On Jul 2, 2014, at 3:59 PM, Louis Talman <email@example.com> wrote:
> Do you mean that when folks talked to your ten-year-old self about your school-work they never said something like "How are you doing at math?" Such identification of arithmetic with mathematics is a deep undercurrent in our society. It has nothing to do with the way the subject is taught. > > On the contrary, the fact that arithmetic is taught as a collection of mysterious algorithms that have no intellectual basis but is identified as mathematics has a great deal to do with the "fear and loathing" of mathematics under discussion here. A misconception of what mathematicians do, popular in our society, is that we sit around adding up big columns of numbers.
First off, I don?t teach arithmetic as a collection of mysterious algorithms that have no intellectual basis, and, for that portion of elementary school that I can even count as school, I don?t remember the experience as being that way at all. My case is a little peculiar because I went to 6 different schools. 1st through 3rd was decent, but 4th through 7th was a black hole, mostly hidden in the back of my mind. 3 different schools, each worst than the last. Not only didn?t I want to go to school, neither did the staff, the teachers, or the principle. In many places, the 60?s and 70?s, if you do not remember, were not a good time to be in school. And in many places today, that is still the case. But that changed in 8th grade, a new school, and algebra. In any event, during the period of time I would call *normal*, I don?t recall arithmetic being mysterious or mindless. But given the crap I went through in 4th through 7th, I?m not doubting your recollection.
I teach arithmetic just as it is, a coherent and fun subject. We do exercises and drills, the pain, but we also solve word problems and puzzles, the gain. Most of the time, especially after the drills, it is gain. 80% of what we learn, we continue to use. This means that we don?t cover a topic on fractions for two weeks and then not see them again for 3 months. We tackle new kinds of numbers as we reach them and once we tackle them we will use them as if they were always there, and the students will feel like they were always there, because they were always there, they just weren?t aware of them yet. The axioms will not be answers to a test question, they will be felt, because that is what axioms are supposed to be. We won?t estimate and round numbers that we haven?t fully felt yet. We will work with repeating decimals, continued fractions, square roots, perfect numbers and many other magical things, like the Collatz conjecture. We will watch an adding machine clank through ! addition, multiplication and division. And near the end of our journey we will start doing problems that are not strictly arithmetic, like these...
No, I can?t teach 10 year olds what mathematicians do. How can anyone? But I guarantee I can teach them how beautiful arithmetic is and feels. Some of it will be pain, but most of it will not. In no place will I need to pretend to teach algebra. We can quibble about the level of formality in the reasoning but I am convinced that this is more of a time to just let it be than to focus on it. In some ways, algebra is the theory behind arithmetic, and in that light, I think a student who fully experienced arithmetic is in a much better position to appreciate algebra.