ABSTRACT
When I first discovered Istvan Banyai’s (1995) books, Zoom and Re-Zoom, I immediately thought, “great visual narratives for ratio and proportion!” We can see now that I had begun, like most of us, in the mode of reading the books as curriculum materials, not as currere. We read the books literally as texts to be plugged into the classroom, losing the jazz of curriculum in the dust. These books are very cool; they lead readers through delightful surprises as each turn of the page requires them to step back and see the previous page in a new context. What first looks like two children on a farm is now seen as a larger child playing with a toy farm of which those two previous children are a part. That child playing with a toy farm turns out to be nothing more than a picture on the cover of a catalog, held by a boy on a ship. Which is simply a picture of a ship on the outside of a bus, in a scene of a city, on a television, viewed by a cowboy, all in a little stamp, on a letter being mailed from somewhere in the South Pacific, seen from an airplane, and so on. Startling transformations of meaning occur with each new page. My students created their own Zoom books. We drew pictures, stepping back a predetermined distance each time. We changed the distances using different formulas, such as twice the distance each time, or three times the distance each time, or increasing the 246distance by the numbers in the Fibonacci sequence, and so on. We video-taped and photographed scenes, stepping back by varying distances. In this way, we developed visual notions of ratio and proportion, and represented them with drawings, videos and photographs, with algebraic notations, with number patterns, and with fractions. Re-Zoom offered newer ideas, because it is clearer in this second book that the perspective of each new page is not just one of stepping back, zooming out, but also turning: one could trace a trajectory path along a complicated hypothetical curve from start to finish. Here, too, we experimented: Can we draw the curve? How do we represent a curve in three dimensions on a piece of paper? Can we use string or pipe cleaners floating in space? (Wikki Stix® worked best.) If we start with pictures, what strategies help us to figure out the curved path in space? If we start with a path, how can we predict what the pictures will look like?
