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It Pays to
Compare
! The Benefits of Contrasting Cases on Students’ Learning of Mathematics
Jon R. Star
1
,
Bethany RittleJohnson
2
,
Kosze Lee
3
,
Jennifer Samson
2
, and
KuoLiang Chang
3
1
Harvard University
,
2
Vanderbilt University
,
3
Michigan State University
Introduction
For
at
least
the
past
20
years,
a
central
tenet
of
reform
pedagogy
in
mathematics has been that students benefit from comparing, reflecting on,
and discussing multiple solution methods (Silver et al., 2005). Case studies
of
expert
mathematics
teachers
emphasize
the
importance
of
students
actively comparing solution methods (e.g., Ball, 1993; Fraivillig, Murphy, &
Fuson,
1999).
Furthermore,
teachers
in
highperforming
countries
such
as
Japan
and
Hong
Kong
often
have
students
produce
and
discuss
multiple
solution methods (Stigler & Hiebert, 1999). While these and other studies
provide
evidence
that
sharing
and
comparing
solution
methods
is
an
important
feature
of
expert
mathematics
teaching,
existing
studies
do
not
directly link this teaching practice to measured student outcomes . We could
find
no
studies that assessed the causal influence of comparing contrasting
methods on student learning gains in mathematics.
There
is
a
robust
literature
in
cognitive
science
that
provides
empirical
support for the benefits of comparing contrasting examples for learning in
other domains, mostly in laboratory settings (e.g., Gentner, Loewenstein, &
Thompson,
2003;
Schwartz
&
Bransford,
1998).
For
example,
college
students who were prompted to compare two business cases by reflecting on
their similarities were much more likely to transfer the solution strategy to a
new
case
than
were
students
who
read
and
reflected
on
the
cases
independently
(Gentner
et
al.,
2003).
Thus,
identifying
similarities
and
differences in multiple examples may be a critical and fundamental pathway
to
flexible,
transferable
knowledge.
However,
this
research
has
not
been
done in mathematics, with K12 students, or in classroom settings.
Current
Study
.
We
evaluated
whether
using
contrasting
cases
of
solution
methods
promoted
greater
learning
in
two
mathematical
domains
(computational estimation and algebra linear equation solving) than studying
these
methods
in
isolation.
The
research
focused
on
three
core
learning
outcomes: (1) problemsolving skill on both familiar and novel problems, (2)
conceptual knowledge of the target domain, and (3) procedural flexibility,
which includes the ability to generate more than one way to solve a problem
and evaluate the relative benefits of different procedures.
Algebra
equation
solving
.
The
transition
from
arithmetic
to
algebra
is
a
notoriously difficult one, and improvements in algebra instruction are greatly
needed
(Kilpatrick
et
al.,
2001).
Algebra
historically
has
represented
students’
first
sustained
exposure
to
the
abstraction
and
symbolism
that
makes
mathematics
powerful
(Kieran,
1992).
Regrettably,
students’
difficulties
in
algebra
have
been
well
documented
in
national
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This note was uploaded on 11/11/2011 for the course MATH 110 taught by Professor Staff during the Winter '08 term at BYU.
 Winter '08
 Staff
 Math

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