nctm presession 2005 1 research issues in developing strategic flexibility: what and how presenters:...

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


Symposium Overview

Audience Task

Introduction

Description of Projects MSU Project UD Project

Challenges of Researching Strategic Flexibility

Discussant’s Comments

Audience Feedback


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.


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]


What counts as different?

Different number of steps 3 lines? 4 lines?

Different sequence of steps Distribute first? Divide first?

Other characteristics?


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…]


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?


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


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


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)


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)


Terms in this Talk

Variations in terms across two projects MSU:• Flexibility• Appropriate or Best

UD:• Strategic Flexibility• Wise/Unwise


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?


MSU Project Description


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


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


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)


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


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


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


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


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?


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


UD Project Description


UD Project Associates

Research Collaborators: Jim Hiebert Yuichi Handa

Instructional Collaborators: Eric Sisofo James Beyers Laurie Goggins


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


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.


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).


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


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:


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.


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


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


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


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


Challenges of Researching Strategic Flexibility


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?


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


Addressing Challenge #1: What strategies are different?

MSU: Different sequence of steps

UD: Different number relationships used


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


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”


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


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”


Reasons for our Choices

MSU: Least mental effort Efficiency/Elegance (Disciplinary values)

UD: Least mental effort Accuracy Conceptual illumination


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



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


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



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


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


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.


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]


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


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

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