Andrés Arrieta pointed me to the following post on Google+.

(The experience of creating circles of interest and sharing posts with them is quite different from making friends on Facebook or joining mailing lists, groups on LinkedIn and other services, or anything else I have seen. Andrés is in my APL & J circle, which gives me the option to see his comments that he shares with one of his circles that includes me. His circles and mine do not have to correlate in any way in order for us to pass ideas around between circles that do not overlap. I don’t even know which of Andrés’s circles he has put me in.)

[The illustration above demonstrates that

1+2+⋯+(*n*−1)=C(*n*, 2)

The left side is the number of yellow dots, each of which corresponds to exactly one selection of two dots from the bottom row of *n* dots. The question then is which children can work out that relationship just from the diagram, the equation, and the definition of the Combinations function C, with no other hints.]

I replied (reformatted with hyperlinks),

This is of interest to those creating lessons for the 2 million+ students in One Laptop Per Child/Sugar Labs programs in dozens of languages. The best-known such proof is of the Pythagorean theorem. There are two versions of this proof in animated GIFs [in Wikimedia]. We are using some others [in the Sugar Labs Turtle Art tutorials]. For example, it is easy to prove visually that the differences between successive square integers are odd integers, and similarly for many other relationships among Figurate Numbers. [in progress.]

The technique of wordless proofs is essentially the same as that of Roy Doty’s Wordless Workshop series.

See also *Proofs without Words: Exercises in Visual Thinking (Classroom Resource Materials)* (Two volumes), by Roger B. Nelsen

Here is a visual proof for odd numbers adding up to square numbers. For example, when we put 5s around the square of side 4, we need 4 for each side, plus one in the corner.

8 8 8 8 8 8 8 8 8 7 7 7 7 7 7 7 8 7 6 6 6 6 6 6 8 7 6 5 5 5 5 5 8 7 6 5 4 4 4 4 8 7 6 5 4 3 3 3 8 7 6 5 4 3 2 2 8 7 6 5 4 3 2 1

One of the great things about visual proofs is that we can program them in Turtle Art, or Logo, or Smalltalk, or J, or Dr. Geo, or a variety of other tools, and we can challenge children to do it, or to invent other visual proofs. We can do some visual proofs in just text, as just above here.

For young children, visual definitions often work better than more abstract definitions in words. For example, the tangent line to a curve defines the direction of the curve at that point. Clear? Or is this better?

And can you tell what the iconic Turtle Art blocks do from the pictures? (The blank block is Clean, which blanks any previous drawing and puts the Turtle at its default starting place in the middle of the screen.)

With this definition, we can immediately prove visually that the direction of the curve at a maximum or minimum is level.

We will have many more of these.