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MAIN IDEAS Most traditional textbook curricula move almost immediately from the skills of counting and set-to numeral matching to addition. The result is that many children count their way completely through the first and second grade. Addition and subtraction facts, mental mathematics, and in general a real sense of numbers should be built on a set of rich relationships for small numbers. It is important to see these relationships in the broader picture of number and computation and to learn effective activities to use with children. Number Sense Number sense is defined in the text as a "good intuition about numbers and their relationships" (Howden, 1989) and flexible, intuitive thinking with numbers: An intuition about numbers and their relationships. This is the first time that the term number sense appears in the book. Here we focus primarily on small numbers. Chapters 13 and 14 discuss number sense with larger whole numbers, Chapter 15 looks at fractions and Chapter 17, decimals. Most K-1 books do a fairly good job of teaching students to count and write numbers. What is more important is to provide students with the opportunities to use these counting abilities to learn more about numbers – to develop number sense. Developing a Collection of Number Relationships Most of attention in this chapter is on activities designed to develop various relationships with numbers. The emphasis is placed on numbers up to about 10 or 12. Experience with children indicates strongly that these relationships can be developed but significant amounts of time are required. Each of the four types of relationships outlined in the text are addressed separately: recognition of spatial relationships in patterned sets, one and two more/less, anchors of five and ten, and most important of all, part-part-whole relationships. There are at least three subsequent chapters where these ideas will be seen again: meanings of the operations, basic facts, and mental computation. It can be argued that development of early number relationships represents a key foundation on which much of number development rests. Extensions to Numbers up to 20 It is not reasonable to simply do the same type of activities for numbers between 12 and 20 that were done for numbers 10 and less. There are approaches that can either extend some of the relationships on smaller numbers to larger numbers or develop new ones. The one-more-than relationship is an obvious example. Students can be helped to connect the relationships between say 6 and 7 to 16 and 17. The anchors of 5 and 10 have obvious extensions to 15 and 20 and subsequently to all numbers ending in 5 and 0. Students can learn that the teens are sets of ten and some more (a special part-part-whole idea) well before they understand the comparatively sophisticated ideas of base-ten place value. The double and near-double relationships are new ideas but important ways to think about larger numbers. Number Sense and the World Reflection on the last two paragraphs will show that number has been discussed in a relatively sterile manner. Number sense also includes an attachment of reality to number. Ten pounds is very different than ten students or ten lima beans. Numbers must be related to measures and quantities. Graphing, measuring, estimating, drawing, building, and in general interacting with the world in terms of number is perhaps as important as the relationships already discussed. The Mental Math Connection The section on early mental computation is included in this number concept chapter for two reasons: to help you realize that you begin the foundations for mental computation as early as grade 1 or even kindergarten; and to attract the attention of those teachers or pre-service teachers who may not be interested in the K-2 span of grades, but who also need to value what goes on in the early grades and understand the foundations of the things they should be working on in grades 3 and above. There are two ideas most clearly connected to the early number relationships. One- and two-more-than (-less-than) is related to one more (or less) ten or the adding and subtracting of tens. The ideas of part-part-whole and 5 and 10 anchors combine as illustrated in the following sequence: Parts of 8 leads to parts of 80 (with tens) leads to parts of 80 using two 5’s (as in 35 and 45) leads to any two parts of 80 and to missing parts of 80 (37 and what makes 80) leads to 80 – 37 in a mental computation.
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