Math Content Part 1

1. Math Content Part 1 Ernestine Saville-BrockJanuary 2013

2. Urgency in the teaching of mathematics • “The United States suffers from innumeracy in its general population, ‘math avoidance’ among high school students, and 50% failure among college calculus students (Reuben Hersh ) • Too many children choose their college major and their career paths based upon how many math courses they need to take. (Boaler, 2008)

3. Urgency in the classroom • Teachers need to see themselves as mathematicians. If we foster environments in which teachers can begin to see mathematics as mathematizing-as constructing mathematical meaning in their lived world-they will be better able to facilitate the journey for the young mathematicians with whom they work.”(Fosnot)

4. Good afternoon, Coaches Good afternoon, Instructional Coaches. We are going to explore developing early number concepts, number sense, and an introduction to early addition and subtraction. Let take a minute and greet at least three others in the room. Activity- Quick Peek Activity

5. Agenda for the session • Afternoon Meeting • Pattern visualization • Quick Peek Game • Number Sense and Counting • Quickwrite • Number Relationships • Counting Stories • Models Matter • Ten Frames • Crazy Mixed Up Numbers • Math Racks • Early addition and subtraction

6. How many dots are there?

7. How many dots are there?

8. Subitizing • Subitizing • The ability to “just see it” • Naming amounts based on patterns not counting • Children must explore quantity before they can count • A fundamental skill in the development of students understanding of number- Baroody, 1987 • A complex skill that needs to be developed and practiced through experiences with patterned sets • Aids students in developing more sophisticated and efficient strategies for counting and learning number combinations.

9. Brain research effecting teaching and learning (Sousa) • Creating and using conceptual subitizing patterns help young students develop the abstract number and arithmetic strategies they will need to master counting. • Information is most likely to be stored if it makes sense and has meaning

10. Brain Research cont’d • Too often, mathematics instruction focuses on skills, knowledge and performance but spends little time on reasoning and deep understanding • Just as phonemic awareness is a prerequisite to learning phonics and becoming a successful reader, developing number sense is a prerequisite for succeeding in mathematics

11. The Power of Groups- Chunk the Dots • Speedy Pictures www.fi.uu.nl/rekenweb/en

12. Number Sense • To achieve in mathematics, students must acquire a good sense of numbers early in their academic career. – Bradley S. Witzel • Quickwrite- What is number sense? • Share ideas • Of the 22 kindergarten CCSS, 14 can be directly linked to elements of number sense.

13. Discussion of Number Sense • Howden, 1989 • Good intuition about numbers and their relationships • Develops gradually as a result of exploring numbers, visualizing them in a variety of contexts, and relating them in ways that re not limited by traditional algorithms • Gersten and Chard, 1999 • A child’s fluidity and flexibility with numbers, the sense of what numbers mean and an ability to perform mental mathematics and to look at the world and make comparsions

14. Number Relationships • Spatial visualization • One and two more; one or two less • Benchmarks of 5 and 10 • Part-part-whole relationship

15. Spatial Relationships – recognizing how many without counting by seeing the visual pattern. • One & Two More, One & Two Less – this is not the ability to count on two or count back two, but instead knowing which numbers are one more or two less than any given number. • Benchmarks of 5 and 10 – since 10 plays such an important role in our number system (and two 5s make up 10), students must know how numbers relate to 5 and 10. • Part-Part-Whole – to conceptualize a number as being made up of two or more parts is the most important relationship to develop. Van De Walle, 2006

16. How do children develop skill with counting and number concepts? • Key Ideas • Counting is the basis of children’s ability to add and subtract • Developing strategies for keeping track of the objects being counted is an important landmark in student learning • The structure of a ten frame can help students develop an understanding of number concepts

17. Counting and Number Sense • Counting is a skill that is easy to take for granted, but it is the foundation for understanding number and learning addition and subtraction. • Two areas of development • Fluency with verbal counting • Ability to accurately and reliably count collections of objects • Skill with the first doesn’t guarantee skill with the second

18. Activity- Counting Stories • Read the student vignettes • Discuss the vignettes with a partner • Sort the vignettes into two piles: • Students with counting fluency • Students who have difficulty counting • Consider the following questions: • Does the student have fluency with verbal counting? What is the evidence? • Does the student have the ability to accurately count each collection of objects? What is the evidence?

19. Counting • Brainstorm a list of tasks or activities that might help students who have not yet mastered counting tasks? • Students develop their understanding of counting and number primarily through practice. Teachers and researchers have found that providing a variety of meaningful counting experiences is the key to helping children develop their counting abilities.

20. Developing Number Concepts Counting builds to a mental visualization of a number line • K K-2 use number paths.

21. Models Matter • To build strong number sense is to introduce models with structures that can develop an understanding of number relationships. And to emphasize important numerical understandings such as organizing with ten. One model is a ten frame. • Ten frame may help students to see numbers in relation to 5 and 10, which is a helpful strategy for gaining fluency with addition and subtraction.

23. Ten Frames • Van de Walle (2009) suggests that teachers should provide students with practice using ten frames rather than teaching procedures.

25. Crazy, Mixed-up Numbers • All students make their ten frames show the same number. • Teacher calls out random numbers between 0-10 • After each number, children change their ten frames to show the new number. ANIMALS ON BOARD by Stuart Murphy

26. Add and subtract within 20, demonstrating fluency for addition and subtraction within 10. Use strategies such as counting on; making ten (e.g., 8 + 6 = 8 + 2 + 4 = 10 + 4 = 14); decomposing a number leading to a ten (e.g., 13 – 4 = 13 – 3 – 1 = 10 – 1 = 9); using the relationship between addition and subtraction (e.g., knowing that 8 + 4 = 12, one knows 12 – 8 = 4); and creating equivalent but easier or known sums (e.g., adding 6 + 7 by creating the known equivalent 6 + 6 + 1 = 12 + 1 = 13). - CCSS

27. Strategies NOT Memorization - Counting on - Making ten - Decomposing a number leading to a ten - Using the relationship between addition and subtraction - Creating equivalent but easier or known sums

28. Counting vs. Collections "Tasks that encourage students to think in collections provide rich opportunities for them to construct abstract mathematical relationships and become powerful problem solvers." --Wheatley & Reynolds, 2010

29. The math rack • is a tool developed at the Freudenthal Institute in the Netherlands by Adrian Treffers to support the natural mathematical development of children • in Dutch means “calculating frame” or “arithmetic rack.” • looks like a counting frame but is designed to move children away from counting each bead. • looks like an abacus but it is not based on place value.

30. Features of the math rack • The beads are red and white. • There are two rows of beads. • There are five red beads and five white beads on the top row, and the same on the bottom. • There are ten beads total on the top row, and ten beads on the bottom row. • There are ten red beads and ten white beads on the rack. • There are twenty beads altogether.

31. How the MathRackCan Help What do you notice? explore the tool and learn the built-in structure before you have them use the tool. Show me On one row, "Show me __." Have them show a certain number. Some may count one-by-one but the structure of the tool allows for more advanced strategies. Flash Forward show a certain number on the MathRack for a few seconds and have them determine which number was flashed.

32. How the MathRack Can Help Show me and Flash Forward builds Spatial and Benchmarks of 5 & 10 Just One More show a certain number on the MathRack and have the kids build theirs to show "one more" or "two less." builds One/Two More or Less Show Me using two rows, "Show me ___." builds Part-Part-Whole Peek-a-Boo cover the right side of the MathRack, push some beads over so kids can see them and ask how many beads are hiding. builds Part-Part-Whole (is a Missing Part activity) and Benchmarks of 5 & 10

33. Quick Images How many beads are there? How do you know?

34. How many beads? Read this side

35. How many beads?

36. How many beads? How do you know?

37. How many beads? How do you know?

38. How many beads? How do you know?

39. How many beads? How do you know?

40. Turn and talk What are all the possible ways children will figure out how many?

41. Developing the landmark strategies • Subitizing • Using the 5-structure • Using the 10-structure • Counting on • Doubles and near-doubles • Compensation • Skip counting • Part/whole

42. Contexts for the MathRack …”mathematical meaning in their lived world” • Bunk beds • Double-decker bus • Bookshelves

43. The Double-Decker Bus

44. Games with the MathRack How many empty seats on top? 3 on top 7 on top 2 on top 8 on top 6 on top Day 5

45. Games with the MathRack • Passenger Pairs matching game: Moving from the bus story to a model of the context

46. Games with the MathRack • Rack Pairs matching game: Moving away from the context

47. Games with the MathRack • Bus Stops game How many on the bus as it pulls away from the bus stop? How do you know? +5 8

48. Games using the MathRack • Bus Stops game How many on the bus as it pulls away from the bus stop? How do you know? - 4 11

49. Fluency and Flexibility • Fluency- efficient and correct • Flexibility- multiple solution strategies determined by the problem Fluency is the by-product of flexibility. Assessing fluency by occasionally using timed tests is acceptable. Using timed tests as an instructional tool to build fluency is ineffective, inefficient, and damaging to student learning.

50. Focus on Relationships • When we focus on relationships, it helps give children flexibility when dealing with their basic facts and extending their knowledge to new task. When we build a child’s number sense it promotes thinking instead of just computing.