Anderson Norton

Continuing Research on Students’ Fraction Schemes

Directly or indirectly, The Fractions Project has launched several research programs in the area of students’ operational development. Research has not been restricted to fractions, but has branched out to proportional reasoning (e.g., Nabors 2003), multiplicative reasoning in general (e.g., Thompson and Saldanha 2003), and the development of early algebra concepts (e.g., Hackenberg accepted). This chapter summarizes current findings and future directions from the growing nexus of related articles and projects, which can be roughly divided into four categories. First, there is an abundance of research on students’ part-whole fraction schemes, much of which preceded The Fractions Project. The reorganization hypothesis contributes to such research by demonstrating how part-whole fraction schemes are based in part on students’ whole number concepts and operations.

Differential effects of learning games on mathematics proficiency

This study examined the effects of a learning game, [The Math App] on the mathematics proficiency of middle school students. For the study, researchers recruited 306 students, Grades 6–8, from two schools in rural southwest Virginia. Over a nine-week period, [The Math App] was deployed as an intervention for investigation. Students were assigned to game intervention treatment, and paper-and-pencil control conditions. For the game intervention condition, students learned fractions concepts by playing [The Math App]. In the analysis, students’ mathematical proficiency levels prior to the intervention were taken into account. Results indicate that students in the game intervention group showed higher mathematics proficiency than those in the paper-and-pencil group. Particularly, the significantly higher performances of intervention groups were noted among 7th graders and inclusion groups. The empirically derived results of the reported study could contribute to the field of educational video game research, which has not reached a consensus on the effects of games on students’ mathematics performance in classroom settings.

The effects of an educational video game on mathematical engagement

In an effort to maximizing success in mathematics, our research team implemented an educational video game in fifth grade mathematics classrooms in five schools in the Eastern US. The educational game was developed by our multi-disciplinary research team to achieve a hypothetical learning trajectory of mathematical thinking of 5th grade students. In this study, we examined overall engagement and three sub-domains of engagement as outcome variables after ten sessions of treatment with fifth grade students. The results showed that both male and female the video game group had slight increases in all engagement levels while students, particularly male, in the paper-and-pencil drill group displayed large decreases in all engagement levels. Implications of the study are 1) more fine-grained evidence of engagement in three sub-domains after implementing an educational video game, and 2) a consideration of gender differences in engagement levels in mathematics in the adoption of a video games.

Learning progression toward a measurement concept of fractions

BackgroundFractions continue to pose a critical challenge for students and their teachers alike. Mathematics education research indicates that the challenge with fractions may stem from the limitations of part-whole concepts of fractions, which is the central focus of fractions curriculum and instruction in the USA. Students’ development of more sophisticated concepts of fractions, beyond the part-whole concept, lays the groundwork for the later study of important mathematical topics, such as algebra, ratios, and proportions, which are foundational understandings for most STEM-related fields. In particular, the Common Core State Standards for Mathematics call for students to develop measurement concepts of fractions. In order to support such concepts, it is important to understand the underlying mental actions that undergird them so that teachers can design appropriate instructional opportunities. In this study, we propose a learning progression for the measurement concept of fractions—one that focuses on students’ mental actions and informs instructional design.ResultsA hierarchy of fraction schemes is charted outlining a progression from part-whole concepts to measurement concepts of fractions: (a) part-whole scheme (PWS), (b) measurement scheme for unit fractions (MSUF), (c) measurement scheme for proper fractions (MSPF), and (d) generalized measurement scheme for fractions (GMSF). These schemes describe concepts with explicit attention to the mental actions that undergird them. A synthesis of previous studies provides empirical evidence to support this learning progression.ConclusionsEvidence from the synthesis of a series of research studies suggests that children’s measurement concept of fractions develops through several distinct developmental stages characterized by the construction of distinct schemes. The mental actions associated with these schemes provide a guide for teachers to design instructional opportunities for children to advance their construction of a measurement concept of fractions. Specifically, the collection of quantitative studies suggest that students need opportunities to engage in activities that support two kinds of coordinations—the coordination of partitioning and iterating, and the coordination of three levels of units inherent in fractions. Instructional implications are discussed with example tasks and activities designed to provoke these coordinations.

Bridging Frameworks for Understanding Numerical Cognition

Journal of Numerical Cognition, 2018, Vol. 4(1), 1–8, doi:10.5964/jnc.v4i1.160 Published (VoR): 2018-06-07. *Corresponding author at: 434 McBryde Hall, Virginia Tech, Blacksburg, VA, 24061-0123, USA. E-mail: norton3@vt.edu This is an open access article distributed under the terms of the Creative Commons Attribution 4.0 International License, CC BY 4.0 (http://creativecommons.org/licenses/by/4.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

A cloud software system for visualization of game-based learning data collected on mobile devices

Digital game-based learning is a type of gameplay with a set of defined learning outcomes. Such gameplays are typically instrumented to collect data for assessing the learning outcomes. However, when the gameplay and data collection take place on a mobile device such as iPad, it becomes very difficult for a teacher to view the collected data on dozens of mobile devices used by students. This paper presents a cloud software system (CSS) under the client-server architecture to remedy this problem. We developed CSS using the Java platform, Enterprise Edition (Java EE) with IBM WebSphere Application Server, IBM DB2, and MongoDB. We also developed an educational iPad game called Taffy Town. Game-based learning data are collected on the iPad during the Taffy Town gameplay and are transmitted to our CSS over the Internet. Players (students) and teachers can login and view dynamically created visualizations of the collected learning data.

Eighty-Eight Thousand, Four Hundred and Eighteen (More) Ways to Fill Space.

As the name implies, space-filling curves possess the fascinating property that they fill regions of two-dimensional (or even higher dimensional) space with the continuous image of a line segment. Until the end of the nineteenth century, mathematicians considered such a feat impossible. How could a continuous function possibly transform a one-dimensional object into a two-dimensional object? In this paper we present Hilbert’s space-filling curve and generalize it to a new class of space-filling curves by establishing a connection with open-faced rook’s tours, which we define as a path that begins in one corner of an m × m grid (such as a chess board) and, traveling vertically and horizontally only, ends in an adjacent corner, having hit each square exactly once. We will see that every open-faced rook’s tour on an m × m chessboard defines a unique space-filling curve, up to symmetry of the square. Thus finding open-faced rooks tours is equivalent to inventing new space-filling curves.