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Ashlock Chapter 2: Error Patterns in Addition and Subtraction Dr. Jill Drake Slide 2 Todays 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. Slide 3 Todays Topics Vocabulary Old and New Vocabulary Re-looking at Operations Chapter 2: Ashlock (2009) Correcting Errors: Demonstration Diagnosing Errors: Case Study Slide 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. Slide 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. Slide 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 Slide 7 Todays Demonstration Slide 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 Slide 9 Error Patterns in Whole Number Operations Addition Slide 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. Slide 11 Gary Slide 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 Ashlocks strategies to help Gary correct his error pattern. o Begin practice. Slide 13 Mike Slide 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 Ashlocks strategies to help Mike correct his error pattern. o Begin practice. Slide 15 Mary Slide 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 Ashlocks strategies to help Mary correct her error pattern. o Begin practice. Slide 17 Carol Slide 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 Ashlocks strategies to help Carol correct her error pattern. o Begin practice. Slide 19 Dorothy Slide 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 Ashlocks strategies to help Dorothy correct her error pattern. o Begin practice. Slide 21 Error Patterns in Whole Number Operations Subtraction Slide 22 For the following errors: 1. Name the error. 2. Is it conceptual, procedural, or both? 3. Name a strategy that could remediate it. Slide 23 Cheryl Slide 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 Ashlocks strategies to help Cheryl correct her error pattern. o Begin practice. Slide 25 George Slide 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 Ashlocks strategies to help George correct his error pattern. o Begin practice. Slide 27 Donna Slide 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 Ashlocks strategies to help Donna correct her error pattern. o Begin practice. Slide 29 Barbara Slide 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 Ashlocks strategies to help Barbara correct her error pattern. o Begin practice. Slide 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. Slide 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 Slide 33 Title Page Slide 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 Slide 35 Section Page Slide 36 Slide 37 Slide 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 Slide 39 Slide 40 Slide 41 Slide 42 Slide 43 Slide 44 Slide 45 Slide 46 Process #3: Evaluating Evaluating Did diagnosis and correction work? Collect Students Work Sample (post-test) Analyze Work Sample for Errors Diagnose Incorrect Responses Determine Effectiveness of Correction Strategy (based on post-tests score) Slide 47 Slide 48 Slide 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? Slide 50 Slide 51 Slide 52 Slide 53 Slide 54 Questions about anything we discussed today? Old vocabulary? New vocabulary? about anything in the course? Slide 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.