1. Ashlock Chapter 2: Error Patterns in Addition and Subtraction Dr. Jill Drake

2. Today’s Warm-up Get in groups of two or three Tell your group what your topic was for your slide from last week. Ask you group members to tell what they remember about your topic. Finally, tell your group members all you can remember about your Chapter 1 topic.

3. Today’s Topics… Vocabulary ◦ Old and New Vocabulary ◦ Re-looking at Operations Chapter 2: Ashlock (2009) ◦ Correcting Errors: Demonstration ◦ Diagnosing Errors: Case Study

4. Old Vocabulary… Procedural Errors – involve errors in skills and/or step-by-step procedures (algorithms) needed to solve mathematical problems Conceptual Errors – involve errors that are caused by the misunderstanding of mathematical ideas such as place value, meaning of operations, and number sense. Both Procedural and Conceptual Errors – involve errors that violations of an algorithm AND a misunderstanding of a mathematical idea.

5. New Vocabulary… Algorithm – ◦ “step-by-step procedures for accomplishing a task, such as solving a problem” (Ashlock, 2006, p. 227). ◦ “a predetermined set of instructions for solving a specific problem in a limited number of steps” (Webster, 1996, p. 34). Meaning of Operations - ◦ Addition – Joining two or more addends together resulting in a number that is larger than all addends. ◦ Subtraction – Separating a smaller quantity from a larger quantity resulting in a quantity that is smaller than the minuend. Place Value - The understanding that the place of a digit tells its value, that a number can be combined and taken apart in different ways (i.e., regrouping), AND that the operations of addition, subtraction, and division take place within places.

6. New Vocabulary… Number Sense - An intuition about numbers: their size AND how reasonable a quantity is once a number operation has occurred. ◦ Use estimation as a strategy for determining whether the answer is reasonable. Properties of Operations ◦ Commutative Property – the order in which two numbers are added does not change the result ◦ Associative Property – the order in which three numbers are added does hot change the result ◦ Distributive Property – adding in parts ◦ Zero Property –  Addition and Subtraction – the result is the non-zero number we started with

7. Today’s Demonstration

8. The Remediation Process Conceptual Only – using manipulatives only, emphasize the concepts being taught Teacher Guided Experiences ▼ Intermediate – identify the error; re-teach procedures for solving problem using the written symbols; use manipulatives (and/or drawings) to support the symbols (the operation and the answer). Teacher Guided Experiences ▼ Procedural Only – identify error (if not already done); re-teach procedures for solving problem using the written symbols; no use of manipulatives. Teacher Guided Experiences Independent Practice (procedural) – allow student to practice procedures away from teacher; once practice is completed, check and give student feedback and decide whether student needs more intermediate work, more procedural only work, or more independent practice. Student-only practice Teacher feedback

9. Error Patterns in Whole Number Operations Addition

10. For the following errors: 1. Name the error: What exactly is the student doing? 2. What misunderstanding might generate this type of error? 3. Is it conceptual, procedural, or both? 4. Find a strategy that could remediate it.

11. Gary

12. Gary What type of Error does Gary have? What strategies does Ashlock recommend for Gary? Whole-Class Practice o Pair with one of your classmates. o Decide who will be Gary and who will be teacher. o Use one of Ashlock’s strategies to help Gary correct his error pattern. o Begin practice.

13. Mike

14. Mike What type of Error does Mike have? What strategies does Ashlock recommend for Mike? Whole-Class Practice o Pair with one of your classmates. o Decide who will be Mike and who will be teacher. o Use one of Ashlock’s strategies to help Mike correct his error pattern. o Begin practice.

15. Mary

16. Mary What type of Error does Mary have? What strategies does Ashlock recommend for Mary? Whole-Class Practice o Pair with one of your classmates. o Decide who will be Mary and who will be teacher. o Use one of Ashlock’s strategies to help Mary correct her error pattern. o Begin practice.

17. Carol

18. Carol What type of Error does Carol have? What strategies does Ashlock recommend for Carol? Whole-Class Practice o Pair with one of your classmates. o Decide who will be Carol and who will be teacher. o Use one of Ashlock’s strategies to help Carol correct her error pattern. o Begin practice.

19. Dorothy

20. Dorothy What type of Error does Dorothy have? What strategies does Ashlock recommend for Dorothy? Whole-Class Practice o Pair with one of your classmates. o Decide who will be Dorothy and who will be teacher. o Use one of Ashlock’s strategies to help Dorothy correct her error pattern. o Begin practice.

21. Error Patterns in Whole Number Operations Subtraction

22. For the following errors: 1. Name the error. 2. Is it conceptual, procedural, or both? 3. Name a strategy that could remediate it.

23. Cheryl

24. Cheryl What type of Error does Cheryl have? What strategies does Ashlock recommend for Cheryl? Whole-Class Practice o Pair with one of your classmates. o Decide who will be Cheryl and who will be teacher. o Use one of Ashlock’s strategies to help Cheryl correct her error pattern. o Begin practice.

25. George

26. George What type of Error does George have? What strategies does Ashlock recommend for George? Whole-Class Practice o Pair with one of your classmates. o Decide who will be George and who will be teacher. o Use one of Ashlock’s strategies to help George correct his error pattern. o Begin practice.

27. Donna

28. Donna What type of Error does Donna have? What strategies does Ashlock recommend for Donna? Whole-Class Practice o Pair with one of your classmates. o Decide who will be Donna and who will be teacher. o Use one of Ashlock’s strategies to help Donna correct her error pattern. o Begin practice.

29. Barbara

30. Barbara What type of Error does Barbara have? What strategies does Ashlock recommend for Barbara? Whole-Class Practice o Pair with one of your classmates. o Decide who will be Barbara and who will be teacher. o Use one of Ashlock’s strategies to help Barbara correct her error pattern. o Begin practice.

31. 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, 2009, p. 7) when correcting errors. Suggested Correctional Approach: Intermediate Step  Teach the meaning of the operation using manipulatives and/or drawings.  As you use the manipulatives/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.

32. Diagnosing and Correcting: Your Case Study There are four processes involved in diagnosing and correcting mathematical errors. These will be documented in your case study. These four processes are: ◦ The Diagnosing Process The Diagnosing Process ◦ The Correcting Process The Correcting Process ◦ The Evaluation Process The Evaluation Process ◦ The Reflection Process The Reflection Process

33. Title Page

34. Process #1: Diagnosing… Diagnosing… Basic Facts Errors ◦ Interview Student  Collect Data ◦ Analyze Data for Errors ◦ Final Diagnosis of Data Whole Number Operations/Algorithm Errors ◦ Collect Data ◦ Analyze Data for Errors ◦ Pre-diagnose Data ◦ Interview Student ◦ Final Diagnosis of Data 2

35. Section Page

38. Process #2: Correcting … Correcting …Correcting … Basic Facts Errors ◦ Teach meaning of operation ◦ Teach and practice number relationship strategies ◦ Work on automaticity Whole Number Operations/Algorithm Errors ◦ Conceptual Only ◦ Intermediate ◦ Procedural Only ◦ Independent Practice 2

46. Process #3: Evaluating… Evaluating… Did diagnosis and correction work? ◦ Collect Student’s Work Sample (post-test) ◦ Analyze Work Sample for Errors ◦ Diagnose Incorrect Responses ◦ Determine Effectiveness of Correction Strategy (based on post-test’s score)

49. Process #4: Reflecting … Reflecting …Reflecting … Is error fixed? If yes, move on to another area of concern and begin diagnosing and correcting process. If no, return to steps in diagnosing and correcting process? Ask yourself: DIAGNOSING: Did I miss a step in the diagnosing process? Did I miss an error? CORRECTING: Did I miss a step in the correcting process? Did I blend correction steps? Did I spend enough time on each step?

54. Questions… …about anything we discussed today? ◦ Old vocabulary? ◦ New vocabulary? …about anything in the course?

55. Ticket Out the Door: 3-2-1 Name three types of error patterns that occur when students are learning to add and subtract. Name two strategies that help remediate addition/subtraction errors. Describe one example of an error that is both conceptual and procedural.