ASSESSMENT AND CORRECTION MATHEMATICS EDUCATION: ECED 4251

1. ASSESSMENT AND CORRECTION MATHEMATICS EDUCATION: ECED 4251 Rosalind Duplechain, PhD University of West Georgia College of Education Whole Number Operations: Part 1 Module 3

2. Basic Structure of PPt • Lecture(slides 3-15) • How the D&C Process works with the Whole Number Operations of Addition and Subtraction • Application(slides 16-17) • See textbook for error patterns associated with whole number operations. • Homework - (See Course Calendar).

3. The D&C Process: Four Sub-processes

4. Sub-process #1: Diagnose • There are five diagnosing steps: • Give a pre-test and make sure student provides all of his/her work. • Analyze errors found on pre-test. • Make a pre-diagnosis of errors. • Interview student. • Make a final diagnosis of errors.

5. About the Pre-test • Give a pre-test and make sure student provides all of his/her work. • For whole numbers and for basic facts use one of May’s diagnostic tests as your pre-test. • Allow only 10 minutes to complete this test. • For fractions, decimals, geometry, measurement, time, or money use a worksheet, a chapter test, or a unit test as your pre-test. • Can use up to 15 problems and no less than 5. • Allow one minute for each problem on test.

6. About Analyzing Errors • Work each problem on the pretest and compare student’s work (step by step) and answer to your work (step by step) and answer. • Ask yourself: • Which problems have been answered incorrectly? • Are there any problems with the correct answer but incorrect work? • For any problem that has incorrect work or an incorrect answer, ask yourself: • What exactly is student doing to get this work (step by step)? • What exactly is student doing to get this answer?

7. About the Pre-diagnosis • Ask yourself: • Are the errors due to problems with… • basic facts, • the algorithm, or • both the basic facts and the algorithm? • If basic facts, which facts might need work? • Copy a blank Basic Facts’ Card (see previous module). Complete this card during interview. • If the algorithm, which step(s) in the algorithm and which mathematical concepts might need work? • Refer to diagnosing checklist for whole numbers.

8. About the Interview • The interview has four goals: • To establish a relationship with the student • Introduce yourself, explain the purpose of your work with the student, and ask student’s permission to work with you. Assuming student is willing to work with you, ask the list of required interview questions (See previous module for this list of questions or see the Supplemental Text titled Drake & Duplechain. Required questions are in red). • To either verify or modify the pre-diagnosis • Say to student: Here is a problem. Show me how you would solve this problem and talk out loud so I can hear what you’re thinking. • To get insight about problems you were unable to pre-diagnose • Say to student: Here is a problem. Show me how you would solve this problem and talk out loud so I can hear what you’re thinking. • If basic facts is a pre-diagnosis, to find out which basic facts need work • Prior to interview, create a set of flash cards for all basic facts within the designated operation. • Jumble all cards so that they are out of order. • During the interview, flash each card one at a time (Remember: Do not flash cards in order. You need them to be jumbled – out of order). • As you flash each card, count to 3 in your mind. • Based on student’s answer to card, place the card in one of three piles: • Pile one: Facts the student knows WITHIN 3 seconds • Pile two: Facts the student does not know. • Pile three: Facts the student knows AFTER 3 seconds • Repeat the process of flashing a card, counting to 3, and placing cards in one of three piles until all of the flash cards in the designated operation have been shown to the student. • After the interview, use the piles to complete the Basic Facts’ Card. • Make sure your completed Basic Facts’ Card has a legend/key for any symbols you used to show each pile.

9. About the Final Diagnosis • Based on my understanding of the pre-test and the interview, these errors are due to problems with… • basic facts, • the algorithm, or • both basic facts and the algorithm? • If basic facts, these are the facts that need work (refer to results of the completed Basic Facts’ Card). • If the algorithm, these steps in the algorithm and these mathematical concepts need work (refer to the results of the appropriate diagnosing checklist).

10. Sub-process #2: Correct Basic Facts Errors Violation of Algorithms What should we do??? • Teach the meaning of the operation. • Teach and practice at least one number strategy for each basic fact that needs work. • Work on automaticity of all basic facts.

11. Correcting Algorithm Errors…

12. Three Types of Mathematical Errors Associated with Algorithms • Procedural Errors – involve errors in skills and/or step-by-step procedures (algorithms) needed to solve mathematical problems • Conceptual Errors – 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 – errors that involve violations of an algorithm AND a misunderstanding of a mathematical idea.

13. More on Sub-process #2: Correct • For Basic Facts, there are three correction steps that must be followed. • Teach the meaning of the operation (addition, subtraction, multiplication, or division). • You will need to provide evidence that you have taught this correction step to your field assignment student. See previous module for how evidence of this correction step needs to look. • Please note the request on this evidence: “Draw me a picture that shows what x means.” • GOAL of this step: Get student to show the meaning of addition, subtraction, multiplication, or division. • Teach and practice at least one number strategy for each basic fact that needs work. • You will need to provide evidence that you have taught this correction step to your field assignment student. See previous module for how evidence of this correction step needs to look. • Number strategies are a more efficient way of counting than counting each and every item in all sets (Van de Walle, 2004). • Notice what is required: List of basic facts being worked on and name of strategy used. • Choose between Plan A and Plan B when you turn in your work. Do not use both plans. • Goal of this step: Get student to provide… • A correct response to fact(s) • A justification of answer (explaining or demonstrating how they got answer using the strategy you taught) • Work on automaticity of all basic facts. • You will need to provide evidence that you have taught this correction step to your field assignment student. See previous module for how evidence of this correction step needs to look. • Any activity or game that helps students achieve a ≤ 3 second response is appropriate. • You will simply need to briefly explain the activity or game you used. • Some common examples include: any competition situation (Around the World, relays, etc), rhymes, songs, repetitive writing of facts, repetitive oral response situations, and speed tests. • GOAL of this step: Get student to provide an accurate answer within 3 seconds.

14. Sub-process #3: Evaluate • Give a post-test and make sure that student provides all of his/her work. • Use the same test you used to collect your pre-data • Allow the same amount of time as you allowed for the pre-test. • Grade student’s work (Aim for at least 85%). • Diagnose all errors and ask yourself: • Are any of these errors like the original errors found on the pre-test? • Are any of these error new – unlike the original errors found on the pre-test?

15. Sub-process #4: Reflect • Use score from post-test to determine what to do next. • If <85%, repeat correction cycle. Student has not mastered a sufficient amount of concepts and skills associated with whole number operations. • If ≥85%, this student is on his/her way to mastery. • Continue to work on number strategies and on automaticity in classroom through learning centers whenever students finished assigned tasks earlier than peers and through drill time (about 5 minutes of every math lesson). • Using written prompts, have students stand and recite fact families/multiplication tables (one family/table per day – about 3 minutes of every math lesson). • As another drill, create a timed drill that requires students to solve, in writing, no more than 15 problems involving these operations (use a combination of test items like those found in May’s tests for whole numbers). Once time is up, quickly model aloud how to solve the problems given. Have students check answers as each problem is modeled aloud. Collect these daily and keep an informal running record of progress (about 12 minutes of every math lesson). • Aside from these learning opportunities, move on to work with more needy students or on other mathematical topics in mathematics curriculum.

16. Application: Error Patterns • Let’s apply what we’ve learned today about the D&C Process to violations of algorithms, and in particular to whole number error patterns. • Mary • Cheryl

17. Application: Guidance Using the error patterns, one at a time, diagnose and plan correction. Refer to previous knowledge, textbook, and other resources as needed. Prepare to justify responses. Diagnosing Checklist for Whole Numbers The procedural error(s): Ask yourself: What exactly is this student doing to get this problem wrong? Basic Facts Violations of Algorithm The conceptual error(s): Ask yourself: What mathematical misunderstandings might cause a student to make this procedural error? Meaning of the Operation in general Meaning of the Operation under specific conditions (i.e., larger quantities) Place Value Properties of Operations Number Sense • Ask yourself: • Which problems are wrong? • What exactly is this student doing to get each problem wrong (i.e., skills and/or steps for solving problem)? • What mathematical misunderstandings might cause a student to make this error? • Given my diagnosis, … • Which corrections steps should apply? • Which specific correction strategies should I use? • Generally speaking, how might my chosen correction strategy look at each phase of the correction process?