1. Math Coaches and the Struggling Learner Valerie Faulkner Coordinating Teacher SpEd Services Wake County Public School System vfaulkner1@wcpss.net 858-1599

2. Agenda The Math – Working on our own skills: How can I improve my own mathematical thinking? The Culture – How is it different to work with a population of students versus a population of teachers? The Instruction – What specific changes can I help teachers implement?

3. The Questions Best practices – How do I get teachers to use them? Number Sense – How do I get teachers to embed it in their instruction? Differentiation – How do I support stronger differentiation in classrooms? Assessment – How do we assess and use it for instruction?

4. The Math Elementary Math is Hard! Are you seeing the STRUCTURES of the arithmetic or still looking for Answers?

5. The Culture Remember – We have a Cultural problem. We tend to think about getting the answers in this lesson, rather than SYSTEMATICALLY working on mathematical structures.

6. Analyzing Student Work: Question posed to me by a colleague names changed to protect the excellent! I was in a 3rd grade math class and the students were working on this problem: Jack had picked 12 apples. Syreeta picked 5 fewer apples than Jack. How many apples did Jack and Syreeta pick altogether?

7. BRAINSTORM: How is this problem difficult for students? SMALL GROUP: How would you support teacher?

8. The word fewer confuses them

9. Research Based Approach # 1 Use number line to ground student in magnitude and quantity. Process the vocabulary through the number line. Develop “diagram literacy” grounded in magnitude Math Talk! Subtext taught: Vocabulary grounded in the number line. Numbers ‘live’ on the number line.

10. Mathematical Problem At Food Lion, butter costs 65 cents per stick. This amount is 2 cents less per stick, than butter at Lowes. If you need to buy 4 sticks of butter, how much will you pay at Lowes? (adapted from Hegarty, Mayer, Green, 1992)

11. Converting a sentence into a mental representation At Food Lion, butter costs 65 cents per stick. So --- Food Lion = 0.65 65 Assignment Sentence: Easy - peasy

12. Relational Sentences: Uh – oh… This amount is 2 cents less per stick, than butter at Lowes. Say What?

13. Cultural Problem The problem is that almost all of the strategies we learned in math pretend that life is filled with nothing but assignment sentences! The only thing that can really help is to Deeply Develop and Understand Arithmetic Relationships – VISUALLY AND STRUCTURALLY. This allows us to make sense of the relational sentences.

14. Building a mental model of the problem situation –The hard part of this problem is NOT the multiplication. It is figuring out the cost of butter at the two stores! FOOD LIONLOWES? 65 LOWES?

15. Younger than, fewer than, fewer, less, less than, more than, help! “More than means add” “Less than means subtract” “For less than take the two numbers and subtract but switch the two numbers”

16. 5 4 3 2 1 10 9 8 7 6 5 4 3 2 1 9 8 7 6 5 4 3 2 1 9 8 7 6 5 4 3 2 1 9 8 7 6

17. Number Worlds Vocabulary Development with Meaning - Not “disembodied”

18. Develop Vocabulary in CONTEXT The number line IS the context The Otter lesson

19. Integrating Yolanda is four years younger than Zelda YZ Z? Z – 4 = Y Y + 4 = Z Z – Y = 4 ?

20. 5 4 3 2 1 10 9 8 7 6 5 4 3 2 1 9 8 7 6 5 4 3 2 1 9 8 7 6 5 4 3 2 1 9 8 7 6

21. Comparison bars and math mountains don't work very well.

22. Develop diagram literacy grounded in magnitude Develop diagram literacy grounded in the structures of the arithmetic operations Develop student understanding that numbers are made of other numbers Subtext taught: Math takes time Respect the depth of Arithmetic: “Profound Understanding of Fundamental Mathematics” Research Based Approach # 2

23. Doug Clements, Julie Sarama Subitizing This is a critical skill and may lay underneath early math number sense difficulties with addition and subtraction.

24. Number Sense and Instructional Choices

26. Math Mountains Total/Sum Jack 14 Syreeta 5

27. Jack Syreeta ? 12 Jack 12 Syreeta 5 Jack – 5 = Syreeta 12 – 5 = Syreeta 12 – 5 = _____ Jack + Syreeta = _____ 12 + ______ = ______ Comparison Bars

28. Structure addition Join and Part-Part Whole –There is something, and you get more of it? –There are two kinds, how many all together? Start Unknown –Some are given away, some are left, how many were there to start? Compare--total unknown - I know one amount and I have some amount more than that--how many do I have? ? ? ? Addition types Adapted from Carpenter, Fennema, Franke, Levi and Empson, 1999, p. 12 in Adding it Up, NRC 2001. Taken Left What did I start with? How many altogether? How many do I have?

29. Structure 3 types of subtraction Ask yourself if a problem is a subtraction problem— Does it fit one of these three types?: –The Classic “Take away” (how many left?) –Comparison (difference between? who has more?) –Deficit/Missing amount (what’s missing?) You’ve got some amount and “take away” from it: What’s left? You compare to see: Who has more or less? You need some more to get where you want to be: What is the missing amount? ? ? ? Subtraction types adapted from Ma and Kessel, Knowing Mathematics, 2003

30. If they have had Number Worlds maybe they would know to count backward from 12 five times.

31. Research Based Approach # 3 Teach through Base-Ten Math Facts ARE important - but, perhaps, for a different reason than we think! Conceptual Repetitions Subtext taught: Our number system is based on ten I can derive answers using ten as my cornerstone

32. Math Facts 8 + 7 Fact Families? Flash Cards? Rewards? Problem? We see this as a one-step building block Cognitively, it is not.

33. Makes 10Left over ________ + _________ = __________

34. Makes 10Left over ____8____ + ____5_____ = _________

35. Makes 10Left over 9 + 7 9+ ( ____ + ____ ) ( 9 + ____ ) + ____ (____) + _____ _________ Demonstrate the Associative Property of Addition

36. Makes 10Left over

37. The Instruction How might The Math affect how you will support instruction?

38. The Components of Number Sense

39. Embedding Number Sense in Instruction How many Therapists does it take to change a lightbulb? How many educators does it take to change a lesson plan?

40. So how do you deal with the CULTURAL issues to get to the INSTRUCTIONAL issues?

41. Staff Development Mini-Modules Create Objective and Measurable ‘challenges’ for your staff to grapple with as colleagues

42. Equality 8 + 5 = ___ + 3

43. When the entire class blows it… The Math Coach Swoops In!

44. Two trucks = one elephant Are two trucks “the same thing” as one elephant?

45. How might Subitizing support instruction in Equality?

46. Video: Does 6 = 6?

47. Remember! Part of Assessing is the Victorious Post-Test! Give the problems again!

48. fluency Can you use a fluency challenge to sneak base-ten into the school?

49. Is it possible that ASSESSMENT can help us with this?

50. Makes 10Left over

51. Makes 10Left over 9 + 7 9+ ( ____ + ____ ) ( 9 + ____ ) + ____ (____) + _____ _________ Demonstrate the Associative Property of Addition

52. Stations, Interventions, Activities to support Meaningful Fluency? Discuss in Small Group

53. Group Discussion Discuss One target issue in your small (2-4 people) group Develop a plan to: Assess Intervene Affect So you are working Together with your teachers!

54. Assess Intervene Affect!

55. Knowledge is the Engine Suggested Books and Articles for Group Study Ma, Liping. Knowing and Teaching Elementary Mathematics Carpenter, Franke, Levi. Thinking Mathematically: Integrating Arithmetic and Algebra in Elementary School Burns, Marilyn. About Teaching Mathematics: A K-8 Resource Carpenter, Hiebert, Fennema, Fuson. Making Sense: Teaching and Learning Mathematics with Understanding Ball, Deborah: Select journal articles by topic (late elem) at: http://www-personal.umich.edu/~dball/articles/inex.htmlhttp://www-personal.umich.edu/~dball/articles/inex.html (articles posted on-line!) Griffin, Sharon Select journal articles by topic (late elem) at: http://www.clark.edu/academicCatalog/facultybio.cfm?id=39 (you can also email me—here articles are not posted on-line, just the titles)http://www.clark.edu/academicCatalog/facultybio.cfm?id=39 Clements, Doug Subitizing: What is it? Pdf on Blackboard Faulkner, Valerie and Cain, Chris Components of Number Sense and How it affected one teacher. Pdf on blackboard