A correspondent asks:

I am curious if you “invented” the function diagrams, or if you know anything about the history of where they come from.

I did not invent them, but I did help to spread them (a little) into secondary education. See my ideas on how to use function diagrams in secondary school, including an overview, worksheets to download, and animations to discuss, on my Web site.

I learned about them many years ago at the Asilomar meeting of the California Math Council, from Professor Martin Flashman, who (among other things) teaches calculus at Humboldt State University. Looking for him on the Web this morning, I find that he calls function diagrams “transformation figures” or “mapping diagrams”.

If I remember correctly, Flashman had learned about function diagrams in a calculus book by Stein (I assume Sherman?). I never saw the calculus book, but I believe the function diagram there is described as a lens, which is the origin of the “magnification” term to refer to rate of change. Flashman took this representation a lot further, applying it to the study of many functions, and making it a central part of what he calls “sensible calculus”.

I got the term “focus” from Abraham Arcavi and other Israeli researchers, who used function diagrams (which they called PAR — parallel axes representation) in their work with math teachers. (Interestingly, as far as I know, not with students.)

When I showed Paul Goldenberg my Cabri function diagrams applets some years ago, he told me he had created a horizontal version, under the name dynagraph, long before I became interested in this. His idea was to give students a tactile experience, allowing them to see what happens to the y when you move the x — a dynamic view, as opposed to looking at the overall diagram. Of course, both approaches have a place. There is a dynagraph implementation in Geometer’s Sketchpad.

If anyone knows more about this history, please let me know!

–Henri

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