Search

Tensions between constructivist learning theories and objectivist learning theories in online courses implicate a need for a theoretical balance. This article describes organizing and enacting tasks that elicit constructivist learning of mathematics in an online mathematics teacher education course. The results focus on a conceptual framework that was developed to explicate this balance and show how the learning goal, evidence of student learning, and the teaching that occurs are built on the teaching philosophy tenet.

This study explored how students develop meaning of functions by building on their understanding of expressions and equations. A teaching experiment using design research was conducted in a sixth-grade classroom. The data was analyzed using a grounded theory approach to provide explanations about why events occurred within this teaching episode and what these events mean in terms of student learning of functions (Corbin & Strauss, 2014; Gravemeijer & Cobb, 2006). The findings revealed that understanding functions involved integrating their understanding of different meanings of variables such as "letters representing changing values" and "letters representing known values" to model the situation using an expression, and seeing linear relationships between the independent and dependent variable through graphing. This paper provides a learning progression for supporting early understandings of functions. We discuss implications for research on students' conceptions of variables and implications for fostering functional thinking.

We discuss the evolution of an instructional unit for supporting prospective elementary and middle grades' teachers' conceptions of fractions as measures. We first studied prospective teachers' conceptions of fractions using a task-based written assessment with discrete, linear, and circle representations. We designed an instructional unit that involved systematically introducing and connecting non-standard circles representations to other representations of fractions. We describe the assessments, learning outcomes of the instructional unit, limitations, and future directions.

Exploring non-Euclidean geometries is a common way for College Geometry instructors to highlight subtleties in Euclidean geometry. The concept of congruence, going undefined or informally defined in multiple axiomatic systems, is particularly susceptible to conflation with the idea of "same measure." Taxicab geometry provides a context in which congruence (defined as a relationship between figures that can be isometrically mapped to each other) and measure can be disambiguated by investigating the effects of transformations in both Euclidean and Taxicab geometry. This paper describes episodes from our classrooms in which students grappled with their understandings of congruence in Euclidean geometry while completing tasks involving Taxicab geometry.

Adolescent and children's concepts of multiplication and fractions have been linked to differences in the number of levels of units they coordinate. In this paper, we discuss relationships between adult students' conceptual structures for coordinating units and their pre-calculus understandings. We conducted interviews and calculus readiness assessments with 27 introductory calculus students at a public university in the United States. Results indicate that students assimilating multiplicative situations with three levels of units performed significantly better on calculus readiness assessments than students assimilating multiplicative situations with fewer than three levels of units. We discuss implications for instruction.

We describe the process of adjusting the balance between computer-based learning and peer interaction in a college algebra course. In our first experimental class, students used the adaptive-learning program ALEKS within an emporium-style format. Comparing student performance in the emporium format class with that in a traditional lecture format class, we found an improvement in procedural skills, but a weakness in the students' conceptual understanding of mathematical ideas. Consequently, we shifted to a blended format, cutting back on the number of ALEKS (procedural) topics and integrating activities that fostered student discourse about mathematics concepts. In our third iteration using ALEKS, we made use of ALEKS-generated data to design peer-to-peer activities that matched student progress.

Previous research has documented schemes and operations that undergird students' understanding of fractions. This prior research was based, in large part, on small-group teaching experiments. However, written assessments are needed in order for teachers and researchers to assess students' ways of operating on a whole-class scale. In this study, scores from written instruments used to assess students' fractions schemes and operations were examined for validity and reliability. Scores from the written assessments were correlated with scores from clinical interviews of 33 sixth graders. Results suggest that the written instruments provide reliable and valid measures for assessing the partitive unit fraction scheme and the splitting operation. However, there is no such evidence regarding the partitive fraction scheme. Implications for teachers and researchers are considered, and possible explanations for the scores from the written assessments related to the partitive fraction scheme are discussed. (Contains 3 tables, 5 figures, and 4 endnotes.)

In general, units coordination refers to the relationships that students can maintain between various units when working within a numerical situation. It is critical that middle school students learn to coordinate three levels of units not only because of their importance in understanding fractions but also because of their implications for multiplicative reasoning and algebra readiness. Students begin coordinating units and developing multiplicative reasoning while learning about whole numbers (Steffe and Olive 2010). Students can then reorganize these units coordinations in establishing knowledge of fractions and algebraic reasoning (Hackenberg and Lee 2015; Steffe 2002). In middle school, educators can expect to find students coordinating units at any of three stages, depending on the number of levels of units they can coordinate at once. In this article, the authors describe how they used variations of a bars task to assess students' stages of units coordination. Using the CandyDepot app, an educational game designed by the Learning Transformation Research Group at Virginia Tech, they adapt the barks task to involve other contexts involving three levels of units. The context involves placing candy bars into bundles and placing bundles into boxes. CandyDepot is an educational game designed to address the critical need for students to develop units coordination. With proper implementation--including opportunities for reflection and discussion--teachers can use the app to support students' units coordination activity across multiple contexts, including fractions knowledge and multiplicative reasoning (Boyce and Norton, under review).

The purpose of this paper is to illustrate cognitive challenges introduced by Common Core State Standards for Mathematics (2010) with regard to conceptualizing fractions. We focus on a strand of standards that appear across grades three through five, which is best represented in grade four, by standard 4.NF.4a: "[Students should] understand a fraction a/b as a multiple of 1/b. For example, use a visual fraction model to represent 5/4 as the product 5 x (1/4), recording the conclusion by the equation 5/4 = 5 x (1/4)." We argue that supporting such conceptualizations is a critical challenge for teachers and researchers to address in US education, but one fraught with conceptual hurdles for students. We adopt a scheme theoretic perspective and share the case study of a sixth-grade student, to illustrate the ways of operating students need to develop for meaningful attainment of the new standards. (Contains 3 tables and 2 figures.)