Transcript Star, J.R., Rittle-Johnson, B., Lee, K., Samson, J

It Pays to Compare! The Benefits of Contrasting Cases on Students’ Learning of Mathematics
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Rittle-Johnson ,
3
Lee ,
2
Samson ,
Bethany
Kosze
Jennifer
and Kuo-Liang
1Harvard University, 2Vanderbilt University, 3Michigan State University
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 high-performing 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.
1. Students in the compare condition made greater gains in
procedural knowledge.
Sequential
Sequential condition
We compared learning from studying contrasting cases (compare group) to
learning from studying sequentially presented solutions (sequential, or
control, group) in the domains of multi-step linear equations (Study 1;
Rittle-Johnson & Star (in press)) and computational estimation (Study 2).
Participants, Study 1: Seventy (36 female) 7th graders and their teacher
Participants, Study 2: Sixty-nine (32 female) 5th graders and their teacher
Procedure: We randomly paired students and assigned them to condition.
Pairs studied worked examples of other students’ solutions and answered
questions about the solutions during a three-day intervention in their intact
math classes. Both conditions were introduced to the same solution
methods and received mini-lectures from the teacher during the
intervention.
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Transfer
Algebra
Familiar
Transfer
Estimation
Compare condition
2. Students in the compare condition made greater gains
in flexibility.
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Sequential
Compare
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Algebra
Estimation
Discussion
Comparing and contrasting alternative solution methods led to
greater gains in procedural knowledge and flexibility, and
comparable gains in conceptual knowledge, compared to studying
multiple methods sequentially. These findings provide direct
empirical support for one common component of reform
mathematics teaching. These studies also suggest that prior
cognitive science research on comparison as a basic learning
mechanism may be generalizable to new domains (algebra and
estimation), a new age group (school-aged children), and a new
setting (the classroom).
These findings were strengthened by our use of random
assignment of students to condition within their regular classroom
context, along with maintenance of a fairly typical classroom
environment. Further, rather than comparing our intervention to
standard classroom practice, which differs from our intervention
on many dimensions, we compared it to a control condition which
was matched on as many dimensions as possible. This allowed us
to evaluate a specific component of effective teaching and
learning.
The current studies are an important first step in providing
experimental evidence for the benefits of comparing alternative
solution methods, but much is yet to be done. In particular, it is
important to evaluate when and how comparison facilitates
learning. We are presently conducting several studies exploring
the effectiveness of different types of comparison, including
comparing solution strategies (the same problem solved in two
different ways), comparing problem types (two different
problems, solved using the same strategy), and comparing
isomorphs (two similar problems, solved using the same strategy).
Our preliminary analyses suggest that the type of comparison that
is most effective appears to depend on prior knowledge and
ability.
References
3. Compare and sequential students achieved similar and
modest gains in conceptual knowledge.
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Sequential condition
Estimation (Study 2)
Compare condition
Computational Estimation. A large majority of students have difficulty doing
simple calculations in their heads or estimating the answers to problems
(e.g., Case & Sowder, 1990; Reys, Bestgen, Rybolt, & Wyatt, 1980). This
disuse or inability to use mental math or estimation is a significant barrier to
using mathematics in everyday life. In addition to being a fundamental, realworld skill, the ability to quickly and accurately perform mental
computations and estimations has two additional benefits: 1) It allows
students to check the reasonableness of their answers found through other
means, and 2) it can help students develop a better understanding of place
value, mathematical operations, and general number sense (Kilpatrick et al,
2001).
Method
Compare
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Algebra (Study 1)
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) problem-solving 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 and
international assessments (Blume & Heckman, 1997; Schmidt et al., 1999).
Current mathematics curricula typically focus on standard procedures for
solving equations, rather than on flexible and meaningful solving of
equations (Kieran, 1992). In contrast, prompting students to solve problems
in multiple ways leads them to greater procedural flexibility (Star & Seifert,
2006).
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Familiar
Flexibility Gain Score (Post - Pre)
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 K-12 students, or in classroom settings.
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Chang
Results
Samples of Intervention Materials
Procedural Gain Score (Post - Pre)
Introduction
Conceptual Gain Score (Post - Pre)
Jon R.
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Star ,
Sequential
Compare
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Algebra
Estimation
Ball, D. L. (1993). With an eye on the mathematical horizon: Dilemmas of teaching elementary school
mathematics. The Elementary School Journal, 93, 373-397.
Blume, G. W., & Heckman, D. S. (1997). What do students know about algebra and functions? In P. A. Kenney &
E. A. Silver (Eds.), Results From the Sixth Mathematics Assessment (pp. 225-277). Reston, VA: National Council
of Teachers of Mathematics.
Case, R., & Sowder, J. T. (1990). The development of computational estimation: A neo-Piagetian analysis.
Cognition and Instruction, 7, 79-104.
Fraivillig, J. L., Murphy, L. A., & Fuson, K. (1999). Advancing children's mathematical thinking in Everyday
Mathematics classrooms. Journal for Research in Mathematics Education, 30, 148-170.
Gentner, D., Loewenstein, J., & Thompson, L. (2003). Learning and transfer: A general role for analogical
encoding. Journal of Educational Psychology, 95(2), 393-405.
Kieran, C. (1992). The learning and teaching of school algebra. In D. Grouws (Ed.), Handbook of Research on
Mathematics Teaching and Learning (pp. 390-419). New York: Simon & Schuster.
Kilpatrick, J., Swafford, J. O., & Findell, B. (Eds.). (2001). Adding it up: Helping children learn mathematics.
Washington DC: National Academy Press.
Lindquist, M. M. (Ed.). (1989). Results from the fourth mathematics assessment of the National Assessment of
Educational Progress. Reston, VA: National Council of Teachers of Mathematics.
Reys, R. W., Bestgen, B., Rybolt, J. F., & Wyatt, J. W. (1980). Identification and characterization of
computational estimation processes used by in-school pupils and out-of-school adults (No. ED 197963).
Washington, DC: National Institute of Education.
Rittle-Johnson, B. & Star, J. (in press). Does comparing solution methods improve conceptual and procedural
knowledge? An experimental study on learning to solve equations. Journal of Educational Psychology.
Schmidt, W. H., McKnight, C. C., Cogan, L. S., Jakwerth, P. M., & Houang, R. T. (1999). Facing the
consequences: Using TIMMS for a closer look at U.S. mathematics and science education. Dordrecht: Kluwer.
Sowder, J. T., & Wheeler, M. M. (1989). The development of concepts and strategies used in computational
estimation. Journal for Research in Mathematics Education, 20, 130-146.
Schwartz, D. L., & Bransford, J. D. (1998). A time for telling. Cognition and Instruction, 16(4), 475-522.
Silver, E. A., Ghousseini, H., Gosen, D., Charalambous, C., & Strawhun, B. (2005). Moving from rhetoric to
praxis: Issues faced by teachers in having students consider multiple solutions for problems in the mathematics
classroom. Journal of Mathematical Behavior, 24, 287-301.
Star, J.R., & Seifert, C. (2006). The development of flexibility in equation solving. Contemporary Educational
Psychology, 31, 280-300.
Stigler, J. W., & Hiebert, J. (1999). The teaching gap: Best ideas from the world's teachers for improving
education in the classroom. New York: Free Press.
Samples of Assessment Items
Algebra (Study 1)
Estimation (Study 2)
Procedural Knowledge 1. Solve: -1/4 (x – 3) = 10
(familiar)
2. Solve: 5(y – 12) = 3(y – 12) + 20
1. Estimate: 12 * 24
2. Estimate: 37 * 17
Procedural Knowledge 3. Solve: 0.25 (t + 3) = 0.5
(transfer)
4. Solve -3(x + 5 + 3x) – 5(x + 5 + 3x) = 24
Flexibility
1. Solve 4(x + 2) = 12 in two different ways.
2. For the equation 2(x + 1) + 4 = 12, identify all possible steps (among 4 given choices) that could be done
next.
3. A student’s first step for solving the equation 3(x + 2) = 12 was x + 2 = 4. What step did the student use
to get from the first step to the second step? Do you think that this way of starting this problem is (a) a very
good way; (b) OK to do, but not a very good way; (c) Not OK to do? Explain your reasoning.
3. Estimate: 1.92 * 5.08
4. Estimate: 148 ÷ 11
1. Estimate 12 * 36 in three different ways.
2. Leo and Steven are estimating 31 * 73. Leo rounds both numbers and multiplies 30 * 70.
Steven multiplies the tens digits, 3█ * 7█ and adds two zeros. Without finding the exact answer,
which estimate is closer to the exact value?
3. Luther and Riley are estimating 172 * 234. Luther rounds both numbers and multiplies 170 *
230. Riley multiplies the hundreds digits 1█ █ * 2█ █ and adds four zeros. Which way to
estimate is easier?
Conceptual Knowledge 1. If m is a positive number, which of these is equivalent to (the same as) m + m + m + m? (Responses are:
4m; m4; 4(m + 1); m + 4)
2. For the two equations 213x + 476 = 984 and 213x + 476 + 4 = 984 + 4, without solving either equation,
what can you say about the answers to these equations? Explain your answer.
1. What does “estimate” mean?
2. Mark and Lakema were asked to estimate 9 * 24. Mark estimated by multiplying 10 * 20 =
200. Lakema estimated by multiplying 10 * 25 = 250. Did Mark use an OK way to estimate the
answer? Did Lakema use an OK way to estimate the answer? (from Sowder & Wheeler, 1989)