To Örebro University

This study contributes to the call for influencing practice by increasing attention to how learning environments can be designed to support learning in statistical inference. We report on a design experiment in secondary school (students 14–16 years old), that resulted in a set of lessons with the learning goal of teaching students how to apply concepts and principles of hypothesis testing for making an inference as to whether or not students in secondary school can taste the difference between two brands of cola soda. The design experiment resulted in four design principles for a simulation-based approach for learning hypothesis testing in secondary school. The design principles highlight the combination of practical and digital simulations of samplings. They stress the need for using random generators that allow for high reliability in collecting sample data and introduce a simulation-based method for determining p-values, i.e. to quantify how likely or surprising a sample result, or a result more extreme, is under a null hypothesis.

The aim of the present study is to explore strategies in enumerating units of three dimensional (3D) arrays. We analyse enumeration strategies of students in grade 3 (ages 8 to 9) in situations of cubical and spherical representations of units of 3D arrays. By exploring students’ strategies in these two situations, we find that difficulties in enu- merating units in 3D arrays can be traced to difficulties in units-locating, with the con- sequence of applying double and triple counting. Our results also indicate that spherical units can serve as perceptual clues in units-locating and in assembling units into rel- evant composites. With input from our findings, we suggest research to investigate the following three hypotheses: (i) spherical units can turn students away from double and triple counting, (ii) spherical units can support students’ units-locating process and their ability to assemble units into relevant composites and (iii) teaching of enumerating 3D arrays should start with spherical units before cubical units.

Professional learning communities (PLC) have increasingly attracted attention in research on teachers' professional development. The aim of this study is to identify contradictions that can occur and be manifested in PLCs in mathematics. Identifying contradictions in PLCs are important, as the identification and resolution of contradictions are crucial to developing PLCs. We have conceptualized PLCs and contradictions within the Cultural Historical Activity Theory. Our data consist of two iterations of interviews with four teacher leader coaches with extensive experience of coaching teacher leaders of PLCs in mathematics. The study distinguishes 26 manifestations of contradictions, taking the overall forms of dilemmas and conflicts. Our results can be used in designing PLCs in mathematics: they can be used to make visible and increase participants' awareness of contradictions involved in PLCs and thereby increase the possibility that the contradictions serve as sources of support rather than obstacles in the development of PLCs in mathematics.

The purpose of this study is to further our understanding of orchestrating math-talk with digital technology. The technology used is common in Swedish mathematics classrooms and involves personal computers, a projector directed towards a whiteboard at the front of the class and software programs for facilitating communication and collective exploration. We use the construct of instrumental orchestration to conceptualize a teacher’s intentional and systematic organization and use of digital technology to guide math-talk in terms of a collective instrumental genesis. We consider math-talk as a matter of inferential reasoning, taking place in the Game of Giving and Asking for Reasons (GoGAR).The combination of instrumental orchestration and inferential reasoning laid the foundation of a design experiment tha taddressed the research question: How can collective inferential reasoning be orchestrated in a technology-enhanced learning environment? The design experiment was conducted in lower-secondary school (students 14–16 years old) and consisted of three lessons on pattern generalization. Each lesson was tested and refined twice by the research team. The design experiment resulted in the emergence of the FlexTech orchestration, which provided teachers and students with opportunities to utilize the flexibility to construct, switch and mark in the orchestration of an instrumental math-GoGAR.

The aim of this study is to investigate task design for differentiated instruction in mathematics in professional learning communities. Based on the cultural-historical activity theory, we conceptualize a professional learning community as an activity system and use the analytical construct of contradictions to give an account of structures that bring forward the teachers’ work. Eight Swedish upper secondary teachers, engaged in designing tasks for differentiated instruction in mixed-ability mathematics classrooms, are studied. The analysis outlines three contradictions, manifested as three dilemmas, and shows how the teachers, by noticing a dilemma and making it an explicit object of inquiry, came to address a diversity of issues related to differentiated instruction in mathematics. For example, the teachers addressed students’ different learning needs, problem-solving activities for amixed-ability classroom, and design of tasks that are challenging for all students. With input from our findings on the three dilemmas, we then discuss the design of a task analysis guide as a means for facilitating the development of a professional learning community that is inquiry-oriented with astrong content focus on designing tasks for differentiated instruction in mixed-ability mathematics classrooms.

This study introduces inferentialism and, particularly, the Game of Giving and Asking for Reasons (GoGAR), as a new theoretical perspective for investigating qualities of procedural and conceptual knowledge in mathematics. The study develops a framework in which procedural knowledge and conceptual knowledge are connected to limited and rich qualities of GoGARs. General characteristics of limited GoGARs are their atomistic, implicit, and noninferential nature, as opposed to rich GoGARs, which are holistic, explicit, and inferential. The mathematical discussions of a Grade 6 class serve the case to show how the framework of procedural and conceptual GoGARs can be used to give an account of qualitative differences in procedural and conceptual knowledge in the teaching of mathematics.

Mathematical reasoning is gaining increasing significance in mathematics education. It has become part of school curricula and the numbers of empirical studies are growing. Researchers investigate mathematical reasoning, yet, what is being under investigation is diverse—which goes along with a diversity of understandings of the term reasoning. The aim of this article is to provide an overview on kinds of mathematical reasoning that are addressed in mathematics education research. We conducted a systematic review focusing on the question: What kinds of reasoning are addressed in empirical research in mathematics education? We pursued this question by searching for articles in the database Web of Science with the term reason* in the title. Based on this search, we used a systematic approach to inductively find kinds of reasoning addressed in empirical research in mathematics education. We found three domain-general kinds of reasoning (e.g., creative reasoning) as well as six domain-specific kinds of reasoning (e.g., algebraic reasoning). The article gives an overview on these different kinds of reasoning both in a domain-general and domain-specific perspective, which may be of value for both research and practice (e.g., school teaching).

This study examines informal hypothesis testing in the context of drawing inferences of underlying probability distributions. Through a small-scale teaching experiment of three lessons, the study explores how fifth-grade students distinguish a non-uniform probability distribution from uniform probability distributions in a data-rich learning environment, and what role processes of data production play in their investigations. The study outlines aspects of students’ informal understanding of hypothesis testing. It shows how students with no formal education can follow the logic that a small difference in samples can be the effect of randomness, while a large difference implies a real difference in the underlying process. The students distinguish the mode and the size of differences in frequencies as signals in data and used these signals to give data-based reasons in processes of informal hypothesis testing. The study also highlights the role of data production and points to a need for further research on the role of data production in an informal approach to the teaching and learning of statistical inference.

The aim of this study is to investigate relationships between note-taking and the orchestrating of math-talk in whole-class teaching. A lesson on (average) velocity in a Swedish Grade 6 has been observed. Taking an inferentialist stance on human understanding, the study conceptualizes teaching and learning from the perspective of how students come to be engaged in the language practice of giving and asking for reasons. The study shows how note-taking supports a teacher-student relationship where the teacher produces content and the students’ participation is reduced to consume content. It shows how note-taking can support descriptive math-talk of concepts and symbols and step-by-step procedural math-talk, connected to the goal of providing students examples of tasks, similar to the tasks in their textbook.