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. My main agenda for my teachers has always been to help them see the importance of these relationships in the broader picture of number sense and computation and to provide them with effective activities to use with children.
From Counting to Number Sense
Number sense is defined in the text as a "good intuition about numbers and their relationships" (Howden, 1989, see T-60) 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 but not the last. Here we focus primarily on small numbers. Chapter 13 discusses flexible computation with whole numbers with Chapter 14 adding the skills of estimationn, Chapter 16 looks at fractions and Chapter 18, decimals. The counting activities described in the early sections of the chapter are there for completeness sake. 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 number - to develop number sense.
Developing a Collection of Number Relationships
I focus most of my efforts in this chapter is on activities designed to develop various relationships on 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. Since the traditional curriculum is so very weak in this regard, it seems most important to make a major point of these relationships with teachers. address separately, each of the four types of relationships outlined in the text: recognition of patterned sets, one and two more/less, anchors of five and ten, and the 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 flexible computation. It can be argued that development of early number relationships represents a key foundation on which much of number development rests. Many teachers of grades 3 to 6 have told me how they have used found it necessary and profitable to develop these same number relationships with their older students whose number foundations were lacking.
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 an 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.
Connections to 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 10 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 rich world of children's literature provides a wonderful connection of reality to number. Many educators place their main emphasis on a reality-based approach to number. To the extent that students do develop number relationships within these contexts, such an approach is excellent. In a teacher education program, these activities are time consuming and there is a danger of losing the forest for the trees. That is, the fun of the activity may overshadow the content of the number relationships you want their students to develop.
The Mental Math Connection
The section on early mental computation is included in this number concept chapter for two reasons. First, I want all teachers to realize that you begin the foundations for mental computation as early as grade 1 or even kindergarten. Second, I want to attract the attention of those teachers or pre-service teachers who may not be interested in the K-2 span of grades. They 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.