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Making Sense of Elementary School
Teachers’ Perspectives on Children’s
Fraction Strategies
Naomi Jessup, Amy Hewitt & Vicki Jacobs
SESSION OVERVIEW
Goal: Explore teachers’ perspectives on children’s fraction
strategies by looking at features used to determine a range
of sophistication
q  Situate our study within the literature
q  Provide an overview of the research project
q  Participate in a research activity
q  Share findings & implications
RESPONSIVE TEACHING
Focus on
Children’s Thinking
q  Foregrounds attention to
children’s thinking
q  Recognizes disciplinary
connections
q  Elicits and builds on
children’s thinking
(Robertson, Scherr, & Hammer, 2016)
Focus on
Teachers’ Thinking
q  Foregrounds attention to
teachers’ thinking
q  Recognizes disciplinary
connections and teachers’
experiences prior to PD
q  Elicits and builds on
teachers’ thinking
Goal of the Study: To understand how teachers
engage with children’s thinking of equal sharing
strategies prior to the start of professional
development
Research Questions
1.  What strategy features are salient to teachers? 2.  How do teachers think about those strategy features
when determining levels of sophistication for equal
sharing strategies?
Responsive Teaching in
Elementary Mathematics
University of Missouri
Susan Empson
University of North Carolina at Greensboro
Vicki Jacobs
Naomi Jessup
Amy Hewitt
University of Texas at Austin
Gladys Krause
D’Anna Pynes
Other Partners: SRI & Teachers Development Group
PROJECT OVERVIEW
q  4-year professional development design study
q  Characterize teaching that is responsive to children’s
mathematical thinking in the domain of fractions q  3 years of PD (total > 150 hours)
o  Summer and academic-year workshop
(8.5 days per year)
o  School-based activities
PD CONTEXT
q  Research frameworks on children’s fraction thinking
q  Artifacts of practice (video clips and student written
work)
q  Instructional Practices
o  Noticing children’s mathematical thinking
o  Supporting and extending children’s mathematical
thinking
(Jacobs & Empson, 2016; Sherin, Jacobs, & Philipp, 2011)
PARTICIPANTS
q  20 teachers in grades 3 – 5 from 2 cohorts
q  Drawn from 3 districts with varying instructional
contexts
q  Range in years of teaching experience and previous PD
on children’s mathematical thinking
PROFESSIONAL DEVELOPMENT TASK
Teachers worked in groups of 2 – 3
q  Reviewed 11 pieces of student work
o  All valid strategies
o  Strategies represented different levels of understanding
q  Organized student work and kept track of your decision
making
o  Ordered from least to most sophisticated
PROFESSIONAL DEVELOPMENT TASK
Problem: 6 children are sharing 16 brownies so that
everyone gets the same amount. How much brownie can
each child have? q  Review 6 pieces of student work
q  Order the strategies from least to most sophisticated
q  Keep track of features used when ordering strategies
TASK DEBRIEF
Problem: 6 children are sharing 16 brownies so that
everyone gets the same amount. How much brownie can
each child have? Were there any particular features within the strategies that
helped you order them?
METHODS
Data Analysis
Analyzed video recordings of 7 group discussions
q  Tracked details in each strategy that groups discussed
q  Compared across strategies to determine salient
strategy features
q  Explored how teachers talked about strategy features q  Compared ordering of strategies across groups
Salient Features of Equal Sharing
Strategies
q 
q 
q 
q 
q 
Teachers
Role of leftovers
Use of representation
Distribution of wholes
Size of parts
Form of final answer
Hiro
Teachers’ Perspectives on Ordering Children’s Strategies
for Equal Sharing Problems
Strategy
Feature
Dominant Perspective Expressed by Teacher Groups
Less Sophisticated
More Sophisticated
Role of
leftovers
v
Use of
representation
v
Distribution
of wholes
v
Size of parts
v
Form of final
answer
v
Role of the number
of sharers in
partitions
Use of fraction terms
and symbols
# of Groups
Expressing
Perspective
(n = 7)
Teachers’ Perspectives on Ordering Children’s Strategies
for Equal Sharing Problems
Strategy
Feature
Role of
leftovers
Dominant Perspective Expressed by Teacher Groups
Less Sophisticated
Ignoring leftovers
More Sophisticated
!
Use of
representation
v
Distribution
of wholes
v
Size of parts
v
Form of final
answer
v
Partitioning leftovers in some
way
Role of the number
of sharers in
partitions
Use of fraction terms
and symbols
# of Groups
Expressing
Perspective
(n = 7)
7
Teachers’ Strategy Feature: Role of the Leftovers
Michael
Angel
Daniel
Hiro
Keisha
Samantha
“So Michael’s strategy [is] going to be your lowest because
[he] doesn’t know what to do with the remainder.”
Teachers’ Perspectives on Ordering Children’s Strategies
for Equal Sharing Problems
Strategy
Feature
Role of
leftovers
Use of
representation
Dominant Perspective Expressed by Teacher Groups
Less Sophisticated
More Sophisticated
Ignoring leftovers
v
Partitioning leftovers in some
way
7
Using pictures to represent
all or some items
!
Partitioning mentally without
representing items
6
Distribution
of wholes
v
Size of parts
v
Form of final
answer
# of Groups
Expressing
Perspective
(n = 7)
v
Role of the number
of sharers in
partitions
Use of fraction terms
and symbols
Teachers’ Strategy Feature: Use of Representation
Hiro
Keisha
“I think she would go [most sophisticated] because she just knew
[the answer]. Hiro didn't do it mentally. Keisha didn't have to split
it, she didn't have to draw it out."
Teachers’ Perspectives on Ordering Children’s Strategies
for Equal Sharing Problems
Strategy
Feature
Role of
leftovers
Use of
representation
Distribution
of wholes
Size of parts
Form of final
answer
Dominant Perspective Expressed by Teacher Groups
Less Sophisticated
More Sophisticated
# of Groups
Expressing
Perspective
(n = 7)
Ignoring leftovers
v
Partitioning leftovers in some
way
7
Using pictures to represent
all or some items
v
Partitioning mentally without
representing items
6
v
Distributing as many wholes
as possible
before
Role
of the number
partitioning extras
5
Partitioning all items before
any distribution
of sharers in
partitions
v
v
Use of fraction terms
and symbols
Teachers’ Strategy Feature: Distribution of the Whole
Hiro
Samantha
T1: Samantha is more sophisticated than Hiro because she could
see the whole pieces.
T2: Yeah, Samantha could see the wholes and then was able to
divide up the parts.
Teachers’ Perspectives on Ordering Children’s Strategies
for Equal Sharing Problems
Strategy
Feature
Role of
leftovers
Dominant Perspective Expressed by Teacher Groups
Less Sophisticated
More Sophisticated
# of Groups
Expressing
Perspective
(n = 7)
Ignoring leftovers
v
Partitioning leftovers in some
way
7
Use of
representation
Using pictures to represent
all or some items
v
Partitioning mentally without
representing items
6
Distribution
of wholes
Partitioning all items before
any distribution
v
Distributing as many wholes
as possible
before
Role
of the number
partitioning
extras
of sharers in
5
v
partitions
Partitioning
items into large
parts regardless of number of
sharers
4
Size of parts
Form of final
answer
Partitioning items using the
number of sharers
v
Use of fraction terms
and symbols
Teachers’ Strategy Feature: Size of Parts
Hiro
Angel
T1: I think Hiro is lower than Angel because Hiro knew that you cut
every brownie into six then counted the sixths.
T2: Yes, this worked, but is this the most practical? Are you going
to want one-sixth of a brownie or are you going to want larger
chunks?
Teachers’ Perspectives on Ordering Children’s Strategies
for Equal Sharing Problems
Strategy
Feature
Role of
leftovers
Dominant Perspective Expressed by Teacher Groups
Less Sophisticated
More Sophisticated
# of Groups
Expressing
Perspective
(n = 7)
Ignoring leftovers
v
Partitioning leftovers in some
way
7
Use of
representation
Using pictures to represent
all or some items
v
Partitioning mentally without
representing items
6
Distribution
of wholes
Partitioning all items before
any distribution
v
Distributing as many wholes
as possible
before
Role
of the number
partitioning
extras
of sharers in
5
v
partitions
Partitioning
items into large
parts regardless of number of
sharers
4
Size of parts
Form of final
answer
Partitioning items using the
number of sharers
Not expressing answer in
simplest form
(e.g., improper fraction)
!
Use of fraction terms
Expressing
answer in
and symbols
simplest form
7
Teachers’ Strategy Feature: Form of Final Answer
Daniel
Samantha
T1: So, since it’s 4/6, [Samantha’s strategy] isn’t as
high as [Daniel’s] 2/3.
T2: Do we need to look at the description? T3: No. We like simplifying the best.
Teachers’ Perspectives on Ordering Children’s Strategies
for Equal Sharing Problems
Strategy
Feature
Role of
leftovers
Dominant Perspective Expressed by Teacher Groups
Less Sophisticated
More Sophisticated
# of Groups
Expressing
Perspective
(n = 7)
Ignoring leftovers
v
Partitioning leftovers in some
way
7
Use of
representation
Using pictures to represent
all or some items
v
Partitioning mentally without
representing items
6
Distribution
of wholes
Partitioning all items before
any distribution
v
Distributing as many wholes
as possible
before
Role
of the number
partitioning
extras
of sharers in
5
v
partitions
Partitioning
items into large
parts regardless of number of
sharers
4
Size of parts
Partitioning items using the
number of sharers
Form of final
answer
Not expressing answer in
simplest form
(e.g., improper fraction)
v
Use of fraction terms
Expressing
answer in simplest
and symbols
form
7
Salient Features of Equal Sharing Strategies
Teachers
Research Framework
(Empson & Levi, 2013)
q  Role of leftovers
q  Role of leftovers
q  Use of
representation
q  Distribution of
wholes
q  Size of parts
q  Form of final answer
q  Use of
Hiro
representation
q  Role of partitioning
based on the
number of sharers
q  Use of fraction
words and notations
Salient features: Similarities and differences
Focus: Individual features vs. set of features
Daniel
Samantha
T1: So, since it’s 4/6, [Samantha’s strategy] isn’t as high
as [Daniel’s] 2/3.
T2: Do we need to look at the description? T3: No. We like simplifying the best.
Hiro
Samantha
T1: Samantha is more sopisticated than Hiro because she could see
the whole pieces.
T2: Yeah, Samantha could see the wholes and then was able to
divide up the parts.
IMPLICATIONS
Researchers and Professional Developers
q  Honor teachers’ knowledge of children’s mathematical
understandings
o  Understand teachers’ starting points
o  Understand teachers’ potential confusions
Teachers
q  Recognize their own initial understandings when making
sense of children’s strategies
FUTURE RESEARCH
Problem Types
q  Equal sharing problems with varied number choices
q  Other types of fraction story problems
Methods
q  Individual teachers
q  Interviews to further probe teachers’ statements
Questions & Discussion