Toward a framework for the development of mathematical knowledge for teaching

Shulman (1986, 1987) coined the term pedagogical content knowledge (PCK) to address what at that time had become increasingly evident—that content knowledge itself was not sufficient for teachers to be successful. Throughout the past two decades, researchers within the field of mathematics teacher education have been expanding the notion of PCK and developing more fine-grained conceptualizations of this knowledge for teaching mathematics. One such conceptualization that shows promise is mathematical knowledge for teaching—mathematical knowledge that is specifically useful in teaching mathematics. While mathematical knowledge for teaching has started to gain attention as an important concept in the mathematics teacher education research community, there is limited understanding of what it is, how one might recognize it, and how it might develop in the minds of teachers. In this article, we propose a framework for studying the development of mathematical knowledge for teaching that is grounded in research in both mathematics education and the learning sciences.

Negotiating the transition between different teaching contexts through shared responsibility and shared reflection

This paper draws on my personal experiences with coteaching and my participation in the research described by Wassell and LaVan (2009). It examines the role of coteaching in the development of structures that afforded opportunities for shared reflection and shared responsibility between stakeholders in the classroom. It also describes how the schema and practices developed through coteaching and cogenerative dialogue helped mediate the transition between my preservice and inservice teaching experiences.

The Role of Doctoral Advisors: A Look at Advising from the Advisor’s Perspective

The doctoral advisor is said to be one of the most important persons—if not the single most critical person—with whom doctoral students will develop a relationship during their doctoral degree programs (Baird 1995). However, we have limited knowledge regarding how doctoral advisors see their roles and responsibilities as advisors. Therefore, through in-depth interviews, we explored the perceptions of 25 exemplary doctoral advisors, who have graduated a large number of doctoral students, about their roles and responsibilities as advisors. We conclude this article with implications for doctoral education.

Peer Helpers in Hungary: A Qualitative Analysis

Hungary is a country in transition that has no real tradition of peer helping. A qualitative study was carried out involving 13 peer helpers of two kinds (a) age-based peers, and (b) way-of-life-based peers (fellow helpers). The motivations for and the processes of becoming a peer helper were analyzed. Results showed the largest difference being that the motivation for becoming an aged-based peer helper tended to involve aspirations for a professional helping career, whereas way-of-life-based peer helpers typically involved recovery narratives. The experiences suggest that a helping system involving civil society (peers) might develop from a state-controlled help system.

The power of Colombian mathematics teachers’ conceptions of social/institutional factors of teaching

In this paper I shall discuss data from a study on Colombian mathematics teachers’ conceptions of their own teaching practices of beginning algebra, which led to the development of a theoretical model of teachers’ thought structures designed as a thinking tool at the initial stage of the study. With a focus on the perspectives of teachers, the study investigated the relationship between the teachers’ conceptions of beginning algebra and their conceptions of their own teaching practices, with a view to unravelling their conceptions of change in their practices. Significant findings which threw light on the aforementioned relationship have been presented in Agudelo-Valderrama, Clarke and Bishop (2007), highlighting a direct association between a teacher’s conceptions of the nature of beginning algebra, the crucial determinants of her/his teaching practice, and her/his attitude to change. After an overview of the study, this paper focuses on specific evidence which clearly shows that in contrast to the strong relationship between a teacher’s conceptions of mathematics and her/his teaching practice, assumed in the theoretical model of teachers’ thought structures, the teachers see a strong relationship between their conceptions of social/institutional factors of teaching and what they do in their teaching. Implications of the findings for teacher education in Colombia are identified.

Views about Physics held by Physics Teachers with Differing Approaches to Teaching Physics

Physics teachers’ approaches to teaching physics are generally considered to be linked to their views about physics. In this qualitative study, the views about physics held by a group of physics teachers whose teaching practice was traditional were explored and compared with the views held by physics teachers who used conceptual change approaches. A particular focus of the study was teachers’ views about the role of mathematics in physics. The findings suggest the traditional teachers saw physics as discovered, close approximations of reality while the conceptual change teachers’ views about physics ranged from a social constructivist perspective to more realist views. However, most teachers did not appear to have given much thought to the nature of physics or physics knowledge, nor to the role of mathematics in physics.

Investigating imagination as a cognitive space for learning mathematics

Our work is inspired by the book Imagining Numbers (particularly the square root of minus fifteen), by Harvard University mathematics professor Barry Mazur (Imagining numbers (particularly the square root of minus fifteen), Farrar, Straus and Giroux, New York, 2003). The work of Mazur led us to question whether the features and steps of Mazur’s re-enactment of the imaginative work of mathematicians could be appropriated pedagogically in a middle-school setting. Our research objectives were to develop the framework of teaching mathematics as a way of imagining and to explore the pedagogical implications of the framework by engaging in an application of it in middle school setting. Findings from our application of the model suggest that the framework presents a novel and important approach to developing mathematical understanding. The model demonstrates in particular the importance of shared visualizations and problem-posing in learning mathematics, as well as imagination as a cognitive space for learning.

Program Assessments: Success Strategies for Three Canadian Teaching Centers

Program assessments are an essential part of the ongoing survival of teaching centers performed by faculty development personnel at institutions of higher education. Little research is available to guide developers in performing these assessments. In this article we describe assessments conducted at three Canadian universities and highlight the theoretical models used to guide the process. Reflections on the strengths and challenges are discussed for each program assessment for the purpose of assisting faculty developers in performing similar program assessments of their faculty development offices.

Is student-centered pedagogy impossible in Hong Kong? The case of inquiry in classrooms

Hong Kong has been actively promoting a student-centered approach to teaching since the 1980s. Despite this effort, students in Hong Kong still tend to be traditional learners who rarely experience and gain from real student-centered learning. While teachers hold a “quantitative” concept of learning and focus on transmitting declarative knowledge to students (Biggs and Watkins, Classroom learning: Educational psychology for the Asian teacher, 1995), students generally practise “rote learning.” Constructive learning models such as inquiry remain little used by students in most Hong Kong classrooms. This article reports a study that examines the feasibility of implementing inquiry method in Kong Kong’s primary classrooms. It analyses the implementation and some outcomes of an inquiry-based project conducted in two local primary schools—a traditional elite Catholic school and a progressive, less-privileged school. Finally, it discusses the contextual factors as well as cultural issues on teachers’ perception and implementation of inquiry in teaching. These factors include the following: impacts of prevailing ideology in the community of Hong Kong, and the top-down policy-making and management by the government.

Student performance and attitudes under formative evaluation by teacher,self and peer evaluators

This study investigated the effects of teacher, self and peer evaluation on preservice teachers’ performance, knowledge and attitudes. Earlier research by the same authors revealed that students made significant improvements in their lesson plans under all three of these conditions, but the teacher-evaluation improved significantly more than the two student-evaluation groups. Therefore, relevant training and practice in the evaluation process were added in this study. All three groups made significant improvements from draft to final version of their plans, and the differences between the teacher-evaluation group and the two student-evaluation groups were non-significant. The authors attribute the stronger performance of the student-evaluation groups to their training on the evaluation task. Students’ overall attitudes were significantly more positive toward teacher evaluation than toward peer evaluation, but did not differ significantly between these two groups and the self-evaluators. Several suggestions are discussed for further improvements in the training of self and peer evaluators.

A scaffolding framework to support the construction of evidence-based arguments among middle school students

Problem-based learning (PBL) is an instructional approach in which students in small groups engage in an authentic, ill-structured problem, and must (1) define, generate and pursue learning issues to understand the problem, (2) develop a possible solution, (3) provide evidence to support their solution, and (4) present their solution and the evidence that supports it (Barrows, How to design a problem-based curriculum for the preclinical years. Springer Publishing, New York, 1985). However, research has shown that novice problem-solvers and learners without deep content knowledge have difficulty developing strong evidence-based arguments (Krajcik et al., J Learn Sci 7:313–350, 1998a; Reiser, J Lear Sci 13(3):273–304, 2004). In this paper, we discuss the components of (e.g., claims and evidence) and processes of making (e.g., define problem and make claim) evidence-based arguments. Furthermore, we review various scaffolding models designed to help students perform various tasks associated with creating evidence-based arguments (e.g., link claims to evidence) and present guidelines for the development of computer-based scaffolds to help middle school students build evidence-based arguments.

Preparing Elementary Preservice Teachers for Urban Elementary Science Classrooms: Challenging Cultural Biases Toward Diverse Students

This study reports the learning of elementary preservice teachers regarding diversity and teaching science in diverse urban elementary classrooms. From participating in a semester-long book club, the preservice teachers reveal their cultural biases, connect and apply their knowledge of diversity, and understand that getting to know their students are important elements for teaching science in diverse classrooms. These 3 things connect in ways that allow the preservice teachers to understand how their cultural biases impede student learning and gain new knowledge of diversity as they change their cultural biases. Implications of this study reveal that preservice teachers need opportunities to reveal, confront, challenge, and change their cultural models and to develop new models for teaching science in urban elementary classrooms.

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Explanations of attitudes to change: Colombian mathematics teachers’ conceptions of the crucial determinants of their teaching practices of beginning algebra

This article arises from a study whose overall purpose was to investigate the relationship between Colombian mathematics teachers’ conceptions of beginning algebra and their conceptions of their own teaching practices. The teachers’ understandings of their teaching practices were explored with a view to unravelling their conceptions of change in their teaching. Focusing on the perspectives of teachers afforded opportunities that exposed the powerful role that the teachers’ conceptions of social/institutional factors of teaching played in their conceptions of their practices. The degree to which the teachers attributed these (external) factors as crucial reasons for what they do in their teaching was the basis of a categorisation of their conceptions of the crucial determinants of their teaching practices into three types. The findings are particularly relevant to our understanding of the stability of mathematics teaching approaches in the Colombian context but have likely implications for a range of international education contexts. Specific implications for the development of the research into teachers’ conceptions of mathematics and its teaching, and for teacher education programmes are presented.

Interactive learning through web-mediated peer review of student science reports

Two studies analyzed impacts of writing and receiving web-mediated peer reviews on revision of research reports by undergraduate science students. After conducting toxicology experiments, 77 students posted draft reports and exchanged double-blind reviews. The first study randomly assigned students to four groups representing full, partial, or no peer review. Students engaging in any aspect of peer review made more revisions than students confined to reviewing their own reports. In the second study, all students engaged in peer review, and the influence of writing versus receiving critiques was analyzed using linear regression. Both studies showed receiving reviews to be more significant than writing them in terms of triggering report revisions. Students valued the peer review experience and credited it with giving them insights about their work. Conclusions address implications for optimal design of online peer review systems and for further research into student learning gains.

Re-mythologizing mathematics through attention to classroom positioning

With our conceptualization of Harré and van Langenhove’s (1999) positioning theory, we draw attention to immanent experience and read transcendent discursive practices through the moment of interaction. We use a series of spatial images as metaphors to analyze the way positioning is conceptualized in current mathematics education literature and the way it may be alternatively conceptualized. This leads us to claim that changing the way mathematics is talked about and changing the stories (or myths) told about mathematics is necessary for efforts to change the way mathematics is done and the way it is taught.

Mathematical belief change in prospective primary teachers

The development and influence of beliefs in teacher education has been a topic of increasing interest for researchers in recent years. This study explores the responses of a group of prospective primary teachers to attempts to facilitate belief change as part of their initial teacher education programme in mathematics. The students’ responses seemed to fall into three categories: non-engagement; building a new set of beliefs and; reforming existing beliefs. In this article the participants’ responses are outlined and illustrated with stories from three individuals. This study suggests that belief reform is complex and fraught with ethical dilemmas. Certainly there is a need for further research in this area, particularly given the pervasive influence of beliefs on teaching practice.

Learning mathematics for teaching in the student teaching experience: two contrasting cases

Student teaching (guided teaching by a prospective teacher under the supervision of an experienced “cooperating” teacher) provides an important opportunity for prospective teachers to increase their understanding of mathematics in and for teaching. The interactions between a student teacher and cooperating teacher provide an obvious mechanism for such learning to occur. We report here on data that is part of a larger study of eight student teacher/cooperating teacher pairs, and the core themes that emerged from their conversations. We focus on two pairs for whom the core conversational themes represent disparate approaches to mathematics in and for teaching. One pair, Blake and Mr. B., focused on controlling student behavior and rarely talked about mathematics for teaching. The other pair, Tara and Mr. T., focused on having students actively participating in the lesson and on mathematics from the students’ point of view. These contrasting experiences suggest that student teaching can have a profound effect on prospective teachers’ understanding of mathematics in and for teaching.

From Peasant to Princess: A Standards Conversion Story with a Twist

The first author, a student in a graduate children’s literature class, designed a project to locate “good” mathematics-based children’s literature selections. However, the reference tools usually consulted (e.g., Books in Print) to locate books by topic were of little help, and those she located under individual mathematics topics were mostly traditional mathematics books rather than good read-aloud selections. Consequently, she perused the university library’s sizeable juvenile collection to find books that would meet her selection criteria. This article describes the influence of two landmark documents for mathematics teaching and learning—Curriculum and Evaluation Standards for School Mathematics (National Council of Teachers of Mathematics [NCTM], 1989) and Principles and Standards for School Mathematics (NCTM, 2000)—as she engaged in the process.