This isn't done yet, but I'll summarize real quick. I see math as a language, and for a teacher to talk to students the way most Calculus books are written (that is, in language you don't understand until you already know it!) just isn't productive. I see the role of a teacher as someone who guides you toward fluency, just like your parents did when you were learning to speak your native language. You have to learn the grammar, syntax, idioms, and just how to "talk" in math, and so, concepts have to be taught in English first. Then and only then can the math labels be put on them. More as I have time, but that's the main idea.
|How dull and prosaic the study of mathematics is made in our schoolbooks; as if the grand science of numbers has been discovered and perfected merely to further the purpose of trade. There is something divine in the science of numbers.... It holds the sea in the hollow of its hand. It measures the earth; it weighs the stars; it illumines the universe; it is law, it is order, it is beauty. And yet we imagine -- that is, most of us -- that its highest end and culminating point is bookkeeping by double entry. It is our way of teaching it which makes it so prosaic.|