Diagnosing and Correcting Mathematical Errors: ECED 4251 Rosalind Duplechain, PhD University of West Georgia College of Education Fractions: Part 1 Module 5

Basic Structure of PPt Opening Activity and Carlos (slides 3-4) Lecture (slides 5-13) How the D&C Process works with Fraction Concepts and Operations Fraction concepts, number sense, and equivalent fractions Application (slides 14-15) See textbook for error patterns associated with fraction concepts and operations. Homework - (See Course Calendar).

Opening Activity: Module 5 Topics to consider What to do about topics? Self-Assessment Using the questions on the left side of this slide, what don’t you “get” about fraction concepts and equivalence? Using what you don’t get, start a question log for fractions. 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 fractions? Using Carlos'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 Carolos’s correction strategies with the correction steps provided in this course. Keep a log of questions you have for me. Fraction Concepts What is a fraction? What is a numerator? What is a denominator? When does the relationship between a numerator and a denominator hold true? What is a proper fraction, improper fraction, and a mixed number? What is the “algorithm”/procedure for symbolically representing a fraction? What can we use to model fractions for children? Equivalence What is an equivalent fraction? What can we used to model this concept for children? What is the algorithm for finding an equivalent fraction with a larger denominator? What is the algorithm for finding an equivalent fraction with the smallest denominator? How do you know when a fraction is in its simplest terms? What property explains the algorithm for finding equivalent fractions?

Fraction Concepts Fractional parts are equal shares or equal-sized portions of a whole or unit. A unit can be an object or a collection of things. A unit is counted as 1. On a number line, the distance form 0 to 1 is the unit. The denominator of a fraction tells how many parts of that size are needed to make the whole. For example: thirds require three parts to make a whole. The denominator is the divisor. Larger denominators = smaller parts; Smaller denominators = larger parts The numerator of a fraction tells how many of the fractional parts are under consideration.

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. “understand and represent commonly used fractions, such as ¼, 1/3, and ½ (Ashlock, 2006, p. 46) less than a 0.5 difference between the estimate and answer when operating with fractions (Van de Walle)

Equivalent Fractions… Two equivalent fractions are two ways of describing … the same amount by using different-sized fractional parts (Van de Walle, 2004, p. 242) the same point on a number line by using different fractions (Ashlock, 2010, p. 68) To create equivalent fractions with larger denominators, we MULTIPLY both the numerator and the denominator by a common whole number factor. Activity Question: Can we use smaller parts (larger denominators) to cover exactly what we have? To create equivalent fractions in the simplest terms (lowest terms), we DIVIDE both the numerator and the denominator by a common whole number factor. Activity Question: What are the largest parts (smaller denominators) we can use to cover exactly what we have (Ashlock, 2010, p. 65)? Simplest terms means that the numerator and denominator have no common whole number factors (Van de Walle, 2004, p. 261) “Reduce” is no longer used because it implies that we are making a fraction smaller when in fact we are only renaming the fraction, not changing its size (Van de Walle, 2004, p. 261) The concept of equivalent fractions is based upon the multiplicative identity property that says that any number multiplied by 1, or divided by 1, remains unchanged (Van de Walle, 2004, p. 261): ¾ x 1 = ¾ x 3/3 = 9/12 ¾ x 1 = ¾ x 5/5 = 15/20

The D&C Process Diagnose Reflect Correct Evaluate

Process #1: Diagnose… Basic Fact Errors and Algorithm Errors Collect Data Analyze Data for Errors Pre-diagnose Data Interview Student Final Diagnosis of Data

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

Process #3: Evaluate… 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)

Process #4: Reflect… Is student’s 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: CORRECTING: Did I miss a step in the correcting process? Did I rush my student by either… blending correction steps? not spending enough time on each step? DIAGNOSING: Did I miss a step in the diagnosing process? Did I miss an error?

Final Notes About Fraction Concepts and Equivalent Fractions Developing Number Sense Fractional-Parts Counting (Van de Walle, 2004, pp. 246-247) Activity 15.3 p. 248 Fraction Number Sense (Van de Walle, 2004, pp. 251-257 Activity 15.6: Zero, One-half, or One Activity 15.7: Close Fractions Activity 15.8: About How Much? Activity 15.9: Ordering Unit Fractions Activity 15.10: Choose, Explain, Test Activity 15.11: Line ‘Em Up

Application Let’s apply what we’ve learned about the D&C Process to violations of algorithms, and in particular to fractions. Carlos Jill

Checklist for Diagnosing Errors: Fraction Concepts and Operations 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? Fraction Concepts Part-Whole Relationship Equal Parts/Fair Shares Equivalent Fractions Multiplicative Identity Property Meaning of Operations in general Meaning of Operations when Fractions are involved Properties Commutative Property Associative Property Zero Property Number Sense

Homework See Course Calendar – Module 5.