Springer Science and Business Media LLC

This article presents a research work aimed to study the role of computational tools in the conceptualization of algebraic properties according to the conceptual framework of instrumental genesis and the Design-Function-Tasks (DeFT) framework of multiple representations at Junior high school level. It concerns a didactical approach which takes advantage of the semiotic representations available by the computational tool ALNuSet in a way that could support students to understand the changing of signs while multiplying inequalities with negative numbers. Forty-eight subjects of the 2nd grade of Greek Junior high school, 13–14 years old, participated in our experiment. We engaged our students in an activity concerning the change of the direction in inequalities when we multiply them by negative quantities, by exploiting the functionalities available in the Algebraic line component of ALNuSet. A qualitative analysis of the data shows that the subjects of the study benefit from ALNuSet in the construction of mental representations and instrumented techniques regarding the handling of signs in inequalities, and, through visualization, seem to conceptualize the property under investigation.

We present here the lessons learned by iteratively designing a tutorial for first-year university students using computer programming to work with mathematical models. Alternating between design and implementation, we used video-taped task interviews and classroom observations to ensure that the design promoted student understanding. The final version of the tutorial we present here has students make their own logarithm function from scratch, using Taylor polynomials. To ensure that the resulting function is accurate and reasonably fast, the students had to understand and apply concepts both from computing and from mathematics. We identify three categories of such concepts and identify three design features that students attended to when demonstrating such understanding. Additionally, we describe four important take-aways from a teaching design point of view that resulted from this iterative design process.

This article investigates how a computer simulator for human interactions and a dyadic system in a pencil-and-paper environment can contribute to helping students with poor motor co-ordination, notably those with dyspraxia, to learn geometry. The aim is to design an alternative way to teach the subject, and to explore its effects on the learning processes. Geometric construction is an important element in the curriculum in primary and secondary school, but is not an end in itself. We draw upon Efraim Fischbein’s work, in particular the link between the figural and conceptual aspects of geometric objects, in order to design a simulator that can execute geometrical constructions for students with motor co-ordination problems, who find drawing difficult. Our initial experiment with students with dyspraxia and their peers showcases the potential benefits of alternating human–human and human–avatar dyads. It serves as a proof of concept and highlights the way in which students appropriate the artifact and construct an instrument in the context of drawing.

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.

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.

In this article, we draw on assemblage theory to investigate how children aged 5 engage with different material surfaces to explore ordinal and relational aspects of number. The children participate in an activity in which they first interact with a strip on the floor, then with a multi-touch iPad application, to work with numbers in expressive ways. Focus is on the physicality and materiality of the activity and the provisional ways that children, surfaces, and number come together. While the notion of assemblage helps us see how movement animates the mathematical activity, we enrich our understanding of the entanglement of children, matter, and number as sustained by coordinated movements, from which numbers emerge as relations.

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.

Can math concepts be experienced through the sensory modality of balance? Balance Board Math (BBM) is a set of pedagogical math activities designed to instantiate mathematical concepts through stimulation to the vestibular sense: an organ in the inner ear that detects our bodily balance and orientation. BBM establishes the different ways children spontaneously rock and move as the basis for inclusively exploring mathematical concepts together across diverse sensory profiles. I describe two activity sets where students explore focal concepts by shifting their balance on rockable balance boards: “the Balance Number Line,” using analog materials to foster understandings of the number line and negative numbers, and “Balance Graphing,” using sensors and a digital display to foster exploration of functions and graphing concepts, including the parameters of trigonometric functions and function addition. I outline proposed ways that engaging with concepts through balance-activating movement can change learners’ mathematical thinking and learning.

The introduction of digital artefacts for the teaching/learning of mathematics raises the issue of “transitioning” between the mathematical knowledge built in digital contexts and that built in a more traditional, non-digital context, and vice versa. In this article, we address this issue by presenting a teaching experiment aimed at fostering the development of the concept of variable in a lower secondary school, through the use of the spreadsheet. We depict a complex network of several intertwined transitions: from the non-digital to the digital context; between different activities within the same digital context; and from the digital to the non-digital context. We highlight the centrality of the teacher’s role in triggering and supporting these transitions, thus fostering the students’ construction of resources for solving the assigned tasks. For our analysis, we propose a conceptualisation of the resources that takes into account the fact that such resources cannot always be established a priori, but that they can emerge during the activity with the digital artefact or that they can even be constructed as an effect of it. Addressing this complexity allows us to deepen insights into how the classroom use of digital artefacts, the process of construction and emergence of resources, both for students and for the teacher, and different types of transitions nurture each other in a productive intertwining.