Fully Worked Examples of Proportion Problems Promote Learning More than Partially Worked or No Examples

• In Event: STEM: Cognition, Motivations, and Interventions

Abstract

Learning to use multiple solutions is a recommended practice for teaching students to be flexible, conceptual mathematical thinkers. Yet teaching multiple strategies is challenging, since students’ cognitive load can be overwhelmed (Chandler & Sweller, 1991). Proportional reasoning is a mathematics domain that is particularly affected since it requires learners to consider key numbers within a problem and also the relationships between these numbers (Hart, 1984).  Providing worked examples may reduce students’ cognitive resource load and allow them to allocate attention to comparing the multiple solutions. At the same time, requiring students to solve examples themselves can be a potent way to ensure that students build understanding, so it is possible that providing students worked examples could impede learning. The study explored competing hypotheses for providing support or not. Furthermore, it focused on what types of worked examples would be most beneficial - fully worked examples (FW) or partially worked examples (PW), where students had to complete a solution that had been only partially filled in.
Study Hypotheses: Providing students with FW examples of different solution strategies would support learning of proportion problems more than opportunities to solve PW examples or problems only (PO).
Participants: 308 students from classrooms in 4-6th grade classrooms with primarily Black and Latinx students in the Chicagoland area. Students were randomly assigned at the classroom level to a worksheet with FW, PW or PO.  
Study Materials. In the FW and PW worksheets, an initial worked example was given, showing the equivalent fraction strategy. Participants were then asked to use that solution on two new problems. Afterwards, they were given a FW or PW worked example with the initial problem again, now showing the equivalent fraction strategy. Participants solved the same problems a second time, using the unit ratio solution. Finally, students applied their knowledge on 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 attempt, set-up, procedure, and solution accuracy.
Figures 1 and 2 show alluvial flow plots to represent changes in student’s attempt to accuracy, revealing that overall FW examples promote use of novel strategies and higher accuracy for both the equivalent fraction strategy, 𝜒2(2, N = 203)=24.16, p<.001) and unit ratio strategy, 𝜒2(1, N=112)=5.92, p=.015) compared to PW example and PO conditions. For the equivalent fraction strategy, students in the FW condition (Figure 1c) attempted the instructed strategy more frequently and accurately compared to those in the PW and PO condition (Figure 1b and 1a). For the unit ratio strategy, students in the FW condition (Figure 2b) attempted the instructed strategy more frequently and accurately compared to those in the WE condition (Figure 2a).
Results suggest that providing full or partial worked-example worksheets encouraged students to attempt novel proportional reasoning strategies, though only the full worked example led to increased procedural skills and transfer, suggesting that in cognitively demanding mathematics, more full worked examples may be particularly advantageous.

Authors

• Elayne Vollman, Lake Forest College
  Presenting Author

• Kelly Marie Trezise, Loughborough University
  Non-Presenting Author

• Lindsey Engle Richland, University of California - Irvine
  Non-Presenting Author