Teacher2Teacher Q&A #1733

Teachers' Lounge Discussion: Long division

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From: Pat Ballew

To: Teacher2Teacher Public Discussion
Date: 2000111612:13:36
Subject: A suggestion on helping students understand

There are lots of algorithms by which one can obtain the answer to a division problem, but to "understand" any of them begins with understanding what division is all about, and that isn't necessarily an easy task. We use division in several ways in applied and abstract mathematics, but at the middle grades I would think it could be translated generally into "how many sets of this much can be formed from that much?". The word quotient for the answer to a division problem is literally the Latin for "How many?". From that approach, if students had difficulty with division, I would begin by asking them to describe situations that might entail 85/23 or other division problems. If the student quickly sees this as "How many classes of 23 can I make from a student body of 85?" or some equally clear explanation, then we can begin to ask them how they would find that answer. At this point I would let them come up with algorithms that met their understanding of "division". If they wanted to use repeated subtraction until less than 23 remained, I would encourage them to do so, and describe the result in the context of the problem, i.e." There will be three classes of 23 and a group of 16 to be otherwise assigned (what you do with the remainder is a social decision, not necessarily a mathematical one) If they have a good understanding of the division process, and you want to work them toward the traditional long division algorithm, at this point you can point out that if we had much larger numbers, it would be necessary to incorporate some bookkeeping into our process to help keep track of what we are doing. If you are just beginning as a teacher, you might keep in mind that many of the algorithms for computation were invented to do arithmetic quickly and in a small space (like a student slate, or a sheet of paper which was expensive in the 16th and 17th century). They most certainly were NOT created to teach the ideas behind the mathematical operations. If understanding is important to you, you may want to seek other algorithms for these, or guide students to create efficient ones of their own. After a while working with their own, if you show them the traditional space conserving algorithms, you can ask the students to explain how it works (a very good question for student teachers to ask each other, "Why does this process produce the right answer?" If this is way off the direction you wanted to go, I apologize, but this is how I would approach the problem you have. In any event, whether you use this approach or another, good luck. -Pat Ballew, for the Teacher2Teacher service

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