Transcript Diagnosing Mathematical Errors of Whole Numbers: Multiplication

Diagnosing Mathematical Errors:
Fractions and Decimals: Addition
and Subtraction
Dr. Jill Drake
College of Education
Today’s Topics…
Quiz
 Vocabulary Review
 Chapter 5: Ashlock (2010)

◦ Diagnosing Errors: Group Work
◦ Correcting Errors: Whole Group

Homework
Fraction Concepts

See Van de Walle (2004), p. 242
◦ 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.
◦ The numerator of a fraction tells how many of the
fractional parts are under consideration.
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).
◦ To create equivalent fractions with larger denominators, we multiply both
the numerator and the denominator by a common whole number factor.
 Question: Can we use smaller parts (larger denominators) to cover exactly what
we have?
 (Activity 15.17 – Van de Walle, p. 260).
◦ To create equivalent fractions in the simplest terms (lowest terms), we
divide both the numerator and the denominator by a common whole
number factor.
 Question: What are the largest parts we can use to cover exactly what we have
(Ashlock, 2006, p. 146)?
 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
property that says that nay number multiplied by, or divided by, 1 remains
unchanged (Van de Walle, 2004, p. 261).
 ¾ x 1 = ¾ x 3/3 = 9/12
Where might a student error in learning
fraction and decimal operations?
Basic Facts (not known)
 Procedural

◦ Algorithm difficulties

Conceptual
◦ Fraction/Decimal Concepts
 Part-Whole Relationship
 Equal Parts/Fair Shares
 Place Value
◦
◦
◦
◦
Equivalent Fractions/Decimals
Meaning of Operations in general
Meaning of Operations when fractions or decimals are involved
Properties




Commutative Property
Associative Property
Zero Property
Multiplicative Identity Property
◦ Number Sense
Demonstrations
Diagnosing Errors

Work with a group of your peers to reach a
consensus about…
◦ Error Type: Conceptual, Procedural or Both?
◦ The procedural error(s)
 Ask yourselves: What exactly is this student doing to get
this problem wrong?
◦ The conceptual error(s)
 Ask yourselves: What mathematical misunderstandings
might cause a student to make this procedural error?
 Fraction Concepts
 Part-Whole Relationship
 Equal Parts/Fair Shares
 Number Sense
Robbie’s Case (A-F-1)
Describe Robbie’s error pattern.
1. Procedural Error:
• Robbie adds the two numerators as the new
numerator. Robbie adds the two denominators as the
new denominator.
2. Conceptual Error
Robbie may not understand the algorithm of fraction
addition.
Fraction Addition:
Correction strategies for Robbie:
1. Conceptual Strategy
•
•
•
Estimation: Estimate answers before computing by using
benchmark numbers such as ½ and 1.
To get students understand the algorithm of fraction addition,
consider a simper task such as 5/8 + 2/8 where no fraction needs
to be changed. Let’s students to explain why 5/8 + 2/8 make 7/8
and relate the reason the algorithm.
To get students to move to common denominator, consider a
simpler task such as 3/8 + 2/4 where only one fraction needs to
be changed. Let’s students to explain why 5/8 + 2/4 make one
whole and there is 1/8 extra. Relate the reason to the need of
common denominator for unlike fractions.
Fraction Addition:
Correction strategies for Robbie
2. Intermediate Strategy
• Use papers or pictures to represent each addend as
fractional parts of a unit region and then divide these
fractional parts into the same size (same denominator)
of parts. Relate the step-by-step representation
procedure to the written algorithm.
3. Procedural Strategy
• Find the common denominator for addends. Change
each addend into an equivalent fraction which has
this common denominator. Add the numerators as the
numerator for the sum.
Diagnosing Errors

Work with a group of your peers to reach a
consensus about…
◦ Error Type: Conceptual, Procedural or Both?
◦ The procedural error(s)
 Ask yourselves: What exactly is this student doing to get
this problem wrong?
◦ The conceptual error(s)
 Ask yourselves: What mathematical misunderstandings
might cause a student to make this procedural error?
 Fraction Concepts
 Part-Whole Relationship
 Equal Parts/Fair Shares
 Number Sense
Dave
Dave’s Case (A-F-2)
•Describe Dave’s error pattern.
1. Procedural Error:
2. Conceptual Error
Fraction Addition:
Correction strategies for Dave:
1. Conceptual Strategy
•
•
•
Estimation: Estimate answers before computing by using
benchmark numbers such as ½ and 1.
To get students understand the algorithm of fraction addition,
consider a simper task such as 5/8 + 2/8 where no fraction needs
to be changed. Let’s students to explain why 5/8 + 2/8 make 7/8
and relate the reason the algorithm.
To get students to move to common denominator, consider a
simpler task such as 3/8 + 2/4 where only one fraction needs to
be changed. Let’s students to explain why 5/8 + 2/4 make one
whole and there is 1/8 extra. Relate the reason to the need of
common denominator for unlike fractions.
Fraction Addition:
Correction strategies for Dave
Intermediate Strategy
• Use papers or pictures to represent each addend as
fractional parts of a unit region and then divide these
fractional parts into the same size (same denominator)
of parts. Relate the step-by-step representation
procedure to the written algorithm.
3. Procedural Strategy
• Find the common denominator for addends. Change
each addend into an equivalent fraction which has
this common denominator. Add the numerators as the
numerator for the sum.
Diagnosing Errors

Work with a group of your peers to reach a
consensus about…
◦ Error Type: Conceptual, Procedural or Both?
◦ The procedural error(s)
 Ask yourselves: What exactly is this student doing to get
this problem wrong?
◦ The conceptual error(s)
 Ask yourselves: What mathematical misunderstandings
might cause a student to make this procedural error?
 Fraction Concepts
 Part-Whole Relationship
 Equal Parts/Fair Shares
 Number Sense
Robin
Robin’s Case (A-F-3)
Describe Robin’s error pattern.
1. Procedural Error:
2. Conceptual Error
Fraction Addition:
Correction strategies for Robin:
1. Conceptual Strategy
•
•
•
Estimation: Estimate answers before computing by using
benchmark numbers such as ½ and 1.
To get students understand the algorithm of fraction addition,
consider a simper task such as 5/8 + 2/8 where no fraction needs
to be changed. Let’s students to explain why 5/8 + 2/8 make 7/8
and relate the reason the algorithm.
To get students to move to common denominator, consider a
simpler task such as 3/8 + 2/4 where only one fraction needs to
be changed. Let’s students to explain why 5/8 + 2/4 make one
whole and there is 1/8 extra. Relate the reason to the need of
common denominator for unlike fractions.
Fraction Addition:
Correction strategies for Robin
Intermediate Strategy
• Use papers or pictures to represent each addend as
fractional parts of a unit region and then divide these
fractional parts into the same size (same denominator)
of parts. Relate the step-by-step representation
procedure to the written algorithm.
3. Procedural Strategy
• Find the common denominator for addends. Change
each addend into an equivalent fraction which has
this common denominator. Add the numerators as the
numerator for the sum.
Diagnosing Errors

Work with a group of your peers to reach a
consensus about…
◦ Error Type: Conceptual, Procedural or Both?
◦ The procedural error(s)
 Ask yourselves: What exactly is this student doing to get
this problem wrong?
◦ The conceptual error(s)
 Ask yourselves: What mathematical misunderstandings
might cause a student to make this procedural error?
 Fraction Concepts
 Part-Whole Relationship
 Equal Parts/Fair Shares
 Number Sense
Andrew
Andrew’s Case (S-F-1)
Describe Andrew’s error pattern.
1. Procedural Error:
2. Conceptual Error:
Fraction Subtraction:
Correction strategies for Andrew
Conceptual
Intermediate
Procedural
Correction Strategies…
Correctional Strategies for Subtraction of
Fractions
◦ See Ashlock’s (2010) text,…
 Andrew’s Correction Strategy pages 82.
◦ See Van de Walle’s (2004) activities…
 Activity 15.4: Mixed-Number Names (p. 249)
 See also pages 257 – 260
 Activity 15.13: Different Fillers
 Activity 15.14: Dot Paper Equivalencies
 Activity 15.15: Group the Counters, Find the Names
 Activity 15.16: Missing-Number Equivalencies
 Activity 15.17: Slicing Squares
Diagnosing Errors

Work with a group of your peers to reach a
consensus about…
◦ Error Type: Conceptual, Procedural or Both?
◦ The procedural error(s)
 Ask yourselves: What exactly is this student doing to get
this problem wrong?
◦ The conceptual error(s)
 Ask yourselves: What mathematical misunderstandings
might cause a student to make this procedural error?
 Fraction Concepts
 Part-Whole Relationship
 Equal Parts/Fair Shares
 Number Sense
Chuck
Chuck’s Case (S-F-2)
Describe Chuck’s error pattern.
1. Procedural Error:
Chuck records the difference between the two
denominators as the new denominator.
2. Conceptual Error
Chuck may not understand the algorithm of fraction
subtraction.
Fraction Subtraction:
Correction strategies for Chuck
Conceptual
Intermediate
Procedural
Correction Strategies…
Correctional Strategies for Subtraction of
Fractions
◦ See Ashlock’s (2010) text,…
 Chuck’s Correction Strategy pages 83.
◦ See Van de Walle’s (2004) activities…
 Activity 15.4: Mixed-Number Names (p. 249)
 See also pages 257 – 260
 Activity 15.13: Different Fillers
 Activity 15.14: Dot Paper Equivalencies
 Activity 15.15: Group the Counters, Find the Names
 Activity 15.16: Missing-Number Equivalencies
 Activity 15.17: Slicing Squares
Case Study

Questions
Non-Basic Facts Correcting Errors…
Conceptual Only – using manipulatives only,
emphasize the concepts being taught
Intermediate – identify the error; re-teach
procedures for solving problem using the written
symbols; use manipulatives (and/or drawings) to
support the symbols (the operation and the
answer).
Procedural Only – identify error (if not already
done); re-teach procedures for solving problem
using the written symbols; no use of
manipulatives.
Independent Practice (procedural) – allow
student to practice procedures away from teacher;
once practice is completed, check and give
student feedback and decide whether student
needs more intermediate work, more procedural
only work, or more independent practice.
Teacher Guided
Experiences
Teacher Guided
Experiences
Teacher Guided
Experiences
Student-only
practice
Teacher feedback
Questions…

Have a blessed week!