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.     

Principal Components Along Quiver Representations

Quiver representations arise naturally in many areas across mathematics. Here we describe an algorithm for calculating the vector space of sections, or compatible assignments of vectors to vertices, of any finite-dimensional representation of a finite quiver. Consequently, we are able to define and compute principal components with respect to quiver representations. These principal components are solutions to constrained optimisation problems defined over the space of sections and are eigenvectors of an associated matrix pencil.

     

Learning to See: Making Sense of the Mathematics of Change in Middle School

In this article, we will describe the results of a study of 6th grade students learning about the mathematics of change. The students in this study worked with software environments for the computer and the graphing calculator that included a simulation of a moving elevator, linked to a graph of its velocity vs. time. We will describe how the students and their teacher negotiated the mathematical meanings of these representations, in interaction with the software and other representational tools available in the classroom. The class developed ways of selectively attending to specific features of stacks of centimeter cubes, hand-drawn graphs, and graphs (labeled velocity vs. time) on the computer screen. In addition, the class became adept at imagining the motions that corresponded to various velocity vs. time graphs. In this article, we describe this development as a process of learning to see mathematical representations of motion. The main question this article addresses is: How do students learn to see mathematical representations in ways that are consistent with the discipline of mathematics? This revised version was published online in July 2006 with corrections to the Cover Date.     

When an argument is the content: Rational number comprehension through conversions across registers

This article reframes previously identified misconceptions about repeating decimals by describing these misconceptions as limited understandings of how mathematics concepts are referenced. In particular, misconceptions about repeating decimals and their quotient of integer representations are recast as limited understandings of mathematics as a discipline that derives its content from representational systems and the denotations they provided. Under this framework, arguments (e.g., proofs) that convert repeating decimals to their quotient of integer representations provide content for “rational number,” which is represented in multiple ways, each offering distinct opportunities for mathematical activity. The notion of an argument as content is illustrated as arguments providing access to a concept. One Grade 8 student’s struggle with understanding rational number is used to illustrate this framework and its implications for teaching and learning.     

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.     

Turnaround Students in High School Mathematics: Constructing Identities of Competence Through Mathematical Worlds

This study investigates prospective secondary teachers’ cognitive difficulties and mathematical ideas involved in making connections among representations. We implemented a three-week teaching unit to help prospective secondary mathematics teachers develop understanding of big ideas that are critical to formulating connections among representations, in the context of conic curves. Qualitative analysis of data showed that most undergraduate mathematics majors and minors in this study struggled with variation, the Cartesian Connection, and other affiliated ideas such as graph as a locus of points. Furthermore, they were unable to identify basic metric relations encoded in algebraic expressions such as the distance between points, which further compounded their difficulties in making connections among representations. We argue that mathematics teacher education needs more focus on these ideas so that their graduates can successfully teach these big ideas in their future instruction.     

Flexible and adaptive use of strategies and representations in mathematics education

The flexible and adaptive use of strategies and representations is part of a cognitive variability, which enables individuals to solve problems quickly and accurately. The development of these abilities is not simply based on growing experience; instead, we can assume that their acquisition is based on complex cognitive processes. How these processes can be described and how these can be fostered through instructional environments are research questions, which are yet to be answered satisfactorily. This special issue on flexible and adaptive use of strategies and representations in mathematics education encompasses contributions of several authors working in this particular field. They present recent research on flexible and adaptive use of strategies or representations based on theoretical and empirical perspectives. Two commentary articles discuss the presented results against the background of existing theories.     

Music as Embodied Mathematics: A Study of a Mutually Informing Affinity

The argument examined in this paper is that music – when approached through making and responding to coherent musical structures,facilitated by multiple, intuitively accessible representations – can become a learning context in which basic mathematical ideas can be elicited and perceived as relevant and important. Students' inquiry into the bases for their perceptions of musical coherence provides a path into the mathematics of ratio,proportion, fractions, and common multiples. Ina similar manner, we conjecture that other topics in mathematics – patterns of change,transformations and invariants – might also expose, illuminate and account for more general organizing structures in music. Drawing on experience with 11–12 year old students working in a software music/math environment, we illustrate the role of multiple representations, multi-media, and the use of multiple sensory modalities in eliciting and developing students' initially implicit knowledge of music and its inherent mathematics. This revised version was published online in July 2006 with corrections to the Cover Date.     

Prospective middle grade mathematics teachers’ knowledge of algebra for teaching

This study examined prospective middle grade mathematics teachers’ knowledge of algebra for teaching with a focus on knowledge for teaching the concept of function. 115 prospective teachers from an interdisciplinary program for mathematics and science middle teacher preparation at a large public university in the USA participated in a survey. It was found that the participants had relatively limited knowledge of algebra for teaching. They also revealed weakness in selecting appropriate perspectives of the concept of function and flexibly using representations of quadratic functions. They made numerous mistakes in solving quadratic or irrational equations and in algebraic manipulation and reasoning. The participants’ weakness in connecting algebraic and graphic representations resulted in their failure to solve quadratic inequalities and to judge the number of roots of quadratic functions. Follow-up interview further revealed the participants’ lack of knowledge in solving problems by integrating algebraic and graphic representations. The implications of these findings for mathematics teacher preparation are discussed.     

Electronic vs. paper textbook presentations of the various aspects of mathematics

Based in part on our work in adapting existing paper textbooks for secondary schools for a digital format, this paper discusses paper form and the various electronic platforms with regard to the presentation of five aspects of mathematics that have roles in mathematics learning in all the grades kindergarten-12: symbolization, deduction, modeling, algorithms, and representations. In moving to digital platforms, each of these aspects of mathematics presents its own challenges and opportunities for both curriculum and instruction, that is, for the content goals and how they connect with students for learning. A combination of paper and electronic presentations may be an optimal solution but some difficulties with such a complex solution are presented.     

Integer number sense and notation: A case study of a student with a mathematics learning disability

Although approximately 6% of students have a mathematics learning disability (MLD) also known as dyscalculia, little is known about how MLD impacts students beyond basic arithmetic. In this study we focused on one mathematical topic foundational to algebra – integer operations – and conducted a videotaped design experiment with one student with MLD. Through 14 teaching episodes we explored the ways in which standard mathematical tools (e.g., symbols, representations) were inaccessible and evaluated the design of alternative tools. Our detailed retrospective analysis revealed that the student had an unconventional understanding of integer quantities and symbolic notation, which resulted in issues of accessibility and persistent difficulties. Deliberate attempts to address inaccessibility revealed nuances in the student’s understanding, and suggests that both number sense and notational issues needed to be addressed in tandem. Implications for instruction are discussed.     

Representational disfluency in algebra: evidence from student gestures and speech

In this investigation, we analyzed US middle school students’ (grades 6–8) gestures and speech during interviews to understand students’ reasoning while interpreting quantitative patterns represented by Cartesian graphs. We studied students’ representational fluency, defined as their abilities to work within and translate among representations. While students translated across representations to address task demands, they also translated to a different representation when reaching an impasse, where the initial representation could not be used to answer a task. During these impasse events, which we call representational disfluencies, three categories of behavior were observed. Some students perceived the graph to be bounded by its physical and numerical limits, and these students were categorized as physically grounded. A second, related, disfluency was categorized as spatially grounded. Students who were classified as spatially grounded exhibited a bounded view of the graph that limited their ability to make far predictions until they physically altered the spatial configuration of the graph by rescaling or extending the axes. Finally, students who recovered from one or more of these disfluencies by translating the quantitative information to alternative but equivalent representations (i.e., exhibiting representational fluency), while retaining the connection back to the linear pattern as graphed, were categorized as interpretatively grounded. Understanding the causes and varieties of representational fluency and disfluency contributes directly to our understanding of mathematics knowledge, learning and adaptive forms of reasoning. These findings also provide implications for mathematics instruction and assessment.     

Cooperative engineering as a specific design-based research

In this paper, we first describe what are the main constituents of what we term cooperative engineering, between teachers and researchers, from an epistemological and theoretical viewpoint. We show that this kind of research tries to dilute the ancient dualisms of Western Thought, and we present some features that may contribute to specifying cooperative engineering within the general paradigm of design-based research. In the second part of the paper, we give an empirical account of such a process of engineering at primary school level in mathematics, by focusing on the co-elaboration, between teachers and researchers, of learning sequences at kindergarten (the Treasures Games, situations of graphical representations) and at grade 1 (The Arithmetic and Comprehension at Elementary School program, building of basic numerical capacities). At the end of the paper, we briefly elaborate on what we consider to be major issues relating to cooperative engineering.     

Mathematics learning and aesthetic production

Some teaching projects in which the learning of mathematics was combined with mainly theatrical productions are reported on. They are related and opposed to an approach of drama in education by Pesci and the proposals of Sinclair for mathematics teaching and beauty. The analysis is based on the distinction between aesthetics as related to beauty or as related to sensual perception. The usefulness of concepts of model and metaphor for the understanding of aesthetic representations of mathematical subject matter is examined. It is claimed that the Peircean concept of the interpretant contributes to a concise analytical approach. The pedagogical attitude is committed to a balanced relationship of scientific and aesthetic values.     

Children's strategy use and interpretations of mathematical representations

This is a summary of research, from an information processing perspective, of children's interpretation and use of strategies and representations for place value, subtraction and addition in the first three years of school. Representations are defined broadly to include concrete embodiments of numbers, symbols for numbers and operations, and combinations of the latter in number sentences and algorithms. The objective was to assess the value and limitations of the use of representations in early mathematics learning and teaching and hence to identify, describe and examine critically some of the strategies and representations that children and teachers use in early mathematics. Children generally chose to use verbal and mental strategies in preference to formal algorithms, and did not want to use analogs unless they could not perform the task in any other way. The latter preference is explained on the basis of the extra demand that use of analogs adds to the cognitive process unless they are used automatically.     

Expanding Notions of “Learning Trajectories” in Mathematics Education

Over the past 20 years learning trajectories and learning progressions have gained prominence in mathematics and science education research. However, use of these representations ranges widely in breadth and depth, often depending on from what discipline they emerge and the type of learning they intend to characterize. Learning trajectories research has spanned from studies of individual student learning of a single concept to trajectories covering a full set of content standards across grade bands. In this article, we discuss important theoretical assumptions that implicitly guide the development and use of learning trajectories and progressions in mathematics education. We argue that diverse theoretical conceptualizations of what it means for a student to “learn” mathematics necessarily both constrains and amplifies what a particular learning trajectory can capture about the development of students’ knowledge.     

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.     

U.S. and Chinese Teachers' Constructing,Knowing, and Evaluating Representations to Teach Mathematics

This study examined U.S. and Chinese teachers' constructing, knowing, and evaluating representations to teach mathematics. All Chinese lesson plans are very similar, because they are all based on the Chinese national unified curriculum in mathematics. However, the U.S. lesson plans are extremely varied, even for those teachers from the same school. The Chinese teachers' lessons are very detailed; the U.S. teachers' lesson plans have exclusively adopted the "outline and worksheet" format. In the Chinese lesson plans, concrete representations are used exclusively to mediate students' understanding of the concept of average. In U.S. lessons, concrete representations are not only used to model the averaging processes to foster students' understanding of the concept, but they are also used to generate data. The U.S. teachers are much more likely than the Chinese teachers to predict drawing and guess-and-check strategies. For some problems, the Chinese teachers are much more likely than are the U.S. teachers to predict algebraic approaches. For the responses using conventional strategies, both the U.S. and Chinese teachers gave them high and almost identical scores. If a response involved a drawing or an estimate of an answer, the Chinese teachers usually gave a relatively lower score, even though the strategy is appropriate for the correct answer, because it is less generalizable. This study contributed to our understanding of the cross-national differences between U.S. and Chinese students' mathematical thinking. It also contributed to our understanding about teachers' beliefs from a cross-cultural perspective.     

U.S. and Chinese Teachers' Constructing, Knowing, and Evaluating Representations to Teach Mathematics

This study examined U.S. and Chinese teachers' constructing, knowing, and evaluating representations to teach mathematics. All Chinese lesson plans are very similar, because they are all based on the Chinese national unified curriculum in mathematics. However, the U.S. lesson plans are extremely varied, even for those teachers from the same school. The Chinese teachers' lessons are very detailed; the U.S. teachers' lesson plans have exclusively adopted the “outline and worksheet” format. In the Chinese lesson plans, concrete representations are used exclusively to mediate students' understanding of the concept of average. In U.S. lessons, concrete representations are not only used to model the averaging processes to foster students' understanding of the concept, but they are also used to generate data. The U.S. teachers are much more likely than the Chinese teachers to predict drawing and guess-and-check strategies. For some problems, the Chinese teachers are much more likely than are the U.S. teachers to predict algebraic approaches. For the responses using conventional strategies, both the U.S. and Chinese teachers gave them high and almost identical scores. If a response involved a drawing or an estimate of an answer, the Chinese teachers usually gave a relatively lower score, even though the strategy is appropriate for the correct answer, because it is less generalizable. This study contributed to our understanding of the cross-national differences between U.S. and Chinese students' mathematical thinking. It also contributed to our understanding about teachers' beliefs from a cross-cultural perspective.