Math

CAREER: Developing Conceptual and Procedural Knowledge: The Roles of Self- and Instructional Explanations

Principal Investigator:

Project Overview

Background & Purpose:

Please see the project webpage: http://www.peabody.vanderbilt.edu/earlyalgebra.xml

This project focuses on evaluating the role of explanation in mathematics learning—specifically “why” explanations, or reasoning about why something is true. Such explanations can be generated by the student to explain new content or provided by the instructor. Although explanations are often used as a learning tool, many factors can determine whether explaining will actually benefit learning, such as the source and timing of these explanations and individual differences such as age. My research group is performing a series of studies to uncover these factors. In doing so, we hope not only to reveal the learning mechanisms that make explanation helpful, but we will also use this knowledge to inform mathematics instruction in the classroom.

Our studies focus on learning early algebra topics such as equivalence and patterns.

Setting:

Private and public schools in a large city in the Southeast.

Research Design:

We have been developing assessments of algebraic knowledge, focusing on knowledge of mathematical equivalence, patterns, and functions. We are using a combination of classical test theory and item-response theory methods, including the use of construct mapping and Rasch modeling. We are also using a cross-sectional approach to look at improvements with grade-level on the measures.

In the intervention phase of the project, we are using causal experimental methods. Our initial interventions use a one-on-one tutoring session, with students randomly assigned to the type of tutoring. For example, in one set of experiments, we are investigating the timing of direct instruction and self-explanation prompts during the tutoring session.

We evaluate student learning using the researcher-designed knowledge assessments that we are validating in the first phase. We use 2 x 2 ANCOVAs to evaluate the effects of tutoring condition on students’ knowledge.

Findings:

1. We have validated a construct map and accompanying assessment for assessing elementary-school children’s knowledge of a key algebraic concept - mathematical equivalence (the idea that two sides of an equation represent the same amount). This assessment provides a more detailed and continuous measure of student knowledge from Grades 2 to 6.

2. We have developed a construct map and assessment for assessing elementary-schol children's knowledge of function tables.

3. We are currently developing a repeating-pattern assessment for use with preschoolers.

4. In our first tutoring study, we found that students understood concepts better if they first explored solving equivalence problems before receiving a lesson on the concept of equivalence, rather than receiving the lesson before solving the problems. These findings highlight the importance of problem exploration for preparing students to learn from direct instruction.

Publications & Presentations:

JOURNAL PUBLICATIONS

Rittle-Johnson, B., Taylor, R., Matthews, P.G., & McEldoon, K. (in press). Assessing Knowledge of Mathematical Equivalence: A Construct Modeling Approach. Journal of Educational Psychology.

CONFERENCE PROCEEDINGS

McEldoon, K. & Rittle-Johnson, B. (2010, October). Assessing Elementary Students' Functional Thinking Skills: The Case of Function Tables. Proceedings of the 2010 annual meeting of the North American Chapters of the International Group for the Psychology of Mathematics Education. Columbus, OH: ERIC Clearinghouse for Science, Mathematics, and Environmental Education.

Rittle-Johnson, B., Matthews, P., Taylor, R. & McEldoon, K. (2010, October). Assessing Knowledge of Mathematical Equivalence: A Construct Modeling Approach. Proceedings of the 2010 annual meeting of the North American Chapters of the International Group for the Psychology of Mathematics Education. Columbus, OH: ERIC Clearinghouse for Science, Mathematics, and Environmental Education.

DeCaro, M. & Rittle-Johnson, B. (2010, August). The Capacity to Discover: Working Memory and the Ability to Use Self-Explanation to Discover Early Algebra Concepts. Proceedings of the Cognitive Science Society 2010 Annual Meeting.

McEldoon, K., Cochrane-Braswell, C., & Rittle-Johnson, B. (2010, August). Effects of Problem Context on Strategy Use within Functional Thinking. To appear in Proceedings of the Cognitive Science Society 2010 Annual Meeting.

CONFERENCE PRESENTATIONS

Matthews, P. G., Rittle-Johnson, B., Taylor, R. S., McEldoon, K. (2010, March). Understanding the Equals Sign as a Gateway to Algebraic Thinking. Paper presented at the 2010 Society for Research on Educational Effectiveness (SREE) conference, Washington DC.

Taylor, R., Rittle-Johnson, B., Matthews, P., & McEldoon, K., (2009, March). Mapping children’s understanding of mathematical equivalence. Paper presented at the 2009 Society for Research on Educational Effectiveness conference, Washington DC.

Other Products:

Assessments available on project website!

On the Web:

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