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Compared to What? How Different Types of Comparison Affect Transfer in MathematicsPowerPoint Presentation

Compared to What? How Different Types of Comparison Affect Transfer in Mathematics

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### 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

- Procedural knowledge: actions for solving problems

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

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

Predicted Outcomes

- Students in compare condition will make greater gains in:
- Procedural knowledge, including
- Success on novel problems
- Flexibility of procedures (e.g. select non-standard procedures; evaluate when to use a procedure)

- Conceptual knowledge (e.g. equivalence, like terms)

- Procedural knowledge, including

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.

Gains in Procedural Knowledge: 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

F(1,31) = 7.51, p < .05

Gains in Conceptual Knowledge

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

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

Compare Solution Methods condition made greatest gains in conceptual knowledge

Gains in Procedural Flexibility: Use of Non-Standard Methods

Greater use of non-standard solution methods in Compare Methods and Problem Type conditions

Gains on Independent Flexibility Measure

No effect of condition

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

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