Transcript Compared to What? How Different Types of Comparison Effect

Compared to What?
How Different Types of
Comparison Affect
Transfer in Mathematics
Bethany Rittle-Johnson
Jon Star
What is Transfer?
• Transfer
– “Ability to extend what has been learned in one context to
new contexts” (Bransford, Brown & Cocking, 2000)
– In mathematics, transfer facilitated by flexible procedural
knowledge and conceptual knowledge
• Two types of knowledge needed in mathematics
– Procedural knowledge: actions for solving problems
• Knowledge of multiple procedures and when to apply them
(Flexibility)
• Extend procedures to a variety of problem types (Procedural
transfer)
– Conceptual knowledge: principles and concepts of a domain
How to Support Transfer:
Comparison
• Cognitive Science: A fundamental
learning mechanism
• Mathematics Education: A key
component of expert teaching
Comparison in Cognitive
Science
• Identifying similarities and differences in
multiple examples is a critical pathway to
flexible, transferable knowledge
– Analogy stories in adults (Gick & Holyoak, 1983; Catrambone & Holyoak,
1989)
– Perceptual Learning in adults (Gibson & Gibson, 1955)
– Negotiation Principles in adults (Gentner, Loewenstein & Thompson,
2003)
– Cognitive Principles in adults (Schwartz & Bransford, 1998)
– Category Learning and Language in preschoolers (Namy &
Gentner, 2002)
– Spatial Mapping in preschoolers (Loewenstein & Gentner, 2001)
– Spatial Categories in infants (Oakes & Ribar, 2005)
Comparison in Mathematics
Education
– “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)
Comparison 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
transfer in mathematics?
• Experimental studies of learning and transfer
in academic domains and settings largely
absent
• Goal of present work
– Investigate whether comparison can support
transfer with student learning to solve equations
– Explore what types of comparison are most
effective
– Experimental studies in real-life classrooms
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
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
Study 1
• Compare condition: Compare and
contrast alternative solution methods vs.
• Sequential condition: Study same
solution methods sequentially
Rittle-Johnson, B. & Star, J.R. (in press). Does comparing solution
methods facilitate conceptual and procedural knowledge? An
experimental study on learning to solve equations. Journal of
Educational Psychology.
Compare Condition
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 nonstandard procedures; evaluate when to use a
procedure)
– Conceptual knowledge (e.g. equivalence,
like terms)
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
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
Gains in Procedural
Knowledge: Equation Solving
45
Compare
Sequential
Post - Pre Gain Score
40
35
30
25
20
15
10
5
0
Familiar
Novel
Equation Solving
F(1, 31) =4.88, p < .05
Gains in Procedural
Flexibility
• Greater use of non-standard solution
methods to solve equations
– Used on 23% vs. 13% of problems,
t(5) = 3.14,p < .05.
Gains on Independent
Flexibility Measure
45
Compare
Sequential
Post - Pre Gain Score
40
35
30
25
20
15
10
5
0
Flexiblity
F(1,31) = 7.51, p < .05
Gains in Conceptual
Knowledge
30
Post - Pre Gain Score
Compare
Sequential
20
10
0
Conceptual
No Difference
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 Study 1
• Comparing alternative solution methods
is more effective than sequential
sharing of multiple methods
– In mathematics, in classrooms
Study 2:
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 problems with the same
solution
– 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)
Study 2 Method
• Participants: 161 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
– Assessment adapted from Study 1
Gains in Procedural
Knowledge
Gains depended on prior conceptual knowledge
Gains in Conceptual
Knowledge
Post - Pre Gain Score
25
20
15
10
5
0
Surface
Problems
Methods
Compare Condition
Compare Solution Methods condition made greatest
gains in conceptual knowledge
Frequency of Use at Posttest
Gains in Procedural Flexibility:
Use of Non-Standard Methods
60
50
40
30
20
10
0
Surface
Problems
Methods
Compare Condition
Greater use of non-standard solution methods in
Compare Methods and Problem Type conditions
Gains on Independent
Flexibility Measure
30
Post - Pre Gain Score
25
20
15
10
5
0
Surface
Problems
Co ndition
No effect of condition
Methods
Summary
• Comparing Solution Methods often
supported the largest gains in
conceptual and procedural knowledge
• However, students with low prior
knowledge may benefit 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 matters
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