The Benefits of Contrasting Cases on Students’ Learning of MathematicsIntroduction For at least the past 20 years, a central tenet of reform pedagogy in mathematics has been that student
Trang 1It Pays to Compare ! The Benefits of Contrasting Cases on Students’ Learning of Mathematics
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 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
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
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)
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, real-world 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
Method
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
Samples of Intervention Materials
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Sequential condition
Algebra (Study 1)
Compare condition
Sequential condition
Estimation (Study 2)
Compare condition
Samples of Assessment Items
Procedural Knowledge
(familiar) 1 Solve: -1/4 (x – 3) = 10 2 Solve: 5(y – 12) = 3(y – 12) + 20 1 Estimate: 12 * 242 Estimate: 37 * 17
Procedural Knowledge
(transfer) 3 Solve: 0.25 (t + 3) = 0.5 4 Solve -3(x + 5 + 3x) – 5(x + 5 + 3x) = 24 3 Estimate: 1.92 * 5.084 Estimate: 148 ÷ 11
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.
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)
Results
2 Students in the compare condition made greater gains
in flexibility.
3 Compare and sequential students achieved similar and modest gains in conceptual knowledge.
0 0.1 0.2 0.3 0.4 0.5
Familiar Transfer Familiar Transfer
Procedural Gain Score (Post - Pre)
Sequential Compare
Algebra Estimation
1 Students in the compare condition made greater gains in procedural knowledge.
0 0.1 0.2 0.3 0.4 0.5
Algebra Estimation
Flexibility Gain Score (Post - Pre)
Sequential Compare
0 0.1 0.2 0.3 0.4 0.5
Algebra Estimation
Conceptual Gain Score (Post - Pre)
Sequential Compare
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
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
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