Transcript Compared to What? How Different Types of Comparison Effect

When it pays to compare:
Benefits of comparison in
mathematics classrooms
Bethany Rittle-Johnson
Jon R. Star
Common Ground:
Comparison
• Cognitive Science: A fundamental
learning mechanism
– This symposium!
• Mathematics Education: A key
component of expert teaching
Comparison in Mathematics
Education
– Compare solution methods
– “You can learn more from solving one
problem in many different ways than you
can from solving many different problems,
each in only one way”
–
(Silver, Ghousseini, Gosen, Charalambous, & Strawhun, p. 288)
Compare Solution Methods
• Expert teachers do it (e.g. Lampert, 1990)
• Reform curriculum advocate for it (e.g.
NCTM, 2000; Fraivillig, Murphy & Fuson, 1999)
• Teachers in higher performing countries
help students do it (Richland, Zur & Holyoak,
2007)
Does comparison support
mathematics learning?
• Experimental studies on comparison in K-12
academic domains and settings largely
absent
• Goals of initial work
– Investigate whether comparing solution methods
facilitates learning in middle-school classrooms
• 7th graders learning to solve equations
• 5th graders learning about computational estimation
Studies 1 & 2
• Compare condition: Compare and
contrast alternative solution methods vs.
• Sequential condition: Study same
solution methods sequentially
Rittle-Johnson, B. & Star, J.R. (2007). Does comparing solution methods
facilitate conceptual and procedural knowledge? An experimental study
on learning to solve equations. Journal of Educational Psychology.
Compare Condition
Equation Solving
Sequential Condition
Predicted Outcomes
• Students in compare condition will make
greater gains in:
– Procedural knowledge, including
• Success on novel problems
• Flexibility of procedures (e.g. select efficient
procedures; evaluate when to use a procedure)
– Conceptual knowledge (e.g. equivalence)
Study 1 Method
• Participants: 70 7th-grade students and their math
teacher
• Design:
– Pretest - Intervention - Posttest
– Replaced 2 lessons in textbook
– Intervention occurred in partner work during 2 1/2 math
classes
Randomly assigned to
Compare or Sequential
condition
Studied worked examples
with partner
Solved practice problems
on own
Knowledge Gains
45
40
35
30
25
20
15
10
5
Compare
Sequential
40
Post - Pre Gain Score
Post - Pre Gain Score
45
Compare
Sequential
35
30
25
20
15
10
5
0
Familiar
F(1, 31) =4.49, p < .05
Novel
Equation Solving
0
Flexiblity
F(1,31) = 7.73, p < .01
Compare condition made greater gains in procedural
knowledge and flexibility; Comparable gains in conceptual
knowledge
Study 2:
Helps in Estimation Too!
• Same findings for 5th graders learning
computational estimation (e.g. About
how much is 34 x 18?)
– Greater procedural knowledge gain
– Greater flexibility
– Similar conceptual knowledge gain
Summary of Studies 1 & 2
• Comparing alternative solution methods
is more effective than sequential
sharing of multiple methods
– In mathematics, in classrooms
My Own Comparison of the
Literatures
• Comparing the cognitive science and
mathematics education literatures
highlighted a potentially important
dimension:
– What is being compared?
Study 3:
Compared to What?
Solution Methods
Problem Types
Surface Features
Compared to What?
• Mathematics Education - Compare solution
methods for the same problem
• Cognitive Science - Compare surface
features of different examples with the same
solution or category structure
– e.g., Dunker’s radiation problem: Providing a solution in 2
stories with different surface features, and prompting for
comparison, greatly increased spontaneous transfer of the
solution (Gick & Holyoak, 1980; 1983; Catrambone & Holyoak, 1989)
– e.g., Providing two exemplars of a novel spatial relation
greatly increased extension of the label to a new exemplar
(Gentner, Christie & Namy)
Similarity May Matter
• Comparing moderately similar examples
is better (Gick & Paterson, 1992; VanderStoep &
Seiffert, 1993)
– But: Comparing highly similar examples is
sometimes better (Reed, 1989; Ross & Kilbane,
1997)
– Comparing highly similar examples can
facilitate success with less similar
examples (Kotovsky & Gentner, 1996; Gentner, Christie
& Namy)
Study 3:
Compared to What?
Solution Methods
•
(M = 3.8 on scale from 1 to 9)
Problem Types
•
(M = 6.6)
Surface Features
•
(M = 8.3)
Predicted Outcomes
• Moderate similarity/dissimilarity is best,
so Compare Solution Methods and
Compare Problem Types groups will
outperform compare surface features
group.
– But, students with low prior knowledge may
benefit from high similarity, and thus learn
more in compare surface features
condition.
Study 3 Method
• Participants: 163 7th & 8th grade students
from 3 schools
• Design:
– Pretest - Intervention - Posttest - Retention
– Replaced 3 lessons in textbook
– Randomly assigned to
• Compare Solution Methods
• Compare Problem Types
• Compare Surface Features
– Intervention occurred in partner work
Conceptual Knowledge
60
Estimated Marginal Mean
55
50
45
40
35
30
Surface
Problems
Methods
Compare Condition
Compare Solution Methods condition made greatest
gains in conceptual knowledge F (2, 154) = 6.10, p = .003)
Flexibility: Flexible
Knowledge of Procedures
65
Estimated Marginal Mean
60
55
50
45
40
35
30
Surface
Problems
Methods
Condition
Solution Methods > Problem Type > Surface Feature
F (2, 154) = 4.95, p = .008)
Flexibility: Use of Efficient
Procedures
Estimated Marginal Mean Use
60
55
50
45
40
35
30
25
20
Surface
Problems
Methods
Compare Condition
Greater use of more efficient solution methods in Compare
Methods and Problem Types conditions
F (2, 135) = 3.35, p = .038)
Procedural Knowledge
Estimated Marginal Mean Use
60
55
50
45
40
35
30
Surface
Problems
Methods
Compare Condition
No effect of condition on familiar or transfer equations
But…
Procedural Knowledge and Prior
Knowledge
Posttest performance depended on prior conceptual knowledge
Explanation Characteristics
• Explanations offered during the intervention:
• Very similar for Compare Solution Methods
and Problem Types:
– Mostly focus on solution methods, and often on
multiple methods
– Most common comparison is of solution steps
– Evaluations usually focus on efficiency of methods
• Compare Surface Features
– more likely to focus on and to compare problem
features
– Evaluations are rare
Summary
• Comparing Solution Methods often
supported the largest gains in
conceptual knowledge and flexibility.
– Comparing Problem Types sometimes as
effective for flexibility.
• However, students with low prior
knowledge may learn equation solving
procedures better from Comparing
Surface Features
Conclusion
• Comparison is an important learning
activity in mathematics
• Careful attention should be paid to:
– What is being compared
– Who is doing the comparing - students’
prior knowledge may matter
Acknowledgements
• For slides, papers or more information,
contact: [email protected]
• Funded by a grant from the Institute for
Education Sciences, US Department of
Education
• Thanks to research assistants at Vanderbilt:
– Holly Harris, Jennifer Samson, Anna Krueger, Heena Ali, Sallie
Baxter, Amy Goodman, Adam Porter, John Murphy, Rose Vick,
Alexander Kmicikewycz, Jacquelyn Beckley and Jacquelyn Jones
• And at Michigan State:
– Kosze Lee, Kuo-Liang Chang, Howard Glasser, Andrea Francis,
Tharanga Wijetunge, Beste Gucler, and Mustafa Demir
Two Equation Solving
Procedures
Method 1
Method 2
3(x + 1) = 15
3(x + 1) = 15
3x + 3 = 15
x+1=5
3x = 12
x=4
x=4
Why Equation Solving?
• Students’ first exposure to abstraction and
symbolism of mathematics
• Area of weakness for US students
– (Blume & Heckman, 1997; Schmidt et al., 1999)
• Multiple procedures are viable
– Some are better than others
– Students tend to learn only one method
Procedural Knowledge
Assessments
• Equation Solving
– Intervention: 1/3(x + 1) = 15
– Posttest Familiar: -1/4 (x – 3) = 10
– Posttest Novel: 0.25(t + 3) = 0.5
• Flexibility
– Solve each equation in two different ways
– Looking at the problem shown above, do you think that this
way of starting to do this problem is a good idea? An ok step
to make? Circle your answer below and explain your
reasoning.
(a) Very good
way
(b)
Ok to do, but not a very
good way
(c) Not OK to do
Conceptual Knowledge
Assessment