Susan B. Empson

A Longitudinal Study of Invention and Understanding in Children's Multidigit Addition and Subtraction.

This 3-year longitudinal study investigated the development of 82 childrens understanding of multidigit number concepts and operations in Grades 1-3. Students were individually interviewed 5 times on a variety of tasks involving base-ten number concepts and addition and subtraction problems. The study provides an existence proof that children can invent strategies for adding and subtracting and illustrates both what that invention affords and the role that different concepts may play in that invention. About 90% of the students used invented strategies. Students who used invented strategies before they learned standard algorithms demonstrated better knowledge of base-ten number concepts and were more successful in extending their knowledge to new situations than were students who initially learned standard algorithms. An understanding of most fundamental mathematics concepts and skills develops over an extended period of time. Although cross-sectional studies can provide snapshots of the development of these concepts at particular points in time, longitudinal studies provide a more complete perspective; however, relatively few studies have traced the development of fundamental mathematics concepts in children over more than a single year. In this paper we report the results of a 3-year longitudinal study of the growth of childrens understanding of addition and subtraction involving multidigit numbers. We focus particularly on childrens construction of invented strategies for adding and subtracting multidigit numbers. The overarching goal of the study was to investigate the role that invented strategies may play in developing an understanding of multidigit addition and subtraction concepts and procedures. We trace the development and use of invented addition and subtraction strategies and examine the relation of these strategies to the development of fundamental knowledge of base-ten number concepts and the use of standard addition and subtraction algorithms. Finally, we consider what the use of invented strategies affords by way of avoiding systematic errors and extending knowledge of basic multidigit operations to new problem situations.

Integration of Technology, Curriculum, and Professional Development for Advancing Middle School Mathematics Three Large-Scale Studies

The authors present three studies (two randomized controlled experiments and one embedded quasi-experiment) designed to evaluate the impact of replacement units targeting student learning of advanced middle school mathematics. The studies evaluated the SimCalc approach, which integrates an interactive representational technology, paper curriculum, and teacher professional development. Each study addressed both replicability of findings and robustness across Texas settings, with varied teacher characteristics (backgrounds, knowledge, attitudes) and student characteristics (demographics, levels of prior mathematics knowledge). Analyses revealed statistically significant main effects, with student-level effect sizes of .63, .50, and .56. These consistent gains support the conclusion that SimCalc is effective in enabling a wide variety of teachers in a diversity of settings to extend student learning to more advanced mathematics.

Low-Performing Students and Teaching Fractions for Understanding: An Interactional Analysis

This article presents an analysis of two low-performing students’ experiences in a firstgrade classroom oriented toward teaching mathematics for understanding. Combining constructs from interactional sociolinguistics and developmental task analysis, I investigate the nature of these students’ participation in classroom discourse about fractions. Pre- and post-instruction interviews documenting learning and analysis of classroom interactions suggest mechanisms of that learning. I propose that three main factors account for these two students’ success: use of tasks that elicited the students’ prior understanding, creation of a variety of participant frameworks (Goffman, 1981) in which the students were treated as mathematically competent, and frequency of opportunities for identity-enhancing interactions.

The Algebraic Nature of Fractions: Developing Relational Thinking in Elementary School

The authors present a new view of the relationship between learning fractions and learning algebra that (1) emphasizes the conceptual continuities between whole-number arithmetic and fractions; and (2) shows how the fundamental properties of operations and equality that form the foundations of algebra are used naturally by children in their strategies for problems involving operating on and with fractions. This view is grounded in empirical research on how algebraic structure emerges in young children’s reasoning. Specifically, the authors argue that there is a broad class of children’s strategies for fraction problems motivated by the same mathematical relationships that are essential to understanding high-school algebra and that these relationships cannot be presented to children as discrete skills or learned as isolated rules. The authors refer to the thinking that guides such strategies as Relational thinking.

Exploratory Study of Informal Strategies for Equal Sharing Problems of Students with Learning Disabilities

Little to no information exists explaining the nature of conceptual gaps in understanding fractions for students with learning disabilities (LD); such information is vital to practitioners seeking to develop instruction or interventions. Many researchers argue such knowledge can be revealed through student’s problem-solving strategies. Despite qualitative differences in thinking and representation use in students with LD that may exist, existing frameworks of student’s strategies for solving fraction problems are not inclusive of students with LD. This exploratory study extends existing literature by documenting the strategies students with LD use when solving fraction problems. Clinical interviews were conducted with 10 students across the third, fourth, and fifth grades (N = 10). Results indicate students with LD used similar strategies as previously reported in research involving non-LD students, although the dominant strategy utilized was less advanced and the range of strategy use was relatively compact. Researchers suggest the nature of conceptual gaps students with LD display in their understanding of fractions originates from a malleable source. Implications for instruction and assessment are presented.

A Principal Components Model of Simcalc Mathworlds

This study examines changes in student knowledge through a pretest/posttest assessment using data from an experimental research project on the effects of dynamic software on student outcomes in 7th-grade classrooms. Through a principal components analysis, we found that students who used the software had test results that clustered concepts differently than students who did not, grouping together test questions involving various representations of rate and proportionality. Students who did not use the software had test results that grouped test questions together based on surface features and proximity. We present a model showing how increased access to dynamic software leads to an increased ability to fuse concepts, where students add to their mental models and link various representations of the same mathematical concepts together.

Responsive Teaching from the Inside Out: Teaching Base Ten to Young Children.

Abstract Decision making during instruction that is responsive to children’s mathematical thinking is examined reflexively by the researcher in the context of teaching second graders. Focus is on exploring how the research base on learning informs teaching decisions that are oriented to building on children’s sound conceptions. The development of four children’s understanding of base ten over a ten-week period is tracked. “The work of teaching orients teachers to constantly consider their next moves” (Jacobs, Lamb, Philipp, & Schapelle, 2011, p. 98).

Scaling Up Innovative Mathematics in the Middle Grades: Case Studies of “Good Enough” Enactments

This set of comparative case studies examined three teachers’ enactments of a mathematics replacement unit to teach concepts of rate and linear function using SimCalc. The cases were drawn from a larger set of 37 teachers in a randomized experimental study that documented a significant main effect on student achievement. The goal was to identify the learning resources that supported students’ opportunities to make conceptual connections and the various configurations of learning resources among classrooms. Content Maps of lessons were created to model the content that was expressed in whole-group recitations. Two of the teachers had high gains, but used different configurations of learning resources to realize them. In contrast, the third teacher had a mean class gain that was below average, despite an MKT score that was among the highest in the treatment sample. In any case, no teacher implemented the curriculum in ways that were completely consistent with SimCalc developers’ vision. Scaling up instructional innovations entails trade-offs. Understanding these trade-offs can help educators recognize successful, “good enough” enactments of curriculum.