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If we want to innovate mathematics education — if we want to achieve something beyond conformity and compliance in mathematics education — these institutional norms need to be challenged. In this paper, I look at the results-first research methodology in which the institutional norms are challenged by the simple goal of increasing student thinking in the classroom. I share the specific results that emerged out of this research and use it as a specific case to argue that real innovation in mathematics education can only occur if we are willing to challenge the institutional norms that have been with us for 150 years.

The development of mathematical reasoning skills has increasingly been of focus for the teaching and learning of mathematics. This research utilizes a teaching simulation using the methodology of scriptwriting, in which prospective teachers are asked to complete a script of a dialogue from a classroom simulation involving fraction multiplication and division with justification, assisting fictional students to work through their difficulties and helping them to justify their reasoning. Such tasks allow for the examination of the prospective teacher moves to support student reasoning through their imagined action and choice of words. Scripts from forty-one prospective primary teachers were examined for the study, and five clusters based on the type of teacher move for supporting student reasoning were found. Overall, the prospective teachers emphasized the elicitation and facilitation of students’ ideas. The cluster analysis, however, provided a nuanced examination of the cohort’s teacher moves. While cluster one saw the highest incident of eliciting teacher moves, albeit only in the low potential category, clusters two and three mostly used facilitating teacher moves, but varied in their use of high and low potential moves. Cluster four concentrated moves on facilitating, eliciting, and responding to student reasoning. Cluster five employed teacher moves from all main categories, with some instances of high potential moves in all categories except extending student reasoning, which can better support reasoning. The prospective mathematics teachers’ scripts and the five clusters that were found during analysis are discussed with implications for future teacher education and the support of building mathematical reasoning.

Student difficulty with mathematics has been a pervasive and systemic problem since the advent of public education—not because students can't learn mathematics, but because, by and large, students can't learn it by being told how to do it. Since the publication of the NCTM Principles and Standards (1998) there has been a concerted effort to change this reality by transitioning to more progressive and student-centered pedagogies. And progress has been made. Yet, something is still missing. Systemically, we are still struggling with high failure rates, low self-efficacy, and massive student disengagement. In this article I look at the results of a research project that was specifically designed to change this reality by finding ways to get students to think. The results of this work shows that there are a collection of teaching practices that any teacher can use to turn their classroom into a space where students think, learn, and succeed

Problem solving and problem posing have long been of interest to the mathematics education community. In this survey paper we first look at some of the seminal moments in the history of research on the important topics. We then use this history to position the state-of-the-art research being done in both problem solving and problem posing, before introducing the presented state-of-the-art developments in problem solving and problem posing. We then use this work as a backdrop against which to introduce the 16 empirical papers that make up this special issue. Together these 16 papers add nuance to what is already known about problem solving and problem posing; this nuance is the result of attending to very specific contexts and purposes in which these activities are embedded. We end the paper by discussing the future directions these fields can take.

Many studies have investigated students’ problem-posing activities. However, there has not been a strong focus on affective aspects in mathematical problem-posing. Also, students’ understanding of ...

For too long, problem solving has been studied as a solitary and isolated activity where individuals make progress based on personal resources acquired from knowledge, previous experiences or moments of illumination; however, in society, and indeed amongst mathematicians, problem solving can be highly collaborative and occurs in spaces where external resources (technology, the internet, social connections etc.) are abundant. In this paper, we present research into what problem solving looks like in classrooms when it is done collaboratively with access to resources that go beyond the knowledge and past experiences of the individual and even the group. Results indicate that in such choice-affluent environments, students will seek out new resources either when their collective or individual resources run low and will do so either by looking or discussing with others. This manuscript also offers a new methodological tool in the form of, what we call, gaze-dialogue transcripts for documenting such resource acquisition.

Changing the beliefs of preservice teachers is a prominent topic in mathematics education. In this paper, we present the results of a study that focuses on the changes in preservice elementary-scho...

This study concerns a professional development course designed and implemented for prospective teachers, centred on a teaching method regarding problem-solving activity, namely, the Thinking Classroom. The study is framed in the theory of cultural transposition, a perspective about the encounter with teaching practices from different cultural/school contexts. Cultural aspects are considered crucial and this encounter between cultures is seen as an opportunity for actors to become aware of their own unthoughts, i.e., some of the ‘invisible’ cultural beliefs about teaching and learning absorbed by their own culture. According to this framework, we present the results from a questionnaire given to all the participants, and two case studies of prospective teachers involved in the professional development, in order to discuss the kind of unthoughts on which they have focused in thinking about this training experience.

In this chapter we focus on emotions and their role in teaching and learning. Through the presentation and discussion of two episodes that took place in the context of teacher education and in-class problem-solving, respectively, we present the two mainstream approaches to emotions in affect-related research. We end the chapter with some methodological considerations and with a sketch of future direction for research on mathematics-related emotions.

The preparation of graduate students to become independent researchers is a central goal of all doctoral programs. In this paper, I present the results of a study looking at how this move to independence is viewed by graduate students enrolled in a mathematics education master’s or PhD program. Results indicate that graduate students have very different views and experiences between the doing of research and the writing up of research. Results also indicate that membership in a community of researchers is central to the transition to independence.

International audience

Three hundred manuscripts on mathematics education are submitted for review to the International Journal of Science and Mathematics Education (IJSME) every year. The vast majority of these are rejected. In many cases, manuscripts that are being rejected are based on good research on interesting topics, but are being rejected because the author has failed to articulate his or her work in ways that reviewers and editors find appealing. This chapter looks closely at what constitutes a good paper and offers guidance for early researchers on how to write for publication in IJSME.

Changes of beliefs do not happen arbitrarily; there are underlying mechanisms that enable the shift from a preexisting belief to a new belief. This study outlines a problem-solving implementation in which we suggest participants experienced shifts in their beliefs. We describe these belief changes and propose a new mechanism for the shift: first-person vicarious experiences. Our results suggest that, despite their initial uncertainties, individuals who agreed to make a change in their practice underwent a change in belief regarding the efficacy of the practice through firsthand experience of its implementation.

Self-efficacy is commonly defined as the belief in one’s abilities to attain a goal or outcome. This has significance in classroom situations where students with low self-efficacy fall into a self-fulfilling feedback loop of low aspirations leading to low performance, leading to even lower aspirations. In this research outline, we present a context in which a whole-class problem-solving implementation interrupted that loop for a student with low self-efficacy in mathematics. We demonstrate this using Bandura’s four sources of self-efficacy and offer further nuance to the original framework.

In this chapter we look at student engagement while doing a modelling task. We present the case of two grade 8 students (age 12–13) and look at their modelling behavior through the double lenses of flow and modelling. During their modelling process, these students experienced an imbalance between the challenges and skills presented to them. Results show that the rebalancing of the challenges and skills were facilitated by the teacher, the students, and the naturally evolving complexities of the task.

In this commentary, I reflect on the chapters by De Simone; Khalil, Lake, and Johnson; and Montoro and Gil (this volume) and how they contribute to our understanding of engagement within mathematics education. I begin by exploring how engagement has been pursued in mathematics education in particular, and in education in general, and argue that each of these instances can be viewed as either an account-of engagement (ways in which participants engage) and/or an accounting-for engagement (and explanations for why they engage in this way). I then apply this duality of accounting-of and accounting-for in my commentary on the three chapters that make up this section.

In this chapter, the engagement of 244 students is measured across three different types of mathematical problems (modelling problems, word problems, and mathematical exercises). We also investigate the potential of teaching modelling problems in changing students’ attitude towards mathematics. This research was conducted with a pre-test, followed by an educational intervention and a post-test. During the educational intervention, two groups of students with different instructional formats attended six sessions of modelling problem solving. The results of this study show that, although, there existed no significant difference in engagement existed between the three types of mathematical problems in the pre-test data, both quantitative and qualitative analysis of the post-test data showed that students generally experienced more engagement on word problems. Results also show that teaching modelling problems improved students’ attitudes towards mathematics in both groups.

For many teachers, the incorporation of problem solving into their practice is often met with difficulty as students who are not accustomed to this form of mathematics teaching struggle with, and resist, these efforts. In this chapter, I present results from two research projects in which I studied the effects of various teaching methods on students’ problem solving. In particular, I look at the affordances of a variety of work spaces on students’ engagement while problem solving. Results indicate that the use of vertical whiteboards is exceptionally conducive to engaging students and eases the introduction of problem solving. Results also indicate that teachers were willing to implement, and did implement, VNPS in their classroom even after only a single workshop.

Research in the affective domain has often been restricted to focused attention on a single affective variable. This is ironic given that we know that affective variables tend to cluster. Perhaps the reason for this is that we lack theories for thinking about affective clusters. In this paper I use Green’s theory of a belief cluster (1971) as the foundation for looking at a new construct—the affect cluster—and how it functions in an experience-rich environment. This proves to be a useful construct in explaining the case of Sara, a girl whose affect around mathematics has been completely changed.

In this chapter I first introduce the notion of a thinking classroom and then present the results of over 10 years of research done on the development and maintenance of thinking classrooms. Using a narrative style I tell the story of how this research began and led first to the notion of a thinking classroom and then to a research project designed to find ways to help teachers build such a classroom. Results show that there are a number of relatively easy to implement teaching practices that can bypass the normative behaviours of almost any classroom and begin the process of developing a thinking classroom.

Engagement in mathematical problem-solving is an aspect of problem-solving that is often overlooked in our efforts to improve students’ problem-solving abilities. In this chapter, I look at these constructs through the lens of Csikszentmihalyi’s theory of flow. Studying the problem-solving habits of students within a problem-solving environment designed to induce flow, I look specifically at student behavior when there is an imbalance between students’ problem-solving skills and the challenge of the task at hand. Results indicate that students have higher than expected perseverance in the face of challenge, higher than expected tolerance in the face of the mundane, and use these as buffers while autonomously correcting the imbalance. Emerging from this research is an extension to Csikszentmihalyi’s theory of flow and support for the teaching methods emerging out of my earlier work on building thinking classrooms.

Students do not always behave the way we intend for them to. Sometimes they substitute the learning opportunities we offer them with behaviours that, on the surface, seem to be conducive to learning but in reality stand in place of learning. In this paper I explore teachers’ beliefs of such behaviour, called proxies for learning, and compare them to teachers’ beliefs of what it means to teach and learn mathematics. Results indicate that if teachers have a problem-solving view of mathematics, then they see such behaviours as their responsibility and theirs to fix. If teachers have a toolbox view of mathematics, they see these behaviours as inevitable and the responsibility of the students.

Pre-service teachers come to mathematics method courses with well-established conceptions of what it means to teach and learn mathematics. Images of teaching reinforced by their own lived experiences shape their pedagogy. This can be problematic for a teacher educator for whom it may be necessary to offer a way of reframing traditional notions of teaching and learning. The research presented here examines that process of reframing. In this study we deliberately introduce a tension in pre-service teachers’ conception of timed drills and examine the resulting process of transition they undergo. Our findings suggest that the introduced tension provided the means for reflection on intent and resulted in a subsequent change in action.

Over the last 15 years, numeracy has become more and more prominent in curriculum initiatives around the world. Yet, the notion of numeracy is still not well defined, and as such, often not well understood by the teachers who are charged with the responsibility of helping our students to develop these skills. In this article I explore the work of a team of mathematics teachers brought together for the purpose of developing a set of numeracy tasks for use within district wide numeracy assessments. Results indicate that these teachers’ experiences designing these tasks, and pilot testing them in their own classrooms, propelled them to make massive changes in their own mathematics teaching practice.