1. ASSESSMENT AND CORRECTION MATHEMATICS EDUCATION: ECED 4251 Rosalind Duplechain University of West Georgia College of Education Decimal Numbers and Operations Module 9

2. Basic Structure of PPt Opening Activity (slide 3) Opening Activity Lecture (slides 4-11) Lecture – Decimal Numbers and Operations – How the D&C Process works with Decimal Numbers and Operations Application (slides 12-13) Application – See textbook for error patterns associated with decimal numbers and their operations (Tonya, Harold, Les, Marsha, & Ted) Homework - (See Course Calendar). Homework

3. Opening ActivityOpening Activity: Module 9 Topics to considerWhat to do about topics? Decimal Concepts and Operations Compare and contrast a decimal number with a whole number. List as many ways as you can find. Using base-10 blocks, how many ways can you represent 2.5 using these place values? Do not assume that you can only use one of each place value. You can use as many as needed. Explain why the following would not be a correct representation of 2.5.. Use, or = to show the relationship between.001,.01,.1, and 1. Justify your answers..001 ____.01 ____.1 ____1 Complete Homework Sheet #10. Self-Assessment Using the questions on the left side of this slide, what don’t you “get” about decimal concepts and operations? Using what you don’t get, start a question log for decimals. Seeking Help Talk to two peers (at least) in hopes of finding answers to your questions. Keep a log of other questions you have for me. Gallery Walk Take the Gallery Walk experience. Building on our past work, how might the D&C Process work with decimals? Using Tonya's error from the textbook,… do the same diagnosing steps apply? If not, which steps don’t apply? Why? do the same correcting steps apply? If not, which steps don’t apply? Why? Compare and contrast Tonya’s correction strategies with the correction steps provided in this course. Keep a log of questions you have for me..

4. Decimal Numbers and Operations Decimals are fractions (and percents) written in a different symbol system. All are merely different names for the same quantity or for the same point on a number line. Despite the interrelatedness of these symbol systems, children see these systems as being very distinct. “The base-10 place-value system extends infinitely in two directions: to tiny values as well as to large values. Between any two place values, the ten-to-one ratio remains the same” (Van de Walle, 2004, p. 280). – The decimal point is a convention that has been developed to indicate the units position to the left of the decimal point that is being counted as singles or ones” (Van de Walle, 2004, p. 280). – Although placing the decimal point is often viewed as a place value concern, it is clearly a procedural concern as well. – Finally, the further away a digit is from the right of a decimal, the smaller its value and vice versa. Like place value of whole numbers, with decimal place value, each place has its own value. Thus, just as ones are added to ones and tens are added to tens, and so on, for whole numbers, with decimal numbers, hundredths are added to hundredths, tenths are added to tenths, ones are added to ones, and so on. Similarly with subtraction, whether whole numbers or decimals, ones are subtracted from ones, hundredths from hundredths, as so on.

5. Decimal Numbers and Operations Children tend to apply what they know about whole number operations to decimal operations. – In some cases, this application is good because it helps students see that the meaning of the operations is always the same. – However, in other cases, this application contributes to a variety of conceptual and procedural errors when operating with decimals, namely, a misplacement of the decimal. In which case, the role of estimating (rounding decimal numbers to the nearest whole number) can help children determine where the decimal point belongs.

6. Decimals and the D&C Process The same four D&C Sub-processes hold for decimal numbers and operations as we used for whole numbers and fractions. – Diagnose Diagnose – Correct Correct – Evaluate Evaluate – Reflect Reflect The diagnosing checklist for decimal numbers and operations is similar to the checklist used for diagnosing whole number operations.checklist

7. Sub-process #1: DiagnoseDiagnose 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

8. Sub-process #2: CorrectCorrect Basic Facts Errors – Teach meaning of operation – Teach and practice number relationship strategies – Work on automaticity Whole Number Operations/Algorithm Errors – Conceptual Only Conceptual Only – Intermediate Intermediate – Procedural Only Procedural Only – Independent Practice Independent Practice

9. Correcting Decimal Errors… Conceptual Only – Using manipulatives/drawings only, show and talk aloud while solving problem. Emphasize ideas related to student’s error. Repeat until student can do alone. Teacher Guided Experiences Intermediate – Using manipulatives/drawings, show and talk aloud while solving problem. Teach and show that algorithm is the step-by-step record of what is being done with manipulatives. Emphasize ideas related to student’s error. Repeat until student can do alone. Teacher Guided Experiences Procedural Only – Using only the algorithm, show and talk aloud while solving problem. Emphasize ideas related to student’s error. Repeat until student can do alone. Teacher Guided Experiences Independent Practice (procedural) – Provide problems for student to solve alone, using only the algorithms. Once practice is completed, teacher checks work. If work earns <85%, teacher repeats correction cycle beginning on either the intermediate level or the procedural only level. Student-only practice Teacher feedback

10. Sub-process #3: EvaluateEvaluate Give a post-test and make sure that student provides all of his/her work. – Assuming you used the correct pre-test, 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?

11. Sub-process #4: ReflectReflect 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 decimal numbers and their 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). As another drill, create a timed drill that requires students to solve, in writing, no more than 15 problems involving these operations (Similar to the way May’s tests are designed, have students perform all operations with whole numbers, fractions, and decimals – no more than 15 problems). Once time is up (no more than 10 minutes), quickly model aloud how to solve each problem. Have students check answers as each problem is modeled aloud. Collect these daily and keep an informal running record of progress (entire drill about no more than 15 minutes of every math lesson). Aside from these learning opportunities (drills), move on to work with more needy students or on other mathematical topics in mathematics curriculum.

12. Application Let’s apply what we’ve learned today about the D&C Process to violations of algorithms, and in particular to error patterns associated with decimal concepts and operations. – Tonya – Harold – Les – Marsha – Ted

13. Diagnosing Checklist: Decimal Numbers and OperationsChecklist The procedural error(s) Ask yourselves: What exactly is this student doing to get this problem wrong? – Basic Facts – Violations of Algorithm The conceptual error(s) Ask yourselves: What mathematical misunderstandings might cause a student to make this procedural error? – Decimals » Can identify and represent decimal numbers – Place Value – Meaning of the Operation in general – Meaning of the Operation under specific conditions (i.e., larger quantities) – Properties of Operations – Number Sense

14. Homework See Course Calendar on CourseDen. – Use Instructional Resource Item #29. – Complete Homework Sheet #9. – Read textbook chapter 4-6 (decimals only). – Use overview sheet for Module 9 as checklist. Complete all items on this checklist.