What Makes Mathematics Teacher Knowledge Specialized?

Contributed by:
Sharp Tutor
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,
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.
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)
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
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,
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
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
 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.
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,
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.
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
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.
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,
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
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.
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).
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.
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