Contributed by:

The purpose of this article is to contribute to the dialogue about the notion of mathematics teacher knowledge, and the question of what makes it specialized. In the first part of the article, central orientations in conceptualizing mathematics teacher knowledge are identified. In the second part of the article, alternative views are provided to each of these orientations that direct attention to underexplored issues about what makes mathematics teacher knowledge specialized.

1.
What Makes Mathematics Teacher Knowledge Specialized?

Offering Alternative Views1

Thorsten Scheiner, University of Hamburg, Hamburg, Germany

Miguel A. Montes, Universidad de Huelva, Huelva, Spain

Juan D. Godino, Universidad de Granada, Granada, Spain

José Carrillo, Universidad de Huelva, Huelva, Spain

Luis R. Pino-Fan, Universidad de Los Lagos, Osorno, Chile

Abstract. The purpose of this article is to contribute to the dialogue about the notion of

mathematics teacher knowledge, and the question of what makes it specialized. In the first

part of the article, central orientations in conceptualizing mathematics teacher knowledge are

identified. In the second part of the article, alternative views are provided to each of these

orientations that direct attention to underexplored issues about what makes mathematics

teacher knowledge specialized. Collectively, these alternative views suggest that

specialization cannot be comprehensively accounted by addressing ‘what’ teachers know, but

rather by accounting for ‘how’ teachers’ knowing comes into being. We conclude that it is not

a kind of knowledge but a style of knowing that signifies specialization in mathematics

teacher knowledge.

Keywords: mathematical knowledge for teaching; pedagogical content knowledge;

specialized knowledge; teacher knowledge; teacher professionalism

Mathematics teacher knowledge has become a fertile research field in mathematics education

(see Ponte & Chapman, 2016). Scholars have considered mathematics teacher knowledge

from multiple perspectives, using various constructs and frameworks to describe and explain

what makes mathematics teacher knowledge specialized1. Despite the relatively short time

that research on teacher knowledge has existed as a field, the literature is currently shaped by

a diversity of conceptualizations of mathematics teacher knowledge (Petrou & Goulding,

2011; Rowland, 2014).

As research on teacher knowledge has moved to a more central role in mathematics education

research (see Ball, Lubienski, & Mewborn, 2001; Even & Ball, 2010; Fennema, & Franke,

1992; Sullivan & Wood, 2008), the search for what signifies the specialization in mathematics

teacher knowledge has been becoming an increasingly important enterprise in the research

field. Recent research has addressed this issue by describing and identifying facets or types of

teacher knowledge that have been considered as crucial for teaching mathematics, and in

obtaining empirical evidence to support these (e.g., Ball, Thames, & Phelps, 2008; Baumert et

Scheiner, T., Montes, M. A., Godino, J. D., Carrillo, J. & Pino-Fan, L. (2017). What Makes Mathe-

matics Teacher Knowledge Specialized? Offering Alternative Views. International Journal of Science

and Mathematics Education. The final publication is available at,

1

Offering Alternative Views1

Thorsten Scheiner, University of Hamburg, Hamburg, Germany

Miguel A. Montes, Universidad de Huelva, Huelva, Spain

Juan D. Godino, Universidad de Granada, Granada, Spain

José Carrillo, Universidad de Huelva, Huelva, Spain

Luis R. Pino-Fan, Universidad de Los Lagos, Osorno, Chile

Abstract. The purpose of this article is to contribute to the dialogue about the notion of

mathematics teacher knowledge, and the question of what makes it specialized. In the first

part of the article, central orientations in conceptualizing mathematics teacher knowledge are

identified. In the second part of the article, alternative views are provided to each of these

orientations that direct attention to underexplored issues about what makes mathematics

teacher knowledge specialized. Collectively, these alternative views suggest that

specialization cannot be comprehensively accounted by addressing ‘what’ teachers know, but

rather by accounting for ‘how’ teachers’ knowing comes into being. We conclude that it is not

a kind of knowledge but a style of knowing that signifies specialization in mathematics

teacher knowledge.

Keywords: mathematical knowledge for teaching; pedagogical content knowledge;

specialized knowledge; teacher knowledge; teacher professionalism

Mathematics teacher knowledge has become a fertile research field in mathematics education

(see Ponte & Chapman, 2016). Scholars have considered mathematics teacher knowledge

from multiple perspectives, using various constructs and frameworks to describe and explain

what makes mathematics teacher knowledge specialized1. Despite the relatively short time

that research on teacher knowledge has existed as a field, the literature is currently shaped by

a diversity of conceptualizations of mathematics teacher knowledge (Petrou & Goulding,

2011; Rowland, 2014).

As research on teacher knowledge has moved to a more central role in mathematics education

research (see Ball, Lubienski, & Mewborn, 2001; Even & Ball, 2010; Fennema, & Franke,

1992; Sullivan & Wood, 2008), the search for what signifies the specialization in mathematics

teacher knowledge has been becoming an increasingly important enterprise in the research

field. Recent research has addressed this issue by describing and identifying facets or types of

teacher knowledge that have been considered as crucial for teaching mathematics, and in

obtaining empirical evidence to support these (e.g., Ball, Thames, & Phelps, 2008; Baumert et

Scheiner, T., Montes, M. A., Godino, J. D., Carrillo, J. & Pino-Fan, L. (2017). What Makes Mathe-

matics Teacher Knowledge Specialized? Offering Alternative Views. International Journal of Science

and Mathematics Education. The final publication is available at,

1

2.
al., 2010; Blömeke, Hsieh, Kaiser, & Schmidt, 2014). As such, the focus tends to be on

(seemingly distinct) facets of knowledge that an individual teacher possesses (knowledge for

teaching) or uses in the classroom (knowledge in teaching). A number of scholars have

pointed to inadequacies in such conceptualizations of teacher knowledge, arguing that they

disregard the deep embeddedness of knowledge in professional activity (Hodgen, 2011) and

ignore the dynamic interactions between different kinds or facets of teacher knowledge

(Hashweh, 2005). Others have argued that the premises on which much research into teacher

knowledge is based depend on assumptions that are rather aligned with transmission views of

teaching (McEwan & Bull, 1991) and, in consequence, are rather asymmetrical to

constructivist viewpoints (Cochran, DeRuiter, & King, 1993). Thus, it is not surprising that

scholars have called for making the assumptions underlying frameworks of teacher

knowledge, teaching, and teacher learning explicit (Lerman, 2017) and for achieving

coherence between research into teacher characteristics and teacher practice (Van Zoest &

Thames, 2013).

This paper aims to make explicit the discussion of what makes mathematics teacher

knowledge specialized, a question that has often been addressed implicitly by several scholars

in various ways and with different emphases. The paper outlines further attempts that reflect

theoretically on this important issue and try to articulate more explicitly what it is, or may be,

that signifies the specialization in mathematics teacher knowledge. The purpose of this paper

is, therefore, twofold: First, we try to elucidate central orientations currently available in the

literature and point to the more serious limitations of the grounds on which they stand.

Second, we provide alternative views that direct attention to underexplored issues about these

We begin this article by briefly discussing previous accounts on what mathematics teacher

knowledge signifies and encompasses, and then take this retrospection as a point of departure

for outlining the limitations of these accounts. Afterwards, we articulate and draw a contrast

with alternative viewpoints that provide a critical stance towards previous accounts but also

provide new ways to think about the issues under consideration. The first perspective

underlines the complex dynamics of the usage and function of mathematics teacher

knowledge in context that calls for specialization as a process of becoming rather than a state

of being. The second perspective points to the epistemological stance inherent in mathematics

teacher knowledge, arguing for the sensitivity for the historical and cognitive geneses of

mathematical insights. The third perspective accentuates the complex interactions of

knowledge facets that generate dynamic structures. Then, we highlight underlying themes and

convergences of these alternative views with regard to specialization in mathematics teacher

knowledge. Finally, we conclude by proposing to construe specialization in mathematics

teacher knowledge as a style of knowing rather than a kind of knowledge.

On the Evolution of Thinking within the Field Regarding Conceptualizing Mathematics

Teacher Knowledge

Research into mathematics teacher knowledge has evolved considerably, especially over the

last three decades. The number of studies in this field has significantly increased, the nature

and scope of the research have expanded, and the frameworks used to guide the study of

mathematics teacher knowledge have become quite diverse. The growing diversity of

frameworks for teacher knowledge testifies to the complexity and multidimensionality of the

research field.

(seemingly distinct) facets of knowledge that an individual teacher possesses (knowledge for

teaching) or uses in the classroom (knowledge in teaching). A number of scholars have

pointed to inadequacies in such conceptualizations of teacher knowledge, arguing that they

disregard the deep embeddedness of knowledge in professional activity (Hodgen, 2011) and

ignore the dynamic interactions between different kinds or facets of teacher knowledge

(Hashweh, 2005). Others have argued that the premises on which much research into teacher

knowledge is based depend on assumptions that are rather aligned with transmission views of

teaching (McEwan & Bull, 1991) and, in consequence, are rather asymmetrical to

constructivist viewpoints (Cochran, DeRuiter, & King, 1993). Thus, it is not surprising that

scholars have called for making the assumptions underlying frameworks of teacher

knowledge, teaching, and teacher learning explicit (Lerman, 2017) and for achieving

coherence between research into teacher characteristics and teacher practice (Van Zoest &

Thames, 2013).

This paper aims to make explicit the discussion of what makes mathematics teacher

knowledge specialized, a question that has often been addressed implicitly by several scholars

in various ways and with different emphases. The paper outlines further attempts that reflect

theoretically on this important issue and try to articulate more explicitly what it is, or may be,

that signifies the specialization in mathematics teacher knowledge. The purpose of this paper

is, therefore, twofold: First, we try to elucidate central orientations currently available in the

literature and point to the more serious limitations of the grounds on which they stand.

Second, we provide alternative views that direct attention to underexplored issues about these

We begin this article by briefly discussing previous accounts on what mathematics teacher

knowledge signifies and encompasses, and then take this retrospection as a point of departure

for outlining the limitations of these accounts. Afterwards, we articulate and draw a contrast

with alternative viewpoints that provide a critical stance towards previous accounts but also

provide new ways to think about the issues under consideration. The first perspective

underlines the complex dynamics of the usage and function of mathematics teacher

knowledge in context that calls for specialization as a process of becoming rather than a state

of being. The second perspective points to the epistemological stance inherent in mathematics

teacher knowledge, arguing for the sensitivity for the historical and cognitive geneses of

mathematical insights. The third perspective accentuates the complex interactions of

knowledge facets that generate dynamic structures. Then, we highlight underlying themes and

convergences of these alternative views with regard to specialization in mathematics teacher

knowledge. Finally, we conclude by proposing to construe specialization in mathematics

teacher knowledge as a style of knowing rather than a kind of knowledge.

On the Evolution of Thinking within the Field Regarding Conceptualizing Mathematics

Teacher Knowledge

Research into mathematics teacher knowledge has evolved considerably, especially over the

last three decades. The number of studies in this field has significantly increased, the nature

and scope of the research have expanded, and the frameworks used to guide the study of

mathematics teacher knowledge have become quite diverse. The growing diversity of

frameworks for teacher knowledge testifies to the complexity and multidimensionality of the

research field.

3.
To identify current views in the literature concerning what makes mathematics teacher

knowledge specialized, we try to briefly sketch the evolution of thinking within the field in

conceptualizing mathematics teacher knowledge. We acknowledge that a great deal of

important detail is lost in the brief sketch of this development. More detailed accounts of this

research can be found elsewhere (see e.g., Kaiser et al., 2017; Kunter et al. 2013; Rowland &

Ruthven, 2011; Schoenfeld & Kilpatrick, 2008). A recent discussion of several research

traditions is provided by Blömeke and Kaiser (2017), in which the same authors arrive at a

complex framework of teacher competence and conceptualize the development of teacher

competence as personally, situationally, and socially determined, as well as embedded in a

professional context.

Our purpose here, however, is to foreground how the field in general has induced particular

attitudes towards what mathematics teacher knowledge signifies. We start by portraying

different dimensions of mathematical knowledge discussed in the literature as being essential

for mathematics teachers. Then, we draw attention to selected contributions that articulate

what particularizes subject matter knowledge for teaching, particularly in reference to

mathematical knowledge for teaching, with an emphasis on the way specialization is

considered. Afterwards, we focus on what is considered as the heart of teaching: the

transformation of subject matter in ways accessible to students, an assumption that underlies

several attempts in conceptualizing mathematics teacher knowledge.

Mathematical Knowledge

The literature foregrounds different aspects of mathematical knowledge as important for

teachers. Shulman (1986), for instance, accredited ―the amount and organization of the

knowledge per se in the mind of the teacher‖ (p. 9), referring to Schwab‘s (1978) distinction

between substantive and syntactic structures of a discipline. Substantive structures are the key

concepts, principles, theories, and explanatory frameworks that guide inquiry in a discipline,

while syntactic structures provide the procedures and mechanisms for the acquisition of

knowledge, and include the canons of evidence and proof. Bromme (1994), then again,

acknowledged that ―school subjects have a ‗life of their own‘ with their own logic; that is, the

meaning of the concepts taught cannot be explained simply by the logic of the respective

scientific disciplines‖ (p. 74). In recognizing school mathematics as a special kind of

mathematics, Bromme (ibid.) suggested school mathematical knowledge and academic

content knowledge as further dimensions of mathematical knowledge. Buchholtz et al. (2013)

set forth a kind of knowledge ―that comprises school mathematics, but goes beyond it and

relates it to the underlying advanced academic mathematics‖ (p. 108). The same authors called

this kind of knowledge, in homage to the pioneering work of Felix Klein, knowledge of

elementary mathematics from an advanced standpoint.

This small selection of a fuller corpus of dimensions of mathematical knowledge already

indicates a critical point to be expanded here: the contributions to dimensions of mathematical

knowledge that teachers know, or should know, is accumulative (or incremental). However, as

Monk (1994) reminds us, ―a good grasp of one‘s subject areas is a necessary but not sufficient

condition for effective teaching‖ (p. 142). We might interpret Monk‘s statement as a call for

additional knowledge, but we might also understand it as a call for a qualitatively different

kind of knowledge.

Subject Matter Knowledge for Teaching (Pedagogical Content Knowledge)

knowledge specialized, we try to briefly sketch the evolution of thinking within the field in

conceptualizing mathematics teacher knowledge. We acknowledge that a great deal of

important detail is lost in the brief sketch of this development. More detailed accounts of this

research can be found elsewhere (see e.g., Kaiser et al., 2017; Kunter et al. 2013; Rowland &

Ruthven, 2011; Schoenfeld & Kilpatrick, 2008). A recent discussion of several research

traditions is provided by Blömeke and Kaiser (2017), in which the same authors arrive at a

complex framework of teacher competence and conceptualize the development of teacher

competence as personally, situationally, and socially determined, as well as embedded in a

professional context.

Our purpose here, however, is to foreground how the field in general has induced particular

attitudes towards what mathematics teacher knowledge signifies. We start by portraying

different dimensions of mathematical knowledge discussed in the literature as being essential

for mathematics teachers. Then, we draw attention to selected contributions that articulate

what particularizes subject matter knowledge for teaching, particularly in reference to

mathematical knowledge for teaching, with an emphasis on the way specialization is

considered. Afterwards, we focus on what is considered as the heart of teaching: the

transformation of subject matter in ways accessible to students, an assumption that underlies

several attempts in conceptualizing mathematics teacher knowledge.

Mathematical Knowledge

The literature foregrounds different aspects of mathematical knowledge as important for

teachers. Shulman (1986), for instance, accredited ―the amount and organization of the

knowledge per se in the mind of the teacher‖ (p. 9), referring to Schwab‘s (1978) distinction

between substantive and syntactic structures of a discipline. Substantive structures are the key

concepts, principles, theories, and explanatory frameworks that guide inquiry in a discipline,

while syntactic structures provide the procedures and mechanisms for the acquisition of

knowledge, and include the canons of evidence and proof. Bromme (1994), then again,

acknowledged that ―school subjects have a ‗life of their own‘ with their own logic; that is, the

meaning of the concepts taught cannot be explained simply by the logic of the respective

scientific disciplines‖ (p. 74). In recognizing school mathematics as a special kind of

mathematics, Bromme (ibid.) suggested school mathematical knowledge and academic

content knowledge as further dimensions of mathematical knowledge. Buchholtz et al. (2013)

set forth a kind of knowledge ―that comprises school mathematics, but goes beyond it and

relates it to the underlying advanced academic mathematics‖ (p. 108). The same authors called

this kind of knowledge, in homage to the pioneering work of Felix Klein, knowledge of

elementary mathematics from an advanced standpoint.

This small selection of a fuller corpus of dimensions of mathematical knowledge already

indicates a critical point to be expanded here: the contributions to dimensions of mathematical

knowledge that teachers know, or should know, is accumulative (or incremental). However, as

Monk (1994) reminds us, ―a good grasp of one‘s subject areas is a necessary but not sufficient

condition for effective teaching‖ (p. 142). We might interpret Monk‘s statement as a call for

additional knowledge, but we might also understand it as a call for a qualitatively different

kind of knowledge.

Subject Matter Knowledge for Teaching (Pedagogical Content Knowledge)

4.
A critical advance in the field was the recognition that teaching entails a specialized kind of

subject matter that is distinct from disciplinary subject matter. Shulman (1986) proposed a

kind of knowledge ―which goes beyond knowledge of subject matter per se to the dimension

of subject matter knowledge for teaching‖ (p. 9, italics in original) that he labeled

pedagogical content knowledge (PCK). Shulman (1986) described PCK as encompassing

―for the most regularly taught topics in one‘s subject area, the most useful forms of [external]

representation of those ideas, the most powerful analogies, illustrations, examples,

explanations, and demonstrations – in a word, the ways of representing and formulating the

subject that make it comprehensible to others […] [and] an understanding of what makes the

learning of specific topics easy or difficult: the conceptions and preconceptions that students

of different ages and backgrounds bring with them to the learning of those most frequently

taught topics and lessons.‖ (p. 9)

In this view, PCK consists of two dimensions: ‗knowledge of representations of subject

matter‘ and ‗knowledge of specific learning difficulties and students‘ conceptions‘. These two

dimensions often served as reference points in thinking about PCK, as Ball (1988), for

instance, assumed ―[…] ‗forms of representation‘ […] to be the crucial substance of

pedagogical content knowledge‖ (p. 166). She then explored the more dynamic aspects of this

idea, examining pre-service teachers‘ pedagogical reasoning in mathematics as the process

whereby they build their knowledge of mathematics teaching and learning. Other scholars in

mathematics education have delineated dimensions of PCK that extended or refined

Shulman‘s original considerations. For instance, Marks (1990) clarified PCK in the context of

mathematics by identifying four dimensions, including knowledge of students‘ understanding,

knowledge of subject matter for instructional purposes, knowledge of media for instruction,

and knowledge of instructional processes.

Shulman (1987) asserted that among multiple knowledge domains for teaching (e.g., content

knowledge, general pedagogical knowledge, curriculum knowledge, knowledge of learners,

etc.), it is PCK that is ―the category most likely to distinguish the understanding of the content

specialist from that of the pedagogue‖ (p. 8). As such, the existence of PCK relies on and

projects the belief in a distinction between the subject matter knowledge of teachers and that

of other subject specialists or scholars (e.g., mathematicians). While the notion of PCK

advocated a position distinguishing teachers‘ and academics‘ subject matter knowledge, the

concept of mathematical knowledge for teaching advocated a position distinguishing

knowledge for teaching mathematics from knowledge for teaching other subjects (such as

physics, biology, or the arts).

Mathematical Knowledge for Teaching

The notion of mathematical knowledge for teaching has become an important point of

departure in thinking about what signifies the specialization in mathematics teacher

knowledge. Various researchers have applied different emphases to this notion, as shall be

seen below. In this realm, it is particularly the Mathematical Knowledge for Teaching (MKT)

framework (e.g., Ball & Bass, 2000; Ball et al., 2008), that has attracted significant research

attention. The MKT framework evolved through the application of a kind of job analysis (Ball

et al., 2008) focusing on the use of knowledge in and for the work of teaching.

The MKT framework defines several sub-domains within two of Shulman‘s (1987) original

knowledge domains: pedagogical content knowledge (PCK) and subject matter knowledge

subject matter that is distinct from disciplinary subject matter. Shulman (1986) proposed a

kind of knowledge ―which goes beyond knowledge of subject matter per se to the dimension

of subject matter knowledge for teaching‖ (p. 9, italics in original) that he labeled

pedagogical content knowledge (PCK). Shulman (1986) described PCK as encompassing

―for the most regularly taught topics in one‘s subject area, the most useful forms of [external]

representation of those ideas, the most powerful analogies, illustrations, examples,

explanations, and demonstrations – in a word, the ways of representing and formulating the

subject that make it comprehensible to others […] [and] an understanding of what makes the

learning of specific topics easy or difficult: the conceptions and preconceptions that students

of different ages and backgrounds bring with them to the learning of those most frequently

taught topics and lessons.‖ (p. 9)

In this view, PCK consists of two dimensions: ‗knowledge of representations of subject

matter‘ and ‗knowledge of specific learning difficulties and students‘ conceptions‘. These two

dimensions often served as reference points in thinking about PCK, as Ball (1988), for

instance, assumed ―[…] ‗forms of representation‘ […] to be the crucial substance of

pedagogical content knowledge‖ (p. 166). She then explored the more dynamic aspects of this

idea, examining pre-service teachers‘ pedagogical reasoning in mathematics as the process

whereby they build their knowledge of mathematics teaching and learning. Other scholars in

mathematics education have delineated dimensions of PCK that extended or refined

Shulman‘s original considerations. For instance, Marks (1990) clarified PCK in the context of

mathematics by identifying four dimensions, including knowledge of students‘ understanding,

knowledge of subject matter for instructional purposes, knowledge of media for instruction,

and knowledge of instructional processes.

Shulman (1987) asserted that among multiple knowledge domains for teaching (e.g., content

knowledge, general pedagogical knowledge, curriculum knowledge, knowledge of learners,

etc.), it is PCK that is ―the category most likely to distinguish the understanding of the content

specialist from that of the pedagogue‖ (p. 8). As such, the existence of PCK relies on and

projects the belief in a distinction between the subject matter knowledge of teachers and that

of other subject specialists or scholars (e.g., mathematicians). While the notion of PCK

advocated a position distinguishing teachers‘ and academics‘ subject matter knowledge, the

concept of mathematical knowledge for teaching advocated a position distinguishing

knowledge for teaching mathematics from knowledge for teaching other subjects (such as

physics, biology, or the arts).

Mathematical Knowledge for Teaching

The notion of mathematical knowledge for teaching has become an important point of

departure in thinking about what signifies the specialization in mathematics teacher

knowledge. Various researchers have applied different emphases to this notion, as shall be

seen below. In this realm, it is particularly the Mathematical Knowledge for Teaching (MKT)

framework (e.g., Ball & Bass, 2000; Ball et al., 2008), that has attracted significant research

attention. The MKT framework evolved through the application of a kind of job analysis (Ball

et al., 2008) focusing on the use of knowledge in and for the work of teaching.

The MKT framework defines several sub-domains within two of Shulman‘s (1987) original

knowledge domains: pedagogical content knowledge (PCK) and subject matter knowledge

5.
(SMK). PCK is divided into knowledge of content and students, knowledge of content and

teaching, and knowledge of curriculum, whilst SMK is divided into common content

knowledge, specialized content knowledge, and knowledge at the mathematical horizon. We

briefly outline four of the six dimensions, excluding horizon content knowledge and

knowledge of curriculum as they have so far not been the primary focus of studies into the

Within PCK, knowledge of content and teaching combines knowing about teaching and

knowing about mathematics, including knowledge of the design of instruction, such as the

knowledge governing the choice of examples to introduce a content item and those used to

take students deeper into it. Knowledge of content and students is the knowledge that

combines knowing about mathematics and knowing about students. It includes knowledge of

common student conceptions and misconceptions about particular mathematical content as

well as the interpretation of students‘ emerging and incomplete thinking.

Within the mathematical knowledge domain, common content knowledge refers to the

mathematical knowledge and skill possessed by any well-educated adult, and certainly by all

mathematicians, which is used in settings other than teaching. Specialized content knowledge,

on the other hand, is defined as mathematical knowledge tailored to the specialized uses that

come up in the work of teaching. It is described as being used by teachers in their work, but

not held by well-educated adults, and is not typically needed for purposes other than teaching.

Ball et al. (2008) noted that teaching may require ―a specialized form of pure subject matter

knowledge‖ (p. 396, italics added):

―pure because it is not mixed with knowledge of students or pedagogy and is thus distinct

from the pedagogical content knowledge identified by Shulman and his colleagues and

specialized because it is not needed or used in settings other than mathematics teaching.‖ (Ball

et al., 2008, p. 396, italics added)

Transforming Subject Matter

These approaches support the assertion that a kind of subject matter knowledge exists that is

qualitatively different from the subject matter knowledge of disciplinary scholars or teachers

of other subjects. The nature of such knowledge, however, is not just a matter of mastering

disciplinary subject matter. From the perspectives presented so far, teachers‘ primary concern

is not with mathematics, but with teaching mathematics. The difference between disciplinary

scholars and educators is, therefore, also seen in the different uses of their knowledge. This

important recognition of the different purposes of disciplinary scholars and teachers

highlights, as Shulman (1987) argued, a unique aspect of teachers‘ professional work: a

teacher must ―transform the content knowledge he or she possesses into forms that are

pedagogically powerful and yet adaptive to the variations in ability and background presented

by the students‖ (Shulman, 1987, p. 15). It is this notion of transforming the subject matter of

an (academic) discipline that highly impacted our thinking about teacher knowledge, but it

seems to have been taken for granted once the picture of knowledge for teaching was defined.

The primary purpose of transformation is to organize, structure, and represent the subject

matter of an (academic) discipline in a form ―that is appropriate for students and peculiar to

the task of teaching‖ (Grossman, Wilson, & Shulman, 1989, p. 32).

The literature on mathematical knowledge for teaching also identifies various discipline-

specific practices of transformation, often described in terms of exemplifying, explaining,

teaching, and knowledge of curriculum, whilst SMK is divided into common content

knowledge, specialized content knowledge, and knowledge at the mathematical horizon. We

briefly outline four of the six dimensions, excluding horizon content knowledge and

knowledge of curriculum as they have so far not been the primary focus of studies into the

Within PCK, knowledge of content and teaching combines knowing about teaching and

knowing about mathematics, including knowledge of the design of instruction, such as the

knowledge governing the choice of examples to introduce a content item and those used to

take students deeper into it. Knowledge of content and students is the knowledge that

combines knowing about mathematics and knowing about students. It includes knowledge of

common student conceptions and misconceptions about particular mathematical content as

well as the interpretation of students‘ emerging and incomplete thinking.

Within the mathematical knowledge domain, common content knowledge refers to the

mathematical knowledge and skill possessed by any well-educated adult, and certainly by all

mathematicians, which is used in settings other than teaching. Specialized content knowledge,

on the other hand, is defined as mathematical knowledge tailored to the specialized uses that

come up in the work of teaching. It is described as being used by teachers in their work, but

not held by well-educated adults, and is not typically needed for purposes other than teaching.

Ball et al. (2008) noted that teaching may require ―a specialized form of pure subject matter

knowledge‖ (p. 396, italics added):

―pure because it is not mixed with knowledge of students or pedagogy and is thus distinct

from the pedagogical content knowledge identified by Shulman and his colleagues and

specialized because it is not needed or used in settings other than mathematics teaching.‖ (Ball

et al., 2008, p. 396, italics added)

Transforming Subject Matter

These approaches support the assertion that a kind of subject matter knowledge exists that is

qualitatively different from the subject matter knowledge of disciplinary scholars or teachers

of other subjects. The nature of such knowledge, however, is not just a matter of mastering

disciplinary subject matter. From the perspectives presented so far, teachers‘ primary concern

is not with mathematics, but with teaching mathematics. The difference between disciplinary

scholars and educators is, therefore, also seen in the different uses of their knowledge. This

important recognition of the different purposes of disciplinary scholars and teachers

highlights, as Shulman (1987) argued, a unique aspect of teachers‘ professional work: a

teacher must ―transform the content knowledge he or she possesses into forms that are

pedagogically powerful and yet adaptive to the variations in ability and background presented

by the students‖ (Shulman, 1987, p. 15). It is this notion of transforming the subject matter of

an (academic) discipline that highly impacted our thinking about teacher knowledge, but it

seems to have been taken for granted once the picture of knowledge for teaching was defined.

The primary purpose of transformation is to organize, structure, and represent the subject

matter of an (academic) discipline in a form ―that is appropriate for students and peculiar to

the task of teaching‖ (Grossman, Wilson, & Shulman, 1989, p. 32).

The literature on mathematical knowledge for teaching also identifies various discipline-

specific practices of transformation, often described in terms of exemplifying, explaining,

6.
decompressing, or simplifying, that converge on teachers‘ core practice of unpacking

mathematics content in ways that are accessible to students (Adler & Davis, 2006; Ball &

Bass, 2000; Ma, 1999). It requires the capacity ―to deconstruct one‘s own mathematical

knowledge into a less polished and final form, where elemental components are accessible and

visible‖ (Ball & Bass, 2000, p. 98). Hodgen (2011), for instance, takes this idea further

arguing that the ―essence of teacher knowledge involves an explicit recognition of this –

‗unpacking‘ the mathematical ideas […], [whereas] doing mathematics only requires an

implicit recognition of this.‖ (pp. 34-35, italics in original).

More recently, the idea of transformation has also been further elaborated by scholars working

in the Knowledge Quartet research program (Rowland, 2009, Rowland, Huckstep, &

Thwaites, 2005), as part of their conceptualization of the classification of situations in which

mathematical knowledge surfaces in teaching. The research group considers transformation as

concerning ―knowledge in action as demonstrated both in planning to teach and in the act of

teaching itself. A central focus is the representation of ideas to learners in the form of

analogies, examples, explanations, and demonstrations‖ (Rowland, 2009, p. 237). This

conceptualization concerns knowledge in action, focusing on teaching activity in the

transmission of content.

Thinking About What Makes Mathematics Teacher Knowledge Specialized: Various

Orientations, Different Responses

As innocent and straightforward as the question What makes mathematics teacher knowledge

specialized? sounds, the research field has found it difficult to provide an explicit answer as

there are various orientations towards teacher knowledge, each with a quite different response

to the question. The previous section briefly outlined the following orientations regarding

what mathematics teacher knowledge signifies: (1) identifying and describing multiple

dimensions of mathematical knowledge (and pedagogical content knowledge), (2) declaring

kinds of subject matter knowledge for teaching that are distinct from subject matter

knowledge per se, and (3) asserting teachers‘ action upon subject matter (that is the

transmission of subject matter in ways accessible to students) as the core task of teaching.

These three orientations seem to indicate different lines of thinking about what makes

mathematics teacher knowledge specialized. Each focuses attention on particular aspects: the

first considers additional knowledge dimensions (quantity), whereas the second turns the

attention towards knowledge that is construed as qualitatively different. These different lines

of thinking seem to be convolved in Shulman‘s idea of transforming subject matter, that is,

the various orientations shape, and are shaped by, our interpretations of Shulman‘s idea of

transforming subject matter.

One might interpret Shulman‘s (1986, 1987) initial writings on teacher knowledge as

indicating a stance in which teachers‘ and disciplinary scholars‘ subject matter knowledge

were differentiated, signifying the existence of a kind of subject matter knowledge for

teaching (held by teachers) that is qualitatively different from subject matter knowledge per se

(held by disciplinary scholars). On the other hand, Ball and her colleagues proposed a more

nuanced differentiation in which subject matter content itself is considered in a way that only

makes sense to mathematics teachers. In other words, while both notions of PCK and

specialized content knowledge indicate the existence of a qualitatively different kind of

knowledge, they differ in where to put emphasis: Shulman‘s notion of PCK puts emphasis on

a kind of knowledge distinctive to teachers (and not to disciplinary scholars) and Ball and her

mathematics content in ways that are accessible to students (Adler & Davis, 2006; Ball &

Bass, 2000; Ma, 1999). It requires the capacity ―to deconstruct one‘s own mathematical

knowledge into a less polished and final form, where elemental components are accessible and

visible‖ (Ball & Bass, 2000, p. 98). Hodgen (2011), for instance, takes this idea further

arguing that the ―essence of teacher knowledge involves an explicit recognition of this –

‗unpacking‘ the mathematical ideas […], [whereas] doing mathematics only requires an

implicit recognition of this.‖ (pp. 34-35, italics in original).

More recently, the idea of transformation has also been further elaborated by scholars working

in the Knowledge Quartet research program (Rowland, 2009, Rowland, Huckstep, &

Thwaites, 2005), as part of their conceptualization of the classification of situations in which

mathematical knowledge surfaces in teaching. The research group considers transformation as

concerning ―knowledge in action as demonstrated both in planning to teach and in the act of

teaching itself. A central focus is the representation of ideas to learners in the form of

analogies, examples, explanations, and demonstrations‖ (Rowland, 2009, p. 237). This

conceptualization concerns knowledge in action, focusing on teaching activity in the

transmission of content.

Thinking About What Makes Mathematics Teacher Knowledge Specialized: Various

Orientations, Different Responses

As innocent and straightforward as the question What makes mathematics teacher knowledge

specialized? sounds, the research field has found it difficult to provide an explicit answer as

there are various orientations towards teacher knowledge, each with a quite different response

to the question. The previous section briefly outlined the following orientations regarding

what mathematics teacher knowledge signifies: (1) identifying and describing multiple

dimensions of mathematical knowledge (and pedagogical content knowledge), (2) declaring

kinds of subject matter knowledge for teaching that are distinct from subject matter

knowledge per se, and (3) asserting teachers‘ action upon subject matter (that is the

transmission of subject matter in ways accessible to students) as the core task of teaching.

These three orientations seem to indicate different lines of thinking about what makes

mathematics teacher knowledge specialized. Each focuses attention on particular aspects: the

first considers additional knowledge dimensions (quantity), whereas the second turns the

attention towards knowledge that is construed as qualitatively different. These different lines

of thinking seem to be convolved in Shulman‘s idea of transforming subject matter, that is,

the various orientations shape, and are shaped by, our interpretations of Shulman‘s idea of

transforming subject matter.

One might interpret Shulman‘s (1986, 1987) initial writings on teacher knowledge as

indicating a stance in which teachers‘ and disciplinary scholars‘ subject matter knowledge

were differentiated, signifying the existence of a kind of subject matter knowledge for

teaching (held by teachers) that is qualitatively different from subject matter knowledge per se

(held by disciplinary scholars). On the other hand, Ball and her colleagues proposed a more

nuanced differentiation in which subject matter content itself is considered in a way that only

makes sense to mathematics teachers. In other words, while both notions of PCK and

specialized content knowledge indicate the existence of a qualitatively different kind of

knowledge, they differ in where to put emphasis: Shulman‘s notion of PCK puts emphasis on

a kind of knowledge distinctive to teachers (and not to disciplinary scholars) and Ball and her

7.
colleagues‘ notion of specialized content knowledge puts emphasis on a kind of knowledge

distinctive to mathematics teachers (and not to teachers of other subjects).

Each of these orientations provides a (partial) response to the question of what signifies

mathematics teacher knowledge. The first orientation calls for the multidimensionality of

mathematical knowledge in particular, and teacher knowledge in general. The second

orientation argues for the qualitative differences between scholars‘ subject matter knowledge

(per se) and teachers‘ subject matter knowledge (for teaching) or the qualitative differences

between knowledge for teaching mathematics and knowledge for teaching other subjects. The

third orientation, underlying and extending the previous one, points to teachers‘ actions upon

subject matter, as manifested in notions such as transforming, unpacking, deconstructing, and

decompressing subject matter.

Correspondingly, we can frame the responses of the various orientations concerning what

makes mathematics teacher knowledge specialized as follows:

mathematics teachers need to know more than the subject matter they teach (additional

knowledge);

mathematics teachers need to know subject matter in a qualitatively different way than

other practitioners of mathematics (mathematicians, physicists, engineers, among

others), and they need to hold a qualitatively different kind of knowledge than teachers

of other subjects (physics teachers, biology teachers, history teachers, among others)

(qualitatively different knowledge); and

mathematics teachers need to know how to organize or structure the subject matter in

ways accessible to students (teaching-oriented action).

These responses, taken together, seem to converge on an understanding that what

mathematics teacher knowledge signifies depends on its distinctiveness or exclusiveness:

mathematics teacher knowledge is construed as knowledge that is needed only for teaching

mathematics, that is, knowledge not required for other purposes than teaching and not needed

for teaching other subjects than mathematics.

Too often when we frame our thinking about what mathematics teacher knowledge signifies

we see ourselves getting caught in the mire of current debates without taking a critical stance

toward the grounds on which they stand. In the present paper, it is intended to take a more

critical stance toward the current state of what the literature implicitly represents as making

mathematics teacher knowledge specialized. To this end, we explicitly identify the more

significant boundaries demarking the outlined orientations, and provide new ways of thinking

about the issue under consideration. Our critique rests on at least three general tendencies that

seem to have been implicit in the current discussion on teacher knowledge:

the field brings up external references in justifying what makes teacher knowledge

specialized (mathematics teachers vs. mathematicians; teaching mathematics vs. teaching

other subjects);

in its consideration of teacher knowledge, the field takes a disciplinary perspective which

is structuralist2 in orientation, arguing from the viewpoint of teaching mathematics; and

distinctive to mathematics teachers (and not to teachers of other subjects).

Each of these orientations provides a (partial) response to the question of what signifies

mathematics teacher knowledge. The first orientation calls for the multidimensionality of

mathematical knowledge in particular, and teacher knowledge in general. The second

orientation argues for the qualitative differences between scholars‘ subject matter knowledge

(per se) and teachers‘ subject matter knowledge (for teaching) or the qualitative differences

between knowledge for teaching mathematics and knowledge for teaching other subjects. The

third orientation, underlying and extending the previous one, points to teachers‘ actions upon

subject matter, as manifested in notions such as transforming, unpacking, deconstructing, and

decompressing subject matter.

Correspondingly, we can frame the responses of the various orientations concerning what

makes mathematics teacher knowledge specialized as follows:

mathematics teachers need to know more than the subject matter they teach (additional

knowledge);

mathematics teachers need to know subject matter in a qualitatively different way than

other practitioners of mathematics (mathematicians, physicists, engineers, among

others), and they need to hold a qualitatively different kind of knowledge than teachers

of other subjects (physics teachers, biology teachers, history teachers, among others)

(qualitatively different knowledge); and

mathematics teachers need to know how to organize or structure the subject matter in

ways accessible to students (teaching-oriented action).

These responses, taken together, seem to converge on an understanding that what

mathematics teacher knowledge signifies depends on its distinctiveness or exclusiveness:

mathematics teacher knowledge is construed as knowledge that is needed only for teaching

mathematics, that is, knowledge not required for other purposes than teaching and not needed

for teaching other subjects than mathematics.

Too often when we frame our thinking about what mathematics teacher knowledge signifies

we see ourselves getting caught in the mire of current debates without taking a critical stance

toward the grounds on which they stand. In the present paper, it is intended to take a more

critical stance toward the current state of what the literature implicitly represents as making

mathematics teacher knowledge specialized. To this end, we explicitly identify the more

significant boundaries demarking the outlined orientations, and provide new ways of thinking

about the issue under consideration. Our critique rests on at least three general tendencies that

seem to have been implicit in the current discussion on teacher knowledge:

the field brings up external references in justifying what makes teacher knowledge

specialized (mathematics teachers vs. mathematicians; teaching mathematics vs. teaching

other subjects);

in its consideration of teacher knowledge, the field takes a disciplinary perspective which

is structuralist2 in orientation, arguing from the viewpoint of teaching mathematics; and

8.
the field has been partly additive, that is, accumulating dimensions of teacher knowledge.

In the following sections, we adopt a critical stance to these general tendencies, around

which we organize our understanding of the question of what makes knowledge for

teaching mathematics specialized. As such we argue for an approach which is:

intrinsic: it dispenses with external reference points, and accounts for specialization as a

process of becoming rather than a state of being;

anthropological-sociocultural: it eschews a restrictive structuralist approach, and instead

underlines the epistemological thread inherent in mathematics teacher knowledge; and

transformative: rather than seeing teacher knowledge as an incremental accumulation of

facets, it accentuates the complex interactions of knowledge within a dynamic structure.

In doing so, we draw on and debate different emerging perspectives that provide critical issues

that are un- or under-addressed in the current literature, and, more importantly, that provide

provocative new avenues for thinking about what makes mathematics teacher knowledge

specialized in ways not yet explicitly articulated.

From an Extrinsic to an Intrinsic Approach

In this section, we adopt a critical stance to a tendency that seems to be common amongst

scholars discussing mathematics teacher knowledge: the tendency of comparing mathematics

teacher knowledge with the knowledge demanded of other professionals (such as

mathematicians, teachers of subjects other than mathematics, etc.). Such an approach is

extrinsically oriented (see Flores, Escudero & Carrillo, 2013) as it takes an external referent

(e.g., mathematicians or teachers of other subjects) as a reference point for comparison. The

explicit purpose of such an approach is to identify the distinctiveness of mathematics teacher

knowledge in relation to someone else‘s knowledge.

Since Shulman (1986) acknowledged teachers as professionals, various scholars in

mathematics education have attempted to identify the distinctiveness of knowledge for

teaching mathematics in comparison with other forms of knowledge. This search took place

primarily by looking outside of mathematics education to provide answers as to what

mathematics teacher knowledge signifies. Researchers articulated ways in which mathematics

teacher knowledge differs from mathematicians‘ knowledge, or how it differs from knowledge

of those who teach subjects other than mathematics. This tendency to look beyond the

discipline, we believe, is a very natural one, particularly when, at the same time, scholars were

searching for an identity for the research field. In relating mathematics teachers to

professionals of other disciplines, scholars were able to determine certain cognitive

dispositions that seemed to be specific for mathematics teachers – aspects of teacher

knowledge that have been referred to as being static, explicit, and objective (in the sense of

being observable). However, it is one thing to make comparisons between mathematics

teacher knowledge and the knowledge pertinent to other professionals, and quite another to

interpret the seemingly distinctive features of teacher knowledge in terms of ‗specialization‘.

Whereas ‗specialization‘ seems to have been understood in terms of distinctiveness, in this

paper, we argue for a different meaning of specialization that allows us to focus our attention

inside and not necessarily outside.

In the following sections, we adopt a critical stance to these general tendencies, around

which we organize our understanding of the question of what makes knowledge for

teaching mathematics specialized. As such we argue for an approach which is:

intrinsic: it dispenses with external reference points, and accounts for specialization as a

process of becoming rather than a state of being;

anthropological-sociocultural: it eschews a restrictive structuralist approach, and instead

underlines the epistemological thread inherent in mathematics teacher knowledge; and

transformative: rather than seeing teacher knowledge as an incremental accumulation of

facets, it accentuates the complex interactions of knowledge within a dynamic structure.

In doing so, we draw on and debate different emerging perspectives that provide critical issues

that are un- or under-addressed in the current literature, and, more importantly, that provide

provocative new avenues for thinking about what makes mathematics teacher knowledge

specialized in ways not yet explicitly articulated.

From an Extrinsic to an Intrinsic Approach

In this section, we adopt a critical stance to a tendency that seems to be common amongst

scholars discussing mathematics teacher knowledge: the tendency of comparing mathematics

teacher knowledge with the knowledge demanded of other professionals (such as

mathematicians, teachers of subjects other than mathematics, etc.). Such an approach is

extrinsically oriented (see Flores, Escudero & Carrillo, 2013) as it takes an external referent

(e.g., mathematicians or teachers of other subjects) as a reference point for comparison. The

explicit purpose of such an approach is to identify the distinctiveness of mathematics teacher

knowledge in relation to someone else‘s knowledge.

Since Shulman (1986) acknowledged teachers as professionals, various scholars in

mathematics education have attempted to identify the distinctiveness of knowledge for

teaching mathematics in comparison with other forms of knowledge. This search took place

primarily by looking outside of mathematics education to provide answers as to what

mathematics teacher knowledge signifies. Researchers articulated ways in which mathematics

teacher knowledge differs from mathematicians‘ knowledge, or how it differs from knowledge

of those who teach subjects other than mathematics. This tendency to look beyond the

discipline, we believe, is a very natural one, particularly when, at the same time, scholars were

searching for an identity for the research field. In relating mathematics teachers to

professionals of other disciplines, scholars were able to determine certain cognitive

dispositions that seemed to be specific for mathematics teachers – aspects of teacher

knowledge that have been referred to as being static, explicit, and objective (in the sense of

being observable). However, it is one thing to make comparisons between mathematics

teacher knowledge and the knowledge pertinent to other professionals, and quite another to

interpret the seemingly distinctive features of teacher knowledge in terms of ‗specialization‘.

Whereas ‗specialization‘ seems to have been understood in terms of distinctiveness, in this

paper, we argue for a different meaning of specialization that allows us to focus our attention

inside and not necessarily outside.

9.
Flores et al. (2013), for instance, identified difficulties in defining the specialized nature of

certain cognitive dispositions when analyzing the knowledge involved in assessing students‘

subtraction strategies. They affirmed that it is debatable whether the knowledge used by a

teacher is exclusive to him or her, or is shared with other practitioners of mathematics. They

focus discussion on certain cognitive dispositions and wonder who else, other than a

mathematics teacher might have such kind of knowledge, thus moving the focal point of the

debate from mathematics teacher knowledge to that of other professionals.

The answers we might gain from such comparisons (mathematics teachers vs.

mathematicians, mathematics teachers vs. teachers of other subjects, etc.) are external to

mathematics education as a discipline, in that they offer justifications that are recognizable

and measurable but neither cognitive (concerning the processes involved in knowledge) nor

epistemological (regarding the nature of knowledge). External referents (such as

mathematicians) might provide useful markers for identifying static traits that differ from

mathematics teachers such as the content of teacher knowledge, that is, what teachers‘

knowledge is about. However, they seem to be inappropriate in accounts of the complex

dynamics of knowledge in use. Rather than framing the discussion of what makes

mathematics teacher knowledge specialized in terms of external referents, we suggest an

account of specialization understood in relation to mathematics teacher knowledge in action.

That is to say, what makes mathematics teacher knowledge specialized is not so much ―what‖

mathematics teachers know (which might indeed differ from other professionals), but ―how‖

mathematics teachers know. This involves a shift away from the content of mathematics

teacher knowledge to its usage and function, that is, how teacher knowledge comes into action

(how it comes into being or how it actualizes). This shift in perspective foregrounds the

context rather than the content.

Instead of an extrinsic perspective, we suggest taking an intrinsic view, that is,

acknowledging the situatedness of mathematics teacher knowledge within the context of

mathematics learning and teaching. Interestingly, Carrillo, Climent, Contreras and Muñoz-

Catalán (2013) have already explicated a framework, termed the Mathematics Teacher’s

Specialized Knowledge (MTSK) framework, which is constructed on, and projects, an

intrinsic perspective whereby the idea of specialization is framed with regard to the

inseparability of knowledge and context. The key to recognizing and making visible what

makes mathematics teacher knowledge specialized lies, we argue, in the context in which the

knowledge comes into being. Contextuality, then, becomes the central concern. Obviously,

that context matters is hardly new nor provocative (see e.g., Fennema & Franke, 1992);

however, the way in which the term is commonly used differs from the point we want to

advance in this paper.

In our view, whether knowledge is specialized or not is a question of whether the knowledge

is contextually adaptive (Hashweh, 2005), that is, functional on a moment-by-moment basis,

and highly sensitive to the changing details of the situation as teachers interact with the

environment and with the students around them. This means, rather than expecting differences

in knowledge (concerning quantity, quality, etc.) based on broad descriptions of context –

such as school vs. scientific environment – the term ―context‖ acquires a very different and

deeper meaning than the ways it has been previously construed. This perspective assumes that

context consists of situations and activities embedded in the learning-teaching complex in the

immediate moment. In consequence, what signifies mathematics teacher knowledge might be

better described (or can be better approached) from within the discipline. In this regard,

certain cognitive dispositions when analyzing the knowledge involved in assessing students‘

subtraction strategies. They affirmed that it is debatable whether the knowledge used by a

teacher is exclusive to him or her, or is shared with other practitioners of mathematics. They

focus discussion on certain cognitive dispositions and wonder who else, other than a

mathematics teacher might have such kind of knowledge, thus moving the focal point of the

debate from mathematics teacher knowledge to that of other professionals.

The answers we might gain from such comparisons (mathematics teachers vs.

mathematicians, mathematics teachers vs. teachers of other subjects, etc.) are external to

mathematics education as a discipline, in that they offer justifications that are recognizable

and measurable but neither cognitive (concerning the processes involved in knowledge) nor

epistemological (regarding the nature of knowledge). External referents (such as

mathematicians) might provide useful markers for identifying static traits that differ from

mathematics teachers such as the content of teacher knowledge, that is, what teachers‘

knowledge is about. However, they seem to be inappropriate in accounts of the complex

dynamics of knowledge in use. Rather than framing the discussion of what makes

mathematics teacher knowledge specialized in terms of external referents, we suggest an

account of specialization understood in relation to mathematics teacher knowledge in action.

That is to say, what makes mathematics teacher knowledge specialized is not so much ―what‖

mathematics teachers know (which might indeed differ from other professionals), but ―how‖

mathematics teachers know. This involves a shift away from the content of mathematics

teacher knowledge to its usage and function, that is, how teacher knowledge comes into action

(how it comes into being or how it actualizes). This shift in perspective foregrounds the

context rather than the content.

Instead of an extrinsic perspective, we suggest taking an intrinsic view, that is,

acknowledging the situatedness of mathematics teacher knowledge within the context of

mathematics learning and teaching. Interestingly, Carrillo, Climent, Contreras and Muñoz-

Catalán (2013) have already explicated a framework, termed the Mathematics Teacher’s

Specialized Knowledge (MTSK) framework, which is constructed on, and projects, an

intrinsic perspective whereby the idea of specialization is framed with regard to the

inseparability of knowledge and context. The key to recognizing and making visible what

makes mathematics teacher knowledge specialized lies, we argue, in the context in which the

knowledge comes into being. Contextuality, then, becomes the central concern. Obviously,

that context matters is hardly new nor provocative (see e.g., Fennema & Franke, 1992);

however, the way in which the term is commonly used differs from the point we want to

advance in this paper.

In our view, whether knowledge is specialized or not is a question of whether the knowledge

is contextually adaptive (Hashweh, 2005), that is, functional on a moment-by-moment basis,

and highly sensitive to the changing details of the situation as teachers interact with the

environment and with the students around them. This means, rather than expecting differences

in knowledge (concerning quantity, quality, etc.) based on broad descriptions of context –

such as school vs. scientific environment – the term ―context‖ acquires a very different and

deeper meaning than the ways it has been previously construed. This perspective assumes that

context consists of situations and activities embedded in the learning-teaching complex in the

immediate moment. In consequence, what signifies mathematics teacher knowledge might be

better described (or can be better approached) from within the discipline. In this regard,

10.
mathematics teacher knowledge is treated not as static traits (that differ from other

professions) but as interpretations of performances that are situated in the immediate context

(see Brown, Danish, Levin, & diSessa, 2016). In this regard, Putnam and Borko (2000) argued

that ―professional knowledge is developed in context, stored together with characteristic

features of classrooms and activities, organised around the tasks that teachers accomplish in

classroom settings, and accessed for use in similar situations‖ (p. 13). As such, a mathematics

teacher‘s action is not a simple display of a static system of some certain knowledge types, but

rather a highly contingent and continually adaptive and proactive response that shapes, and is

shaped by, the environment in which the teacher interacts.

In other words, it is not about being but about becoming, that is, it is less about static

dispositions or traits differentiable from those of other professions and more about the

complex dynamics of the usage and function of knowledge in context. Mathematics teacher

knowledge becomes specialized in its adaptive function in response to the dynamics and

complexities in which it comes into being.

From a Structuralist to an Anthropological-Sociocultural Approach

In this section, we adopt a critical stance to the disciplinary approach to teacher knowledge, an

approach that is primarily structuralist in orientation and that argues from the viewpoint of

teaching mathematics rather than from the standpoint of learning mathematics. We argue

against a restrictive structuralist perspective that relies on, and projects, a reductionist

understanding of knowing and learning, in which knowledge is construed as independent of

the knower. Instead we argue for an anthropological-sociocultural perspective that accounts

for the evolving nature of mathematical meaning in the learning process.

Shulman (1987) declared that subject matter knowledge per se ―must be transformed in some

manner if they are to be taught. To reason one‘s way through an act of teaching is to think

one‘s way from the subject matter as understood by the teacher into the minds and

motivations of learners‖ (p. 16). Generally speaking, the central task of teaching is considered

as transforming subject matter knowledge into a form in which it is teachable to particular

learners. This transformation of the subject matter is, according to Shulman (1987), heavily, if

not wholly, determined by the disciplinary subject matter as the primary source of information

for teaching and the principal route to informed decisions about instruction. Gudmundsdottir

(1991) described this transformation as a ―reorganization [of content knowledge] that derives

from a disciplinary orientation‖ (p. 412) and Grossman et al. (1989) designated it as

―translat[ing] knowledge of subject matter into instructional representations‖ (p. 32). As

mentioned above, scholars in the field of mathematics education have recommended several

discipline-specific practices of transformation that aim to unpack mathematics content in ways

accessible to students: elementarizing, exemplifying, decompressing, and simplifying, among

other. In this view, teachers must be able to take apart mathematical concepts, operations and

strategies so as to enable students to gain access to the thought processes and ideas that they

represent. Students, on the other hand, are considered as putting together the constituent

pieces of those mathematical concepts, operation and strategies. Such assertions rely on, and

project, a reductionist understanding of the knowing and learning processes; an understanding

in which the knowing and learning processes are construed as putting together what teachers

intentionally picked apart. This view not only distorts the complexity of the processes of

knowing and learning mathematics, but also advocates the assumption that knowledge is

independent of the knower.

professions) but as interpretations of performances that are situated in the immediate context

(see Brown, Danish, Levin, & diSessa, 2016). In this regard, Putnam and Borko (2000) argued

that ―professional knowledge is developed in context, stored together with characteristic

features of classrooms and activities, organised around the tasks that teachers accomplish in

classroom settings, and accessed for use in similar situations‖ (p. 13). As such, a mathematics

teacher‘s action is not a simple display of a static system of some certain knowledge types, but

rather a highly contingent and continually adaptive and proactive response that shapes, and is

shaped by, the environment in which the teacher interacts.

In other words, it is not about being but about becoming, that is, it is less about static

dispositions or traits differentiable from those of other professions and more about the

complex dynamics of the usage and function of knowledge in context. Mathematics teacher

knowledge becomes specialized in its adaptive function in response to the dynamics and

complexities in which it comes into being.

From a Structuralist to an Anthropological-Sociocultural Approach

In this section, we adopt a critical stance to the disciplinary approach to teacher knowledge, an

approach that is primarily structuralist in orientation and that argues from the viewpoint of

teaching mathematics rather than from the standpoint of learning mathematics. We argue

against a restrictive structuralist perspective that relies on, and projects, a reductionist

understanding of knowing and learning, in which knowledge is construed as independent of

the knower. Instead we argue for an anthropological-sociocultural perspective that accounts

for the evolving nature of mathematical meaning in the learning process.

Shulman (1987) declared that subject matter knowledge per se ―must be transformed in some

manner if they are to be taught. To reason one‘s way through an act of teaching is to think

one‘s way from the subject matter as understood by the teacher into the minds and

motivations of learners‖ (p. 16). Generally speaking, the central task of teaching is considered

as transforming subject matter knowledge into a form in which it is teachable to particular

learners. This transformation of the subject matter is, according to Shulman (1987), heavily, if

not wholly, determined by the disciplinary subject matter as the primary source of information

for teaching and the principal route to informed decisions about instruction. Gudmundsdottir

(1991) described this transformation as a ―reorganization [of content knowledge] that derives

from a disciplinary orientation‖ (p. 412) and Grossman et al. (1989) designated it as

―translat[ing] knowledge of subject matter into instructional representations‖ (p. 32). As

mentioned above, scholars in the field of mathematics education have recommended several

discipline-specific practices of transformation that aim to unpack mathematics content in ways

accessible to students: elementarizing, exemplifying, decompressing, and simplifying, among

other. In this view, teachers must be able to take apart mathematical concepts, operations and

strategies so as to enable students to gain access to the thought processes and ideas that they

represent. Students, on the other hand, are considered as putting together the constituent

pieces of those mathematical concepts, operation and strategies. Such assertions rely on, and

project, a reductionist understanding of the knowing and learning processes; an understanding

in which the knowing and learning processes are construed as putting together what teachers

intentionally picked apart. This view not only distorts the complexity of the processes of

knowing and learning mathematics, but also advocates the assumption that knowledge is

independent of the knower.

11.
Some general approaches in mathematics education have challenged reductionist views on

knowing and learning, including, but not limited to, Gestaltism, constructivism, problem-

solving, socio-culturalism, and complexity thinking. Here we follow anthropological-

sociocultural perspectives, which, rather than consider knowledge as an object that exists

apart from the individual, acknowledge the co-implicated nature of knowledge, knower and

context. In this perspective, particular emphasis is given to the genesis of mathematical

knowing and learning by accounting for historical and cognitive evolutions, dynamics, and

changes. In this view, knowledge is considered a process rather than an object (see e.g.,

Radford, 2013) – to acknowledge the complex dynamics in knowing mathematics.

For instance, the Didactic Mathematical Knowledge (DMK) framework (Pino-Fan, Assis

& Castro, 2015) is grounded in an onto-semiotic perspective of mathematical knowledge and

instruction (Font, Godino & Gallardo, 2013; Godino, Batanero & Font, 2007). As such, the

framework is rooted in anthropological-sociocultural assumptions about mathematical

knowledge (where mathematics is understood as a human activity), and takes up the

ontological assumption of a diversity of mathematical objects as well as the semiotic

assumption of a plurality of languages and meanings. The DMK framework, similar to other

proposals (e.g., Ernest, 1989), relies on, and projects, assumptions that transcends realistic-

Platonic positions on the nature of mathematics and foregrounds an anthropological

conception of mathematics. That is, teachers have to recognize the emergence of concepts,

procedures, and propositions from mathematical practices, and attribute a central role to the

various languages and artifacts involved in such practices. The applications – the use of

mathematics as a cultural reality in itself to solve real-life or mathematical problems –

promote a variety of meanings for mathematical objects, which must be progressively

articulated in the learning process. Such a view acknowledges the embodied meanings of

mathematical concepts that evolve in the learning process. The DMK framework particularly

foregrounds an epistemic facet of teachers‘ didactical-mathematical knowledge which,

according to Godino, Font, Wilhelmi and Lurduy (2011), interacts with other knowledge

facets (affective, cognitive, ecological, interactional, and mediational). Consequently, the

attentiveness (or mindfulness) to epistemological issues (such as the nature of mathematics

and mathematics learning) is illuminated. From this perspective, teachers‘ sensitivity towards

the epistemic genesis of mathematics and mathematics learning becomes a central aspect of

what mathematics teacher knowledge signifies.

In short, an anthropological-sociocultural perspective acknowledges knowledge as an

evolving process rather than a more or less static object that exists independent of the knower.

In this view, not only the interaction between knowledge, knower, and context is highlighted,

but also the historical and cognitive genesis of mathematical meanings. Thus, what makes

mathematics teacher knowledge specialized is not the accumulation of dstinct facets of

knowledge, but the teachers‘ stance towards knowledge, in the light of the historical and

cognitive geneses of mathematical insights. This perspective calls for a shift in thinking about

teachers‘ core tasks: the teachers‘ focus should not be on acting upon subject matter by re-

structuring, re-interpreting, re-configuring, and re-building mathematical concepts to make

them accessible to students, but instead on the complex interactions between students and

subject matter. That is, the key is not teachers‘ capacity to unpack mathematics, but their

capacity to unpack students‘ ways of understanding in order to make students‘ ways of

mathematical thinking visible.

This is not to be understood as dichotomizing teachers‘ capacity for unpacking mathematics

knowing and learning, including, but not limited to, Gestaltism, constructivism, problem-

solving, socio-culturalism, and complexity thinking. Here we follow anthropological-

sociocultural perspectives, which, rather than consider knowledge as an object that exists

apart from the individual, acknowledge the co-implicated nature of knowledge, knower and

context. In this perspective, particular emphasis is given to the genesis of mathematical

knowing and learning by accounting for historical and cognitive evolutions, dynamics, and

changes. In this view, knowledge is considered a process rather than an object (see e.g.,

Radford, 2013) – to acknowledge the complex dynamics in knowing mathematics.

For instance, the Didactic Mathematical Knowledge (DMK) framework (Pino-Fan, Assis

& Castro, 2015) is grounded in an onto-semiotic perspective of mathematical knowledge and

instruction (Font, Godino & Gallardo, 2013; Godino, Batanero & Font, 2007). As such, the

framework is rooted in anthropological-sociocultural assumptions about mathematical

knowledge (where mathematics is understood as a human activity), and takes up the

ontological assumption of a diversity of mathematical objects as well as the semiotic

assumption of a plurality of languages and meanings. The DMK framework, similar to other

proposals (e.g., Ernest, 1989), relies on, and projects, assumptions that transcends realistic-

Platonic positions on the nature of mathematics and foregrounds an anthropological

conception of mathematics. That is, teachers have to recognize the emergence of concepts,

procedures, and propositions from mathematical practices, and attribute a central role to the

various languages and artifacts involved in such practices. The applications – the use of

mathematics as a cultural reality in itself to solve real-life or mathematical problems –

promote a variety of meanings for mathematical objects, which must be progressively

articulated in the learning process. Such a view acknowledges the embodied meanings of

mathematical concepts that evolve in the learning process. The DMK framework particularly

foregrounds an epistemic facet of teachers‘ didactical-mathematical knowledge which,

according to Godino, Font, Wilhelmi and Lurduy (2011), interacts with other knowledge

facets (affective, cognitive, ecological, interactional, and mediational). Consequently, the

attentiveness (or mindfulness) to epistemological issues (such as the nature of mathematics

and mathematics learning) is illuminated. From this perspective, teachers‘ sensitivity towards

the epistemic genesis of mathematics and mathematics learning becomes a central aspect of

what mathematics teacher knowledge signifies.

In short, an anthropological-sociocultural perspective acknowledges knowledge as an

evolving process rather than a more or less static object that exists independent of the knower.

In this view, not only the interaction between knowledge, knower, and context is highlighted,

but also the historical and cognitive genesis of mathematical meanings. Thus, what makes

mathematics teacher knowledge specialized is not the accumulation of dstinct facets of

knowledge, but the teachers‘ stance towards knowledge, in the light of the historical and

cognitive geneses of mathematical insights. This perspective calls for a shift in thinking about

teachers‘ core tasks: the teachers‘ focus should not be on acting upon subject matter by re-

structuring, re-interpreting, re-configuring, and re-building mathematical concepts to make

them accessible to students, but instead on the complex interactions between students and

subject matter. That is, the key is not teachers‘ capacity to unpack mathematics, but their

capacity to unpack students‘ ways of understanding in order to make students‘ ways of

mathematical thinking visible.

This is not to be understood as dichotomizing teachers‘ capacity for unpacking mathematics

12.
and their capacity for unpacking students‘ understandings, but to re-emphasize that teaching is

not merely a top-down approach of transposing subject matter to the students but a bottom-up

approach of students constructing mathematical ideas that are used as points of departure in

the teaching-leaning complex.

From an Additive to a Transformative Approach

In this section, we adopt a critical stance to another apparently widespread tendency that

seems to have implicitly driven recent discussions on teacher knowledge: the tendency

towards atomizing teacher knowledge for the sake of accumulating distinct and refined

dimensions of teacher knowledge. We argue for a transformative approach that goes beyond a

merely incremental approach to facets of knowledge by turning back to Shulman‘s idea of

blending knowledge facets.

The last three decades have been colored by various attempts to capture what mathematics

teacher knowledge is about and what it entails. Research studies started out by distinguishing,

refining, and adding to various dimensions of knowledge regarded as critical for teaching

mathematics. Since then we have accumulated a considerable number of, often

indistinguishable (see Silverman & Thompson, 2008), knowledge dimensions that, taken

together, seem to provide a more refined picture of the multidimensionality of teacher

knowledge. This undertaking allowed scholars to order, structure, and, most important,

simplify the complexity of teacher knowledge, to reduce it to its observable and measurable

The approach relies on the assumption that a full understanding of teacher knowledge should

emerge from a detailed analysis of each of its parts. It is believed that the complexity of

teacher knowledge can be studied by dissecting it into its smallest parts (knowledge facets,

types, etc.), and that these knowledge units are the basis, or the fundamental particles, of what

mathematics teacher knowledge signifies. Following these lines of thinking, reflections on

mathematics teacher knowledge emphasize the nature of these parts – paying little attention to

transformations that arise when knowledge elements are blended.

Instead of dividing and thinking in terms of multiple, distinct sub-categories of teacher

knowledge, our disposition is to take a broader view that sees teacher knowledge as an organic

Interestingly, Shulman (1987) already described PCK as ―that special amalgam of content and

pedagogy that is uniquely the province of teachers, their own special form of professional

understanding‖ (p. 8, italics added). Here, Shulman understood PCK not as the summation or

accumulation of content knowledge and pedagogical knowledge: ―[…] just knowing the

content well was really important, just knowing general pedagogy was really important and

yet when you add the two together, you didn‘t get the teacher‖ (Shulman, cit. in Berry,

Loughran, & van Driel, 2008, p. 1274). Rather, the amalgamation of content and pedagogy

means ―the blending of content and pedagogy‖ (Shulman, 1987, p. 8, italics added) into a new

kind of knowledge that is distinctively and qualitatively different from the knowledge

dimensions from which it was constructed. However, by proposing PCK as the amalgam of

content and pedagogy without accounting for the complex interactions between these and

other knowledge facets, Shulman left the task of further clarifying the blending process to

other scholars.

not merely a top-down approach of transposing subject matter to the students but a bottom-up

approach of students constructing mathematical ideas that are used as points of departure in

the teaching-leaning complex.

From an Additive to a Transformative Approach

In this section, we adopt a critical stance to another apparently widespread tendency that

seems to have implicitly driven recent discussions on teacher knowledge: the tendency

towards atomizing teacher knowledge for the sake of accumulating distinct and refined

dimensions of teacher knowledge. We argue for a transformative approach that goes beyond a

merely incremental approach to facets of knowledge by turning back to Shulman‘s idea of

blending knowledge facets.

The last three decades have been colored by various attempts to capture what mathematics

teacher knowledge is about and what it entails. Research studies started out by distinguishing,

refining, and adding to various dimensions of knowledge regarded as critical for teaching

mathematics. Since then we have accumulated a considerable number of, often

indistinguishable (see Silverman & Thompson, 2008), knowledge dimensions that, taken

together, seem to provide a more refined picture of the multidimensionality of teacher

knowledge. This undertaking allowed scholars to order, structure, and, most important,

simplify the complexity of teacher knowledge, to reduce it to its observable and measurable

The approach relies on the assumption that a full understanding of teacher knowledge should

emerge from a detailed analysis of each of its parts. It is believed that the complexity of

teacher knowledge can be studied by dissecting it into its smallest parts (knowledge facets,

types, etc.), and that these knowledge units are the basis, or the fundamental particles, of what

mathematics teacher knowledge signifies. Following these lines of thinking, reflections on

mathematics teacher knowledge emphasize the nature of these parts – paying little attention to

transformations that arise when knowledge elements are blended.

Instead of dividing and thinking in terms of multiple, distinct sub-categories of teacher

knowledge, our disposition is to take a broader view that sees teacher knowledge as an organic

Interestingly, Shulman (1987) already described PCK as ―that special amalgam of content and

pedagogy that is uniquely the province of teachers, their own special form of professional

understanding‖ (p. 8, italics added). Here, Shulman understood PCK not as the summation or

accumulation of content knowledge and pedagogical knowledge: ―[…] just knowing the

content well was really important, just knowing general pedagogy was really important and

yet when you add the two together, you didn‘t get the teacher‖ (Shulman, cit. in Berry,

Loughran, & van Driel, 2008, p. 1274). Rather, the amalgamation of content and pedagogy

means ―the blending of content and pedagogy‖ (Shulman, 1987, p. 8, italics added) into a new

kind of knowledge that is distinctively and qualitatively different from the knowledge

dimensions from which it was constructed. However, by proposing PCK as the amalgam of

content and pedagogy without accounting for the complex interactions between these and

other knowledge facets, Shulman left the task of further clarifying the blending process to

other scholars.

13.
Surprisingly, though many scholars paraphrased Shulman‘s idea of amalgamation, they almost

always took the result of blending knowledge domains (that is, according to Shulman, PCK)

as given and often considered it as static (for a critique, see Hashweh, 2005). In other words,

many scholars ignored the complex dynamics of blending, a high interaction of knowledge

facets that forms new structure not evident in the previous facets.

To the best of our knowledge, blending seems to be an undertheorized phenomenon in

research on teacher knowledge. Recently, Scheiner (2015) has suggested construing teacher

knowledge as a complex, dynamic system of various knowledge atoms, which are understood

as blends of different knowledge facets. The idea of ‗knowledge atom‘ shares similarities with

Sherin‘s (2002) idea of ‗content knowledge complexes‘ construed as ―tightly integrated

structures containing [pieces of] both subject matter knowledge and pedagogical content

knowledge‖ (p. 125) repeatedly accessed during instruction. Scheiner (2015) proposed that

teacher knowledge is dynamic not simply because it evolves dynamically (which it does), but

because it forms dynamically: teacher knowledge is dynamically emergent from the

interactions of knowledge facets. This interaction of knowledge facets is in the nature of what

Fauconnier and Turner (2002) described as conceptual blending. In technical terms, blending

is a process of conceptual mapping and integration, a mental operation for combining frames

or models in integration networks that leads to new meaning, global insights, and conceptual

compression (see Fauconnier & Turner, 2002). The essence of conceptual blending is to

construct a partial match, called cross-space mapping, between frames from established

domains (known as inputs), to project selectively from those inputs into a novel hybrid frame

(a blend or blended model), comprised of structure from each of its inputs, as well as a unique

structure of its own (emergent structure). Crucially, the inputs are not just projected wholesale

into the blend, but a combination of the processes of projection, completion, and elaboration

(or ‗running‘ the blend) ―develops emergent structure that is not in the inputs‖ (Fauconnier &

Turner, 2002, p. 42). The point we want to make here is that knowledge facets interact

dynamically to form emergent structures. Not only do new elements arise in the blend that are

not evident in either input domain on its own, but blending accounts also for the

interdependencies of knowledge dimensions: the production of a blend is recursive, in the

sense, that blends depend on previous blends.

Scheiner‘s (2015) proposal of teacher knowledge as a complex, dynamic system of various

knowledge atoms attempts a dialectic between atomistic and holistic views of teacher

knowledge. It puts the refinements of teacher knowledge identified and gained through

atomistic approaches together into a complex system of blends that – as a whole – is more

than the sum of its parts.

In a nutshell, a complex system perspective regards teacher knowledge as dynamically

emergent and dimensions of teacher knowledge as being organically interrelated. It

emphasizes that various knowledge facets are in constant dialogue with each other, inform

each other, and interact dynamically to form emergent structures. Thus, the key relies not on

accumulating types of teacher knowledge but on blending knowledge facets that emerge

dynamically. Accumulating teacher knowledge facets is additive (or complementary), but

blending is transformative.

In the three previous sections, we have critically appraised what the current literature

implicitly represents as making mathematics teacher knowledge specialized. In each section,

always took the result of blending knowledge domains (that is, according to Shulman, PCK)

as given and often considered it as static (for a critique, see Hashweh, 2005). In other words,

many scholars ignored the complex dynamics of blending, a high interaction of knowledge

facets that forms new structure not evident in the previous facets.

To the best of our knowledge, blending seems to be an undertheorized phenomenon in

research on teacher knowledge. Recently, Scheiner (2015) has suggested construing teacher

knowledge as a complex, dynamic system of various knowledge atoms, which are understood

as blends of different knowledge facets. The idea of ‗knowledge atom‘ shares similarities with

Sherin‘s (2002) idea of ‗content knowledge complexes‘ construed as ―tightly integrated

structures containing [pieces of] both subject matter knowledge and pedagogical content

knowledge‖ (p. 125) repeatedly accessed during instruction. Scheiner (2015) proposed that

teacher knowledge is dynamic not simply because it evolves dynamically (which it does), but

because it forms dynamically: teacher knowledge is dynamically emergent from the

interactions of knowledge facets. This interaction of knowledge facets is in the nature of what

Fauconnier and Turner (2002) described as conceptual blending. In technical terms, blending

is a process of conceptual mapping and integration, a mental operation for combining frames

or models in integration networks that leads to new meaning, global insights, and conceptual

compression (see Fauconnier & Turner, 2002). The essence of conceptual blending is to

construct a partial match, called cross-space mapping, between frames from established

domains (known as inputs), to project selectively from those inputs into a novel hybrid frame

(a blend or blended model), comprised of structure from each of its inputs, as well as a unique

structure of its own (emergent structure). Crucially, the inputs are not just projected wholesale

into the blend, but a combination of the processes of projection, completion, and elaboration

(or ‗running‘ the blend) ―develops emergent structure that is not in the inputs‖ (Fauconnier &

Turner, 2002, p. 42). The point we want to make here is that knowledge facets interact

dynamically to form emergent structures. Not only do new elements arise in the blend that are

not evident in either input domain on its own, but blending accounts also for the

interdependencies of knowledge dimensions: the production of a blend is recursive, in the

sense, that blends depend on previous blends.

Scheiner‘s (2015) proposal of teacher knowledge as a complex, dynamic system of various

knowledge atoms attempts a dialectic between atomistic and holistic views of teacher

knowledge. It puts the refinements of teacher knowledge identified and gained through

atomistic approaches together into a complex system of blends that – as a whole – is more

than the sum of its parts.

In a nutshell, a complex system perspective regards teacher knowledge as dynamically

emergent and dimensions of teacher knowledge as being organically interrelated. It

emphasizes that various knowledge facets are in constant dialogue with each other, inform

each other, and interact dynamically to form emergent structures. Thus, the key relies not on

accumulating types of teacher knowledge but on blending knowledge facets that emerge

dynamically. Accumulating teacher knowledge facets is additive (or complementary), but

blending is transformative.

In the three previous sections, we have critically appraised what the current literature

implicitly represents as making mathematics teacher knowledge specialized. In each section,

14.
we have tried to make explicit the more serious limitations of the grounds on which at least

three general tendencies stand, and which seem to have been inherent in the current discussion

on teacher knowledge. Each section provides provocative new ways of thinking about the

issue under consideration.

First, we called for an account of specialization that comes from the inside rather that the

outside (such as comparisons with professionals working in other disciplines). In recognizing

the situated nature of mathematics teacher knowledge in the immediate context, the complex

dynamics of the usage and function of knowledge in the immediate context can be underlined.

As such, specialization is not a state of being but a process of becoming: mathematics teacher

knowledge becomes specialized in its adaptive function in response to the dynamics and

complexities in which it comes into being.

Second, we argued that an account of specialization cannot be provided with itemisation of

mathematics teacher knowledge, but rather through teachers‘ epistemological stance toward

knowledge and the sensitivity for the historical and cognitive geneses of mathematical

insights. Going beyond a structuralist viewpoint, in which the teacher‘s task is considered to

be unpacking the subject matter of mathematics, we encouraged the view of teachers

unpacking students‘ understandings to make students‘ ways of mathematical thinking explicit.

Third, we argued that an account of specialization lies not in the sum of the parts of

mathematics teacher knowledge but in its organic whole, that is, various knowledge facets

constantly in dialogue with each other, informing each other, and interacting dynamically to

form emergent structures. We proposed a complex system perspective that construes

mathematics teacher knowledge as blends of various knowledge facets that emerge dynamic

On the one hand, these alternative views point to several aspects that scholars attempted to

encompass in their use of the notion of knowing rather than knowledge: knowledge is usually

treated as static, explicit, and objective, whereas what is described as knowing is seen as

dynamic, tacit, and contextualized (see Adler, 1998; Ponte, 1994). However, the alternative

views outlined above foreground aspects that might contribute further to the discussion of

knowledge versus knowing. First, whereas knowledge has been debated as either existing

independently of the knower (the realist viewpoint) or only existing in the mind of the knower

(the relativist viewpoint), with the term knowing we can signal the inseparability of

knowledge and knower. That is, it makes no sense to talk about something being known

without also talking about who knows it (and under which circumstances). Second, what is

called knowledge is usually perceived as a state of being (or product), whereas what is

described as knowing is seen as an emergent process – a process of becoming. However, this

is not a call for a distinction between product and process, since the main point is seen in the

complex dynamics underpinning the stability of established knowledge (see Davis & Simmt,

2006). It implies the dynamic character of knower, knowledge, and context such that all three

are changing and evolving over time. This means knowing is not just situated in place – that

is, it is contextual and embedded in the practices of teaching (Adler, 1998) – but also situated

with respect to time and other factors, given that the context of knowing is similarly dynamic

and changing over time. That knowing is situated with regard to time, place and other factors

implies that it cannot be reduced to some observable and measurable by-products. The whole

venture is to understand mathematics teacher knowing as it is, as it comes into being, as it

works in the immediate context; that is, to take a holistic (rather than a reductionist) view that

acknowledges mathematics teacher knowing as highly personal, embodied, enacted, and

three general tendencies stand, and which seem to have been inherent in the current discussion

on teacher knowledge. Each section provides provocative new ways of thinking about the

issue under consideration.

First, we called for an account of specialization that comes from the inside rather that the

outside (such as comparisons with professionals working in other disciplines). In recognizing

the situated nature of mathematics teacher knowledge in the immediate context, the complex

dynamics of the usage and function of knowledge in the immediate context can be underlined.

As such, specialization is not a state of being but a process of becoming: mathematics teacher

knowledge becomes specialized in its adaptive function in response to the dynamics and

complexities in which it comes into being.

Second, we argued that an account of specialization cannot be provided with itemisation of

mathematics teacher knowledge, but rather through teachers‘ epistemological stance toward

knowledge and the sensitivity for the historical and cognitive geneses of mathematical

insights. Going beyond a structuralist viewpoint, in which the teacher‘s task is considered to

be unpacking the subject matter of mathematics, we encouraged the view of teachers

unpacking students‘ understandings to make students‘ ways of mathematical thinking explicit.

Third, we argued that an account of specialization lies not in the sum of the parts of

mathematics teacher knowledge but in its organic whole, that is, various knowledge facets

constantly in dialogue with each other, informing each other, and interacting dynamically to

form emergent structures. We proposed a complex system perspective that construes

mathematics teacher knowledge as blends of various knowledge facets that emerge dynamic

On the one hand, these alternative views point to several aspects that scholars attempted to

encompass in their use of the notion of knowing rather than knowledge: knowledge is usually

treated as static, explicit, and objective, whereas what is described as knowing is seen as

dynamic, tacit, and contextualized (see Adler, 1998; Ponte, 1994). However, the alternative

views outlined above foreground aspects that might contribute further to the discussion of

knowledge versus knowing. First, whereas knowledge has been debated as either existing

independently of the knower (the realist viewpoint) or only existing in the mind of the knower

(the relativist viewpoint), with the term knowing we can signal the inseparability of

knowledge and knower. That is, it makes no sense to talk about something being known

without also talking about who knows it (and under which circumstances). Second, what is

called knowledge is usually perceived as a state of being (or product), whereas what is

described as knowing is seen as an emergent process – a process of becoming. However, this

is not a call for a distinction between product and process, since the main point is seen in the

complex dynamics underpinning the stability of established knowledge (see Davis & Simmt,

2006). It implies the dynamic character of knower, knowledge, and context such that all three

are changing and evolving over time. This means knowing is not just situated in place – that

is, it is contextual and embedded in the practices of teaching (Adler, 1998) – but also situated

with respect to time and other factors, given that the context of knowing is similarly dynamic

and changing over time. That knowing is situated with regard to time, place and other factors

implies that it cannot be reduced to some observable and measurable by-products. The whole

venture is to understand mathematics teacher knowing as it is, as it comes into being, as it

works in the immediate context; that is, to take a holistic (rather than a reductionist) view that

acknowledges mathematics teacher knowing as highly personal, embodied, enacted, and

15.
performed. Any approach toward what makes teacher knowledge specialized must deal with

this complex whole rather than with piecemeal facets or types of knowledge (see Beswick,

Callingham, & Watson, 2012).3 Of course, such sensibilities are not entirely new. They might

be argued to have been represented in the discourses of different movements of thought such

as cognitive approaches and situated approaches (see Kaiser et al., 2017), as well as other

discourses. However, the view advanced here takes the discussion to realms that often cast

knowing and knowledge as oppositional.

On the other hand, and more importantly, the alternative viewpoints converge on the

understanding that it is not a kind of knowledge but a style of knowing that accounts for

specialization in mathematics teacher knowledge. To elaborate this aspect in more detail: In

the past, the focus was primarily on knowledge about/of/for/in the discipline. This resulted in

multiple descriptions and distinctions, such as knowledge about mathematics versus

knowledge of mathematics, or mathematical knowledge for teaching as opposed to

mathematical knowledge in teaching, and knowledge for teaching mathematics in

contradistinction to knowledge in teaching mathematics, all primarily concerned with the

question of ‗what‘ mathematics teachers know. In this regard, comparisons such as

mathematics teachers versus mathematicians or mathematics teachers versus teachers of other

subjects were assumed to be decisive, as it was believed that it was the kind of knowledge –

whether quantitatively or qualitatively different – that set mathematics teachers apart from

other professionals. However, the alternative views discussed above consider the yet unsettled

question of ‗how‘ teachers knowing comes into being rather than pointing to the question of

‗what‘ teachers know. This brings to the fore the complex, dynamic usage, function, and

interaction of mathematics teacher knowing, a position that goes beyond accounts that

primarily address kinds of teacher knowledge. We intend to enunciate this shift in perspective

by calling for attention to mathematics teachers‘ styles of knowing rather than merely teachers‘

kinds of knowledge. We believe that this shift in perspective is critical as it provides a new

light on the discussion of the nature of mathematics teacher knowledge that allows us to better

integrate knowledge and action. It articulates mathematics teacher knowledge more as a

mindset rather than as some static traits or dispositions. To cast this idea in a term, we suggest

a fine distinction in thinking about the issues under consideration: knowledge about/of/for/in a

discipline and disciplinary knowing. Knowledge about/of/for/in the discipline prompts the

question of different kinds of knowledge, while disciplinary knowing prommpts the question

of a style of knowing that is a function of particular activities, orientations, and dynamics

recognizably disciplinary. From this perspective, we argue that it is mathematics educational

knowing that signifies specialization in mathematics teacher knowledge.

Mathematics teacher knowing is a mysterious phenomenon indeed. To acknowledge this

mystery is not to mystify mathematics teacher knowing, but to express our recognition of the

exquisite complexity of how mathematics teacher knowing comes into being. Breaking up the

complex nature of teacher knowledge for the sake of insights leads to atomizing our

understanding, our thinking, of what makes mathematics teacher knowledge specialized. Such

insights are themselves fragmented, not holistic. The piecemeal, atomistic, analytic approach

(as advocated in the past) does not work in relation to the complex usage, function, and

interaction of teacher knowing. Any approach toward what makes teacher knowledge

specialized must deal with the complex whole rather than with some piecemeal facets or types

of teacher knowledge.

this complex whole rather than with piecemeal facets or types of knowledge (see Beswick,

Callingham, & Watson, 2012).3 Of course, such sensibilities are not entirely new. They might

be argued to have been represented in the discourses of different movements of thought such

as cognitive approaches and situated approaches (see Kaiser et al., 2017), as well as other

discourses. However, the view advanced here takes the discussion to realms that often cast

knowing and knowledge as oppositional.

On the other hand, and more importantly, the alternative viewpoints converge on the

understanding that it is not a kind of knowledge but a style of knowing that accounts for

specialization in mathematics teacher knowledge. To elaborate this aspect in more detail: In

the past, the focus was primarily on knowledge about/of/for/in the discipline. This resulted in

multiple descriptions and distinctions, such as knowledge about mathematics versus

knowledge of mathematics, or mathematical knowledge for teaching as opposed to

mathematical knowledge in teaching, and knowledge for teaching mathematics in

contradistinction to knowledge in teaching mathematics, all primarily concerned with the

question of ‗what‘ mathematics teachers know. In this regard, comparisons such as

mathematics teachers versus mathematicians or mathematics teachers versus teachers of other

subjects were assumed to be decisive, as it was believed that it was the kind of knowledge –

whether quantitatively or qualitatively different – that set mathematics teachers apart from

other professionals. However, the alternative views discussed above consider the yet unsettled

question of ‗how‘ teachers knowing comes into being rather than pointing to the question of

‗what‘ teachers know. This brings to the fore the complex, dynamic usage, function, and

interaction of mathematics teacher knowing, a position that goes beyond accounts that

primarily address kinds of teacher knowledge. We intend to enunciate this shift in perspective

by calling for attention to mathematics teachers‘ styles of knowing rather than merely teachers‘

kinds of knowledge. We believe that this shift in perspective is critical as it provides a new

light on the discussion of the nature of mathematics teacher knowledge that allows us to better

integrate knowledge and action. It articulates mathematics teacher knowledge more as a

mindset rather than as some static traits or dispositions. To cast this idea in a term, we suggest

a fine distinction in thinking about the issues under consideration: knowledge about/of/for/in a

discipline and disciplinary knowing. Knowledge about/of/for/in the discipline prompts the

question of different kinds of knowledge, while disciplinary knowing prommpts the question

of a style of knowing that is a function of particular activities, orientations, and dynamics

recognizably disciplinary. From this perspective, we argue that it is mathematics educational

knowing that signifies specialization in mathematics teacher knowledge.

Mathematics teacher knowing is a mysterious phenomenon indeed. To acknowledge this

mystery is not to mystify mathematics teacher knowing, but to express our recognition of the

exquisite complexity of how mathematics teacher knowing comes into being. Breaking up the

complex nature of teacher knowledge for the sake of insights leads to atomizing our

understanding, our thinking, of what makes mathematics teacher knowledge specialized. Such

insights are themselves fragmented, not holistic. The piecemeal, atomistic, analytic approach

(as advocated in the past) does not work in relation to the complex usage, function, and

interaction of teacher knowing. Any approach toward what makes teacher knowledge

specialized must deal with the complex whole rather than with some piecemeal facets or types

of teacher knowledge.

16.
In this paper, new avenues for theoretical reflection on some of the major orientations and

tendencies in the field of mathematics teacher knowledge were outlined. These reflections

were not intended to exhaust the object of consideration, but to include those approaches,

initiatives, and theoretical insights that might prompt re-thinking about what mathematics

teacher knowledge signifies.

We explained that the question of what makes teacher knowledge specialized cannot be

comprehensively answered by only addressing ―what‖ teachers know, but we need to account

for ―how‖ teachers knowing comes into being. The alternative views discussed in the paper

bring to the foreground that it is not a kind of knowledge but a style of knowing that accounts

for specialization in mathematics teacher knowledge. Such style of knowing is not a state of

being but a process of becoming – the becoming of a mathematics educational mindset.

This call for a style of knowing is rather different from what normally receives emphasis in

discussion of mathematics teacher knowledge. We hypothesize that considering specialization

as a style of knowing (rather than a kind of knowledge) can have far-reaching consequences

not only for conceptualizing mathematics teacher knowledge.

With respect to mathematics teacher education, for instance, considering specialization as a

style of knowing (rather than a kind of knowledge) advocates a holistic approach to

mathematics teacher education programs, criticizing the separate acquirement of different

kinds of knowledge (generally acquired from different academic departments such as

mathematics, education, psychology, among others). Mathematics teacher education programs

should be deliberately designed in an integrated fashion to support teachers in blending

insights from various disciplines including, but not limited to, mathematics, education, and

psychology, thereby creating novel styles of knowing that empowers teachers to reshape the

way they view their own profession. It is reasonable to assume that such styles of knowing

develop gradually, rooted in authentic activities and within a community of individuals

engaged in inquiry and practice (see Putnam & Borko, 2000). Further, a shift toward a style of

knowing is expected to affect researchers‘ and educators‘ perceptions of teachers‘ professional

identity, as the path to a mathematics educational mindset is a journey, not a proclamation.

This would mean giving up deficit-oriented discussions on teacher knowledge in terms of

identifying and fixing teachers‘ lack of knowledge (Askew, 2008). The central concern for

future research, then, is to understand those mindsets, which underpin any authentic form of

mathematics educational knowing. It is hoped that this call for a style of knowing offers a new

vision of what makes mathematics teacher knowledge specialized.

We prefer using the term ‗specialized‘ instead of ‗special‘ with respect to mathematics

teacher knowledge. The latter implies the assertion of a quality of teacher knowledge that is

distinguishable from something. We use the term ‗specialized‘ to indicate a quality of

mathematics teacher knowledge that comes into being when enacted.

We use the term structuralism (or structuralist) in a broad sense as described by Bourdieu

(1989): ―By structuralism or structuralist, I mean that there exist, within the social world itself

and not only within symbolic systems (language, myths, etc.), objective structures independent

of the consciousness and will of agents, which are capable of guiding and constraining their

practices or their representations‖ (p. 14).

tendencies in the field of mathematics teacher knowledge were outlined. These reflections

were not intended to exhaust the object of consideration, but to include those approaches,

initiatives, and theoretical insights that might prompt re-thinking about what mathematics

teacher knowledge signifies.

We explained that the question of what makes teacher knowledge specialized cannot be

comprehensively answered by only addressing ―what‖ teachers know, but we need to account

for ―how‖ teachers knowing comes into being. The alternative views discussed in the paper

bring to the foreground that it is not a kind of knowledge but a style of knowing that accounts

for specialization in mathematics teacher knowledge. Such style of knowing is not a state of

being but a process of becoming – the becoming of a mathematics educational mindset.

This call for a style of knowing is rather different from what normally receives emphasis in

discussion of mathematics teacher knowledge. We hypothesize that considering specialization

as a style of knowing (rather than a kind of knowledge) can have far-reaching consequences

not only for conceptualizing mathematics teacher knowledge.

With respect to mathematics teacher education, for instance, considering specialization as a

style of knowing (rather than a kind of knowledge) advocates a holistic approach to

mathematics teacher education programs, criticizing the separate acquirement of different

kinds of knowledge (generally acquired from different academic departments such as

mathematics, education, psychology, among others). Mathematics teacher education programs

should be deliberately designed in an integrated fashion to support teachers in blending

insights from various disciplines including, but not limited to, mathematics, education, and

psychology, thereby creating novel styles of knowing that empowers teachers to reshape the

way they view their own profession. It is reasonable to assume that such styles of knowing

develop gradually, rooted in authentic activities and within a community of individuals

engaged in inquiry and practice (see Putnam & Borko, 2000). Further, a shift toward a style of

knowing is expected to affect researchers‘ and educators‘ perceptions of teachers‘ professional

identity, as the path to a mathematics educational mindset is a journey, not a proclamation.

This would mean giving up deficit-oriented discussions on teacher knowledge in terms of

identifying and fixing teachers‘ lack of knowledge (Askew, 2008). The central concern for

future research, then, is to understand those mindsets, which underpin any authentic form of

mathematics educational knowing. It is hoped that this call for a style of knowing offers a new

vision of what makes mathematics teacher knowledge specialized.

We prefer using the term ‗specialized‘ instead of ‗special‘ with respect to mathematics

teacher knowledge. The latter implies the assertion of a quality of teacher knowledge that is

distinguishable from something. We use the term ‗specialized‘ to indicate a quality of

mathematics teacher knowledge that comes into being when enacted.

We use the term structuralism (or structuralist) in a broad sense as described by Bourdieu

(1989): ―By structuralism or structuralist, I mean that there exist, within the social world itself

and not only within symbolic systems (language, myths, etc.), objective structures independent

of the consciousness and will of agents, which are capable of guiding and constraining their

practices or their representations‖ (p. 14).

17.
Notice that we do not construe the relationship between knowing and knowledge as

contradictory but rather as dialectical. In terms of the onto-semiotic approach there is no

mathematical practice without objects, or objects without practice, which is equivalent to the

issues of knowing and knowledge discussed here.

Acknowledgments. Writing was done while the first author, Thorsten Scheiner, was a Klaus

Murmann Fellow of the Foundation of German Business and completed while he was recipient of the

Research Excellent Scholarship of Macquarie University. This work was supported, in part, by grant

number EDU2013-44047-P (Spanish Ministry of Economy and Competitiveness) to José Carrillo and

Miguel A. Montes, EDU2016- 74848-P (FEDER, AEI) to Juan D. Godino, and FONDECYT

Nº11150014 (CONICYT, Chile) to Luis R. Pino-Fan.

Adler, J. (1998). Lights and limits: Recontextualising Lave and Wenger to theorise knowledge

of teaching and learning school mathematics. In A.Watson (Ed.), Situated cognition and

the learning of mathematics (pp. 161–177). Oxford, UK: CMER.

Adler, J., & Davis, Z. (2006). Opening another black box: Research mathematics for teaching

in mathematics teacher education. Journal for Research in Mathematics Education,

37(4), 270–296.

Askew, M. (2008). Mathematical discipline knowledge requirements for prospective primary

teachers, and the structure and teaching approaches of programs designed to develop that

knowledge. In P. Sullivan & T. Wood (Eds.), The international handbook of

mathematics teacher education. Volume 1: Knowledge and beliefs in mathematics

teaching and teaching development (pp. 13–35). Rotterdam, The Netherlands: Sense.

Ball, D. L. (1988). Knowledge and reasoning in mathematical pedagogy: Examining what

prospective teachers bring to teacher education (Unpublished doctoral dissertation).

Michigan State University, East Lansing.

Ball, D. L., & Bass, H. (2000). Interweaving content and pedagogy in teaching and learning to

teach: Knowing and using mathematics. In J. Boaler (Ed.), Multiple perspectives on the

teaching and learning of mathematics (pp. 83–104). Greenwich, CT: JAI/Albex.

Ball, D. L., Thames, M. H., & Phelps, G. (2008). Content knowledge for teaching: What

makes it special? Journal of Teacher Education, 59(5), 389–407.

Baumert, J., Kunter, M., Blum, W., Brunner, M., Voss, T., Jordan, A., . . . Tsai, Y.-M. (2010).

Teachers‘ mathematical knowledge, cognitive activation in the classroom, and student

progress. American Educational Research Journal, 47(1), 133–180.

Berry, A., Loughran, J., & van Driel, J. H. (2008). Revisiting the roots of pedagogical content

knowledge. International Journal of Science Education, 30(10), 1271–1279.

Beswick, K., Callingham, R., &Watson, J. (2012). The nature and development of middle

school mathematics teachers‘ knowledge. Journal of Mathematics Teacher Education,

15(2), 131–157.

Blömeke, S., Hsieh, F.-J., Kaiser, G., & Schmidt, W. H. (Eds.). (2014). International

perspectives on teacher knowledge, beliefs and opportunities to learn. TEDS-M results.

Dordrecht, The Netherlands: Springer.

Blömeke, S., & Kaiser, G. (2017). Understanding the development of teachers‘ professional

competencies as personally, situationally, and socially determined. In D. J. Clandinin &

J. Husu (Eds.), International handbook of research on teacher education (pp. 783-802).

Thousand Oakes, CA: Sage.

Bromme, R. (1994). Beyond subject matter: A psychological topology of teachers‘

professional knowledge. In R. Biehler, R. W. Scholz, R. Strässer, & B. Winkelmann

contradictory but rather as dialectical. In terms of the onto-semiotic approach there is no

mathematical practice without objects, or objects without practice, which is equivalent to the

issues of knowing and knowledge discussed here.

Acknowledgments. Writing was done while the first author, Thorsten Scheiner, was a Klaus

Murmann Fellow of the Foundation of German Business and completed while he was recipient of the

Research Excellent Scholarship of Macquarie University. This work was supported, in part, by grant

number EDU2013-44047-P (Spanish Ministry of Economy and Competitiveness) to José Carrillo and

Miguel A. Montes, EDU2016- 74848-P (FEDER, AEI) to Juan D. Godino, and FONDECYT

Nº11150014 (CONICYT, Chile) to Luis R. Pino-Fan.

Adler, J. (1998). Lights and limits: Recontextualising Lave and Wenger to theorise knowledge

of teaching and learning school mathematics. In A.Watson (Ed.), Situated cognition and

the learning of mathematics (pp. 161–177). Oxford, UK: CMER.

Adler, J., & Davis, Z. (2006). Opening another black box: Research mathematics for teaching

in mathematics teacher education. Journal for Research in Mathematics Education,

37(4), 270–296.

Askew, M. (2008). Mathematical discipline knowledge requirements for prospective primary

teachers, and the structure and teaching approaches of programs designed to develop that

knowledge. In P. Sullivan & T. Wood (Eds.), The international handbook of

mathematics teacher education. Volume 1: Knowledge and beliefs in mathematics

teaching and teaching development (pp. 13–35). Rotterdam, The Netherlands: Sense.

Ball, D. L. (1988). Knowledge and reasoning in mathematical pedagogy: Examining what

prospective teachers bring to teacher education (Unpublished doctoral dissertation).

Michigan State University, East Lansing.

Ball, D. L., & Bass, H. (2000). Interweaving content and pedagogy in teaching and learning to

teach: Knowing and using mathematics. In J. Boaler (Ed.), Multiple perspectives on the

teaching and learning of mathematics (pp. 83–104). Greenwich, CT: JAI/Albex.

Ball, D. L., Thames, M. H., & Phelps, G. (2008). Content knowledge for teaching: What

makes it special? Journal of Teacher Education, 59(5), 389–407.

Baumert, J., Kunter, M., Blum, W., Brunner, M., Voss, T., Jordan, A., . . . Tsai, Y.-M. (2010).

Teachers‘ mathematical knowledge, cognitive activation in the classroom, and student

progress. American Educational Research Journal, 47(1), 133–180.

Berry, A., Loughran, J., & van Driel, J. H. (2008). Revisiting the roots of pedagogical content

knowledge. International Journal of Science Education, 30(10), 1271–1279.

Beswick, K., Callingham, R., &Watson, J. (2012). The nature and development of middle

school mathematics teachers‘ knowledge. Journal of Mathematics Teacher Education,

15(2), 131–157.

Blömeke, S., Hsieh, F.-J., Kaiser, G., & Schmidt, W. H. (Eds.). (2014). International

perspectives on teacher knowledge, beliefs and opportunities to learn. TEDS-M results.

Dordrecht, The Netherlands: Springer.

Blömeke, S., & Kaiser, G. (2017). Understanding the development of teachers‘ professional

competencies as personally, situationally, and socially determined. In D. J. Clandinin &

J. Husu (Eds.), International handbook of research on teacher education (pp. 783-802).

Thousand Oakes, CA: Sage.

Bromme, R. (1994). Beyond subject matter: A psychological topology of teachers‘

professional knowledge. In R. Biehler, R. W. Scholz, R. Strässer, & B. Winkelmann

18.
(Eds.), Mathematics didactics as a scientific discipline: The state of the art (pp. 73–88).

Dordrecht, The Netherlands: Kluwer.

Buchholtz, N., Leung, F. K., Ding, L., Kaiser, G., Park, K., & Schwarz, B. (2013). Future

mathematics teachers‘ professional knowledge of elementary mathematics from an

advanced standpoint. ZDM—The International Journal of Mathematics Education,

45(1), 107–120.

Carrillo, J., Climent, N., Contreras L. C., & Muñoz-Catalán, M. C. (2013). Determining

specialised knowledge for mathematics teaching. In B. Ubuz, Ç. Haser & M. Mariotti

(Eds.), Proceedings of the Eighth Congress of the European Society for Research in

Mathematics Education (pp. 2985–2994). Antalya, Turkey: ERME.

Cochran, K. F., DeRuiter, J. A., & King, R. A. (1993). Pedagogical content knowing: An

integrative model for teacher preparation. Journal of Teacher Education, 44, 263–272.

Davis, B., & Simmt, E. (2006). Mathematics-for-teaching: An ongoing investigation of the

mathematics that teachers (need to) know. Educational Studies in Mathematics, 61(3),

293–319.

Ernest, P. (1989). The knowledge, beliefs and attitudes of the mathematics teacher: A model.

Journal of Education for Teaching, 15(1), 13–33.

Even, R., & Ball, D. L. (Eds.). (2010). The professional education and development of

teachers of mathematics: The 15th ICMI study. New York, NY: Springer.

Fauconnier, G., & Turner, M. (2002). The way we think: Conceptual blending and the mind’s

hidden complexities. New York, NY: Basic Books.

Fennema, E., & Franke, M. (1992). Teachers‘ knowledge and its impact. In D. Grouws (Ed.),

Handbook of research on mathematics teaching and learning (pp. 147–163). New York,

NY: Macmillan.

Flores, E., Escudero, D., & Carrillo, J., (2013). A theoretical review of specialised content

knowledge. In B. Ubuz, Ç. Haser & M.Mariotti (Eds.), Proceedings of the Eighth

Congress of the European Society for Research in Mathematics Education (pp. 3055–

3064). Antalya, Turkey: ERME.

Font, V., Godino, J. D., & Gallardo, J. (2013). The emergence of objects from mathematical

practices. Educational Studies in Mathematics, 82, 97–124.

Godino, J. D., Batanero, C., & Font, V. (2007). The onto-semiotic approach to research in

mathematics education. ZDM – The International Journal on Mathematics Education,

39(1-2), 127–135.

Godino, J. D., Font, V., Wilhelmi, M. R., & Lurduy, O. (2011). Why is the learning of

elementary arithmetic concepts difficult? Semiotic tools for understanding the nature of

mathematical objects. Educational Studies in Mathematics, 77(2/3), 247–265.

Grossman, P. L., Wilson, S. M., & Shulman, L. S. (1989). Teachers of substance: Subject

matter knowledge for teaching. In M. C. Reynolds (Ed.), Knowledge base for the

beginning teacher (pp. 23–36). Elmsford, NY: Pergamon Press.

Hashweh, M. Z. (2005). Teacher pedagogical constructions: A reconfiguration of pedagogical

content knowledge. Teachers and Teaching, 11(3), 273–292.

Hodgen, J. (2011). Knowing and identity: A situated theory of mathematics knowledge in

teaching. In T. Rowland & K. Ruthven (Eds.), Mathematical knowledge in teaching (pp.

27–42). Dordrecht, The Netherlands: Springer.

Kaiser, G., Blömeke, S., König, J., Busse, A., Döhrmann, M., & Hoth, J. (2017). Professional

competencies of (prospective) mathematics teachers—cognitive versus situated

approaches. Educational Studies in Mathematics, 94, 161–182.

Kunter, M., Baumert, J., Blum, W., Klusmann, U., Krauss, S., & Neubrand, M. (Eds.). (2013).

Cognitive activation in the mathematics classroom and professional competence of

Dordrecht, The Netherlands: Kluwer.

Buchholtz, N., Leung, F. K., Ding, L., Kaiser, G., Park, K., & Schwarz, B. (2013). Future

mathematics teachers‘ professional knowledge of elementary mathematics from an

advanced standpoint. ZDM—The International Journal of Mathematics Education,

45(1), 107–120.

Carrillo, J., Climent, N., Contreras L. C., & Muñoz-Catalán, M. C. (2013). Determining

specialised knowledge for mathematics teaching. In B. Ubuz, Ç. Haser & M. Mariotti

(Eds.), Proceedings of the Eighth Congress of the European Society for Research in

Mathematics Education (pp. 2985–2994). Antalya, Turkey: ERME.

Cochran, K. F., DeRuiter, J. A., & King, R. A. (1993). Pedagogical content knowing: An

integrative model for teacher preparation. Journal of Teacher Education, 44, 263–272.

Davis, B., & Simmt, E. (2006). Mathematics-for-teaching: An ongoing investigation of the

mathematics that teachers (need to) know. Educational Studies in Mathematics, 61(3),

293–319.

Ernest, P. (1989). The knowledge, beliefs and attitudes of the mathematics teacher: A model.

Journal of Education for Teaching, 15(1), 13–33.

Even, R., & Ball, D. L. (Eds.). (2010). The professional education and development of

teachers of mathematics: The 15th ICMI study. New York, NY: Springer.

Fauconnier, G., & Turner, M. (2002). The way we think: Conceptual blending and the mind’s

hidden complexities. New York, NY: Basic Books.

Fennema, E., & Franke, M. (1992). Teachers‘ knowledge and its impact. In D. Grouws (Ed.),

Handbook of research on mathematics teaching and learning (pp. 147–163). New York,

NY: Macmillan.

Flores, E., Escudero, D., & Carrillo, J., (2013). A theoretical review of specialised content

knowledge. In B. Ubuz, Ç. Haser & M.Mariotti (Eds.), Proceedings of the Eighth

Congress of the European Society for Research in Mathematics Education (pp. 3055–

3064). Antalya, Turkey: ERME.

Font, V., Godino, J. D., & Gallardo, J. (2013). The emergence of objects from mathematical

practices. Educational Studies in Mathematics, 82, 97–124.

Godino, J. D., Batanero, C., & Font, V. (2007). The onto-semiotic approach to research in

mathematics education. ZDM – The International Journal on Mathematics Education,

39(1-2), 127–135.

Godino, J. D., Font, V., Wilhelmi, M. R., & Lurduy, O. (2011). Why is the learning of

elementary arithmetic concepts difficult? Semiotic tools for understanding the nature of

mathematical objects. Educational Studies in Mathematics, 77(2/3), 247–265.

Grossman, P. L., Wilson, S. M., & Shulman, L. S. (1989). Teachers of substance: Subject

matter knowledge for teaching. In M. C. Reynolds (Ed.), Knowledge base for the

beginning teacher (pp. 23–36). Elmsford, NY: Pergamon Press.

Hashweh, M. Z. (2005). Teacher pedagogical constructions: A reconfiguration of pedagogical

content knowledge. Teachers and Teaching, 11(3), 273–292.

Hodgen, J. (2011). Knowing and identity: A situated theory of mathematics knowledge in

teaching. In T. Rowland & K. Ruthven (Eds.), Mathematical knowledge in teaching (pp.

27–42). Dordrecht, The Netherlands: Springer.

Kaiser, G., Blömeke, S., König, J., Busse, A., Döhrmann, M., & Hoth, J. (2017). Professional

competencies of (prospective) mathematics teachers—cognitive versus situated

approaches. Educational Studies in Mathematics, 94, 161–182.

Kunter, M., Baumert, J., Blum, W., Klusmann, U., Krauss, S., & Neubrand, M. (Eds.). (2013).

Cognitive activation in the mathematics classroom and professional competence of

19.
teachers: Results from the COACTIV project. New York, NY: Springer.

Lerman, S. (2013). Theories in practice: Mathematics teaching and mathematics teacher

education. ZDM— The International Journal on Mathematics Education, 43(3), 623–

631.

Ma, L. (1999). Knowing and teaching elementary mathematics: Teachers’ understanding of

fundamental mathematics in China and the United States. Mahwah, NJ: Lawrence

Erlbaum.

Marks, R. (1990). Pedagogical content knowledge: From a mathematical case to a modified

conception. Journal of Teacher Education, 41(3), 3–11.

McEwan, H., & Bull, B. (1991). The pedagogic nature of subject matter knowledge. American

Educational Research Journal, 28(2), 316–334.

Monk, D. H. (1994). Subject area preparation of secondary mathematics and science teachers

and student achievement. Economics of Education Review, 13(2), 125–145.

Petrou, M., & Goulding, M. (2011). Conceptualising teachers‘ mathematical knowledge in

teaching. In T. Rowland & K. Ruthven (Eds.), Mathematical knowledge in teaching (pp.

9–25). Dordrecht, The Netherlands: Springer.

Pino-Fan, L., Assis, A., & Castro, W. F. (2015). Towards a methodology for the

characterization of teachers‘ didactic-mathematical knowledge. Eurasia Journal of

Mathematics, Science, & Technology Education, 11(6), 1429–1456.

Ponte, J. P. (1994). Mathematics teachers‘ professional knowledge. In J. P. Ponte & J. F.

Matos (Eds.), Proceedings of the 18th conference of the International Group for the

Psychology of Mathematics Education (Vol. 1, pp. 195–210). Lisbon, Portugal: PME.

Ponte, J. P., & Chapman, O. (2016). Prospective mathematics teachers‘ learning and

knowledge for teaching. In L. D. English & D. Kirshner (Eds.), Handbook of

international research in mathematics education (3rd ed., pp. 275–296). New York, NY:

Routledge.

Putnam, T. T., & Borko, H. (2000).What do new views of knowledge and thinking have to say

about research on teacher learning? Educational Researcher, 29(2), 4–15.

Rowland, T. (2009). Developing primary mathematics teaching: Reflecting on practice with

the knowledge quartet. London, UK: Sage.

Rowland, T. (2014). Frameworks for conceptualizing mathematics teacher knowledge. In S.

Lerman (Ed.), Encyclopedia of mathematics education (pp. 235–238). Dordrecht, The

Netherlands: Springer.

Rowland, T., Huckstep, P., & Thwaites, A. (2005). Elementary teachers‘ mathematics subject

knowledge: The knowledge quartet and the case of Naomi. Journal of Mathematics

Teacher Education, 8(3), 255–281.

Rowland, T., & Ruthven, K. (Eds.). (2011). Mathematical knowledge in teaching. Dordrecht,

The Netherlands: Springer.

Scheiner, T. (2015). Shifting the emphasis toward a structural description of (mathematics)

teachers‘ knowledge. In K. Bewick, T. Muir, & J. Wells (Eds.). Proceedings of the 39th

conference of the International Group for the Psychology of Mathematics Education

(Vol. 4, pp. 129–136). Hobart, Australia: PME.

Schoenfeld, A. H., & Kilpatrick, J. (2008). Toward a theory of proficiency in teaching

mathematics. In D. Tirosh & T. Wood (Eds.), International handbook of mathematics

teacher education. Vol. 2: Tools and processes in mathematics teacher education (pp.

321–354). Rotterdam, The Netherlands: Sense Publishers.

Schwab, J. J. (1978). Education and the structure of the disciplines. In I. Westbury & N. J.

Wilkof (Eds.), Science, curriculum, and liberal education (pp. 229–272). Chicago, IL:

University of Chicago Press. (Original work published 1961).

Lerman, S. (2013). Theories in practice: Mathematics teaching and mathematics teacher

education. ZDM— The International Journal on Mathematics Education, 43(3), 623–

631.

Ma, L. (1999). Knowing and teaching elementary mathematics: Teachers’ understanding of

fundamental mathematics in China and the United States. Mahwah, NJ: Lawrence

Erlbaum.

Marks, R. (1990). Pedagogical content knowledge: From a mathematical case to a modified

conception. Journal of Teacher Education, 41(3), 3–11.

McEwan, H., & Bull, B. (1991). The pedagogic nature of subject matter knowledge. American

Educational Research Journal, 28(2), 316–334.

Monk, D. H. (1994). Subject area preparation of secondary mathematics and science teachers

and student achievement. Economics of Education Review, 13(2), 125–145.

Petrou, M., & Goulding, M. (2011). Conceptualising teachers‘ mathematical knowledge in

teaching. In T. Rowland & K. Ruthven (Eds.), Mathematical knowledge in teaching (pp.

9–25). Dordrecht, The Netherlands: Springer.

Pino-Fan, L., Assis, A., & Castro, W. F. (2015). Towards a methodology for the

characterization of teachers‘ didactic-mathematical knowledge. Eurasia Journal of

Mathematics, Science, & Technology Education, 11(6), 1429–1456.

Ponte, J. P. (1994). Mathematics teachers‘ professional knowledge. In J. P. Ponte & J. F.

Matos (Eds.), Proceedings of the 18th conference of the International Group for the

Psychology of Mathematics Education (Vol. 1, pp. 195–210). Lisbon, Portugal: PME.

Ponte, J. P., & Chapman, O. (2016). Prospective mathematics teachers‘ learning and

knowledge for teaching. In L. D. English & D. Kirshner (Eds.), Handbook of

international research in mathematics education (3rd ed., pp. 275–296). New York, NY:

Routledge.

Putnam, T. T., & Borko, H. (2000).What do new views of knowledge and thinking have to say

about research on teacher learning? Educational Researcher, 29(2), 4–15.

Rowland, T. (2009). Developing primary mathematics teaching: Reflecting on practice with

the knowledge quartet. London, UK: Sage.

Rowland, T. (2014). Frameworks for conceptualizing mathematics teacher knowledge. In S.

Lerman (Ed.), Encyclopedia of mathematics education (pp. 235–238). Dordrecht, The

Netherlands: Springer.

Rowland, T., Huckstep, P., & Thwaites, A. (2005). Elementary teachers‘ mathematics subject

knowledge: The knowledge quartet and the case of Naomi. Journal of Mathematics

Teacher Education, 8(3), 255–281.

Rowland, T., & Ruthven, K. (Eds.). (2011). Mathematical knowledge in teaching. Dordrecht,

The Netherlands: Springer.

Scheiner, T. (2015). Shifting the emphasis toward a structural description of (mathematics)

teachers‘ knowledge. In K. Bewick, T. Muir, & J. Wells (Eds.). Proceedings of the 39th

conference of the International Group for the Psychology of Mathematics Education

(Vol. 4, pp. 129–136). Hobart, Australia: PME.

Schoenfeld, A. H., & Kilpatrick, J. (2008). Toward a theory of proficiency in teaching

mathematics. In D. Tirosh & T. Wood (Eds.), International handbook of mathematics

teacher education. Vol. 2: Tools and processes in mathematics teacher education (pp.

321–354). Rotterdam, The Netherlands: Sense Publishers.

Schwab, J. J. (1978). Education and the structure of the disciplines. In I. Westbury & N. J.

Wilkof (Eds.), Science, curriculum, and liberal education (pp. 229–272). Chicago, IL:

University of Chicago Press. (Original work published 1961).

20.
Sherin, M. G. (2002). When teaching becomes learning. Cognition and Instruction, 20(2),

119–150.

Shulman, L. S. (1986). Those who understand: Knowledge growth in teaching. Educational

Researcher, 15(2), 4–14.

Shulman, L. S. (1987). Knowledge and teaching: Foundations of the new reform. Harvard

Educational Review, 57, 1–22.

Silverman, J., & Thompson, P. W. (2008). Toward a framework for the development of

mathematical knowledge for teaching. Journal of Mathematics Teacher Education,

11(6), 499–511.

Sullivan, P., & Wood, T. (Eds.). (2008). The international handbook of mathematics teacher

education. Volume 1: Knowledge and beliefs in mathematics teaching and teaching

development. Rotterdam, The Netherlands: Sense.

Van Zoest, L., & Thames, M. (2013). Building coherence in research on mathematics teacher

characteristics by developing practice-based approaches. ZDM—The International

Journal on Mathematics Education, 45(4), 583–594.

119–150.

Shulman, L. S. (1986). Those who understand: Knowledge growth in teaching. Educational

Researcher, 15(2), 4–14.

Shulman, L. S. (1987). Knowledge and teaching: Foundations of the new reform. Harvard

Educational Review, 57, 1–22.

Silverman, J., & Thompson, P. W. (2008). Toward a framework for the development of

mathematical knowledge for teaching. Journal of Mathematics Teacher Education,

11(6), 499–511.

Sullivan, P., & Wood, T. (Eds.). (2008). The international handbook of mathematics teacher

education. Volume 1: Knowledge and beliefs in mathematics teaching and teaching

development. Rotterdam, The Netherlands: Sense.

Van Zoest, L., & Thames, M. (2013). Building coherence in research on mathematics teacher

characteristics by developing practice-based approaches. ZDM—The International

Journal on Mathematics Education, 45(4), 583–594.