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In this article we propose the notion of differentiation from an advanced standpoint as a teaching strategy, particularly valuable for working with students with low prior attainment. The notion arose from an enactivist analysis of the work of three teachers, engaged in action research in their own classrooms. All three teachers chose to teach their students (who were aged 15-16) topics that are usually only offered to those with relatively high prior attainment in mathematics. No intermediate or bridging topics were offered, instead, these teachers found ways to differentiate work for their classes, from this advanced standpoint. There is tentative evidence of students experiencing their relationship to mathematics in new ways, recognising they were doing "A-grade" work, and of gains in their attainment. There is also evidence of the teachers' own surprise at what their students could achieve.

The goal of this study is to characterize how pre-service teachers recognize students' reasoning, based in the unitizing process, when students solve ratio comparison problems and, how they consider the unitizing process as a Key Development Understanding to propose teaching decisions. Ninety-one pre-service primary teachers interpreted primary school students' answers to a ratio comparison problem and proposed teaching decisions to improve students' reasoning. Findings indicate that pre-service teachers recognized features of the unitizing process as a Key Development Understanding but had difficulties to propose teaching decisions addressed to improve the students' mathematical reasoning. These findings display the role played by specific mathematical content domain when pre-service teachers make teaching decisions considering the students' mathematical reasoning.

A problem identified in the literature around task design is the persistence of a gap between teacher intention and student activity. We show how principles designed around the making of distinctions and having an explicit language of mathematical thinking can eliminate the "gap" by guiding teacher planning, teacher actions in the classroom and student activity in the classroom. We show how our task design principles have developed during the time of our collaboration, over a period of 20 years, across several research projects. We argue for the importance, in task design, of an explicit theory of change and an explicit image of mathematical thinking, where the theory of change is applicable to researchers, teachers and students.

This paper focusses on how researching is done through reflections about, or at a meta-level to, the practice over time of an enactivist mathematics education researcher. How are the key concepts of enactivist theory ("ZDM Mathematics Education," doi: 10.1007/s11858-014-0634-7, 2015) applied? This paper begins by giving an autobiographical account of the author's engagement with enactivist ideas and the development of enactivist research projects. The rest of the paper then discusses principles of the design of enactivist followed by four themes of learning, observing, interviewing and find-ing(s). The spelling, find-ing(s), draws attention to the findings of enactivist research being processes not objects. In the case of the collaborative research group used as an exemplar throughout the paper, for instance, the find-ing(s) shed light onto the journeys of professional development travelled by the members of the group as they develop their teaching.

We illustrate and exemplify how the idea of reflection is framed by the enactive concept of "deliberate analysis". In keeping with this frame, we do not attempt to define reflection but rather work on the question of "how do we do reflecting?" within such a frame. We set out our enactivist theoretical stance, in particular pointing to implications for how we can learn from experience and showing the role of "deliberate analysis". We then describe, drawing on education literature, what is generally seen as the purpose of reflection and review some existing conceptualizations in mathematics education, pointing out where we draw distinctions. To illustrate how we do reflecting, we offer excerpts from two lessons of an expert teacher and the writing of a prospective teacher. We exemplify how reflecting as deliberate analysis leads to a way of working with teachers supporting them in handling multiple views and ambiguity, their actions being contingent upon their students' actions in learning mathematics.

This article presents the authors' personal accounts and their experiences in working on mathematics using "arms." "Arms" is an idea that first appeared as a program written by John Warwick and David Wooldridge in an ATM publication "Some Lessons in Mathematics with a Microcomputer," 1983. The introduction to "Arms" in the book states that it is a program for the study of locus. It goes on to say that two rotating arms are displayed on the screen, the tips of these arms are joined by an invisible elastic band on which a dot is marked. As the arms rotate the path of the dot is plotted on the screen. Here, the authors split their attention between working on the mathematics and paying attention to how it was being worked on by the group.

We describe the "somatic marker hypothesis" proposed by Damasio (1996) to account for the ability of most people to make decisions quickly and continually in the course of their lives. We relate this hypothesis to two other theoretical constructs, emotional orientations and purposes, which we have used in our research on students' reasoning and teachers' decision making. Given that somatic markers are a part of unconscious mental activity, they cannot be observed by introspective reflection. How then can we research something we cannot see? Beginning with the hypothesis that somatic markers influence actions, we observe, particularly, the actions of student teachers, teachers and children in mathematics classrooms at points where they make decisions. This process is illustrated through examples both of teaching and learning in mathematics, and through the account (see Op't Eynde and Hannula, this issue) of "Frank" reflecting on his decision-making in mathematical activity. We use the case of Frank to illustrate some differences between viewing mathematical activity from our perspective and from those of some other contributors to this special issue. The connections between emotional orientations, somatic markers and purposes are further illustrated by two examples drawn from our research into teacher development and students' reasoning processes.

In this article, Laurinda Brown and Alf Coles relate to each other their own observations from each other's teaching styles. Brown relates that she was quite unsuccessful when she used Cole's strategy of using chanting when she introduced the Gattegno chart to her class. She realizes that could have raised the level of complexity of her lessons to her class until she could find something to work on at their level. On the other hand, Coles relates that he had been impressed about how Brown seemed to "get alongside" with her students. He also relates how he had been impressed upon Brown's dealing with her low-attaining students and with how she leaves her students energized about their work after 1-1 interaction.