Interactive video: A bridge between motion and math

This paper examines the characteristics of interactive digitized video as a medium in which motion is presented to students learning graphical representations. We situate graphs of motion as early topics in learning calculus, the bugaboo of many math students. In comparing video to both everyday perceptions and mathematical representations, we construct a conceptual framework that compares these three contexts along several dimensions: object extent, scale, time, and space. We then examine one student's experience constructing graphs of her own design from a video image and describe her work in the context of the our conceptual framework. To further specify the unique characteristics of video, we compare it as a medium with that of computer simulations of motion, in particular as studied by diSessa et al. (1991).The authors are listed in alphabetical order.     

Potential scenarios for Internet use in the mathematics classroom

Research on the influence of multiple representations in mathematics education gained new momentum when personal computers and software started to become available in the mid-1980s. It became much easier for students who were not fond of algebraic representations to work with concepts such as function using graphs or tables. Research on how students use such software showed that they shaped the tools to their own needs, resulting in an intershaping relationship in which tools shape the way students know at the same time the students shape the tools and influence the design of the next generation of tools. This kind of research led to the theoretical perspective presented in this paper: knowledge is constructed by collectives of humans-with-media. In this paper, I will discuss how media have shaped the notions of problem and knowledge, and a parallel will be developed between the way that software has brought new possibilities to mathematics education and the changes that the Internet may bring to mathematics education. This paper is, therefore, a discussion about the future of mathematics education. Potential scenarios for the future of mathematics education, if the Internet becomes accepted in the classroom, will be discussed.     

Teachers’ views on dynamically linked multiple representations, pedagogical practices and students’ understanding of mathematics using TI-Nspire in Scottish secondary schools

Do teachers find that the use of dynamically linked multiple representations enhances their students’ relational understanding of the mathematics involved in their lessons and what evidence do they provide to support their findings? Throughout session 2008–2009, this empirical research project involved six Scottish secondary schools, two mathematics teachers from each school and students from different ages and stages. Teachers used TI-Nspire PC software and students the TI-Nspire handheld technology. This technology is specifically designed to allow dynamically linked multiple representations of mathematical concepts such that pupils can observe links between cause and effect in different representations such as dynamic geometry, graphs, lists and spreadsheets. The teachers were convinced that the use of multiple representations of mathematical concepts enhanced their students’ relational understanding of these concepts, provided evidence to support their argument and described changes in their classroom pedagogy.     

Mathematics teaching practices with technology that support conceptual understanding for Latino/a students

We analyze how three seventh grade mathematics teachers from a majority Latino/a, linguistically diverse region of Texas taught the same lesson on interpreting graphs of motion as part of the Scaling Up SimCalc study (Roschelle et al., 2010). The students of two of the teachers made strong learning gains as measured by a curriculum-aligned assessment, while the students of the third teacher were less successful. To investigate these different outcomes, we compare the teaching practices in each classroom, focusing on the teachers’ use of class time and instructional format, their use of mathematical discourse practices in whole-class discussions, and their responses to student contributions. We show that the more successful teachers allowed time for students to use the curriculum and software and discuss it with peers, that they used formal mathematical discourse along with less formal language, and that they responded to student errors using higher-level moves. We conclude by discussing implications for teachers and mathematics educators, with special attention to issues related to the mathematics education of Latinos/as.     

Developing mathematical modelling skills: The role of CAS

Mathematical modelling as one component of problem solving is an important part of the mathematics curriculum and problem solving skills are often the most quoted generic skills that should be developed as an outcome of a programme of mathematics in school, college and university. Often there is a tension between mathematics seen at all levels as ‘a body of knowledge’ to be delivered at all costs and mathematics seen as a set of critical thinking and questioning skills. In this era of powerful software on hand-held and computer technologies there is an opportunity to review the procedures and rules that form the ‘body of knowledge’ that have been the central focus of the mathematics curriculum for over one hundred years. With technology we can spend less time on the traditional skills and create time for problem solving skills. We propose that mathematics software in general and CAS in particular provides opportunities for students to focus on the formulation and interpretation phases of the mathematical modelling process. Exploring the effect of parameters in a mathematical model is an important skill in mathematics and students often have difficulties in identifying the different role of variables and parameters This is an important part of validating a mathematical model formulated to describe, a real world situation. We illustrate how learning these skills can be enhanced by presenting and analysing the solution of two optimisation problems.     

Visage—Visualization of algorithms in discrete mathematics

Which route should the garbage collectors' truck take? Just a simple question, but also the starting point of an exciting mathematics class. The only “hardware” you need is a city map, given on a sheet of paper or on the computer screen. Then lively discussions will take place in the classroom on how to find an optimal routing for the truck. The aim of this activity is to develop an algorithm that constructs Eulerian tours in graphs and to learn about graphs and their properties. This teaching sequence, and those stemming from discrete mathematics, in particular combinatorial optimization, are ideal for training problem solving skills and modeling—general competencies that, influenced by the German National Standards, are finding their way into curricula. In this article, we investigate how computers can help in providing individual teaching tools for students. Within the Visage project we focus on electronic activities that can enhance explorations with graphs and guide studients even if the teacher is not available—without taking away freedom and creativity. The software package is embedded into a standard DGS, and it offers many pre-built and teacher-customizable tools in the area of graph algorithms. Until now, there are no complete didactical concepts for teaching graph algorithms, in particular using new media. We see a huge potential in our methods, and the topic is highly requested on part of the teachers, as it introduces a modern and highly relevant part of mathematics into the curriculum.     

Using interactive algebra software to support a discourse community

In this work we studied the impact of using NuCalc, an interactive computer algebra software, on the development of a discourse community in a college level mathematics class. Qualitative and quantitative data were collected over the course of 3 weeks of instruction. We examined the influence of the software on: group interactions; the mathematical investigations of learners; and the teacher’s interactions with students. Data points to four distinct ways in which the presence of NuCalc positively impacted the learning community we studied: (1) it served as a tool for extending students’ mathematical thinking, (2) it motivated students’ engagement in group discourse, (3) it became a tool for mediating discourse, (4) it became a catalyst for refining the culture of classroom, shifting the patterns of interactions between the teacher and learners.     

On the use of computational tools to promote students’ mathematical thinking

This column will publish short (from just a few paragraphs to ten or so pages), lively and intriguing computer-related mathematics vignettes. These vignettes or snapshots should illustrate ways in which computer environments have transformed the practice of mathematics or mathematics pedagogy. They could also include puzzles or brain-teasers involving the use of computers or computational theory. Snapshots are subject to peer review. In this snapshot students employ dynamic geometry software to find great mathematical richness around a seemingly simple question about rectangles.

Editor: Uri Wilensky

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Attention to meaning by algebra teachers

Non-attendance to meaning by students is a prevalent phenomenon in school mathematics. Our goal is to investigate features of instruction that might account for this phenomenon. Drawing on a case study of two high school algebra teachers, we cite episodes from the classroom to illustrate particular teaching actions that de-emphasize meaning. We categorize these actions as pertaining to (a) purpose of new concepts, (b) distinctions in mathematics, (c) mathematical terminology, and (d) mathematical symbols. The specificity of the actions that we identify allows us to suggest several conjectures as to the impact of the teaching practices observed on student learning: that students will develop the belief that mathematics involves executing standard procedures much more than meaning and reasoning, that students will come to see mathematical definitions and results as coincidental or arbitrary, and that students’ treatment of symbols will be largely non-referential.     

Representation in Computational Environments: Epistemological and Social Distance

Computational environments have the potential to provide new representational resources and new ways of supporting teaching and learning of mathematics. In this paper, we seek to characterize relationships between the representations offered by particular technologies and other representations commonly available in the classroom context, using the notion of ‘distance’. Distance between representations in different media may be epistemological, affecting the nature of the mathematical concepts available to students, or may be social, affecting pedagogic relationships in the classroom and the ease with which the technology may be adopted in particular classroom or national contexts. We illustrate these notions through examples taken from cross-experimentation of computational environments in national contexts different from those in which they were developed. Implications for the design and dissemination of computational environments for use in learning mathematics are discussed.     

Motion, speed, and other ideas that “should be put in books”

This paper focuses on a portion of a research project involving a group of inner-city middle school students who used SimCalc simulation software over the course of an entire school year to investigate ideas relating to graphical representations of motion and speed. The classroom environment was one in which students openly defended and justified their thinking as they actively explored and solved rich mathematical problems. The activities, generally speaking, involved functions that were intended to tap students’ real world intuitions as well as prior mathematical skills and understandings about speed, motion, and other graphical representations that underlie the mathematics of motion. Results indicate that these students did build ideas related to those concepts. This paper will provide documentation of the ways in which these students interpreted graphical representations involving linear and quadratic functions that are associated with constant and linearly changing velocities, respectively.     

Using comics-based representations of teaching, and technology, to bring practice to teacher education courses

This article situates comic-based representations of teaching in the long history of tensions between theory and practice in teacher education. The article argues that comics can be semiotic resources in learning to teach and suggests how information technologies can support experiences with comics in university mathematics methods courses that (a) help learners see the mathematical work of teaching in lessons they observe, (b) allow candidates to explore tactical decision-making in teaching, and (c) support preservice teachers in rehearsing classroom interactions.     

About the use of the TI-92 for an open learning approach to power functions

In this report we present the results of a teaching study introducing the concept “power function” using a graphing calculator. The focus of our attention is on the development of the understanding of 15–16 year-old mathematics students. In the centre of our interest is their learning through graphs of power functions by discovering the properties of graphs. Our report presents the mathematical and social constructivist background together with a new deliberately constructivist approach beginning the teaching experiment with an open question. The students' cognitive and intuitive strategies and their attitudes towards computer algebra are described.     

Examining Secondary Mathematics Teachers' Opportunities to Develop Mathematically in Professional Learning Communities

To make progress toward ambitious and equitable goals for students’ mathematical development, teachers need opportunities to develop specialized ways of knowing mathematics such as mathematical knowledge for teaching (MKT) for their work with students in the classroom. Professional learning communities (PLCs) are a common model used to support focused teacher collaboration and, in turn, foster teacher development, instructional improvement, and student outcomes. However, there is a lack of specificity in what is known about teachers’ work in PLCs and what teachers can gain from those experiences, despite broad claims of their benefit. We discuss an investigation of the work of secondary mathematics teachers in PLCs at two high schools to describe and explicate possible opportunities for teachers to develop the mathematical knowledge needed for the work of teaching and the ways in which these opportunities may be pursued or hindered. The findings show that, without pointed focus on mathematical content, opportunities to develop MKT can be rare, even among mathematics teachers. Two detailed images of teacher discussion are shared to highlight these claims. This article contributes to the ongoing discussion about the affordances and limitations of PLCs for mathematics teachers, considerations for their use, and how they can be supported.     

Preparing K‐8 Preservice Teachers in a Content Course for Standards‐Based Mathematics Pedagogy

Many K–8 preservice teachers have not experienced learning mathematics in a standards‐based classroom. This article describes a mathematics content course designed to provide preservice teachers experiences in learning mathematics that will help build a solid foundation for a standards‐based methods course. The content course focuses on developing preservice teachers' mathematical knowledge, as well as helping them realize what it means to learn mathematics that is taught using the pedagogy in the Principles and Standards for School Mathematics ( National Council of Teachers of Mathematics, 2000 ). Furthermore, findings are presented from a study on this course that describe students' pre‐ and postcourse beliefs, attitudes, and perceptions of what it means to learn and teach mathematics. These findings provide evidence that the students in the study are beginning to understand what is meant by a standards‐based classroom. Data were collected from surveys and interviews. Quotes from the students who aspire to be elementary teachers are used throughout the article to support the points.     

Middle School Mathematics Teachers Learning to Teach With Calculators and Computers

This paper reports the results of a project in which experienced middle grades mathematics teachers immersed themselves in calculator and computer use for both doing and teaching mathematics and prepared themselves as leaders for communicating their knowledge to colleagues. Project evaluation included formal observation of students while they used technology in learning mathematics. Classroom observation data suggested that computers hold somewhat more attraction for students than calculators. Overall, students in all 13 classes, independent of the type of technology used, were observed to be off-task 3% of the time. These data suggested a classroom environment in which the teacher worked hard to engage students in mathematical activity. The fact that students were observed off-task so little is encouraging. The difference in off-task behaviors for calculators versus computers suggests that different technologies will indeed have different effects on students. It appears that the introduction of technologies in classrooms altered the ways teachers taught.     

Emergent modeling: discrete graphs to support the understanding of change and velocity

In this paper we focus on an instructional sequence that aims at supporting students in their learning of the basic principles of rate of change and velocity. The conjectured process of teaching and learning is supposed to ensure that the mathematical and physical concepts will be rooted in students’ understanding of everyday-life situations. Students’ inventions are supported by carefully planned activities and tools that fit their reasoning. The central design heuristic of the instructional sequence is emergent modeling. We created an educational setting in three tenth grade classrooms to investigate students’ learning with this sequence. The design research is carried out in order to contribute to a local instruction theory on calculus. Classroom events and computer activities are video-taped, group work is audio-taped and student materials are collected. Qualitative analyses show that with the emergent modeling approach, the basic principles of calculus can be developed from students’ reasoning on motion, when they are supported by discrete graphs.