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I agree with you, Mr. Thatcher. Students must master basic algebra skills and procedures to access higher-level mathematics. They must master basic arithmetic skills and procedures to access algebra. My questions are these: 1. What do students need in order to master arithmetic skills and procedures? and 2. Who should have the option to access higher-level mathematics? If the answer to the first question is "the ability to memorize lists of facts and procedures" then the answer to the second question will be "only those who have the ability to memorize lists of facts and procedures." For a long time, it was acceptable for the small cross-section of the population that had this ability to memorize to go on to pursue higher-level mathematics. The rest of us? We each said, “Oh, me? No, I’m no good at math.” And everyone was ok with that. Our disagreement lies in the answers to these questions, Mr. Thatcher.I believe that the answer to the first question is this: what students need in order to access basic arithmetic skills is a solid understanding of the mathematical principles that support those skills. Indeed, we are all born with basic mathematical intuition that supports the construction of more complex mathematical understandings, given the opportunities to do so. Memorization--or more accurately, automaticity--is ideally coupled with context and comprehension, so that students have repertoires of efficient strategies to choose from. Certainly the old “borrow and carry” algorithm is not the most efficient for a problem such as 2000-178. Whyever do such a thing, when one could simply change 2000 to 1999 and 178 to 177, and THEN subtract? The difference is the same, and no borrowing was necessary. The trick is finding teachers--generally memorizers themselves, during their own school years--who know and understand these strategies well enough to teach them and support students in constructing the mathematics behind them. “Doing math” does not equate to “spewing math.”My answer to the second question is that ALL students should have the option to access higher-level mathematics, which is why teachers continue to work so diligently to figure out how to help ALL students--not only those who are natural memorizers. As long as students believe that good mathematicians are those who are able to memorize the myriad seemingly unrelated facts and procedures downloaded by lecturing teachers, then the others of us, those who just can’t seem to get all those darn disconnected and crazy rules and procedures straight, will continue to say “Oh me? No, I’m no good at math.” And everyone will be ok with that.
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