Comparing Levels of Completion Within Worked Examples: Benefits of Full Support for Proportional Problem Solving

• In Event: Using Examples to Help Students Learn Mathematics: How, When, and With Whom to Use Worked Examples

Abstract

Learning to understand and use multiple solutions to a problem is a recommended practice for teaching students to be flexible, conceptual mathematical thinkers, yet teaching learners to use new strategies is challenging. This study examines whether incorporating support from worked examples provides an opportunity for facilitating this process, specifically testing the role of varying levels of support. Theoretically, providing worked examples should reduce students’ cognitive resource load and allow them to allocate attention to the broader conceptual frame of the mathematics problem, gaining a deeper understanding of multiple solution opportunities (Chandler & Sweller, 1991; Sweller, 1999).

Considering cognitive load is particularly important when students grapple with a solution or problem-type that requires more complex reasoning. This study focuses specifically on proportional thinking, which is relational at heart, meaning that learners have to consider not only the key numbers within a problem but also their relationships, which provides a higher cognitive load than some mathematical content areas (Kaput & West, 1994; Lamon, 1993; Lo & Watanabe, 1997).

The current study examined the effects of administering worksheets that included Fully Worked Examples (FW), Partial Worked Examples (PW), or Problems Only (PO) (No worked examples provided) when introducing proportional thinking strategies. Worksheets were randomly assigned within classrooms to 308 4th, 5th, and 6th graders from schools with primarily underrepresented minority or low income youth. Two students’ strategies were provided in the worked examples: equivalent fractions and unit ratio. All packets began with one problem and students either studied a fully worked example or completed the procedures to solve a partial worked example. Students then were asked to use that solution strategy on two near transfer problems, followed by the same procedure for unit ratio. Finally, students applied their knowledge in far transfer problems. The PO packet contained the same problems, and students were told to use any strategy they preferred. Problem solutions were then coded for strategy, set-up, comparisons, procedure, and solution.

Data overall revealed benefits for worked examples, with particularly high accuracy in the FW conditions. Students in the FW and PW conditions were equally likely to attempt the correct strategy for near-transfer problems, t(1,124) = 1.31, p = .193, though problem solving accuracy was highest for students in the FW condition. There was no difference between PO and PW conditions, F(2,173) = 8.94, p <.001. Students in the FW and PW conditions were 14% more likely to use a proportional strategy than PO students. Further analysis revealed that students in the PO condition did use the same equivalent strategy as in the worked examples, but for this subset, students were less likely to set up the fractions explicitly, and were less likely to use long division set-up in unit ratio strategy, compared to students in either of the worked example groups.

Results suggest that providing full or partial worked-example worksheets encouraged students to attempt proportional reasoning strategies, though only the full worked example led to increased transfer, suggesting that in cognitively demanding mathematics, more complete worked examples may be particularly advantageous.

Authors

• Elayne Teska, University of Chicago

• Kelley Trezise, University of Chicago

• Lyndsey Richland, University of California - Irvine