Rosalind Duplechain, PhD

1
Diagnosing and Correcting Mathematical Errors
ECED 4251

• Rosalind Duplechain, PhD
• University of West Georgia
• College of Education
• Fractions Part 3
• Module 7

2
Basic Structure of PPt

• Lecture (slides 3-11)
• How the DC Process works with Fraction Concepts
  and Operations
• Fraction concepts, number sense, and equivalent
  fractions
• Application (slide 12-13)
• See textbook for error patterns associated with
  fraction concepts and operations.
• Homework - (See Course Calendar).

3
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.

4
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)

5
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

6
The DC Process
Diagnose
Correct
Reflect
Evaluate
7
Process 1 Diagnose

• Basic Fact Errors and Algorithm Errors
• Collect Data
• Analyze Data for Errors
• Pre-diagnose Data
• Interview Student
• Final Diagnosis of Data

8
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

9
Process 3 Evaluate

• 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)

10
Process 4 Reflect

Is students 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?

11
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

12
Application

• Lets apply what weve learned about the DC
  Process to violations of algorithms, and in
  particular to fractions.
• Dan
• Linda

13
Application Guidance

• Using work samples, diagnose and plan correction.
  Refer to previous knowledge, textbook, and other
  resources as needed. Prepare to justify responses.

• 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)?
• Refer to diagnosing checklist for fractions
  (right side of this slide)
• What mathematical misunderstandings might cause a
  student to make this error?
• Refer to diagnosing checklist for fractions
  (right side of this slide)

• Diagnosing Checklist for Fractions
• 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

14
Homework

• See Course Calendar Module 7.