Abstract: This study focusses on high school students’ written discourse about their experiences in a dynamic interactive digital environment in which functions were represented in one dimension, as dynagraphs, that are digital artifacts in which the independent variable can be acted upon and its movement causes the variation of the dependent variable. After the introduction of the notion of Dynamic Interactive Mediators within the theory of Commognition, we analyze and classify students’ written productions describing their experience with the dynagraphs. We present this classification as a tool of analysis that allows us to gain insight into how their writing reflects the temporal and dynamic dimensions of their experience with the dynagraphs. This tool is used to analyze 11 excerpts; finally, epistemological, cognitive and didactic implications of this tool are discussed. PubDate: 2019-10-17

Abstract: For any convex quadrilateral, joining each vertex to the mid-point of the next-but-one edge in a clockwise direction produces an inner quadrilateral (as does doing so in a counter-clockwise direction). In many cases, a dynamic geometry measurement of the ratio of the area of the outer quadrilateral to the area of the inner one appears to be 5:1. It turns out, however, that this is due to rounding. We generalise the construction by replacing mid-points by more general ratios, finding the maximum and minimum values of the area ratio and determining the conditions on the original quadrilateral that achieve those two extremes. PubDate: 2019-10-08

Abstract: Straightedge-and-compass construction problems in Euclidean geometry are known, at times, to be unsolvable. In such cases, additional mathematical tools, such as the Archimedean spiral, may yield a solution. Dynamic geometry (DG) environments suggest adding another mathematical tool as an alternative to the classical approach to geometric constructions: Continuous variation of one of the parameters of the problem. Adoption of this approach raises both mathematical and didactic issues for discussion. This paper presents several solved problems of triangle construction given data that includes angle bisectors. Some of these problems have been shown to be otherwise unsolvable; others may be solvable, but, to the author’s best knowledge, have not been solved using only classical tools. Presented for solution to a special group of prospective teachers, these problems provided them with the experience of facing the need to justify a DG-based solution mathematically, distinguish a ‘dynamic illustration’ from a self-consistent algorithm and refer to mathematical constraints or ‘rules of the game’ – altogether a challenging and fruitful learning opportunity. PubDate: 2019-08-01

Abstract: This article describes the findings of a pilot study that used computer simulations to broaden urban children’s opportunities to learn and participate in science, technology, engineering and mathematics (STEM). Culturally responsive instructional practices were used to engage urban children in mathematical reasoning and science process skills to create computer simulations. In this study, African-American and Latinx students’ self-efficacy in technology and twenty-first-century skills, as well as attitudes toward STEM and STEM careers, were examined using the context of critical race theory. Due to the small sample size, a non-parametric test was performed. The results revealed significant differences from pre-test to post-test on the constructs of twenty-first-century skills, science attitude and engineering careers. The effect sizes were moderate. Qualitative data revealed the instructor engaged in four out of six elements associated with culturally responsive instruction. Future studies should examine how instructors’ use of sociopolitical consciousness and funds of knowledge influences underrepresented students’ interest in and motivation to learn about STEM. PubDate: 2019-08-01

Abstract: My interest here is in the way in which a single task can develop, gaining both in mathematical richness and in pedagogic purpose. This leads me to consider pedagogical affordances which can turn tasks from having rich potential into tasks used richly. The result is a domain of variations and extensions, together with possible pedagogical actions which could come to mind for a teacher while their students explore. As these tasks evolved, I experienced moments of struggle myself, trying to keep track of my (often rather clumsy) notation. This gave me confidence that the tasks might be useful to teachers and would-be teachers for the opportunity afforded to experience struggles parallel to those experienced by students working either on similar tasks or on the initial task itself. This, in turn, can sensitise teachers to something of what their learners may experience, inspiring them to pose tasks in a way which avoids the didactic transposition (Chevallard 1985). PubDate: 2019-08-01

Abstract: This Snapshot paper presents a case study of Auden, a student aged 5 years and 6 months, who interacted with a touchscreen App called TouchCounts. This App is designed to support children’s activity around counting. I use Sfard’s commognitive framework to examine how Auden thinks and learns about number, and show how his exploration of number helped me understand the challenges children can face when moving from identifying subsequent numbers as they appear in the natural counting sequence to identifying numbers that appear before and after each other. PubDate: 2019-08-01

Abstract: This article describes construction processes of mathematical meaning of the function–derivative relationship, as it is studied graphically with a dynamic digital artifact. The discussion centres on a case study involving one student during his interaction with the artifact. He was asked to explain the connection between two linked dynamic graphs: the graph of a function and the graph of its derivative function. The study was guided by the semiotic mediation approach, which treats artifacts as fundamental to cognition and views learning as the evolution from meanings connected to the use of a certain artifact to those recognizable as mathematical, that is, connected directly to the mathematical object. In the course of three rounds of data analysis, the student was shown to progress from a point-specific view to an interval one, and to move toward a construction of the meaning of the derivative as a function. The actions of the student and his interactions with the artifact that enabled him to construct the mathematical meanings of the function–derivative relationship are identified and described. PubDate: 2019-07-08

Abstract: The teaching and learning of functions has generally been considered a problematic area of school mathematics. Also, in Norway, functions have been identified as a difficult topic for many students. In this article, we address some of the difficulties that may arise when introducing the concept of time–distance graphs and discuss misconceptions that can be avoided with our approach. We report on a case study undertaken within the international project FaSMEd (Formative assessment in Science and Mathematics Education). Two groups of grade 6 students were introduced to time-distance graphs for the first time. This happened at an earlier stage than normally dictated by the national curriculum. We show how students indicated they were learning mathematics and developing graph sense through a couple of tasks utilizing motion-sensor technology. We will see how students express connections with translating movement into language about graphs and how they translate graphs into physical movement. PubDate: 2019-05-10

Abstract: We develop a way to determine the period of periodic data quite precisely using dynamic graphing and the modulo operator. PubDate: 2019-04-01

Abstract: It is an increasingly common phenomenon that elementary school students are using mobile applications (apps) in their mathematics classrooms. Classroom teachers, who are using apps, require a tool, or a set of tools, to help them determine whether or not apps are appropriate and how enhanced educational outcomes can be achieved via their use. In this article we investigate whether Artifact Centric Activity Theory (ACAT) can be used to create a useful tool for evaluating apps, present a review guide based on the theory and test it using a randomly selected geometry app [Pattern Shapes] built upon different (if any at all) design principles. In doing so we broaden the scope of ACAT by investigating a geometry app that has additional requirements in terms of accuracy of external representations, and depictions of mathematical properties (e.g. reflections and rotations), than is the case for place value concepts in [Place Value Chart] which was created using ACAT principles and has been the primary app evaluated using ACAT. We further expand the use of ACAT via an independent assessment of a second app [Click the Cube] by a novice, using the ACAT review guide. Based on our latest research, we argue that ACAT is highly useful for evaluating any mathematics app and this is a critical contribution if the evaluation of apps is to move beyond academic circles and start to impact student learning and teacher pedagogy in mathematics. PubDate: 2019-04-01

Abstract: On-line math videos for student learning are abundant; yet they are surprisingly uniform in their expository mode of presentation and their emphasis on procedural skill. In response, we created an alternative model of on-line math videos that are dialogue-intensive and focus on the development of mathematical meaning and problem solving. Each video shows a pair of students (called the talent) next to their mathematical inscriptions, which allows other students viewing the videos (called vicarious learners) to see both the talent and their work clearly. In this exploratory study, we investigated how eleven pairs of vicarious learners approached a research session in which they were asked to watch these videos and then solve a related mathematical task. We found qualitatively different ways that the vicarious learner pairs interpreted the goal of the research session, which, in turn, constrained their behavior when solving the task and getting information from the videos. Specifically, by drawing upon board-game theory, we inferred that the vicarious learners engaged in four different games: (a) the Definition game, (b) the Concept Image game, (c) the Procedure game and (d) the Video Expert game. The particular game that vicarious learners played had consequences for the productivity of their mathematical work. This article explores the significance and implications of these results. PubDate: 2019-04-01

Abstract: Learning mathematics requires students to become fluent in some of the discipline-specific ways of communicating through words, symbols and diagrams, creating difficulties for many learners to engage with the subject. Computer programming environments have inspired efforts to support students’ communication, by making abstract ideas more tangible and concrete within such environments. In this article, I examine students’ challenges in interpreting symbols and diagrams when using a visual programming environment to create representations of distance and speed. The environment presented difficulties related to understanding the meanings of symbols and how they fit together, as well as using symbols to create visual representations. In overcoming these challenges students created representations that, although they contained less information than traditional mathematical ones, were nonetheless meaningful to the students who created them. These findings add complexity to existing research on the potential of programming environments for learning mathematics, and suggest a potential to re-envision what technical mathematical discourse might look like in interdisciplinary settings. PubDate: 2019-04-01

Abstract: This study contributes insights into how task design with different elements of guidance may influence students’ utilization of dynamic software for problem solving and reasoning. It compared students’ solving of two tasks with different designs supported by the dynamic software GeoGebra. Data analysed examined students’ approaches to utilizing GeoGebra, the characteristics of their reasoning and their ability to prove the validity of their solutions after solving the problems. The results showed that students who solved the task with less guidance (without instructions about a specific solving method) were better able to utilize GeoGebra’s potential to support their reasoning and problem solving. These students reasoned more creatively and presented more advanced proofs for their solutions than the more guided ones. PubDate: 2019-03-21

Abstract: Statistics is a challenging subject for many university students. In addition to dedicated methods of didactics of statistics, adaptive educational technologies can also offer a promising approach to target this challenge. Inspectable student models provide students with information about their mastery of the domain, thus triggering reflection and supporting the planning of subsequent study steps. In this article, we investigate the question of whether insights from didactics of statistics can be combined with inspectable student models and examine if the two can reinforce each other. Five inspectable student models were implemented within five didactically grounded online statistics modules, which were offered to 160 Social Sciences students as part of their first-year university statistics course. The student models were evaluated using several methods. Learning curve analysis and predictive validity analysis examined the quality of the student models from the technical point of view, while a questionnaire and a task analysis provided a didactical perspective. The results suggest that students appreciated the overall design, but the learning curve analysis revealed several weaknesses in the implemented domain structure. The task analysis revealed four underlying problems that help to explain these weaknesses. Addressing these problems improved both the predictive validity of the adjusted student models and the quality of the instructional modules themselves. These results provide insight into how inspectable student models and didactics of statistics can augment each other in the design of rich instructional modules for statistics. PubDate: 2018-12-01

Abstract: Proof by contradiction presents various difficulties for students relating especially to the formulation and interpretation of a negation, the managing of impossible mathematical objects, and the acceptability of the validity of the statement once a contradiction has been reached from its negation. This article discusses how a Dynamic Geometry Environment (DGE) can contribute to students’ argumentation processes when trying to explain contradictions. Four cases are presented and analysed, involving students from high school, as well as undergraduate and graduate students. The approach of the analyses makes use of a symbolic logical chain and the notion of pseudo-object. Such analyses lead to a hypothesis, that experiencing a pseudo-object during an exploration can foster DGE-supported processes of argumentation culminating in geometrical proofs by contradiction, while the lack of experience of a pseudo-object may hinder such processes. If this hypothesis is confirmed by further studies, we foresee important didactical implications since it sheds light on the transition from students’ DGE-based argumentations to proofs by contradiction. PubDate: 2018-12-01

Abstract: This study documents the development of a Framework (the Dynamic Geometry Task Analysis Framework) to be used to indicate the relative quality of tasks produced for dynamic geometry software. Its purpose is to assist curriculum writers and teachers in evaluating and creating dynamic geometry tasks. To produce it, numerous tasks submitted by secondary mathematics teachers as part of a year-long professional development program were analyzed, before creating three dynamic geometry mathematics tasks that, according to the Framework, were ranked as low, medium and high in quality. Semi-structured interviews with twelve high school students were conducted and analyzed to examine relationships between the quality of tasks as specified by the Framework and the quality of student argumentation. Results showed the Framework effectively reflects task quality based on student mathematical activity and argumentation. PubDate: 2018-12-01

Abstract: With louder and more widespread calls to include computer programming as a core element of school curriculum, global efforts to define innovative and distinct coding curricula are underway. We take a different tack in this paper, one oriented by an investigation of the common ground between learning to program and learning mathematics. We observed 9- and 10-year-olds as they learned to build and program Lego Mindstorms EV3 robots over 4 days, attending in particular to the ways that programming robots to move might support the development and integration of powerful instantiations of number, arithmetic and multiplication. Our findings suggest that children’s understanding of number, and their transitions from additive to multiplicative thinking, can be powerfully supported by engaging in practical tasks rather than practice exercises. PubDate: 2018-12-01

Abstract: Since the 1960s, a few, yet very influential, educational researchers have investigated how computer programming can be used to foster mathematics learning. However, since the term ‘computational thinking’ was popularised by Jeannette Wing in 2006, the number of studies in this area has grown substantially. In this article, we present a systematic analysis of literature linking mathematics education to computational thinking in an attempt to quantify the breadth and depth of existing work in the area. Our analysis indicates that many studies: (1) originate from computer science academics rather than education experts; (2) involve mathematics but mainly concentrate on teaching programming skills; (3) present small-scale research designs on self-reported attitudes or beliefs; (4) rarely deal with concepts in mathematical domain areas such as probability, statistics, measurement or functions. Thus, we conclude that there are opportunities for rigorous research designs reporting on observable learning outcomes, explicitly targeting mathematics, conducted by multidisciplinary teams, and focusing on less-explored domain areas. We believe that these opportunities should be investigated, in order to provide a broader evidence base for developing meaningful digital learning experiences in mathematics for school-aged children. PubDate: 2018-04-01

Abstract: In order to facilitate the development of a model of two children’s topological ideas, I required a tool that could support fundamental topological representations and transformations so that their reasoning could be made visible and further developed. In this article I document the theoretically and conceptually oriented design process through which I developed a microworld for topological equivalence that I used in my teaching experiments. Emphasis is given to key design decisions I arrived at as I aimed to design an exploratory environment that could be used to engage and advance children’s reasoning about topological ideas. PubDate: 2018-04-01

Abstract: Spoken and written language change when students (age 14–15) work with digital tools in the mathematics classroom. The digital tool not only offers new experiences and actions that directly influence language, but also provides lexical expressions such as buttons or technical expressions that students use. This article studies language used in the mathematics classroom empirically when working with dynamic geometry software (DGS), especially in the context of geometrical constructions. The results show the variety of different lexical phenomena in the students’ use of language. The use of such lexical expressions also mirrors the use of the digital tool itself. Qualitative analysis of the videotaped paired task-based clinical interviews shows both obstacles and potentials of the lexical expressions that students use when working with dynamic geometry software. PubDate: 2018-04-01