Transforming the mathematical practices of learners and teachers through digital technology

This paper argues that mathematical knowledge, and its related pedagogy, is inextricably linked to the tools in which the knowledge is expressed. The focus is on digital tools and the different roles they play in shaping mathematical meanings and in transforming the mathematical practices of learners and teachers. Six categories of digital tool-use that distinguish their differing potential are presented: i. dynamic and graphical tools, ii. tools that outsource processing power, iii. tools that offer new representational infrastructures for mathematics, iv. tools that help to bridge the gap between school mathematics and the students’ world; v. tools that exploit high-bandwidth connectivity to support mathematics learning; and vi. tools that offer intelligent support for the teacher when their students engage in exploratory learning with digital technologies Following exemplification of each category, the paper ends with some reflections on the progress of research in this area and identifies some remaining challenges

A computational lens on design research

In this commentary, we briefly review the collective effort of design researchers to weave theory with empirical results, in order to gain a better understanding of the processes of learning. We seek to respond to this challenging agenda by centring on the evolution of one sub-field: namely that which involves investigations within a constructionist framework of learning with carefully designed computational tools. We argue that these studies, specifically those where children were learning to program, were early adopters of the Design Research methodology and offer a useful lens through which to focus on the current field

Designing a programming-based approach for modelling scientific phenomena

We describe an iteratively designed sequence of activities involving the modelling of 1- dimensional collisions between moving objects based on programming in ToonTalk. Students aged 13-14 in two settings (London and Cyprus) investigated a number of collision situations, classified into six classes based on the relative velocities and masses of the colliding objects. We describe iterations of the system in which students engaged in a repeating cycle of activity for each collision class: prediction of object behaviour from given collision conditions, observation of a relevant video clip, building a model to represent the phenomena, testing, validating and refining their model, and publishing it ? together with comments ? on our web-based collaboration system, WebReports. Students were encouraged to consider the limitations of their current model, with the aim that they would eventually appreciate the benefit of constructing a general model that would work for all collision classes, rather than a different model for each class. We describe how our intention to engage students with the underlying concepts of conservation, closed systems and system states was instantiated in the activity design, and how the modelling activities afforded an alternative representational framework to traditional algebraic description

A teacher’s use of dynamic digital technology to address students’ misconception about additive strategies for geometric similarity

Research has well documented that students develop a significant misconception associated with the incorrect use of additive strategies when engaging with geometric similarity (GS) tasks. Since dynamic digital technology (DDT) has the potential to support students in addressing this misconception, teachers can exploit the affordances of DDT in the classroom to accomplish it. The aim of this paper is to explore how and why a secondary mathematics teacher uses DDT in the classroom to promote students’ understanding of why additive strategies are inappropriate to use for GS tasks. Drawing on the data collected, through classroom observations and post-lesson teacher interviews, the research findings indicate that the dynamic and visual nature of DDT can be used to help students realise the inappropriateness of the use of additive strategies for GS tasks

Bridging Primary Programming and Mathematics: some findings of design research in England

In this paper we present the background, aims and methodology of the ScratchMaths (SM) project, which has designed curriculum materials and professional development (PD) to support mathematical learning through programming for pupils aged between 9 and 11 years. The project was framed by the particular context of computing in the English education system alongside the long history of research and development in programming and mathematics. In this paper, we present a “framework for action” (diSessa and Cobb 2004) following design research that looked to develop an evidence-based curriculum intervention around carefully chosen mathematical and computational concepts. As a first step in teasing out factors for successful implementation and addressing any gap between our design intentions and teacher delivery, we focus on two key foundational concepts within the SM curriculum: the concept of algorithm and of 360-degree total turn. We found that our intervention as a whole enabled teachers with different backgrounds and levels of confidence to tailor the delivery of the SM in ways that can make these challenging concepts more accessible for both themselves and their pupils

Mathematics and digital technology: challenges and examples from design research

Mathematics is a ubiquitous and vital substrate on which our culture is built. Yet this fact is seldom fully exploited in educational contexts. The first step must, in our view, be to open the black box of invisible mathematics to more people, (see Hoyles, 2015). A key challenge for task design and an organising design principle is to exploit digital technology to reveal more of what mathematics actually is; first, by offering a glimpse of the mathematical models underlying a given (and carefully chosen) phenomenon; and second, by fostering an approach to mathematical tasks that transcends the purely procedural. We describe in this paper how we have attempted to address these challenges

Microworlds, Constructionism and Mathematics = Micromundos, Construccionismo y Matemáticas

En este artículo, esbozamos la idea del construccionismo y cómo puede ser puesta en práctica a través del diseño de “micromundos”, islas aisladas y accesibles de actividad, dónde se encuentren pepitas de conocimiento relevante a través de herramientas y secuencias de actividades especialmente diseñadas y con pedagogías adecuadamente orientadas. En la segunda parte del artículo, describimos el diseño, implementación y evaluación de una intervención construccionista, ScratchMaths, introducida en Inglaterra, país en el que la computación es obligatoria para todos los niveles educativos (de los 5 a los 16 años). Este estudio de caso pone de manifiesto la tensión entre la fidelidad de una innovación al implementarla y su adaptación por parte de los profesores, especialmente en el contexto de las matemáticas, la cuál es una materia muy exigente tanto para los docentes como para los alumnos

Beyond jam sandwiches and cups of tea: An exploration of primary pupils' algorithm‐evaluation strategies

The long-standing debate into the potential benefit of developing mathematical thinking skills through learning to program has been reignited with the widespread introduction of programming in schools across many countries, including England where it is a statutory requirement for all pupils to be taught programming from five years old. Algorithm is introduced early in the English computing curriculum, yet, there is limited knowledge of how young pupils view this concept. This paper explores pupils’ (aged 10-11) understandings of algorithm following their engagement with one year of ScratchMaths (SM), a curriculum designed to develop computational and mathematical thinking skills through learning to program. 181 pupils from six schools undertook a set of written tasks to assess their interpretations and evaluations of different algorithms that solve the same problem, with a subset of these pupils subsequently interviewed to probe their understandings in greater depth. We discuss the different approaches identified, the evaluation criteria they used and the aspects of the concept that pupils found intuitive or challenging, such as simplification and abstraction. The paper ends with some reflections on the implications of the research, concluding with a set of recommendations for pedagogy in developing primary pupils’ algorithmic thinking