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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.

BackgroundAutologous engineered skin substitutes comprised of keratinocytes, fibroblasts, and biopolymers can serve as an adjunctive treatment for excised burns. However, engineered skin lacks a vascular plexus at the time of grafting, leading to slower vascularization and reduced rates of engraftment compared with autograft. Hypothetically, vascularization of engineered skin grafts can be improved by treatment with proangiogenic agents at the time of grafting. Epoxyeicosatrienoic acids (EETs) are cytochrome P450 metabolites of arachidonic acid that are inactivated by soluble epoxide hydrolase (sEH). EETs have multiple biological activities and have been shown to promote angiogenesis. Inhibitors of sEH (sEHIs) represent attractive therapeutic agents because they increase endogenous EET levels. We investigated sEHI administration, alone or combined with EET treatment, for improved vascularization of engineered skin after grafting to mice.MethodsEngineered skin substitutes, prepared using primary human fibroblasts and keratinocytes, were grafted to full-thickness surgical wounds in immunodeficient mice. Mice were treated with the sEHI 1-trifluoromethoxyphenyl-3-(1-propionylpiperidin-4-yl) urea (TPPU), which was administered in drinking water throughout the study period, with or without topical EET treatment, and were compared with vehicle-treated controls. Vascularization was quantified by image analysis of CD31-positive areas in tissue sections.ResultsAt 2 weeks after grafting, significantly increased vascularization was observed in the TPPU and TPPU + EET groups compared with controls, with no evidence of toxicity.ConclusionsThe results suggest that sEH inhibition can increase vascularization of engineered skin grafts after transplantation, which may contribute to enhanced engraftment and improved treatment of full-thickness wounds.

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.)

This document titled "Units Construction and Coordination: Past, Present, and Future" provides an in-depth exploration of the concept of units construction and coordination in mathematics learning. It discusses the historical contributions of Jean Piaget and Ernst von Glasersfeld to the field and emphasizes the importance of units construction and coordination in understanding mathematical reasoning. The document offers various interpretations of units coordination, including place value operations and relationships between measurement units. It also examines the stages of units coordination and its relationship to other mathematical concepts such as fractions and algebra. The document includes summaries of five articles that further investigate units coordination in different contexts, such as algebraic reasoning, fractions knowledge, and teacher education. Overall, this document provides valuable insights into the complex nature of units construction and coordination in mathematics education. [Extracted from the article]

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.)