Connecting engagement and focus in pedagogic task design

Digital Experiences in Mathematics Education, 2016

In 2011, the European Mathematical Society stated that a third pillar of scientific inquiry of complex systems has emerged in the form of a combination of modelling, simulation, optimization and visualization. This paper presents a naturalistic case study of an undergraduate student enrolled in a sequence of three programming, project-based mathematics courses at Brock University (Canada) where students learn to design, program, and use interactive environments for the investigation of a mathematics concept, theorem, conjecture, or a real-world situation. We argue that these courses instituted in 2001 aim to develop students' proficiency in the third pillar. Findings from the study suggest that the student engaged in constructionist experiences for learning mathematics through her 14 mathematical programming-based projects, and that she appropriated computer programming as an instrument for engagement in the third pillar.

International Journal of Computers for Mathematical Learning, 2010

Our focus is on the design of systems (pedagogical, technical, social) that encourage mathematical abstraction, a process we refer to as designing for abstraction. In this paper, we draw on detailed design experiments from our research on children's understanding about chance and distribution to represent this work as a case study in designing for abstraction. Through the case study, we elaborate a number of design heuristics that we claim are also identifiable in the broader literature on designing for mathematical abstraction. Our previous work on the micro-evolution of mathematical knowledge indicated that new mathematical abstractions are routinely forged in activity with available tools and representations, coordinated with relatively naïve unstructured knowledge. In this paper, we identify the role of design in steering the micro-evolution of knowledge towards the focus of the designer's aspirations. A significant finding from the current analysis is the identification of a heuristic in designing for abstraction that requires the intentional blurring of the key mathematical concepts with the tools whose use might foster the construction of that abstraction. It is commonly recognized that meaningful design constructs emerge from careful analysis of children's activity in relation to the designer's own framework for mathematical abstraction. The case study in this paper emphasizes the insufficiency of such a model for the relationship between epistemology and design. In fact, the case study characterises the dialectic relationship between epistemological analysis and design, in which the theoretical foundations of designing for abstraction and for the micro-evolution of mathematical knowledge can co-emerge.

Mathematics Education Research Journal, 2017

Inquiry-based learning (IBL) is a pedagogical approach in which students address complex, ill-structured problems set in authentic contexts. While IBL is gaining ground in Australia as an instructional practice, there has been little research that considers implications for student motivation and engagement. Expectancy-value theory (Eccles and Wigfield 2002) provides a framework through which children's beliefs about their mathematical competency and their expectation of success are able to be examined and interpreted, alongside students' perceptions of task value. In this paper, Eccles and Wigfield's expectancy-value model has been adopted as a lens to examine a complete unit of mathematical inquiry as undertaken with a class of 9-10-year-old students. Data were sourced from a unit (∼10 lessons) based on geometry and geometrical reasoning. The units were videotaped in full, transcribed, and along with field notes and student work samples, subjected to theoretical coding using the dimensions of Eccles and Wigfield's model. The findings provide insight into aspects of IBL that may impact student motivation and engagement. The study is limited to a single unit; however, the results provide a depth of insight into IBL in practice while identifying features of IBL that may be instrumental in bringing about increased motivation and engagement of students in mathematics. Identifying potentially motivating aspects of IBL enable these to be integrated and more closely studied in IBL practises.

ZDM

Children have limited exposure to statistical concepts and processes, yet researchers have highlighted multiple benefits of experiences in which they design and/or engage informally with statistical modelling. A study was conducted with a classroom in which students developed and utilised data-based models to respond to the inquiry question, Which origami animal jumps the furthest? The students used hat plots and box plots in Tinkerplots to make sense of variability in comparing distributions of their data and to support them to write justified conclusions of their findings. The study relied on classroom video and student artefacts to analyse aspects of the students' modelling experiences which exposed them to powerful statistical ideas, such as key repeatable structures and dispositions in statistics. Three principles-purpose, process and prediction-are highlighted as ways in which the problem context, statistical structures and inquiry dispositions and cycle extended students' opportunities to reason in sophisticated ways appropriate for their age. The research question under investigation was, How can an emphasis on purpose, process and prediction be implemented to support children's statistical modelling? The principles illustrated in the study may provide a simple framework for teachers and researchers to develop statistical modelling practices and norms at the school level.

Mathematics Education Research Journal, 2017

Constructionism, best known as the framework for action underpinning Seymour Papert's work with Logo, has stressed the importance of engaging students in creating their own products. Noss and Hoyles have argued that such activity enables students to participate increasingly in a web of connections to further their activity. Ainley and Pratt have elaborated that learning is best facilitated when the student is engaged in a purposeful activity that leads to appreciation of the power of mathematical ideas. Constructionism gives prominence to how the learner's logical reasoning and emotion-driven reasons for engagement are inseparable. We argue that the dependence of constructionism upon the orienting framework of constructivism fails to provide sufficient theoretical underpinning for these ideas. We therefore propose an alternative orienting framework, in which learning takes place through initiation into the space of reasons, such that a person's thoughts, actions and feelings are increasingly open to critique and justification. We argue that knowing as responsiveness to reasons encompasses not only the powerful ideas of mathematics and disciplinary knowledge of modes of enquiry but also the extralogical, such as in feelings of the aesthetic, control, excitement, elegance and efficiency. We discuss the implication that mathematics educators deeply consider the learner's reasons for purposeful activity and design settings in which these reasons can be made public and open to critique.

International Handbook of Research in Statistics Education, 2017

The goal of this article is to draw attention to the need to think about learning environments and their design as a way of considering how sustainable change in the learning and teaching of statistics can be supported. The goal is not to advocate one particular approach to the design of learning environments, but rather to raise awareness of the need to consider this lens in statistics education. We first present the rationale for the importance of a focus on learning environments for statistics education. We consider six design considerations for statistics learning environments. Finally, we discuss implications and emerging directions and goals for further implementation and research.

International Journal of Science Education, 2017

This randomised controlled trial used a mixed-methods approach to investigate the frequency and how elementary teachers integrated engineering design (ED) principles into their science instruction following professional development (PD). The ED components of the PD were aligned with Cunningham and Carlsen's [(2014). Teaching engineering practices. Journal of Science Teacher Education, 25, 197-210] guidelines for ED PD and promoted inclusion of ED within science teaching. The treatment group included 219 teachers from 83 schools. Participants in the control group included 145 teachers from 60 schools in a mid-Atlantic state. Data sources, including lesson overviews and videotaped classroom observations, were analysed quantitatively to determine the frequency of ED integration and qualitatively to describe how teachers incorporated ED into instruction after attending the PD. Results indicated more participants who attended the PD (55%) incorporated ED into instruction compared with the control participants (24%), χ 2 (1, n = 401) = 33.225, p < .001, r f = 0.308. Treatment and control teachers taught similar science content (p's > .05) through ED lessons. In ED lessons, students typically conducted research and created and tested initial designs. The results suggest the PD supported teachers in implementing ED into their science instruction and support the efficacy of using Cunningham and Carlsen's (2014) guidelines to inform ED PD design.

Mathematics Education Research Journal, 2014

Proportional reasoning as the capacity to compare situations in relative (multiplicative) rather than absolute (additive) terms is an important outcome of primary school mathematics. Research suggests that students tend to see comparative situations in additive rather than multiplicative terms and this thinking can influence their capacity for proportional reasoning in later years. In this paper, excerpts from a classroom case study of a fourth-grade classroom (students aged 9) are presented as they address an inquiry problem that required proportional reasoning. As the inquiry unfolded, students' additive strategies were progressively seen to shift to proportional thinking to enable them to answer the question that guided their inquiry. In wrestling with the challenges they encountered, their emerging proportional reasoning was supported by the inquiry model used to provide a structure, a classroom culture of inquiry and argumentation, and the proportionality embedded in the problem context.

Task Design In Mathematics Education, 2015

Mathematical Thinking and Learning, 2019

While various studies suggest that informal statistical inference (ISI) can be developed by young students, more research is needed to translate this claim into a well-founded learning trajectory (LT). As a contribution, this paper presents the results of a cycle of design research that focuses on the design, implementation, and evaluation of the first part of an LT for ISI, in which ninth-grade students (N = 20) are introduced to the key concepts of sample, frequency distribution, and simulated sampling distribution. The results show that an LT starting from repeated sampling with a black box may support the accessibility of these concepts, as these students were able to make inferences with the frequency distribution from repeated samples as well as with corresponding simulated sampling distributions. This suggests a promising way to make ISI more accessible to students.

Frontiers in Education

This paper attempts to demonstrate the usefulness of the linkage data from two international large-scale assessment studies, Teaching and Learning International Survey 2013 (TALIS) 2013 and Programme for International Student Assessment (PISA) 2012, in examining the effects of schools. Data from seven educational systems are used to link, and four critical issues with five selection criteria are applied to the data selected. The linking dataset facilitates the investigation of mathematics performance while considering individual learner characteristics, mathematics teacher variables in the classroom environment and the school-level variables. We extend the new avenue of research by developing a linked database geared to the specific mathematics teaching and learning domain to reflect the school mathematics educational environment. The case study using Singapore linkage data demonstrated the feasibility and potential of exploring school effectiveness. In Singapore, schools with teach...

Investigations in mathematics learning, 2012

The central role that tasks play in student learning has become clearer in the last decade. As a result of calls for a structured design process, task design has become one of the main fields of investigation in mathematics education. Here, we propose five design considerations (deciding on the mathematical content, choosing the context, deciding on the level of structural openness, choosing representations, and arranging a task sequence within an activity). We show how these considerations influenced the design of an activity, and compare students' actual work while solving this particular activity and the designers' considerations.

ZDM, 2017

In this Special Issue, we aim to extend thinking around mathematical task design by focusing on a complex, layered relationship between task designers, teachers, and students. Researchers have had increasing interest in investigating a student perspective on task design, resulting in a student theme at the International Commission on Mathematical Instruction (ICMI) study on mathematical task design and a Research Forum on mathematical tasks and the student, at a meeting of the International group for the Psychology of Mathematics Education (PME) . In the PME Research Forum, Clarke et al. (2014, p. 119) brought together a collection of research studies to "equip researchers to better situate the student within research on instructional task design". Looking ahead, Ainley and Margolinas (2015) called for research using "alternative methods to understand student perspectives more fully, particularly in the context of innovative task design" (p. 137). Together, the collection of papers in this Special Issue represents a response to this call. Many contributors to this Special Issue attended the student theme group at the ICMI study on mathematical task design. In this Special Issue, contributors have further developed research presented at the ICMI study, extending in their thinking and practice, enriched by the work of each other. Some contributors have an explicitly methodological aim and, for others, the focus is more on innovative task design, analyzed or approached with a student perspective. We articulate a student perspective on task design in mathematics education, foregrounding a dynamic relationship between intentions of task designers, teachers, and students. We organize this survey paper into four main sections. First, we characterize a student perspective on task design. Second, we provide theoretical perspectives that we use as tools to account for different facets of task design from a Abstract We articulate a student perspective on task design in mathematics education, foregrounding a dynamic relationship between intentions of task designers, teachers, and students. First, we characterize a student perspective on task design. Second, we provide theoretical perspectives that we use as tools to account for different facets of task design from a student perspective. Third, we elaborate on "tensions of intentions" between designers, teachers, and students when considering the use of tasks to promote students' mathematical learning. Fourth, we address mathematical tasks and the student: characterizing task context from a student perspective, contrasting reflective and emergent task design, and discussing theoretical and methodological approaches that take into account the student perspective. We conclude with implications for research and practice.