Learners’ Difficulties with Quantitative Units in Algebraic Word Problems and the Teacher’s Interpretation of those Difficulties

This study examines 8th grade students’ coordination of quantitative units arising from word problems that can be solved via a set of equations that are reducible to a single equation with a single unknown. Along with Unit-Coordination, Quantitative Unit Conservation also emerges as a necessary construct in dealing with such problems. We base our analysis within a framework of quantitative reasoning (Thompson, 1988; 1989; 1993; 1995) and a theory of children’s units-coordination with different levels of units (Steffe 1994) that both encompass and are extended by these two constructs. Our data consist of videotaped classroom lessons, student interviews, and teacher interviews. Ongoing analyses of these data were conducted during the teaching sequence. A retrospective analysis using constant comparison methodology was then undertaken during which the classroom video, related student interviews, and teacher interviews were revisited many times in order to generate a thematic analysis. Our results indicate that the identification and coordination of the units involved in the problem situation are critical aspects of quantitative reasoning and need to be emphasized in the teaching-learning process. We also concluded that unit coordination and unit conservation are cognitive prerequisites for constructing a meaningful algebraic equation when reasoning quantitatively about a situation.     

Meta-Representation in an Algebra I Classroom

We describe how 1 Algebra I teacher and her 8th-grade students used meta-representational knowledge when generating and evaluating equations to solve word problems. Analyzing data from a sequence of 4 lessons, we found that the teacher and her students used criteria for evaluating equations, in addition to other types of knowledge (e.g., different interpretations of the equal sign) previously reported in the literature. Moreover, the teacher and her students had trouble understanding one another's proposed algebraic models of problem situations due to differences in the criteria that each applied, and this impeded learning. These findings (a) extend an accumulating body of evidence for the role meta-representation plays in mathematics and science learning and (b) add a new dimension to researchers' growing understanding of what teachers must know in order to teach algebra and other complex mathematics and science topics effectively.     

Facilitating preservice teachers’ development of mathematics knowledge for teaching arithmetic operations

Mathematical tasks, centered on arithmetic word problems, are discussed as the basis of an approach to facilitate preservice elementary teachers’ development of mathematical knowledge for teaching arithmetic operations. The approach consists of three groups of tasks that allow students to reflect on their initial knowledge, explore arithmetic word problem situations from multiple perspectives, and apply their resulting knowledge, respectively. Findings from the investigation of the approach indicate that the tasks can provide an effective and meaningful basis to help preservice teachers to develop deeper understandings of the arithmetic operations and their relationships and pedagogical-mathematical knowledge for teaching the arithmetic operations conceptually.     

Multilevel Analysis of Multiple-baseline Data Evaluating Precision Teaching as an Intervention for Improving Fluency in Foundational Reading Skills for at Risk Readers

In this article, multiple-baseline across participants designs were used to evaluate the impact of a precision teaching (PT) program, within a Tier 2 Response to Intervention framework, targeting fluency in foundational reading skills with at risk kindergarten readers. Thirteen multiple-baseline design experiments that included participation from 35 kindergarten students were included in the current analysis. By combining data from a series of multiple-baseline studies using a multilevel model, we analyzed outcomes to determine an estimated treatment effect. The results indicate that across participants and across studies, PT showed an average increase of 15 correct responses in foundational reading skills per minute. This immediate influence on rates of core reading skills was statistically significant. Moreover, the treatment had a significant positive effect on the time trend indicating that rates in foundational reading skills increased more across time during the treatment phase compared to the baseline phase. The study provides evidence that the obtained rates in foundational reading skills at the end of the intervention were retained after removing the intervention. From the outcomes of the multilevel model, PT can be considered as a promising Tier 2 intervention to increase reading fluency with individuals who are at risk of reading failure.     

Inservice Teacher Development in Mathematical Problem Solving

A humanistic perspective provided the basis for a problem-solving oriented teacher inservice program. The program was designed to provide opportunities that allowed elementary teachers to focus on personal experience as a way of achieving self-understanding and a way of reconstructing their personal meanings about problem solving and problem-solving instruction. Impact of the program was studied through interviews with the six participants and observations of their teaching. The results indicated that the program had a positive effect on the participants' beliefs about and teaching of problem solving. This revised version was published online in July 2006 with corrections to the Cover Date.