Strategy Flexibility Matters for
Student Mathematics
Achievement: A Meta-Analysis
Kelley Durkin
Bethany Rittle-Johnson
Vanderbilt University, United States
Jon R. Star
Harvard University, United States
Defining Strategy Flexibility
• Simplest definition:
Knowing more than one strategy for solving a
particular type of problem (e.g., Heirdsfield & Cooper, 2002)
• Most complex definition:
Being able to use a variety of strategies and
information from the problem context, the
learner’s environment, and the sociocultural
context to select the most appropriate problem
solving procedure (e.g., Verschaffel, Luwel, Torbeyns, & Van
Dooren, 2007)
2
Recent Focus on Strategy Flexibility
• Previously, flexibility rarely measured as an
instructional outcome (Star, 2005).
• Standardized tests in the U.S. include sections on:
– Concepts
– Procedures
– Problem solving
– But not flexibility
• Recently, flexibility examined as a separate
outcome (Star, 2007; Verschaffel et al., 2007).
3
Importance of Strategy Flexibility
• Helps adapt existing procedures to unfamiliar
problems (e.g., Blöte, Van der Burg, & Klein, 2001)
• Greater understanding of domain concepts (e.g.,
Hiebert & Wearne, 1996)
• Crucial component of expertise in problem
solving (Dowker, 1992; Dowker, Flood, Griffiths, Harris, & Hook, 1996;
Star & Newton, 2009)
4
Current Study
• Is strategy flexibility related to other
mathematical constructs?
– Conceptual Knowledge
• Success recognizing and explaining key domain
concepts (Carpenter et al., 1998; Hiebert & Wearne, 1996)
– Procedural Knowledge
• Success executing action sequences to solve problems
(Hiebert & Wearne, 1996; Rittle-Johnson, Siegler, & Alibali, 2001)
– General Mathematics Achievement
• Meta-analysis of our past work
5
Our Definition of Strategy
Flexibility
• Knowing multiple strategies and their relative
efficiencies (Flexibility Knowledge)
AND
• Adapting strategy choice to specific problem
features (Flexible Use)
(e.g., Blöte et al., 2001; National Research Council, 2001; Rittle-Johnson &
Star, 2007)
6
Method Overview
• Selected Studies
• Measures
• Analysis Strategies
7
Included Studies
Study Authors
Rittle-Johnson & Star
Year
2007
Topic
N
Equation Solving 70
Grade
7
Star & Rittle-Johnson
2008
Equation Solving 155
6
Rittle-Johnson & Star
2009
Equation Solving 162
7&8
Rittle-Johnson, Star, &
Durkin
Star et al.
2009
Equation Solving 236
7&8
2009
Estimation
65
5
Star & Rittle-Johnson
2009
Estimation
157
5&6
Rittle-Johnson, Star, &
Durkin
Schneider, Rittle-Johnson &
Star
2011
Equation Solving 198
8
2011
Equation Solving 293
7&8
8
Measures
•
•
•
•
•
Flexibility Knowledge
Flexible Use
Conceptual Knowledge
Procedural Knowledge
Standardized Tests
9
Measures
• Flexibility Knowledge
– Knowing multiple procedures and the relative
efficiency of the procedures
5(x + 3) + 6 = 5(x + 3) + 2x
6 = 2x
a. What step did the student use to get from the first line to
the second line?
b. Do you think that this is a good way to start this problem?
Circle One:
(a) a very good way
(b) OK to do, but not a very good way
(c) Not OK to do
c. Explain your reasoning.
10
Measures
• Flexible Use
– Students using the most appropriate strategy
depending on problem features
3(h + 2) + 4(h + 2) = 35
7(h + 2) = 35
• Sometimes know a more appropriate strategy
for solving a problem before actually use it
(Blöte et al., 2001; Siegler & Crowley, 1994)
11
Measures
• Conceptual Knowledge
– Ability to recognize and explain key domain concepts
Which of the following is a like term to (could be
combined with) 7(j + 4)?
(a) 7(j + 10) (b) 7(p + 4) (c) j (d) 2(j + 4) (e) a and d
• Procedural Knowledge
– Ability to execute action sequences to solve problems
3(h + 2) + 4(h + 2) = 35
12
Measures
• Standardized Tests
National Tests
• Comprehensive Testing Program (CTP)
• Measures of Academic Progress (MAP)
State Tests
• Massachusetts Comprehensive Assessment System (MCAS)
• Tennessee Comprehensive Assessment Program (TCAP)
• Collected scores from school records
13
Coding and Analysis Strategies
• Calculated correlation between each pair of
outcomes for each study
• Fischer’s z to transform correlations to get effect
sizes, ESr, for each study (Lipsey & Wilson, 2001).
• The mean correlation effect size was calculated using
a random effects model.
14
Results
• Mean correlations between outcomes
Flexibility
Knowledge
Flexible
Use
Conceptual
Knowledge
Flexibility
Knowledge
Flexible Conceptual
Use
Knowledge
Procedural Standardized
Knowledge Test
1
.635
.563
.610
.535
1
.541
.627
.404
1
.544
.520
1
.475
Procedural
Knowledge
Note: All correlations were significant (p < .001)
15
Results
• Mean correlations between outcomes
Flexibility
Knowledge
Flexible
Use
Conceptual
Knowledge
Flexibility
Knowledge
Flexible Conceptual
Use
Knowledge
Procedural Standardized
Knowledge Test
1
.635
.563
.610
.535
1
.541
.627
.404
1
.544
.520
1
.475
Procedural
Knowledge
Note: All correlations were significant (p < .001)
16
Results
• Mean correlations between outcomes
Flexibility
Knowledge
Flexible
Use
Conceptual
Knowledge
Flexibility
Knowledge
Flexible Conceptual
Use
Knowledge
1
.635
.563Flexibility knowledge
.610
.535
and
flexible
use strongly related
1
.541
.627
.404
1
.544
.520
1
.475
Procedural
Knowledge
Note: All correlations were significant (p < .001)
Procedural Standardized
Knowledge Test
17
Results
• Mean correlations between outcomes
Flexibility
Knowledge
Flexible
Use
Conceptual
Knowledge
Flexibility
Knowledge
Flexible Conceptual Procedural Standardized
Use
Knowledge Knowledge Test
1
.635
.563
.610
.535
1
.541
.627
.404
1
.544
.520
1
.475
Procedural
Knowledge
Note: All correlations were significant (p < .001)
18
Results
• Mean correlations between outcomes
Flexibility
Knowledge
Flexibility
Knowledge
Flexible
Use
Conceptual
Knowledge
Flexible Conceptual Procedural Standardized
Use
Knowledge Knowledge Test
1 Conceptual .635
knowledge had
moderately strong
relationships to
1
flexibility
.563
.610
.535
.541
.627
.404
1
.544
.520
1
.475
Procedural
Knowledge
Note: All correlations were significant (p < .001)
19
Results
• Mean correlations between outcomes
Flexibility
Knowledge
Flexibility
Knowledge
Flexible
Use
Conceptual
Knowledge
Procedural
Knowledge
Flexible Conceptual Procedural Standardized
Use
Knowledge Knowledge Test
1 Conceptual .635
knowledge had
moderately strong
relationships to
1
flexibility
Procedural
knowledge had
moderately strong
relationships to
flexibility
.563
.610
.535
.541
.627
.404
1
.544
.520
1
.475
Note: All correlations were significant (p < .001)
20
Results
• Mean correlations between outcomes
Flexibility
Knowledge
Flexible
Use
Conceptual
Knowledge
Procedural
Knowledge
Flexibility
Knowledge
Flexible Conceptual Procedural Standardized
Use
Knowledge Knowledge Test
1
.635
.563
.610
.535
1
.541
.627
.404
.544
.520
1
.475
Similar to correlation 1
between conceptual
and procedural
knowledge
Note: All correlations were significant (p < .001)
21
Results
• Mean correlations between outcomes
Flexibility
Knowledge
Flexible
Use
Conceptual
Knowledge
Flexibility
Knowledge
Flexible Conceptual Procedural Standardized
Use
Knowledge Knowledge Test
1
.635
.563
.610
.535
1
.541
.627
.404
1
.544
.520
1
.475
Procedural
Knowledge
Note: All correlations were significant (p < .001)
22
Results
• Mean correlations between outcomes
Flexibility
Knowledge
Flexible
Use
Conceptual
Knowledge
Flexibility
Knowledge
Flexible Conceptual Procedural Standardized
Use
Knowledge Knowledge Test
1
.635
.563test measures
.610
Standardized
significantly correlated with
flexibility
1
.541
.627
1
Procedural
Knowledge
Note: All correlations were significant (p < .001)
.535
.404
.544
.520
1
.475
23
Results
• Mean correlations between outcomes
Flexibility
Knowledge
Flexible
Use
Conceptual
Knowledge
Flexibility
Knowledge
Flexible Conceptual Procedural Standardized
Use
Knowledge Knowledge Test
1
.635
.563test measures
.610
Standardized
significantly correlated with
flexibility
1
.541
.627
Standardized test measures
significantly correlated with
other outcomes
1
.544
Procedural
Knowledge
Note: All correlations were significant (p < .001)
1
.535
.404
.520
.475
24
Results
• Mean correlations between outcomes
Flexibility
Knowledge
Flexible
Use
Conceptual
Knowledge
Procedural
Knowledge
Flexibility
Knowledge
Flexible Conceptual Procedural Standardized
Use
Knowledge Knowledge Test
1
.635
.563test measures
.610
Standardized
significantly correlated with
flexibility
1
.541
.627
Standardized test measures
significantly correlated with
other outcomes
1
.544
Correlations between flexibility
and standardized tests 1similar
to other correlations
Note: All correlations were significant (p < .001)
.535
.404
.520
.475
25
3 Main Findings
• Flexibility knowledge and flexible use are
separate constructs
• Flexibility is related to other constructs
• Standardized tests relate to flexibility as well
as they relate to other constructs
26
The Construct of Strategy Flexibility
• May be important to measure flexible use and
flexibility knowledge separately.
– Appears measures of knowledge and use are
tapping different aspects of flexibility.
• Conceptual and procedural knowledge are
related to flexibility (Schneider et al., 2011).
27
Relation to Standardized Tests
• Standardized test scores relate to flexibility just as
well as they relate to conceptual and procedural
knowledge.
• Teachers can feel pressured to teach to the test,
and the lack of flexibility items on assessments
could lead to less time on flexibility in the
classroom.
• Push for standardized tests to include items that
assess flexibility.
• Flexibility a valued outcome when evaluating
interventions.
28
Conclusion
• Strategy flexibility is important for developing
expertise and efficient problem solving
• Need to measure and encourage students’
strategy flexibility in the future
29
Acknowledgements
• E-mail: kelley.durkin@vanderbilt.edu
• Visit our Contrasting Cases Website at
http://gseacademic.harvard.edu/contrastingcases/index
.html for more information
• Thanks to the Children’s Learning Lab at Vanderbilt
University
• Funded by a grant from the Institute for Education
Sciences, U.S. Department of Education
– The opinions expressed are those of the authors and do not
represent views of the U.S. Department of Education.
30
30
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