Numeration and Addition/Subtraction Errors

1. Numeration and Addition/Subtraction Errors February 8, 2007 Slides 8-14 are from last week!

2. Housekeeping • Task Project • Diagnostic Project • Dr. Polly’s Research Project • Rubric Project • We’ll discuss tonight

3. Opening Tasks • Postage Stamps • Soccer Scheduling • Read the task. Before you start… • Possible approaches? • Manipulatives needed?

4. Solutions • Postage Stamps

5. Opening Tasks • Soccer Scheduling

6. Student Work Samples • Postage Stamps • Errors? • If we were there during implementation • Questions to check for understanding • Questions to help students overcome mistakes

7. Student Work Samples • Soccer Scheduling • Errors? • If we were there during implementation • Questions to check for understanding • Questions to help students overcome mistakes

8. Assessment Recap and Review • Purposes and Examples of…. • Norm-referenced • Criterion-referenced • Performance-based • Are these summative, formative or both? • Why?

9. Numeration • Development of Number Sense • At what age do you think students develop a sense of numeration?

10. Numeration Tools: Dot Cards What do you see? How are these the same? How are they different? Other arrangements of this quantity?

11. Numeration Tools: Dot Cards • Composing and decomposing numbers • Multiple representations are needed • Students commonly learn before school… • Domino configurations • Linear configurations • Exposure to numbers should… • Enhance their prior knowledge (sound familiar ) • Include regular and irregular configurations of numbers

12. Numeration Tools: Dot Cards • Composing and decomposing numbers • Multiple representations are needed • Students commonly learn before school… • Domino configurations • Linear configurations • Exposure to numbers should… • Enhance their prior knowledge (sound familiar ) • Include regular and irregular configurations of numbers

13. Numeration Tools: Quick Count • Let’s play a few rounds!

14. Numeration Resources • Ten Frames- link • Matching game • 20 cards: 10 with red circles, 10 with blue circles • Each card has a different number of circles (0 to 10) • Find your partner to fill the 10 frame • Person with 8 blue circles has to find the person with 2 blue circles

15. Worth considering • What role do PWC and DM tasks have in regards to helping students develop a deeper understanding of numeration and number sense? • In a class of 22 Kindergarten children how can you design instruction to meet all 22 children’s needs effectively? • In a 3rd grade class students still lack an understanding of number, which is impeding their understanding of 3rd grade content. How can you effectively teach these children?

16. Addition and Subtraction Tasks Solve each problem. How are they similar? Different? Monica has six pens. Tyrisha gives her five more. How many does Monica now have? Pedro has 11 pieces of candy. He only had six before going to the store. How many did he buy? Kobe’s mom gave him five dollars. He now has eleven dollars. How much money did he have before?

17. Structures of Addition Problems How are these problems similar? How are these problems different?

18. Initial- the starting value Result- the “sum” or “total” Change- the amount added to the initial value Structures of Addition Problems 5 + 4 = 9

19. Structures of Addition Tasks • Join tasks • Problems where items or groups are combined • Result unknown • Traditional addition problem, e.g. 4 + 5 = 9 • Change unknown • The second addend is missing, e.g. 4 + = 9 • Initial unknown • The first addend is missing, e.g. + 5 = 9 • Research shows that Join are easiest and Initial Unknown are the hardest.

20. Structures of Addition Problems • Write four addition/subtraction story tasks. Include at least one of each type of problem. • Pair up and exchange tasks • Identify the type of task • Model each task in two of these ways • Cuisenaire rods • Base-10 blocks • Drawing a picture • Jot down how you modeled the task.

21. Structures of Addition Tasks • Which was the most difficult to write? • Which was the most difficult to model? • Which manipulatives or approaches were most appropriate?

22. Adding and Subtracting Errors • Two types of knowledge • Operative • Knowledge of action (operation) • Figurative • Linguistic knowledge (“figure of speech”)

23. Addition and Subtracting- Concrete Errors • Misuse of the manipulatives • Miscounting the manipulatives • Counting 1-to-1correspondence • Counting from zero • Counting from one • Counting on • Misapplication of Action Language • I had four cookies. Today my Mom gave me more. Now I have nine. • What do you think students’ most common error is?

24. Addition- Representational Errors (picture of base-10 blocks) • Common incorrect answers: • 14 • 1,004 • 5

25. Addition- Representational Errors • Place Value Chart • Regrouping errors • Regrouping everywhere as opposed to only one place • Regrouping in the wrong spot • Not using zero as a place holder • 103 gets recorded as 13 • Reading the place value chart correctly • Reading the digits backwards • Off by a digit: 124 gets read as 1,240

26. Addition- Symbolic Errors • Failing to complete the addition • Treating each column separately with no regrouping • Adding ones to tens and tens to hundreds • Omitting a regrouped decade from a computation

27. Addition- Symbolic Errors • Regrouping when not necessary • Regrouping twice- to tens and hundreds when it should only happen once • Regrouping without using place value • Always moving one to the left • Regrouping the ones place instead of the decade (tens place)

28. Addition- Symbolic Errors • Basic Facts • Problematic • 7+6, 7+9 • 8+6 • Start with +1, +2, doubles, doubles +1, doubles +2, combinations that add to 10

29. Symbolic Errors- Gary (p. 102) 7 + 8 = 14 7 + 6 = 12 8 + 6 = 13 What would he do for… 7 + 7 = 8 + 5 = What is Gary doing incorrectly? Suggested modifications?

30. Mary (p. 103) What error is occurring? Are any of them correct? How would she solve? 254 618 +535+782 Possible ways to help her?

31. Subtraction- Concrete Errors • Similar to addition, also… • Adding instead of taking away manipulatives • Carrying (trading with the base-10 blocks)

32. Subtraction- Errors • Misuse of Action Language The shuttle astronauts have been aloft for 16 hours. How many more hours will it take to complete one full 24 hour day in space? 16 + __ = 24 16 + 24 = 40. Here, students will say the answer is 40. Operative- knowing the action, “what to do” Figurative- interpreting and reading the language of the problem

33. Subtraction- Errors • Confusion between subtraction and addition • Pre-operational behavior • Strategies • Give students a variety of subtraction problems • Tasks that include open number sentences • 17 + ___ = 35 • 35 - ___ - 17

34. Subtraction- Errors • Strategies • Estimation, check the reasonableness of answers • Use a variety of manipulatives • Counters, base-10 blocks, chips • Cuisenaire rods, abacus • Reconstruct action language with actions

35. Subtraction- Errors • Renaming errors • Explicit trade: 1 hundred=10 tens, 1 ten=10 ones • Regrouping in the wrong place • 1 hundred = 10 ones?? • Modifications? • Base-10 blocks and place value charts • Bundled straws (calendar manipulative)

36. Subtraction- Symbolic Errors

37. Best Practices for Add/Subtract Errors • Experience with a variety of problem solving situations • Practice translating problems into open number sentences • Promotes active communication about mathematics • Simultaneously fostering understanding of quantitative relations. • Incorporating checking for accuracy and reasonableness • Multiple representations, uses of manipulatives • “Reconstruct" with student the correspondence between action language and actions on objects

38. Housekeeping • For next week…

39. Index Card • Questions from tonight?