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Objectives:

This project aims to develop and test effective mathematics interventions in close partnership with mathematics education practitioners; to date, this work has been carried out in approximately 10 school districts over the past decade. So far, the partnership has produced two sets of worksheet-based practice materials: AlgebraByExample and MathByExample.

Theoretical Framework:

This work draws on a set of laboratory-tested cognitive science principles for learning: the Worked Example principle (e.g., Sweller, 1999), the Self-Explanation principle (e.g., Chi, 2000), and the Learning from Errors principle (e.g., Siegler, 2002). Empirical evidence has amassed supporting the effectiveness of having students explain why correct problem solutions are correct (e.g., Hilbert, Renkl, Kessler, & Reiss, 2008) as well as why incorrect problem solutions are incorrect (e.g., Durkin & Rittle-Johnson, 2012), but prior to our work in this partnership very little work had been conducted in real-world classrooms.

Methods:

Over the past decade, we conducted a series of design experiments and randomized-controlled trials in which over 1500 late elementary and middle/high school students in dozens of classrooms from our partner districts have been given experimental worksheets in place of typical practice assignments in their elementary math or algebra courses. In the byExample worksheets, we replaced half of the practice problems that would appear in a typical assignment with correct or incorrect examples of problem solutions for the students to study and explain. Study length varied from 4 weeks (one content unit) to full school years.

Materials:

Students were tested on their conceptual understanding of course material (e.g., their understanding of the meaning of conceptual features within the problems they are being taught to solve), procedural skill (e.g., ability to carry out procedures to solve problems in their courses), and performance on released items from grade-appropriate standardized achievement tests.

Results:

Results have indicated that students who complete byExample assignments in their Algebra or Mathematics courses learn more than students who complete typical practice assignments; these benefits have emerged for conceptual understanding, procedural skill, and performance on standardized test items. The greatest benefit often emerges for students who struggle to learn in mathematics; these struggling students tend to receive particular benefits for their conceptual understanding.

Significance:

Results from the present study confirmed that explaining correct and incorrect examples as part of practice sessions can be an effective instructional activity for students in middle/high school algebra and upper elementary school mathematics. Interestingly, even though byExample assignments by definition contained half of the problem-solving practice that was included in the typical assignments, students gained just as much or more procedural knowledge after completing the byExample assignments. It is important to note that this work was conducted in the context of design-based research in which teachers and researchers were both heavily involved with decision-making about not only the content of the assignments, but also the structure and format, in order to make sure the worked example approach was translated carefully for real-world classroom use.

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