Encyclopedia of Mathematics Education

Mathematics Teacher Educator as Learner M433

teachers’ learning can take place in their class- version of action research by teachers is practiced M
rooms influenced by interventions of their in Japan within the framework of “lesson study”
colleagues or often – as research shows – by (see, e.g., Hart et al. 2011). In general, teacher
their own interventions (e.g., see Chapman educators who participate directly or indirectly in
2008) or in the field where it does not only such cases of teachers’ action research are
apply knowledge that has been generated within afforded opportunities to learn in and from these
the university, but much more, it generates “local experiences.
knowledge” that could not be generated outside
the practice. Thus, this kind of research is mostly Cross-References
process oriented and context bounded, generated
through continuous interaction and communica- ▶ Education of Mathematics Teacher Educators
tion with practice. Intervention research tries to ▶ Inquiry-Based Mathematics Education
overcome the institutionalized division of labor
between science and practice. It aims both References
at balancing the interests in developing and
understanding and at balancing the wish to par- Adler J, Ball D, Krainer K, Lin F-L, Novotna´ J (2005)
ticularize and generalize. Action research as Mirror images of an emerging field: researching math-
intervention research done by practitioners them- ematics teacher education. Educ Stud Math
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action research, see, e.g., Elliott 1991). Teachers investigate their work. An introduction to
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Worldwide, there is an increasing number of Routledge, London/New York. [German original:
initiatives in mathematics education based on Altrichter H, Posch P (1990) Lehrer erforschen ihren
action research or intervention research. How- Unterricht. Klinkhardt, Bad Heilbrunn. Chinese trans-
ever, most of them are related to teachers’ action lation 1997, Yuan-Liou, Taipeh]
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Adler 1996; papers in JMTE 6(2) and 9(3); Benke Benke G, Hospesova´ A, Ticha´ M (2008) The use of
et al. 2008; Kieran et al. 2013). In some cases, action research in teacher education. In: Krainer K,
even the traditional role names (teachers vs. Wood T (eds) International handbook of mathemat-
researchers) are changed in order to express that ics teacher education, vol 3, Participants in mathe-
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Mathematics (LCM) project (Jaworski et al.
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(“both of whom are also researchers”). There are a cognitive perspective. In: Jaworski B, Wood
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their (evidence-based) experiences in reflecting education, vol 4, The mathematics teacher educator as
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papers – written by teachers for teachers – have pp 110–129
been gathered since the 1980s within the context
of programs like PFL (see, e.g., Krainer 1998) Chapman O (2011) Elementary school teachers’ growth in
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The most extensive and nationally widespread Cochran-Smith M (2003) Learning and unlearning: the
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19:5–28

Crawford K, Adler J (1996) Teachers as researchers in
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Keitel C, Kilpatrick J, Laborde C (eds) International
handbook of mathematics education (part 2). Kluwer,
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Elliott J (1991) Action research for educational change.
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M 434 Mathematics Teacher Identity

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of mathematics teacher-educators: growth through
Even R, Ball DL (eds) (2009) The professional education practice. J Math Teach Educ 7(1):5–32
and development of teachers of mathematics – the 15th
ICMI study. Springer, New York Mathematics Teacher Identity

Hart LC, Alston A, Murata A (eds) (2011) Lesson study Mellony Graven1 and Stephen Lerman2
research and practice in mathematics education. 1Rhodes University, Grahamstown, South Africa
Learning together. Springer, Dordrecht 2Department of Education, Centre for
Mathematics Education, London South Bank
Jaworski B, Wood T (eds) (2008) International handbook University, London, UK
of mathematics teacher education, vol 3, The mathe-
matics teacher educator as a developing professional. Keywords
Sense, Rotterdam
Teacher identity; Specialisation; Pedagogic
Jaworski B, Fuglestad A-B, Bjuland R, Breiteig T, identity
Goodchild S, Grevholm B (2007) Learning communi-
ties in mathematics. Caspar Forlag, Bergen Definition

Kieran C, Krainer, K & Shaughnessy, JM (2013) Linking Mathematics teacher identity (MTI) is commonly
research to practice: teachers as key stakeholders in “defined” or conceptualized in recent publica-
mathematics education research. In: Clements MA, tions of the mathematics education research
Bishop AJ, Keitel C, Kilpatrick J, Leung FKS (eds) community as ways of being, becoming, and
Third international handbook of mathematics educa- belonging, as developing trajectories, and in
tion. Springer, New York, pp 361–392 narrative and discursive terms.

Krainer K (1998) Some considerations on problems and Characteristics
perspectives of mathematics teacher in-service educa-
tion. In: Alsina C, Alvarez JM, Hodgson B, Laborde C, A Brief History
Perez A (eds) The 8th international congress on math- The concept of identity can be traced to Mead
ematical education (ICME 8), selected lectures. S.A.E. (1934) and Erikson (1968), the former seeing
M. Thales, Sevilla, pp 303–321 identity as developed in interaction with the envi-
ronment, and thus multiple, though it appears
Krainer K (2003) Teams, communities & networks. more unified to the individual (Lerman 2012).
J Math Teac Educ 6(2):93–105. Editorial The latter saw identity as something that
develops throughout one’s life and is seen as
Krainer K, Llinares S (2010) Mathematics teacher educa- more unified. The study of teacher identity is
tion. In: Peterson P, Baker E, McGaw B (eds) Interna- more recent. Perspectives focus on images of
tional encyclopedia of education, vol 7. Elsevier, self (Nias 1989) as determining how teachers
Oxford, pp 702–705 develop or on roles (Goodson and Cole 1994).
One can argue that societal expectations and
Krainer K, Zehetmeier S (2013) Inquiry-based learning
for students, teachers, researchers, and representatives
of educational administration and policy: reflections
on a nation-wide initiative fostering educational inno-
vations. ZDM Int J Math Educ 45(6):875–886.

Llinares S, Krainer K (2006) Mathematics (student)
teachers and teacher educators as learners. In: Gutie´r-
rez A, Boero P (eds) Handbook of research on the
psychology of mathematics education. Past, present
and future. Sense, Rotterdam, pp 429–459

Pegg J, Krainer K (2008) Studies on regional and national
reform initiatives as a means to improve mathematics
teaching and learning at scale. In: Krainer K, Wood
T (eds) International handbook of mathematics teacher
education, vol 3, Participants in mathematics teacher
education: individuals, teams, communities and net-
works. Sense, Rotterdam, pp 255–280

Russell T, Korthagen F (eds) (1995) Teachers who teach
teachers: reflections on teacher education. Falmer
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Scho¨n D (1983) The reflective practitioner: how profes-
sionals think in action. Temple-Smith, London

Swennen A, van der Klink M (eds) (2009) Becoming
a teacher educator. Springer, Dordrecht

Mathematics Teacher Identity M435

perceptions and at the same time the teacher’s relation to mathematics teacher support (e.g., M
own sense of what matters to them play key roles Hodgen and Askew 2007; Lerman 2012). In con-
in teachers’ professional identity. Beijaard et al. texts where mathematics teacher morale is low
(2004) argue, in their review, that 1988 saw the and teacher identities are portrayed as mathemat-
emergence of teacher identity as a research field. ically deficient (supported by poor results on
Perhaps the key impetus for this focus on identity international studies such as TIMMS), this frame-
in mathematics education can be attributed to work opens the space for teacher education to
Jean Lave and Etienne Wenger who wrote, “We focus explicitly on the reauthoring of negative
have argued that, from the perspective we have and damaging narratives (e.g., Graven 2012).
developed here, learning and a sense of identity
are inseparable: They are aspects of the same Clusters of issues illuminated in recent
phenomenon.” (Lave and Wenger 1991, p. 115). published mathematics teacher education research
include:
Research on teacher identity has gained • Discipline specialization is widely considered
prominence over the past decade. Special issues
of teacher education journals focusing on to be highly significant in teacher identity
teacher identity attest to this (e.g., Teaching and both generally and in mathematics teacher
Teacher Education 21, 2005; Teacher Education research specifically (Hodgen and Askew
Quarterly, June 2008). Related issues revealed 2007). Relating teacher identity and teacher
across the literature include “the problem of emotion is argued by some to be particularly
defining the concept; the place of the self, and important in relation to mathematics teacher
related issues of agency, emotion, narrative and identity where many teachers teach the
discourse; the role of reflection; and the influence subject without disciplinary specialization in
of contextual factors” (Beauchamp and Thomas their teacher training and with histories of
2009, p. 175). As Grootenboer and Zevenbergen negative experiences of learning mathematics
(2008, p. 243) argue, in relation to understanding within their own schooling (Graven 2004;
the learning of mathematics, identity is “a unify- Hodgen and Askew 2007; Grootenboer and
ing and connective concept that brings together Zevenbergen 2008; Lerman 2012). For other
elements such as life histories, affective qualities literature outside of mathematics education
and cognitive dimensions.” that connects teacher identity with researching
teacher emotions, see the special issue no. 21
Despite increasing engagement with mathe- of Teaching and Teacher Education which
matical learning and identity, many have argued focused on this relationship (e.g., Zembylas
that researchers working with the notion of 2005).
mathematics teacher identity have not clearly • Research into mathematics teacher identities
defined and operationalized the notion (e.g., often deals separately with primary
Sfard and Prusak 2005). MTI is increasingly nonspecialist teachers, who teach across sub-
accepted as a dynamic rather than a fixed con- jects, and with secondary teachers, who teach
struct even while debates continue as to whether only or predominantly mathematics and who
an individual has one identity with multiple may or may not have specialized in mathemat-
aspects or multiple identities (see Grootenboer ics in their preservice education. The nature of
and Ballantyne 2010). These post-structuralist the way in which the discipline specificity
interpretations of identity are more “contingent of mathematics influences evolving teacher
and fragile than previously thought and thus open identities differs in relation to whether one
to reconstruction” (Zembylas 2005, p. 936). The is identified as a generalist teacher or
agency of the teacher to reconstruct or reauthor a mathematics teacher. While it can be inter-
her story in relation to participation within class- nationally accepted that many more secondary
room, pre- and in-service practices in which mathematics teachers have discipline specific
teachers participate, is a central opportunity for training in their preservice studies, the extent
much of the literature focused on identity in to which this is the case differs across

M 436 Mathematics Teacher Identity

countries. As Grootenboer and Zevenbergen • A growing area of research in MTI explores
(2008) point out, depending on the extent the relationship between mathematics
of the shortage of qualified mathematics “teacher change” or teacher learning and rad-
teachers, secondary school mathematics clas- ical curriculum change (e.g., Schifter 1996;
ses are often taught by nonspecialist teachers. Van Zoest and Bohl 2005). Similarly research
Shortage of qualified mathematics teachers is beginning to investigate the relationship
can be particularly high for developing coun- between teacher identity/positions and the
tries (for example in South Africa). In this increasing use of national standardized assess-
respect research into supporting such teachers ments across various contexts (Morgan et al.
to strengthen their mathematics teacher 2002). As in Lasky’s (2005) research across
identities becomes important. Graven (2004) subject teachers this research often points
engages with how teachers in a longitudinal to disjuncture between mathematics teacher
mathematics education in-service program identities and expectations (and contradictory
transformed their identities from accidental messages) of reform mandates and to the ways
“teachers of mathematics” (they specialized in which these constrain teacher identities.
in other subjects in their training) to “profes- As Wenger (1998) notes, national education
sional mathematics teachers” with long term departments can design roles but they cannot
trajectories of further learning and increasing design (local) identities of teachers.
specialization in the subject. Research also In this respect the work of Bernstein becomes
tends to deal separately with either preservice
or in-service teachers as the way in which useful in providing a more macro perspective on
identities emerge for these groups of teachers teacher identity and the way policy, curriculum,
differs in relation to the different practices in and assessment practices shape this. His work com-
which they participate. plements localized case study analyses of identity
• Mathematics teacher identity is also increas- within teacher communities with a broader concept
ingly being considered in relation to studies of identity connected to macro structures of power
researching mathematics teacher retention. and control. Bernstein’s model (1996) shows:
The ICME-12 Discussion Group (DG11)
on teacher retention included as a key how the distribution of power and principles of
theme the notion of identity and mathemat- control translate into pedagogic codes and their
ics teacher retention and several of the modalities. I have also shown how these codes are
papers presented in this DG highlighted the acquired and so shape consciousness. In this way,
role of strengthened professional identities, a connection has been made between macro struc-
increasing sense of belonging and develop- tures of power and control and the micro process of
ing leadership identities and trajectories the formation of pedagogic consciousness. (p. 37).
as enabling factors contributing to teacher
retention. Presenters in this discussion Bernstein first introduced the concept of iden-
group were from the USA, South Africa, tity in 1971 (Bernstein and Solomon 1999). This
Israel, New Zealand, Norway, and India. analysis did not focus on identity in terms of
Research on mathematics teacher identity regulation and realization in practice but rather
seems to be of particular interest in these on identity in terms of the “construction of
countries as well as in the UK and Australia identity modalities and their change within an
(see reference list). Similarly research into institutional level” (p. 271). Thus Bernstein
the relationship between teacher identity approaches identity from a broader systemic
and sustaining commitment to teaching level, which of course impacts on enabling and
(more generally than only for mathematics constraining the emergence of localized individual
teaching) has been argued across US and teacher identities. Bernstein’s notion of “Projected
Australian contexts (e.g., Day et al. 2005). Pedagogic Identities” (Bernstein and Solomon
1999) provides a way of analyzing macro pro-
moted identities within contemporary curriculum
change which is the context within which teacher

Mathematics Teacher Identity M437

roles are elaborated in curriculum documents. Erikson EH (1968) Identity, youth and crisis. Norton, M
South African and British Mathematics Educators New York
have particularly drawn on the work of Bernstein
to analyze positions available to teachers within Goodson IF, Cole AL (1994) Exploring the teacher’s
often contradictory and shifting “official” dis- professional knowledge: constructing identity and
courses (see, e.g., Parker and Adler 2005; Graven community. Teach Educ Q 21(1):85–105
2002; Morgan et al. 2002).
Graven M (2002) Coping with New Mathematics Teacher
Conclusion roles in a contradictory context of curriculum change.
Math Educ 12(2):21–27
The references seem to suggest that “identity” as
an alternative way of identifying teacher learning Graven M (2004) Investigating mathematics teacher
is not necessarily a global perspective. Research learning within an in-service community of practice:
on international interpretations of the relevance of the centrality of confidence. Educ Stud Math
the notion is needed. At the same time the notion is 57:177–211
ubiquitous in the social sciences, and mathematics
education researchers working with “identity” Graven M (2012) Changing the story: teacher education
need to specify how they are using the term, through re-authoring their narratives. In: Day C (ed)
what the sources are for their perspectives, and The Routledge international handbook of teacher
the relevance for the teaching and learning of and school development. Routledge, Abingdon,
mathematics. pp 127–138

Cross-References Grootenboer P, Ballantyne J (2010) Mathematics
teachers: negotiating professional and discipline iden-
▶ Communities of Practice in Mathematics tities. In: Sparrow L, Kissane B, Hurst C (eds) Shaping
Teacher Education the future of mathematics education: proceedings of
the 33rd annual conference of the Mathematics
▶ Teacher Beliefs, Attitudes, and Self-Efficacy Education Research Group of Australasia. Fremantle,
in Mathematics Education MERGA, pp 225–232

References Grootenboer P, Zevenbergen R (2008) Identity as a lens to
understanding learning mathematics: developing
Beauchamp C, Thomas L (2009) Understanding teacher a model. In Goos M, Brown R, Makar K (eds) Pro-
identity: an overview of issues in the literature and ceedings of the 31st annual conference of the Mathe-
implications for teacher education. Cam J Educ matics Education Research Group of Australasia.
39(2):175–189 MERGA, Australia, pp 243–249

Beijaard D, Meijer PC, Verloop N (2004) Reconsidering Hodgen J, Askew M (2007) Emotion, identity and learn-
research on teachers’ professional identity. Teach ing: becoming a primary mathematics teacher. Oxf
Teach Educ 20:107–128 Rev Educ 33(4):469–487

Bernstein B (1996) Pedagogy, symbolic control and iden- Lasky S (2005) A sociocultural approach to understanding
tity: theory, research, critique. Taylor and Frances, teacher identity, agency and professional vulnerability
London in a context of secondary school reform. Teach Teach
Educ 21:899–916
Bernstein B, Solomon J (1999) Pedagogy, identity and the
construction of a theory of symbolic control: basil Lave J, Wenger E (1991) Situated learning: legitimate
Bernstein questioned by Joseph Solomon. Br J Soc peripheral participation. Cambridge University Press,
Educ 20(2):265–279 New York

Day C, Elliot B, Kington A (2005) Reform, standards and Lerman S (2012) Agency and identity: mathematics
teacher identity: challenges of sustaining commitment. teachers’ stories of overcoming disadvantage. In:
Teach Teach Educ 21:563–577 Tso T-Y (ed) Proceedings of the 36th conference
of the international group for the psychology of
mathematics education, vol 3. Department of
Mathematics National Taiwan University, Taiwan,
pp 99–106

Mead GJ (1934) Mind, self and society. University of
Chicago Press, Chicago

Morgan C, Tsatsaroni A, Lerman S (2002) Mathematics
teachers’ positions and practices in discourses of
assessment. Br J Soc Educ 23(3):445–461

Nias J (1989) Teaching and the self. In: Holly ML,
McLoughlin CS (eds) Perspective on teacher
professional development. Falmer Press, London,
pp 151–171

Parker D, Adler J (2005) Constraint or catalyst? The
regulation of teacher education in South Africa.
J Educ, 36: 59–78

M 438 Mathematics Teachers and Curricula

Schifter D (ed) (1996) What’s happening in the math (Clandinin and Connelly 1992). These different
class? vol 2, Reconstructing professional identities. meanings have defined several roles of teachers in
Teachers College Press, New York mathematics curriculum development. Regarding
these roles, the relationship between teachers and
Sfard A, Prusak A (2005) Telling identities: in search of an curricula can be described as the history of a shift
analytic tool for investigating learning as a culturally from teachers as curriculum users to teachers with
shaped activity. Educ Res 34(4):14–22 roles as curriculum interpreters and/or curriculum
makers. Whereas the former view assumes curric-
Van Zoest L, Bohl J (2005) Mathematics teacher identity: ula to be “teacher-proof,” the latter includes
a framework for understanding secondary school teachers’ activities like reflecting, negotiating issues
mathematics teachers’ learning through practice. of curricula, and disseminating to their peers. This
Teach Dev 9(3):315–345 shift mirrors acknowledgement of the centrality of
the teacher in curricula issues in particular
Wenger E (1998) Communities of practice: learning, (Clarke et al. 1996; Hershkowitz et al. 2002;
meaning, and identity. Cambridge University Press, Lappan et al. 2012) and viewing teachers as key
New York stakeholders of educational change in general
(Krainer 2011). These meanings are located along
Zembylas M (2005) Discursive practices, genealogies, a continuum from a view of curricula as fixed,
and emotional rules: a poststructuralist view on embodying discernible and complete images of
emotion and identity in teaching. Teach Teach Educ practice, to a view of curricula guidelines as possible
21:935–948 influencing forces in the construction of practice.

Mathematics Teachers and Curricula In the 1970s, Stenhouse (1975) defined curric-
ulum as “an attempt to communicate the essential
Salvador Llinares1, Konrad Krainer2 and principles and features of an educational proposal
Laurinda Brown3 in such a form that it is open to critical scrutiny
1Facultad de Educacio´n, University of Alicante, and capable of effective translation into practice”
Alicante, Spain (p. 4). The teacher is central to this translation into
2School of Education, Alpen-Adria-Universit€at practice. A model that is commonly used for anal-
Klagenfurt, Klagenfurt, Austria ysis in mathematics education sees curricula as
3Graduate School of Education, University of located at three levels: the intended curriculum (at
Bristol, Bristol, UK the system level, the proposal), the implemented
curriculum (at the class level, the teacher’s role),
Keywords and the attained curriculum (at the student level,
the learning that takes place) (Clarke et al. 1996).
Collaborative work; Curricula development;
Mathematics teachers; Production of materials; Focusing on the implemented curriculum,
Text books; Professional development of Stenhouse began the “teachers as researchers”
teachers movement. He believed that the “development
of teaching strategies can never be a priori.
Definition and Historical Background New strategies [principled actions] must be
worked out by groups of teachers collaborating
The word curriculum has had several meanings within a research and development framework
over time and has been interpreted broadly not [. . .] grounded in the study of classroom practice”
only as a project about what should be learned (p. 25). The development of this idea in the
by students but, in the context of teachers and mathematics education field illustrated the com-
curriculum, as all the experiences which occur plexity of teaching and the key roles played by
within a classroom. These different meanings are teachers in students’ learning, underlining the
grounded in different assumptions about teaching importance of teachers’ processes of interpreta-
and the nature of interactions of the teacher tion of curricula materials (Zack et al. 1997).
with ideas that support curriculum guidelines

Mathematics Teachers and Curricula M439

Different Cultures Shaping Different Forms of a long tradition of teachers developing curricula M
Interaction Between Teachers and Curricula materials in collaborative groups.
The relation between teacher and curricula
depends on internal and external influences. For In the United Kingdom in the late 1970s and
example, teachers frame their approach to early 1980s, Philip Waterhouse’s research (2001,
curricula differently dependent on their concep- updated by Chris Dickinson), supported by the
tions of different components of curricula (Lloyd Nuffield Foundation, led to the founding of
1999) and/or through the different structures of a number of curricula development organizations
professional development initiatives. Locally, called Resources for Learning Development
teachers’ knowledge and pedagogical beliefs are Units. In these units, the mathematics editor
influences as they engage with curricula mate- (one of a cross-curricular team of editors) worked
rials. Furthermore, the content and form of cur- with groups of not more than ten teachers,
ricula materials influence the ways in which facilitating their work on either developing mate-
teachers interpret, evaluate, and adapt these rials related to government initiatives or from
materials considering their students’ responses perceived needs of teachers themselves. The
and needs in a specific institutional context. explicit focus for the teachers was on the
Globally, countries have different curricular tra- development and then production of materials
ditions shaping different conditions for teachers’ that had been tried out in their classrooms.
roles in curriculum development. Thus, the diver- However, the implicit focus of the editor was on
sity of cultures and features of each country’s the professional development of those teachers in
system generates different modes of interaction the groups. It was still important for the materials
between teachers and curricula, as well as differ- to be designed, printed, and distributed.
ent needs and trends in teacher professional
development related to curriculum reform Also, in France, since the 1970s, the IREM
(Clarke et al. 1996). The main elements which network has functioned on the basis of mixed
have been proved to affect the relation between groups of academics, mathematicians, and
teachers and curricula are, for instance, the dis- teachers inquiring, experimenting in classrooms,
tance that usually exists between the intended producing innovative curriculum material, and
curriculum and the implemented curriculum; organizing teacher professional development
whatever the level of detail and prescription sessions relying on their experience (cf. www.
of the curriculum description, for years after univ-irem.fr/ ). In recent views of how teachers
curriculum reform, the implemented curriculum interact with, draw on, refer to, and are
remains a subtle composition of the old and the influenced by curricula resources, teachers are
new, differing between one teacher and another. challenged to express their professional knowl-
In this sense, curricula are related with teacher edge keeping a balance between the needs of
practice, and curricula change is linked to their specific classrooms and their conceptions.
how teachers continuously further develop or In many countries, as mathematics education
change their current practice, in particular with research has matured, there is increasingly
regard to teaching and assessment and profes- development of curricula materials by teachers
sional development initiatives (Krainer and themselves working collaboratively in groups,
Llinares 2010). possibly in association with researchers, and
the organization of teacher professional develop-
Teachers and Curricula Within a Collaborative ment around such collaborative actions based on
Perspective a more developed idea of the teacher.
From this view of interaction between teacher and
curriculum, curricula development initiatives are Barbara Jaworski, working in Norway from
a context for teacher professional development 2003 to 2010, has led research projects in part-
reconstructing wisdom through inquiry. There is nership with teachers to investigate “Learning
Communities in Mathematics” and “Teaching
Better Mathematics” (see, e.g., Kieran et al.
2013). In Canada, led by Michael Fullan, there is

M 440 Mathematics Teachers and Curricula

a large-scale project supporting professional devel- focus was initiating purposeful pedagogical
opment of teachers through curriculum reform in change through involving teachers in rich
literacy and numeracy based on in-school collabo- professional learning experiences. The motiva-
rative groupings of teachers attending a central tion for these initiatives was a perceived
“fair” to present their inquiry work once a year. deficiency in students’ knowledge of mathemat-
This project, Reach Every Student, energizing ics (and science) understood as the attained
Ontario Education, works on the attained curricu- curriculum. In all of these programs, collabora-
lum through the implemented one and has led to tion, communication, and partnerships played
Fullan’s (2008) book Six Secrets of Change. a major role among teachers and university staff
members of the program and within these groups.
More recently, with the spread of ideas through In these programs, the teachers were seen not
international conferences, meetings, and research only as participants but crucial change agents
collaborations, ideas such as the Japanese “lesson who were regarded as collaborators and
study” have spread widely (Alston 2011). Lesson experts (Pegg and Krainer 2008). This view of
study is a professional development process that teachers as change agents emerged from the close
teachers engage in to systematically examine their collaboration among groups of stakeholders
practice. It is considered to be a means of supporting and the different forms of communications that
the dissemination of documents like standards, developed.
benchmarks, and nationally validated curricula.
Other collaborative groups take many forms, fre- Open Questions
quently facilitated by a university academic or The relationship between teacher and curricula
sometimes with university mathematicians and defines a set of open questions in different
mathematics teacher educators forming part of the realms. These questions are linked to the fact
group. These multiple views define distinctive that the relationship between teachers and
professional development pathways through curric- curricula is moving, due to a diversity of
ula reforms. These pathways influence teachers’ factors: the increasing autonomy and power
professional identities and work practices. given to teachers regarding curriculum design
and implementation in some countries at least,
Social perspectives on the role of teachers in the development of collaborative practices
curricula reforms are being reported by Kieran and networks in teachers’ communities, the
and others (2013), where the major focus is on the evolution of relationships between researchers
role and nature of teachers’ interactions within and teachers, the explosion of curriculum
a group of teachers. From this perspective, teachers resources and their easier accessibility thanks
are motivated by collaborative inquiry activities to the Internet. . . . So, some open questions
(teams, communities, and networks) aiming at are:
interpreting and implementing curricula materials, 1. What are the implications of the school-based
“participation with.” How do teachers actively
engage or collaborate with curricular resources partial transfer of power in curriculum
(Remillard 2005)? How do teachers collaborate decision-making to teachers based on
with other groups of participants (Pegg and Krainer teachers’ practical, personal reflective experi-
2008)? Both engagements must be understood in ence and networks?
light of their particular local and global contexts. 2. What role do collegial networks play in how
ideas about curricula change are shared (e.g.,
Teachers’ learning through collaborative using electronic communications, practical
inquiry activities, contextualized in curriculum coaching)?
development initiatives, has allowed the contex- 3. How are new kinds of practices and teaching
tual conditions in which curriculum is objectives emerging as a consequence of new
implemented in different traditions to be made resources influencing the relation between
explicit. Pegg and Krainer (2008) reported exam- teacher and curricula?
ples of large-scale projects involving national
reform initiatives in mathematics where the

Mathematization as Social Process M441

4. How can reform initiatives cope with the Kieran C, Krainer K, Shaughnessy JM (2013) M
balance between national frameworks for Linking research to practice: teachers as key
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views on curricula as negotiated between the J, Leung FKS (eds) Third international handbook
teachers of one school? of mathematics education. Springer, New York,
pp 361–392
5. What role students play in bringing in ideas
related to curricula (e.g., starting topics based Krainer K (2011) Teachers as stakeholders in mathematics
on students’ interests, questions)? education research. In: Ubuz B (ed) Proceedings of
the 35th conference of the international group for the
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Keywords
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Hadas N, Resnick T, Tabach M, Schwarz B (2002) Mathematization; Demathematization; Mathe-
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international research in mathematics education. Critical mathematics education
Lawrence Erlbaum Associates, Mahwah, pp 657–694

M 442 Mathematization as Social Process

Definition characterize the mathematization of society in
the following way: “Mathematics has penetrated
Mathematization refers to the formatting of pro- many parts of our lives. It has capitalised on its
duction, decision-making, economic manage- abstract consideration of number, space, time,
ment, means of communication, schemes for pattern, structure, and its deductive course of
surveilling and control, war power, medical tech- argument, thus gaining an enormous descriptive,
niques, etc. by means of mathematical insight and predictive and prescriptive power” (p. 19).
techniques.
However, most often the mathematics that is
Mathematization provides a particular chal- brought into action is operating beneath the
lenge for mathematics education as it becomes surface of the practice. At the supermarket there
important to develop a critical position to math- is no mathematics in sight. In this sense,
ematical rationality as well as new approaches to as also emphasized by Jablonka and Gellert
the construction of meaning. (2007), a demathematization is accompanying
a mathematization.

Characteristics There Is Mathematics Everywhere
Mathematization and the accompanying
Mathematization and Demathematization demathematization have a tremendous impact on
The notions of mathematization and demathema- all forms of practices. Mathematics-based technol-
tization, the claim that there is mathematics ogy is found everywhere.
everywhere, and mathematics in action are
addressed, before we get to the challenges that One can see the modern computer as
mathematics education is going to face. a materialized mathematical construct. Certainly
the computer plays a defining part of a huge range
It is easy to do shopping in a supermarket. of technologies. It is defining for the formation of
One puts a lot of things into the trolley and pushes databases and for the processing of information
it to the checkout desk. Here an electronic device and knowledge.
used by the cashier makes a pling-pling-pling
melody, and the total to be paid is shown. Processes of production are continuously taking
One gets out the credit card, and after a few new forms due to new possibilities for automatiza-
movements by the fingers, one has bought what- tion, which in turn can be considered a materialized
ever. No mathematics in this operation. mathematical algorithm. Any form of production –
being of TV sets, mobile phones, kitchen utensils,
However, if we look at the technologies that cars, shoes, whatever – represents a certain com-
are configuring the practice of shopping, one position of automatic processes and manual labor.
finds an extremely large amount of advanced However, this composition is always changing due
mathematics being brought in operation: The to new technologies, new needs for controlling
items are coded and the codes are read mechan- the production process, new conditions for
ically; the codes are connected to a database outsourcing, and new salary demands. Crucial for
containing the prices of all items; the prices are such changes is not only the development of
added up; the credit card is read; the amount is mathematics-based technologies of automatization
subtracted from the bank account associated to but also of mathematics-based procedures for
the credit card; security matters are observed; decision-making.
schemes for coding and decoding are taking
place. In general mathematical techniques have
a huge impact on management and
We have to do with a mathematized daily decision-making. As an indication, one can
practice, and we are immersed in such practices. think of the magnitude of cost-benefit analyses.
We live in a mathematized society (see Such analyses are crucial, in order not only to
Keitel et al. 1993, for an initial discussion of identify new strategies for production and
such processes). Gellert and Jablonka (2009) marketing but to decision-making in general.

Mathematization as Social Process M443

Complex cost-benefit analyses depend on the be used for interpreting processes of mathemati- M
calculation power that can be executed by the zation (see, for instance, Christensen et al. 2009;
computer. The accompanying assumption is that Skovsmose 2009, 2010, in print; Yasukawa et al.
a pro et contra argumentation can be turned into in print). Mathematics in action can be character-
a straightforward calculation. This approach to ized in terms of the following issues:
decision-making often embraces an ideology of
certainty claiming that mathematics represents Technological imagination refers to the con-
objectivity and neutrality. Thus in decision- ceptualization of technological possibilities. We
making we find an example not only of a broad can think of technology of all kinds: design and
application of mathematical techniques but also construction of machines, artifacts, tools, robots,
an impact of ideological assumptions associated automatic processes, networks, etc.; decision-
with mathematics. making concerning management, advertising,
investments, etc.; and organization with respect
Mathematics-based technologies play crucial to production, surveillance, communication,
roles in different domains, and we can think money processing, etc. In all such domains math-
of medicine as an example. Here we find ematics-based technological imagination has
mathematics-based technologies for making diag- been put into operation. A paradigmatic example
noses, for defining normality, for conducting is the conceptualization of the computer in terms
a treatment, and for completing a surgical operation. of the Turing machine. Even certain limits of
Furthermore, the validation of medical research is computational calculations were identified before
closely related to mathematics. Thus any new type any experimentation was completed. One can
of medical treatment needs to be carefully also think of the conceptualization of the Internet,
documented, and statistics is crucial for doing this. of new schemes for surveilling and robotting
(see, for instance, Skovsmose 2012), and of new
Not only medicine but also modern warfare is approaches in cryptography (see, for instance,
mathematized. As an example one can consider Skovsmose and Yasukawa 2009). In all such
the drone, the unmanned aircraft, which has been cases mathematics is essential for identifying
used by the USA, for instance, in the war in new possibilities.
Afghanistan. The operation of the drone includes
a range of mathematics brought in action. Hypothetical reasoning addresses conse-
The identification of a target includes complex quences of not-yet-realized technological con-
algorithms for pattern recognition. The operation structions and initiatives. Reasoning of the form
of a drone can only take place through the most “if p then q, although p is not the case” is essential
sophisticated channels of communication, which to any kind of technological enterprise. Such hypo-
in turn must be protected by advanced cryptogra- thetical reasoning is most often model based: one
phy. Channels of communication as well as tries to grasp implications of a new technological
cryptography are completely mathematized. The construct by investigating a mathematical repre-
decision of whether to fire or not is based on cost- sentation (model) of the construct. Hypothetical
benefit analyses: Which target has been identi- reasoning makes part of decision-making about
fied? How significant is the target? What is the where to build an atomic power plant, what
probability that the target has been identified investment to make, what outsourcing to
correctly? What is the probability that other peo- make, etc. In all such cases one tries to provide
ple might be killed? What is the price of the a forecasting and to investigate possible
missile? Mathematics is operating in the middle scenarios using mathematical models. Naturally
of this military logic. a mathematical representation is principally
different from the construct itself, and the real-
Mathematics in Action life implication might turn out to be very different
The notion of mathematics in action – that can be from calculated implications. Accompanied by
seen as a further development of “formatting (mischievous) mathematics-based hypothetical
power of mathematics” (Skovsmose 1994) – can reasoning, we are entering the risk society.

M 444 Mathematization as Social Process

Legitimation or justification refers to possible acknowledging the complexity of mathematics in
validations of technological actions. While the action such celebration cannot be sustained. Math-
notion of justification includes an assumption that ematics in action has to be addressed critically in all
some degree of logical honesty has been exercised, its different instantiations. Like any form of action,
the notion of legitimation does not include such an mathematics in action may have any kind of qual-
assumption. In fact, mathematics in action might ities, such as being productive, risky, dangerous,
blur any distinction between justification and legit- benevolent, expensive, dubious, promising, and
imation. When brought into effect, a mathematical brutal. It is crucial for any mathematics education
model can serve any kind of interests. to provide conditions for reflecting critically on any
form of mathematics in action.
Realization refers to the phenomenon that math-
ematics itself comes to be part of reality, as was the This is a challenge to mathematics education
case at the supermarket. A mathematical model both as an educational practice and research. It
becomes part of our environment. Our lifeworld is becomes important to investigate mathematics in
formed through techniques as well as through action as part of complex sociopolitical processes.
discourses emerging from mathematics. Real-life Such investigations have been developed with
practices become formed through mathematics in reference to ethnomathematical studies, but many
action. It is this phenomenon that has been referred more issues are waiting for being addressed (see,
to as the formatting power of mathematics. for instance, D’Ambrosio’s 2012 presentation of
a broad concept of social justice).
Elimination of responsibility might occur
when ethical issues related to implemented action Due to processes of mathematization and
are removed from the general discourse about not least to the accompanying processes of
technological initiatives. Mathematics in action demathematization, one has to face new challenges
seems to be missing an acting subject. As in creating meaningful activities in the classroom.
a consequence, mathematics-based actions easily Experiences of meaning have to do with experiences
appear to be conducted in an ethical vacuum. of relationships. How can we construct classroom
They might appear to be determined by some activities that, on the one hand, acknowledge the
“objective” authority as they represent a logical complex mathematization of social practices and,
necessity provided by mathematics. However, on the other hand, acknowledge the profound
the “objectivity” of mathematics is a myth that demathematization of such practices? This general
needs to be challenged. issue has to be interpreted with reference to particu-
lar groups of students in particular sociopolitical
Mathematics in action includes features of imag- contexts (see, for instance, Gutstein 2012).
ination, hypothetical reasoning, legitimation, justi-
fication, realization including a demathematization To break from any general celebration of
of many practices, as well as an elimination of mathematics, to search for new dimensions of
responsibility. Mathematics in action represents meaningful mathematics education, and to open
a tremendous knowledge-power dynamics. for critical reflections on any form of mathemat-
ics in action are general concerns of critical
New Challenges mathematics education (see also “▶ Critical
Mathematics in action brings about several Mathematics Education” in this Encyclopedia).
challenges to mathematics education of which
I want to mention some. Cross-References

Over centuries mathematics has been celebrated ▶ Critical Mathematics Education
as crucial for obtaining insight into nature, as being ▶ Critical Thinking in Mathematics Education
decisive for technological development, and as ▶ Dialogic Teaching and Learning in
being a pure science. Consistent or not, these
assumptions form a general celebration of mathe- Mathematics Education
matics. This celebration can be seen as almost ▶ Mathematical Literacy
a defining part of modernity. However, by

Metacognition M445

References Metacognition

Christensen OR, Skovsmose O, Yasukawa K (2009) The Gloria Stillman
mathematical state of the world: Explorations into the Faculty of Education, Australian Catholic
characteristics of mathematical descriptions. In: University, Ballarat Campus (Aquinas), Ballarat,
Sriraman B, Goodchild S (eds) Relatively and philo- VIC, Australia
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65th birthday. Information Age, Charlotte, pp 83–96 Keywords

D’Ambrosio U (2012) A broad concept of social justice. Metamemory; Metacognitive knowledge;
In: Wager AA, Stinson DW (eds) Teaching mathemat- Metacognitive experiences; Metacognitive
ics for social justice: conversations with mathematics strategies
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ics, Reston, pp 201–213 Definition M

Gellert U, Jablonka E (2009) The demathematising effect of Any knowledge or cognitive activity that takes as
technology: calling for critical competence. In: Ernest P, its object, or monitors, or regulates any aspect of
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education. Information Age, Charlotte, pp 19–24 thinking about, one’s own thinking.

Gutstein E (2012) Mathematics as a weapon in a struggle. Characteristics
In: Skovsmose O, Greer B (eds) Opening the cage:
critique and politics of mathematics education. Although the construct, metacognition, is used
Sense, Rotterdam, pp 23–48 quite widely and researched in various fields of
psychology and education, its history is relatively
Jablonka E (2010) Reflections on mathematical model- short beginning with the early work of John Flavell
ling. In: Alrø H, Ravn O, Valero P (eds) Critical on metamemory in the 1970s. Metamemory was
mathematics education: past, present and future. a global concept encompassing a person’s
Sense, Rotterdam, pp 89–100 knowledge of “all possible aspects of information
storage and retrieval” (Schneider and Artelt 2010).
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demathematisation. In: Gellert U, Jablonka E (eds) nitive monitoring has underpinned much of the
Mathematization and de-mathematization: social, research on metacognition since he first articulated
philosophical and educational ramifications. Sense, it. It was a revised version of his taxonomy of
Rotterdam, pp 1–18 metamemory that he had developed with Wellman
(Flavell and Wellman 1977). According to his
Keitel C, Kotzmann E, Skovsmose O (1993) Beyond the model, a person’s ability to control “a wide variety
tunnel vision: analysing the relationship between of cognitive enterprises occurs through the actions
mathematics, society and technology. In: Keitel C, and interactions among four classes of phenomena:
Ruthven K (eds) Learning from computers: mathemat- (a) metacognitive knowledge, (b) metacognitive
ics education and technology. Springer, Berlin, experiences, (c) goals (or tasks), and (d) actions
pp 243–279 (or strategies)” (p. 906). Metacognitive knowledge
incorporates three interacting categories of knowl-
Skovsmose O (1994) Towards a philosophy of critical edge, namely, personal, task, and strategy
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Skovsmose O (2009) In doubt – about language, mathe-
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Skovsmose O (2010) Symbolic power and mathematics.
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Critical issues in mathematics education. Information
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Yasukawa K, Skovsmose O, Ravn O (in print) Scripting the
world in mathematics and its ethical implications. In:
Ernest P, Sriraman B (eds) Critical mathematics
education: theory and praxis. Information Age, Charlotte

M 446 Metacognition

knowledge. It involves one’s (a) sensitivity to (Desoete and Veenman 2006). The field has firmly
knowing how and when to apply selected forms established the foundations of the construct and by
and depths of cognitive processing appropriately to building on these foundations, several researchers
a given situation (similar to subsequent definitions have extended Flavell’s work usefully and there is
of partly what is called procedural metacognitive an expanding body of knowledge in the area.
knowledge), (b) intuitions about intra-individual The elements of his model have been extended
and inter-individual differences in terms of beliefs, by others (e.g., elaborations of metacognitive
feelings, and ideas, (c) knowledge about task experiences, see Efklides 2001, 2002) or are the
demands which govern the choice of processed subject of debate (e.g., motivational and emotional
information, and (d) a stored repertoire of the nature knowledge as a component of metacognitive
and utility of cognitive strategies for attaining knowledge, see Op ‘t Eynde et al. 2006). Subse-
cognitive goals. The first of these is mostly implicit quently, it has led to many theoretical elaborations,
knowledge, whereas the remaining three are interventions, and ascertaining studies in mathemat-
explicit, conscious knowledge. Metacognitive ics education research (Schneider and Artelt 2010).
experiences are any conscious cognitive or
affective experiences which control or regulate cog- Flavell did not expect metacognition to be evi-
nitive activity. Achieving metacognitive goals are dent in students before Piaget’s stage of formal
the objectives of any metacognitive activity. operational thought, but more recent work by
Metacognitive strategies are used to regulate and others has shown that preschool children already
monitor cognitive processes and thus achieve start to develop metacognitive awareness. Work in
metacognitive goals. developmental and educational psychology as
well as mathematics education has shown that
In the two decades that followed when Flavell metacognitive ability, that is, the ability to gain-
and his colleagues had initiated research into fully apply metacognitive knowledge and strate-
metacognition (Flavell 1976, 1979, 1981), the gies, develops slowly over the years of schooling
use of the term became a buzzword resulting in and there is room for improvement in both adoles-
an extensive array of constructs with subtle cence and adulthood. Furthermore, studying the
differences in meaning all referred to as metacog- developmental trajectory of metacognitive exper-
nition (Weinert and Kluwe 1987). This work was tise in mathematics entails examining both fre-
primarily in the area of metacognitive research on quency of use and the level of adequacy of
reading; however, from the early 1980s, work in utilization of metacognition. Higher frequency of
mathematics education had begun mainly related use does not necessarily imply higher quality
to problem solving (Lester and Garofalo 1982) of application, with several researchers reporting
particularly inspired by Schoenfeld (1983, 1985, such phenomena as metacognitive vandalism,
1987) and Garofalo and Lester (1985). Cognition metacognitive mirage and metacognitive misdirec-
and metacognition were often difficult to distin- tion. Metacognitive vandalism occurs when the
guish in practice, so Garofalo and Lester (1985) response to a perceived metacognitive trigger
proposed an operational definition distinguishing (“red flag”) involves taking drastic and destructive
cognition and metacognition which clearly demar- actions that not only fail to address the difficulty but
cates the two, namely, cognition is “involved in also could change the nature of the task being
doing,” whereas metacognition is “involved in undertaken. Metacognitive mirage results when
choosing and planning what to do and monitoring unnecessary actions are engaged in, because
what is being done” (p. 164). This has been used a difficulty has been perceived, but in reality, it
subsequently by many researchers to be able to does not exist. Metacognitive misdirection is
delineate the two. the relatively common situation where there is
a potentially relevant but inappropriate response
Today, the majority of researchers in to a metacognitive trigger that is purely inadequacy
metacognitive research in mathematics education on the part of the task solver not deliberate vandal-
have returned to the roots of the term and ism. Recent research shows that as metacognitive
share Flavell’s early definition and elaborations

Metaphors in Mathematics Education M447

abilities in mathematics develop, not only is there Schneider W, Artelt C (2010) Metacognition and
increased usage but also the quality of that usage mathematics education. ZDM Int J Math Educ
increases. 40(2):149–161

The popularity of the metacognition construct Schoenfeld A (1983) Episodes and executive decisions in
stems from the belief that it is a crucial part of mathematical problem solving. In: Lesh R, Landau M
everyday reasoning, social interaction as occurs (eds) Acquisition of mathematics concepts and
in whole class and small group work and more processes. Academic, New York, pp 345–395
complex cognitive tasks such as mathematical
problem solving, problem finding and posing, Schoenfeld A (1985) Making sense of “outloud” problem
mathematical modeling, investigation, and solving protocols. J Math Behav 4(2):171–191
inquiry based learning.
Schoenfeld A (1987) What’s the fuss about metacognition?
In: Schoenfeld A (ed) Cognitive science and mathematics
education. Erlbaum, Hillsdale, pp 189–215

Weinert FE, Kluwe RH (eds) (1987) Metacognition,
motivation, and understanding. Hillsdale, Erlbaum

Cross-References

▶ Problem Solving in Mathematics Education Metaphors in Mathematics
Education

References Jorge Soto-Andrade M
Department of Mathematics, Faculty of Science
Desoete A, Veenman M (eds) (2006) Metacognition in and Centre for Advanced Research in Education,
mathematics education. Nova Science, New York University of Chile, Santiago, Chile

Efklides A (2001) Metacognitive experiences in problem Keywords
solving: metacognition, motivation, and self-regulation.
In: Efklides A, Kuhl J, Sorrentino RM (eds) Trends and Metaphor; Conceptual metaphor; Metaphoring;
prospects in motivation research. Kluwer, Dordrecht, Reification; Embodied cognition; Gestures;
pp 297–323 Analogy; Representations

Efklides A (2002) The systematic nature of metacognitive Definition
experiences: feelings, judgements, and their interrela-
tions. In: Izaute M, Chambres P, Marescaux PJ (eds) Etymologically metaphor means “transfer,” from
Metacognition: process, function, and use. Kluwer, the Greek meta (trans) + pherein (to carry).
Dordrecht, pp 19–34 Metaphor is in fact “transfer of meaning.”

Flavell JH (1976) Metacognitive aspects of problem Introduction
solving. In: Resnick LB (ed) The nature of
intelligence. Hillsdale, Erlbaum, pp 231–235 Metaphors are very likely as old as humankind.
Recall Indra’s net, a 2,500-year-old Buddhist
Flavell JH (1979) Metacognition and cognitive metaphor of dependent origination and intercon-
monitoring: a new area of cognitive-developmental nectedness (Cook 1977; Capra 1982), consisting
inquiry. Am Psychol 34:906–911 of an infinite network of pearls, each one
reflecting all others, in a never-ending process
Flavell JH (1981) Cognitive monitoring. In: Dickson of reflections of reflections, highly appreciated
W (ed) Children’s oral communication skills. by mathematicians (Mumford et al. 2002).
Academic, New York, pp 35–60

Flavell JH, Wellman HM (1977) Metamemory. In: Kail R,
Hagen J (eds) Perspectives on the development of
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Garofalo J, Lester F (1985) Metacognition, cognitive
monitoring and mathematical performance. J Res
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education. Nova Science, New York, pp 83–101

M 448 Metaphors in Mathematics Education

It was Aristotle, however, with his taxonomic resource rather than a temporary scaffold
genius, who first christened and characterized becoming later a “dead metaphor” (Chiu 2000).
metaphors c. 350 BC in his Poetics: “Metaphor We find also theory-constitutive metaphors that
consists in giving the thing a name that belongs do not “worn out” like literary metaphors and
to something else; the transference being either provide us with heuristics and guide our research
from genus to species, or from species to genus, (Boyd 1993; Lakoff and Nu´n˜ez 1997). Recall the
or from species to species, on the grounds of “tree of life” metaphor in Darwin’s theory
analogy” (Aristotle 1984, 21:1457b). Interest- of evolution or the “encapsulation metaphor”
ingly for education, Aristotle added: in Dubinsky’s APOS theory (Dubinsky and
McDonald 2001).
The greatest thing by far is to be a master of
metaphor. It is the one thing that cannot be learned In the field of mathematics education proper,
from others; it is also a sign of genius, since a good it has been progressively recognized during the
metaphor implies an eye for resemblance. (loc. cit. last decades (e.g., Chiu 2000, 2001; van
21:1459a) Dormolen 1991; Edwards 2005; English 1997;
Ferrara 2003; Gentner 1982, 1983; Lakoff and
But time has not passed in vain since Aristotle. Nu´n˜ez 2000; Parzysz et al. 2007; Pimm 1987;
Widespread agreement has been reached Presmeg 1997; Sfard 1994, 1997, 2009;
(Richards 1936; Black 1962, 1979; Ortony Soto-Andrade 2006, 2007) that metaphors are
1993; Ricoeur 1977; Reddy 1993; Gibbs 2008, powerful cognitive tools that help us in grasping
2008; Indurkhya 1992, 2006; Johnson and Lakoff or building new mathematical concepts, as well
2003; Lakoff and Nu´n˜ez 2000; Wu 2001; Sfard as in solving problems in an efficient and friendly
1994, 1997, 2009) that metaphor serves as the way: “metaphors we calculate by” (Bills 2003).
often unknowing foundation for human thought
(Gibbs 2008) since our ordinary conceptual According to Lakoff and Nu´n˜ez (2000),
system, in terms of which we both think and act, (conceptual) metaphors appear as mappings
is fundamentally metaphorical in nature (Johnson from a source domain into a target domain,
and Lakoff 2003). carrying the inferential structure of the first
domain into the one of the second, enabling us
Characteristics to understand the latter, often more abstract and
opaque, in terms of the former, more down-to-
Metaphors for Metaphor earth and transparent. In the classical example
“There is no non metaphorical standpoint from “A teacher is a gardener,” the source is
which one could look upon metaphor” remarked gardening, and the target is education.
Ricoeur (1977). To Bruner (1986) “Metaphors
are crutches to help us to get up the abstract Figure 1 maps metaphors, analogies, and
mountain,” but “once up we throw them away representations and their relationships (Soto-
(even hide them) . . . (p. 48). Empirical evidence Andrade 2007).
suggests however that metaphor is a permanent
We thus see metaphor as bringing the target
concept into being rather than just shedding
a new light on an already existing notion, as

Target domain: Analogy Target domain:
higher, more abstract higher, more abstract

Metaphors in Metaphor
Mathematics Education,
Fig. 1 A topographic Representation Metaphor Representation
metaphor for metaphors,
representations, and Metaphor Metaphor
analogies Analogy
Source domain: Source domain:
down-to-earth down-to-earth

Metaphors in Mathematics Education M449

representation usually does, whereas analogy involving haptic-kinesthetic experiences. The M
states a similarity between two concepts already underlying mechanism of cross-domain map-
constructed (Sfard 1997). Since new concepts pings may explain how abstract concepts can
arise from a crossbreeding of several metaphors emerge in brains that evolved to steer the body
rather than from a single one, multiple meta- through the physical, social, and cultural world
phors, as well as the ability to transiting between (Coulson 2008). It has been proposed that acquir-
them, may be necessary for the learner to make ing metaphoric items might be facilitated by
sense of a new concept (Sfard 2009). Teaching acting them out, as in total physical response
with multiple metaphors, as an antidote to learning (Low 2008).
unwanted entailments of one single metaphor,
has been recommended (e.g., Low 2008; Sfard The didactical chasm existing between the
2009; Chiu 2000, 2001). ubiquitous motion metaphors in the teaching of
calculus and the static and timeless character of
Metaphor and Reification current formal definitions (Kaput 1979) is in fact
Sfard (1994) named reification the metaphorical bridged by the often unconscious gestures (Yoon
creation of abstract entities, seen as the transition et al. 2011) that lecturers enact in real time while
from an operational to a structural mode of speaking and thinking in an instructional context
thinking. Experientially, the sudden appearance (Nu´n˜ez 2008). So gestures inform mathematics
of reification is an “aha!” moment, the birth of education better than traditional disembodied
a metaphor that brings a mathematical concept mathematics (Nu´n˜ez 2007).
into existence. Reification is however a double-
edged sword: Its poietic (generating) edge brings Metaphors for Teaching and Learning
abstract ideas into being, and its constraining When confronted with the metaphor “teaching
edge bounds our imagination and understanding is transmitting knowledge,” many teachers say:
within the confines of our former experience and This is not a metaphor, teaching is transmitting
conceptions (Sfard 2009). This “metaphorical knowledge! What else could it be? Unperceived
constraint” (Sfard 1997) explains why it is not here is the “Acquisition Metaphor,” dominant in
quite true that anybody can invent anything, mathematics education, that sees learning as acquir-
anywhere, anytime, and why metaphors are ing an accumulated commodity. The alternative,
often “conceptual recycling.” For instance, the complementary, metaphor is the Participation
construction of complex numbers was hindered Metaphor: learning as participation (Sfard 1998).
for a long time by overprojection of the metaphor Plutarch agreed when he said “A mind is a fire to be
“number is quantity” until the new metaphor kindled, not a vessel to be filled” (Sfard, 2009).
“imaginary numbers live in another dimension”
installed them in the “complex plane.” “To Educational Metaphors
understand a new concept, I must create an appro- Grounding and linking metaphors are used in
priate metaphor. . .” says one of the mathemati- forming mathematical ideas (Lakoff and Nu´n˜ez
cians interviewed by Sfard (1994). 2000). The former “ground” our understanding of
mathematics in familiar domains of experience, the
Metaphor, Embodied Cognition, latter link one branch of mathematics to another.
and Gestures
Contemporary evidence from cognitive neurosci- Lakoff and Nu´n˜ez (1997) point out that often
ence shows that our brains process literal and mathematics teachers attempt to concoct ad hoc
metaphorical versions of a concept in the same extensions of grounding metaphors beyond their
localization (Knops et al. 2009; Sapolsky 2010). natural domain, like “helium balloons” or “anti-
Gibbs and Mattlock (2008) show that real and matter objects” for negative numbers. Although
imagined body movements help people create the grounding “motion metaphor” extends better
embodied simulations of metaphorical meanings to negative numbers: À3 steps means walking
backwards 3 steps and multiplying by À1 is
turning around, they consider this extension

M 450 Metaphors in Mathematics Education

Metaphors in Mathematics Education, Fig. 2 Two metaphors for commutativity of multiplication

a forced “educational metaphor.” For an explicit “Area metaphor” and “Branching metaphor”
account of such educational metaphors, see Chiu for multiplication (Soto-Andrade 2007) are
(1996, 2000, 2001). Negative numbers arise more illustrated in Fig. 2.
naturally, however, via flows in a graph:
A “negative flow” of 3 units from agent A to In the area metaphor, commutativity is per-
agent B “is” a usual flow of 3 units from B to A. ceived as invariance of area under rotation. We
“see” that 2 Â 3 ¼ 3 Â 2, without counting and
Metaphoring (Metaphorical Thinking) knowing that it is 6. In the branching metaphor,
in Mathematics Education commutativity is less obvious unless this meta-
Presmeg (2004) studied idiosyncratic metaphors phor becomes a “met-before” (McGowen and
spontaneously generated by students in problem- Tall 2010) because you know trees very well.
solving as well as their influence on their sense Our trees also suggest a “hydraulic metaphor,”
making. Students generating their own metaphors useful to grasp multiplication of fractions: A litre
increase their critical thinking, questioning, and of water drains evenly from the tree apex, through
problem-solving skills (Low 2008). There are the ducts. Then 1/6 appears as 1/3 of 1/2 in the left
however potential pitfalls occasioned by invalid tree and also as 1/2 of 1/3 in the right tree. Our
inferences and overgeneralization. hydraulic metaphor enables us to see the “two
sides of the multiplicative coin”: 2 Â 3 is bigger
Building on their embodied prior knowledge, but 1/2 Â 1/3 is smaller than both factors. It also
students can understand difficult concepts opens up the way to a deeper metaphor for
metaphorically (Lakoff and Nu´n˜ez 1997). Explicit multiplication: “multiplication is concatenation”,
examples have been given by Chiu (2000, 2001), a generating metaphor for category theory in
e.g., students using their knowledge of motion to mathematics.
make sense of static polygons through the “poly-
gons are paths” metaphor, and so “seeing” that the On the Metaphorical Nature of Mathematics
sum of the exterior angles is a whole turn and that Lakoff and Nu´n˜ez’s claim that mathematics
exterior angles are more “natural” than interior consists entirely of conceptual metaphors has
angles! “Polygons are enclosures between crossing stirred controversy among mathematicians and
sticks” elicits different approaches. Source under- mathematics educators. Dubinsky (1999) sug-
standing overcomes age to determine metaphoring gests that formalism can be more effective than
capacity, since 13-month infants can already metaphor for constructing meaning. Goldin
metaphorize (Chiu 2000). Also, a person’s prior (1998, 2001) warns that the extreme view that
(nonmetaphorical) target understanding can curtail all thought is metaphorical will be no more
or block metaphoring (loc. cit.). helpful than earlier views that it was proposi-
tional and finds that Lakoff and Nu´n˜ez’s
Examples of Metaphors for Multiplication “ultrarelativistism” dismisses perennial values
Chiu (2000) indicates the following: central to mathematics education like mathemat-
ical truth and processes of abstraction, reasoning,
“Multiplication A Â B is replacing the original and proof among others (Goldin 2003).
A pieces by B replications of them.”
However some distinguished mathematicians
“Multiplication A Â B is cutting each of the dissent. Manin (2007), referring to Metaphor and
current A objects into B pieces.”

Metaphors in Mathematics Education M451

Proof, complains about the imbalance between processes suggest further research on M
various basic values which is produced by the comparing visual, auditory, and kinesthetic
emphasis on proof (just one of the mathematical metaphors.
genres) that works against values like “activi-
ties”, “beauty” and “understanding”, essential in How can teachers facilitate the emergence
high school teaching and later, neglecting which of idiosyncratic metaphors in the students?
a teacher or professor tragically fails. He also
claims that controverted Thom’s Catastrophe May idiosyncratic metaphors be voltaic arcs
Theory “is one of the developed mathematical that spring when didactical tension is high
metaphors and should only be judged as such”. enough in the classroom?
Thom himself complains that “analogy, since
positivism, has been considered as a remain of How and where do students learn relevant
magical thinking, to be condemned absolutely, metaphors: from teachers, textbooks, or sources
being nowadays hardly considered as more than outside of the classroom?
a rhetorical figure (Thom, 1994). He sees catas-
trophe theory as a pioneering theory of analogy How can we facilitate students’ transiting
and points out that narrow minded scientists between metaphors?
objecting the theory because it provides nothing
more than analogies and metaphors, do not How can teaching trigger change in students’
realize that they are stating its true purpose: to metaphors?
classify all possible types of analogical situations
(Porte, 2013). What roles should the teacher play in meta-
phor teaching?
The preface to Mumford et al. (2002) reads:
“Our dream is that this book will reveal to our What happens when there is a mismatch
readers that mathematics is not alien and remote between teacher and student’s metaphors?
but just a very human exploration of the patterns
of the world, one which thrives on play and Do experts continue using the same metaphors
surprise and beauty.” as novices? If yes, do they use them in the
same way?
McGowen and Tall (2010) argue that even
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M 454 Misconceptions and Alternative Conceptions in Mathematics Education

Characteristics resilient or pervasive when one tries to get rid of it.
The reason why misconceptions are stubborn is that
Research on misconceptions in mathematics and they are viable, useful, workable, or functional in
science commenced in the mid-1970s, with the other domains or contexts. Therefore, it is important
science education community researching the for teachers not only to treat misconceptions with
area much more vigorously. This research care- equal importance to mathematical concepts but also
fully rejected the tabula rasa assumption that chil- to identify what exactly the misconception is in the
dren enter school without preconceptions about learning context and to clarify the relationship
a concept or topic that a teacher tries to teach in between the misconception and the mathematical
class. The first international seminar Misconcep- concept to be taught. In other words, the teacher
tions and Educational Strategies in Science and needs to construct the task for the lesson taking the
Mathematics was held at Cornell University, Ith- misconception into consideration in order to resolve
aca, NY, in 1983, with researchers from all over the conflict between the misconception and the
the world gathering to present research papers in mathematical concept. By doing this the lesson
this area – although the majority of research papers may open up a new pathway to children’s deeper
were in the field of science education. and wider understanding of the mathematical con-
cept to be taught.
In mathematics education, according to
Confrey (1987), research on misconceptions So far many misconceptions have been identi-
began with the work of researchers such as fied at the elementary and secondary levels, how-
Erlwanger (1975), Davis (1976), and Ginsburg ever only a few of them are considered for
(1976), who pioneered work focusing on stu- inclusion in actual teaching situations. While very
dents’ conceptions. In the proceedings of the few of these are incorporated in mathematics text-
second seminar: Misconceptions and Educational books, one exception is the misconception that
Strategies in Science and Mathematics, Confrey figures with the same perimeter have the same
(1987) used constructivism as a framework for area. For example, Takahashi (2006) describes
a deep analysis of research on misconceptions. an activity used in a fourth-grade Japanese
Almost two decades later, Confrey and Kazak textbook to introduce the formula for the
(2006) identified examples of misconceptions area of a rectangle that asks students to com-
which have been extensively discussed by pare the areas of carefully chosen figures that
the mathematics education community – for have the same perimeter – for example, 3 cm
example, “Multiplication makes bigger, division  5 cm and 4 cm  4 cm rectangles.
makes smaller,” “The graph as a picture of
the path of an object,” “Adding equal amounts Further research is needed to develop how
to numerators and denominators preserves to incorporate misconceptions into textbook or
proportionality,” and “longer decimal number teaching materials in order to not only resolve
are bigger, so the 1.217 > 1.3” (pp. 306–307). the misconception but also to deepen and
Concerning decimals, a longitudinal study by expand children’s understanding of mathematical
Stacey (2005) showed that this misconception concepts.
is persistent and pervasive across age and
educational experience. In another extensive Cross-References
study, Ryan and Williams (2007) examined
a variety of misconceptions among 4–15-year- ▶ Concept Development in Mathematics
old students in number, space and measurement, Education
algebra, probability, and statistics, as well as
preservice teachers’ mathematics subject matter ▶ Constructivism in Mathematics Education
knowledge of these areas. ▶ Constructivist Teaching Experiment
▶ Learning Environments in Mathematics
From the teacher’s perspective, a misconception
is not a trivial error that is easy to fix, but rather it is Education
▶ Situated Cognition in Mathematics Education

Models of In-Service Mathematics Teacher Education Professional Development M455

References where emphases are placed. The basic organizer M
is around teacher decision making since effective
Confrey J (1987) Misconceptions across subject matter: classroom teaching is essentially about planning
science, mathematics and programming. In: Novak JD experiences that engage students in activities that
(ed) Proceedings of the second international seminar: are mathematically rich, relevant, accessible, and
misconceptions and educational strategies in science the management of the learning that results. As
and mathematics. Cornell University, Ithaca, pp 81–106 Zaslavsky and Sullivan (2011) propose, educat-
ing practising teachers involves facilitating
Confrey J, Kazak S (2006) A thirty-year reflection on growth from “uncritical perspectives on teaching
constructivism in mathematics education in PME. In: and learning to more knowledgeable, adaptable,
Gutierrez A, Boero P (eds) Handbook of research on judicious, insightful, resourceful, reflective and
the psychology of mathematics education: past, pre- competent professionals ready to address the
sent and future. Sense, Rotterdam, pp 305–345 challenges of teaching” (p. 1).

Davis R (1976) The children’s mathematics project: the The entry is structured around an adaptation of
syracuse/illinois component. J Child Behav 1:32–58 the Clark and Peterson (1986) schematic in which
three background factors – specifically teacher
Erlwanger S (1975) Case studies of children’s conceptions knowledge; the constraints they anticipate they
of mathematics: 1. J Child Behav 1(3):157–268 will experience; and their attitudes, beliefs, and
self-goals – influence each other and together
Ginsburg H (1976) The children’s mathematical project: inform teachers’ intentions to act and ultimately
an overview of the Cornell component. J Child Behav their classroom actions. Because the schematic
1(1):7–31 essentially connects background considerations
with practice, it is ideal for structuring the
Ryan J, Williams J (2007) Children’s mathematics 4–15: education of practising mathematics teachers.
learning from errors and misconceptions. Open Uni-
versity Press, Maidenhead The first of these background factors refers
to teacher knowledge. A model informing the
Stacey K (2005) Travelling the road to expertise: design of practising teacher education directed
a longitudinal study of learning. In: Chick HL, Vincent at improving their knowledge was proposed by
JL (eds) Proceedings of the 29th conference of the Hill et al. (2008) in which there were two major
international group for the psychology of mathematics categories: subject matter knowledge and peda-
education, vol 1. University of Melbourne, Melbourne, gogical content knowledge. Hill et al. described
pp 19–36 Subject Matter Knowledge as consisting of com-
mon content knowledge, specialized content
Takahashi A (2006) Characteristics of Japanese mathe- knowledge, and knowledge at the mathematical
matics lessons. Tsukuba J Educ Stud Math 25:37–44 horizon. For each of these, the emphasis is on
developing in teachers the capacity not only to
Watson JM (2011) Foundations for improving statistical learn any new mathematics they need but also to
literacy. Stat J IAOS 27:197–204. doi:10.3233/ view the mathematics they know in new ways.
SJI20110728 Generally, both of these orientations are facili-
tated by connecting this learning to the further
Models of In-Service Mathematics development of their pedagogical content knowl-
Teacher Education Professional edge. Hill et al. argued that Pedagogical Content
Development Knowledge includes knowledge of content and
teaching, knowledge of content and students,
Peter Sullivan and knowledge of curriculum. In addressing
Faculty of Education, Monash University, knowledge of content and teaching, Zaslavsky
Monash, VIC, Australia and Sullivan (2011) proposed focusing teacher
learning on experiences such as those involving
Essentially ongoing improvement in learning is
connected to the knowledge of the teacher.
This knowledge can be about the mathematics
they will teach, communicating that mathemat-
ics, finding out what students know and what
they find difficult to learn, and managing the
classroom to maximize the learning of students.
This entry is about approaches to in-service
mathematics teacher education and highlighting

M 456 Models of In-Service Mathematics Teacher Education Professional Development

comparing and contrasting between and across learn mathematics or whether such learning is
topics to identify patterns and make connections, just for some (Hannula 2004). Also important is
designing and solving problems for use in their whether teachers see their own and students’
classrooms, fostering awareness of similarities achievement as incremental and amenable to
and differences between tasks and concepts, and improvement through effort (Dweck 2000).
developing the capacity of teachers to adapt Teacher education can include experiences that
successful experiences to match new content. address this by, for example, examining forms of
Knowledge of content and students is primarily affirmation, studying tasks that foster inclusion,
about the effective use of data to inform planning and developing awareness of threats such as self-
and teaching. Essentially, the goal is to examine fulfilling prophecy effects (Brophy 1983).
what students know, as distinct from what they
do not. In terms of knowledge of curriculum, Having formed intentions, teachers act in
Sullivan et al. (2012) described processes where classrooms. Rather than compartmentalizing the
teachers evaluate resources, draw on the elements of the background factors described
experience of colleagues, analyze assessment above, it is preferable that the education of prac-
data to make judgments on what the students tising teachers incorporate all elements together,
know, and interpret curriculum documents to a suitable framework for which is the study of
identify important ideas (Charles 2005) as the practice. The most famous example of teacher
first level of knowing the curriculum. The subse- learning from the study of practice is Japanese
quent levels involve selecting, sequencing, and Lesson Study which is widely reported in the
adapting experiences for the students, followed Japanese context (e.g., Fernandez and Yoshida
by planning the teaching. All of these can inform 2004; Inoue 2010) and has been adapted to West-
the design of practising teacher education. ern contexts (e.g., Lewis et al. 2004). Other
examples of learning through the study of
The second background factor refers to the practice include realistic simulations offered by
constraints that teachers anticipate they may con- videotaped study of exemplary lessons (Clarke
front. Such constraints can be exacerbated by the and Hollingsworth 2000); interactive study of
socioeconomic, cultural, or language background recorded exemplars (e.g., Merseth and Lacey
of the students, geographic factors, and gender. 1993); case methods of teaching dilemmas that
A further constraint is the diversity of readiness problematize aspects of teaching (e.g., Stein et al.
that teachers experience in all classes, even those 2000); and Learning Study which is similar to
grouped to maximize homogeneity. Sullivan Japanese Lesson Study but which focuses on
et al. (2006) described a planning framework that student learning (Runesson et al. 2011).
includes accessible tasks, explicit pedagogies, and
specific enabling prompts for students experienc- An associated factor is the need for
ing difficulty. Such prompts involve slightly low- effective school-based leadership of the math-
ering an aspect of the task demand, such as the form ematics teachers. If the focus is on sustainable,
of representation, the size of the number, or the collaborative school-based approaches to
number of steps, so that a student experiencing improving teaching, this needs active and sen-
difficulties can proceed at that new level; and then sitive leadership. Such leaders can be assisted
if successful can proceed with the original to study processes for leadership, as well
task. Teacher educators can encourage practising as developing their confidence to lead the
teachers to examine the existence and sources of aspects of planning, teaching, and assessment
constraints and strategies that can be effective in described above.
overcoming those constraints.
References
The third background factor includes
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Dweck CS (2000) Self theories: their role in motivation,
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investigations. For the profession of teaching
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through lesson study. J Math Teach Educ opportunities and exploring the students’
14(1):5–23 understanding or adaptive strategies of fostering
mathematical competency require not only math-
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(Shulman 1986; Ball et al. 2008; Bromme
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the pedagogical promise of hypermedia and case and procedural components (e.g., Baumert et al.
methods in teacher education. Teach Teach Educ 2010; Ball et al. 2008), as well as prescriptive
9(3):283–299 views and epistemological orientations (e.g.,
Pajares 1992; McLeod 1989; To¨rner 2002); it
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M 458 Models of Preservice Mathematics Teacher Education

ranges from rather global components (cf. To¨rner video-based work (e.g., Sherin and Han 2003;
2002) to content-specific or even classroom Seago 2004; Dreher and Kuntze 2012;
situation-specific components (Kuntze 2012; Kuntze 2006), or work with lesson transcripts.
Lerman 1990). For several decades, approaches such as
“microteaching” (e.g., Klinzing 2002) had
The goal of developing such a multifaceted emphasized forms of teacher training centered
professional knowledge underpins the signifi- in practicing routines for specific instructional
cance of specific and structured environments situations. Seen under today’s perspective, the
for initial professional learning. However, it is latter approach tends to underemphasize the
widely agreed that models of preservice teacher goal of supporting reflective competencies of
education have to be seen as subcomponents in prospective teachers which tend to be trans-
the larger context of continued professional ferable across contents and across specific
learning throughout the whole working period classroom situations (Tillema 2000).
of teachers rather than being considered as an • Developing competencies of instruction- and
accomplished level of qualification. Even though content-related reflection is a major goal in
these models of preservice teacher education are preservice teacher education. Accordingly,
framed by various institutional contexts and learning opportunities such as the analysis
influenced by different cultural environments and the design of mathematical tasks (e.g.,
(Leung et al. 2006; Bishop 1988), the following Sullivan et al. 2009, cf. Biza et al. 2007), the
fundamental aspects which are faced by many exploration of overarching ideas linked to
such models of preservice teacher education mathematical contents or content domains
may be considered: (Kuntze et al. 2011), or the analysis of
• Theoretical pedagogical content knowledge videotaped classroom situations (Sherin and
Han 2003; Reusser 2005; Kuntze et al. 2008)
is essential for designing opportunities of are integrated in models of preservice mathe-
rich conceptual learning in the classroom. matics education, supporting preservice
Hence, in models of preservice teacher educa- teachers to build up reflective competencies
tion, theoretical knowledge such as knowl- or to become “reflective practitioners” (e.g.,
edge about dealing with representations or Smith 2003; Atkinson 2012).
knowledge about frequent misconceptions of The scenarios mentioned above indicate that
learners (cf. Ball 1993) is being supported in there are a wide variety of possible models of
particular methodological formats which preservice teacher education, as it has also been
may take the form, e.g., of lectures, seminars, observed in comparative studies of institutional
or focused interventions accompanying frameworks (Ko¨nig et al. 2011; Tatto et al. 2008).
a learning-on-the job phase (Lin and Cooney In contrast, research on the effectiveness of dif-
2001). ferent models of preservice teacher education is
• Linking theory to practice is a crucial still relatively scarce. Studies like TEDS-M
challenge of models of preservice teacher (Tatto et al. 2008) constitute a step into this
education. The relevance of professional direction and set the stage for follow-up research
knowledge for acting and reacting in the not only in processes of professional learning in
classroom is asserted to be supported by an the settings of specific models of preservice
integration of theoretical knowledge with teacher education but also into effects of specific
instructional practice. In models of preservice professional learning environments, as they can
teacher education, this challenge is addressed be explored in quasi-experimental studies. In
by methodological approaches such as school addition to a variety of existing qualitative case
internships, frequently with accompanying studies, especially quantitative evidence about
seminars and elements of coaching (cf. Joyce models of preservice teacher education is still
and Showers 1982; Staub 2001; Kuntze needed (cf. Adler et al. 2005). Such evidence
et al. 2009), and specific approaches such as
lesson study (Takahashi and Yoshida 2004),

Models of Preservice Mathematics Teacher Education M459

from future research should systematically iden- numbers: an integration of research. Erlbaum, M
tify characteristics of effective preservice teacher Hillsdale, pp 157–196
education. Moreover empirical research about Ball D, Thames MH, Phelps G (2008) Content knowledge
models of preservice teacher education should for teaching: what makes it special? J Teach Educ
give insight how characteristics of effective 59(5):389–407
professional development for in-service mathe- Baumert J, Kunter M, Blum W, Brunner M, Voss T,
matics teachers (Lipowsky 2004) may translate Jordan A et al (2010) Teachers’ mathematical knowl-
into the context of the work with preservice edge, cognitive activation in the classroom, and stu-
teachers, which differs from professional devel- dent progress. Am Educ Res J 47(1):133–180
opment of in-service teachers (da Ponte 2001). Bishop AJ (1988) Mathematics education in its cultural
context. Educ Stud Math 19:179–191
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biological drives and incentive in the first satisfaction, and positive learning outcomes. M
decades of the twentieth century (see Brownell Lower performing demographic populations
1939 for a good review of this perspective as tend to show more external and unstable attribu-
applied to education). Following the tenets of tional patterns. These appear to be caused by
classical and operant (instrumental) conditioning, systematic educational biases (Kloosterman
it was found that if a reinforcer was provided for 1988; Pedro et al. 1981; Weiner 1980).
successfully completing a behavior, the probabil-
ity of that behavior occurring in the future under Goal Theory
similar circumstances would increase. Addition- Goal theories focus on the stated and unstated
ally, Thorndike found that the intensity of the reasons people have for engaging in mathemati-
behavior would increase as a function of the cal tasks. Goals can focus on Learning (also
reinforcement value (1927). These general theo- called Mastery), Ego (also called Performance),
ries of the use of incentives to motivate student or Work Avoidance. People with learning goals
learning dominated educational theory roughly tend to define success as improvement of their
until the middle of the 1960s. performance or knowledge. Working towards
these kinds of goals shows results in the valuation
They are still valuable to educators today, of challenge, better metacognitive awareness,
particularly in the use of behavior modification and improved learning than people with ego
techniques, which regulate the use of rewards and goals. Work avoidance goals are debilitating,
other reinforcers contingent upon the learner’s psychologically, as they result from learned
successive approximation of the desired helplessness and other negative attributional
behavioral outcomes, which could be successful patterns (Wolters 2004; Covington 2000; Gentile
skill attainment or increase in positive self- and Monaco 1986).
statements to reduce math anxiety and so on
(Bettinger 2008). Intrinsic Motivation and Interest
The level of interest a student has in mathematics,
Since the mid-1960s, research on motiva- the more effort he or she is willing to put out, the
tion in the psychology of learning has focused more he or she thinks the activity is enjoyable,
on six different, but not distinct, theoretical and the more they are willing to persist in the
constructs: Attributions, Goal Theory, Intrinsic face of difficulties (Middleton 1995; Middleton
Motivation, Self-Regulated Learning, Social and Spanias 1999; Middleton and Toluk 1999).
Motivation, and Affect. These factors grew Intrinsic Motivation and Interest theories
out of a general cognitive tradition in psychol- have shown that mathematical tasks can be
ogy but recently have begun to explain the designed to improve the probability that
impact of social forces, particularly classroom a person will exhibit task-specific interest
communities and teacher-student relationships and that this task-specific interest, over time,
on student enjoyment and engagement in math- can be nurtured into long-term valuation of
ematical subject matter (see Middleton and mathematics and its applications (Hidi and
Spanias 1999 for a review comparing these Renninger 2006; Ko¨ller et al. 2001; Cordova
perspectives). and Lepper 1996).

Attribution Theory Self-Regulated Learning
Learners’ beliefs about the causes of their Taken together, these primary theoretical per-
successes and failures in mathematics determine spectives can be organized under a larger
motivation based on the locus of the cause (inter- umbrella concept: Self-Regulated Learning
nal or external to the learner) and its stability (SRL). Internal, stable attributions are a natural
(stable or unstable). Productive motivational outcome of Learning Goals, and Interest is
attributions tend to focus on internal, stable a natural outcome of internal, stable, attributions.
causes (like ability and effort) for success as
these lead to increased persistence, self-efficacy,

M 462 Motivation in Mathematics Learning

Each of these perspectives contributes to the Cross-References
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design classroom environments, tasks, and
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Eccles and Wigfield 2002; Wolters and ▶ Equity and Access in Mathematics Education
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▶ Mathematics Teacher Identity

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M

N

Noticing of Mathematics Teachers professional vision (1994), Mason’s (2002)
discipline of noticing, and research on expertise.
Randolph Philipp1, Victoria R. Jacobs2 and
Miriam Gamoran Sherin3 Conceptualizations and Contributions
1San Diego State University, San Diego, of Mathematics Teacher Noticing
CA, USA
2The University of North Carolina at Greensboro, Teachers, in particular, have always been
Greensboro, NC, USA confronted with a “blooming, buzzing confusion
3Northwestern University, Evanston, IL, USA of sensory data” (B. Sherin and Star in
M.G. Sherin et al. 2011), and so they need to
Keywords find ways to distinguish between more produc-
tive and less productive noticing. This task
Attention; Learning to notice; Noticing; Teacher has been made even more complex as mathe-
professional development; Student conceptions; matics teaching has increasingly become asso-
Teacher education ciated with in-the-moment decisions whereby
teachers take into account the variety of
Definition students’ conceptions that arise. Thus, the con-
ceptualization and study of teacher noticing
Noticing is a term used in everyday language to contributes to national efforts to decompose
indicate the act of observing or recognizing the practice of teaching into specific compo-
something, and people engage in this activity nents that might be studied and learned
regularly while they navigate a perceptually (Grossman et al. 2009).
complex world. At the same time, individual
professions have strategic ways of noticing, Current conceptualizations of mathematics
and understanding and promoting productive teacher noticing have generally been associated
noticing by mathematics teachers has become with two components: attending and making
a growing area of inquiry among researchers sense. Researchers differ on what constitutes
(For a compilation, see Sherin et al. 2011). making sense, with some focusing exclusively
The field currently embraces a range of concep- on teachers’ interpretations of events whereas
tualizations of noticing, but many researchers use others also include consideration of teachers’
as their foundation Goodwin’s ideas about instructional responses. For example, Jacobs
et al. (2010) consider instructional responses
in conceptualizing professional noticing of

S. Lerman (ed.), Encyclopedia of Mathematics Education, DOI 10.1007/978-94-007-4978-8,
# Springer Science+Business Media Dordrecht 2014

N 466 Number Lines in Mathematics Education

children’s mathematical thinking as comprised of Cross-References
three interrelated skills: (a) attending to chil-
dren’s strategies, (b) interpreting children’s ▶ Frameworks for Conceptualizing Mathematics
understandings, and (c) deciding how to respond Teacher Knowledge
on the basis of children’s understandings.
Another difference in how researchers conceptu- ▶ Mathematics Teacher Educator as Learner
alize noticing is whether the focus is on ▶ Questioning in Mathematics Education
documenting everything teachers find notewor- ▶ Teacher Beliefs, Attitudes, and Self-Efficacy
thy or documenting whether teachers notice
particular aspects of instruction identified as in Mathematics Education
important by researchers, such as students’ math-
ematical thinking or specific mathematical References
content knowledge.
Goodwin C (1994) Professional vision. Am Anthropol
Noticing differs from constructs such as 96:606–633
knowledge and beliefs because noticing names
an interactive, practice-based process rather Grossman P, Compton C, Igra D, Ronfeldt M, Shahan E,
than a category of cognitive resource. Specif- Williamson P (2009) Teaching practice: a cross-
ically, the focus of mathematics-teacher professional perspective. Teach Coll Rec 111:
noticing is on how teachers interact with 2055–2100
a mathematical instructional situation, and
this practice-based nature of noticing makes Jacobs VR, Lamb LLC, Philipp RA (2010) Professional
it complex and thereby challenging to develop. noticing of children’s mathematical thinking. J Res
However, professional development has been Math Educ 41:169–202
found to support teachers while they learn to
notice differently. One particularly promising Mason J (2002) Researching your own practice: the disci-
approach to enhancing mathematics-teacher pline of noticing. Routledge Falmer, London
noticing has been through the work of video
clubs (e.g., M.G. Sherin and van Es 2009), Sherin MG, van Es EA (2009) Effects of video club
whereby teachers collaboratively view and participation on teachers’ professional vision.
analyze classroom videos. J Teach Educ 60:20–37

Learning to notice in new, more sophisticated, Sherin MG, Jacobs VR, Philipp RA (eds) (2011) Mathe-
ways supports teachers while they learn to teach matics teacher noticing: seeing through teachers’ eyes.
more effectively. As such, learning to notice pro- Routledge/Taylor & Francis, New York
ductively in an instructional setting is an impor-
tant, but often hidden, teaching skill. Sherin et al. Number Lines in Mathematics
(2011) noted that classrooms are too complex Education
for teachers to ever be able to notice every-
thing before responding. They suggested that Koeno Gravemeijer
instead of focusing on all possible contingen- Eindhoven School of Education, Eindhoven
cies, professional developers focus on ways of University of Technology, Eindhoven,
helping teachers develop new understandings The Netherlands
of their learning environments so that they can
make more informed instructional decisions. Keywords
In this way, teachers’ changing practices are
driven by enhanced teacher noticing whereby Addition and subtraction; Mental arithmetic;
they are “seeing and making sense differently Visualization; Instruction theory; Modeling
of things that are happening in the classroom”
(p. 11). Characteristics

Number lines figure prominently in mathematics
education. They may take various shapes and

Number Lines in Mathematics Education N467
Number Lines in Mathematics Education, Fig. 1 Bead string

Number Lines in Mathematics Education, Fig. 2 Jumps on the bead string

forms, from a clothesline with number cards using multiples of ten as reference points, both N
in the early grades, to straight lines on paper for identifying given numbers of beads (e.g.,
representing rational numbers or integers. 63 ¼ 6 Â 10 + 3 or 68 ¼ 7 Â 10 À 2) and for
Number lines may feature all numbers under adding and subtracting beads. Adding 30 to 42,
consideration or just a selection, depending on for instance, may be carried out via “jumps of
the function the number line has to fulfill. The ten”: “42 + 10 ¼ 52, 52 + 10 ¼ 62, 62 + 10 ¼ 72”
1st-grade number cards, for instance, are to sup- (see Fig. 2).
port the learning of the number sequence.
Whereas a more schematized number line may Next the activities with the bead string are
be used to illuminate the structure and magnitude symbolized on a number line, where small arcs
of rational numbers and decimals. In this signify jumps of one and bigger arcs jumps of ten.
contribution we will focus on the empty number In this manner the number line may start to
line, which is kept even more sparse than the function as a way of scaffolding ten-referenced
latter in order to fulfill its role as a specifically strategies for addition and subtraction up to 100.
designed instructional tool. And the students may start curtailing the jumps in
various manners (see Fig. 3), a method which can
The Empty Number Line be expanded to numbers up to 1,000 (Selter 1998).
The idea of using the empty number line as Research showed that the empty number
a means of support for adding and subtracting line is a powerful model for instruction
numbers up to 100 was introduced by Whitney (Klein et al. 1998).
(1985) and elaborated and publicized by Treffers
(1991), who linked it to the so-called domain- Flexible Solution Strategies
specific instruction theory for realistic mathemat- Note that number line the students start with is
ics education (RME) (see also Gravemeijer literally empty, and the students only mark the
2004). In doing so, he also adopted Whitney’s numbers that play a role in their calculation. The
suggestion of using a bead string, consisting of marks on the number line emanate from the
100 beads that are grouped in a pattern of ten student’s own thinking. This allows for a wide
dark beads, ten light beads, ten dark beads, variety of flexible solution strategies – which are
etc. (see Fig. 1), as a precursor to the number line. compatible with a group of solution strategies
that students develop spontaneously. Research
The activities with the bead string consist of shows that the informal strategies students
counting beads (starting from the left and mark- develop to solve addition and subtraction prob-
ing the total with a clothespin), incrementing, lems up to 100 fall in two broad categories,
decrementing, and comparing numbers of beads. “splitting tens and ones” and “counting in
The rationale for those activities is that students jumps” (Beishuizen 1993). An instance of split-
will start to use the color structure of the bead ting tens and ones would be solving 44 + 37 ¼ . . .,
string, by curtailing the counting of beads to for example, via 40 + 30 ¼ 70; 4 + 7 ¼ 11; and
counting by tens and ones. Students may start 70 + 11 ¼ 81. Counting in jumps would involve

N 468 Number Lines in Mathematics Education
3 10 10 10 10 4
37 40 50 60 70 80 84

10 10 10 10 7
37 47 57 67 77 84

7 40

37 47 84

Number Lines in Mathematics Education, Fig. 3 Various strategies for 37 + 47 on the number line

solutions such as 44 + 37 ¼ . . .; 44 + 30 ¼ 74; partitioning, jumps of 10, and relating subtraction
74 + 7 ¼ 81 or 44 + 37 ¼ . . .; 44 + 6 ¼ 50; and addition. This program has been integrated in
50 + 10 ¼ 60; 60 + 10 ¼ 70; 70 + 10 ¼ 80; and a teaching and learning trajectory for calculation
80 + 1 ¼ 81. According to Beishuizen (1993), with whole numbers in primary school in the
procedures based on splitting lead to more errors, Netherlands (Heuvel-Panhuizen 2001). The latter
than solution procedures that are based on advices to start with counting in jumps in grade 2
curtailed counting. Other researchers found that and to expand the repertoire of mental calculation
students tend to come up with a wide variety of techniques in grade 3 with de split method and
counting solutions when confronted with “linear- “flexible” or “varying” strategies.
type” context problems (see Gravemeijer 2004).
Capitalizing on counting strategies therefore An Alternative Approach
fits the reform mathematics idea of supporting Parallel to this, an alternative approach has been
students in constructing their own mathematical developed in which the bead string is replaced by
knowledge. a series of measuring tools in an interactive
inquiry classroom culture setting (Stephan et al.
Further Elaboration 2003; Gravemeijer 2004). This approach gives
Treffer’s approach with the bead string as precur- priority to unitizing and to developing a network
sor to the empty number line is further elaborated of number relations. The rationale of the focus on
in an instructional program that aims at teaching number relations is that the students’ knowledge
flexible solution strategies via a process of of number relations forms the basis for what –
progressive schematizing, which proceeds along from an observer’s point of view – looks like the
three levels of schematizing: informal/contextu- application of strategies. While what the students
alized; semiformal/model supported; and formal/ actually do is combining number facts which are
arithmetical. This process is supported by the ready to hand to them, in order to derive new
training of subskills. The program consists of number facts. According to this view, the
two parts, “numbers” and “operations with construction of a network of number relations
numbers.” The former addresses the basic skills involves a shift from numbers that signify count-
of counting, ordering and localizing, and able objects for the students, to numbers as enti-
jumping to given numbers. The latter addresses ties in and of themselves. This idea is further
complementary skills, such as addition to 10, elaborated with the emergent modeling design

Number Lines in Mathematics Education N469

heuristic (Gravemeijer 1999) in design experi- original approach, the actions on the number N
ment in Nashville (Stephan et al. 2003). Here line are expected to signify corresponding activ-
the choice for measuring is dictated by the ambi- ities on the bead string. In the sequence that is
guity of the numbers on the number line. On the based on linear measurement, a series of transi-
one hand, the numbers refer to quantities, and, tions take place in which activities with new tools
on the other hand, they refer to positions on have to fulfill the imagery criterion: first, when
the number line. Most addition and subtraction the activity of measuring various lengths by iter-
problems that the students have to solve deal with ating some measurement unit is curtailed to mea-
quantities, while the solution methods involve the suring with tens & ones; next, when the activity
order of the numbers in the number sequence. of iterating tens & ones is modeled with a ruler;
Linear measurement offers the opportunity to then, when the activity shifts from measuring to
address this ambiguity. A number on a ruler reasoning about measures while incrementing,
also signifies both a position and a quantity or decrementing or comparing lengths; thereafter,
a magnitude. And students may develop a deeper when the arithmetical solution methods that
understanding of the relation between the two, may be supported by referring to the decimal
when they come to see the activity of measuring structure of the ruler are symbolized with arcs
as the accumulation of distance. The latter on an empty number line; and finally, when this
implies that each number word used in the activ- more abstract representation is used as a way of
ity of iterating signifies the total measure of the scaffolding and as a way of communicating solu-
distance measured until that moment. From an tion methods for all sorts of addition and subtrac-
emergent modeling perspective, the notion of tion problems.
a ruler can be construed as an overarching
model. The ruler may be seen as a curtailment Effect Studies
of iterating a measurement unit and thus emerge Most Dutch primary school textbooks are compat-
as a model of iterating some measurement unit, ible with the way the empty number line approach
which is superseded by the empty number line as is elaborated in the “teaching and learning
a more abstract ruler that functions as a model for trajectory for calculation with whole numbers in
mathematical reasoning with numbers up to 100. primary school in the Netherlands” (Heuvel-
Panhuizen 2001) that was mentioned earlier. The
As a caveat, however, it should be noted that results of national surveys halfway primary school,
ample care has to be taken to avoid that the empty nevertheless, show that the Dutch students are not
number line is seen as a simplified picture of as proficient in subtracting two-digit numbers as
a ruler. Instead, the jumps on the number line might have been expected (Kraemer 2011).
have to be perceived as means of describing A follow-up study on the solution procedures of
one’s arithmetical thinking. In contrast to the students (Kraemer 2011) shows that jumping is
ruler, the empty number line should not be seen used frequently, and with good results, but the
as proportional. For trying to strive for an exact other methods generate many wrong answers. His
proportional representation would severely ham- data further reveal a strong tendency to solve con-
per flexible use of the number line. textual problems in two directions, via direct sub-
traction or indirect addition, and bare sums
Imagery primarily in one direction, direct subtraction,
An issue of concern is what the number line and which is not always efficient. Kraemer (2011)
its precursors signify for the students. A teaching argues that the identified patterns suggest the stu-
experiment in which the number line was not dents use what works for them. This is initially the
preceded by a bead string or a ruler showed the combination of jumping and “subtract strategies.”
importance of “imagery.” To come to grips with Over time, however, they start trying to combine
a new tool, the students have to be able to see an these strategies with split and reasoning proce-
earlier activity with earlier tools in the activities dures. Then they run into problems because they
with the new tool (Gravemeijer 1999). In the

N 470 Number Teaching and Learning

0 ¼ ½ ¾ 1 tank References

0 15 30 45 60 liters Beishuizen M (1993) Mental strategies and materials or
models for addition and subtraction up to 100 in Dutch
Number Lines in Mathematics Education, Fig. 4 The second grades. J Res Math Educ 24(4):294–323
amount of fuel in a gasoline tank
Gravemeijer K (1999) How emergent models may foster
still miss important conceptual and instrumental the constitution of formal mathematics. Math Think
building blocks for splitting and more Learn 1(2):155–177
sophisticated reasoning with numbers up to 100.
Gravemeijer K (2004) Learning trajectories and local
From these findings, we may conclude that instruction theories as means of support for teachers
careful attention has to be paid to fostering a in reform mathematics education. Math Think Learn
conceptual understanding of splitting strategies 6(2):105–128
and variable strategies and of the relations
between the various strategies. Which is to show Klein AS, Beishuizen M, Treffers A (1998) The empty
that the empty number line can be a powerful tool, number line in Dutch second grades: realistic versus
but its success is very dependent of the way it is gradual program design. J Res Math Educ
embedded in a broader instructional setting. 29(4):443–464

The Double Number Line Kraemer J-M (2011) Oplossingsmethoden voor aftrekken
Although the empty number line is well researched, tot 100. Dissertatie, Cito, Arnhem
a variant of it, the double number line, has not
gotten that much attention. The double number Selter C (1998) Building on children’s mathematics –
line can be used as a means of support for coordi- a teaching experiment in grade three. Educ Stud
nating two units of measure. This is particularly Math 36(1):1–27
useful in the domains of fractions and percentages,
where the units are often linked to numerosities or Stephan M, Bowers J, Cobb P, Gravemeijer K (eds)
magnitudes (van Galen et al. 2008). Here we may (2003) Supporting students’ development of measur-
think, for instance, of reasoning about the content ing conceptions: analyzing students’ learning in social
of a petrol tank which can hold 60 l: half a tank context, Journal for research in mathematics education
contains 30 l, ¼ tank half of that, and ¾ tank the monograph no. 12. National Council of Teachers
sum of the latter two (see Fig. 4). of Mathematics, Reston

Further research is needed to establish Treffers A (1991) Didactical background of
whether working with the double number a mathematics program for primary education. In:
line can – similarly to the empty number line – Streefland L (ed) Realistic mathematics education in
effectively foster more formal forms of primary school. Cd-b Press, Utrecht, pp 21–57
mathematical reasoning.
Whitney H (1985) Taking responsibility in school mathe-
matics education. In: Streefland L (ed) Proceedings of
the ninth international conference for the psychology
of mathematics education, vol 2. OW&OC, Utrecht

van den Heuvel-Panhuizen M (ed) (2001) Children learn
mathematics. Freudenthal Institute, Utrecht Univer-
sity/SLO, Utrecht/Enschede

van Galen F, Feijs E, Gravemeijer K, Figueiredo N, van
Herpen E, Keijzer R (2008) Fractions, percentages, dec-
imals and proportions: a learning-teaching trajectory for
grade 4, 5 and 6. Sense Publishers, Rotterdam

Number Teaching and Learning

Cross-References Demetra Pitta-Pantazi
Department of Education, University of Cyprus,
▶ Curriculum Resources and Textbooks in Nicosia, Cyprus
Mathematics Education
Keywords
▶ Manipulatives in Mathematics Education
▶ Mathematical Representations Arithmetic; Irrational numbers; Natural numbers;
▶ Semiotics in Mathematics Education Negative numbers; Rational numbers
▶ Visualization and Learning in Mathematics

Education

Number Teaching and Learning N471

Characteristics Natural Numbers, Operations, and Estimation N
Research on numbers has been under the focus of
“Numbers” is one of the most important strands researchers since the end of the nineteenth century
in the mathematics curricula worldwide. and has attracted enormous research attention.
According to Verschaffel, Greer and De Corte Dewey (1898) was one of the first researchers who
(2007), there are several reasons for this: provided an analysis of early number and presented
(a) The teaching of numbers is worldwide, and methods for teaching arithmetic. A few years later,
number operations and applications are Thorndike (1922) published a book “The Psychol-
connected and used in real life; (b) it relates and ogy of Arithmetic” where he presented the nature,
constitutes a foundation for all other topics in measurement, and construction of arithmetical
mathematics; and (c) it is one of the first abilities. Since then the learning and teaching of
topics students are formally taught in school, numbers has been a fundamental stream of mathe-
and students’ disposition to mathematics often matics education research. Until the 1950s most of
depends on this. Because of its importance and the work on numerical understanding concen-
its nature, number learning and teaching has trated mainly on natural numbers, number
attracted enormous attention by mathematics sequence, counting, and subitizing. After the
education researchers, experimental psycholo- 1950s one of the most dominant theoretical and
gists, cognitive psychologists, developmental methodological approaches that guided research
psychologists, and neuroscientists. Through the in numbers was Piaget’s theory, which suggested
years, a better understanding has been developed that the construction of natural numbers is based
regarding the components that constitute numer- on logical reasoning abilities (e.g., conservation
ical understanding, the nature of its learning and of number, class inclusion, transitivity property,
development, the learning environments that and seriation). The Piagetian tradition tended to
facilitate this learning as well as appropriate disregard counting and subitizing. A second
tools for assessing the learning and teaching of theoretical approach that guided research was the
numbers. counting-based approach which suggested that
numerical concepts evolve from counting skills
Natural Numbers which individuals develop through the quantifica-
Researchers identified three main strands tion process (Bergeron and Herscovics 1990).
of research on numbers: the behaviorist, the cog-
nitive, and the situative (Greeno, et al. 1996). Most of the studies that flourished from the
Research on numbers has been mostly cognitive 1980s until the 1990s were mostly cognitive and
(Bergeron and Herscovics 1990; Verschaffel rather local (Bergeron and Herscovics 1990). They
et al. 2006, 2007), but since the 1990s the influ- described students’ development, their strategies,
ence of situative theories such as social construc- and misconceptions as well as their conceptual
tivism, ethnomathematics, and situated cognition structures of whole number concepts and opera-
made a strong impact (Verschaffel et al. 2006). tions (Verschaffel et al. 2006). Research has
Recently, the prevailing view is that cognitive shown that children’s understanding of numbers
and situated methodologies (e.g., tests, clinical and their operations progresses successively in
research procedures, experimental teaching, more abstract, complex, and general conceptual
design experiments, computer simulations, structures (Fuson 1992). Researchers also identified
action research) may be combined to give a better three main categories of strategies that students use
picture of the various phenomena. In this when solving one-step addition and subtraction
section we will examine the way in which word problems: direct modeling with physical
research on natural numbers has evolved over objects, verbal counting (counting all, counting
the years. This evolution also applies for research on, or counting back), and mental strategies
on rational and in some extent of other (derived facts and known facts). Students’ strate-
number systems. gies were also explored in multiplication and divi-
sion although less extensively than in addition and

N 472 Number Teaching and Learning

subtraction. Researchers suggested that develop- mathematical learning; (c) and the acquisition of
ment in subtraction and division progresses in numerical knowledge out of school and ways in
a different way from that of addition and which this knowledge may be exploited and used in
multiplication. the classrooms. Several researchers examined the
impact that the environment where individuals
Researchers also explored students’ abilities in grow and act may have on their abilities with
operations with multi-digit numbers. These studies numbers (e.g., ethnomathematics).
suggested that a number of students often make
procedural mistakes in algorithms since they get After the 1990s, the field of cognitive
confused by the multistep procedures, while in neuroscience also started making links to mathe-
other cases they have poor conceptual understand- matics education research. Neuroscientific research
ing of place value, grouping, and ungrouping. examined students’ mental structures of numbers
Researchers also claimed that without efficient and the way in which individuals internally repre-
knowledge of basic number facts, students are sent and process numbers. The idea was that brain
bound to have difficulties in multi-digit oral and activation (e.g., using fMRI) might provide us with
written arithmetic (Kilpatrick et al. 2001). The a more detailed picture of the cognitive sub pro-
relationship of strategies, principles, and number cesses that have an effect on mathematical thinking
facts was also examined. One of the findings of and learning.
these studies was that different strategies may be
employed by students when dealing with different After 2000 an increased research interest
numbers (Kilpatrick et al. 2001). has been shown by mathematics educators on
early childhood understanding of numbers and
Special attention was also given to the abilities, their operations (Sarama and Clements 2009).
difficulties, and strategies of students with Researchers argued that young children’s informal
learning difficulties in numbers and their opera- mathematical knowledge is strong, wide, and
tions as well as of the appropriate teaching advanced. Researchers developed level of cognitive
approaches for these students (e.g., Baroody 1999). progressions, in various number domains, with the
use of learning trajectories (Sarama and Clements
Apart from the emphasis on number operations, 2009). Other research studies demonstrated
current reform documents call for emphasis on that young children need to be engaged
estimation. Three types of estimation were identi- in sophisticated, purposeful, and meaningful
fied: numerosity, computational, and measurement mathematical activities which will support
(Sowder 1992). Although research on estimation is the development of various strategies (Sarama
rather limited, researchers seem to agree that esti- and Clements 2009) and students’ conceptual
mation is complex and difficult for students and understanding of number.
often for adults. It develops over time and individ-
uals use either self-invented or taught strategies to Numbers and Problem Solving
respond to estimation tasks. In the 1980s–1990s, word problems followed
a cognitive approach with emphasis on students’
After the 1990s research on numbers was strategies, errors, and mental structures. Three
affected by the situated theoretical perspective and types of additive number problems were
more specifically the emerging theoretical frame- identified (change, combine, and compare), two
works of social constructivism, ethnomathematics, types of multiplicative problems (asymmetric
and situated cognition. According to Verschaffel (equal grouping, multiplicative comparison, rate)
et al. (2006) numerous studies focused on (a) the and symmetric (area, Cartesian product) (Greer
design, implementation, and evaluation of instruc- 1992)), and two types of division problems
tional programs, such as Realistic Mathematics (partitioning and measurement). Researchers also
Education and the impact of the socio- looked into intuitive models that may affect stu-
mathematical norms on mathematical learning; dents’ responses such as multiplication being
(b) teachers content knowledge, pedagogical con- repeated addition and division as partition.
tent knowledge, actions and beliefs in the learning
of numbers, and their impact on students’

Number Teaching and Learning N473

In their review Verschaffel et al. (2006) argued Number Sense N
that research on problem solving focused on In recent years curricula reforms use extensively
different aspects each time: (a) on conceptual the term “number sense” and consider it a major
schemas that students possess while solving such essential outcome of school curricula. Although,
problems, (b) on students’ strategies when dealing its importance in mathematics curricula is recog-
with numbers in the context of mathematical nized; its usefulness in research is controversial
problems, (c) on the different heuristic and (Verschaffel et al. 2007). This arises from the fact
metacognitive skills in the solution of numerical that there is no catholic acceptance of what this
problems, and (d) on problem posing. However, term involves. Most often the term “number
after the 1990s, it became apparent that the cogni- sense” encompasses the (a) use of different
tive perspective which guided this initial research representations of numbers, (b) identification of
was mainly related to problems which were not relative and absolute magnitudes of numbers, (c)
authentic. At this point the impact of situative use system of benchmarks, (d) composition and
theoretical perspective became stronger, and decomposition of numbers, (e) conceptual under-
researchers started investigating students’ miscon- standing of operations, (f) estimation, (g) mental
ceptions based on social, cultural, affective, and computations, as well as (h) the judgment about
metacognitive factors such as, students’ informal the reasonableness of results. Based on the descrip-
knowledge, teachers’ mathematical and pedagog- tions and on the components of “number sense,”
ical knowledge, and teaching approaches. This there have been several attempts to construct
move also led to an increased interest in the intro- tools to measure number sense. Furthermore,
duction of modeling problems from as early as researchers also focused on designing intervention
primary schools as well “emergent modeling” programs and examining their impact.
activities related to numbers.
Rational Numbers
Arithmetic and Other Mathematical Domains There is a lot of research on students’ understand-
Early in the twentieth century, the teaching of ing of rational numbers at different levels (from
arithmetic was restricted to performing the young learners to prospective and in-service
standard operations and algorithms. In the 1980s teachers). These studies are mainly epistemologi-
emphasis was given to the procedural and con- cal, cognitive, and situative. Most of them concen-
ceptual understanding of numbers (Hiebert trated on the various interpretations and
1986). This emphasis continued well into the representations of rational numbers, students’ abil-
1990s. At this point the reform of mathematics ities, and in a smaller extent on instructional pro-
curricula also yielded a shift towards the grams. Recently Confrey, Maloney, and Nguyen
development of students’ understanding of (2008) identified eight major areas of research on
numbers and emphasized the importance for rational numbers: (1) fractions; (2) multiplication
students to investigate the relationships, patterns, and division; (3) ratio, proportion, and rate;
and connections. Extensive attention started to (4) area; (5) decimals and percent; (6) probability;
emerge regarding the connections between (7) partitioning; and (8) similarity and scaling.
numbers and algebra. A number of researchers Based on this synthesis, they concluded that ratio-
claimed that arithmetic is essentially algebraic nal number is a complex concept and its teaching
and can set the ground for formal algebra. needs major revisions, especially regarding the
At the same time, they argued that algebra sequence of the topics taught.
can strengthen the understanding of arithmetic
structure (Verschaffel et al. 2007). Special Epistemological and Cognitive View Rational
emphasis was also given to the connections of Numbers Learning
numbers to other areas of mathematics such as Research studies that had taken an epistemolog-
measurement, geometry, and probability but ical view tried to clarify the nature of rational
especially data handling. number and its subconstructs. Kieren (1976)

N 474 Number Teaching and Learning

was the first to propose that fractions consist of Furthermore, research also indicated that
four subconstructs: measure, ratio, quotient, and individuals often have a procedural understand-
operator. Later Behr, Lesh, Post, and Silver ing of fraction operations which is attributed to
(1983) extended this model and proposed that the reliance of mechanical learning of rules. For
part-whole/partitioning is posited a fundamental instance, young students accept the representa-
subconstruct underlying the other four subconstructs tion of “a” parts of “b” unequal parts as fractions
previously suggested by Kieren (1976). According or that in the division of fractions one needs to
to researchers none of the subconstructs can stand reverse the second fraction and multiply. In
alone. Each construct allows the consideration of addition to this, researchers seem to agree out of
rational numbers from a different perspective. the four operations, division of fraction is the
Other studies discussed the different cognitive struc- most difficult for individuals to understand.
tures needed to understand the various subconstructs
of rational numbers. Several researchers, interna- Teaching of Rational Numbers
tional curricula, and textbooks are in favor of the According to Behr et al. (1992) until 1992 few
inclusion of multiple fraction subconstructs and research studies specifically targeted teaching of
argued that students benefit from this. rational numbers. Lamon (2007) argued that this
was due to the fact that the research domain includ-
Studies on rational numbers which are ing rational numbers, fractions, ratios, and propor-
cognitive in nature concentrated on the investiga- tions had not reached a level of maturity which
tion of the cognitive structures children bring into could inform teaching practices. A number of
the understanding of rational numbers and the researchers (e.g., Lamon 2007; Confrey et al.
way in which these cognitive structures develop 2008) designed intervention programs by identify-
when the children are formally introduced to ing learning trajectories and then tested their results.
rational numbers. Such cognitive studies also
concentrated on the development of the concep- The use of manipulatives (concrete or virtual)
tual understanding of fractions and the obstacles and of multiple representations are considered as
to this learning. For instance, a number of studies very important in the teaching of fractions and
concentrated on the way that the conceptualiza- especially in teaching operations with fractions.
tion of whole numbers may affect students under- Research has shown positive impact of the use of
standing of rational numbers and make sense of visual representations on students’ conceptual
decimal and fractions notations (Streefland understanding of fractions. A widely used repre-
1991). Students often do not interpret fractions sentation is the area model representation (Saxe
as numbers but view fractions as two numbers et al. 2007). However, according to Saxe et al.
with a line between them. When adding fractions, (2007), area models have some limitations. These
they often add the numerators and denominators researchers (Saxe et al. 2007) conducted research
or are unable to order fractions from smaller to studies and designed programs in order to inves-
larger (e.g., Behr et al. 1992). Regarding the tigate the way in which number lines can help
decimal representation of fractions, young students develop understanding of the fraction
students often believe that decimal numbers concepts and their properties. The use of number
have a predictable order and that decimals with line was also acknowledged as very important for
more digits after the decimal point are larger than the understanding of decimal. In addition, there is
decimals with fewer digits after the decimal also considerable research on the investigation of
point. Steffe and Olive (2010) have a rather dif- virtual representations and more extensively of
ferent view. They argue that the mental digital technologies in the learning of fractions.
operations necessary for the understanding of
whole number should not be viewed as an obsta- Negative Numbers
cle to fractions understanding but as a foundation The concept of negative numbers is introduced
for fractional understanding (reorganization when students have already learned to work with
hypothesis). natural numbers. As a result, when the teaching of

Number Teaching and Learning N475

negative numbers begins, some properties focused on this topic. The concept of irrational N
concerning natural numbers turn out to be numbers is considered as one of the most difficult
conflicting. Fischbein (1987) claimed that two intu- concepts in mathematics, especially since it does
itive obstacles affect students’ understanding of not present discrete countable quantities but
negative numbers. First, the concept of negative refers to continuous quantities. It is a by-product
numbers is intuitively contradictory to the concept of logical deduction and cannot be captured by
of positive numbers defined as quantifiable entities. our senses. Most studies emphasize the deficien-
Secondly, negative numbers are a “by-product of cies in students’ and teachers’ understanding of
mathematical calculations and not the symbolic irrational numbers; for instance, their difficulty to
expression of existing properties” (Fischbein 1987, provide appropriate definition for irrational num-
p. 101). Another main obstacle identified is the ber or to recognize whether a number is rational
difficulty to see the number line as one thing or irrational. Students often develop the under-
(unified number line) where the value of numbers standing of natural, rational, and irrational num-
is supported and not as two opposite semi-lines bers as different systems and are unable to see
(divided line) where only the magnitude of numbers them in a flexible whole (Zazkis and Sirotic
is supported. Students have to realize the difference 2010). This is often a source of students’ difficul-
between the magnitude and the value of the number. ties and misconceptions. Other research studies
Despite the difficulties students might face while showed that students’ and prospective teachers’
dealing with negative numbers, there is also evi- difficulties may also arise from the discreteness
dence that students have intuitive knowledge about of natural numbers, which is a barrier to under-
negative numbers and in some cases are able to standing the dense structure of the rational and
perform operations with negative numbers before irrational numbers. Attempts were also made
formal instruction. Thus, recently a number of to develop instructional material for secondary
researchers claimed that addition and/or subtraction school students, preservice, and in-service
with negative numbers may be introduced from teachers.
younger ages if appropriate models are used.
Still there is also evident that students face difficulty Cross-References
to move from concrete operations to formal
operations. ▶ Affect in Mathematics Education
▶ Algorithms
Regarding the teaching of negative numbers, ▶ Data Handling and Statistics Teaching and
there is a long-standing debate whether they
should be introduced through models (such as Learning
number lines, elevators, or the annihilation/ ▶ Early Algebra Teaching and Learning
creation model where two-color counters are ▶ Early Childhood Mathematics Education
used) (Verschaffel et al. 2006) or as formal abstrac- ▶ Ethnomathematics
tions (Fischbein 1987). Most of the researchers ▶ Information and Communication Technology
seem to adopt the model approach. There is no
consensus regarding the model or representation (ICT) Affordances in Mathematics Education
which is most effective as well as the number of ▶ Intuition in Mathematics Education
different models (multiplicity or not) that should be ▶ Learning Difficulties, Special Needs, and
used. Opinions are also conflicting regarding the
use of these models and whether they should be Mathematics Learning
used only during the introduction of negative num- ▶ Manipulatives in Mathematics Education
bers or all the way through the teaching of integers. ▶ Mathematical Modelling and Applications in

Irrational Numbers Education
Despite the importance of irrational numbers, ▶ Mathematical Representations
only a small number of research studies have ▶ Metacognition
▶ Misconceptions and Alternative Conceptions

in Mathematics Education

N 476 Number Teaching and Learning

▶ Number Lines in Mathematics Education Greer B (1992) Multiplication and division as models of
▶ Pedagogical Content Knowledge in situations. In: Grouws DA (ed) Handbook of research
on mathematics teaching and learning. Macmillan,
Mathematics Education New York, pp 276–295
▶ Probability Teaching and Learning
▶ Problem Solving in Mathematics Education Hiebert J (1986) Conceptual and procedural knowledge:
▶ Realistic Mathematics Education the case of mathematics. Erlbaum, Hillsdale
▶ Shape and Space – Geometry Teaching and
Kieren TE (1976) On the mathematical, cognitive, and
Learning instructional foundations of rational numbers. In:
▶ Situated Cognition in Mathematics Education Lesh R (ed) Number and measurement: papers from
▶ Sociomathematical Norms in Mathematics a research workshop. ERIC/SMEAC, Columbus,
pp 101–144
Education
▶ Teacher Beliefs, Attitudes, and Self-Efficacy Kilpatrick J, Swafford J, Findell B (2001) Adding it up:
helping children learn mathematics. National
in Mathematics Education Academy Press, Washington, DC
▶ Word Problems in Mathematics Education
Lamon SJ (2007) Rational numbers and proportional
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ME, Elliott PC (eds) The learning of mathematics:
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concepts. In: Lesh R, Landau M (eds) Acquisition of of Mathematics, Reston, pp 221–237
mathematics concepts and processes. Academic Press,
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Grouws DA (ed) Handbook of research on mathemat-
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(eds) Mathematics and cognition: a research synthesis
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a educational approach. Reidel, Dordrecht concepts and operations. In: Lester FK (ed) Second
handbook of research on mathematics teaching and
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and learning. In: Berliner DC, Calfee RC (eds) pp 51–82
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pp 1–27

P

Pedagogical Content Knowledge in the particular form of content knowledge that
Mathematics Education embodies the aspects of content most germane to
its teachability. (p. 9)

Hamsa Venkat and Jill Adler Immediate and widespread interest in the
University of the Witwatersrand, Johannesburg, notion rested on Shulman’s claim that PCK, com-
South Africa bined with subject knowledge and curriculum
knowledge, formed critical knowledge bases
Keywords for understanding and improving subject-specific
teaching. While subject matter knowledge (SMK)
Mathematical knowledge for teaching; Profes- and PCK are frequently dealt with together in
sional knowledge; Lee Shulman, Deborah Ball research studies, interest and contestation in the
boundary lead to separate but related entries for
Characteristics them in this encyclopedia (see SMK entry). PCK
studies in mathematics education indicate attempts
Intense focus on the notion of “pedagogical content at (a) sharpening theorizations of PCK, (b) mea-
knowledge” (PCK) within teacher education is suring PCK, and (c) using notions of PCK to build
attributed to Lee Shulman’s 1985 AERA Presiden- practical skills within teacher education or combi-
tial address (Shulman 1986) in which he referred to nations of these elements. This entry summarizes
PCK as the “special amalgam of content and key work across these groups.
pedagogy” central to the teaching of subject matter.
His widely cited follow-up paper (Shulman 1987) Theorizations of PCK
elaborated PCK as follows: Key writings in the category of sharpening theo-
rizations of PCK examine both the boundary
the most powerful analogies, illustrations, exam- between PCK and the broader field of subject-
ples, explanations, and demonstrations — [. . .] the related knowledge – sometimes referred to as
most useful ways of representing and formulating “mathematics knowledge for teaching” (MKT) –
the subject that make it comprehensible to others.... and inwards at subcategories within PCK.
Pedagogical content knowledge also includes an
understanding of what makes the learning of Deborah Ball and the Michigan research group
specific topics easy or difficult: the conceptions sharpened the distinctions between content knowl-
and preconceptions that students of different ages edge and PCK in their theorization based on the
and backgrounds bring with them. . . (p. 7) classroom practices of expert teachers: “subject
matter knowledge” (SMK) broke down into

S. Lerman (ed.), Encyclopedia of Mathematics Education, DOI 10.1007/978-94-007-4978-8,
# Springer Science+Business Media Dordrecht 2014

P 478 Pedagogical Content Knowledge in Mathematics Education

common content knowledge (CCK), specialized that were administered to teachers, with data
content knowledge (SCK), and horizon knowledge collected on their elementary grade classes’
and PCK into knowledge of content and students learning backgrounds and learning gains across
(KCS), knowledge of content and teaching, and a year. Hill et al.’s (2005) analysis showed con-
knowledge of curriculum (Ball et al. 2008). tent knowledge measures across the common and
specialized categories as significantly associated
Critiques of work drawing from Shulman’s with learning gains. While Ball’s group concep-
categorizations argue that the “static” conceptual- tualizes CCK and SCK as part of content
ization of MKT with separate components is knowledge, the descriptions of SCK that are
unhelpful in relation to the interactive and provided – e.g., understanding of representations
dynamic nature of MKT. Centrally, these critiques and explanations – fall within other writers’
argue that MKT is better interpreted as an attribute conceptualizations of PCK.
of pedagogic practices in specific contexts and
related to specific mathematical ideas, rather than Baumert et al. (2010), noting the absence
a generalized attribute of the teacher. Fennema of direct attention to teaching in Ball
and Franke’s (1992) conceptualization of MKT et al.’s measurement-oriented work, developed
as constituted by knowledge of mathematics, the COACTIV framework that distinguished
combined with PCK comprised of elements of content knowledge from PCK and examined the
knowledge of learners’ mathematical cognition, relationships between content knowledge, PCK,
pedagogical knowledge, and beliefs views this classroom teaching, and student learning gains
combination as a taxonomy that can identify the in Germany. In the COACTIV (Professional
“context-specific knowledge” of a teacher, rather Competence of Teachers, Cognitively Activating
than a more generalized picture of the teacher’s Instruction, and the Development of Students’
MKT. Rowland et al. (2003) similarly emphasize, Mathematical Literacy) model (focused on
in their “Knowledge Quartet” formulation secondary mathematics teaching), content knowl-
consisting of Foundation, Transformation, Con- edge is understood as “a profound mathematical
nection, and Contingency knowledge (the latter understanding of the mathematics taught at
three relating to PCK), that the profile of MKT school” (p. 142), and PCK is subdivided into
produced is a categorization of teaching situations, knowledge of mathematical tasks as instructional
rather than of teachers. tools, knowledge of students’ thinking and assess-
ment of understanding, and knowledge of multiple
While all of these models were developed representations and explanations of mathematical
from studies of practice, Fennema and Franke problems. With this distinction, separate content
and Rowland et al.’s models include a beliefs knowledge and PCK open response items were
component – which does not feature in Ball developed and administered to nearly 200 teachers
et al.’s conceptualization. in different tracks of the German schooling
system. Mathematics test performance data were
Other studies have looked at PCK in alterna- gathered from over 4,000 students in these
tive formulations (e.g., Silverman and Thompson teachers’ classes. Instructional quality was mea-
2008), with the notion of “connections” within sured through three data sources. The first
mathematics and with learning (Askew et al. encompassed selected class, homework, test and
1997; Ma 1999) seen as critical. Petrou and examinations tasks, and the degree of alignment
Goulding (2011) provide an overview of key between assessment tasks and the Grade 10 cur-
writings in the MKT field. riculum. The second source considered the extent
of individual learning support, measured through
Measuring PCK student rating scales. The third source examined
Ball’s research group shifted their attention into classroom management as degree of agreement
measuring MKT to verify assumptions about its between teacher and student perceptions about
relationship to teaching quality and student disciplinary climate.
learning. The group developed multiple choice
items based on specific MKT subcomponents

Pedagogical Content Knowledge in Mathematics Education P479

Baumert et al.’s findings suggested that Cross-References P
their theoretical division of content knowledge
and PCK was empirically distinguishable, ▶ Mathematical Knowledge for Teaching
with their PCK variable showing more sub-
stantial associations with student achievement References
and instructional quality than their content
knowledge variable. Askew M, Brown M, Rhodes V, Johnson DC, Wiliam D
(1997) Effective teachers of numeracy. Report of
Using PCK to Support the Development of a study carried out for the Teacher Training Agency
Pedagogic Practice 1995–96 by the School of Education, King’s College
The third category of PCK literature links to London. Teacher Training Agency, London
studies of teacher development using PCK frame-
works. This strand often uses longitudinal case Askew M, Venkat H, Mathews C (2012) Coherence and
study methodologies. consistency in South African primary mathematics
lessons. In: Tso T-Y (ed) 36th conference of the
Fennema and Franke and Rowland’s MKT international group for the psychology of mathematics
models have associated development-focused education: opportunities to learn in mathematics
studies. Turner and Rowland (2011) provide education, vol 2. National Taiwan Normal University,
examples of the Knowledge Quartet’s use in Taipei, pp 27–34
England to stimulate development of teaching,
and Fennema and Franke, with colleagues, have Ball D, Thames MH, Phelps G (2008) Content knowledge
produced studies on the longevity of the PCK fr teaching: what makes it special? J Teach Educ
aspects presented within professional develop- 59(5):389–407
ment programs.
Baumert J, Kunter M, Blum W, Brunner M, Voss T,
This category too contains other studies draw- Jordan A, Tsai Y (2010) Teachers’ mathematical
ing on aspects of PCK. Kinach (2002) focuses on knowledge, cognitive activation in the classroom,
secondary mathematics teachers’ development of and student progress. Am Educ Res J 47(1):133–180
instructional explanations – a key feature of PCK
across different formulations. Learning studies Fennema E, Franke ML (1992) Teachers’ knowledge and
interventions (Lo and Pong 2005) focus on its impact. In: Grouws DA (ed) Handbook of research
building teachers’ awareness of the relationship on mathematics teaching and learning. Macmillan,
between particular objects of learning and New York, pp 147–164
students’ work with these objects – a feature of
the KCS terrain. Hill H, Rowan B, Ball D (2005) Effects of teachers’
mathematical knowledge for teaching on student
Emerging Directions achievement. Am Educ Res J 42(2):371–406
Emerging work questions the assumption of basic
coherence and connection in MKT that underlies Kinach BM (2002) A cognitive strategy for developing
much of the PCK writing (Silverman and pedagogical content knowledge in the secondary
Thompson 2008). Qualitative case studies of mathematics methods course: toward a model of
classroom teaching detail inferences relating effective practice. Teach Teach Educ 18:51–71
to PCK (and SMK), and the consequences for
mathematics learning in contexts of pedagogic Lo ML, Pong WY (2005) Catering for individual
fragmentation and disconnections are beginning differences: building on variation. In: Lo ML, Pong
to feature (Askew et al. 2012). WY, Pakey CPM (eds) For each and everyone: catering
for individual differences through learning studies.
PCK as a field therefore continues to thrive, in Hong Kong University Press, Hong Kong, pp 9–26
spite of ongoing differences in nomenclature,
underlying views about specific sub-elements, Ma L (1999) Knowing and teaching elementary mathe-
and the nature of their interaction. matics: teachers’ understandings of fundamental
mathematics in China and the United States. Lawrence
Erlbaum Associates, Mahwah

Petrou M, Goulding M (2011) Conceptualising teachers’
mathematical knowledge in teaching. In: Rowland T,
Ruthven K (eds) Mathematical knowledge in teaching.
Springer, Dordrecht, pp 9–25

Rowland T, Huckstep P, Thwaites A (2003) The
knowledge quartet. In: Williams J (ed) Proceedings
of the British Society for Research into Learning
Mathematics. 23(3), November

Shulman LS (1986) Those who understand: knowledge
growth in teaching. Educ Res 15(2):4–14

Shulman LS (1987) Knowledge and teaching: foundations
of the new reform. Harv Educ Rev 57(1):1–22

P 480 Policy Debates in Mathematics Education

Silverman J, Thompson PW (2008) Toward a framework (including sets, functions, matrices, vectors),
for the development of mathematical knowledge for statistics and probability, computer mathematics
teaching. J Math Teach Educ 11:499–511 (including base arithmetic), and modern geome-
try (transformation geometry, topological graph
Turner F, Rowland T (2011) The knowledge quartet as an theory). The launch of Sputnik, the first earth
organising framework for developing and deepening orbiting satellite, by the Soviet Union in 1957,
teachers’ mathematical knowledge. In: Rowland T, during the Cold War led to fears that the USA and
Ruthven K (eds) Mathematical knowledge in teaching. UK were being overtaken in technology and in
Springer, Dordrecht, pp 195–212 mathematics and science education by the
Soviets. Government funding became available,
Policy Debates in Mathematics especially in the USA, to extend projects mod-
Education ernizing the mathematics curriculum in a bid to
broaden and improve students’ knowledge of
Paul Ernest mathematics, such as the Madison Project in
School of Education, University of Exeter, 1957 and The School Mathematics Study Group
Exeter, Devon, UK in 1958 in the USA. In the UK independent cur-
riculum projects emerged, including the School
Definition Mathematics Project (SMP) in 1961 and Nuffield
Primary Mathematics in 1964. These projects did
Policy in mathematics education concerns the not cause much controversy at the national policy
nature and shape of the mathematics curriculum, levels although there was a concern by parents
that is, the course of study in mathematics of that they did not understand the New Math their
a school or college. This is the teaching sequence children were learning. The relatively muted
for the subject as planned and experienced by the debates concerned the changing content of the
learner. Four aspects can be distinguished, and mathematics curriculum rather than its pedagogy
these are the focuses of policy debates: or assessment.
1. The aims, goals, and overall philosophy of the
In the mid to late 1960s onwards a new debate
curriculum emerged about discovery learning. In the UK
2. The planned mathematical content and its the Schools Council Curriculum Report No. 1
(Biggs 1965) on the teaching and learning
sequencing, as in a syllabus of mathematics in primary school proposed
3. The pedagogy employed by teachers practical approaches and “discovery learning”
4. The assessment system as the most effective ways of teaching mathemat-
ics. Sixty-five percent of all primary teachers in
History the UK read Biggs (1965), and it had a significant
impact. Discovery learning was a central part of
The New Math debate of the late 1950s to the the 1957 Madison Project developed by Robert
mid-1960s was primarily about the content of the B. Davis. This and similar developments led to
mathematics curriculum. At that time traditional a major policy debate on discovery learning. Is
school mathematics did not incorporate any discovery learning the most effective way to
modern topics. The content consisted primarily learn mathematics? Proponents of discovery
of arithmetic at elementary school, plus contrasted it with rote learning. Self-evidently
traditional algebra, Euclidean geometry, and rote learning cannot be the best way to learn
trigonometry at high school. The New Math cur- all but the simplest mathematical facts and
riculum broadened the elementary curriculum to skills since it means simply “learning by heart.”
include other aspects of mathematics, and high However, educational psychologist Ausubel
school mathematics incorporated modern algebra (1968) argued successfully that discovery and
rote learning are not part of a continuum but on

Policy Debates in Mathematics Education P481

two orthogonal axes defined by pairs of Analytical Framework P
opposites: meaningful versus rote learning and
reception versus discovery learning. Meaningful The British government developed and installed
learning is linked to existing knowledge; it is the first legally binding National Curriculum in
relational and conceptual. Rote learning is arbi- 1988 for all students age 5–16 years in all state
trary, verbatim, and disconnected – unrelated to schools (excluding Scotland). The debate over
other existing knowledge of the learner. Knowl- the mathematics part of National Curriculum in
edge learned by reception comes already formu- became a heated contest between different social
lated and is acquired through communication, interest groups. Ernest (1991) analyzed this as a
such as in expository teaching or reading. Ausubel contest between five different groups with different
distinguishes this from discovered knowledge broad ranging ideologies of education, the aims,
that has to be formulated by the learner herself. and orientation of which are summarized in Table 1
(In the full treatment there are 14 different ideo-
The promotion of discovery learning led to logical components for each of these 5 groups).
heated debate on both sides of the Atlantic.
Shulman and Keislar (1966) offered a review, but These different social groups were engaged in
to this day the evidence remains equivocal. This a struggle for control over the National Curricu-
debate was primarily about pedagogy – how best to lum in mathematics, since the late 1980s (Brown
teach mathematics. But underneath this debate one 1996). In brief, the outcome of this contest was that
can discern battle lines being drawn between the first three more reactionary groups managed to
a child-centered, progressive ideology of education win a place for their aims in the curriculum. The
with roots going back to Rousseau, Montessori, fourth group (progressive educators) reconciled
Dewey, and a traditionalist teacher- and knowl- themselves with the inclusion of a personal knowl-
edge-centered ideology of education favored by edge-application dimension, namely, the processes
some mathematicians and university academics. of “Using and Applying mathematics,” constitut-
ing one of the National Curriculum assessment
The mid-1970s saw the birth of the back-to- targets. However instead of representing progres-
basics movement promoting basic arithmetical sive self-realization through creativity aims
skills as the central goal of the teaching through mathematics, this component embodies
and learning of mathematics for the majority. utilitarian aims: the practical skills of being able
This was a reaction to the progressivism of the to apply mathematics to solve work-related prob-
previous decade, most clearly defined in the aims lems with mathematics. Despite this concession
of the Industrial Trainers group mentioned below, over the nature of the process element included in
and became an important plank of the traditionalist the curriculum, the scope of the element has been
position on the mathematics curriculum. reduced over successive revisions that have
occurred in the subsequent 20 years and has largely
The early 1980s led to a further entrenchment been eliminated. The fifth group, the public educa-
in the progressive/traditional controversy. In the tors, found their aims played no part in the National
USA the influential National Council of Teachers Curriculum. The outcome of the process is
of Mathematics (NCTM) recommended that a largely utilitarian mathematics curriculum devel-
“Problem solving must be the focus of school oping general or specialist mathematics skills and
mathematics in the 1980s” (1980, pp. 2–4). capabilities, which are either decontextualized –
In the UK the Cockcroft Inquiry (1982) equipping the learner with useful tools – or
recommended problem solving and investiga- which are applied to practical problems.
tional work be included in mathematics for all The contest between the interest groups was an
students. Thus the debate remained at the level of ideological one, concerning not only all four
pedagogy but shifted to problem solving. aspects of curriculum but also about deeper epis-
temological theories on the nature of mathematics
The progressivist versus traditionalist debate and the nature of learning.
was born anew in the late 1980s (UK) and the
1990s (USA) but now encompassed the whole
mathematics curriculum on a national basis.

P 482 Policy Debates in Mathematics Education

Policy Debates in Mathematics Education, Table 1 Five interest groups and their aims for mathematics teaching
(based on Ernest 1991)

Interest group Social location Orientation Mathematical aims
1. Industrial
trainers Radical New Right conservative Authoritarian, Acquiring basic mathematical skills and
politicians and petty bourgeois basic skills numeracy and social training in obedience
2. Technological centered
pragmatists Meritocratic industry-centered
industrialists, managers, etc., New Industry and Learning basic skills and learning to solve
3. Old Humanist Labor work centered practical problems with mathematics and
mathematicians Conservative mathematicians information technology
preserving rigor of proof and purity
4. Progressive of mathematics Pure Understanding and capability in advanced
educators Professionals, liberal educators, mathematics mathematics, with some appreciation of
5. Public welfare state supporters centered mathematics
educators Democratic socialists and radical
reformers concerned with social Child-centered Gaining confidence, creativity, and
justice and inequality progressivist self-expression through maths

Empowerment Empowerment of learners as critical and
and social justice mathematically literate citizens in society
concerns

During the period following the introduction them for themselves. This initiated the savage
of the National Curriculum in mathematics, debate in the USA called the Math Wars (Klein
pressure from various groups continued to be 2007).
exerted to shift the emphasis of the curriculum.
Mathematicians who can often be characterized The Standards influenced a generation of new
as belonging to the Old Humanist grouping mathematics textbooks in the 1990s, often funded
published a report entitled Tackling the Mathe- by the National Science Foundation. Although
matics Problem (London Mathematical Society widely praised by mathematics educators, partic-
1995), commissioned by professional mathemati- ularly in California, concerned parents formed
cal organizations. This criticized the inclusion grassroots organizations to object and to pressure
of “time-consuming activities (investigations, schools to use other textbooks. Reform texts were
problem solving, data surveys, etc.)” at the expense criticized for diminished content and lack of
of “core” technique and technical fluency. Further- attention to basic skills and an emphasis on
more, it claimed many of these activities are poorly progressive pedagogy based on constructivist
focused and can obscure the underlying mathemat- learning theory. Critics in the debate derided
ics. This criticism parallels that heard in the “math mathematics programs as “dumbed-down” and
wars” debate in the USA. described the genre as “fuzzy math.”

“Math Wars” In 1997 Senator Robert Byrd joined the debate
by making searing criticisms of the mathematics
In the USA the National Council of Teachers of education reform movement from the Senate floor
Mathematics (NCTM) published its so-called focusing on the inclusion of political and social
Standards document in 1989 recommending justice dimensions in one mathematics textbook.
a “Reform”-based (progressive) mathematics cur- In the spreading and increasingly polarized debate,
riculum for the whole country. This emphasized the issues spread from traditional versus progressive
problem solving and constructivist learning theory. content and pedagogy to left versus right political
The latter is not just discovery learning under a new orientations and traditional objectivist versus con-
name because constructivists acknowledge that structivist (relativist) epistemology and philosophy
learners need to be presented with representations of mathematics. This way the debate took on
of existing mathematical knowledge to reconstruct aspects of the parallel “science wars” also taking
place, primarily in the USA. This is the heated
debate between scientific realists, who argued that
objective scientific knowledge is real and true, and

Policy Debates in Mathematics Education P483

sociologists of science. The latter questioned scien- mathematics or the child should be the central P
tific objectivity and argued that all knowledge is focus of the curriculum. Proponents of a child-
socially constructed. This is an insoluble epistemo- centered curriculum promote general pedagogy
logical dispute that has persisted at least since the in teacher education as opposed to a specifically
time of Socrates in philosophical debates between mathematics pedagogy with its associated
skeptics and dogmatists. Nevertheless, it fanned the emphasis on teachers’ pedagogical content
flames of the Math Wars debate. knowledge in mathematics.

In 1999 the US Department of Education The spread of policy debates has also become
released a report designating 10 mathematics much wider following the impact of international
programs as “exemplary” or “promising.” Several assessment projects such as TIMSS. Politicians
of the programs had been singled out for criticism sometimes blame what is perceived to be poor
by mathematicians and parents. Almost immedi- national performance levels in mathematics on
ately an open letter to Secretary of Education Rich- one or other aspect of the curriculum. Unfortu-
ard Riley was published calling on him to withdraw nately policy debates too often become
these recommendations. Over 200 university math- politicized and drift away from the central issues
ematicians signed their names to this letter and of determining the best mathematics curriculum
included seven Nobel laureates and winners of the for students. In becoming polarized, the debates
Fields Medal. This letter was repeatedly used by become controversies that propel policy swings
traditionalists in the debate to criticize Reform from one extreme to the other, like a pendulum.
mathematics, and in 2004 NCTM President Johnny Ernest (1989) noted this pattern, but regrettably
Lott posted a strongly worded denunciation of the the pendulum-like swings from one extreme posi-
letter on the NCTM website. tion to the opposite continue unabated to this day.

In 2006, President Bush was stirred into action The fruitlessness of swings from traditional
by the heated controversy and created the National to progressive pedagogy in mathematics is
Mathematics Advisory Panel to examine and sum- illustrated in an exemplary piece of research by
marize the scientific evidence related to the teach- Askew et al. (1997). This project studied the belief
ing and learning of mathematics. In their 2008 sets and teaching practices of primary school
report, they concluded that recommendations that teachers and their correlation with students’
instruction should be entirely “student centered” or numeracy scores over a period of 6 months.
“teacher directed” are not supported by research. Three belief sets and approaches to teaching
High-quality research, they claimed, does not sup- numeracy were identified in the teachers:
port the exclusive use of either approach. The 1. Connectionist beliefs: valuing students’
Panel called for an end to the Math Wars, although
its recommendations were still the subject of crit- methods and teaching with emphasis on
icism, especially from within the mathematics edu- establishing connections in mathematics
cation community for its comparison of extreme (mathematics and learner centered)
forms of teaching and for the criteria used to deter- 2. Transmission beliefs: primacy of teaching and
mine “high-quality” research. view of maths as collection of separate rou-
tines and procedures (traditionalist)
Defusing the Debates 3. Discovery beliefs: primacy of learning and
view of mathematics as being discovered by
Policy debates have raged over the mathematics students (progressivist)
curriculum throughout the past 50 years. They The classes of teachers with a connectionist
have been strongest in the USA and UK but orientation made the greatest gains, so teaching
have occurred elsewhere in the world as well. In for connectedness were measurably the most
Norway, for example, there is a much more effective methods. This included attending to
muted but still heated debate as to whether and valuing students’ methods as well as teaching
with an emphasis on establishing connections in
mathematics. Traditional transmission beliefs