ERME column

In this contribution we introduce three classical theoretical stances within the field of mathematics education regarding representations. Our aim is to highlight what we consider to be an interesting shift in how representations are conceived and studied in the field of mathematics education, and how this could impact both the practice of teaching and learning mathematics, and on further theorizing mathematical representation. We also indicate potential directions in which to develop ways to talk about newer forms of dynamic interactive representation

EXTERNAL REPRESENTATION FLEXIBILITY OF DOMAIN AND RANGE OF FUNCTION

This study attempts to analyze pre-service secondary mathematics teachers’ flexibility of external representations of domain and range of functions. To reach the purpose, a task consisted of thirty question items were designed. Participants of the study were thirty-eight Indonesian pre-service secondary mathematics teachers attending mathematics education department at one private university in Jakarta, Indonesia. Based on the analysis participants written responses, this paper revealed participants’ difficulties in providing a proper and consistent definition of the concept of domain and range of functions. We also disclosed the participants’ lack of flexibility in doing translation among representations under the concept of domain and range of function. In general, participants written responses to the task did not provide evidence of a solid understanding of domain and range. There are several implications of these findings offered for secondary mathematics teacher education’s program

The effect on learners strategies of varying computer-based representations: evidence from gazes, actions, utterances and sketches

Computer-based Multiple External Representations (MERs) have been found in some cases to help and in others to hinder the learning process. This thesis examines how varying the external representations that are presented in a computer environment influences the strategies that learners choose when tackling mathematics tasks. It has been noted (Ainsworth, 2006) that learners fail to transfer insights from one representation to another. Previous work analysing video data of learners' problem-solving with computer-based MERs emphasises the need to identify which representation is being considered by a learner as utterances are made, and to examine more closely learners' movement between representations. This research focuses on the relationship between strategy and representation during learners' problem solving. A set of analytical techniques was developed to characterise learner strategies, to identify how different computer-based MERs influence strategy choices, and to explore how these choices change over the course of task completion. Rich data were collected using a variety of technologies: learners' shifts in attention were recorded using an unobtrusive eye-tracking device and screen capture software; keyboard and mouse actions were logged automatically; utterances and gestures were video recorded; notes and sketches were recorded in real-time using a Tablet PC. This research suggests how integrated analysis of learners' gazes, actions, writing, sketches and utterances can better illuminate subtle cognitive strategies. The study involved completion of three tasks by eighteen participants using multiple mathematical representations (numbers, graphs and algebra) presented in different computer-based 'instantiations': Static (non-moving, non-changing, non-Interactive); Dynamic (capable of animation following keyboard inputs); Interactive (directly manipulable using a mouse). Having computer-based MERs available to learners provides an opportunity to use representations with which they are comfortable. A detailed analysis showed that both representation and instantiation have an impact on strategy choice. It identified differences in expression of inferences, construction of visual images, and attention to representations between different types of instantiation. One of the important findings of the research is that learners are less likely to use imagining strategies when representational instantiation is Interactive. These results may provide some explanation of how interactivity helps or hinders learners' understanding of multiple representations

An Analysis of Interactive Learning Environments for Arithmetic and Algebra Through an Integrative Perspective

International audienceThe analysis presented in this article tries to obtain a global view of the field of interactive learning environments (ILE) dedicated to arithmetic and algebra. As preliminaries, a brief overview of evaluation methods focusing on educational software is given and a short description of ten ILEs concerned by the study is provided as a kind of a state-of-the-art. Then the methodology of ILEs analysis developed in the TELMA project is explained consisting in the design and the refinement of an analysis grid and its use on the ten ILEs is mentioned. Next, a first level analysis of results leading to a compiled, analytic and synthetic view of the ILEs available and/or missing functionalities is given. A second level of the analysis is also proposed, with two concise representations of the ILEs, composed of graphical representations of the previous results, leading to a 3D map of ILEs dedicated to arithmetic and algebra. This map provides, as promised, a global view of the field and permits to define five sorts of ILEs according to two criteria: the first one is teacher-oriented and concerns usages enabled by the ILE; the second one is student-oriented and concerns control provided by the ILE to accomplish such usages

An Analysis of Interactive Learning Environments for Arithmetic and Algebra Through an Integrative Perspective

International audienceThe analysis presented in this article tries to obtain a global view of the field of interactive learning environments (ILE) dedicated to arithmetic and algebra. As preliminaries, a brief overview of evaluation methods focusing on educational software is given and a short description of ten ILEs concerned by the study is provided as a kind of a state-of-the-art. Then the methodology of ILEs analysis developed in the TELMA project is explained consisting in the design and the refinement of an analysis grid and its use on the ten ILEs is mentioned. Next, a first level analysis of results leading to a compiled, analytic and synthetic view of the ILEs available and/or missing functionalities is given. A second level of the analysis is also proposed, with two concise representations of the ILEs, composed of graphical representations of the previous results, leading to a 3D map of ILEs dedicated to arithmetic and algebra. This map provides, as promised, a global view of the field and permits to define five sorts of ILEs according to two criteria: the first one is teacher-oriented and concerns usages enabled by the ILE; the second one is student-oriented and concerns control provided by the ILE to accomplish such usages

UTILIZING SEMIOTIC PERSPECTIVE TO INVESTIGATE ALGEBRA II STUDENTS’ EXPOSURE TO AND USE OF MULTIPLE REPRESENTATIONS IN UNDERSTANDING ALGEBRAIC CONCEPTS

The study employed Ernest (2006) Theory of Semiotic Systems to investigate the use of and exposure to multiple representations in a 10th grade algebra II suburban high school class located in the southeastern region of the United States. The purpose of this exploratory case study (Yin, 2014) was to investigate the role of multiple representations in influencing and facilitating algebra II students’ conceptual understanding of piece-wise function, absolute-value functions, and quadratic functions. This study attempted to answer the following question: How does the use of and exposure to multiple representations influence algebra II students’ understanding and transfer of algebraic concepts? Furthermore, the following sub-questions assisted in developing a deeper understanding of the question: a) how does exposure to and use of multiple representations influence students’ identification of their pseudo-conceptual understanding of algebraic concepts?; b) how does exposure to and use of multiple representations influence students’ transition from pseudo-conceptual to conceptual understanding?; c) how does exposure to and use of multiple representations influence students’ transfer of their conceptual understanding to other related concepts? Understanding the notion of pseudo-conceptual understanding in algebra is significant in providing a tool for examining the veracity of algebra students’ conceptual understanding, where teachers have to consistently examine if students accurately understand the meanings of the mathematical signs that they are constantly using. The following data collection techniques were utilized: a) classroom observation, b) task based interviews, and c) study of documents. The unit of analysis was students’ verbal and written responses to task questions. Three themes emerged from the analysis of in this study: (a) re-imaging of conceptual understanding; (b) reflective approach to understanding and using mathematical signs; and (c) representational versatility in the use of mathematical signs. Findings from this study will contribute to the body of knowledge needed in research on understanding and assessing algebra students’ conceptual understanding of mathematics. In particular the findings from the study will contribute to the literature on understanding; the process of algebraic concepts knowledge acquisition, and the challenges that algebra students have with comprehension of algebraic concepts (Knuth, 2000: Zaslavsky et al., 2002)

Utilizing Microsoft Mathematics in Teaching and Learning Calculus

The experimental design was conducted to investigate the use of Microsoft Mathematics, free software made by Microsoft Corporation, in teaching and learning Calculus. This paper reports results from experimental study details on implementation of Microsoft Mathematics in Calculus, students\u27 achievement and the effects of the use of Microsoft Mathematics on students\u27 attitudes in relation to such experience. Two classes of the students from the first year student in Universitas Serang Raya were participated in the study. This study found that students who taught by using Microsoft Mathematics had higher achievement and has a positive effect on students\u27 confidence of mathematics

Using dialogue to learn math in the LeActiveMath project

We describe a tutorial dialogue system under development that assists students in learning how to differentiate equations. The system uses deep natural language understanding and generation to both interpret students ’ utterances and automatically generate a response that is both mathematically correct and adapted pedagogically and linguistically to the local dialogue context. A domain reasoner provides the necessary knowledge about how students should approach math problems as well as their (in)correctness, while a dialogue manager directs pedagogical strategies and keeps track of what needs to be done to keep the dialogue moving along.