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 other 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.
Now comes the question. I hypothesize that children growing up knowing these concepts since kindergarten, and using them throughout their education, will have far less difficulty with calculus calculations and proofs when the time comes, compared with those who come to it cold in high school, lacking this visual and physical intuition. It will take time to do the experiments properly, but the stakes are great enough to justify putting considerable resources into it. Starting, of course, with an implementation of all of these ideas and more in Smalltalk, Python/NumPy, or TurtleArt.