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NCTM Presession 2005 1
Research Issues inDeveloping Strategic Flexibility:
What and How
Presenters:Christine Carrino Gorowara University of
DelawareDawn Berk University of DelawareChristina Poetzl University of DelawareJon R. Star Michigan State UniversitySusan B. Taber Rowan University
Discussant:John K. Lannin University of Missouri-
Columbia
NCTM Presession 2005 2
Symposium Overview
Audience Task
Introduction
Description of Projects MSU Project UD Project
Challenges of Researching Strategic Flexibility
Discussant’s Comments
Audience Feedback
NCTM Presession 2005 3
Audience Task
Problem #1: Find x if 4(x + 5) = 80.
Problem #2: Joan drove 200 miles in 3.5 hours. How far can she drive in 14 hours?
For each problem—
Solve, using any strategy you like. Solve again, using a different strategy. Determine which strategy is better, and why. Try to change the problem so that the other
strategy is now better.
NCTM Presession 2005 4
Problem #1: Find x if 4(x + 5) = 80
4(x + 5) = 80
x + 5 = 20
x = 15
4(x + 5) = 79
x + 5 = 79/4
x = 59/4
Strategy 1:
Strategy 2:
Strategy 1 with changed problem:
Strategy 2 with changed problem:
4(x + 5) = 80
4x + 20 = 80
4x = 60
x = 15
4(x + 5) = 79
4x + 20 = 79
4x = 59
x = 59/4
[Find x if 4(x + 5) = 79]
NCTM Presession 2005 5
What counts as different?
Different number of steps 3 lines? 4 lines?
Different sequence of steps Distribute first? Divide first?
Other characteristics?
NCTM Presession 2005 6
Problem #2: Joan drove 200 miles in 3.5 hours. How far can she drive in 14 hours?
? x 3.5 hours = 14 hours
4 x 3.5 hours = 14 hours
4 x 200 miles = 800 miles
Joan drives 200/3.5 mph.
In 14 hours, she drives 14 x 200/3.5 = 800 miles.
? x 5 = 14 hours
14/5 x 5 = 14 hours
14/5 x 200 miles = 560 miles
Strategy 1:
Strategy 2:
Strategy 1, with changed problem:
Strategy 2, with changed problem:
Joan drives 200/5 mph, or 40 mph.
In 14 hours, she drives 14 x 40 miles, or 560 miles.
[Joan drove 200 miles in 5 hours…]
NCTM Presession 2005 7
What counts as different?
Different number of steps Different multiplicative relationships used
Scale factor between two sets of hours vs. scale factor between hours and miles
Other characteristics?
NCTM Presession 2005 8
Introduction
Many problems can be solved with a variety of strategies
Important goal for students is to develop flexibility in the use of strategies, which means that they: Know multiple strategies for solving a class of
problems Select from among those strategies the most
appropriate for solving a particular problem Audience task demonstrated your flexibility
NCTM Presession 2005 9
Defining Flexibility
Proficiency in executing a range of strategies
AND Ability and disposition to choose wisely
among those strategies with respect to a particular goal
NCTM Presession 2005 10
Related Work
Our notion of “flexibility” is related to but distinct from other terms Adaptive expertise (Baroody & Dowker)
Procedural fluency, strategic competence (NRC, 2001)
Conceptual knowledge, procedural knowledge (Hiebert, 1986)
Informed by research on: Strategy development in developmental psychology
(e.g., Siegler) Problem solving (e.g., Schoenfeld, Silver)
Strategy choice in arithmetic (e.g., Baroody, Fuson, Carpenter)
NCTM Presession 2005 11
Importance of Flexibility
When flexible, students are more successful on transfer problems (e.g., Resnick, 1980; Schwartz & Martin, 2004; Carpenter et al, 1998)
When not flexible, teachers are less likely to promote flexibility in their students (Hines and McMahon, 2005)
NCTM Presession 2005 12
Terms in this Talk
Variations in terms across two projects MSU:• Flexibility• Appropriate or Best
UD:• Strategic Flexibility• Wise/Unwise
NCTM Presession 2005 13
Challenges (Preview)
#1: What strategies are different?
#2: What strategies are “best”?
#3: How can we tell when a student is flexible?
#4: How do we develop strategic flexibility?
NCTM Presession 2005 14
MSU Project Description
NCTM Presession 2005 15
MSU Project Team
PI: Jon Star Graduate research assistants at MSU:
Howard Glasser Mustafa Demir Kosze Lee Beste Gucler Kuo-Liang Chang
Collaborator: Bethany Rittle-Johnson, Vanderbilt University
NCTM Presession 2005 16
My Research Paradigm
Work with students with minimal knowledge of strategies in problem class
Provide brief instruction with no worked-out examples and no strategic instruction
Provide minimal feedback Observe what strategies develop Conduct problem solving interviews to explore
rationales behind strategy choices Implement and evaluate instructional
interventions
NCTM Presession 2005 17
Instructional Interventions
Alternative ordering task Students asked to re-solve previously completed
problems using a different ordering of steps (Star, 2001; 2002)
Explicit strategy instruction Strategy instruction is provided after students have
achieved basic fluency and differentiated domain knowledge (Schwartz & Bransford,1998)
NCTM Presession 2005 18
Method
134 6th graders (83 girls, 51 boys) 5 1-hour classes in one week (Mon - Fri) Class size 8 to 15 students; worked individually Pre-test (Mon), post-test (Fri) Domain was linear equation solving
3(x + 1) = 12
2(x + 3) + 4(x + 3) = 24
9(x + 2) + 3(x + 2) + 3 = 18x + 9
NCTM Presession 2005 19
Instruction
30 minute benchmark lesson Combine like terms, add to both sides, multiply to both
sides, distribute How to use each step individually
Not shown how to chain together steps No strategic or goal-oriented instruction
Not told when to use a step No worked examples of solved equations
NCTM Presession 2005 20
Alternative Ordering Treatment
Random assignment by class AO treatment vs. AO control
Solve this problem again, but using a different ordering of steps
AO control group solved new but isomorphic problem3(x + 1) = 12
4(x + 2) = 24
NCTM Presession 2005 21
Explicit Strategy Instruction
Random assignment by class Strategy instruction vs. no strategy instruction
At start of 2nd problem solving class, 3 worked examples presented to strategy instruction classes “This is the way I solve this equation.” Problems solved with atypical, ‘better’ strategy No notes taken by students
Total time was 8 minutes of supplemental instruction
NCTM Presession 2005 22
Assessing Flexibility
When I work on equations, I always use the same steps, in the same order (true/false)
Figure out ALL possible NEXT steps that can be done on2(x + 3) + 6 + 4x + 8 = 4(x + 2) + 6x + 2x
Use the “combine like terms” step on2(x + 1) + 5(x + 1) = 14
Given this partially solved equation, what step did the student use to go from the first line to the second line?
NCTM Presession 2005 23
Results
AO Treatment students significantly more flexible 54% treatment 41% control
Strategy instruction led to significantly more flexibility 53% strategy instruction 45% no strategy instruction
No significant interaction effect
NCTM Presession 2005 24
UD Project Description
NCTM Presession 2005 25
UD Project Associates
Research Collaborators: Jim Hiebert Yuichi Handa
Instructional Collaborators: Eric Sisofo James Beyers Laurie Goggins
NCTM Presession 2005 26
Context of the Study
Domain: Missing-value proportion problems
Participants: Pre-service K-8 teachers (n = 148) Familiar with missing-value proportion problems Many identified cross-multiplication as THE strategy for
solving missing-value proportion problems
Setting: Mathematics content course Semester-long focus on rational number concepts 4 proportional reasoning lessons taught at end of
semester
NCTM Presession 2005 27
Example of Students’ ThinkingIt takes 9 minutes to read 10 pages. How many minutes will it take to read 500 pages?
The criteria the group used to decide "best" strategy was instructive. They seemed to choose strategies that had a more formal appearance or ones that were similar to what someone remembered having learned before.
In solving one problem, a student suggested early in the conversation that:
because 10 pages would take 9 minutes to read, and because 500 pages is 50 times that number of pages, 500 pages should take 50 times longer to read, and 50 x 9 = 450, so it would take 450 minutes to read 500
pages.
NCTM Presession 2005 28
Example of Students’ Thinking (cont’d)
After the student explained this several times so the others understood, the group wondered whether this was right. They had convinced themselves that the answer was right but wondered whether this is what they should be doing. "I'm not sure this is right because I don't think it's even a method." "Yeah, it seems too easy." "I think we should do it another way.”
The group ended up writing a proportion and using cross-multiplication, congratulating the student who thought of this because they all agreed this looked much better. They were surprised to find they got the same answer both ways (even though they seemed convinced that the first strategy had given them the right answer).
NCTM Presession 2005 29
Instructional Goals
Develop a recognition of, appreciation for, and ability to use multiple strategies Cross-multiplication Unit rate Scale factor Scaling up/down
Develop an ability to analyze a given problem and choose “wisely” from among a range of strategies Both computational and conceptual benefits
NCTM Presession 2005 30
ExamplesIt takes 9 minutes to read 10 pages. How many minutes will it take to read 500 pages?
x 50 x 50500 pages: x minutes
Scale Factor Strategy:10 pages: 9 minutes
500 pages: 450 minutes
= 450 minutes
500 pages x 9/10 minutes per page
9/10 minutes per (1) page Unit Rate Strategy:
NCTM Presession 2005 31
Instructional Design
Multiple strategies were identified and named.
Students were asked to compare and contrast strategies for solving a given problem.
Students were encouraged to choose strategies that capitalized on particular number relationships in the problem.
NCTM Presession 2005 32
Data Collection
Pre/Post Tests (n = 148), Delayed Post Test (n = 53) 6 items: Solve using 1 strategy 2 items: Solve using 2 strategies
Pre/Post Interviews (n = 22) 4 items: Solve using 1 strategy 1 item: Solve, given first step of strategy 1 item: Choose “best” among 3 worked
solutions
NCTM Presession 2005 33
Coding Scheme
Correctness Three measures of flexibility
Number of strategies used across problems
“Wise” choice of strategy on a given problem
Ability to solve same problem using multiple strategies
NCTM Presession 2005 34
Results
Significant increase in:
Number of problems solved correctly
Number of strategies used across all problems
Number of problems on which a “wise” strategy was used
Use of multiple strategies on a given problem
NCTM Presession 2005 35
Comparing our Projects
MSU UDInstructional
GoalsNo discussion of strategy choice
“Wise” use an explicit focus
Populations 6th-graders Prospective teachers
Students’ Prior
Knowledge
No prior instruction on symbolic approaches
Competence in domain
Students’ Prior Values
No knowledge of established values
Value for “formal” strategies
NCTM Presession 2005 36
Challenges of Researching Strategic Flexibility
NCTM Presession 2005 37
Challenges
#1: What strategies are different?
#2: What strategies are “best”?
#3: How can we tell when a student is flexible?
#4: How do we develop strategic flexibility?
NCTM Presession 2005 38
Challenge #1: What strategies are different? Different ways of representing the solution (e.g.,
graphically vs. symbolically)
Number of lines/steps
Different sequence of steps
Steps ‘chunked’ vs. not ‘chunked’
Different structural elements (e.g., number relationships) used
NCTM Presession 2005 39
Addressing Challenge #1: What strategies are different?
MSU: Different sequence of steps
UD: Different number relationships used
NCTM Presession 2005 40
Joan drove 200 miles in 3.5 hours. How far can she drive in 14 hours?
145.3200 x=
4
4
Joan will drive 4 times as many hours.
This means she can drive 4 times as many miles, or 4 x 200 = 800 miles
Joan drove 200/3.5 miles in one hour
This means she can drive 14 x 200/3.5 miles in 14 hours. 145.3
200 x=
14 (200) = 3.5 (x) 145.3200 x=
Scale Factor
Cross-Multiplication
Unit Rate
200/3.5
200/3.5
NCTM Presession 2005 41
Reasons for our Choices MSU: Different sequence of steps
Relatively simple way to begin looking at multiple strategies
Bottom-up classification of “different”
UD: Different number relationships Different computations result Different concepts about proportions are
illuminated (e.g.,that the scale factors are equal, that the unit rates are equal, that the cross-products are equal)
Top-down classification of “different”
NCTM Presession 2005 42
Challenge #2:
What strategies are “best”? For what purpose(s)?
Speed Accuracy Generalizability Preference Elegance Conceptual illumination
For whose purpose(s)? Learner Instructor/Researcher Discipline
NCTM Presession 2005 43
Addressing Challenge #2:
What strategies are “best”?
MSU: Strategies with fewer steps Strategies with low cognitive load Bottom-up classification of “best”
UD: Strategies taking advantage of simple (whole-number)
multiplicative relationships between related values Top-down classification of “best”
NCTM Presession 2005 44
Reasons for our Choices
MSU: Least mental effort Efficiency/Elegance (Disciplinary values)
UD: Least mental effort Accuracy Conceptual illumination
NCTM Presession 2005 45
Challenge #3:
How can we tell when a student is flexible?
Competence vs. Performance: False Negative? Lack of variety in strategies does not necessarily
indicate an inability to use multiple strategies
Compliance vs. Disposition: False Positive? Greater variety of strategies following instruction
does not necessarily indicate a disposition to be flexible
NCTM Presession 2005 46
Addressing Challenge #3:
How can we tell when a student is flexible?
MSU: Competence vs. Performance: Creative
assessments Compliance vs. Disposition: Not so much of an
issue UD:
Compliance vs. Disposition: Assessments over time
Competence vs. Performance: Not so much of an issue
NCTM Presession 2005 47
Challenge #4:
How do we develop strategic flexibility? Awareness of other strategies and competence
in executing other strategies are necessary, but not sufficient
Prior knowledge may play a role
NCTM Presession 2005 48
Addressing Challenge #4:
How do we develop strategic flexibility?
Draw attention to strategy choice…
Manipulate problem features Problems must be complex enough to have multiple
different solutions, yet simple enough so that the students can solve them
Structure of problem and number choice should highlight appropriateness of various strategies
NCTM Presession 2005 49
Addressing Challenge #4 (continued):
How do we develop strategic flexibility?
Draw attention to strategy choice…
Name strategies or strategy elements Legitimizing effect
• "I'm not sure this is right because I don't think it's even a method."
Reifying effect• For example: performance of students in AO treatment
NCTM Presession 2005 50
Closing Thoughts
Studying strategic flexibility involves making a set of judgments about what counts as different and what counts as best.
Measuring strategic flexibility involves measuring students’ perceptions, motivations, and intentions in addition to strategy choice, and/or measuring strategy choice over time.
Developing strategic flexibility involves shifting students’ focus from the solution product to the solution process.
NCTM Presession 2005 51
Thank You! Christine Carrino Gorowara [email protected]
Dawn Berk [email protected]
Christina Poetzl [email protected]
Jon R. Star [email protected]
Susan B. Taber [email protected]
NCTM Presession 2005 52
What counts as different?[Examples with Original Problem]
Different relationships between the given values being used:
Strategy 1 uses the relationship between number of hours already driven and number of hours to be driven.
4 (# hours already driven) = # hours to be driven, so
4 (# miles already driven) = # miles to be driven
Strategy 2 uses the relationship between number of hours already driven and number of miles already driven.
200/3.5 (# hours already driven) = # miles already driven, so
200/3.5 (# hours to be driven) = # miles to be driven
NCTM Presession 2005 53
What counts as different?[Examples with Changed Problem]
Different relationships between the given values being used:
Strategy 1 uses the relationship between number of hours already driven and number of hours to be driven.
14/5 (# hours already driven) = # hours to be driven, so
14/4 (# miles already driven) = # miles to be driven
Strategy 2 uses the relationship between number of hours already driven and number of miles already driven.
40 (# hours already driven) = # miles already driven, so40 (# hours to be driven) = # miles to be driven