# ELEMENTARY & MIDDLE SCHOOL MATHEMATICS

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1 ELEMENTARY & MIDDLE SCHOOL MATHEMATICS Teaching Developmentally, 5/E 2003 John A.Van de Walle X Bookstore ISBN Visit to contact your local Allyn & Bacon/Longman representative. sam p l e c h a p t e r The pages of this Sample Chapter may have slight variations in final published form. Allyn & Bacon 75 Arlington St., Suite 300 Boston, MA

2 c h a p t e r 9 Developing Early Number Concepts and Number Sense Number is a complex and multifaceted concept. A rich understanding of number, a relational understanding, involves many different ideas, relationships, and skills. Children come to school with many ideas about number. These ideas should be built upon as we work with children and help them develop new relationships. It is sad to see the large number of students in grades 4, 5, and above who essentially know little more about number than how to count. It takes time and lots of experiences for children to develop a full understanding of number that will grow and enhance all of the further number-related concepts of the school years. This chapter looks at the development of number ideas for numbers up to about 20. These foundational ideas can all be extended to larger numbers, operations, basic facts, and computation. Big Ideas 1. Counting tells how many things are in a set. When counting a set of objects, the last word in the counting sequence names the quantity for that set. 2. Numbers are related to each other through a variety of number relationships. The number 7, for example, is more than 4, two less than 9, composed of 3 and 4 as well as 2 and 5, is three away from 10, and can be quickly recognized in several patterned arrangements of dots. These ideas further extend to an understanding of 17, 57, and Number concepts are intimately tied to the world around us. Application of number relationships to the real world marks the beginning of making sense of the world in a mathematical manner. Content Connections Early number development is related to other areas in the curriculum in two ways: content that interacts with and enhances the development of number and content that is directly affected by how well early number concepts have been developed. Measurement, data, and operation meanings fall in the first category. Basic facts, place value, and computation fall in the second. Operations (Chapter 10): As children solve story problems for any of the four operations, they count on, count back, make and count groups, and make comparisons. In the process, they form new relationships and methods of working with numbers. Measurement (Chapter 19): The determination of measures of length, height, size, or weight is an important use of number for the young child. Measurement involves meaningful counting and comparing (number relationships) and connects number to the world in which the child lives. Data (Chapter 21): Data, like measurement, involve counts and comparisons to both aid in developing number and connecting it to the real world. Basic Facts (Chapter 11): A strategy or way of thinking about nearly every one of the addition and subtraction facts can be directly traced to number relationships. Place Value and Computation (Chapters 12 and 13): Many of the ideas that contribute to computational fluency and flexibility with numbers are clear extensions of how numbers are related to ten and how numbers can be taken apart and recombined in different ways. 115

7 120 CHAPTER 9 Developing Early Number Concepts and Number Sense Anchors or benchmarks of 5 and 10: Since 10 plays such a large role in our numeration system and because two fives make up 10, it is very useful to develop relationships for the numbers 1 to 10 to the important anchors of 5 and 10. Part-part-whole relationships: To conceptualize a number as being made up of two or more parts is the most important relationship that can be developed about numbers. For example, 7 can be thought of as a set of 3 and a set of 4 or a set of 2 and a set of 5. Spatial Relationships Five (learned pattern) Six (3 and 3) Seven (6 and 1 more) One More / Two More / One Less / Two Less 1 MORE LESS 2 MORE 2 LESS The principal tool that children will use as they construct these relationships is the one number tool they possess: counting. Initially, then, you will notice a lot of counting, and you may wonder if you are making progress. Have patience! Counting will become less and less necessary as children construct these new relationships and begin to use the more powerful ideas. Spatial Relationships: Patterned Set Recognition Many children learn to recognize the dot arrangements on standard dice due to the many games they have played that use dice. Similar instant recognition can be developed for other patterns as well. The activities suggested here encourage reflective thinking about the patterns so that the relationships will be constructed. Quantities up to 10 can be known and named without the routine of counting. This can then aid in counting on (from a known patterned set) or learning combinations of numbers (seeing a pattern of two known smaller patterns). A good set of materials to use in pattern recognition activities is a set of dot plates. These can be made using small paper plates and the peel-off dots commonly available in office supply stores. A reasonable collection of patterns is shown in Figure Anchors to 5 and Five and three more Two away from ten 6 Part-Part-Whole 7 8 Six and three is nine FIGURE 9.4 Four relationships to be developed involving small numbers. FIGURE 9.5 A useful collection of dot patterns for dot plates.

12 Relationships Among Numbers 1 Through Wooden Cubes Toothpicks With the list on the board, overhead, or worksheet, children can take turns selecting the two numbers that make the whole. As with all problem-solving activities, children should be challenged to justify their answers. 4 and 2 3 and 3 4 and 2 or 3 and 3 5 and 1 1 and 2 and 3 or 3 and 3 Pattern Blocks 2 and 2 and 2 Squares 3 and 3 Missing-Part Activities Missing-part activities require some way for a part to be hidden or unknown. Usually this is done with two children working together or else in a teacher-directed manner with the class. Again, the focus of the activity remains on a single designated quantity as the whole. The next four activities illustrate variations of this important idea. 4 and 2 or 2 and 2 and 2 Make arrangements of wooden cubes. Make designs with pattern blocks. It is a good idea to use only one or two shapes at a time. Make designs with flat toothpicks. These can be dipped in white glue and placed on small squares of construction paper to create a permanent record. Make designs with touching squares or triangles. Cut a large supply of small squares or triangles out of construction paper. These can also be pasted down. It is both fun and useful to challenge children to see their designs in different ways, producing different number combinations. In Figure 9.9, decide how children look at the designs to get the combinations listed under each. The following activity is strictly symbolic. However, children should use counters if they feel they need to. ACTIVITY 9.19 Two out of Three Make lists of three numbers, two of which total the whole that children are focusing on. Here is an example list for the number 5: and 3 and 2 or 1 and 5 FIGURE 9.9 Designs for 6. 2 and 4 or 3 and 3 ACTIVITY 9.20 Covered Parts A set of counters equal to the target amount is counted out, and the rest are put aside. One child places the counters under a margarine tub or piece of tagboard. The child then pulls some out into view. (This amount could be none, all, or any amount in between.) For example, if 6 is the whole and 4 are showing, the other child says, Four and two is six. If there is hesitation or if the hidden part is unknown, the hidden part is immediately shown (see Figure 9.10). Four and two is six. Six minus four is two or Four and two is six. 6 I have 6? (You need 3 more.) I WISH I HAD 6. I have FIGURE 9.10 Missing-part activities. Counters under tub Flip the flap on a missing part card. (You need 1 more.)

14 Relationships Among Numbers 1 Through ACTIVITY 9.25 Dot-Card Trains Make a long row of dot cards from 0 up to 9, then go back again to 1, then up, and so on. Alternatively, begin with 0 or 1 and make a two-more/two-less train. ACTIVITY 9.26 Difference War Besides dealing out the cards to the two players as in regular War, prepare a pile of about 50 counters. On each play, the players turn over their cards as usual. The player with the greater number of dots wins as many counters from the pile as the difference between the two cards. The players keep their cards. The game is over when the counter pile runs out. The player with the most counters wins the game. FIGURE 9.11 Dot cards can be made using the Blackline Masters. Dot Card Activities Many good number development activities involve more than one of the relationships discussed so far. As children learn about ten-frames, patterned sets, and other relationships, the dot cards in the Blackline Masters provide a wealth of activities (see Figure 9.11). The cards contain dot patterns, patterns that require counting, combinations of two and three simple patterns, and ten-frames with standard as well as unusual placements of dots. When children use these cards for almost any activity that involves number concepts, the cards make them think about numbers in many different ways. The dot cards add another dimension to many of the activities already described and can be used effectively in the following activities. ACTIVITY 9.24 Double War The game of Double War (Kamii, 1985) is played like war, but on each play, both players turn up two cards instead of one. The winner is the one with the larger total number. Children playing the game can use many different number relationships to determine the winner without actually finding the total number of dots. ACTIVITY 9.27 Number Sandwiches Select a number between 5 and 12, and find combinations of two cards that total that number. With the two cards students make a sandwich with the dot side out. When they have found at least ten sandwiches, the next challenge is to name the number on the other slice of the sandwich. The sandwich is turned over to confirm. The same pairs can then be used again to name the hidden part. Assessment Notes The four types of number relationships (spatial representations, one and two more or less than, 5 and 10 anchors, and partwhole relationships) provide an excellent reference for assessing where your students are relative to number concepts. If you have station activities for these relationships, careful observations alone will tell a lot about students number concepts. For a more careful assessment, each relationship can be assessed separately in a one-on-one setting, taking only a few minutes. With a set of dot plates and a set of ten-frame cards you can quickly check which dot patterns children recognize without counting and whether they recognize quantities on tenframes. To check the one and two more or less than relationships, simply write a few numbers on a sheet of paper. Point to a number and have the child tell you the number that is two less than this number, varying the specific request with different numbers. It is not necessary to check every possibility. For part-whole relationships, use a missing-part assessment similar to Activity 9.20 ( Covered Parts ) on p Begin

15 128 CHAPTER 9 Developing Early Number Concepts and Number Sense Context Investigations in Number, Data, and Space Grade K, How Many in All? Investigation 2: Six Tiles The activities in this investigation take place over one to two weeks near the end of the year but could be done earlier. The entire investigation focuses on the number 6. The authors explain that 6 is an amount most kindergarten children can count, even early in the year. Children need more than one hand to show 6. It provides enough two-part combinations to be interesting, and it is an amount that can be visually recalled and manipulated. Furthermore, most kindergarten children will turn 6 years old during the year. Task Descriptions In Cover Up, the teacher draws an arrangement of six squares on a grid so that all can see. For example: or Students are told to look at the drawings and discuss different ways they could remember what they see. Then the picture is covered up and students try to describe the drawing. The discussion is focused on ideas that get at the parts of the picture, not just that there are six squares. Cover Up is repeated with several different drawings. Six Tiles in All is a follow-up activity in which students paste down six squares on grid paper in as many ways as they can find. Squares have to touch on a full side. The different arrangements are shared and the teacher helps students see number combinations for 6 as in the teacher From Investigations in Number, Data, and Space: Kindergarten by Marlene Kliman et al. Copyright 1998 by Dale Seymour Publications. Reprinted by permission of Pearson Education, Inc. notes shown here. This investigation provides just one opportunity to naturally introduce the symbolism for addition as children say the combinations they see. The authors point out that it is tempting to think that the activities in this unit could profitably be repeated for other numbers as well. Their experience, however, is that students get bored doing the same thing over and over. These ideas are important for other numbers, but students need different contexts and materials to maintain interest. with a number you believe the child has mastered, say, 5. Have the child count out that many counters into your open hand. Close your hand around the counters and confirm that she knows how many are hidden there. Then remove some and show them in the palm of your other hand. (See Figure 9.12.) Ask the child, How many are hidden? Repeat with different FIGURE 9.12 A missing-part number assessment. Eight in all. How many are hidden? amounts removed, although it is only necessary to check three or four missing parts for each number. If the child responds quickly and correctly and is clearly not counting in any way, call that a mastered number. If a number is mastered, repeat the entire process with the next higher number. Continue until the child begins to stumble. In early first grade you will find a range of mastered numbers from 4 to 7 or 8. By spring of the first grade, most children should have mastered numbers to 10. The Investigations in Number, Data, and Space curriculum for kindergarten integrates number development throughout most of the year. Four of seven units focus on the development of number, including the unit on data and measurement. The activity described in this excerpt provides some flavor of the Investigations program s early approach to number.

20 Reflections on Chapter and 5 more: 2 from 5 to get to 10 and 3 more is 13. So and 5: 2 more to get to 70 and 3 is 73. One more than 6 is 7. One more ten is 7 tens. Ten more than 60 is 70. (a) (b) 6 and 2 is 8 so and 20 is 80. (c) 45 and 35 is 80. FIGURE 9.15 Extending early number relationships to mental computation activities. Reflections on Chapter 9 Writing to Learn 1. Describe an activity that deals with a basic concept of number that does not require an understanding of counting. Explain the purpose of this activity. 2. What things must a child be able to do in order to count a set accurately? 3. When does a child have the cardinality principle? How can you tell if this principle has been acquired? 4. Describe an activity that is a set-to-numeral match activity. What ideas must a child have to do these activities meaningfully and correctly? 5. How can Real Counting On (Activity 9.8, p. 119) be used as an assessment to determine if children understand counting on or are still in a transitional stage? 6. What are the four types of relationships that have been described for small numbers? Explain briefly what each of these means, and suggest at least one activity for each. 7. Describe a missing-part activity. What should happen if the child trying to give the missing part does not know it? 8. How can a teacher assess each of the four number relationships? Give special attention to part-whole ideas. 9. How can a calculator be used to develop early counting ideas connected with number? How can a calculator be used to help

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