Reconceptualizing Mathematics Part 1: Reasoning About Numbers and Quantities Judith Sowder, Larry Sowder, Susan Nickerson CHAPTER 3 – UNDERSTANDING WHOLE NUMBER OPERATIONS 1 © 2010 by W. H. Freeman and Company. All rights reserved. 3-2 ACTIVITY 3-3 3-4 EXAMPLE 3-5 EXAMPLE What would the diagram look like? 3-6 ACTIVITY 3-7 3.1 3-8 A problem situation that calls for addition often describes one quantity being physically, or actively, put together with another quantity. The term joining is also often used. Example: 3-9 Another type of problem that involves addition involves conceptually (versus physically or literally) placing objects together. Example: 3-10 3-11 Example: 3-12 3-13 Example: 3-14 Example: 3-15 3-16 Example: 3-17 ACTIVITY Pair up with someone in the class and… a) classify each of the problem situations below (e.g., comparison subtraction): b) write problem situations that illustrate four of the different views presented for addition or subtraction. 3-18 3.2 Addition is a combining that is done actively or conceptually. Subtraction is classified in one of three ways: take-away, comparison, and missing addend. It is important that a teacher illustrate all the ways that addition and subtraction come into play when giving examples of their use in story problems. Elementary school teachers often only illustrate the “take away” case when introducing subtraction examples. Students should be asked to use addition and subtraction in both the discrete and continuous case of quantities. 3-19 Children who understand place value and have not yet learned a standard way to subtract often have very unique ways of undertaking subtraction. Prepare to be surprised!! 3-20 ACTIVITY Consider the work of nine second-graders as displayed on the following slide, all of whom were asked to solve 364 – 79 in written form without calculators or base-ten blocks. a) Which students clearly understand what they are doing, which ones might understand, and which do not understand? b) Describe in writing the steps the students followed to complete their work. 3-21 3-22 Is any one of these methods of subtraction better than the others? Why or why not? Which methods could be more easily understood? Are there any that should not be taught? Why or why not? 3-23 3.3 Children do not think about math the same way adults do. Adults have a wider range of experience with numbers. But consider that part of the reason is that we had limited opportunities to explore numbers and the meanings of operations. We became “set in our ways” so to speak. Young children are often inquisitive and this can be used to their advantage. Coming up with novel ways to solve problems often brings with it a deeper, more conceptually based understanding. The use of traditional algorithms can still be taught to children, but only after children have the opportunity to explore meanings for themselves. 3-24 3-25 DISCUSSION Researchers have found that success on the second problem is up to 40% less than the first, even among adults. Many solvers think they should divide or subtract on the second problem. Why would that happen? 3-26 Example: 3-27 3-28 3-29 Example: Let’s say that Jim needs to bake 2/3 of seven pounds of meat. We represent this as 2/3 7. Note that the multiplication sign represents the word “of” here in this fractional part example. Here the unit or whole is the seven pounds of meat. 3-30 3-31 3-32 Activity Suppose Jane has three different colors of turtle-necks, four different sweaters, and three different pairs of slacks. Assuming they are all color compatible, and that she will always wear one of each, how many different ways can she dress? How could this be represented with a diagram? 3-33 Activity 3-34 PROPERTIES OF MULTIPLICATION Commutative Property of Multiplication: mn = nm (for numbers m and n) Associative Property of Multiplication: (pq)r = p(qr) (for numbers p, q, r) Distributive Property of Multiplication: c (d + e) = cd + ce (for numbers c, d, e) 3-35 3.4 continued…. 3-36 3-37 Example: 3-38 Example: 3-39 3-40 3-41 ACTIVITY Hint: In which case will repeatedly subtracting the 2.5 yield the correct response? In which case must the miles instead be rationed out to totally separate “objects”? Carefully discuss this activity until the difference is clear. 3-42 DISCUSSION Write two word problems for each of the two divisions shown below. One problem should reflect the use of repeated subtraction, and the other should reflect sharing equally. a) 85 ÷ 7 b) 4.8 ÷ 2 3-43 Example: 3-44 DISCUSSION 3-45 3.5 3-46 3-47 DISCUSSION Discuss what you think each child was thinking about in terms of their approach to the problem based on their work given on the following slide. 3-48 3-49 DISCUSSION Below is the work of two seven-year-olds. Neither of these children knew the standard division algorithm, but still got by quite well without it. You are to study each method and tell what each child was thinking : 3-50 DISCUSSION Examine and discuss each of these student’s thinking process as well: 3-51 3.6 3-52 The idea of developing number sense is crucial. Too often, when focusing only on “getting the right answer,” children lose a lot of what it means to actually make sense of mathematics. 3-53 In a document from the National Research Council called Everyone Counts, it is stated that, “The major objective of elementary school mathematics should be to develop number sense.” 3-54 Unfortunately many teachers feel that disadvantaged children from inner-city and remedial schools would do better to have a rigid approach to mathematics that does not involve as much exploration. Research does not support this position. Research instead has shown that instruction that emphasizes meaning and understanding is highly beneficial to all students. Also, in the case of all students, number sense is something that develops gradually, over time. It is something that therefore needs to permeate all of mathematics teaching, not just to some cases of it. 3-55 3.7 3-56 3-57