1. Multiplication and Division of Whole Numbers Ashlock Chapter 3

2. Warm-up: Fill in the boxes 37 x5 93

3. Class Check In Case Study Questions?Case Study Questions? Case Study MeetingsCase Study Meetings

4. Multiplication What conceptual knowledge does a child need to have to successfully multiply whole numbers?What conceptual knowledge does a child need to have to successfully multiply whole numbers? They need to know the meaning of :They need to know the meaning of : –Whole numbers –Place value –Addition –Multiplication –Regrouping –??????????

5. Multiplication What procedural knowledge does a child need to have to successfully multiply whole numbers?What procedural knowledge does a child need to have to successfully multiply whole numbers? They need to know how to :They need to know how to : –Write whole numbers –Follow steps of one and two or more digit addition –Follow steps of Multiplication –Follow the steps of Regrouping

6. What kinds of Error Patterns are found in relationship to Multiplying Whole Numbers? Conceptual The child does not understand –whole numbers –place value –addition –Multiplication –Order of operations –Regrouping Procedural The child does not correctly –write whole numbers –Follow steps of one and two or more digit addition –Follow steps of Multiplication –Follow the steps of Regrouping

7. Division What conceptual knowledge does a child need to have to successfully multiply whole numbers?What conceptual knowledge does a child need to have to successfully multiply whole numbers? They need to know the meaning of :They need to know the meaning of : –Whole numbers 9 (including 0) –Place value –Subtraction –Division –Order of operations –Borrowing

8. Division What procedural knowledge does a child need to have to successfully division whole numbers?What procedural knowledge does a child need to have to successfully division whole numbers? They need to know how to :They need to know how to : – write whole numbers –Follow steps of one and two or more digit subtraction –Follow step of Division –Follow the step of Borrowing

9. What kinds of Error Patterns are found in relationship to Dividing Whole Numbers? Conceptual The child does not understand –whole numbers –place value –Subtraction –Division –Order of operations –Regrouping – Procedural The child does not correctly –write whole numbers –Follow steps of one and two or more addition –Follow step of Multiplication –Follow the step of Regrouping

10. Demonstration

11. Let’s Practice For each of the following errors answer the following questions:For each of the following errors answer the following questions: –What is the error pattern? –Is the error conceptual, procedural or both? –Which one of Ashlock’s strategies would best remediate this error?

12. Error Patterns in Whole Number Operations Multiplication

13. Joe

14. Correction Strategies for Multiplication 1. Conceptual Only Strategy? 2. Intermediate Strategy? 3. Procedural Only Strategy?

15. Doug

16. Correction Strategies for Multiplication 1. Conceptual Only Strategy? 2. Intermediate Strategy? 3. Procedural Only Strategy?

17. 17 Bob

18. Correction Strategies for Multiplication 1. Conceptual Only Strategy? 2. Intermediate Strategy? 3. Procedural Only Strategy?

19. Error Patterns in Whole Number Operations Division

20. Jim

21. Correction Strategies for Division 1. Conceptual Only Strategy? 2. Intermediate Strategy? 3. Procedural Only Strategy?

22. Gail

23. Correction Strategies for Division 1. Conceptual Only Strategy? 2. Intermediate Strategy? 3. Procedural Only Strategy?

24. John

25. Correction Strategies for Division 1. Conceptual Only Strategy? 2. Intermediate Strategy? 3. Procedural Only Strategy?

26. Correcting Errors… Discussing Correctional Strategies… Remember: Many procedural errors occur because students do not have an adequate understanding of the concepts that underlie these procedures. For this reason “procedural knowledge must be tied to conceptual knowledge” (Ashlock, 2006, p. 47) when correcting errors.

27. For Error Patterns that are both conceptual and procedural… Suggested Correctional Approach: Intermediate Step Teach the meaning of the operation using manipulatives and/or drawings.Teach the meaning of the operation using manipulatives and/or drawings. AS you use the manipulatives or drawings, record what is done step-by- step. This written record needs to be the algorithm.AS you use the manipulatives or drawings, record what is done step-by- step. This written record needs to be the algorithm. Both of these steps are meant to be done simultaneously for each problem.Both of these steps are meant to be done simultaneously for each problem.

28. Preparing for Your Case Study Read information in Course Materials folder on WebCTRead information in Course Materials folder on WebCT

29. Questions… …about anything we discussed today?…about anything we discussed today? –Old vocabulary? –New vocabulary? …about anything in the course?…about anything in the course?

30. Final Thoughts… Remember:Remember: –Diagnosing and Correcting Mathematical Errors involve four processes: the Diagnosing Process, the Correcting Process, the Evaluating Process, and the Reflection Process. –Use your diagnosis to guide your instruction. –Interviews and observations of students are important tools for diagnosticians. They can help rule out and/or reveal reasons for errors. –When diagnosing errors related to whole numbers, refer to the list of errors related to whole number operations. –Many procedural errors occur because students do not have an adequate understanding of the concepts that underlie these procedures. For this reason “procedural knowledge must be tied to conceptual knowledge” (Ashlock, 2006, p. 47) when correcting errors.

31. Homework… Read: Ashlock (2009) – Chapter 4Read: Ashlock (2009) – Chapter 4 Begin to Prepare for Case Study MeetingBegin to Prepare for Case Study Meeting Research Articles are dueResearch Articles are due