Comparison Studies: Contrasting Cases:
Comparing multiple examples typically supports learning and transfer in laboratory studies and is considered a key feature of high quality mathematics instruction (e.g., Ball, 1993; Gentner, Loewenstein, & Thompson, 2003; Gick & Holyoak, 1983; Silver, Ghousseini, Gosen, Charalambous, & Strawhun, 2005). We have been experimentally evaluating whether comparison supports conceptual knowledge, procedural transfer and flexibility in mathematics. In a series of studies, comparison aided learning in the domains of algebraic equation solving, computational estimation (e.g. estimating the answer to 95 x 87), and decimal fractions. This research is supported by the Institute of Education Sciences, U.S. Department of Education.
In our first study, middle-school students’ learning of algebraic equation solving improved after comparing solution methods of worked examples (Rittle-Johnson & Star, 2007). We wanted to see if comparison would be beneficial in another domain, so we worked with younger students on computational estimation. These results indicated that students also benefit more from comparison than from seeing examples one at a time when learning computational estimation. Since comparison again improved student learning, we wanted to examine what kinds of comparisons would be most beneficial. Students scored higher on measures of procedural flexibility and conceptual knowledge when they compared solution methods, followed by comparing different problems with the same solution method (Rittle-Johnson & Star, in press).
However, could different types of comparison be more beneficial for those students with less prior knowledge? Students with low prior knowledge of algebra benefited most from comparing different problem types or from seeing examples sequentially, while students with higher prior algebra knowledge benefited most from comparing solution methods. Thus what kind of comparisons you use and when you use them matter depending on students’ prior knowledge. In addition, we examined whether comparison could only be useful if done with correct examples, or if comparison with incorrect examples would be beneficial as well. In the domain of decimal fractions, students who compared both correct and marked incorrect examples used more correct procedures and remembered more correct concepts than students who only compared correct examples. This indicates that use of comparison with incorrect examples helps students beyond the general benefits of comparison. Overall, comparison is a useful instructional strategy for mathematics learning.
On going work is evaluating what should be compared and when in the learning process different types of comparison are most useful.
CAREER Study: Developing Conceptual and Procedural Knowledge: The Roles of Self- and Instructional Explanations:
Competence in mathematics rests on children developing and linking their conceptual understanding and procedural skill (Hiebert, 1986). However, competing theories have been proposed regarding the developmental and pedagogical relations between the two types of knowledge. A majority focus on which type of knowledge develops first or is more important.
Contrary to these perspectives, I have found that conceptual and procedural knowledge can develop in an iterative process, such that improvements in one type of knowledge lead to improvements in the other, which in turn lead to improvements in the first (Rittle-Johnson & Alibali, 1999; Rittle-Johnson, Siegler, & Alibali, 2001). In this proposal, I develop an information-processing framework for the iterative development of conceptual and procedural knowledge. In particular, I focus on the roles of self- and instructional explanations for supporting knowledge growth. Self-explanation has emerged as a pervasive and domain general learning activity, and I propose that it helps to support conceptual and procedural knowledge growth. However, I contend that these self-generated explanations need not be discovered by the individual; rather, instructional explanations provide an important source of knowledge and can augment the benefits of self-explanation.
The proposed series of studies focuses on the development of conceptual and procedural knowledge for two core algebra topics in elementary school: mathematical equivalence and sequential patterns. I propose to a) examine and improve the reliability and discriminant validity of measures of conceptual and procedural knowledge for each topic, b) examine the roles of self- and instructional explanations as sources of conceptual and procedural knowledge growth, and c) explore who benefits most from prompts to self-explain. I also propose to design and teach an undergraduate course on the development of mathematical cognition that incorporates findings from these studies and includes weekly tutoring of struggling children.
*Currently, we are evaluating whether different types of instruction influence explanation quality and learning.
Principal Investigator: Dr. Bethany Rittle-Johnson
Here at the Children's Learning Lab, we study how children learn and how to improve their learning. Specifically, we explore children’s learning of key concepts and problem solving strategies in academic domains such as mathematics. We are also interested in the application of learning research to educational interventions.
Students
Undergraduate students may work in the Children's Learning Lab for course credit. A weekly commitment of 3 hours per credit hour is required. Students gain valuable research experience that is relevant to many future careers. It's also fun! Please contact Dr. Bethany Rittle-Johnson for more information.
Prospective Graduate students are encouraged to contact Dr. Rittle-Johnson directly and to access the Psychology at Vanderbilt website (http://www.peabody.vanderbilt.edu/psychology/) for information about the graduate program in developmental psychology.
