I teach mathematics so that a student understands the motivation for the topics, the definitions of the concepts, the reasoning behind the procedures, and how to correctly use all that knowledge to think about and solve the various problems that arise. I call this "old-school"; it is how I learned the subject decades ago.
When taught this way, mathematics is one coherent whole, rather than the grab-bag of unrelated tricks that is presented in the schools. It is artificial to distinguish between the various subjects, such as elementary math, pre-algebra, algebra 1, algebra 2, etc. These are all just parts of the continuum of understanding that forms mathematics. Many people think that math is difficult; the big secret is that math is actually the easiest subject, because it all makes sense. Once you understand the ideas behind it, you can reason it out for yourself.
I emphasize a solid grounding in the basics; I have helped many students in advanced subjects whose real difficulty turned out to be a poor understanding of what negative numbers and fractions actually are, and how to work with them. Clarity in the fundamentals pays great dividends later on.
I have a BA in pure mathematics from UC Berkeley. I work with Professor Emeritus Hung-Hsi Wu there on various aspects of the effort to fix the math curriculum in the United States, in which he has been a central player for over two decades. I edited his book "Understanding Numbers in Elementary Mathematics," which is the text used in the summer workshop for teachers at the Mathematical Sciences Research Institute (MSRI). I also work in various capacities at several math research journals.
The quote "Know how, and also know why" is from Liping Ma's "Knowing and Teaching Elementary Mathematics." I highly recommend this book to anyone who wants to understand why the teaching of mathematics in the US is in such a dismal state.
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