I had a small epiphany in my kitchen this afternoon, cutting copy paper into ever-smaller strips. I didn’t like the EveryDayMath “jumping” thing and I don’t want to just hand over the algorithm.
What does it mean when we say, “what is one third of one fourth?” What does that look like and how can we represent it? If I cut a strip off the bottom of a piece of copy paper, and call it My Whole, then cut another just as long, but cut it further into three pieces, these are thirds. Cutting another Whole into thirds, and then each of those pieces into four pieces, I end up with twelve pieces, and each of those is one Fourth of one Third. Playing and cutting, I can figure out the a/b of any other a/b, as long as I keep the b’s small so I don’t end up with 81 super-thin slivers of paper or whatever.
I want my students to know the “why” and “how”, not just a mechanical reflex. I want them to see these parts in their heads, fitting together and moving about.