Developing Students\u27 Representational Fluency Using Virtual and Physical Algebra Balances

Both virtual and physical manipulatives are reported as effective learning tools when used with different groups of students in a variety of contexts to learn mathematical content. The use of multiple representations and the flexibility to translate among those representational forms facilitates students\u27 learning and has the potential to deepen their understanding. This classroom project involved two groups of third-grade students in a week-long unit focusing on algebraic relationships. The purpose of the unit was to engage students with different algebraic models and encourage students to use informal strategies to represent their relational thinking. The paper highlights examples of these student representations as evidence of the children\u27s developing algebraic thinking. Result from the pre- and post-test measures showed that students in the physical and virtual manipulative environments gained significantly in achievement and showed flexibility in translating and representing their understanding in multiple representations: manipulative model, pictorial, numeric and word problems. The researchers recorded field notes, interviewed students, and videotaped class sessions in order to identify unique features of the learning environments. The virtual environment had unique features that promoted student thinking such as: a) explicit linking of visual and symbolic modes; b) guided step-by-step support in algorithmic processes; and c) immediate feedback and self-checking system. In the physical environment, some unique features were: a) tactile features; b) opportunities for invented strategies; and c) mental mathematics. These results show that although the different manipulative models had different features, both the physical and virtual environments were effective in supporting students\u27 learning and encouraging relational thinking and algebraic reasoning

Third Graders Learn about Fractions Using Virtual Manipulatives: A Classroom study

With recent advances in computer technology, it is no surprise that the manipulation of objects in mathematics classrooms now includes the manipulation of objects on the computer screen. These objects, referred to as virtual manipulatives, are essentially replicas of physical manipulatives placed on the World Wide Web in the form of computer applets with additional advantageous features. The purpose of this project was to explore the effects of using several virtual manipulative ma·nip·u·la·tive adj. Serving, tending, or having the power to manipulate. n. Any of various objects designed to be moved or arranged by hand as a means of developing motor skills or understanding abstractions, especially in computer applets for instruction during a fraction unit in a third-grade classroom. The participants in this study were 19 third-grade students. During a two-week unit on fractions, students interacted with several virtual manipulative applets in a computer lab. Data sources in the project included a pre and posttest post·test n. A test given after a lesson or a period of instruction to determine what the students have learned. of students\u27 conceptual knowledge, a pre and posttest of students\u27 procedural computation Computation is a general term for any type of information processing that can be represented mathematically. This includes phenomena ranging from simple calculations to human thinking. ..... Click the link for more information. skills, student interviews, and a student attitudes survey. The results indicated a statistically significant improvement in students\u27 posttest scores on a test of conceptual knowledge, and a significant relationship between students\u27 scores on the posttests of conceptual knowledge and procedural knowledge Procedural knowledge is the knowledge exercised in the performance of some task. See below for the specific meaning of this term in cognitive psychology and intellectual property law. ..... Click the link for more information.. Student interviews and attitude surveys indicated that the virtual manipulatives (1) helped students in this class learn more about fractions by providing immediate and specific feedback, (2) were easier and faster to use than paper-and-pencil methods, and (3) enhanced students\u27 enjoyment while learning mathematics

Learning Mathematics with Technology: The Influence of Virtual Manipulatives on Different Achievement Groups

This study examined the influence of virtual manipulatives on different achievement groups during a teaching experiment in four fifth-grade classrooms. During a two-week unit focusing on two rational number concepts (fraction equivalence and fraction addition with unlike denominators) one low achieving, two average achieving, and one high achieving group participated in two instructional treatments (three groups used virtual manipulatives and one group used physical manipulatives). Data sources included pre- and post-tests of students’ mathematical content knowledge and videotapes of classroom sessions. Results of paired samples t-tests examining the three groups using virtual manipulatives indicated a statistically significant overall gain following the treatment. Follow-up paired samples individual t-tests on the low, average, and high achieving groups indicated a statistically significant gain for students in the low achieving group, but only numerical gains for students in the average and high achieving groups. There were no significant differences between the average achieving student groups in the virtual manipulatives and physical manipulatives treatments. Qualitative data gathered during the study indicated that the different achievement groups experienced the virtual manipulatives in different ways, with the high achieving group recognizing patterns quickly and transitioning to the use of symbols, while the average and low achieving groups relied heavily on pictorial representations as they methodically worked step-by-step through processes and procedures with mathematical symbols

Learning to Question: Categories of Questioning Used by Preservice Teachers During Diagnostic Mathematics Interviews

Developing appropriate questioning techniques is an important part of mathematics teaching and assessment. This study examined the questioning strategies used by 48 preservice teachers during one-on-one diagnostic mathematics interviews with children. Each participant conducted an audiotaped interview with one child, followed by an analysis and reflection of the interview. Data were analyzed to develop general categories of questions used by the preservice teachers. These categories included: 1) checklisting, 2) instructing rather than assessing, and 3) probing and follow-up questions. The analyses and reflections completed by preservice teachers indicated that using the diagnostic interview format allowed them to recognize and reflect on effective questioning techniques. Through an examination of these categories of questions, we offer suggestions for teaching the skill of mathematics questioning in preservice teacher education courses

Exploring the Phenomenon of Distance in Children\u27s Interactions with Touchscreen Digital Mathematics Games

This study examines the construct of distance – the degree of difficulty of interacting with something – as part of activity involving children using touchscreen digital games to learn mathematics. Ten fifth-grade children engaged in video-recorded semi-structured task-based interviews in which they used two touchscreen digital mathematics games on a touchscreen tablet and responded to semi-structured follow-up questions. Qualitative data analysis was iterative, featuring analytic memoing and eclectic coding techniques to identify themes related to distance. In advanced coding stages, magnitude coding was used to characterize the degree of distance present. Findings provide evidence of the presence of distance, changes in distance, and interactions between distance types throughout the activity. In particular, both mathematical distance and technological distance were present, changed in various ways, and often influenced each other. Implications include the relevance of distance for designing, implementing, and researching educational technology

Tools for cognition: Student free access to manipulative materials in controlversus autonomy-oriented middle grades teachers’ classrooms

This study investigated how middle grades students provided with free access to manipulative materials use these mathematical tools in classrooms where their teachers are identified as Control-Oriented and Autonomy-Oriented. Also of interest in this investigation was how Control-Oriented and Autonomy-Oriented teachers administered the free access treatment in their classrooms. A Pre -- Post -1 Post -2 design was used with two treatments. During Treatment 1, teachers used the manipulatives for mathematics instruction using the strategies learned in the summer professional development workshop. During Treatment 2, teachers provided students with free access to the manipulative materials. Results indicated teachers\u27 control orientations--control versus autonomy--were significantly different. The study also reveals that when students are allowed some measure of control in the selection and use of manipulative materials given the time to overcome their initial apprehension, they will spontaneously and selectively use these materials effectively as appropriate mathematical tools to mediate learning. Contains 20 references

Revisiting the Definition of a Virtual Manipulative

In 2002, Moyer, Bolyard and Spikell defined a virtual manipulative as an “an interactive, Web-based visual representation of a dynamic object that presents opportunities for constructing mathematical knowledge” (p. 373). The purpose of this chapter is to revisit, clarify and update the definition of a virtual manipulative. After clarifying what a virtual manipulative is and what it is not, we propose an updated definition for virtual manipulative: an interactive, technology-enabled visual representation of a dynamic mathematical object, including all of the programmable features that allow it to be manipulated, that presents opportunities for constructing mathematical knowledge. The chapter describes the characteristics of five of the most common virtual manipulative environments in use in education: single-representation, multi-representation, tutorial, gaming and simulation

Learning Mathematics with Technology: The Influence of Virtual Manipulatives on Different Achievement Groups

This study examined the influence of virtual manipulatives on different achievement groups during a teaching experiment in four fifth-grade classrooms. During a two-week unit focusing on two rational number concepts (fraction equivalence and fraction addition with unlike denominators) one low achieving, two average achieving, and one high achieving group participated in two instructional treatments (three groups used virtual manipulatives and one group used physical manipulatives). Data sources included pre- and post-tests of students’ mathematical content knowledge and videotapes of classroom sessions. Results of paired samples t-tests examining the three groups using virtual manipulatives indicated a statistically significant overall gain following the treatment. Follow-up paired samples individual t-tests on the low, average, and high achieving groups indicated a statistically significant gain for students in the low achieving group, but only numerical gains for students in the average and high achieving groups. There were no significant differences between the average achieving student groups in the virtual manipulatives and physical manipulatives treatments. Qualitative data gathered during the study indicated that the different achievement groups experienced the virtual manipulatives in different ways, with the high achieving group recognizing patterns quickly and transitioning to the use of symbols, while the average and low achieving groups relied heavily on pictorial representations as they methodically worked stepby- step through processes and procedures with mathematical symbols

A Review of the Literature on Mathematics and Science Teacher Quality

A large body of literature exists that examines teacher quality characteristics and the relationship of indicators of those characteristics to teacher effectiveness. This existing research literature broadly views teacher quality research without illuminating specific areas of teacher quality, such as mathematics and science. In an effort to focus the literature base for researchers and policymakers more narrowly, this review specifically examines teacher quality as it relates to mathematics and science teaching and learning. The review highlights key policy and practitioner perspectives, provides a focused synthesis on current research findings on mathematics and science teacher quality, and suggests areas of research that are limited in the literature