Research concerns

Teachers’ beliefs, views and conceptions. Undoubtedly this is one of the most popular research areas in mathematics teacher education. Probably the interest of the community in investigating mathematics teachers' beliefs and conceptions is associated with the prevailing idea that teachers’

beliefs and conceptions inform and define their teaching practices (Skott, 2009). This could explain why there is a great interest in identifying teachers’ beliefs, conceptions and views about different aspects of their teaching. This could also be the origin of the effort made by some researchers to modify and develop these entities in order to positively impact teaching practice (see for example Lavy & Shriki, 2008;

Grootenboer, 2008; Potari & Georgiadou-Kabouridis, 2009).

The interest in this research area has not decreased over the ten-year period covered by the review; on the contrary, researchers' interests in this area have become more specialised and their research reports and studies reflect this specialisation: we can find studies related to teachers’ beliefs about their role as mathematics teacher (Lloyd, 2005); beliefs about the concept of computational estimation (Alajmi, 2009); beliefs about gender and the use of computers for mathematical learning (Forgasz, 2006);

beliefs about a new educational reform (Gooya, 2007), teachers’ views of mathematics (Sterenberg, 2008; Kaasila, Hannula, Laine & Pehkonen, 2008), etc.

Although research on teachers’ beliefs may seem very diverse, there are prevailing trends. According to Philipp (2007), research on mathematics teachers’ beliefs is focused upon: (1) understanding teachers’ beliefs; (2) investigating the relationship between teachers’ beliefs and practices; and (3) changing teachers’ beliefs (p. 306).

Teachers’ practices. This is another dominant research area in mathematics teacher education. Primarily, researchers in this area are trying to characterise the actions that the teacher performs within the classroom, and understand what are the factors shaping and promoting their development. In my opinion, the interest in this aspect of teachers' professional life is due to the fact that many researchers in the community believe that the most prominent part of teachers' professional work is done in classrooms (see for example Krainer & Gofree, 1999, p. 294). These kind of studies report different aspects of teaching practice within the classroom, for example, how teachers make real-world connections in their classrooms (Gainsburg, 2008); the types of questions asked during their lessons (Sahin & Kulm, 2008); the way teachers manage their time during a particular lesson (Assude, 2005); teachers’ role in promoting collaboration among a heterogeneous group of students (Staples, 2008) or teachers’ choice of examples in the classroom (Zodik & Zaslavsky, 2008).

It is important to note that a small group of researchers has begun to focus on the work done by mathematics teachers outside the classroom.

They are particularly focused in the kind of resources used by teachers in order to define the content of their lessons or develop themselves as educators. The argument for focusing on the interaction between a teacher and the external resources she uses to plan her lessons is that this type of activity is at the core of a teacher's professional activity and development (see Gueudet & Trouche, 2009, p. 199). Another example of this type of work is Nicol & Crespo (2006). In their research they analyze how elementary pre-service teachers interpret and use curriculum materials (particularly textbooks) in their lesson planning. These researchers suggest that this type of analysis provide teacher educators with opportunities to help pre-service teachers to consider the strengths and weaknesses of their

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particular adaptations and designs from mathematical, curricular, and pedagogical perspectives. It also provides teacher educators with opportunities to gain insight into what pre-service teachers find important and how we might help them learn to select and pose mathematical tasks that engage students mathematically (p. 352).

Teachers' knowledge and skills. At the centre of this research area the following question is found: what kind of knowledge and skills does a person need in order to be a “good” mathematics teacher? There are many studies that underline the importance of mathematical knowledge (for example Sirotic & Zaskis, 2007; Leikin & Levav-Waynberg, 2007); but there is widespread recognition that to possess mathematical knowledge is a necessary, but not a sufficient condition for being a good mathematics teacher. It is argued that other kinds of knowledge and skills are required, such as mathematical knowledge for teaching or mathematical pedagogy (Silverman & Thompson, 2008; Koirala, Davis & Johnson, 2008);

knowledge of students’ cognition in mathematics (Carpenter & Fennema, 1992) and attention-dependent knowledge or awareness (Ainley &

Luntley, 2007; Mason 1998; Mason, 2008). Indeed, mathematics teaching is a complex job that requires very specialised knowledge and skills. I think the following quotation captures such complexity:

“It's one thing to know that 307 minus 168 equals 139; it is another thing to be able understand why a third grader might think that 261 is the right answer. Mathematicians need to understand a problem only for themselves; math teachers need both to know the math and to know how 30 different minds might understand (or misunderstand) it. Then they need to take each mind from no getting to mastery. And they need to do this in 45 minutes or less” (Green, 2010, March 2).

There are some theoretical models that try to capture what are the necessary skills to become a proficient or competent mathematics teacher (see Kilpatrick, 2004; Niss, 2004). Among the skills covered by these models we can find the ability to collaborate with colleagues and parents concerning mathematics teaching and its conditions, and planning effective instruction and solving problems that arise during instruction.

I think the discussion about mathematics teachers’ knowledge should be shaped by the context in which the teacher develops his or her work. In other words, I think there must be some basic knowledge and skills that any mathematics teacher should have, but I also believe there are other skills and abilities that are especially needed in particular contexts. Just as Adler (2000) has pointed out: What knowledge bases [are necessary] for teaching culturally and linguistically diverse learners? And for teaching across urban and rural, under-resourced schools? (p. 210).

My impression is that the current tendency is to avoid seeing the components, skills or knowledge that make up a “good mathematics teaching” as divided and disconnected elements (see for example Bergsten

& Grevholm, 2005). Researchers now are thinking on the possible balances and the connections between them.

The relationship between theory and practice. The relationship between theory and practice is an academic consideration that has been present in the mathematics teacher educators’ community for many years. One concern that is at the heart of this discussion is that theoretical knowledge (the one produced by researchers) is usually perceived as something different and disconnected from practical knowledge (the one that teachers acquire through their experience). Researchers are trying to show that both types of knowledge are mutually informed, but they are also trying to

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explain the nature of this relationship, how to support it, and what its consequences are. When doing this review, the first article I came across which addressed this aspect was Jaworski (1999). She mentions that one of the causes of the problematic relationship between theory and practice is that educational theories are seen not to take account of the conditions and constraints of learners and classrooms that affect teachers and teaching (p.

184).

It is notable that the discussion of the relationship between theory and practice has been of particular interest to the CERME community of teacher educators. In fact at the CERME 3 conference a thematic group called “Inter-Relating Theory and Practice in Mathematics Teacher Education” was organised (see Jaworski, Serrazina, Koop, & Krainer, 2004). One of the conclusions of this working group was that more collaboration between teacher educators and teachers was needed in order to strengthen the relationship between theory and practice. This collaborative trend is reflected in the special issue also entitled “Inter- Relating Theory and Practice in Mathematics Teacher Education” which was published in the Journal of Mathematics Teacher Education (year 2007, volume 9, number 2). In this issue the papers written by Scherer &

Steinbring (2007) and Jaworski (2007) report results of research projects that were developed through a close collaboration between researchers and teachers. This type of collaborative research in which teachers are regarded as professionals investigating their own practice, is known as action research and challenges the assumption that knowledge is separate from and superior to practice. The production of local knowledge is seen equally important as general knowledge. (Krainer, 2006, p. 213).

It seems to me that the relationship between theory and practice will remain one of the trends in mathematics education research in the coming

years for two reasons: firstly, there are different aspects of the relationship between theory and practice that can be studied, that is to say, it is a fertile area of research. For instance, as I will argue in chapter 7, it is possible (and worthwhile) to continue exploring the use of didactical theories and other products of the mathematics education research as tools for the development of teachers (see for example Even, 2003; Tsamir, 2008); or to make explicit and confront the different views about what it means to provide a research-based teacher education (see for example Grevholm, 2004a). The second reason is that the discussion on how to address the relationship between theory and practice is still alive in recent international reports (see for example Grevholm & Ball, 2008, p. 268; Even

& Ball, 2009, p. 3). I interpret this as an indication that the community of teacher educators continues to be interested in seeking ways of reducing the gap between research and practice.

Reflective thinking. Under the label of reflective thinking I have grouped all the research that deals with teachers or teacher educators reflecting on and learning from their own practices and experiences. This kind of research has been strongly influenced by the work of Dewey (1933) and Schön (1983), and it has remained in constant development over the ten years covered by this review.

It is clear that there is general agreement in the community of mathematics teacher educators on considering reflection as a key element in the education and development of mathematics teachers (see Lerman, 2001; Llinares & Krainer, 2006; Sowder, 2007; Chapman, 2008; Schoenfeld

& Kilpatrick, 2008). Nevertheless, we can also see that the meanings attributed to the concept of reflection are varied. In the literature one can find a variety of terms such as reflective thinking, reflective stance, critical

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reflection, joint reflection, self-reflection, etc. that refers to different nuances and meanings of the concept of reflection. As Mason & Spence (1999) have stated: “[T]he term reflection has become too broad and diffuse in meaning to carry significance in itself” (p. 153). Due to the key role that the concept of reflection plays in my own research, I have separately analysed some of the meanings attributed to the concept of reflection in the literature. The reader will find this analysis in the next chapter of the dissertation.