1. Using Visualization to Develop Children's Number Sense and Problem Solving Skills in Grades K-3 Mathematics (Part 1) LouAnn Lovin, Ph.D. Mathematics Education James Madison University

2. Number Sense What is number sense? Turn to a neighbor and share your thoughts. Lovin NESA Spring 2012 2

3. Number Sense “…good intuition about numbers and their relationships.” It develops gradually as a result of exploring numbers, visualizing them in a variety of contexts, and relating them in ways that are not limited by traditional algorithms” (Howden, 1989). “Two hallmarks of number sense are flexible strategy use and the ability to look at a computation problem and play with the numbers to solve with an efficient strategy” (Cameron, Hersch, Fosnot, 2004, p. 5). Flexibility in thinking about numbers and their relationships. Lovin NESA Spring 2012 3 Developing number sense through problem solving.

4. A picture is worth a thousand words…. Lovin NESA Spring 2012 4

5. Do you see what I see? Cat or mouse? A face or an Eskimo? An old man’s face or two lovers kissing? Not everyone sees what you may see. Lovin NESA Spring 2012 5

6. What do you see? Everyone does not necessarily hear/see/interpret experiences the way you do. www.couriermail.com.au/lifestyle/left-brain-v-right-brain-test/story- e6frer4f-1111114604318 Lovin NESA Spring 2012 6

7. Manipulatives…Hands-On… Concrete…Visual Lovin NESA Spring 2012 7

8. T: Is four-eighths greater than or less than four- fourths? J: (thinking to himself) Now that’s a silly question. Four-eighths has to be more because eight is more than four. (He looks at the student, L, next to him who has drawn the following picture.) Yup. That’s what I was thinking. Ball, D. L. (1992). Magical hopes: Manipulatives and the reform of mathematics education (Adobe PDF). American Educator, 16(2), 14-18, 46-47.Magical hopes: Manipulatives and the reform of mathematics education (Adobe PDF). Lovin NESA Spring 2012 8

9. But because he knows he was supposed to show his answer in terms of fraction bars, J lines up two fraction bars and is surprised by the result: Ball, D. L. (1992). Magical hopes: Manipulatives and the reform of mathematics education (Adobe PDF). American Educator, 16(2), 14-18, 46-47.Magical hopes: Manipulatives and the reform of mathematics education (Adobe PDF). J: (He wonders) Four fourths is more? T: Four fourths means the whole thing is shaded in. J: (Thinks) This is what I have in front of me. But it doesn’t quite make sense, because the pieces of one bar are much bigger than the pieces of the other one. So, what’s wrong with L’s drawing? Lovin NESA Spring 2012 9

10. T: Which is more – three thirds or five fifths? J: (Moves two fraction bars in front of him and sees that both have all the pieces shaded.) J: (Thinks) Five fifths is more, though, because there are more pieces. Ball, D. L. (1992). Magical hopes: Manipulatives and the reform of mathematics education (Adobe PDF). American Educator, 16(2), 14-18, 46-47.Magical hopes: Manipulatives and the reform of mathematics education (Adobe PDF). This student is struggling to figure out what he should pay attention to about the fraction models: is it the number of pieces that are shaded? The size of the pieces that are shaded? How much of the bar is shaded? The length of the bar itself? He’s not “seeing” what the teacher wants him to “see.” Lovin NESA Spring 2012 10

11. Base Ten Pieces and Number 10 20 30 40 4 3 2 1 Adult’s perspective: 31 Lovin NESA Spring 2012 11

12. What quantity does this “show”? Is it 4? Could it be 2/3? (set model for fractions) ? Lovin NESA Spring 2012 12

13. Manipulatives are Thinker Toys, Communicators Hands-on AND minds-on The math is not “in” the manipulative. The math is constructed in the learner’s head and imposed on the manipulative/model. What do I see? What do my students see?. Lovin NESA Spring 2012 13

14. The Doubting Teacher Do they “see” what I “see”? How do I know? Lovin NESA Spring 2012 14

15. Visualization strategies to make significant ideas explicit Color Coding Visual Cuing Highlighting (talking about, pointing out) significant ideas in students’ work. 48 + 36 70 +14 84 48 + 36 = ? Lovin NESA Spring 2012 15 Perimeter Area All Over ⅓

16. Number Sense “…good intuition about numbers and their relationships.” It develops gradually as a result of exploring numbers, visualizing them in a variety of contexts, and relating them in ways that are not limited by traditional algorithms” (Howden, 1989). “Two hallmarks of number sense are flexible strategy use and the ability to look at a computation problem and play with the numbers to solve with an efficient strategy” (Cameron, Hersch, Fosnot, 2004, p. 5). Flexibility in thinking about numbers and their relationships. Lovin NESA Spring 2012 16

17. Spatial Relationships Anchors or benchmarks of 5 and 10 Part-part-whole Relationships Children Can Develop With Numbers Addition and subtraction are embedded in these relationships. Lovin NESA Spring 2012 17

18. Five and Ten Frames Yes? No? Virtual five and ten frames: Five illuminations.nctm.org/ActivityDetail.aspx?ID=74 Ten illuminations.nctm.org/activitydetail.aspx?id=75illuminations.nctm.org/ActivityDetail.aspx?ID=74 illuminations.nctm.org/activitydetail.aspx?id=75 Lovin NESA Spring 2012 18

19. Five-Frame Tell-About Lovin NESA Spring 2012 19

20. Number Medley First, have all children show the same number on their ten-frame. Then call out random numbers between 0 and 10. After each number, the children change their ten-frames to show the new number. How do you decide how to change your ten-frame? Lovin NESA Spring 2012 20

21. How Many Dots? Lovin NESA Spring 2012 21

22. How Many Dots? Lovin NESA Spring 2012 22

23. How Many Dots? Do you see…. Spatial Relationships Anchors or benchmarks of 5 and 10 Part-part-whole Addition/subtraction Lovin NESA Spring 2012 23

24. My Kids Can: 2 nd grade video (15 mins) Notice the teacher’s comments and questions to the students’ responses. How is he using visualization to help his students track on significant ideas? What ideas is he trying to make explicit to the students? Notice the opportunities students have to share their reasoning. Lovin NESA Spring 2012 24

25. Counting What skills/concepts are involved in counting? Turn to a neighbor and share your thoughts. Lovin NESA Spring 2012 25

26. Stage 0: Emergent Counting - Cannot count visible items. Either does not know the number words or cannot coordinate the number words with items (one-to-one correspondence). Stage 1: Perceptual Counting- Can count perceived items but not those in screened collections. This may involve seeing, hearing, or feeling items. Stage 2: Figurative Counting- Can count the items in a screened collection but counting typically includes what adults might regard as redundant activity. For example, when presented with two screened collections, told how many in each collection and asked how many in all, the child will count from “one” instead of counting on. Stage 3: Initial Number Sequence- Uses counting-on rather than counting from “one” to solve addition tasks. Wright, R., Martland. J, Stafford, A., & Stanger, G. (2006). Teaching Number: Advancing Children’s Skills and Strategies. London: Sage. Stages of Early Arithmetic Learning (SEAL) Lovin NESA Spring 2012 26

27. Counting versus Non-Counting Correct and consistent (forward) number sequence One-to-one correspondence Cardinality (the last number said represents how many in all) Conservation (no matter how objects are arranged or counted, the cardinality of the set is constant) Lovin NESA Spring 2012 27

28. Moose Tracks! (Karma Wilson, 2006) Use one board. Play using one centimeter cube as your “marker” and two dice. Take turns rolling the dice and moving your marker. Think about: How does a game such as Moose Tracks help a child with counting? Counting Board has separate spaces for each “moose track”. (one- to-one correspondence) Counting dots on dice (one-to-one correspondence; cardinality; conservation) Lovin NESA Spring 2012 28

29. Play a second time using alternating colors of several pieces as your “marker”. Think about: How does this modification of the game help a child with counting? Lovin NESA Spring 2012 29

30. Ten Frames Dot Arrays Billboards – go by quickly (Fosnot & Dolk, 2001) Quick Images Lovin NESA Spring 2012 30

31. How many dogs do you see? How do you know? Dog Billboard Lovin NESA Spring 2012 31

32. Forces students to find other ways to determine how many other than counting. You can arrange the images in ways to capitalize on students’ innate ability to subitize (to determine an amount without counting – quantities of 5 or less) to help them move beyond counting by ones. There are multiple ways to perceive the images – flexible thinking – builds different relationships. Quick Images Lovin NESA Spring 2012 32

33. Your task is to analyze the Billboards to determine the mathematical ideas that the Billboards are intended to bring out. Dog Billboard Task Analysis Lovin NESA Spring 2012 33

34. Why frogs? Frog Billboards Lovin NESA Spring 2012 34

35. Both use small groups to encourage subitizing and counting on. Both encourage looking for part-part-whole relationships. Ten frame lends itself to building towards benchmarks of five and ten. When working on subitizing smaller amounts (four and less) billboards may be better to start with. Comparing Ten Frames & Billboards Lovin NESA Spring 2012 35

36. Numbers What do you think about when you think of… The number 7? The number 13? The number 18? Lovin NESA Spring 2012 36

37. Add Mentally As you add these numbers mentally, be aware of the strategy you use to determine the answer. 8 + 7 Lovin NESA Spring 2012 37

38. 8 + 7 = ? What mental adjustments did you make as you solved this problem? Double 8, subtract 1? (8 + 8 = 16; 16 - 1 = 15) Double 7, add 1? (7 + 7 = 14; 14 + 1 = 15) Make 10, add 5? (8 + 2 = 10; 10 + 5 = 15) Make 10 another way? (7 + 3 = 10; 10 + 5 = 15) Other strategies? Lovin NESA Spring 2012 38

39. Question… If we use these strategies as adults, how can we teach them explicitly to young children? Lovin NESA Spring 2012 39

40. This is a rekenrek (Arithmetic Rack) What do you notice? Lovin NESA Spring 2012 40

41. What is the Rekenrek? A tool that combines key features of other manipulative models like counters, the number line, and base-10 models. It is comprised of two strings of 10 beads each, strategically broken into groups of five. It entices students to think in groups of 5 and 10. The structure of the rekenrek offers visual images for young learners, encouraging them to “see” numbers within other numbers…building from groups of 5 and 10; subitizing. Lovin NESA Spring 2012 41

42. . With the rekenrek, young learners learn quickly to “see” the number 7 in two distinct parts: One group of 5, and 2 more. 5 and 2 more Lovin NESA Spring 2012 42

43. Similarly, 13 is seen as one group of 10 (5 red and 5 white), and three more. Or as one group of 10 (5 red on top and 5 red on bottom), and three more. A group of 10 3 more A group of 10 & 3 more Lovin NESA Spring 2012 43

44. Making a rekenrek Materials: A small foam board rectangle (cardboard) Pipe cleaners/String 20 beads (10 red, 10 white) (or two different colors) For younger children, one string with 10 beads may be sufficient. Lovin NESA Spring 2012 44

45. Making a rekenrek Step 1: Cut 4 small slits in the foam board. Lovin NESA Spring 2012 45

46. Making a rekenrek Step 2: Stringing the beads Each row is 10 beads: 5 red, 5 white (Make two rows) Two pipe cleaners: Bend back the end of the pipe cleaner and string the beads, 5 red and 5 white, onto each pipe cleaner. Lovin NESA Spring 2012 46

47. Making a rekenrek Step 3: Strings on foam board Slip the ends of the pipe cleaner through the slits on the foam board so that the beads are on the front of the foam board, and the pipe cleaner is bent onto the back side to secure the “string”. Lovin NESA Spring 2012 47

48. See and Slide An Introduction activity to the Rekenrek I will choose a number between 1 and 10. Think how you will move that number of beads in only one slide. Lovin NESA Spring 2012 48

49. See and Slide II An Introduction activity to the Rekenrek Let’s choose a number between 11 and 20 now. You are to use no more than two slides to show numbers larger than 10. Lovin NESA Spring 2012 49

50. Building a Number Building a Number An Introduction activity to the Rekenrek We are going to work together to build numbers using both rows. Let’s build the number 5. I will start. I will push 2 beads over on the top row. Now you do the same. How many beads do you need to push over on the bottom row to make 5? Sample sequence “Let’s make 8. I start with 5. How many more?” “Let’s make 9. I start with 6. How many more?” “Let’s make 6. I start with 3. How many more?” Lovin NESA Spring 2012 50

51. Children’s Books The Double Decker Bus The Sleepover Lovin NESA Spring 2012 51

52. Teaching episodes Quick Images ( How do you see 13? ) (2:21) Notice the questions the teacher asks. Lovin NESA Spring 2012 52

53. Teaching episodes Notice the questions the teacher asks; after watching the videos, what do we know about these students? Notice the questions the teacher asks; after watching the videos, what do we know about these students? Quick Images (13) Quick Images (13) Quick Images (13) Quick Images (13) Lovin NESA Spring 2012 53

54. Teaching episodes Small groups ( making combos of 10 ) (1:09, 1:50; 3:00) Notice the questions the teacher asks. After watching the videos, what do we know about these students? Lovin NESA Spring 2012 54

55. Combos of ten 1-1 counter Lovin NESA Spring 2012 55

56. Combos of ten Guess and Check Lovin NESA Spring 2012 56

57. Combos of ten Patterns, 1-1 Counter Lovin NESA Spring 2012 57

58. Teaching episodes Mini lesson of addition strategies (doubles, near doubles, fast ten, etc.) (2:18) Notice the questions the teacher asks. After watching the videos, what do we know about these students? Lovin NESA Spring 2012 58

59. Addition strategies Lovin NESA Spring 2012 59

60. The rekenrek The Rekenrek is a powerful tool that supports children to develop/reinforce cardinality (visualization of groupings), develop one-to-one counting (organizes the count), allows those who still need to count by ones to do so, but also helps children to build towards counting on, visualize and build number relationships, and work flexibly with numbers by encouraging decomposition strategies. Lovin NESA Spring 2012 60

61. Memorization vs automaticity Unlike drill and practice worksheets and flashcards, the rekenrek supports even the youngest learners with the visual models they need to develop number relationships and automaticity with number facts. Learning number facts is more than memorization! Lovin NESA Spring 2012 61

62. Lovin NESA Spring 2012 62 Take a minute and write down two things you are thinking about from this morning’s session. Share with a neighbor.

63. Take Aways Visualization can be helpful….BUT we can interpret the same visual differently - and differently from others. (Visual) Number Relationships Spatial Relationships Benchmarks of 5 and 10 Part-part-whole Quick Images to help children build mental images of number (subitizing) REKENREKS Rule!!!! Lovin NESA Spring 2012 63