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

2. 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).

3. 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?

5. 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.

6. 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)

7. 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

8. The D&C Process
Diagnose
Reflect
Correct
Evaluate

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

10. 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

11. 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)

12. 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?

13. Final Notes About Fraction Concepts and Equivalent Fractions
Developing Number Sense
Fractional-Parts Counting (Van de Walle, 2004, pp )
Activity 15.3 p. 248
Fraction Number Sense (Van de Walle, 2004, pp
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

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

15. 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

16. Homework
See Course Calendar – Module 5.