Backward transfer refers to the influence on prior knowledge of the acquisition and generalisation of new knowledge. Studies of backward transfer of mathematical knowledge have focused on content that is closely related in time and in curricular sequencing. Employing the notion of thickening understanding, we describe instances of transfer that…

This paper introduces a model built upon the Method of Varying Inquiry, offering a didactical approach to problem posing and solving activities that stimulates inquiry-based learning in mathematics classrooms. The model combines the inquiry-based framework with the variation theory and with specific didactical and theoretical elements (the…

Many universities throughout the world apply student-centered approaches. Common to these is that projects, problems, challenges etc. stem from "real life". Mathematics is an effective tool to solve problems, but in this paper, we discuss how we reconcile the fact that mathematics is both a pure and an applied discipline with…

When a Year 7 student physically reacted to a prompt of another student by anxiously drumming the desk with his ruler, exclaiming "uuuuhh", the initial thought of the observing researcher, Laura, was: "this is an interesting account". This started a reflective journey of first applying robust research methodologies to the…

We present a synthesis of the Onto-semiotic Approach (OSA) theoretical system to mathematical knowledge and instruction, while highlighting the problems, principles and research methods that are addressed in this approach and considering the didactics of mathematics as a scientific and technological discipline. We suggest that Didactics should…

In this writing, I report on how two 12-year old children used fraction strips to add 1/2 and 2/5. In their interaction with the tool, I look for the emergence of the tool-mediated Zone of Proximal Development to analyse the knowing that become available to the children. In thinking about this interaction, I ask what is the role of the more…

We aim to open up discussion about the intertwined roles of teachers and tasks that involve students communicating about mathematics when working in groups. Over many years we have observed, researched and ourselves have taught students working on mathematics in groups and find that it is often easier to pay attention to the forms of communication…

We describe how a group of secondary mathematics teachers posed and solved a real-world problem. The problem was posed by teachers on a sandbar off the coast of an island in the Bahamas, where panoramic views of water meeting sky spurred the teachers to wonder how far away they could see. We analyze how the teachers translated the natural…

Advancement of self-regulation during complex problem solving and the development of strategical reasoning are among the central educational goals linked to 21st century skills. In this paper we introduce the notion of "Stepped Tasks", which are specially designed in Top-Down structure to achieve these goals in mathematics instruction.…

This theoretical paper suggests identity as a nexus of research on affect and discourse in mathematical learning. It broadens Sfard and Prusak's (2005) discursive definition of identity by building on an analytical framework that examines positioning of students at three levels: the objects described, the interactions achieved, and the alignment…

Originating from interviews with mathematics colleagues, written accounts of mathematicians engaging with mathematics, and Wes's reflections on his own mathematical work, we describe a process that we call mathematical foresight: the imagining of a resolution to a mathematical situation and a path to that resolution. In a sense, mathematical…

The application of ethnomathematics and mathematical modelling allow us to see a different reality and give us insight into mathematics accomplished holistically. In this context, a pedagogical action that connects ethnomathematics and the cultural aspects of mathematical modelling with its academic features is referred to as ethnomodelling. This…

The notion of temporal range is introduced and discussed in this paper. Two dimensions of temporal range are identified: mathematical temporality relating to mathematical ideas, their precursors and horizons; and a mathematical learning temporality where what students say/do provides the ground on which future learning can be built. These…

Identifying mathematical and didactical conditions under which mathematics learners can encounter an intellectual need for defining and proving is recognized as a challenging research enterprise. This paper presents a particular configuration of conditions under which a group of pre-service mathematics teachers successfully constructed a…

Current conceptualizations of knowing and learning tend to separate the knower from others, the world they know, and themselves. In this article, we offer "relationality" as an alternative to such conceptualizations of mathematical knowing. We begin with the perspective of Maturana and Varela to articulate some of its problems and our alternative.…