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II. Learning & comparison: Cognitive science
research There is a great deal of cognitive research
I. Abstract This poster presents a research program that showing
seeks to
the benefits of comparison for learning in young
improve educational practice and student learning in children (e.g., Loewenstein & Gentner, 2001; Oakes &
mathematics by developing, implementing, and
Ribar, 2005) and adults (e.g., Gentner, Loewnstein, &
testing curriculum materials built on findings from
Thompson, 2003; Namy & Gentner, 2002). Yet, little of
cognitive science. We convey the process of
this type of research has been done in classrooms.
conducting experimental classroom studies built on
Building on these findings, we engaged in small-scale
lab-based cognitive science research and the
experimental classroom studies to explore the
subsequent design and implementation of a
benefits of comparison for students’ learning of
supplemental first-year algebra curriculum based on
mathematics, focusing on equation solving.
comparison.
III. Experimental classroom research Our
experimental research showed positive effects of
Two-year randomized controlled trial
564RITTLE-JOHNSON AND STARIV. Curriculum Design and Development During 2008-2009, we worked with a small
group of expert teachers to transform our experimental materials (see
Figure 1) into a supplementary Algebra I curriculum that embodied the principles derived from previous
experimental research (see Figure 2).
•  Side-by-side comparison •
Labeled solution steps •  Prompts
to identify similarities & differences
Figure 1. Experimental comparison materials from Rittle-Johnson & Star (2007)
Which is better?
comparison on student learning in controlled settings.
th
and 8th
Alex and Morgan were asked to solve
•  “Infuse” comparison into first-year algebra classes •
Facilitate comparison of and reflection on multiple
First I gave the two
fractions the same
denominator.
Then I subtracted the
fractions.
First I multiplied
both sides of the
equation by the
least common
multiple of the
denominators,
which is 20.
Then I multiplied by 20
on both sides.
205 " #$ %x 4! x& ’=
!2(20)
Then I simplified
both sides of the
equation.
I simplified both sides of
the equation
to get the answer.
Then I combined like
terms to get
the answer.
x 20= !2(20)
Which is better?
•  About 80 “worked example pairs” •
Characters: Alex and Morgan •  Four
comparison types
•  Which is better? •
Why does it work? •  How
do they differ? •  Which is
correct? •  Discussion
phases
•  Understand, compare, make connections •  Help
teachers facilitate comparison conversations •
Supplementary materials
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First I multiplied
both sides of the
equation by the
least common
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denominator.
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by 20 as a first step? * Why did Morgan find a common denominator as a first step?
3.1.2
•
Beyo
nd
linear
equa
tion
solvin
g to
inclu
de
Alge
bra I
cont
ent
Figure
1.
Sample
pages
from
interven
tion
packet
for (A)
compare
and (B)
sequenti
al
conditio
ns.
Pilot testing
Curriculum design
Experimental classroom
studies
• 7th grade students
• Worked examples
side-by-side • Compare
side-by-side
or reflect sequentially
• Comparison condition:
Greater procedural
knowledge and
• grade students
• What you compare
7matters
• Solution methods:
largest gains in
conceptual knowledge
and flexibility
• Problem types: Support
both to a lesser extent
Alex’s “eliminate the fractions” way!
(20)
Morgan’s “find common denominators” way
strategies
How doesfcomparison affect student learning in
real
l classroom environments?
Cognitive science
research
e
VI. Randomized controlled
x
trials
Our revised materials are currently being tested
i
using
b
Alex and Morgan were asked to solve
V. Pilot testing During the 2009-2010 school year, we
worked with 12
i
about 80 first-year algebra
l
classrooms
across Massachusetts.
i
Morgan’s “find common denominators” way
t
y
middle and high school teachers to test our materials in
classrooms.
Then I subtractedThen
the I simplified
fractions.
both sides of the
equation.
Figure 1. Sample pages from intervention packet for (A) compare and (B) sequential conditions.
fractions first or by finding the
common denominators first. You
get the same answer using both
• One-week summer PD • Comparison
activities • Create own worked
example pairs • Model teaching with
comparison
• Tasks • Use materials 1-2 x per week
• Videotape 2 x per month • Submit log
after material use (e.g., time spent, student
learning, teacher satisfaction)
• Student assessments • Feedback
• Ongoing feedback throughout the year
• Student and teacher end-of-year
When solving equation with
fractions as coefficients, you can
des of the equation by
the LCM of the
s
t
a
r
t
b
y
m
u
l
t
i
p
l
y
i
n
g
b
o
t
h
s
i
•  New comp
versus incorre
F
i
r
s
t
* What are
some
similarities
and
differences
between
Alex's and
Morgan's
ways? *
Which way is
easier, Alex's
way or
Morgan's
way? Why?
I simplified both sides
of the equation
to get the
answer.
interviews
Based on our pilot year feedback, we expanded our
curriculum to include 150 worked example pairs. We
h
e
ns the same
Figure 2. Worked example from comparison curricu
t
w
o
I
f
r
a
c
t
i
o
g
a
v
e
t
There was also a packet of 12 practice problems. The problems
were isomorphic to the equations used in the worked
examples, and the same practice problems were used for both
conditions. Three brief homework assignments were
developed, primarily using problems in the students’ regular
textbook, and homework was the same for both conditions.
Assessment. The same assessment was used as an individual
pretest and posttest. It was designed to assess procedural
knowledge, flexibility, and conceptual knowledge. Sample
items of each knowledge type are shown in Table 2. The
procedural knowledge items were four familiar equations (one
of each type presented during the intervention) and four novel,
transfer equations (e.g., a problem that included three terms
within parentheses). There were six flexibility items designed
to tap three components of flexibility—the abilities to
generate, recognize, and evaluate multiple solution methods
for the same problem. There were six conceptual
Figure 3. Take-away page excerpt from the revised curriculum
experimental research grounded in cognitive
science has substantially improved educational
extending the benefits of learning through
* What are some similarities and differences between Alex's and Morgan's ways? * Which way is easier, Alex's way or
comparison to authentic classroom settings.
Morgan's way? Why? We are currently finishing data collection
of videos, logs, and student assessments for the first
year of the RCT. We will continue to collect data
during the 2011-2012 school year.
relevance of its findings to classroom settings. They
note that lab findings cannot alone improve
classroom practice, but that “controlled
practice,” (p. 184). We hope to illustrate one
way that building on experimental research can
improve educational practice and student
learning, by
3.
1.
2
Before you start solving a problem
you can look at the problem first
and try to see which way might be
easier. !
* Why did Alex multiply each term by 20 as a
first step? * Why did Morgan find a common
denominator as a first step?
also added a second page for each (see Figure
3).
VII. Conclusion
References separate packet for each of the two days of partner work;
(2000)
methods. Multiplying both sides of
on both sides.
Atkinson, Derry, Renkl, and Wortham
the equation by the LCM of the
the first two problem types in Table 1 were presented in the first packet, and
the third and fourth problem types in Table 1 were presented in a second
packet. In the sequential packets, there were 24 equations, the 12 equations
from the compare condition and an isomorphic equation for each that was
identical in form and varied only in the particular numbers. The same acknowledge the gap between mathematics
solution methods were presented as in the compare condition, but each research in controlled laboratory settings and the
worked example was presented on a separate sheet. Thus, exposure to
multiple solution methods was equivalent across the two conditions. As in the
compare condition, steps were labeled or students needed to fill in the
appropriate label. At the bottom of each page was one question prompting
students to reflect on that solution. The number of reflection questions (24)
was the same across the two conditions. A pair of sample pages from the
packet is shown in Panel B of Figure 1.
Then I multiplied=by
20
!2(20)
fractions first might be easier
because it eliminates the fractions. !
Then I combined
(20)x 20
like terms to get
the answer.
Atkinson, R. K., Derry, S. J., Renkl, A., & Wortham, D. (2000). Learning from e
Educational Research, 70(2), 181-214. Gentner, D., Loewnstein, J., & Thom
Journal of Educational Psychology, 95(2), 393-408. Loewenstein, J., & Ge
far mappings. Journal of Cognition and Development, 2, 189– 219. Namy,
children’s use of comparison in category learning. Journal of Experimenta
comparison of infants’ categorization in paired and successive presentati
Does comparing solution methods facilitate conceptual and procedural k
Educational Psychology, 99(3), 561-574. Rittle-Johnson, B. & Star, J. R. (20
knowledge and procedural flexibility for equation solving. Journal of Educ
the two days of partner work; the first two problem
and the third and fourth problem types in Table 1 w
packets, there were 24 equations, the 12 equations
equation for each that was identical in form and va
methods were presented as in the compare conditio
separate sheet. Thus, exposure to multiple solution
in the compare condition, steps were labeled or stu
bottom of each page was one question prompting s
reflection questions (24) was the same across the tw
is shown in Panel B of Figure 1.separate packet for
problem types in Table 1 were presented in the firs
Table 1 were presented in a second packet. In the s
equations from the compare condition and an isom
varied only in the particular numbers. The same so
condition, but each worked example was presented
solution methods was equivalent across the two co
labeled or students needed to fill in the appropriate
prompting students to reflect on that solution. The
across the two conditions. A pair of sample pages f