Encyclopedia of Mathematics Education

C 78 Communities of Practice in Mathematics Education

Communities of Inquiry Didacticians inquiring with teachers
in Mathematics Teacher to promote professional development...
Education, Fig. 1 Three
layers of inquiry in Teachers engaging
mathematics teaching in professional inquiry ...
development
Students engaging in
inquiry in mathematics

in the classroom

to learn more about creating mathematical
opportunities for students

and learn more about practical implications
and issues for mathematical
development

Cross-References Communities of Practice
in Mathematics Education
▶ Communities of Practice in Mathematics
Education Ellice A. Forman
University of Pittsburgh, Pittsburgh, PA, USA
▶ Inquiry-Based Mathematics Education
▶ Mathematics Teacher Education Organization,

Curriculum, and Outcomes

References Keywords

Cochran Smith M, Lytle SL (1999) Relationships Ethnomathematics; Informal learning; Situated
of knowledge and practice: teacher learning in cognition
communities’. In: Iran-Nejad A, Pearson PD (eds)
Review of research in education. American Educa- Definition and Originators
tional Research Association, Washington, DC,
pp 249–305 Communities of practice (CoP) are an important
component of an emerging social theory of
Ernest P (1991) The philosophy of mathematics educa- learning. Lave and Wenger (1991) originally
tion. Falmer Press, London envisioned this social learning theory as a way
to deepen and extend the notion of situated
Jaworski B (2006) Theory and practice in mathematics learning that occurs in traditional craft appren-
teaching development: critical inquiry as a mode of ticeships, contexts in which education occurs
learning in teaching. J Math Teach Educ outside of formal schools. Drawing upon
9(2):187–211 evidence from ethnographic investigations of
apprenticeships in a range of settings (e.g., tailor-
Jaworski B, Wood T (eds) (2008) The mathematics ing), they have frequently argued that it is
teacher educator as a developing professional, vol 4,
International handbook of mathematics teacher
education. Sense, Rotterdam, pp 335–361

Wells G (1999) Dialogic inquiry. Cambridge University
Press, Cambridge

Wenger E (1998) Communities of practice. Cambridge
University Press, Cambridge

Communities of Practice in Mathematics Education C79

important to separate learning from formal school as well as social products (e.g., tools, language, C
contexts to understand that most human activities laws) in order to help newcomers master the
involve some form of teaching and learning. important practices of their community. In
Wenger (1998) argued that CoP’s two compo- addition, social theories of learning are needed
nents (community and practice) are inherently to address some of the fundamental quandaries of
connected by three dimensions: “(1) mutual educational research and practice (Sfard 2008).
engagement; (2) a joint enterprise; (3) a shared These enduring dilemmas include the unwilling-
repertoire” (p. 73). One important aim of a CoP is ness of some students to expend enough energy
the negotiation of meaning among participants. to master difficult mathematical concepts and
This is one way to differentiate groups of people the puzzling discrepancy in performance on in-
who live or work in the same location from other school and out-of-school mathematical problems.
groups who are actively involved in communi-
cating with each other about important issues and History of Use
working together towards common goals. Lave’s (1988, 2011) own empirical research
Another important aspect of CoP is that learning began with a focus on mathematical proficiency
may be demonstrated by changes in the personal in out-of-school settings (tailoring garments).
identities of the community members. Changes in She initially chose situated cognition tasks that
identity are accompanied by increasing participa- required mathematical computations so that she
tion in the valued practices of this particular could more easily compare them with school-like
CoP as newcomers become old-timers in the tasks. Similar work in ethnomathematics was
community. conducted by other colleagues for a range of
cultural activities (e.g., selling candy on the
Characteristics street) (Nunes et al. 1993). One recurrent finding
of this research has been that children, adoles-
How the Problem Was Identified and Why cents, and adults can demonstrate higher levels of
Social theories of learning have a long history in mathematical proficiency in their out-of-school
psychology (Cole 1996). Nevertheless, more activities than in school, even when the
experimental and reductionist theories were the actual mathematical computations are the same
predominant form of psychology until the late (Forman 2003). Another finding was that social
twentieth century. The reemergence of social processes (e.g., guided participation) and cultural
theories of learning has occurred in numerous tools (e.g., currency) were important resources
places, such as discursive psychology (Harre´ for people as they solved mathematical problems
and Gillett 1994), as well as in mathematics outside of school (Saxe 1991). This research
education (Lerman 2001; van Oers 2001). The forces one to question the validity of formal
reasons why we need a social learning theory in assessments of mathematical proficiency and
mathematics education have been outlined by to wonder how mathematical concepts and
Sfard (1998). She contrasted two key metaphors: procedures are developed in everyday contexts
learning as acquisition versus learning as partic- of work and play. Many of these investigators
ipation. Most research conducted during the last began to question the basic assumptions of our
century in mathematics education used the acqui- individual learning theories and turn their
sition metaphor. In contrast, the participation attention to developing new social theories of
metaphor shifts the focus from individual owner- learning, like those proposed by Lave, Wenger,
ship of skills or ideas to the notion that learners Saxe, and Sfard. It also led them to rely, to an
are fundamentally social beings who live and increasing degree on ethnographic studies of
work as members of communities. Teaching everyday life (Lave 2011).
and learning within CoP depend upon social
processes (collaboration or expert guidance) When Lave and others began their research in
the early 1970s, it had limited impact on research
in schools. This has changed as mathematics

C 80 Communities of Practice in Mathematics Education

educators have begun to use this research to Perspectives on Issue in Different
improve school instruction, curriculum develop- Cultures/Places
ment, and teacher education (e.g., Nasir and The origin of CoP in ethnomathematics means
Cobb 2007). For example, Cobb and Hodge that the earliest research was conducted in
(2002) use the notion of CoP to propose that we cultural settings very different from those of
investigate at least two types of mathematical European or North American classrooms such
communities in our work: local (home, school, as Brazil, Liberia, Mexico, and Papua New
neighborhood) and broader (district, state, Guinea. In addition to a broad range of national
national, international). In the classroom, all of settings, this ethnographic work focused on the
these types of communities affect students’ mathematical reasoning that occurred in the daily
access to high levels of mathematical reasoning lives of poor and middle-class people who may or
and problem solving as well as their own sense of may not be enrolled in formal educational
identity as capable mathematical learners. Cobb institutions. After it began to be applied to school
and Hodge suggest that we try to understand and classrooms, many of its research sites were
acknowledge these discontinuities in students’ located in Europe or North America (e.g., Seeger
communities that can impact their motivation, et al. 1998). Thus, unlike many educational inno-
self-image, and school achievement. Building, vations, the study of CoP began in impoverished
in part, on the work of Luis Moll and his locations and later spread to wealthy settings. In
colleagues (Gonzalez et al. 2004), Cobb and addition, the methods of ethnography previously
Hodge recommend that teachers view their used to study the work and play lives of people in
students’ families and neighborhood communi- impoverished communities were then applied to
ties as sources of important funds of knowledge classrooms where experimental methods are
that can be accessed in school. By viewing home more often used (Lave 2011).
proficiencies in a positive light (as funds of
knowledge) instead of deficits (limited formal Gaps that Need to Be Filled
education), these educators propose that the In Wenger’s (1998) expanded formulation of
discontinuities between classroom and home CoP, he clarified its dynamic properties. Forms
CoP be minimized and destigmatized, making it of mutual engagement change over time within
easier for students to engage more fully in learn- any community. Collective goals evolve as dif-
ing about mathematics and other subjects. ferent interpretations clash and new understand-
ings are negotiated. As this occurs, new tools are
Unfortunately, many students from impoverished created and modified, new vocabulary developed,
neighborhoods experience discontinuities between and new routines and narratives invented. Only
their home and peer communities and those of recently have ethnographers like Lave and Saxe
schools. As a result, researchers such as Nasir been able to take a long view of the CoP they
(2007) have tried to forge new connections originally studied in the 1970s and 1980s. As
between students’ out-of-school lives (e.g., when a result they have been able to deepen their
they use mathematics to win at basketball) and the theoretical positions and reconceptualize their
mathematics they are taught in school. One key methods. Clearly this is one area where the
component of these connections is the need to theoretical formulations of CoP are ahead of the
engage students in the process of viewing them- empirical results in mathematics classrooms.
selves as capable learners of mathematics and
establishing a new identity as successful mathe- Another area that has been growing in mathe-
matics problem solvers (Martin 2007). Finally, matics education is the study of identity in math-
Cobb and Hodge (2002) remind us that resolving ematics classrooms (e.g., Martin 2007). Although
the tensions between local communities of practice this work is also hampered by the limited time
for students may also involve acts of imagination span studied, early work suggests that repeated
when young people work towards a revised and experiences with different classroom CoP may
expanded identity for themselves in the future. result in negative or positive reactions to their

Communities of Practice in Mathematics Education C81

instructional experiences in mathematics by Harre R, Gillett G (1994) The discursive mind. Sage, C
students (Boaler and Greeno 2000). The develop- London
ment of a person’s mathematical identity may
build slowly over time in homes, communities, Lampert M (1990) When the problem is not the question
and schools through recurrent processes such as and the solution is not the answer: mathematical
social positioning by parents, teachers, and peers knowing and teaching. Am Educ Res J 27:29–63
(Yamakawa et al. 2009).
Lave J (1988) Cognition in practice: mind, mathematics
Finally, an area of rich growth in mathematics and culture in everyday life. Cambridge University
education is the attempt to align classroom com- Press, New York
munities with those of professional mathemati-
cians (Lampert 1990). For example, studies of Lave J (2011) Apprenticeship in critical ethnographic
argumentation and proof in K-12 classrooms practice. University of Chicago Press, Chicago
indicate that even young children are capable of
articulating their reasons for their mathematical Lave J, Wenger E (1991) Situated learning: legitimate
decisions and defending those positions when peripheral participation. Cambridge University Press,
carefully guided by an experienced teacher New York
(O’Connor 2001). In addition, current research
in other content areas such as history or science Lerman S (2001) Cultural, discursive psychology:
shows that argumentation can be productively a sociocultural approach to studying the teaching
fostered across the curriculum as well as in and learning of mathematics. Educ Stud Math
different grades. 46(1–3):87–113

Cross-References Martin DB (2007) Mathematics learning and participation
in the African American context: the co-construction
▶ Ethnomathematics of identity in two intersecting realms of experience. In:
▶ Situated Cognition in Mathematics Education Nasir NS, Cobb P (eds) Improving access to
mathematics: diversity and equity in the classroom.
References Teachers College Press, New York, pp 146–158

Boaler J, Greeno JG (2000) Identity, agency, and knowing Nasir NS (2007) Identity, goals, and learning: the case of
in mathematics worlds. In: Boaler J (ed) Multiple basketball mathematics. In: Nasir NS, Cobb P (eds)
perspectives on mathematics teaching and learning. Improving access to mathematics: diversity and equity
Ablex, Westport, pp 171–200 in the classroom. Teachers College Press, New York,
pp 132–145
Cobb P, Hodge LL (2002) A relational perspective on
issues of cultural diversity and equity as they play out Nasir NS, Cobb P (eds) (2007) Improving access to
in the mathematics classroom. Math Think Learn mathematics: diversity and equity in the classroom.
4(2–3):249–284 Teachers College Press, New York

Cole M (1996) Cultural psychology: a once and future Nunes T, Schliemann AD, Carraher DW (1993) Street
discipline. Belknap Press of Harvard University Press, mathematics and school mathematics. Cambridge
Cambridge, MA University Press, Cambridge

Forman EA (2003) A sociocultural approach to mathe- O’Connor MC (2001) “Can any fraction be turned into
matics reform: speaking, inscribing, and doing mathe- a decimal?” A case study of a mathematical group
matics within communities of practice. In: Kilpatrick discussion. Educ Stud Math 46(1–3):143–185
J, Martin WG, Schifter D (eds) A research companion
to the principles and standards for school mathematics. Saxe GB (1991) Culture and cognitive development.
National Council of Teachers of Mathematics, Reston, Lawrence Erlbaum, Hillsdale
pp 333–352
Seeger F, Voigt J, Waschescio U (1998) The culture of the
Gonzalez N, Moll LC, Amanti C (eds) (2004) Funds of mathematics classroom. Cambridge University Press,
knowledge: theorizing practices in households, New York
communities, and classrooms. Lawrence Erlbaum
Associates, Mahwah Sfard A (1998) On two metaphors for learning and the
dangers of choosing just one. Educ Res 27(2):4–13

Sfard A (2008) Thinking as communicating: human
development, the growth of discourse, and
mathematizing. Cambridge University Press,
New York

van Oers B (2001) Educational forms of initiation in
mathematical culture. Educ Stud Math 46(1–3):59–85

Wenger E (1998) Communities of practice: learning,
meaning, and identity. Cambridge University Press,
New York

Yamakawa Y, Forman EA, Ansell E (2009) The role of
positioning in constructing an identity in a third
grade mathematics classroom. In: Kumpulainen K,
Hmelo-Silver CE, Cesar M (eds) Investigating class-
room interaction: methodologies in action. Sense,
Rotterdam, pp 179–202

C 82 Communities of Practice in Mathematics Teacher Education

Communities of Practice in participants in social practices that shape
Mathematics Teacher Education identities.

Merrilyn Goos To further analyze the concepts of identity and
School of Education, The University of community of practice, Wenger (1998) proposed
Queensland, Brisbane, QLD, Australia a more elaborated social theory of learning that
integrates four components – meaning (learning
Keywords as experience), practice (learning as doing), com-
munity (learning as belonging), and identity
Mathematics teacher education; Communities of (learning as becoming). Wenger explained that
practice; Identity communities of practice are everywhere – in
people’s workplaces, families, and leisure pur-
Definition suits, as well as in educational institutions. Most
people belong to multiple communities of prac-
Communities of practice in mathematics tice at any one time and will be members of
teacher education are informed by a theory of different communities throughout their lives.
learning as social participation, in which teacher His theory has been applied to organizational
learning and development are conceptualized as learning as well as learning in schools and other
increasing participation in social practices that formal educational settings.
develop an identity as a teacher.
Communities of Practice as a
Framework for Understanding
Mathematics Teacher Learning
and Development

Background Social theories of learning are now well
established in research on mathematics educa-
The idea of learning in a community of practice tion. Lerman (2000) discussed the development
grew from Jean Lave’s and Etienne Wenger’s of “the social turn” in mathematics education
research on learning in apprenticeship contexts research and proposed that social theories
(Lave 1988; Lave and Wenger 1991). Drawing drawing on community of practice models
on their ethnographic observations of apprentices provide insights into the complexities of teacher
learning different trades, Lave and Wenger learning and development. From this perspective,
developed a theory of learning as social practice learning to teach involves developing an identity
to describe how novices come to participate in as a teacher through increasing participation in
the practices of a community. These researchers the practices of a professional community
introduced the term “legitimate peripheral (Lerman 2001). At the time of publication of
participation” to explain how apprentices, as Lerman’s (2001) review chapter on research
newcomers, are gradually included in the com- perspectives on mathematics teacher education,
munity through modified forms of participation there were few studies drawing on Lave’s and
that are accessible to potential members working Wenger’s ideas. Reviewing the same field
alongside master practitioners. Although social 5 years later, Llinares and Krainer (2006)
practice theory aimed to offer a perspective on noted increasing interest in using the idea of
learning in out-of-school settings, Lave (1996) a community of practice to conceptualize
afterwards argued that apprenticeship research learning to teach mathematics. Such studies can
also has implications for both learning and teach- be classified along several dimensions, according
ing in schools and for students and teachers as to their focus on:

Communities of Practice in Mathematics Teacher Education C83

1. Preservice teacher education or the different contexts in which prospective and C
professional learning and development of beginning teachers’ learning occurs – the univer-
practicing teachers sity teacher education program, the practicum,
and the early years of professional experience
2. Face-to-face or online interaction (or a (Llinares and Krainer 2006). One of the more
combination of both) common discontinuities is evident in the diffi-
culty many beginning teachers experience in sus-
3. Questions about how a community of taining the innovative practices they learn about
practice is formed and sustained compared in their university courses. This observation can
with questions about the effectiveness of be explained by acknowledging that prospective
communities of practice in promoting teacher and beginning teachers participate in separate
learning communities – one based in the university and
Research has been informed by the two key the other in school – which often have different
regimes of accountability that regulate what
conceptual strands of Wenger’s (1998) social counts as “good teaching.”
practice theory. One of these strands is related
to the idea of learning as increasing participation Researchers have also investigated how partici-
in socially situated practices and the other to pation in online communities of practice supports
learning as developing an identity in the context the learning of prospective and practicing teachers
of a community of practice. of mathematics, and insights into principles
informing the design of such communities are
Learning as Participation in Practices beginning to emerge (Goos and Geiger 2012).
With regard to participation in practices, Wenger Some caution is needed in interpreting the findings
describes three dimensions that give coherence to of these studies, since few present evidence that a
communities of practice: mutual engagement of community of practice has actually been formed:
participants, negotiation of a joint enterprise, and for example, by analyzing the extent of mutual
development of a shared repertoire of resources engagement, how a joint enterprise is negotiated,
for creating meaning. Mutuality of engagement and whether a shared repertoire of meaning-making
need not require homogeneity, since productive resources is developed by participants (Goos and
relationships arise from diversity and these may Bennison 2008). Nevertheless, studies of online
involve tensions, disagreements, and conflicts. communities of practice demonstrate that technol-
Participants negotiate a joint enterprise, finding ogy-mediated collaboration does more than simply
ways to do things together that coordinate increase the amount of knowledge produced by
their complementary expertise. This negotiation teachers; it also leads to qualitatively different
gives rise to regimes of mutual accountability forms of knowledge and different relationships
that regulate participation, whereby members between participants.
work out who is responsible for what and to
whom, what is important and what can safely Learning as Developing an Identity
be ignored, and how to act and speak appro- With regard to identity development, Wenger
priately. The joint enterprise is linked to the wrote of different modes of belonging to
larger social system in which the community a community of practice through engagement,
is nested. Such communities have a common imagination, and alignment. Beyond actually
cultural and historical heritage, and it is engaging in practice, people can extrapolate
through the sharing and reconstruction of this from their experience to imagine new possibili-
repertoire of resources that individuals come ties for the self and the social world. Alignment,
to define their relationships with each other in the third mode of belonging, refers to coordinat-
the context of the community. ing one’s practices to contribute to the larger
enterprise or social system. Alignment can
This aspect of Wenger’s theory has been used
to investigate discontinuities that may be experi-
enced in learning to teach mathematics in the

C 84 Communities of Practice in Mathematics Teacher Education

amplify the effects of a practice and increase the to understand the role of teacher educators in
scale of belonging experienced by community shaping teachers’ learning.
members, but it can also reinforce normative
expectations of practice that leave people Cross-References
powerless to negotiate identities.
▶ Communities of Inquiry in Mathematics
Research into teacher identity development in Teacher Education
communities of practice is perhaps less advanced
than studies that analyze evidence of changing ▶ Communities of Practice in Mathematics
participation in the practices of a community. Education
This may be due to a lack of well-developed
theories of identity that can inform research ▶ Mathematics Teacher as Learner
designs and provide convincing evidence ▶ Mathematics Teacher Identity
that identities have changed. Jaworski’s (2006) ▶ Professional Learning Communities in
work on identity formation in mathematics
teacher education proposes a conceptual shift Mathematics Education
from learning within a community of practice
to forming a community of inquiry. The References
distinguishing characteristic of a community of
inquiry is reflexivity, in that participants critically Goos M, Bennison A (2008) Developing a communal
reflect on the activities of the community in identity as beginning teachers of mathematics:
developing and reconstructing their practice. emergence of an online community of practice.
This requires a mode of belonging that Jaworski J Math Teach Educ 11(1):41–60
calls “critical alignment” – adopting a critically
questioning stance in order to avoid perpetuating Goos M, Geiger V (2012) Connecting social perspectives
undesirable normative states of activity. on mathematics teacher education in online
environments. ZDM Int J Math Educ 44:705–715
Issues for Future Research
Graven M, Lerman S (2003) Book review of Wenger
Elements of Wenger’s social practice theory E (1998) Communities of practice: learning, meaning
resonate with current ways of understanding and identity. Cambridge University Press, Cambridge,
teachers’ learning, and this may explain why UK. J Math Teach Educ 6:185–194
his ideas have been taken up so readily by
researchers in mathematics teacher education. Jaworski B (2006) Theory and practice in mathematics
Nevertheless, the notion of situated learning in teaching development: critical inquiry as a mode of
a community of practice composed of experts and learning in teaching. J Math Teach Educ 9:187–211
novices was not originally focused on school
classrooms, nor on pedagogy, and so caution is Lave J (1988) Cognition in practice. Cambridge
needed in applying this perspective on learning as University Press, New York
an informal and tacit process to learning in formal
education settings, including preservice and Lave J (1996) Teaching, as learning, in practice. Mind
in-service teacher education (Graven and Lerman Cult Act 3(3):149–164
2003). Wenger’s model was developed from
studying learning in apprenticeship contexts, Lave J, Wenger E (1991) Situated learning: legitimate
where teaching is incidental rather than peripheral participation. Cambridge University Press,
deliberate and planned, as in university-based New York
teacher education. It remains to be seen whether
community of practice approaches can be applied Lerman S (2000) The social turn in mathematics education
research. In: Boaler J (ed) Multiple perspectives on
mathematics teaching and learning. Ablex, Westport,
pp 19–44

Lerman S (2001) A review of research perspectives on
mathematics teacher education. In: Lin F-L, Cooney T
(eds) Making sense of mathematics teacher education:
past, present and future. Kluwer, Dordrecht, pp 33–52

Llinares S, Krainer K (2006) Mathematics (student)
teachers and teacher educators as learners. In:
Gutierrez A, Boero P (eds) Handbook of research on
the psychology of mathematics education: past, pre-
sent and future. Sense, Rotterdam, pp 429–459

Wenger E (1998) Communities of practice: learning,
meaning, and identity. Cambridge University Press,
New York

Competency Frameworks in Mathematics Education C85

Competency Frameworks in They are as follows: general cognitive competen- C
Mathematics Education cies, specialized cognitive competencies, the
competence-performance model, modifications of
Jeremy Kilpatrick the competence-performance model, cognitive
University of Georgia, Athens, GA, USA competencies and motivational action tendencies,
objective and subjective competence concepts,
Keywords and action competence. Competency frameworks
in mathematics education fall primarily into
Competence; Conceptual framework; Taxonomy; Weinert’s specialized-cognitive-competencies
Subject matter; Mental process category, but they also overlap some of the
other categories.
Definition
The progenitor of competency frameworks in
A structural plan for organizing the cognitive mathematics education is Bloom’s (1956)
skills and abilities used in learning and doing Taxonomy of Educational Objectives, which
mathematics. attempted to lay out, in a neutral way, the
cognitive goals of any school subject. The main
Characteristics categories were knowledge, comprehension,
application, analysis, synthesis, and evaluation.
The concept of competence is one of the most These categories were criticized by mathematics
elusive in the educational literature. Writers often educators such as Hans Freudenthal and Chris
use the term competence or competency and Ormell as being especially ill suited to the subject
assume they and their readers know what it of mathematics (see Kilpatrick 1993 on the
means. But arriving at a simple definition is critiques as well as some antecedents of Bloom’s
a challenging matter. Dictionaries give such work). Various alternative taxonomies have
definitions as “the state or quality of being subsequently been proposed for school mathe-
adequately or well qualified”; “the ability matics (see Trista´n and Molgado 2006,
to do something successfully or efficiently”; pp. 163–169, for examples). Further, Bloom’s
“possession of required skill, knowledge, taxonomy has been revised (Anderson and
qualification, or capacity”; “a specific range Krathwohl 2001) to separate the knowledge
of skill, knowledge, or ability”; and “the dimension (factual, conceptual, procedural, and
scope of a person’s or group’s knowledge or metacognitive) from the cognitive process
ability.” Competence seems to possess a host of dimension (remember, understand, apply, ana-
near synonyms: ability, capability, cognizance, lyze, evaluate, and create), which does address
effectuality, efficacy, efficiency, knowledge, one of the complaints of mathematics educators
mastery, proficiency, skill, and talent – the list that the original taxonomy neglected content in
goes on. favor of process. But the revision nonetheless
fails to address such criticisms as the isolation
Arriving at a common denotation across of objectives from any context, the low
different usages in social science is even more placement of understanding in the hierarchy of
difficult. “There are many different theoretical processes, and the failure to address important
approaches, but no single conceptual framework” mathematical processes such as representing,
(Weinert 2001, p. 46). Weinert identifies seven conjecturing, and proving.
different ways that “competence has been defined,
described, or interpreted theoretically” (p. 46). Whether organized as a taxonomy, with an
explicit ordering of categories, or simply as an
arbitrary listing of topics, a competency frame-
work for mathematics may include a breakdown
of the subject along with the mental processes
used to address the subject, or it may simply

C 86 Competency Frameworks in Mathematics Education

treat those processes alone, leaving the mathe- committee addressed the following question:
matical content unanalyzed. An example of the What does it mean to master mathematics?
former is the model of outcomes for secondary They identified eight competencies, which fell
school mathematics proposed by James Wilson into two groups. The first four address the ability
(cited by Trista´n and Molgado 2006, p. 165). In to ask and answer questions in and with
that model, mathematical content is divided mathematics:
into number systems, algebra, and geometry; 1. Thinking mathematically
cognitive behaviors are divided into computa- 2. Posing and solving mathematical problems
tion, comprehension, application, and analysis; 3. Modeling mathematically
and affective behaviors are either interests and 4. Reasoning mathematically
attitudes or appreciation. Another example is
provided by the framework proposed for The second four address the ability to deal
the Third International Mathematics and Science with and manage mathematical language and
Study (TIMSS; Robitaille et al., 1933, tools:
Appendix A). The main content categories are 5. Representing mathematical entities
numbers; measurement; geometry (position, 6. Handling mathematical symbols and
visualization, and shape; symmetry, congruence,
and similarity); proportionality; functions, rela- formalisms
tions, and equations; data representation, prob- 7. Communicating in, with, and about mathematics
ability, and statistics; elementary analysis; 8. Making use of aids and tools
validation and structure; and other content
(informatics). The performance expectations are Niss (2003) observes that each of these com-
knowing, using routine procedures, investigating petencies has both an analytic and a productive
and problem solving, mathematical reasoning, side. The analytic side involves understanding
and communicating. and examining the mathematics, whereas the
productive side involves carrying it out. Each
Other competency frameworks, like that of competency can be developed and used only by
Bloom’s (1956) taxonomy, do not treat different dealing with specific subject matter, but the
aspects of mathematical content separately but choice of curriculum topics is not thereby deter-
instead attend primarily to the mental processes mined. The competencies, though specific to
used to do mathematics, whether the results mathematics, cut across the subject and can be
of those processes are termed abilities, addressed in multiple ways.
achievements, activities, behaviors, performances,
practices, proficiencies, or skills. Examples include The KOM project also found it necessary to
the five strands of mathematical proficiency iden- focus on mathematics as a discipline. The project
tified by the Mathematics Learning Study of the committee identified three kinds of “overview
US National Research Council – conceptual and judgment” that students should develop
understanding, procedural fluency, strategic com- through their study of mathematics: its actual
petence, adaptive reasoning, and productive dispo- application, its historical development, and its
sition – and the five components of mathematical special nature. Like the competencies, these qual-
problem-solving ability identified in the Singapore ities are both specific to mathematics and general
mathematics framework: concepts, skills, pro- in scope.
cesses, attitudes, and metacognition (see Kilpatrick
2009, for details of these frameworks). Niss (2003) observes that the competencies
and the three kinds of overview and judgment
A final example of a competency framework can be used: (a) normatively, to set outcomes
in mathematics is provided by the KOM for school mathematics; (b) descriptively, to
project (Niss 2003), which was charged with characterize mathematics teaching and learning;
spearheading the reform of mathematics in the and (c) metacognitively, to help teachers and
Danish education system. The KOM project students monitor and control what they are teach-
ing or learning. These three usages apply as well
to the other competency frameworks developed
for mathematics.

Complexity in Mathematics Education C87

Regardless of whether a competency frame- Trista´n A, Molgado D (2006) Compendio de taxonom´ıas: C
work is hierarchical and regardless of whether it Clasificaciones para los aprendizajes de los dominios
addresses topic areas in mathematics, its primary educativos [Compendium of taxonomies: classifica-
use will be normative. Competency frameworks tions for learning in educational domains]. Instituto
are designed to demonstrate to the user that learn- de Evaluacio´n e Ingenier´ıa Avanzada, San Luis Potosi
ing mathematics is more than acquiring an array
of facts and that doing mathematics is more than Weinert FE (2001) Concept of competence: a conceptual
carrying out well-rehearsed procedures. School clarification. In: Rychen DS, Salganik LH (eds)
mathematics is sometimes portrayed as a simple Defining and selecting key competencies. Hogrefe &
contest between knowledge and skill. Compe- Huber, Seattle, pp 45–65
tency frameworks attempt to shift that portrayal
to a more nuanced portrait of a field in which Complexity in Mathematics
a variety of competences need to be developed. Education

Cross-References Brent Davis1 and Elaine Simmt2
1University of Calgary, Calgary, AB, Canada
▶ Bloom’s Taxonomy in Mathematics Education 2Faculty of Education, University of Alberta,
▶ Competency Frameworks in Mathematics Edmonton, Canada

Education Keywords
▶ Frameworks for Conceptualizing Mathematics
Complexity science; Adaptive learning systems;
Teacher Knowledge Collective knowledge-producing systems
▶ International Comparative Studies in

Mathematics: An Overview

References Definition

Anderson LW, Krathwohl DR (eds) (2001) A taxonomy Over the past half century, “complex systems”
for learning, teaching, and assessing: a revision of perspectives have risen to prominence across
Bloom’s taxonomy of educational objectives. many academic domains in both the sciences and
Longman, New York the humanities. Mathematics was among the orig-
inating domains of complexity research. Education
Bloom BS (ed) (1956) Taxonomy of educational has been a relative latecomer, and so perhaps not
objectives, handbook I: cognitive domain. McKay, surprisingly, mathematics education researchers
New York have been leading the way in the field.

Kilpatrick J (1993) The chain and the arrow: from the There is no unified definition of complexity,
history of mathematics assessment. In: Niss M (ed) principally because formulations emerge from the
Investigations into assessment in mathematics study of specific phenomena. One thus finds quite
education: an ICME Study. Kluwer, Dordrecht, focused definitions in such fields as mathematics
pp 31–46 and software engineering, more indistinct mean-
ings in chemistry and biology, and quite flexible
Kilpatrick J (2009) The mathematics teacher and interpretations in the social sciences (cf. Mitchell
curriculum change. PNA: Rev Investig Dida´ct Mat 2009). Because mathematics education reaches
3:107–121 across several domains, conceptions of complexity
within the field vary from the precise to the vague,
Niss MA (2003) Mathematical competencies and the depending on how and where the notion is taken up.
learning of mathematics: the Danish KOM project.
In: Gagatsis A, Papastavridis S (eds) Third Mediterra- Diverse interpretations do collect around
nean conference on mathematical education – Athens, a few key qualities, however. In particular,
Hellas, 3-4-5 January 2003. Hellenic Mathematical
Society, Athens, pp 116–124

Robitaille DF, Schmidt WH, Raizen S, McKnight C,
Britton E, Nicol C (1993) Curriculum frameworks for
mathematics and science. TIMSS Monograph No. 1.
Pacific Educational Press, Vancouver

C 88 Complexity in Mathematics Education

complex systems adapt and are thus distinguish- another, it is impossible to study one of these
able from complicated, mechanical systems. phenomena without studying all of the others.
A complicated system is one that comprises
many interacting components and whose global This is a sensibility that has been well
character can be adequately described and represented in the mathematics education research
predicted by applying laws of physics. literature for decades in the form of varied theories
A complex system comprises many interacting of learning. Among others, radical constructivism,
agents – and those agents, in turn, may comprise sociocultural theories of learning, embodied, and
many interacting subagents – presenting the pos- critical theories can all be read as instances of
sibility of global behaviors that are rooted in but complexity theories. That is, they all invoke
that cannot be reduced to the actions or qualities bodily metaphors, systemic concerns, evolutionary
of the constituting agents. In other words, dynamics, emergent possibility, and self-
a complex system is better described by using maintaining properties.
Darwinian principles than Newtonian ones. It is
thus that each complex phenomenon must be This is not to assert some manner of hidden
studied in its own right. For each complex unity, uniformity to the theories just mentioned. On the
new laws emerge that cannot be anticipated contrary, much of their value is to be found in their
or explained strictly by reference to prior, diversity. As illustrated in Fig. 1, when learning
subsequent, or similar systems. Popularly cited phenomena of interest to mathematics educators
examples include anthills, economies, and brains, are understood as nested systems, a range of theo-
which are more than the collected activities of ries become necessary to grapple with the many
ants, consumers, and neurons. In brief, whereas issues the field must address. Sophisticated and
the opposite of complicated is simple, opposites effective mathematics pedagogies demand simi-
of complex include reducible and decomposable. larly sophisticated insights into the complex
Hence, prominent efforts toward a coherent, dynamics of knowing and learning. More signifi-
unified description of complexity revolve around cantly, perhaps, by introducing the time frames of
such terms as emergent, noncompressible, meaningful systemic transformation into discus-
self-organizing, context-sensitive, and adaptive. sions of individual knowing and collective knowl-
edge, complexity not only enables but compels
The balance of this discussion is organized a consideration of the manners in which knowers
around four categories of usage within mathemat- and systems of knowledge are co-implicated
ics education – namely, complexity as a theoretical (Davis and Simmt 2006).
discourse, a historical discourse, a disciplinary dis-
course, and a pragmatic discourse. Complexity as a Historical Discourse
School mathematics curricula are commonly
Characteristics presented as a-historical and a-cultural. Contra
this perception, complexity research offers an
Complexity as a Theoretical Discourse instance of emergent mathematics that has arisen
Among educationists interested in complexity, and that is evolving in a readily perceptible
there is frequent resonance with the notions that time frame. As an example of what it describes –
a complex system is one that knows (i.e., per- a self-organizing, emergent coherence –
ceives, acts, engages, and develops) and/or learns complexity offers a site to study and interrogate
(adapts, evolves, maintains self-coherence, etc.). the nature of mathematics, interrupting assump-
This interpretation reaches across many systems tions of fixed and received knowledge.
that are of interest among educators, including
physiological, personal, social, institutional, To elaborate, the study of complexity in math-
epistemological, cultural, and ecological sys- ematics reaches back the late nineteenth century
tems. Unfolding from and enfolding in one when Poincare´ conjectured about the three-body
problem in mechanics. Working qualitatively,
from intuition Poincare´ recognized the problem
of thinking about complex systems with the

Complexity in Mathematics Education C89

C

Complexity in Mathematics Education, Fig. 1 Some of the nested complex systems of interest to mathematics
educators

assumptions and mathematics of linearity conferences, and think tanks such as the Santa
(Bell 1937). The computational power of mathe- Fe Institute, a research center dedicated to all
matics was limited to the calculus of the time; matters of complexity science.
however, enabled by digital technologies of the
second half of the twentieth century, such In brief, the emergence of complexity as a field
problems became tractable and the investigation of study foregrounds that mathematics might be
of dynamical systems began to flourish. productively viewed as a humanity. More provoc-
atively, the emergence of a mathematics of
With computers, experimental mathematics implicatedness and entanglement alongside the
was born, and the study of dynamical systems led rise of a more sophisticated understanding of
to new areas in mathematics. For the first time, it humanity’s relationship to the more-than-human
was easy and quick to consider the behavior of world might be taken as an indication of the
a function over time by computing thousands and ecological character of mathematics knowledge.
hundreds of thousands of iterations of the function.
Numerical results were readily converted into Complexity as a Disciplinary Discourse
graphical representations (the Lorenz attractor, A common criticism of contemporary grade school
Julia sets, bifurcation diagrams) which in turn mathematics curriculum is that little of its content
inspired a new generation of mathematicians, is reflective of mathematics developed after the
scientists, and human scientists to think differently sixteenth or seventeenth centuries, when publicly
about complex dynamical systems. funded and mandatory education spread across
Europe. A deeper criticism is that the mathematics
The mid-twentieth century brought about included in most preuniversity curricula is fitted
great insights into features of dynamical systems to a particular worldview of cause-effect and
that had been overlooked. As mathematicians and linear relationships. Both these concerns might be
physical and computer scientists were exploring addressed by incorporating complexity-based
dynamical systems (e.g., Smale, Prigogine, content into programs of study.
Lorenz, Holland), their work and the work of
biologists began to intersect. Emerging out of Linear mathematics held sway at the time of the
that activity were interdisciplinary workshops, emergence of the modern school – that is, during

C 90 Complexity in Mathematics Education

the Scientific and Industrial Revolutions – because New curriculum in mathematics is emerging.
it lent itself to calculations that could be done by More profoundly, when, how, who, and where
hand. Put differently, linear mathematics was first we teach are also being impacted by the pres-
championed and taught for pragmatic reasons, not ence of complexity sensibilities in education
because it was seen to offer accurate depictions of because they are a means to nurture emergent
reality. Descartes, Newton, and their contempo- possibility.
raries were well aware of nonlinear phenomena.
However, because of the intractability of many Complexity as a Pragmatic Discourse
nonlinear calculations, when they arose they were To recap, complexity has emerged in education as
routinely replaced by linear approximations. As a set of mathematical tools for analyzing
textbooks omitted nonlinear accounts, generations phenomena, as a theoretical frame for interpreting
of students were exposed to over-simplified, line- activity of adaptive and emergent systems, as
arized versions of natural phenomena. Ultimately a new sensibility for orienting oneself to the
that exposure contributed to a resilient worldview world, and for considering the conditions for emer-
of a clockwork reality. However, the advent of gent possibilities leading to more productive,
powerful computing technologies over the past “intelligent” classrooms.
half century has helped to restore an appreciation
of the relentless nonlinearity of the universe. That In the last of these roles, complexity might be
is, the power of digital technologies have not just regarded as the pragmatic discourse – and of the
opened up new vistas of calculation, they have applications of complexity discussed here, this one
triggered epistemic shifts as they contribute to may have the most potential for affecting school
redefinitions of what counts as possible and what mathematics by offering guidance for structuring
is expressible (Hoyles and Ness 2008, p. 89). learning contexts. In particular, complexity offers
direct advice for organizing classrooms to
With the ready access to similar technologies in support the individual-and-collective generation
most school classrooms within a culture of ubiqui- of insight – by, for example, nurturing the common
tous computation, some (e.g., Jacobson and experiences and other redundancies of learners
Wilensky 2006) have advocated for the inclusion while making space for specialist roles, varied
of such topics as computer-based modeling and interpretations, and other diversities. Another strat-
simulation languages, including networked collab- egy is to utilize digital technologies that offer
orative simulations (see Kaput Center for Research environments for (and systemic memories of)
and Innovation in STEM Education). In this vein, assembling interpretations, strategies, solutions,
complexity is understood as a digitally enabled, evaluations, and judgments.
modeling-based branch of mathematics, creating
space in secondary and tertiary education for new Mobile digital technologies have occasioned
themes such as recursive functions, fractal geome- other kinds of learning opportunities that lend
try, and modeling of complex phenomena with themselves to both the sorts of analysis and the
mathematical tools such as iteration, cobwebbing, sorts of advice afforded by complexity research.
and phase diagrams. Others (e.g., English 2006; For example, within platforms such as wikis,
Lesh and Doerr 2003) have advocated for similarly Twitter, and Facebook, students can organize
themed content, but in a less calculation-dependent along language and interest lines while they con-
format, arguing that the shift in sensibility from nect with and elaborate the contributions of their
linearity to complexity is more important than the peers. In a more extreme frame, the emergence of
development of the computational competencies massive online open-learning courses (MOOCs)
necessary for sophisticated modeling. In either represents an interesting new example of the
case, the imperative is to provide learners with impact a complexity sensibility can have in the
access to the tools of complexity, along with educational context as they invite large numbers
its affiliated domains of fractal geometry, chaos of participants to engage with the thinking of
theory, and dynamic modeling. experts. It is not without irony that, within
a complexity frame, even the most denigrated of

Concept Development in Mathematics Education C91

teaching strategies – the large lecture – can serve Davis B, Simmt E (2006) Mathematics-for-teaching: an C
as a critical part of a vibrant, knowledge- ongoing investigation of the mathematics that teachers
producing system when coupled to connectivity, (need to) know. Educ Stud Math 61(3):293–319
playback, feedback, and other aspects of a digital
environment. English L (2006) Mathematical modeling in the primary
school: children’s construction of a consumer guide.
As complexity becomes more prominent in Educ Stud Math 62(3):303–329
educational discourses and entrenches in the
infrastructure of “classrooms,” mathematics edu- Hoyles C, Noss R (2008) Next steps in implementing
cation can move from a culture of cooperation to Kaput’s research programme. Educ Stud Math
one of collaboration, and that has entailments for 68(2):85–94
the outcomes of schooling – articulated in, e.g.,
movements from generalist preparation to spe- Jacobson M, Wilensky U (2006) Complex systems in
cialist expertise, from independent workers to education: scientific and educational importance and
team-based workplaces, and from individual research challenges for the learning sciences. J Learn
knowing to social action. Sci 15(1):11–34

Lesh R, Doerr H (eds) (2003) Beyond constructivism:
models and modeling perspectives on mathematics
problem solving learning and teaching. Lawrence
Erlbaum Associates, Mahwah

Mitchell M (2009) Complexity: a guided tour. Oxford
University Press, Oxford, UK

Cross-References

▶ Activity Theory in Mathematics Education Concept Development in
▶ Calculus Teaching and Learning Mathematics Education
▶ Collaborative Learning in Mathematics
Shlomo Vinner
Education Hebrew University of Jerusalem Science
▶ Critical Mathematics Education Teaching Department, Faculty of Science,
▶ Critical Thinking in Mathematics Education Jerusalem, Israel
▶ Enactivist Theories
▶ Functions Learning and Teaching Keywords
▶ Gender in Mathematics Education
▶ Information and Communication Technology Notion; Concept; Concept formation in babies;
Concrete object; Similarities; Generalization;
(ICT) Affordances in Mathematics Education Ostensive definitions; Mathematical definitions;
▶ Metaphors in Mathematics Education Intuitive; Concept image; Concept definition;
▶ Mathematical Modelling and Applications in Stereotypical examples; System 1 and system 2;
Pseudo-analytical; Pseudo-conceptual; Mathe-
Education matical objects; Mathematical mind
▶ Mathematics Teacher Education Organization,
Characteristics
Curriculum, and Outcomes
▶ Situated Cognition in Mathematics Education Concept formation and development in general is
▶ Theories of Learning Mathematics an extremely complicated topic in cognitive psy-
▶ Types of Technology in Mathematics chology. There exists a huge literature about it,
classical and current. Among the classical works
Education on it, one can mention for instance, Piaget and
▶ Zone of Proximal Development in Inhelder (1958) and Vygotsky (1986). However,
this issue is restricted to concept formation and
Mathematics Education

References

Bell ET (1937) Men of mathematics: the lives and
achievements of the great mathematicians from Zeno
to Poincare´. Simon and Schuster, New York

C 92 Concept Development in Mathematics Education

development in mathematics. Nevertheless, it is chairs presented to her or him in the past. The
suggested not to isolate mathematical concept second mechanism is the one which distinguishes
formation and development from concept forma- differences. The mind distinguishes that a certain
tion and development in general. object is not similar to the chairs which were
presented to the baby in the past, and therefore,
One terminological clarification should be the baby is not supposed to say “chair” when an
made before the main discussion. When dealing object that is not a chair is presented to him or her
with concepts, very often, also the term “notion” by the adult. Mistakes about the acquired concept
is involved. A notion is a lingual entity – a word, might occur because of two reasons. An object,
a word combination (written or pronounced); it which is not a chair (say a small table), appears to
can also be a symbol. A concept is the meaning the baby (or even to an adult) like a chair. In this
associated in our mind with a notion. It is an idea case, the object will be considered as an element
in our mind. Thus, a notion is a concept name. of the class of all chairs while, in fact, it is not an
There might be concepts without names and for element of this class. The second reason for mis-
sure there are meaningless notions, but discussing takes is that an object that is really a chair will not
them requires subtleties which are absolutely be identified as a chair because of its weird shape.
irrelevant to this context. In many discussions Thus, an object which was supposed to be an
people do not bother to distinguish between element of the class is excluded from it. More
notions and concepts, and thus the word “notion” examples of this type are the following:
becomes ambiguous. The ambiguity is easily sometimes, babies consider dogs as cats and
resolved by the context. vice versa. These are intelligent mistakes because
there are some similarities between dogs and cats.
As recommended above, it will be more useful They are both animals; sometimes they even have
not to disconnect mathematical concept forma- similar size (in the case of small dogs) and so on.
tion from concept formation in general, and
therefore, let us start our discussion with an The above process which leads, in our mind, to
example of concept formation in babies. How the construction of the set of all possible objects
do we teach them, for instance, the concept of to which the concept name can be applied is
chair? The common practice is to point at various a kind of generalization. Thus, generalizations
chairs in various contexts and to say “chair.” are involved in the formation of any given
Amazingly enough, after some repetitions, the concept. Therefore, concepts can be considered
babies understand that the word “chair” is as generalizations.
supposed to be related to chairs, which occur to
them in their daily experience, and when being The actions by means of which we try to teach
asked “what is this?” they understand that they children concepts of chair are called ostensive
are supposed to say “chair.” Later on, they will definitions. Of course, only narrow class of con-
imitate the entire ritual on their own initiative. cepts can be acquired by means of ostensive
They will point at chairs and say “chair.” I would definitions. Other concepts are acquired by
like to make a theoretical claim here by saying means of explanations which can be considered
that, seemingly, they have constructed in their at this stage as definitions. Among these concepts
mind the class of all possible chairs. Namely, I can point, for instance, at a forest, a school,
a concept is formed in their mind, and whenever work, hunger and so on. When I say definitions
a concrete object is presented to them, they will at this stage, I do not mean definitions which are
be able to decide whether it is a chair or not. Of similar, or even seemingly similar to rigorous
course, some mistakes can occur in that concept mathematical definitions. The only restriction
formation process. It is because in this process, on these definitions is that familiar concepts will
two cognitive mechanisms are involved. The first be used in order to explain a non-familiar
mechanism is the one that identifies similarities. concept. Otherwise, the explanation is useless.
The mind distinguishes that one particular chair (This restriction, by the way, holds also for
presented to the baby is similar to some particular mathematical definitions, where new concepts

Concept Development in Mathematics Education C93

are defined by means of previously defined from the natural intuitive mode of thinking C
concepts or by primary concepts.) In definitions according to which the child’s intellectual
which we use in non-technical context in order to development takes place. The major problem is
teach concepts, we can use examples. For that mathematical thinking is shaped by rigorous
instance, in order to define furniture, we can rules, and in order to think mathematically, chil-
say: A chair is furniture, a bed is furniture, and dren, as well as adults, should be aware of these
tables, desks, and couches are furniture. rules while thinking in mathematical contexts.
One crucial difficulty in mathematical thinking
The description which was just given deals is that mathematical concepts are strictly deter-
with the primary stage of concept formation. mined by their definitions. In the course of their
However, concept formation in ordinary lan- mathematical studies, children, quite often, are
guage is by far more complicated and very presented to mathematical notions with which
often, contrary to the mathematical language, they were familiar from their past experience.
ends up in a vague notion. Take, for instance, For instance, in Kindergarten they are shown
again, the notion of furniture. The child, when some geometrical figures such as squares and
facing an object which was not previously rectangles. The adjacent sides of the rectangle
introduced to him or to her as furniture, should which are shown to the children in Kindergarten
decide whether this object is furniture or not. He have always different length. Thus, the set of
or she may face difficulties doing it. Also adults all possible rectangles which is constructed in
might have similar difficulties. This is only one the child’s mind includes only rectangles, the
example out of many which demonstrates the adjacent sides of which have different length. In
complexity of concept formation in the child’s the third grade, in many countries, a definition
mind as well as in the adult’s mind. There are of a rectangle is presented to the child. It is
even greater complexities when concept a quadrangle which has 4 right angles. According
formation of abstract nouns, adjectives, verbs, to this definition, a square is also a rectangle.
and adverbs is involved. Nevertheless, despite Thus, a conflict may be formed in the child’s
that complexity, the majority of children acquire mind between the suggested definition and the
language at an impressive level by the age of six concept he or she already has about rectangles.
(an elementary level is acquired already at the age The concept the child has in mind was formed by
of three). The cognitive processes associated with the set of examples and the properties of these
the child’s acquisition of language are discussed examples which were presented to the child. It
in details in cognitive psychology, linguistics, and was suggested (Vinner 1983) to call it the concept
philosophy of language. One illuminating source image of that notion. Thus, in the above case of
which is relevant to this issue is Quine’s (1964) the rectangle, there is a conflict between the
“Word and object.” However, a detailed concept image and the concept definition. On
discussion of these processes is not within the the other hand, quite often some concepts are
scope of this issue. introduced to the learner by means of formal
definitions. For instance, an altitude in a triangle.
In addition to the language acquisition, the However, a formal definition, generally, remains
child acquires also broad knowledge about the meaningless unless it is associated with some
world. He or she knows that when it rains, it is examples. The examples can be given by
cloudy, they know that dogs bark and so on and a teacher or by a textbook, or they can be formed
so forth. In short, they know infinitely many by the learners themselves. The first examples
other facts about their environment. And again, which are associated with the concept have
it is obtained in a miraculous way, smoothly a crucial impact on the concept image. Unfortu-
without any apparent difficulties. Things, how- nately, quite often, in mathematical thinking,
ever, become awkward when it gets to mathemat- when a task is given to students, in order to
ics. One possible reason for things becoming carry it out, they consult their concept image
awkward in mathematics is that, in many cases,
mathematical thinking is essentially different

C 94 Concept Development in Mathematics Education

and forget to consult the concept definition. It the correspondence which assigns to every living
turns out that, in many cases, there are critical creature its mother), even then, the stereotypical
examples which shape the concept image. In concept image of a function is that of an algebraic
some cases, these are the first examples which formula, as claimed above.
are introduced to the learner. For instance, in the
case of the altitude (a segment which is drawn A plausible explanation to these phenomena
from one vertex or the triangle and it is perpen- can be given in terms of the psychological theory
dicular to the opposite side of this vertex or to about system 1 and system 2. Psychologists,
its continuation), it is pedagogically reasonable nowadays, speak about two cognitive systems
to give examples of altitudes in acute angle which they call system 1 and system 2. It sounds
triangles. Later on, in order to form the appropri- as if there are different parts in our brain which
ate concept image of an altitude, the teacher, as produce different kinds of thinking. However,
well as the textbook, should give examples of this interpretation is wrong. The correct way to
altitudes from vertices of acute angles in an look at system 1 and system 2 is to consider them
obtuse angle triangle. However, before this as thinking modes. This is summarized very
stage of the teaching takes place, the concept clearly in Stanovich (1999, p. 145). System 1 is
image of the altitude was shaped by the stereo- characterized there by the following adjectives:
typical examples of altitudes in an acute angle associative, tacit, implicit, inflexible, relatively
triangle (sometimes, even by the stereotypical fast, holistic, and automatic. System 2 is
examples of altitudes which are perpendicular to characterized by: analytical, explicit, rational,
a horizontal side of a triangle). Thus, when the controlled, and relatively slow. Thus, notions
learners face a geometrical problem about that were used by mathematics educators in the
altitudes which do not meet the stereotypes in past can be related now to system 1 or system 2,
their concept image, they are stuck. It does not and therefore this terminology is richer than the
occur to them to consult the concept definition of previously suggested notions. Fischbein (1987)
the altitude, and if it does occur, they usually spoke about intuition and this can be considered
recall the first part of the definition (“a segment as system 1. Skemp (1979) spoke about two
which is drawn from one vertex or the triangle systems which he called delta one and delta two.
and it is perpendicular to the opposite side of They can be considered as intuitive and reflective
this vertex”) and forget the additional phrase in or using the new terminology, system 1 and
the definition (“or to its continuation”). Two system 2, respectively. Vinner (1997) used the
additional examples of this kind are the follow- notions pseudo-analytical and pseudo-conceptual
ing: (1) At the junior high level, in geometry, which can be considered as system 1.
when a quadrangle is defined as a particular
case of a polygon (a quadrangle is a polygon In mathematical contexts the required think-
which has 4 sides), the learners have difficulties ing mode is that of system 2. This requirement
to accept a concave quadrangle or a quadrangle presents some serious difficulties to many peo-
that intersects itself as quadrangles. (2) At the ple (children and adults) since, most of the
high school level, when a formal definition of a time, thought processes are carried out within
function is given to the students, eventually, the system 1. Also, in many people, because of
stereotypical concept image of a function is that various reasons, system 2 has not been devel-
of an algebraic formula. A common formal def- oped to the extent which is required for math-
inition of a function can be the following one: ematical thinking in particular and for rational
a correspondence between two non-empty sets thinking in general. Nevertheless, in many
which assigns to every element in the first set contexts, learners succeed in carrying out
(the domain) exactly one element in the second mathematical tasks which are presented to
set (the range). Even if some non-mathematical them by using system 1. This fact does not
examples are given to the students (for instance, encourage them to become aware of the
need to use system 2 while carrying out
mathematical tasks.

Concept Development in Mathematics Education C95

When discussing concept development in (Peano’s Arithmetic, Euclidean Geometry, C
mathematical thinking, it is worthwhile to Set Theory, Group Theory, etc.). Also, in
mention also some concepts which can be more advanced mathematical thinking, we con-
classified as metacognitive concepts. Such ceive mathematical objects (numbers, functions,
concepts are algorithm, heuristics, and proof. geometrical figures in Euclidean geometry, etc.)
While studying mathematics, the learners face as abstract objects. All these require thought
many situations in which they or their teachers processes within system 2. However, it should
use algorithms, heuristics, and proof. However, be emphasized that all the above concept devel-
usually, the notions “algorithm” and “heuristics” opments do not occur simultaneously. They
are not introduced to the learners in their school also do not occur in all students who study
mathematics. Some of them will be exposed to mathematics. One should take many mathematics
them in college, in case they choose to take courses and solve a lot of mathematical problems
certain advanced mathematics courses. As to the in order to achieve that level. Those who do it
notion of proof, in spite of the fact that this notion should have special interest in mathematics or
is mentioned a lot in school mathematics (espe- what can be called mathematical curiosity. It
cially in geometry), the majority of students do not requires, what some people call, a mathematical
fully understand it. Many of them try to identify mind. Is it genetic (Devlin 2000) or
mathematical proof by its superficial characteris- acquired? At this point we have reached
tics. They do it without understanding the logical a huge domain of psychological research
reasoning associated with these characteristics. A which is far beyond the scope of this particular
meaningless use of symbols and verbal expres- encyclopedic issue.
sions as “therefore,” “it follows,” and “if. . . then”
is considered by many students as a mathematical Cross-References
proof (See for instance Healy and Hoyles 1998). It
turns out that it takes a lot of mathematical expe- ▶ Abstraction in Mathematics Education
rience until meaningless verbal rituals (as in the ▶ Concept Development in Mathematics
case of the baby acquiring the concept of chair)
become meaningful thought processes. And how Education
do we know that the learners use the above verbal ▶ Critical Thinking in Mathematics Education
expressions meaningfully? We assume so because ▶ Intuition in Mathematics Education
their use of these expressions is in absolute ▶ Mathematical Proof, Argumentation, and
agreement with the way we, mathematicians and
mathematics educators, use them. Reasoning
▶ Metacognition
Another important aspect of mathematical ▶ Problem Solving in Mathematics Education
concept development is the understanding that ▶ Theories of Learning Mathematics
certain mathematical concepts are related to ▶ Values in Mathematics Education
each other. Here comes the idea of structure. ▶ Zone of Proximal Development in
For instance, from triangles, quadrangles,
pentagons, and hexagons, we reach the concept Mathematics Education
of a polygon. From the general concept of quad-
rangles, we approach to trapezoids, parallelo- References
grams, rhombus, rectangles, and squares, and
we realize there all kinds of class inclusions. Devlin K (2000) The math gene. Basic Books, New York
Thus, we distinguish partial order in the set of Fischbein E (1987) Intuition in science and mathematics –
mathematical concepts. Finally, and this is
perhaps the ultimate stage of mathematical con- an educational approach. Reidel Publishing Company,
cept development, we conceive mathematics as Dordrecht
a collection of various deductive structures Healy L, Hoyles C (1998) Justifying and proving in school
mathematics. Technical report, University of London,
Institute of Education

C 96 Constructivism in Mathematics Education

Piaget J, Inhelder B (1958) The growth of logical thinking (some people say individual or psychological)
from childhood to adolescence. Basic Books, and social constructivism. Within each there is
New York also a range of positions. While radical and social
constructivism will be discussed in a later sec-
Quine WVO (1964) Word and object. The MIT Press, tion, it should be noted that both schools are
Cambridge, MA grounded in a strong skeptical stance regarding
reality and truth: Knowledge cannot be thought of
Skemp R (1979) Intelligence, learning and action: as a copy of an external reality, and claims of
a foundation for theory and practice in education. truth cannot be grounded in claims about reality.
Wiley, Chichester, W. Sussex
The justification of this stance toward
Stanovich KE (1999) Who is rational. Lawrence Erlbaum knowledge, truth, and reality, first voiced by the
Associates, Mahwah skeptics of ancient Greece, is that to verify that
one’s knowledge is correct, or that what one
Vinner S (1983) Concept definition, concept image and knows is true, one would need access to reality
the notion of function. Int J Math Educ Sci Technol by means other than one’s knowledge of it. The
14(3):293–305 importance of this skeptical stance for mathemat-
ics educators is to remind them that students have
Vinner S (1997) The pseudo-conceptual and the pseudo- their own mathematical realities that teachers and
analytical thought processes in mathematics learning. researchers can understand only via models of
Educ Stud Math 34:97–129 them (Steffe et al. 1983, 1988).

Vygotsky L (1986) Thought and language (English Constructivism did not begin within
translation). MIT Press, Cambridge, MA mathematics education. Its allure to mathematics
educators is rooted in their long evolving rejection
Constructivism in Mathematics of Thorndike’s associationism (Thorndike 1922;
Education Thorndike et al. 1923) and Skinner’s behaviorism
(Skinner 1972). Thorndike’s stance was that learn-
Patrick W. Thompson ing happens by forming associations between stim-
Department of Mathematics and Statistics, uli and appropriate responses. To design instruction
Arizona State University, Tempe, AZ, USA from Thorndike’s perspective meant to arrange
proper stimuli in a proper order and have students
Keywords respond appropriately to those stimuli repeatedly.
The behaviorist stance that mathematics educators
Epistemology; Social constructivism; Radical found most objectionable evolved from Skinner’s
constructivism; Knowledge; Reality; Truth; claim that all human behavior is due to environ-
Objectivity mental forces. From a behaviorist perspective, to
say that children participate in their own learning,
Background aside from being the recipient of instructional
actions, is nonsense. Skinner stated his position
Constructivism is an epistemological stance clearly:
regarding the nature of human knowledge,
having roots in the writings of Epicurus, Science . . . has simply discovered and used subtle
Lucretius, Vico, Berkeley, Hume, and Kant. forces which, acting upon a mechanism, give it the
Modern constructivism also contains traces of direction and apparent spontaneity which make it
pragmatism (Peirce, Baldwin, and Dewey). In seem alive. (Skinner 1972, p. 3)
mathematics education the greatest influences
are due to Piaget, Vygotsky, and von Glasersfeld. Behaviorism’s influence on psychology, and
See Confrey and Kazak (2006) and Steffe and thereby its indirect influence on mathematics
Kieren (1994) for related historical accounts of education, was also reflected in two stances that
constructivism in mathematics education. were counter to mathematics educators’ growing
awareness of learning in classrooms. The first
There are two principle schools of thought
within constructivism: radical constructivism

Constructivism in Mathematics Education C97

stance was that children’s learning could be championed Piaget’s notions of schema, assimila- C
studied in laboratory settings that have no resem- tion, accommodation, equilibration, and reflection
blance to environments in which learning as ways to conceptualize students’ mathematical
actually happens. The second stance was that thinking as having an internal coherence. Piaget’s
researchers could adopt the perspective of method of clinical interviews also was attractive to
a universal knower. This second stance was evi- researchers of students’ learning. However, until
dent in Simon and Newell’s highly influential 1974 mathematics educators were interested in
information processing psychology, in which Piaget’s writings largely because they thought of
they separated a problem’s “task environment” his work as “developmental psychology” or “child
from the problem solver’s “problem space.” psychology,” with implications for children’s
learning. It was in 1974, at a conference at the
We must distinguish, therefore, between the task University of Georgia, that Piaget’s work was rec-
environment—the omniscient observer’s way of ognized in mathematics education as a new field,
describing the actual problem “out there”— and one that leveraged children’s cognitive development
the problem space—the way a particular subject to study the growth of knowledge. Smock (1974)
represents the task in order to work on it. (Simon wrote of constructivism’s implications for instruc-
and Newell 1971, p. 151) tion, not psychology’s implications for instruction.
Glasersfeld (1974) wrote of Piaget’s genetic epis-
Objections to this distinction were twofold: temology as a theory of knowledge, not as a theory
Psychologists considered themselves to be of cognitive development. The 1974 Georgia
Simon and Newell’s omniscient observers conference is the first occasion this writer could
(having access to problems “out there”), and find where “constructivism” was used to describe
students’ understandings of the problem were the epistemological stance toward mathematical
reduced to a subset of an observer’s understand- knowing that characterizes constructivism in
ing. This stance among psychologists had the mathematics education today.
effect, in the eyes of mathematics educators, of
blinding them to students’ ways of thinking that Acceptance of constructivism in mathematics
did not conform to psychologists’ preconceptions education was not without controversy. Disputes
(Thompson 1982; Cobb 1987). Erlwanger (1973) sometimes emerged from competing visions
revealed vividly the negative consequences of of desired student learning, such as students’
behaviorist approaches to mathematics education performance on accepted measures of compe-
in his case study of a successful student in tency (Gagne´ 1977, 1983) versus attendance to
a behaviorist individualized program who the quality of students’ mathematics (Steffe and
succeeded by inventing mathematically invalid Blake 1983), and others emerged from different
rules to overcome inconsistencies between his conceptions of teaching effectiveness (Brophy
answers and an answer key. 1986; Confrey 1986). Additional objections
to constructivism were in reaction to its funda-
The gradual release of mathematics education mental aversion to the idea of truth as
from the clutches of behaviorism, and infusions of a correspondence between knowledge and reality
insights from Polya’s writings on problem solving (Kilpatrick 1987).
(Polya 1945, 1954, 1962), opened mathematics
education to new ways of thinking about student Radical and Social Constructivism in
learning and the importance of student thinking. Mathematics Education
Confrey and Kazak (2006) described the influence
of research on problem solving, misconceptions, Radical constructivism is based on two tenets:
and conceptual development of mathematical ideas “(1) Knowledge is not passively received but
as precursors to the emergence of constructivism in actively built up by the cognizing subject;
mathematics education. (2) the function of cognition is adaptive and

Piaget’s writings had a growing influence in
mathematics education once English translations
became available. In England, Skemp (1961, 1962)

C 98 Constructivism in Mathematics Education

serves the organization of the experiential action.” This interpretation would fit nicely
world, not the discovery of ontological reality” with the finding that adults mimic infants’
(Glasersfeld 1989, p. 114). Glasersfeld’s use of speech abundantly (Fernald 1992; Schachner
“radical” is in the sense of fundamental – that and Hannon 2011). Glasersfeld and Piaget
cognition is “a constitutive activity which, might have thought that adults’ imitative speech
alone, is responsible for every type or kind acts, once children recognize them as imitations,
of structure an organism comes to know” provide occasions for children to have a sense
(Glasersfeld 1974, p. 10). that they can influence actions of others through
verbal behavior. This interpretation also would
Social constructivism is the stance that history fit well with Bauersfeld’s (1980, 1988, 1995)
and culture precede and preform individual understanding of communication as a reflexive
knowledge. As Vygotsky famously stated, interchange among mutually oriented individ-
“Every function in the child’s cultural develop- uals: “The [conversation] is constituted at every
ment appears twice: first, on the social level, and moment through the interaction of reflective sub-
later, on the individual level; first between people jects” (Bauersfeld 1980, p. 30 italics in original).
. . ., then inside the child” (Vygotsky 1978, p. 57).
Paul Ernest (1991, 1994, 1998) introduced the
The difference between radical and social term social constructivism to mathematics edu-
constructivism can be seen through contrasting cation, distinguishing between two forms of it.
interpretations of the following event. Vygotsky One form begins with a radical constructivist
(1978) illustrated his meaning of internalization – perspective and then accounts for human interac-
“the internal reconstruction of an external tion in terms of mutual interpretation and adap-
operation” – by describing the development of tation (Bauersfeld 1980, 1988, 1992). Glasersfeld
pointing: (1995) considered this as just radical constructiv-
ism. The other, building from Vygotsky’s notion
The child attempts to grasp an object placed of cultural regeneration, introduced the idea of
beyond his reach; his hands, stretched toward that mathematical objectivity as a social construct.
object, remain poised in the air. His fingers make
grasping movements. At this initial state pointing is Social constructivism links subjective and objec-
represented by the child’s movement, which seems tive knowledge in a cycle in which each contributes
to be pointing to an object—that and nothing more. to the renewal of the other. In this cycle, the path
When the mother comes to the child’s aid followed by new mathematical knowledge is from
and realizes his movement indicates something, subjective knowledge (the personal creation of an
the situation changes fundamentally. Pointing individual), via publication to objective knowledge
becomes a gesture for others. The child’s unsuc- (by intersubjective scrutiny, reformulation, and
cessful attempt engenders a reaction not from the acceptance). Objective knowledge is internalized
object he seeks but from another person [sic]. and reconstructed by individuals, during the learn-
Consequently, the primary meaning of that unsuc- ing of mathematics, to become the individuals’
cessful grasping movement is established by others subjective knowledge. Using this knowledge, indi-
[italics added]. (Vygotsky 1978, p. 56) viduals create and publish new mathematical
knowledge, thereby completing the cycle.
Vygotsky clearly meant that meanings (Ernest 1991, p. 43)
originate in society and are transmitted via social
interaction to children. Glasersfeld and Piaget Ernest focused on objectivity of adult mathe-
would have listened agreeably to Vygotsky’s matics. He did not address the matter of how
tale – until the last sentence. They instead children’s mathematics comes into being or how
would have described the child as making it might grow into something like an adult’s
a connection between his attempted grasping mathematics.
action and someone fetching what he wanted.
Had it been the pet dog bringing the desired Radical and social constructivists differ some-
item, it would have made little difference to the what in the theoretical work they ask of construc-
child in regard to the practical consequences of tivism. Radical constructivists concentrate on
his action. Rather, the child realized, in a sense, understanding learners’ mathematical realities and
“Look at what I can make others do with this the internal mechanisms by which they change.

Constructivism in Mathematics Education C99

They conceive, to varying degrees, of learners in discussions on the work that theories in mathe- C
social settings, concentrating on the sense that matics education must do – they must contribute
learners make of them. They try to put themselves to our ability to improve the learning and teach-
in the learner’s place when analyzing an interaction. ing of mathematics. Cobb et al. first reminded
Social constructivists focus on social and cultural the field that, from any perspective, what
mathematical and pedagogical practices and attend happens in mathematics classrooms is important
to individuals’ internalization of them. They for students’ mathematical learning. Thus,
conceive of learners in social settings, concentrat- a theoretical perspective that can capture more,
ing, to various degrees, on learners’ participation and more salient, aspects for mathematics learn-
in them. They take the stances, however, of an ing (including participating in practices) is the
observer of social interactions and that social more powerful theory. With a focus on the need
practices predate individuals’ participation. to understand, explain, and design events within
classrooms, they recognized that there are indeed
Conflicts between radical and social construc- social dimensions to mathematics learning and
tivism tend to come from two sources: (1) differ- there are psychological aspects to participating
ences in meanings of truth and objectivity and in practices and that researchers must be able to
their sources and (2) misunderstandings and view classrooms from either perspective while
miscommunications between people holding holding the other as an active background:
contrasting positions. The matter of (1) will be “[W]e have proposed the metaphor of mathemat-
addressed below. Regarding (2), Lerman (1996) ics as an evolving social practice that is consti-
claimed that radical constructivism was internally tuted by, and does not exist apart from, the
incoherent: How could radical constructivism constructive activities of individuals” (Cobb
explain agreement when persons evidently agree- et al. 1992, p. 28, italics added).
ing create their own realities? Steffe and Thomp-
son (2000a) replied that interaction was at the core Cobb et al.’s perspective is entirely consistent
of Piaget’s genetic epistemology and thus the idea with theories of emergence in complex systems
of intersubjectivity was entirely coherent with rad- (Schelling 1978; Eppstein and Axtell 1996;
ical constructivism. The core of the misunderstand- Resnick 1997; Davis and Simmt 2003) when
ing was that Lerman on the one hand and Steffe and taken with Maturana’s statement that “anything
Thompson on the other had different meanings for said is said by an observer” (Maturana 1987).
“intersubjectivity.” Lerman meant “agreement of Practices, as stable patterns of social interaction,
meanings” – same or similar meanings. Steffe and exist in the eyes of an observer who sees them.
Thompson meant “nonconflicting mutual interpre- The theoretician who understands the behavior
tations,” which might actually entail nonagreement of a complex system as entailing simultaneously
of meanings of which the interacting individuals both microprocesses and macrobehavior is
are unaware. Thus, Lerman’s objection was valid better positioned to affect macrobehavior
relative to the meaning of intersubjectivity he pre- (by influencing microprocesses) than one who
sumed. Lerman on one side and Steffe and Thomp- sees just one or the other. It is important to note
son on the other were in a state of intersubjectivity that this notion of emergence is not the same as
(in the radical constructivist sense) even though Ernest’s notion of objectivity as described
they publicly disagreed. They each presumed they above.
understood what the other meant when in fact each
understanding of the other’s position was faulty. Truth and Objectivity

Other tensions arose because of interlocutors’ Radical constructivists take the strong position
different objectives. Some mathematics educa- that children have mathematical realities
tors focused on understanding individual’s math- that do not overlap an adult’s mathematics
ematical realities. Others focused on the social (Steffe et al. 1983; Steffe and Thompson 2000).
context of learning. Cobb, Yackel, and Wood
(1992) diffused these tensions by refocusing

C 100 Constructivism in Mathematics Education

Social constructivists (of Ernest’s second type) • Conceptual analysis of mathematical thinking
take this as pedagogical solipsism. and mathematical ideas is a prominent and
widely used analytic tool (Smith et al. 1993;
The implications of [radical constructivism] are Glasersfeld 1995; Behr et al. 1997; Thompson
that individual knowers can construct truth that 2000; Lobato et al. 2012).
needs no corroboration from outside of the knower,
making possible any number of “truths.” Consider • What used to be thought of as practice is now
the pedagogical puzzles this creates. What is the conceived as repeated experience. Practice
teacher trying to teach students if they are all busy focuses on repeated behavior. Repeated
constructing their own private worlds? What are experience focuses on repeated reasoning,
the grounds for getting the world right? Why even which can vary in principled ways from
care whether these worlds agree? (Howe and Berv setting to setting (Cooper 1991; Harel 2008a, b).
2000, pp. 32–33)
• Constructivism has clear and operationalized
Howe and Berv made explicit the social con- implications for the design of instruction
structivist stance that there is a “right” world to be (Confrey 1990; Simon 1995; Steffe and
got – the world of socially constructed meanings. D’Ambrosio 1995; Forman 1996; Thompson
They also revealed their unawareness that, from its 2002) and assessment (Carlson et al. 2010;
very beginning, radical constructivism addressed Kersting et al. 2012).
what “negotiation” could mean in its framework
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an attempt to understand children’s numerical Students’ mathematical concepts and opera- C
thinking and how that thinking might change rather tions constitute first-order models, which are
than to rely on models that were developed outside models that students construct to organize, com-
of mathematics education for purposes other than prehend, and control their own experience
educating children (e.g., Piaget and Szeminska (Steffe et al. 1983, p. xvi). Through a process of
1952; McLellan and Dewey 1895; Brownell conceptual analysis (Glasersfeld 1995), teacher/
1928). The use of the constructivist teaching exper- researchers construct models of students’
iment in the United State was buttressed by ver- mathematical concepts and operations to explain
sions of the teaching experiment methodology that what students say and do. These second-order
were being used already by researchers in the models (Steffe et al. 1983, p. xvi) are called
Academy of Pedagogical Sciences in the then mathematics of students and students’ first-order
Union of Soviet Socialist Republics (Wirszup and models are called simply students’ mathematics.
Kilpatrick 1975–1978). The work at the Academy While teacher/researchers may write about the
of Pedagogical Sciences provided academic schemes and operations that constitute these
respectability for what was then a major departure second-order models as if they are identical to
in the practice of research in mathematics educa- students’ mathematics, these constructs, in fact,
tion in the United States, not only in terms of are a construction of the researcher that only
research methods but more crucially in terms of references students’ mathematics. Conceptual
the research orientation of the methodology. analysis is based on the belief that mathematics
In El’konin’s (1967) assessment of Vygotsky’s is a product of the functioning of human
(1978) research, the essential function of intelligence (Piaget 1980), so the mathematics
a teaching experiment is the production of models of students is a legitimate mathematics to the
of student thinking and changes in it: extent that teacher/researchers can find rational
grounds to explain what students say and do.
Unfortunately, it is still rare to meet with the
interpretation of Vygotsky’s research as modeling, The overarching goal of the teacher/
rather than empirically studying, developmental researchers who use the methodology is to
processes. (El’konin 1967, p. 36) establish the mathematics of students as a con-
ceptual foundation of students’ mathematics edu-
Similarly, the primary purpose of constructivist cation (Steffe and Wiegel 1992; Steffe 2012).
teaching experiments is to construct explanations The mathematics of students opens the way to
of students’ mathematical concepts and operations ground school mathematics in the history of
and changes in them. Without experiences of stu- how it is generated by students in the context
dents’ mathematics afforded by teaching, there of teaching. This way of regarding school
would be no basis for coming to understand the mathematics casts it as a living subject rather
mathematical concepts and operations students than as a subject of being (Steffe 2007).
construct or even for suspecting that these concepts
and operations may be distinctly different from Characteristics: The Elements of
those of teacher/researchers. The necessity to attri- Constructivist Teaching Experiments
bute mathematical concepts and operations to stu-
dents that are independent of those of teacher/ Teaching Episodes
researchers has been captured by Ackermann A constructivist teaching experiment involves
(1995) in speaking of human relations: a sequence of teaching episodes (Hunting
1983; Steffe 1983). A teaching episode includes a
In human relations, it is vital to attribute autonomy teacher/researcher, one or more students, a witness
to others and to things—to celebrate their existence of the teaching episodes, and a method of record-
independently from our current interaction with ing what transpires during the episodes. These
them. This is true even if an attribution (of exis- records can be used in preparing subsequent
tence) is a mental construct. We can literally rob
others of their identity if we deny them an existence
beyond our current interests (p. 343).

C 104 Constructivist Teaching Experiment

episodes as well as in conducting conceptual course of a teaching experiment as well.
analyses of teaching episodes either during or However, one does not embark on the intensive
after the experiment. work of a constructivist teaching experiment
without having initial research hypotheses to test.
Exploratory Teaching
Any teacher/researcher who hasn’t conducted The research hypotheses one formulates prior
a teaching experiment but who wishes to do so to a teaching experiment guide the initial selec-
should first engage in exploratory teaching tion of the students and the teacher/researcher’s
(Steffe and Thompson 2000). It is important that overall general intentions. The teacher/researcher
the teacher/researcher becomes acquainted, at an does his or her best to set these initial hypotheses
experiential level, with students’ ways and means aside during the course of the teaching episodes
of operating in whatever domain of mathematical and focus on promoting the greatest progress
concepts and operations are of interest. In explor- possible in all participating students. The
atory teaching, the teacher/researcher attempts to intention of teacher/researcher is for the students
put aside his or her own concepts and operations to test the research hypotheses by means of how
and not insist that the students learn what he or they differentiate themselves in the trajectory of
she knows (Norton and D’Ambrosio 2008). teaching interactions (Steffe 1992; Steffe and
Otherwise, the teacher/researcher might become Tzur 1994). A teacher/researcher returns to the
caught in what Stolzenberg (1984) called initial research hypotheses retrospectively after
a “trap” – focusing on the mathematics the completing the teaching episodes. This method –
teacher/researcher takes as given instead of setting research hypotheses aside and focusing on
focusing on exploring students’ ways and means what actually happens in teaching episodes – is
of operating. The teacher/researcher’s mathemat- basic in the ontogenetic justification of school
ical concepts and operations can be orienting, but mathematics.
they should not be regarded, initially at least,
as constituting what students should learn until Generating and Testing Working Hypotheses.
they are modified to include at least aspects of In addition to formulating and testing initial
a mathematics of students (Steffe 1991a). research hypotheses, another modus operandi in
a teaching experiment is for a teacher/researcher
Meanings of “Experiment” to generate and test hypotheses during the teach-
Testing Initial Research Hypotheses. One goal ing episodes. Often, these hypotheses are con-
of exploratory teaching is to identify essential ceived “on the fly,” a phrase Ackermann (1995)
differences in students’ ways and means of oper- used to describe how hypotheses are formulated
ating within the chosen context in order to estab- in clinical interviews. Frequently, they are for-
lish initial research hypotheses for the teaching mulated between teaching episodes as well.
experiment (Steffe et al. 1983). These differences A teacher/researcher, through reviewing the
are essential in establishing the constructivist records of one or more earlier teaching episodes,
teaching experiment as involving an “experi- may formulate hypotheses to be tested in the
ment” in a scientific sense. The established next episode (Hackenberg 2010). In a teaching
differences can be used to place students in episode, the students’ language and actions are
experimental groups and the research hypothesis a source of perturbation for the teacher/researcher.
is that the differences between the students in the It is the job of the teacher/researcher to continually
different experimental groups would become postulate possible meanings that lie behind stu-
quite large over the period of time the students dents’ language and actions. It is in this way that
participate in the experiment and that the students students guide the teacher/researcher. The teacher/
within the groups would remain essentially alike researcher may have a set of hypotheses to test
(Steffe and Cobb 1988). Considerable hypothesis before a teaching episode and a sequence of situa-
building and testing must happen during the tions planned to test the hypotheses. But because of
students’ unanticipated ways and means of operat-
ing as well as their unexpected mistakes, a teacher/

Constructivist Teaching Experiment C105

researcher may be forced to abandon these hypoth- The focus of the clinician [teacher] is to understand C
eses while interacting with the students and to the originality of [the child’s] reasoning, to
create new hypotheses and situations on the describe its coherence, and to probe its robustness
spot (Norton 2008). The teacher/researchers also or fragility in a variety of contexts. (p. 346)
might interpret the anticipated language and
actions of the students in ways that were unex- Meanings of Teaching in a Teaching
pected prior to teaching. These impromptu inter- Experiment
pretations are insights that would be unlikely to
happen in the absence of direct, longitudinal Learning how to interact with students through
interaction with the students in the context effective teaching actions is a central issue in any
of teaching interactions. Here, again, the teaching experiment (Steffe and Tzur 1994). If
teacher/researcher is obliged to formulate new teacher/researchers knew ahead of time how to
hypotheses and to formulate situations of learn- interact with the selected students and what the
ing to test them (Tzur 1999). outcomes of those interactions might be, there
would be little reason for conducting a teaching
Living, Experiential Models of Students’ experiment (Steffe and Cobb 1983). There are
Mathematics essentially two types of interaction engaged in by
teacher/researchers in a teaching experiment:
Through generating and testing hypotheses, responsive and intuitive interactions and analytical
boundaries of the students’ ways and means of interactions.
operating – where the students make what to
a teacher/researcher are essential mistakes – can Responsive and Intuitive Interaction
be formulated (Steffe and Thompson 2000). In responsive and intuitive interactions, teacher/
These essential mistakes are of the same nature researchers are usually not explicitly aware of how
as those Piaget found in his studies of children, or why they interact as they do. In this role, teacher/
and a teacher/researcher uses them for essentially researchers are agents of interaction and they strive
the same purpose he did. They are observable to harmonize themselves with the students with
when students fail to make viable adaptations whom they are working to the extent that they
when interacting in a medium. Operations and “lose” themselves in their interactions. They make
meanings a teacher/researcher imputes to stu- no intentional distinctions between their knowledge
dents constitute what are called living, experien- and the students’ knowledge, and, experientially,
tial models of students’ mathematics. Essential everything is the students’ knowledge as they strive
mistakes can be thought of as illuminating to feel at one with them. In essence, they become the
the boundaries of what kinds of adaptations students and attempt to think as they do (Thompson
a living, experiential model can currently make 1982, 1991; van Manen 1991). Teacher/researchers
in these operations and meanings. These bound- do not adopt this stance at the beginning of
aries are usually fuzzy, and what might be placed a teaching experiment only. Rather, they maintain
just inside or just outside them is always a source it throughout the experiment whenever appropriate.
of tension and often leads to creative efforts on By interacting with students in a responsive and an
the part of a teacher/researcher. What students intuitive way, the goal of teacher/researchers is to
can do is understood better if what they cannot engage the students in supportive, nonevaluative
do is also understood. It also helps to understand mathematical interactivity.
what a student can do if it is understand what
other students, whose knowledge is judged to be Analytical Interaction
at a higher or lower level, can do (Steffe and When teacher/researchers turn to analytical inter-
Olive 2010). In this, we are in accordance action, they “step out” of their role in responsive/
with Ackermann (1995) that: intuitive interaction and become observers as well.
As first-order observers, teacher/researchers focus

C 106 Constructivist Teaching Experiment

on analyzing students’ thinking in ongoing inter- When this happens, the observer may help
action (Steffe and Wiegel 1996). All of the teacher/ a teacher/researcher both to understand the student
researchers’ attention and energy is absorbed in and to posit further interaction. There are also
trying to think like the students and produce and occasions when the observer might make an inter-
then experience mathematical realities that are pretation of a student’s actions that is different
intersubjective with theirs. The teacher/researchers from that of a teacher/researcher for any one of
probes and teaching actions are not to foment several reasons. For example, the observer might
adaptation in the students but in themselves. catch important elements of a student’s actions that
When investigating student learning, teacher/ apparently are missed by a teacher/researcher. In
researchers become second-order observers, any case, the witness should suggest but not
which Maturana (1978) explained as “the demand specific teaching interventions.
observer’s ability . . . to operate as external to
the situation in which he or she is, and thus be Retrospective Conceptual Analysis
an observer of his or her circumstance as an
observer” (p. 61). As second-order observers, Conceptual analysis is intensified during the
teacher/researchers focus on the accommoda- period of retrospective analysis of the public
tions they might engender in the students’ ways records of the teaching episodes, which is
and means of operating (Steffe 1991b). They a critical part of the methodology. Through
become aware of how they interact and of the analyzing the corpus of video records, the
consequences of interacting in a particular way. teacher/researchers conduct a historical analysis
Assuming the role of a second-order observer is of the living, experiential models of students’
essential in investigating student learning in mathematics throughout the period of time the
a way that explicitly as well as implicitly takes teaching episodes were conducted. The activity
into account the mathematical knowledge of the of model building that was present throughout the
teacher/researchers as well as the knowledge of teaching episodes is foregrounded, and concepts
the students (Steffe and Wiegel 1996). in the core of a constructivist research program
like assimilation, accommodation, scheme (von
The Role of a Witness of the Teaching Glasersfeld 1981), cognitive and mathematical
Episodes play, communication, spontaneous development
A teacher/researcher should expect to encounter (Piaget 1964), interaction (von Foerster 1984),
students operating in unanticipated and apparently mental operation (von Glasersfeld 1987), and
novel ways as well as their making unexpected self-regulation emerge in the form of specific
mistakes and becoming unable to operate. In and concrete explanations of students’ mathemat-
these cases, it is often helpful to be able to appeal ical activity. In this regard, the modeling process
to an observer of a teaching episode for an alterna- in which we engage is compatible with how
tive interpretation of events. Being immersed in Maturana (1978) regards scientific explanation:
interaction, a teacher/researcher may not be able
to act as a second-order observer and step out of the As scientists, we want to provide explanations for
interaction, reflect on it, and take further action on the phenomena we observe. That is, we want to
that basis. In order to do so, a teacher/researcher propose conceptual or concrete systems that can be
would have to “be” in the interaction and outside of deemed intentionally isomorphic to the systems
it, which can be difficult. It is quite impossible to that generate the observed phenomena. (p. 29)
achieve this if there are no conceptual elements
available to the teacher/researcher from past teach- However, in the case of a teaching experiment,
ing experiments that can be used in interpreting we seek models that fit within our living, experien-
the current situation. The result is that teacher/ tial models of students’ mathematics without
researchers usually react to surprising behavior by claiming isomorphism because we have no access
switching to a more intuitive mode of interaction. to students’ mathematical realities outside of our
own ways and means of operating when bringing

Constructivist Teaching Experiment C107

the students’ mathematics forth. So, we cannot Appendix: Example Studies Using C
get outside our observations to check if our Teaching Experiment Methodology
conceptual constructs are isomorphic to stu-
dents’ mathematics. But we can and do establish Battista MT (1999) Fifth graders’ enumeration of
viable ways and means of thinking that fit within cubes in 3D arrays: conceptual progress in an
the experiential constraints that we established inquiry-based classroom. J Res Math Educ
when interacting with the students in teaching 30(4):417–448
episodes (Steffe 1988, 1994; Norton and
Wilkins 2010). Cobb P (1995) Mathematics learning and small
group interactions: four case studies. In: Cobb
Since the time of its emergence, the construc- P, Bauersfeld H (eds) Emergence of mathemat-
tivist teaching experiment has been widely used ical meaning: interaction in classroom cultures.
in investigations of students’ mathematics as well Lawrence Erlbaum Associates, Hillsdale,
as in investigations of mathematics teaching pp 25–129
(cf. Appendix for sample studies). It has also
been adapted to fit within related research pro- Cobb P (1996) Constructivism and activity the-
grams (e.g., Cobb 2000; Confrey and Lachance ory: a consideration of their similarities and
2000; Simon et al. 2010). differences as they relate to mathematics
education. In: Mansfield H, Patemen N,
Acknowledgment We would like to thank Dr. Anderson Bednarz N (eds) Mathematics for tomorrow’s
Norton for his insightful comments on an earlier version of young children: international perspectives on
this paper. curriculum. Kluwer, Dordrecht, pp 10–56

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C 110 Creativity in Mathematics Education

mathematical creativity implies mathematical our perception of the world. Feldman,
giftedness, but the reverse is not necessarily true Cziksentmihalyi, and Gardner (1994) writes:
(Sriraman 2005). Usiskin’s (2000) eight tiered “the achievement of something remarkable and
hierarchy of creativity and giftedness in mathe- new, something which transforms and changes
matics further shed some light of this view of the a field of endeavor in a significant way . . . the
relationship between creativity and giftedness in kinds of things that people do that change the
professional mathematics. In his model, mathe- world.” Ordinary, or everyday, creativity is
matically gifted individuals such as professional, more relevant in a regular school setting.
working mathematicians are at level five, while Feldhusen (2006) describes little c as: “Wherever
creative mathematicians are at level six and there is a need to make, create, imagine, produce,
seven. However, the relationship between gifted- or design anew what did not exist before – to
ness and creativity has been the subject of much innovate – there is adaptive or creative behavior,
controversy (Leikin 2008; Sternberg and O’Hara sometimes called ‘small c.” Investigation into the
1999) as some see creativity as part of an overall concept of creativity also distinguishes between
concept of giftedness (Renzulli 1986). In this creativity as either domain specific or domain
entry the relationship between mathematical general (Kaufman and Beghetto 2009).
creativity and giftedness and ability will be
looked at through a synthesis of some recent Whether or not creativity is domain specific or
articles published in ZDM. First, the concepts of domain general, or if you look at ordinary or
giftedness, ability, and creativity will be extraordinary creativity, most definitions of
discussed. Second, common themes from the creativity include some aspect of usefulness and
three articles will be synthesized that capture novelty (Sternberg 1999; Plucker and Beghetto
the main ideas in the studies. Lastly, the synthesis 2004; Mayer 1999). What is useful and novel
will be situated into the more generally framed depends on the context of the creative process
research in psychology. of an individual. The criteria for useful and
novel in professional arts would differ signifi-
Theory cantly from what is deemed useful and novel in
a mathematics class in lower secondary school.
Creativity There is therefore a factor of relativeness to cre-
One of the main challenges in investigating math- ativity. For a professional artist, some new,
ematical creativity is the lack of a clear and groundbreaking technique, product, or process
accepted definition of the term mathematical that changes his or her field in some significant
creativity and creativity itself. Previous examina- way would be creative, but for a mathematics
tions of the literature have concluded that there is student in lower secondary school, an unusual
no universally accepted definition of either crea- solution to a problem could be creative. Mathe-
tivity or mathematical creativity (Sriraman 2005; matical creativity in a K-12 setting can as such be
Mann 2005). Treffinger et al. (2002) write, for defined as the process that results in a novel solu-
instance, that there are more than 100 contempo- tion or idea to a mathematical problem or the
rary definitions of mathematical creativity. formulation of new questions (Sriraman 2005).
Nevertheless, there are certain parameters agreed
upon in the literature that helps narrow down Giftedness
the concept of creativity. Most investigations of For decades giftedness was equated with concept
creativity take one of two directions: extraordi- of intelligence or IQ (Renzulli 2005; Brown et al.
nary creativity, known as big C, or everyday 2005; Coleman and Cross 2005). Terman (1925)
creativity, known as little c (Kaufman and claimed that gifted individuals are those who
Beghetto 2009). Extraordinary creativity refers score at the top 1 % of the population on the
to exceptional knowledge or products that change Stanford-Binet test. This understanding of
giftedness has survived to this day in some
conceptions. However, most researchers now

Creativity in Mathematics Education C111

view giftedness as a more multifaceted concept in classrooms, and Kim et al. (2003) state that C
which intelligence is but one of several aspects traditional tests rarely identify mathematical
(Renzulli 2005). One example is Renzulli’s creativity. Hong and Aqui (2004) compared
(1986) three-ring model of giftedness. In an cognitive and motivational characteristics of
attempt to capture the many facets of giftedness, high school students who were academically
Renzulli presented giftedness as an interaction gifted in math, creatively talented in math, and
between above-average ability, creativity, and non-gifted. The authors found that the creatively
task commitment. He went on to separate talented students used more cognitive strategies
giftedness into two categories: schoolhouse gift- than the academically gifted students. These
edness and creative productive giftedness. The findings indicate that mathematical ability and
former refers to the ease of acquiring knowledge mathematical attainment in a K-12 setting are
and taking standardized tests. The latter involves not necessarily synonymous.
creating new products and processes, which
Renzulli thought was often overlooked in school In the online thefreedictionary.com, ability is
settings. Many researchers support this notion defined as “the quality of being able to do some-
that creativity should be included in the concep- thing, especially the physical, mental, financial,
tion of giftedness in any area (Miller 2012). or legal power to accomplish something.” Attain-
ment is defined as “Something, such as an accom-
For this entry, giftedness will be looked at in plishment or achievement, that is attained.” The
the domain of mathematics, as Csikszentmihalyi key difference is that ability points to a potential
(2000) pointed out the field-dependent character to do something, while attainment refers to some-
of the concept of giftedness. Due to the lack of thing that has been accomplished. In the field of
a conceptual clarity regarding giftedness and the mathematics, mathematical ability then refers to
heterogeneity of the gifted population, both in the ability to do mathematics and not the ability
general and in mathematics, identification of to do well on mathematics attainment tests in
gifted students has varied (Kontoyianni et al. school. In order to de facto define mathematical
2011). Instead, prominent characteristics of ability, mathematics itself has to be defined. It is
giftedness in mathematics are found in the beyond the scope of this entry to discuss what
research literature. Krutetskii (1976) noted in mathematics itself is (for a K-12 setting, see,
his investigation of gifted students in mathemat- for instance, NCTM 2000 or Niss 1999), so
ics a number of characteristic features: ability for mathematical ability will simply be defined as
logical thought with respect to quantitative and the ability to do mathematics.
spatial relationships, number and letter symbols;
the ability for rapid and broad generalization of Conceptual Relationships
mathematical relations and operations, flexibility In some recent ZDM articles the concept of math-
of mental processes and mathematical memory. ematical creativity is linked to other concepts
Similar features of mathematical giftedness have through statistical and qualitative investigation.
been proposed by other researchers (see for In Kattou et al. (2013), the relationship
instance Sriraman 2005). between mathematical ability and mathematical
creativity was investigated quantitatively with
Ability the use of a mathematical ability test and
Often, mathematical ability has been seen as a mathematical creativity test. Data were col-
equivalent to mathematical attainment and to lected by administering the two tests to 359 ele-
some degree, there is some truth to that notion. mentary school students. The authors concluded,
There is a statistical relationship between using confirmatory factor analysis, that mathe-
academic attainment in mathematics and high matical creativity is a subcomponent of mathe-
mathematical ability (Benbow and Arjmand matical ability. Mathematical ability was
1990). However, Ching (1997) discovered that measured by 29 items in the following categories:
hidden talent go largely unnoticed in typical quantitative ability, causal ability, spatial ability,

C 112 Creativity in Mathematics Education

qualitative ability, and inductive/deductive beliefs as a possible explanation for the observed
ability. The operationalization of mathematical inconsistencies. Teacher-directed conceptions of
ability was based on the assumption that mathe- creativity are associated with surface beliefs and
matical ability is a multidimensional construct student-directed conceptions of creativity are
and Krutetskii’s (1976) classification of gifted- associated with deep beliefs. The authors go on
ness in mathematics. Mathematical creativity to state that student-oriented conceptions of
was measured with five open-ended multiple creativity are more of a mathematical nature and
solution tasks that were assessed on the basis of this attention enables teachers to be more flexible
fluency, flexibility, and originality (Leikin 2007). during their lessons.

Pitta et al. (2013) investigated the relationship These three articles all focus on different char-
between mathematical creativity and cognitive acteristics of mathematical creativity. They do
styles. Mathematical creativity was measured sim- not explicitly investigate the relationship
ilarly to Kattou et al. (2013). A mathematical between giftedness, ability, and creativity. Nev-
creativity test consisting of five tasks was given ertheless, there are certain similarities that might
to 96 prospective primary school teachers and was be inferred on a more structural level, and the
assessed on the basis of fluency, flexibility, and studies add to our overall understanding of gift-
originality. Cognitive style was measured with the edness, creativity, and ability in mathematics.
Object-Spatial Imagery and Verbal Questionnaire Certain cognitive styles, mathematical ability,
(OSIVQ) with respect to three styles: spatial, and types of beliefs are all found to predict and
object, and verbal. Using multiple regression, the have a relationship with mathematical creativity.
authors conclude that spatial and object styles Student-directed conception of mathematical
were significant predictors of mathematical creativity as a deep belief, spatial cognitive
creativity, while verbal style was not significant. style, and general mathematical ability are all
Spatial cognitive style was positively related to linked to mathematical creativity. As a concept,
mathematical creativity, while object cognitive mathematical creativity does not exist in
style was negatively related to mathematical crea- a vacuum. The literature synthesized in this
tivity. Furthermore, spatial cognitive style was entry suggests that certain features and factors
positively related to fluency, flexibility, and orig- are required for mathematical creativity to arise.
inality, while object cognitive style was negatively
related to originality and verbal cognitive style Although no explicit relationships between
was negatively related to flexibility. mathematical ability, cognitive styles, and beliefs
are explored in the three studies, an underlying
Using an entirely different methodology, link may be inferred from them. Pitta et al. (2013)
Lev-Zamir and Leikin (2013) analyzed two point out that previous research has found that
teachers’ declarative conceptions about mathe- spatial cognitive style can be beneficial for phys-
matical creativity in teaching and conceptions- ics, mechanical engineering, and mathematics
in-action seen in their lessons. The authors write tasks (see, for instance, Kozhevnikov et al.
that while declarative statements about mathe- 2005). In Lev-Zamir and Leikin’s study (2013),
matical creativity in teaching may seem similar, the teacher with the deep student-directed con-
their conceptions-in-action could differ vastly. In ceptions of mathematical creativity had a much
the study, the two teachers used much of the same stronger mathematical background than the other
terminology when relating to originality and flex- teacher. Both spatial cognitive style and deep
ibility in teaching. However, there was a large student-directed conceptions of mathematical
gap between the teachers’ declarations and creativity are therefore conceivably connected
actions. One of the teachers displayed a lack of to mathematical ability and knowledge. Kattou
flexibility in the classroom, while the interaction et al. (2013) found a strong correlational relation-
between the other teacher and her students ship between mathematical ability and mathe-
displayed flexibility. The authors point out the matical creativity. If certain cognitive styles and
distinction between deep beliefs and surface types of beliefs are connected to mathematical

Creativity in Mathematics Education C113

ability, and mathematical ability is linked to distinguish individuals into different levels of C
mathematical creativity, it stands to reason that mathematical creativity according to some other
mathematical ability, mathematical creativity, quality or ability.
certain cognitive styles, and types of beliefs are
all linked. Implications for Teaching
Although only one of the articles (Kattou et al.
Who Are Creative? 2013) makes explicit recommendations for math-
Closely related to conceptual relationships ematics teaching, the implications of the three
between mathematical creativity and other con- articles are on some levels related. Kattou et al.
cepts is the question of “who are mathematically conclude that the encouragement of mathemati-
creative?” Can individuals be distinguished into cal creativity is important for further develop-
separate groups according to their mathematical ment of students’ mathematical ability. More
creativity, and what characterizes these groups? importantly, they write, teachers should not
Kattou et al. (2013) clustered students into three limit their teaching to spatial conception, arith-
subgroups: low, average, and high mathematical metic, and proper use of methods and operations.
ability. The high-ability students were also Teachers should recognize the importance of
highly creative students, the average-ability stu- creative thinking in the classroom. This is closely
dents had an average performance on the math- related to what Lev-Zamir and Leikin (2013)
ematical creativity test, while low-ability conclude. Teachers who hold a mathematically
students have a low creative potential in student-oriented conception of mathematical
mathematics. Pitta et al. (2013) classified the creativity were found to be more flexible during
prospective teachers as spatial visualizers, lessons and stimulate students’ mathematical
object visualizers, or verbalizers. The spatial creativity. In other words, with a student-oriented
visualizers scored higher on the mathematical conception of mathematical creativity, teachers
creativity test than both other groups. In the will to a greater degree be able to recognize and
third article examined here, the conceptions of encourage creative mathematical thinking during
creativity of only two teachers were investigated their lessons.
(Lev-Zamir and Leikin 2013). As such, it is
difficult to generalize any finding. Nevertheless, The third article (Pitta et al. 2013) did not
the authors point out the different mathematical make any explicit recommendations or implica-
backgrounds of the two teachers and how the tions for teaching mathematics. However, as they
teacher with the stronger mathematical back- investigated prospective teachers’ mathematical
ground has deeper beliefs regarding mathemati- creativity, the results and conclusions may be
cal creativity. relevant for teaching when seen in a broader per-
spective. Pitta et al. found that spatial visualizers
All three studies, through different methodol- had a statistical significant higher creative
ogies, can be said to cluster individuals according performance than other teachers. The observed
to their level of mathematical creativity. As with differences were related to the different strategies
conceptual relationships, the findings of the employed by the spatial visualizers, object
three studies synthesized in this entry cannot be visualizers, and verbal visualizers. The spatial
unified explicitly. The three studies investigated visualizers employed more flexible and analytic
different aspects of mathematical creativity, strategies to tasks. This allowed them to be more
using different methodologies. Instead, the creative and provide more, different, and unique
findings have to be looked at from a more general solutions. In light of Lev-Zamir and Leikin
and systemic perspective. That means instead of (2013) and Kattou et al. (2013) conclusions, the
looking at what the specific characteristics of question becomes whether a flexible and analytic
mathematically creative individuals are, the approach to mathematics tasks translates into an
focus is that there are characteristics of mathe- analytic and flexible approach to mathematics
matically creative individuals. All three studies teaching. If that is the case, then flexible and

C 114 Creativity in Mathematics Education

creative teaching is also related to spatial cogni- classified as nonintellective requisites. Flexible
tive style. However, as Pitta et al. (2013) ask, it is teaching that stimulates mathematical creativity
unknown whether having a spatial cognitive style falls under the category of environmental sup-
is a result of experience or inborn abilities. They port. As such, the observations in the studies
go on to recommend further investigation to see if synthesized here are in many ways analogous
prospective teachers can be trained to use their to research into general creativity and
spatial visualization. It may lead to enhanced giftedness.
spatial imagery and consequently facilitate math-
ematical creativity, possibly also in their mathe- Similarly, the dynamic theory of giftedness
matics teaching. (Babaeva 1999), which emphasizes the social
aspects of the development in giftedness, can
Giftedness and Creativity in Psychology also provide a theoretical perspective on the
observations synthesized in this entry. This theory
The research into the field of general creativity consists of three principles that explain the devel-
focuses on four different variables: person, pro- opment of giftedness: (a) an obstacle for positive
cess, product, and press. The person category growth is introduced, (b) a process to overcome
highlights the internal cognitive characteristics the obstacle, and (c) alteration and incorporation
of individuals. The process category looks at the of the experience (Miller 2012). Kattou et al.
internal process that takes place during a creative (2013) point out how mathematical creativity
activity. Product focuses on the characteristics of is essential for the growth of overall mathematical
products thought to be creative. Last, the press ability (or giftedness), while Lev-Zamir and
category explores the ways environmental factors Leikin (2013) show how challenging mathemati-
can influence creativity (Taylor 1988). The arti- cal problems and flexible teaching can help the
cles recently published in ZDM focused their development of mathematical creativity. Both
research primarily into the person and press com- studies show the dynamic aspect of mathematical
ponent of mathematical creativity. In other creativity, in the sense that it evolves and is
words: what characterizes the mathematically influenced by other external factors.
creative individual and how can mathematical
creativity be developed in the classroom. Cross-References

Mathematical creativity is linked to and ▶ Giftedness and High Ability in Mathematics
influenced by ability, beliefs, cognitive style,
and the classroom environment (Lev-Zamir and References
Leikin 2013; Pitta et al. 2013; Kattou et al. 2013).
These findings are analogous to much of the Babaeva JD (1999) A dynamic approach to giftedness:
research into general creativity and giftedness. Theory and practice. High Ability Studies, 10(1):51–68
The star model of Abraham Tannenbaum (2003)
conceptualizes giftedness into five elements, Benbow CP, Arjmand O (1990) Predictors of high
some of which are seen in the studies synthesized academic achievement in mathematics and science
in this entry: (a) superior general intellect, (b) by mathematically talented students: a longitudinal
distinctive special aptitudes, (c) nonintellective study. J Educ Psychol 82(3):430
requisites, (d) environmental supports, and (e)
chance. Creativity is here included in the Brown SW, Renzulli JS, Gubbins EJ, Siegle D, Zhang W,
nonintellective requisites. Mathematical ability Chen C (2005) Assumptions underlying the identifica-
would be placed in the distinctive special apti- tion of gifted and talented students. Gifted Child
tudes, as it portrays to domain specific abilities, Q 49:68–79
while both beliefs and cognitive styles would be
Ching TP (1997) An experiment to discover mathematical
talent in a primary school in Kampong Air. ZDM
29(3):94–96

Coleman LJ, Cross TL (2005) Being gifted in school,
2nd edn. Prufrock Press, Waco, TX

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teenagers prepare for the world of work. Basic In: Sternberg RJ (ed) Handbook of creativity
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Kattou M, Kontoyianni K, Pitta-Pantazi, Christou C
(2013) Connecting mathematical creativity to mathe- Plucker J, Beghetto R (2004) Why creativity is domain
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C 116 Critical Mathematics Education

Critical Mathematics Education grammar of instrumental reason. How could one
imagine any form of emancipatory interests being
Ole Skovsmose associated to this subject?
Department of Learning and Philosophy, Aalborg
University, Aalborg, DK, Denmark Steps into Critical Mathematics Education
Although there were no well-defined theoretical
Keywords frameworks to draw on, there were from
the beginning of the 1970s many attempts in
Mathematics education for social justice; Critical formulating a critical mathematics education.
mathematics education; Ethnomathematics; Let me mention some publications.
Mathematics in action; Students’ foregrounds;
Mathemacy; Landscapes of investigation; The book Elementarmathematik: Lernen fu€r
Mathematization; Dialogic teaching and learning die Praxis (Elementary mathematics: Learning
for the praxis) by Peter Damerow, Ulla Elwitz,
Definition Christine Keitel, and Ju€rgen Zimmer from 1974
was crucial for the development of critical
Critical mathematics education can be character- mathematics education in a German context.
ized in terms of concerns: to address social In the article “Pl€adoyer f€ur einen problemor-
exclusion and suppression, to work for social ientierten Mathematikunterrich in emanzipa-
justice in whatever form possible, to try open torisher Absicht” (“Plea for a problem-oriented
new possibilities for students, and to address mathematics education with an emancipatory
critically mathematics in all its forms and aim”) from 1975, Dieter Volk emphasized that
application. it is possible to establish mathematics education
as a critical education. The book Indlæring som
Characteristics social proces (Learning as a social process) by
Stieg Mellin-Olsen was published in 1977. It
Critical Education provided an opening of the political dimension
Inspired by the students’ movement, a New Left, of mathematics education, a dimension that
peace movements, feminism, antiracism, and was further explored in Mellin-Olsen (1987).
critical education proliferated. A huge amount Indlæring som social proces was crucial for the
of literature became published, not least in development of critical mathematics education in
Germany, and certainly the work of Paulo Freire the Scandinavian context. An important
was recognized as crucial for formulating radical overview of Mellin-Olsen’s work is found in
educational approaches. Kirfel and Linde´n (2010). Dieter Volk’s Kritische
Stichwo¨rter zum Mathematikunterricht (Critical
However, critical education was far from notions for mathematics education) from 1979
expressing any interest in mathematics. In fact, provided a broad overview of what could be
with reference to the Frankfurt School, mathe- called the first wave in critical mathematics
matics was considered almost an obstruction to education. Soon after followed, in Danish,
critical education. Thus, Habermas, Marcuse, and Skovsmose (1980, 1981a, b).
many others associated instrumental reason with,
on the one hand, domination, and, on the other Marilyn Frankenstein (1983) provided
hand, the rationality cultivated by natural science an important connection between critical
and mathematics. Mathematics appeared as the approaches in mathematics education and the
outlook of Freire, and in doing so she was the
first in English to formulate a critical mathemat-
ics education (see also Frankenstein 1989).
Around 1990, together with Arthur Powell
and several others, she formed the critical

Critical Mathematics Education C117

mathematics education group, emphasizing the form of mathematics: everyday mathematics, C
importance of establishing a united concept of engineering mathematics, academic mathemat-
critique and mathematics (see Frankenstein ics, and ethnomathematics.
2012; Powell 2012). Skovsmose (1994) provided • Students. To a critical mathematics education,
an interpretation of critical mathematics it is important to consider students’ interests,
education and Skovsmose (2012) a historical expectations, hopes, aspirations, and motives.
perspective. Thus, Frankenstein (2012) emphasizes the
importance of respecting student knowledge.
Critical mathematics education developed The notion of students’ foregrounds has been
rapidly in different directions. As a consequence, suggested in order to conceptualize students’
the very notion of critical mathematics education perspectives and interests (see, for instance,
came to refer to a broad range of approaches, such Skovsmose 2011). A foreground is defined
as mathematics education for social justice through very many parameters having to do
(see, for instance, Sriraman 2008; Penteado and with economic conditions, social-economic
Skovsmose 2009; Gutstein 2012), pedagogy of processes of inclusion and exclusion, cultural
dialogue and conflict (Vithal 2003), responsive values and traditions, public discourses, and
mathematics education (Greer et al. 2009), and, racism. However, a foreground is, as well,
naturally, critical mathematics education. Many defined through the person’s experiences
ethnomathematical studies also link closely with of possibilities and obstructions. It is a preoc-
critical mathematics education (see, for instance, cupation of critical mathematics education to
D’Ambrosio 2006; Knijnik 1996; Powell and acknowledge the variety of students’ fore-
Frankenstein 1997). grounds and to develop a mathematics educa-
tion that might provide new possibilities for
Some Issues in Critical Mathematics the students. The importance of recognizing
Education students’ interest has always been a concern of
Critical mathematics education can be character- critical mathematics education.
ized in terms of concerns, and let me mention • Teachers. As it is important to consider the
some related to mathematics, students, teachers, students’ interests, it is important to consider
and society: the teachers’ interests and working conditions
• Mathematics can be brought in action in as well. Taken more generally, educational sys-
tems are structured by the most complex sets of
technology, production, automatization, deci- regulations, traditions, and restrictions, which
sion making, management, economic transac- one can refer to as the “logic of schooling.”
tion, daily routines, information procession, This “logic” reflects (if not represents) the eco-
communication, security procedures, etc. In nomic order of today, and to a certain degree it
fact, mathematics in action plays a part in all determines what can take place in the classroom.
spheres of life. It is a concern of a critical math- It forms the teachers’ working conditions. It
ematics education to address mathematics in its becomes important to consider the space of pos-
very many different forms of applications and sibilities that might be left open by this
practices. There are no qualities, like objectivity logic. These considerations have to do with the
and neutrality, that automatically can be associ- micro–macro (classroom-society) analyses as in
ated to mathematics. Mathematics-based actions particular addressed by Paola Valero (see, for
can have all kind of qualities, being risky, instance, Valero 2009). Naturally, these com-
reliable, dangerous, suspicious, misleading, ments apply not only to the teachers’ working
expensive, brutal, profit generating, etc. Mathe- conditions but also to the students’ conditions
matics-based action can serve any kind of for learning. While the concern about the stu-
interest. As with any form of action, so dents’ interests has been part of critical
also mathematics in action is in need of
being carefully criticized. This applies to any

C 118 Critical Mathematics Education

mathematics education right from the beginning, explored in many different directions.
a direct influence from the students’ movements, Addressing equity also represents concerns of
the explicit concern about teaching conditions is critical mathematics education, and the
a more recent development of critical mathemat- discussion of social justice and equity bring
ics education. us to address processes of inclusion and
• Society can be changed. This is the most exclusion. Social exclusion can take the most
general claim made in politics. It is the explicit brutal forms being based on violent discourses
claim of any activism. And it is as well integrating racism, sexism, and hostility
a concern of critical mathematics education. towards “foreigners” or “immigrants.” Such
Following Freire’s formulations, Gutstein discourses might label groups of people as
(2006) emphasizes that one can develop being “disposable,” “a burden,” or “nonpro-
a mathematics education which makes it ductive,” given the economic order of today. It
possible for students to come to read and is a concern of critical mathematics education
write the world: “read it,” in the sense that it to address any form of social exclusion. As
becomes possible to interpret the world filled an example I can refer to Martin (2009).
with numbers, diagrams, figures, and mathe- However, social inclusion might also
matics, and “write it,” in the sense that is represent a questionable process: it could
becomes possible to make changes. However, mean an inclusion into the capitalist mode
a warning has been formulated: one cannot of production and consumption. So critical
talk about making sociopolitical changes mathematics education needs to address
without acknowledge the conditions for inclusion–exclusion as contested processes.
making changes (see, for instance, Pais However, many forms of inclusion–exclusion
2012). Thus, the logic of schooling could have until now not been discussed profoundly
obstruct many aspirations of critical mathe- in mathematics education: the conditions of
matics education. Anyway, I find that it blind students, deaf students, and students
makes good sense to articulate a mathematics with different handicaps – in other words,
education for social justice, not least in a most students with particular rights. However,
unjust society. such issues are now being addressed in the
research environment created by the Lulu
Some Notions in Critical Mathematics Healy and Miriam Goody Penteado in Brazil.
Education Such initiatives bring new dimensions to
Notions such as social justice, mathemacy, critical mathematics education.
dialogue, and uncertainty together with many • Mathemacy is closely related to literacy,
others are important for formulating concerns of as formulated by Freire, being a competence
critical mathematics education. In fact we have to in reading and writing the world. Thus,
concern ourselves with clusters of notions of D’Ambrosio (1998) has presented a
which I highlight only a few: “New Trivium for the Era of Technology” in
• Social justice. Critical mathematics education terms of literacy, matheracy, and technoracy.
Anna Chronaki (2010) provided a multifaceted
includes a concern for addressing any form of interpretation of mathemacy, and in this way it is
suppression and exploitation. As already emphasized that this concept needs to be
indicated, there is no guarantee that an educa- reworked, reinterpreted, and redeveloped in
tional approach might in fact be successful in a never ending process. Different other notions
bringing about any justice. Still, working for have, however, been used as well for these
social justice is a principal concern of critical complex competences, including mathematical
mathematics education. Naturally, it needs to literacy and mathematical agency. Eva Jablonka
be recognized that “social justice” is a most (2003) provides a clarifying presentation of
open concept, the meaning of which can be

Critical Mathematics Education C119

mathematical literacy, showing how this very Valero, (Eds.) (2010), Wager, A. A. and Stinson, C
notion plays a part in different discourses, D. W. (Eds.) (2012); and Skovsmose and Greer
including some which hardly represent critical (Eds.) (2012). Looking a bit into the future much
mathematics education. The notion of mathe- more is on its way. Let me just refer to some
matical agency helps to emphasize the impor- doctoral studies in progress that I am familiar
tance of developing a capacity not only with with. Denival Biotto Filho is addressing students
respect to understanding and reflection but also in precarious situations and in particular their
with respect to acting. foregrounds. Raquel Milani explores further the
• Dialogue. Not least due to the inspiration from notion of dialogue, while Renato Marcone
Freire, the notion of dialogue has played an addresses the notion of inclusion–exclusion,
important role in the formulation of critical emphasizing that we do not have to do with
mathematics education. Dialogic teaching and a straightforward good-bad duality. Inclusion
learning has been presented as one way of could also mean an inclusion into the most
developing broader critical competences related questionable social practices.
to mathematics. Dialogic teaching and learning
concerns forms of interaction in the classroom. Critical mathematics education is an ongoing
It can be seen as an attempt to break at least endeavor. And naturally we have to remember
some features of the logic of schooling. Dia- that as well the very notion of critical mathemat-
logic teaching and learning can be seen as a way ics education is contested. There are very many
of establishing conditions for establishing different educational endeavors that address
mathemacy (or mathematical literacy, or math- critical issues in mathematics education that do
ematical agency). Problem-based learning and not explicitly refer to critical mathematics
project work can also be seen as way of framing education. And this is exactly as it should be as
a dialogic teaching and learning. the concerns of critical mathematics cannot
• Uncertainty. Critique cannot be any dogmatic be limited by choice of terminology.
exercise, in the sense that it can be based on
any well-defined foundation. One cannot take Cross-References
as given any particular theoretical basis for
critical mathematics education; it is always ▶ Critical Thinking in Mathematics Education
in need of critique (see, for instance, Ernest ▶ Dialogic Teaching and Learning in
2010). In particular one cannot assume
any specific interpretation of social justice, Mathematics Education
mathemacy, inclusion–exclusion, dialogue, ▶ Mathematical Literacy
critique, etc. They are all contested concepts. ▶ Mathematization as Social Process
There is no particular definition of, say, social
justice that one can take as a given. We have to References
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Critical Thinking in Mathematics Education C121

Critical Thinking in Mathematics change. In Asian traditions derived from C
Education the Ma˜dhyamika Buddhist philosophy, critical
deconstruction is a method of examining possible
Eva Jablonka alternative standpoints on an issue, which might
Department of Education and Professional amount to finding self-contradictions in all of
Studies, King’s College London, London, UK them (Fenner 1994). When combined with medi-
tation, the deconstruction provides for the student
Keywords a path towards spiritual insight as it amounts to
a freeing from any form of dogmatism. The posi-
Logical thinking; Argumentation; Deductive rea- tion coincides with some postmodern critiques of
soning; Mathematical problem solving; Critique; purely intellectual perspectives that lack contact
Mathematical literacy; Critical judgment; Goals with experience and is echoed in some European
of mathematics education traditions of skepticism (Garfield 1990). Hence,
paradoxical deconstruction appears more radical
Characteristics than CT as it includes overcoming the methods
and frames of reference of previous thinking and
Educational psychologists frame critical thinking of purely intellectual plausibility.
(CT) as a set of generic thinking and reasoning
skills, including a disposition for using them, as The role assigned to CT in mathematics
well as a commitment to using the outcomes of CT education includes CT as a by-product of
as a basis for decision-making and problem solv- mathematics learning, as an explicit goal of
ing. In such descriptions, CT is established as a mathematics education, as a condition for math-
general standard for making judgments and deci- ematical problem solving, as well as critical
sions. Some descriptions of CT activities and engagement with issues of social, political, and
skills include a sense for fairness and the assess- environmental relevance by means of mathemati-
ment of practical consequences of decisions as cal modeling and statistics. Such engagement can
characteristics of CT (e.g., Paul and Elder 2001). include a critique of the very role mathematics
This assumes autonomous subjects who share plays in these contexts. In the mathematics educa-
a common frame of reference for representation tion literature, explicit reference to CT as defined
of facts and ideas, for their communication, as in educational psychology is not very widespread,
well as for appropriate (morally “good”) action. but general mathematical problem-solving skills
Important is also the difference as to what extent are commonly associated with critical thinking,
a critical examination of the criteria for CT is even though such association remains under-the-
included in the definition: If education for CT is orized. On the other hand, the notion of critique,
conceptualized as instilling a belief in a more or rather than CT, is employed in the mathematics
less fixed and shared system of skills and criteria education literature in various programs related to
for judgment, including associated values, then it critical mathematics education.
seems to contradict its very goal. If, on the other
hand, education for CT aims at overcoming poten- Critical Thinking and Mathematical
tially limiting frames of reference, then it needs to Reasoning
allow for transcending the very criteria assumed Mathematical argumentation features prominently
for legitimate “critical” judgment. The dimension as an example of disciplined reasoning based on
of not following rules and developing a fantasy for clear and concise language, questioning of assump-
alternatives connects CT with creativity and tions, and appreciation of logical inference for
deriving conclusions. These features of mathemat-
ical reasoning have been contrasted with intuition,
associative reasoning, justification by example, or
induction from observation. While the latter are

C 122 Critical Thinking in Mathematics Education

also important aspects of mathematical inquiry, there is a large overlap of literature on mathemat-
a focus on logic is directed towards extinguishing ical reasoning, problem solving, and CT.
subjective elements from judgments and it is the
essence of deductive reasoning. Underpinned by There is agreement that CT does not automat-
the values of rationalism and objectivity, reasoning ically emerge as a by-product of any mathematics
with an emphasis on logical inference is opposed to curriculum, but only with a pedagogy that draws
intuition and epiphany as a source of knowledge on students’ contributions and affords processes
and viewed as the counterinsurance against dog- of reasoning and questioning when students col-
matism and opportunism. lectively engage in intellectually challenging
tasks. Fawcett (1938), for example, suggested
The enhancement of students’ general reason- that teachers (in geometry instruction) should
ing capacity has for quite some time been seen as make use of students’ disposition for critical
a by-product of engagement with mathematics. thinking and that this capacity can be harnessed
Francis Bacon (1605), for example, wrote that it and cultivated by an appropriate choice of
would “remedy and cure many defects in the wit pedagogy. Reflective thinking practices could
and faculties intellectual. For if the wit be too be enacted when drawing the students’ attention
dull, they [the mathematics] sharpen it; if too to the need for clear definition of key terms in
wandering, they fix it; if too inherent in the statements, for examination of alleged evidence,
sense, they abstract it” (VIII (2)). Even though for exposition of assumptions behind their
this promotion of mathematics education is based beliefs, and for evaluation of arguments and
on its alleged value for developing generic conclusions. Fawcett’s teaching experiments
thinking or reasoning skills, these skills are not included the critical examination of everyday
called “critical thinking.” notions. A more recent example of a pedagogical
approach with a focus on argumentation is the
Historically, the notion of critique was tied to organization of a “scientific debate” in the math-
the tradition of rhetoric and critical evaluation of ematics classroom (Legrand 2001), where stu-
texts. Only through the expansion of the function dents in an open discussion defend their own
of critique towards general enlightenment, ideas about a conjecture, which may be prepared
critique became a generic figure of thinking, argu- by the teacher or emerge spontaneously during
ing, and reasoning. This more general notion tran- class work.
scends what is usually associated with accuracy
and rigor in mathematical reasoning. Accord- While cultivating some form of discipline-
ingly, CT in mathematics education not only is transcending CT has long been promoted by
conceptualized as evaluating rigor in definitions mathematics educators, explicit reference to CT
and logical consistency of arguments but also is not very common in official mathematics
includes attention to informal logic and heuristics, curriculum documents internationally. For
to the point of identifying problem-solving skills example, “critical thinking” is not mentioned in
with CT (e.g., O’Daffer and Thomquist 1993). the US Common Core Standards for Mathematics
Applebaum and Leikin (2007), for example, see (Common Core State Standards Initiative 2010).
the faculty of recognizing contradictory informa- However, in older recommendations from the US
tion and inconsistent data in mathematics tasks as National Council of Teachers of Mathematics,
a demonstration of CT. mention of “critical thinking” is made in relation
to creating a classroom atmosphere that fosters it
However, as most notions of CT include an (NCTM 1989). A comparative analysis of asso-
awareness of the subject doing it, neither a mere ciations made between mathematics education
application of logical inference nor successful and CT in international curriculum documents
application of mathematical problem-solving remains a research desideratum.
skills would reasonably be labeled as CT. But
as a consequence of often identifying CT Notions of CT in mathematics education with
with general mathematical reasoning processes a focus on argumentation and reasoning skills
embedded in mathematical problem solving, have in common that the critical competence

Critical Thinking in Mathematics Education C123

they promote is directed towards claims, state- any application of mathematics. This is not to C
ments, hypotheses, or theories (“texts”), but do dismiss rational inquiry, but aims at expanding
include neither a critique of the social realities, in rationality beyond instrumentality through inclu-
which these texts are produced, nor a critique of sion of moral and political thought. Such an expan-
the categories, in which these texts describe real- sion is seen as necessary by those who see purely
ities. As it is about learning how to think, but not formally defined CT as ultimately self-destructive
what to think about, this notion of CT can be and hence not emancipatory.
taken to implicate a form of thinking without
emotional or moral commitment. However, the Limitations of Developing CT Through
perspective includes the idea that the same prin- Mathematics Education
ciples that guide critical scientific inquiry could The take-up of poststructuralist and psychoana-
also guide successful problem solving in social lytic theories by mathematics educators has
and moral matters and this would lead to afforded contributions that hold CT up for
improvement of society, an idea that was, for scrutiny. Based on the postmodern acknowledg-
example, shared by Dewey (Stallman 2003). ment that all forms of reasoning are only
Education for CT is then by its nature legitimized through the power of some groups in
emancipatory. society, and in line with critics who see applied
mathematics as the essence of instrumental reason,
Critical Thinking and Applications of an enculturation of students into a form of
Mathematics CT embedded in mathematical reasoning must
For those who see scientific standards of reason- be seen as disempowering. As it excludes
ing as limited, the enculturation of students into imagination, fantasy, emotion, and the particular
a form of CT derived from these standards alone and metaphoric content of problems, this form of
cannot be emancipatory. Such a view is based on CT is seen as antithetical to political thinking or
a critique of Enlightenment’s scientific image of social commitment (Walkerdine 1988; Pimm
the world. The critique provided by the philoso- 1990; Walshaw 2003; Ernest 2010). Hence,
phers of the Frankfurt School is taken up in var- the point has been made that mathematics
ious projects of critical mathematics education education, if conceptualized as enculturation into
and critical mathematical literacy. This critique dispassionate reason and analysis, limits critique
is based on the argument that useful things are rather than affording it and it might lead to political
conflated with calculable things and thus formal apathy.
reasoning based on quantification, which is made
possible through the use of mathematics, is Further Unresolved Issues
purely instrumental reasoning. Mathematics edu- Engaging students in collaborative CT and
cators have pointed out that reliance on mathemat- reasoning in mathematics classrooms assumes
ical models implicates a particular worldview and some kind of an ideal democratic classroom envi-
mathematics education should widen its perspec- ronment, in which students are communicating
tive and take critically into account ethical and freely. However, classrooms can hardly be seen
social dimensions (e.g., Steiner 1988). In order to as ideal speech communities. Depending on their
cultivate CT in the mathematics classroom, reflec- backgrounds and educational biographies, students
tion not only of methodological standards of math- will not be equally able to express their thoughts
ematical models but also of the nature of these and not all will be guaranteed an audience. Further,
standards themselves, as well as of the larger social the teacher usually has the authority to phrase the
contexts within which mathematical models are questions for discussion and, as a representative of
used, has been suggested (e.g., Skovsmose 1989; the institution, has the obligation to assess students’
Keitel et al. 1993; Jablonka 1997; Appelbaum and contributions. Thus, even if a will to cultivate some
Davila 2009; Fish and Persaud 2012). Such a view form of critical reasoning in the mathematics class-
is based on acknowledging the interested nature of room might be shared amongst mathematics

C 124 Critical Thinking in Mathematics Education

educators, more attention to the social, cultural, References
and institutional conditions under which this is
supposed to take place needs to be provided by Appelbaum P, Davila E (2009) Math education and social
those who frame CT as an offshoot of mathemati- justice: gatekeepers, politics and teacher agency. In:
cal reasoning. Further, taxonomies of CT skills, Ernest P, Greer B, Sriraman B (eds) Critical issues in
phrased as metacognitive activities, run the risk of mathematics education. Information Age, Charlotte,
suggesting to treat these explicitly as learning pp 375–394
objectives, including the assessment of the extent
to which individual students use them. Such a Applebaum M, Leikin R (2007) Looking back at the
didactical reification of CT into measurable learn- beginning: critical thinking in solving unrealistic prob-
ing outcomes implicates a form of dogmatism and lems. Mont Math Enthus 4(2):258–265
contradicts the very notion of CT.
Bacon F (1605) Of the proficience and advancement of
The antithetical character of the views of what learning, divine and human. Second Book (transcribed
it means to be critical held by those who see CT as from the 1893 Cassell & Company edition by David
a mere habit of thought that can be cultivated Price. Available at: http://www.gutenberg.org/dirs/
through mathematical problem solving, on the etext04/adlr10h.htm
one hand, and mathematics educators inspired
by critical theory and critical pedagogy, on the Common Core State Standards Initiative (2010) Mathe-
other hand, needs further exploration. matics standards. http://www.corestandards.org/Math.
Accessed 20 July 2013
Attempts to describe universal elements of
critical reasoning, which are neither domain nor Ernest P (2010) The scope and limits of critical mathe-
context specific, reflect the idea of rationality itself, matics education. In: Alrø H, Ravn O, Valero P (eds)
the standards of which are viewed by many as best Critical mathematics education: past, present and
modeled by mathematical and scientific inquiry. future. Sense Publishers, Rotterdam, pp 65–87
The extent to which this conception of rationality
is culturally biased and implicitly devalues other Fawcett HP (1938) The nature of proof. Bureau of
“rationalities” has been discussed by mathematics Publications, Columbia, New York City. University
educators, but the implications for mathematics (Re-printed by the National Council of Teachers of
education remain under-theorized. Mathematics in 1995)

Cross-References Fenner P (1994) Spiritual inquiry in Buddhism. ReVision
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▶ Argumentation in Mathematics Education
▶ Authority and Mathematics Education Fish M, Persaud A (2012) (Re)presenting critical mathe-
▶ Critical Mathematics Education matical thinking through sociopolitical narratives as
▶ Dialogic Teaching and Learning in mathematics texts. In: Hickman H, Porfilio BJ (eds)
The new politics of the textbook. Sense Publishers,
Mathematics Education Rotterdam, pp 89–110
▶ Didactic Contract in Mathematics Education
▶ Logic in Mathematics Education Garfield JL (1990) Epoche and s´u¯nyata¯: skepticism east
▶ Mathematical Modelling and Applications in and west. Philos East West 40(3):285–307

Education Jablonka E (1997) What makes a model effective and
▶ Mathematical Proof, Argumentation, and useful (or not)? In: Blum W, Huntley I, Houston SK,
Neill N (eds) Teaching and learning mathematical
Reasoning modelling: innovation, investigation and applications.
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▶ Problem Solving in Mathematics Education Keitel C, Kotzmann E, Skovsmose O (1993) Beyond the
▶ Questioning in Mathematics Education tunnel vision: analyzing the relationship between
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Legrand M (2001) Scientific debate in mathematics
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National Council of Teachers of Mathematics (NCTM)
(1989) Curriculum and evaluation standards for school
mathematics. National Council of Teachers of
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O’Daffer PG, Thomquist B (1993) Critical thinking, math-
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Paul R, Elder L (2001) The miniature guide to critical perspectives to mathematics education developed C
thinking concepts and tools. Foundation for Critical since the late 1980s (Bishop 1988) and in cultural
Thinking Press, Dillon Beach approaches to mathematical cognition (Cole
1996). However, until recently issues of cultural
Pimm D (1990) Mathematical versus political awareness: diversity were considered to be out there in other
some political dangers inherent in the teaching of non-Western cultures or to be issues of marginal-
mathematics. In: Noss R, Brown A, Dowling P, ized and poor groups in society. Globalization
Drake P, Harris M, Hoyles C et al (eds) Political dimen- changed this perspective. With changes in
sions of mathematics education: action and critique. communication, technologies, and unprecedented
Institute of Education, University of London, London levels of migration, cultures have become increas-
ingly complex, connected, and heterogeneous.
Skovsmose O (1989) Models and reflective knowledge. One of the major impacts on education has been
Zentralblatt fu€r Didaktik der Mathematik 89(1):3–8 a substantial change in the cultural and ethnic
composition of the school population.
Stallman J (2003) John Dewey’s new humanism and liberal
education for the 21st century. Educ Cult 20(2):18–22 Schools and classrooms become places where
teachers, students, and parents are exposed to
Steiner H-G (1988) Theory of mathematics education and have to respond to many types of cultural
and implications for scholarship. In: Steiner H-G, differences. For many these differences are
Vermandel A (eds) Foundations and methodology of resources enriching the learning opportunities
the discipline mathematics education, didactics of and environments. For many others, diversity is
mathematics. In: Proceedings of the second tme experienced as a problem, which is reflected in
conference, Bielefeld-Antwerpen, pp 5–20 school achievement (Secada 1995). The issues
cultural diversity poses to education have many
Walkerdine V (1988) The mastery of reason: cognitive facets and have been approached from different
development and the production of rationality. perspectives in social sciences (De Haan and
Routledge, London Elbers 2008). Conceptions of culture and the
role of culture in psychological development
Walshaw M (2003) Democratic education under scrutiny: inform these perspectives. Examining culture
connections between mathematics education and as a way of life of specific cultural groups has
feminist political discourses. Philos Math Educ J 17. contributed to the understanding of cultural
http://people.exeter.ac.uk/PErnest/pome17/contents.htm discontinuities between schools and the home
background of the students. In this perspective,
Cultural Diversity in Mathematics the emphasis has been on the shared cultural
Education practices of the group. A more recent perspective
focuses on more dynamic aspects of culture, i.e.,
Guida de Abreu on the way a person experiences participation
Psychology Department, Oxford Brookes in multiple practices, and the production of
University, Oxford, UK new cultural knowledge, meaning, and identities.
Mathematics education research draws on these
Keywords perspectives but also considers issues that are
specific to mathematics learning (Cobb and
Cultural identity and learning; Vygotsky; Home Hodge 2002; Nasir and Cobb 2007; Abreu
and school mathematics; Immigrant students; 2008; Gorgorio´ and Abreu 2009).
Minority students; Cultural discontinuity;
Sociocultural approaches Here the focus is on the development of ideas
that examine mathematics as a form of cultural
Introduction knowledge (Bishop 1988; Asher 2008) and learn-
ing as a socioculturally mediated process
Cultural diversity in mathematics education is a (Vygotsky 1978). These ideas offer a critique to
widely used expression to discuss questions
around why students from different cultural, eth-
nic, social, economic, and linguistic groups per-
form differently in their school mathematics.
These questions are not new in cultural

C 126 Cultural Diversity in Mathematics Education

approaches that locate the sources of diversity investigations in Brazil by Nunes, Schliemann,
in the autonomous individual mind. More and Carraher (1993). In a series of studies that
importantly, sociocultural approaches have con- started with street children, Nunes and her col-
tributed to rethinking cultural diversity as “rela- leagues examined differences between school
tional” and “multilayered” phenomena, which mathematics and out-of-school mathematics.
can be studied from different angles (Cobb and Their findings added support to the notion that
Hodge 2002; De Haan and Elbers 2008). Empir- mathematical thinking was mediated by cultural
ical research following these approaches has tools, such as oral and written arithmetic. The
evolved from an examination of diversity within society studies also highlighted the situ-
between cultural groups, i.e., the nature of math- ated nature of mathematical cognition.
ematical knowledge specific to cultural prac- Depending on the context of the practice, the
tices, to an examination of the person as same person may draw on different cultural
a participant in specific sociocultural practices. tools; they can call on an oral method to solve
a shopping problem and a written method to solve
Diversity and Uses of Cultural a school problem.
Mathematical Tools
A driving force for researching the impact of How cultural tools mediate mathematical
cultural diversity in mathematics education has thinking and learning continues to be a key aspect
been to understand why certain cultural groups in investigations in culturally diverse classrooms.
experience difficulties in school mathematics. In Research with minority and immigrant students
the culture-free view of mathematics, poor perfor- in different countries shows that the students
mance in school mathematics was explained in learned often to use different forms of mathemat-
terms of deficits, namely, cognitive deficits that ics at home and at school (Bishop 2002; Gorgorio´
could be the result of cultural deficits. However, et al. 2002; Abreu 2008). Similarly, research with
since the 1980s, this view has become untenable. parents shows that they refer often to differences
Researchers exploring the difficulties non- in their methods and the ones their children are
Western children, such as the Kpelle children in being taught in school. To sum up, research
Liberia, experienced with Western-like mathe- shows that students from culturally diverse back-
matics introduced with schooling (Cole 1996) grounds are exposed often to different cultural
realized that their difficulties could not be tools in different contexts of mathematical
explained by cognitive deficits or cultural deficits. practices. It also suggests that many students
They discovered that differences in mathematical experience cultural discontinuities in their
thinking could be linked to the tools used as medi- transitions between contexts of mathematical
ators. Thus, for instance, the performance in a practices. A cultural discontinuity perspective
mathematical task, such as estimating length, offers only a partial account of the impact of
was linked to the use of a specific cultural measur- diversity, however. The fact that students from
ing system. With the advance of cultural research similar home cultural groups perform differently
and the view of mathematics and cognition as at school requires research to consider other
cultural phenomena, alternative explanations of aspects of diversity. A fruitful way of continuing
poor performance in school mathematics have to explore the different impacts of diversity in
been put forward in terms of cultural differences. school mathematical learning focuses on how
the person as a participant in mathematical prac-
Drawing on the insights from examining tices makes sense of their experiences. The per-
the mathematics of particular cultural groups son here can be, for example, an immigrant
research moved to explore cultural differences student in a mathematics classroom, a parent
within societies, which is still the major focus of that supports their children with their school
current research on cultural diversity in mathe- homework, and a teacher that is confronted with
matics education. A classic example of this students from cultural backgrounds they are not
research is the “street mathematics” familiar with. Here the focus turns to culture as

Cultural Diversity in Mathematics Education C127

being reconstructed in contexts of practices, and identity (Crafter and Abreu 2010). Identities, as C
issues of identity and social representations are socially constructed, can then be conceptualized
foregrounded. as powerful mediators in the way diversities are
being constructed in the context of school prac-
Diversity and Cultural and Mathematical tices. Indeed, studies examining other types of
Identities diversity, such as gender, have also implied sim-
Many studies with immigrant and minority ilar processes (Boaler 2007).
students have now illustrated that they become
aware of the differences between their home Studies with immigrant students with a history
culture and their school practices (Bishop 2002; of success in their school mathematical learning
see also ▶ Immigrant Students in Mathematics in their home country are also particularly inter-
Education). Accounts from parents of their esting to illustrate the intersection of identities.
experiences of supporting their children’s school Firstly, the difficulties of these students cannot be
mathematics at home (e.g., homework) also easily attributed to the individual mathematical
illustrate the salience of differences between home ability as they have a personal history of being
and school mathematics. These could be experi- “good mathematics students.” Secondly, in this
enced in terms of (a) the content of school mathe- case the cultural diversity is already internalized
matics and in the strategies used for calculations, (b) as part of the student’s previous schooling. These
the methods of teaching and the tools used in teach- students’ positive school mathematical identities
ing (e.g., methods for learning times tables, use of get disrupted when they receive low grades in the
calculators), (c) the language in which they learned host country school mathematics. Suddenly, the
and felt confident doing mathematics, and (d) the students’ common representation that mathemat-
parents’ and the children’s school mathematical ics is just about numbers and formulae and that
identities. Though all the dimensions are important, these are the same everywhere is challenged. It is
this research shows that identities take a priority revealing that young people from different
in the way the parents organize their practices immigrant backgrounds and going to school in
to support their children. The societal and different countries report similar experiences
institutional valorization of mathematical practices (e.g., Portuguese students in England; Ecuador-
plays a role on this process (Abreu 2008). ian students in Catalonia, Spain). This can be
interpreted as evidence that when a student joins
Recent studies also show that students talk a mathematical classroom in a new cultural
about differences in relation to how they perceive context, their participation is mediated by repre-
their home cultural identities as intersecting with sentations of what counts as mathematical knowl-
their school mathematical learning. Studies with edge. These examples illustrate a culture-free
students from minority ethnic backgrounds in view of mathematics that is still predominant in
England whose parents had been schooled in many educational systems but that could be det-
other countries show that differences between rimental to immigrant students’ academic math-
school mathematical practices at home and at ematical careers. Having shown that issues of
school have implications on their mathematical diversity are very salient in the experiences of
identities. For example, some students report try- students and their parents, the next section briefly
ing to separate home and school, i.e., to use the examines teachers’ representations.
“home way” at home and the “school way” at
school. The reason provided for the separation is Diversity and Teachers’ Social
that they do not feel that the home ways are Representations of Cultural Differences
valued at school. Other students simply claim In many schools, teachers, who have trained to
that their parents do not know or that their knowl- teach monolingual and monocultural students
edge is old fashioned. In both cases, the construc- from their own culture, teach students who may
tion of a positive school mathematical identity speak a different language and come from cul-
involves suppressing the home mathematical tures they are not familiar with. However, in