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Using Visualization to Develop Children's Number Sense and Problem Solving Skills in Grades K-3 Mathematics (Part 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. Number Sense. What is number sense? Turn to a neighbor and share your thoughts. Number Sense.

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Using Visualization to Develop Children's Number Sense and Problem Solving Skills in Grades K-3 Mathematics (Part 1

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  1. Using Visualization to Develop Children's Number Sense andProblem Solving Skillsin 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

  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. Developing number sense through problem solving. Lovin NESA Spring 2012

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

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

  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

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

  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. Lovin NESA Spring 2012

  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: 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? Ball, D. L. (1992).  Magical hopes:  Manipulatives and the reform of mathematics education (Adobe PDF).American Educator, 16(2), 14-18, 46-47. Lovin NESA Spring 2012

  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. 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.” Ball, D. L. (1992).  Magical hopes:  Manipulatives and the reform of mathematics education (Adobe PDF). American Educator, 16(2), 14-18, 46-47. Lovin NESA Spring 2012

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

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

  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

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

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

  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

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

  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=75 Lovin NESA Spring 2012

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

  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

  21. How Many Dots? Lovin NESA Spring 2012

  22. How Many Dots? Lovin NESA Spring 2012

  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

  24. My Kids Can: 2nd grade video (15 mins) • Notice the teacher’s comments and questionsto 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

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

  26. Stages of Early Arithmetic Learning (SEAL) 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. Lovin NESA Spring 2012

  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

  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

  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

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

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

  32. Quick Images • 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. Lovin NESA Spring 2012

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

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

  35. Comparing Ten Frames & Billboards • 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. Lovin NESA Spring 2012

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

  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

  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

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

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

  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

  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

  43. A group of 10 3 more • 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 Lovin NESA Spring 2012

  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

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

  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

  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

  48. See and SlideAn 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

  49. See and Slide IIAn 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

  50. Building a NumberAn 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

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