CHAPTER 4 – SOME CONVENTIONAL WAYS OF COMPUTING Reconceptualizing Mathematics Part 1: Reasoning About Numbers and Quantities Judith Sowder, Larry Sowder, Susan Nickerson © 2010 by W. H. Freeman and Company. All rights reserved. 1

An algorithm is a sensible step-by-step procedure for carrying out some operation. 4-2

EXAMPLE Some children understand algorithms better when they are represented with base-ten materials or drawings: 4-3 continued….

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ACTIVITY Act out the following calculations using base-ten blocks or drawings, whichever is appropriate. As in the previous example, record the steps taken numerically as well. The numeric work should always be linked to the blocks or drawings. 4-5

Multiplication and division algorithms are often more difficult for children to understand. However, if children have a more conceptual understanding of place value, they can think of multiplication in terms of partial products. Example: 4-6

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ACTIVITY Below are examples of a series of division algorithms for 472 ÷ 37 that lead to the standard algorithm that you probably use. Can you see what has been done in each case below? 4-8

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ACTIVITY Try dividing.18 by 1.5 using each of the techniques demonstrated for 472 ÷ 37. Use knowledge gained in the previous activity to account for “moving the decimal point.” 4-10

Discussion 4-12

The standard algorithms are called “standard” because children are taught to do them the same way. The algorithms came into existence as people attempted to become as efficient and speedy as possible in their calculations. Some of the algorithms we use are not standard in all parts of the world. Discussion Could the use of algorithms actually inhibit true conceptualization? If so, describe how that could be possible. 4-13

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Calculators are ubiquitous, and are much faster than any human calculator. But efficiency and automaticity should be reconsidered as valid reasons for teaching arithmetic skills. Paper-and-pencil calculations are clearly a vital part of learning for good reason. But perhaps, now, with the calculators in hand, the emphasis on doing paper and pencil calculations quickly is not needed. This could lead to a further emphasis, at the pencil-and-paper stage, of why the procedures in the standard algorithm lead to correct answers. 4-15

DISCUSSION 4-16

4-17 Keep in mind that students often don’t understand enough about arithmetic operations to be able to choose the suitable operation to be applied to a real world problem; in such cases a calculator is useless.

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