How do students’ behaviors relate to the growth of their mathematical ideas?

The purpose of this study is to analyze the relationship between student behaviors and the growth of mathematical ideas (using the Pirie-Kieren model). This analysis was accomplished through a series of case studies, involving middle school students of varying ability levels, who were investigating a combinatorics problem in after-school problem-solving sessions. The results suggest that certain types of student behaviors appear to be associated with the growth of ideas and emerge in specific patterns. More specifically, as understanding grows, there is a general shift from behaviors such as students questioning each other, explaining and using their own and others’ ideas toward behaviors involving the setting up of hypothetical situations, linking of representations and connecting of contexts. Recognizing that certain types of student behaviors tend to emerge in specific layers of the Pirie-Kieren model can be important in helping us to understand the development of mathematical ideas in children.     

Graphical Representations of Speed: Obstacles Preservice K‐8 Teachers Experience

This study examines the difficulties college students experience when creating and interpreting graphs in which speed is one of the variables. Nineteen students, all preservice elementary or middle school teachers, completed an upper‐level course exploring algebraic concepts. Although all of these preservice teachers had previously completed several mathematics courses, including calculus, they demonstrated widespread misconceptions about the variable speed. This study identifies four cognitive obstacles held by the students, provides excerpts of their graphical constructions and verbal interpretations, and discusses potential causes for the confusion. In particular, misconceptions arose when students interpreted the behavior and nature of speed within a graphical context, as well as in situations where they were required to construct a graph involving speed as a variable. The study concludes by offering implications for the teaching and learning of speed and its interpretation within a graphical setting.     

Exploring mathematical connections of pre-university students through tasks involving rates of change

This paper reports the results of a research exploring the mathematical connections of pre-university students while they solving tasks which involving rates of change. We assume mathematical connections as a cognitive process through which a person finds real relationships between two or more ideas, concepts, definitions, theorems, procedures, representations or meanings or with other disciplines or the real-world. Four tasks were proposed to the 33 pre-university students that participated in this research; the central concept of the first task is the slope, the last three tasks contain concepts like velocity, speed and acceleration. Task-based interviews were conducted to collect data and later analysed with thematic analysis. Results showed most of the students made mathematical connections of the procedural type, the mathematical connections of the common features type are made in smaller quantities and the mathematical connection of the generalization type is scarcely made. Furthermore, students considered slope as a concept disconnected from velocity, speed and acceleration.     

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.     

Instructional Supports for Representational Fluency in Solving Linear Equations with Computer Algebra Systems and Paper‐and‐Pencil

This research addresses the issue of how to support students' representational fluency—the ability to create, move within, translate across, and derive meaning from external representations of mathematical ideas. The context of solving linear equations in a combined computer algebra system (CAS) and paper‐and‐pencil classroom environment is targeted as a rich and pressing context to study this issue. We report results of a collaborative teaching experiment in which we designed for and tested a functions approach to solving equations with ninth‐grade algebra students, and link to results of semi‐structured interviews with students before and after the experiment. Results of analyzing the five‐week experiment include instructional supports for students' representational fluency in solving linear equations: (a) sequencing the use of graphs, tables, and CAS feedback prior to formal symbolic transpositions, (b) connecting solutions to equations across representations, and (c) encouraging understanding of equations as equivalence relations that are sometimes, always, or never true. While some students' change in sophistication of representational fluency helps substantiate the productive nature of these supports, other students' persistent struggles raise questions of how to address the diverse needs of learners in complex learning environments involving multiple tool‐based representations.     

Complex calculators in the classroom: theoretical and practical reflections on teaching pre-calculus

University and older school students following scientific courses now use complex calculators with graphical, numerical and symbolic capabilities. In this context, the design of lessons for 11th grade pre-calculus students was a stimulating challenge.In the design of lessons, emphasising the role of mediation of calculators and the development of schemes of use in an 'instrumental genesis' was productive. Techniques, often discarded in teaching with technology, were viewed as a means to connect task to theories. Teaching techniques of use of a complex calculator in relation with 'traditional' techniques was considered to help students to develop instrumental and paper/pencil schemes, rich in mathematical meanings and to give sense to symbolic calculations as well as graphical and numerical approaches.The paper looks at tasks and techniques to help students to develop an appropriate instrumental genesis for algebra and functions, and to prepare for calculus. It then focuses on the potential of the calculator for connecting enactive representations and theoretical calculus. Finally, it looks at strategies to help students to experiment with symbolic concepts in calculus.This revised version was published online in September 2005 with corrections to the Cover Date.     

Intra-mathematical connections made by high school students in performing Calculus tasks

In this article, we report the results of research that explores the intra-mathematical connections that high school students make when they solve Calculus tasks, in particular those involving the derivative and the integral. We consider mathematical connections as a cognitive process through which a person relates or associates two or more ideas, concepts, definitions, theorems, procedures, representations and meanings among themselves, with other disciplines or with real life. Task-based interviews were used to collect data and thematic analysis was used to analyze them. Through the analysis of the productions of the 25 participants, we identified 223 intra-mathematical connections. The data allowed us to establish a mathematical connections system which contributes to the understanding of higher concepts, in our case, the Fundamental Theorem of Calculus. We found mathematical connections of the types: different representations, procedural, features, reversibility and meaning as a connection.     

Characterizing Interaction with Visual Mathematical Representations

This paper presents a characterization of computer-based interactions by which learners can explore and investigate visual mathematical representations (VMRs). VMRs (e.g., geometric structures, graphs, and diagrams) refer to graphical representations that visually encode properties and relationships of mathematical structures and concepts. Currently, most mathematical tools provide methods by which a learner can interact with these representations. Interaction, in such cases, mediates between the VMR and the thinking, reasoning, and intentions of the learner, and is often intended to support the cognitive tasks that the learner may want to perform on or with the representation. This paper brings together a diverse set of interaction techniques and categorizes and describes them according to their common characteristics, goals, intended benefits, and features. In this way, this paper aims to provide a preliminary framework to help designers of mathematical cognitive tools in their selection and analysis of different interaction techniques as well as to foster the design of more innovative interactive mathematical tools. An effort is made to demonstrate how the different interaction techniques developed in the context of other disciplines (e.g., information visualization) can support a diverse set of mathematical tasks and activities involving VMRs.     

On the use of history of mathematics: an introduction to Galileo's study of free fall motion

In this paper, we report on an experimental activity for discussing the concepts of speed, instantaneous speed and acceleration, generally introduced in first year university courses of calculus or physics. Rather than developing the ideas of calculus and using them to explain these basic concepts for the study of motion, we led 82 first year university students through Galileo's experiments designed to investigate the motion of falling bodies, and his geometrical explanation of his results, via simple dynamic geometric applets designed with GeoGebra. Our goal was to enhance the students’ development of mathematical thinking. Through a scholarship of teaching and learning study design, we captured data from students before, during and after the activity. Findings suggest that the historical development presented to the students helped to show the growth and evolution of the ideas and made visible authentic ways of thinking mathematically. Importantly, the activity prompted students to question and rethink what they knew about speed and acceleration, and also to appreciate the novel concepts of instantaneous speed and acceleration at which Galileo arrived.     

High‐School Students' Approaches to Solving Algebra Problems that are Posed Symbolically: Results from an Interview Study

A cross‐curricular structured‐probe task‐based clinical interview study with 44 pairs of third year high‐school mathematics students, most of whom were high achieving, was conducted to investigate their approaches to a variety of algebra problems. This paper presents results from three problems that were posed in symbolic form. Two problems are TIMSS items (a linear inequality and an equation involving square roots). The other problem involves square roots. We found that the majority of student pairs used symbol manipulation when solving the problems, and while many students seemed to prefer symbolic over graphical and tabular representations in their first attempt at solving the problems, we found that it was common for student pairs to use more than one strategy throughout the course of their solving. Students' use of graphing calculators to solve the problems is discussed.     

On multiindices and multivariables presentation of the Voigt functions

This paper aims at presenting multiindices and multivariables study of the unified (or generalized) Voigt functions which play an important rôle in the several diverse field of physics such as astrophysical spectroscopy and the theory of neutron reactions. Some expressions (representations) of these functions are given in terms of familiar special functions of multivariables. Further representations and series expansions involving multidimensional classical polynomials (Laguerre and Hermite) of mathematical physics are established.     

THE INFLUENCE OF TEACHERS’ PRESENTATIONS ON PUPILS’ MENTAL REPRESENTATIONS

Teachers use a variety of external representations to communicate mathematical ideas to their pupils. This paper reports a preliminary study of the internal mental representations that 6- and 7- year-old pupils form as a result of their interactions with the teacher's verbal, written, pictorial and concrete material representations, involving two-digit numbers and operations on them. The results presented here concern the picture-like mental representations that pupils use in performing two-digit calculations mentally. The evidence suggests that pupils seldom spontaneously visualise teachers’ representations or attempt mental manipulation of visual images to help with calculation. Pupils can, however, have mental representations which reproduce some aspects of the teachers’ representations.     

Derivation and Visualization of the Binomial Theorem

The binomial theorem presents us with the opportunity to weave many different mathematical strands into one lesson. It has a fascinating history — the study of which leads to a better understanding of how mathematics evolved. In this paper, we have involved computer graphics, geometry, algebra and combinatorics in the derivation of the binomial theorem. The study of functions with finite domains and ranges helps students understand some of the more subtle properties of functions which have the set of real numbers for their domain and range. These are the functions which they study to the exclusion of all others in high school and in their first two years in college. We believe that the lesson presented in this paper encourages students to express mathematical ideas in the vernacular, one of the major standards recommended by the National Council of Teachers of Mathematics (NCTM).This revised version was published online in September 2005 with corrections to the Cover Date.     

Meaning and Formalism in Mathematics

This essay is an exploration of possible sources (psychological, not mathematical) of mathematical ideas. After a short discussion of plationism and constructivism, there is a brief review of some suggestions for these sources that have been put forward by various researcher (including this author). These include: mental representations, deductive reasoning, metaphors, natural language, and writing computer programs.The problem is then recast in terms of the relation between meaning and formalism. On one hand, formalism can be seen as a tool for expressing meaning that is already present in an individual's mind. On the other hand, and the discussion of this point is the main contribution of this paper, it is not only possible, but a standard activity of mathematicians, to use formalism to construct meaning and this can also be a source of mathematical ideas.Although using formalism to construct meaning is a very difficult method for students to learn, it may be that this is the only route to learning large portions of mathematics at the upper high school and tertiary levels. The essay ends with an outline of a pedagogical strategy for helping students travel this route.This revised version was published online in September 2005 with corrections to the Cover Date.     

The mathematical beliefs and behavior of high school students: Insights from a longitudinal study

There is a documented need for more research on the mathematical beliefs of students below college. In particular, there is a need for more studies on how the mathematical beliefs of these students impact their mathematical behavior in challenging mathematical tasks. This study examines the beliefs on mathematical learning of five high school students and the students’ mathematical behavior in a challenging probability task. The students were participants in an after-school, classroom-based, longitudinal study on students’ development of mathematical ideas funded by the United States National Science Foundation. The results show that particular educational experiences can alter results from previous studies on the mathematical beliefs and behavior of students below college, some of which have been used to justify non-reform pedagogical approaches in mathematics classrooms. Implications for classroom practice and ideas for future research are discussed.     

Students’ reflections on their learning experiences: lessons from a longitudinal study on the development of mathematical ideas and reasoning

This paper characterizes the views on mathematical learning of five high school students based on the students’ reflections on their mathematical experiences in a longitudinal study that focused on the development of mathematical ideas and reasoning in particular research conditions. The students’ views are presented according to five themes about learning which describe the students’ views on the nature of knowledge and what it means to know, source of knowledge, motivation to engage in learning, certainty in knowing, and how the students’ views vary with particular areas of mathematical activity. The study addresses the need for more research on epistemological beliefs of students below college age. In particular, the results provide evidence that challenge the existing assumption that, prior to college, students exhibit naïve epistemological beliefs.     

“Playing the game” of story problems: Coordinating situation-based reasoning with algebraic representation

This study critically examines a key justification used by educational stakeholders for placing mathematics in context –the idea that contextualization provides students with access to mathematical ideas. We present interviews of 24 ninth grade students from a low-performing urban school solving algebra story problems, some of which were personalized to their experiences. Using a situated cognition framework, we discuss how students use informal strategies and situational knowledge when solving story problems, as well how they engage in non-coordinative reasoning where situation-based reasoning is disconnected from symbol-based reasoning and other problem-solving actions. Results suggest that if contextualization is going to provide students with access to algebraic ideas, supports need to be put in place for students to make connections between formal algebraic representation, informal arithmetic-based reasoning, and situational knowledge.     

VISUALISATION AND THE INFLUENCE OF TECHNOLOGY IN ‘A’ LEVEL MATHEMATICS: A CLASSROOM INVESTIGATION

The study reported here is part of a wider study, which aims to investigate the potential of the graphical calculator for mediating the development of students’ abilities to visualise the graphs of functions at GCE Advanced level. This paper focuses on how the graphical calculator influenced six particular students’ work with functions. Initial results have illuminated ways in which the technology can have a positive impact on students’ visualisation capabilities. It is proposed that visual thinking forms a significant part of many students’ mathematical reasoning, enabling students to derive richer meaning from given problems. It is suggested further that use of the technology mediates the development of students’ visual capacities, by helping to highlight the links between complementary modes of representation.     

Why do U.S. and Chinese students think differently in mathematical problem solving?: Impact of early algebra learning and teachers’ beliefs

This paper reports two studies that examined the impact of early algebra learning and teachers’ beliefs on U.S. and Chinese students’ thinking. The first study examined the extent to which U.S. and Chinese students’ selection of solution strategies and representations is related to their opportunity to learn algebra. The second study examined the impact of teachers’ beliefs on their students’ thinking through analyzing U.S. and Chinese teachers’ scoring of student responses. The results of the first study showed that, for the U.S. sample, students who have formally learned algebraic concepts are as likely to use visual representations as those who have not formally learned algebraic concepts in their problem solving. For the Chinese sample, students rarely used visual representations whether or not they had formally learned algebraic concepts. The findings of the second study clearly showed that U.S. and Chinese teachers view students’ responses involving concrete strategies and visual representations differently. Moreover, although both U.S. and Chinese teachers value responses involving more generalized strategies and symbolic representations equally high, Chinese teachers expect 6th graders to use the generalized strategies to solve problems while U.S. teachers do not. The research reported in this paper contributed to our understanding of the differences between U.S. and Chinese students’ mathematical thinking. This research also established the feasibility of using teachers’ scoring of student responses as an alternative and effective way of examining teachers’ beliefs.