Encyclopedia of Mathematics Education

P 484 Political Perspectives in Mathematics Education

and practices were not shown to be as effective. Ernest P (1991) The philosophy of mathematics
Likewise, discovery beliefs and practices were education. The Falmer Press, London
equally ineffective, refuting the progressivist
claim that the teaching and learning of mathemat- Klein D (2007) A quarter century of US “math wars” and
ics by discovery is the most effective approach. political partisanship. BSHM Bull J Br Soc Hist Math
Of course Askew et al. (1997) only report a small- 22(1):22–33
scale, in-depth study of about 20 teachers and
must be viewed with caution and needs replica- London Mathematical Society (1995) Tackling the
tion. Nevertheless its results illustrate the futility mathematics problem. LMS, London
of policy debates becoming overly ideological
and losing contact with empirical measures of NCTM (1980) An agenda for action. National Council of
effectiveness from properly conducted research. Teachers of Mathematics, Reston

NCTM (1989) Curriculum and evaluation standards for
school mathematics. National Council of Teachers of
Mathematics, Reston

Shulman LS, Keislar ER (eds) (1966) Learning by
discovery: a critical appraisal. Rand McNally, Chicago

Cross-References Political Perspectives in
Mathematics Education
▶ Authority and Mathematics Education
▶ Critical Mathematics Education Paola Valero
▶ Dialogic Teaching and Learning in Department of Learning and Philosophy, Aalborg
University, Aalborg, Denmark
Mathematics Education
▶ History of Mathematics Teaching and Keywords

Learning Power; Politics; Modernity; Neutrality of mathe-
▶ Inquiry-Based Mathematics Education matics; Social rationalities; In(ex)clusion;
▶ Mathematical Literacy Mathematics for all; Credit system; Subjectivity
▶ Political Perspectives in Mathematics
Definition
Education
▶ Realistic Mathematics Education
▶ Recontextualization in Mathematics Education
▶ Teacher-Centered Teaching in Mathematics

Education

References A political perspective in mathematics education
is a way of looking at how mathematics, educa-
Askew M, Brown M, Rhodes V, Johnson D, Wiliam tion, and society relate to power. It stands on the
D (1997) Effective teachers of numeracy, final report. critical recognition that mathematics is not only
King’s College, University of London, London important in society due to its exceptional, intrin-
sic characteristics as the purest and most power-
Ausubel DP (1968) Educational psychology, a cognitive ful form of abstract thinking but also and
view. Holt, Rinehart and Winston, New York foremost, because of its functionality in the con-
stitution of the dominant cultural project of
Biggs EE (ed) (1965) Mathematics in primary schools Modernity. Thus, it assumes that the teaching
(Curriculum Bulletin 1). Her Majesty’s Stationery and learning of mathematics are not neutral prac-
Office, London tices but that they insert people – be it children,
youth, teachers, and adults – in socially valued
Brown M (1996) The context of the research – the evolu- mathematical rationalities and forms of knowing.
tion of the National Curriculum for mathematics. In: Such insertion is part of larger processes of selec-
Johnson DC, Millett A (eds) Implementing the tion of people that schooling operates in society.
mathematics national curriculum: policy, politics and It results in differential positioning of inclusion or
practice. Paul Chapman, London, pp 1–28

Cockcroft WH (chair) (1982) Mathematics counts. Her
Majesty’s Stationery Office, London

Ernest P (ed) (1989) Mathematics teaching: the state of the
art. The Falmer Press, London

Political Perspectives in Mathematics Education P485

exclusion of learners in relation to access to “awareness” does not constitute the center of a P
socially privileged resources such as further edu- political approach since there is a distinction
cation, labor market, and cultural goods. between being sympathetic to how mathematics
education relates to political processes of different
History type and making power in mathematics education
the focus of one’s research. In other words, not all
The political perspectives of mathematics educa- people who express a political sympathy actually
tion became a concern for teachers and study the political in mathematics education
researchers in the 1980s. While the change from (Gutierrez 2013; Valero 2004).
the nineteenth to the twentieth centuries was
a time of inclusion of mathematics in growing, With this central distinction in mind, it is pos-
massive, national education systems around the sible to differentiate a variety of political perspec-
world, the change from the twentieth to the tives, some that could be called weak in the sense
twenty-first centuries has been a time for focusing that they make a connection between mathemat-
on the justifications for the privileged role of ics education and power but do not concentrate on
mathematics in educational systems at all levels. the study of it as a constituent of mathematics
The apparent failure of the New Math movement education but rather as a result or a simply asso-
in different industrialized countries allowed to ciated factor. Strong political approaches in math-
raise concerns about the need for mathematics ematics education are a variety of perspectives
teaching and learning that could reach as many that do have a central interest in understanding
students as possible and not only a selected few mathematics education as political practices.
(Damerow et al. 1984). Questions of how math-
ematics education could be studied from perspec- Weak Political Perspectives
tives that allowed moving beyond the boundaries
of the mathematical contents in the school cur- A general characteristic of weak political
riculum started to be raised. In mathematics edu- perspectives in mathematics education is the
cation, the first book published in English as part adherence to some of the positive features
of an international collection, containing the attributed to mathematics and mathematics edu-
word politics in the title, was “The politics of cation, particularly those that have to do with
mathematics education” by Stieg Mellin-Olsen people’s empowerment and social and economic
(1987). However, “The mastery of reason: progress. More often than not, these views
Cognitive development and the production of assume some kind of intrinsic goodness of
rationality” by Valerie Walkerdine (1988) is mathematics and mathematics education that is
a seminal work in critical psychology discussing transferred to teachers and learners alike through
how school mathematics education subjectifies good and appropriate education practices. In
children through inscribing in them and in the decade of the 1980s and fully in the 1990s,
society, in general, specific notions of the rational the broadening of views on what constitutes
child and of abstract thinking. mathematics education allowed for formulations
of the aims of school mathematics in relation to
The political concern and involvement of many the response to social challenges of changing
mathematics educators in their teaching and societies and, in particular, in response to the
research practice was also an initial entry that consolidation of democracy. It was possibility to
allowed sensitivity and awareness for searching enunciate the idea that, as part of a global policy
how mathematics education could be “political” of “Education for all” by UNESCO, mathematics
(Lerman 2000). Such political awareness on issues education had to contribute to the competence of
such as how mathematics has played a role as citizens, but also to open access for all students.
gatekeeper to entry in further education, for exam- In many countries, both at national policy level
ple, has been important. However, a political and at the level of researchers and teachers, there

P 486 Political Perspectives in Mathematics Education

was a growing concern for mathematics for all in relation to how mathematics is a formatting
and mathematics for equity and inclusion. The power in society through its immersion in the
study of how different groups – women, linguists, creation of scientific and technological structures
ethnic or religious minorities, and particular that operate in society (Christensen et al. 2008). It
racial groups – of students systematically under- also studies the processes of exclusion and differ-
achieve and how to remediate that situation grew entiation of students when mathematics education
extensively. Part of the weak political approaches practices reproduce the position of class and dis-
also includes studies of how mathematics educa- advantage of students (Frankenstein 1995); and
tion practices are shaped by educational policies. when such reproduction is part of the way (school),
South Africa, given the transition from apartheid mathematics is given meaning in public discourses
to democracy at the beginning of the 1990s, has and popular culture (Appelbaum 1995). It also
been a particularly interesting national case offers possibilities for rethinking practices when
where deep changes of policy had been studied democracy is thought as a central element of math-
to see how and why mathematics education in ematics education (Skovsmose and Valero 2008).
primary and secondary school is transforming to
contribute – or not – to the construction of a new The study of the political in the alignment of
society. Many of these studies have a weak polit- relation to the alignment of mathematics educa-
ical approach in the sense that they are justified tion practices with Capitalism is also a recent and
and operate on some political assumptions on strong political reading of mathematics education
mathematics education and its role in society, that offers a critical perspective on the material,
but intend to study appropriate pedagogies and economic significance of having success in
not how pedagogies in themselves effect the mathematics education. Both educational prac-
exclusion that the programs intend to remediate. tices (Baldino and Cabral 2006) and research
practices (Lundin 2012; Pais 2012) lock students
Strong Political Perspectives in a credit system where success in mathematics
represents value.
Strong political perspectives in mathematics
education problematize the assumed neutrality In the USA, and as a reaction to endemic
of mathematical knowledge and provide new operation of race as a strong element in the classi-
interpretations of mathematics education as prac- fication of people’s access to cultural and economic
tices of power. Ethnomathematics can be read as resources, the recontextualization of critical race
a political perspective in mathematics education theories into mathematics education has provided
in its challenge to the supremacy of Eurocentric new understandings of mathematics education as a
understandings of mathematics and mathematical particular instance of a White-dominant cultural
practices. The strong political perspectives of space that operates exclusion from educational suc-
ethnomathematics is presented in studies that cess for African American learners (Martin 2011),
not only argue for how the mathematical prac- as well as for Latino(a)s (Gutie´rrez 2012).
tices of different cultural groups – not only indig-
enous or ethnic groups but also professional The recontextualization of poststructural
groups – are of epistemological importance and theories in mathematics education has also led
value but also how some of those cultural prac- to the study of power in relation to the historical
tices are inserted in the calculations of power so construction of Modern subjectivities. The
that they can construct a regime of truth around effects of power in the bodies and minds of
themselves and thus gain a privileged positioning students and teachers (Walshaw 2010), as well
in front of other practices (Knijnik 2012). as in the public and media discourses on mathe-
matics (Moreau et al. 2010), are studied in an
Critical mathematics education as a wide and attempt to provide insights into how the mathe-
varied political approach takes the study of power matical rationality that is at the core of different
technologies in society shapes the meeting
between individuals and their culture. Even
though most research concentrates on the issue

Poststructuralist and Psychoanalytic Approaches in Mathematics Education P487

of identity construction and subjectivity, some Lundin S (2012) Hating school, loving mathematics: on P
studies attempting cultural histories of mathemat- the ideological function of critique and reform in
ics as part of Modern, massive educational mathematics education. Educ Stud Math 80:73–85
systems are also broadening this type of political
perspective (Popkewitz 2004; Valero et al. 2012). Martin DB (2011) What does quality mean in the context
of white institutional space? In: Atweh B, Graven M,
Cross-References Secada W, Valero P (eds) Mapping equity and quality
in mathematics education. Springer, New York,
▶ Equity and Access in Mathematics Education pp 437–450
▶ Mathematization as Social Process
▶ Policy Debates in Mathematics Education Mellin-Olsen S (1987) The politics of mathematics
▶ Poststructuralist and Psychoanalytic education. Kluwer, Dordrecht

Approaches in Mathematics Education Moreau MP, Mendick H, Epstein D (2010) Constructions
▶ Socioeconomic Class in Mathematics of mathematicians in popular culture and learners’
narratives: a study of mathematical and non-
Education mathematical subjectivities. Camb J Educ 40:25–38
▶ Sociological Approaches in Mathematics
Pais A (2012) A critical approach to equity. In: Skosvmose
Education O, Greer B (eds) Opening the cage. Critique and politics
of mathematics education. Sense, Rotterdam, pp 49–92
References
Popkewitz TS (2004) The alchemy of the mathematics
Appelbaum PM (1995) Popular culture, educational curriculum: inscriptions and the fabrication of the
discourse, and mathematics. State University of child. Am Educ Res J 41:3–34
New York, New York
Skovsmose O, Valero P (2008) Democratic access to
Baldino R, Cabral T (2006) Inclusion and diversity from powerful mathematical ideas. In: English LD (ed)
Hegelylacan point of view: do we desire our desire for Handbook of international research in mathematics
change? Int J Sci Math Educ 4:19–43 education. Directions for the 21st century, 2nd edn.
Erlbaum, Mahwah, pp 415–438
Christensen OR, Skosvmose O, Yasukawa K (2008) The
mathematical state of the world – explorations into the Valero P (2004) Socio-political perspectives on mathe-
characteristics of mathematical descriptions. ALEX- matics education. In: Valero P, Zevenbergen R (eds)
ANDRIA. Revista de Educac¸a˜o em Cieˆncia Researching the socio-political dimensions of mathe-
e Tecnologia 1:77–90 matics education: issues of power in theory and
methodology. Kluwer, Boston, pp 5–24
Damerow P, Dunkley M, Nebres B, Werry B (eds)
(1984) Mathematics for all. UNESCO, Paris Valero P, Garc´ıa G, Camelo F, Mancera G, Romero J
(2012) Mathematics education and the dignity of
Frankenstein M (1995) Equity in mathematics being. Pythagoras 33(2). http://dx.doi.org/10.4102/
education: class in the world outside the class. In: pythagoras.v33i2.171
Fennema E, Adajian L (eds) New directions for equity
in mathematics education. Cambridge University, Walkerdine V (1988) The mastery of reason: cognitive
Cambridge, pp 165–190 development and the production of rationality.
Routledge, London
Gutie´rrez R (2012) Context matters: how should we
conceptualize equity in mathematics education? In: Walshaw M (ed) (2010) Unpacking pedagogies. New per-
Herbel-Eisenmann B, Choppin J, Wagner D, Pimm spectives for mathematics. Information Age, Charlotte
D (eds) Equity in discourse for mathematics education.
Springer, Dordrecht, pp 17–33 Poststructuralist and Psychoanalytic
Approaches in Mathematics
Gutierrez R (2013) The sociopolitical turn in mathematics Education
education. J Res Math Educ 44(1):37–68
Margaret Walshaw
Knijnik G (2012) Differentially positioned language School of Curriculum & Pedagogy, College of
games: ethnomathematics from a philosophical Education, Massey University, Palmerston
perspective. Educ Stud Math 80:87–100 North, New Zealand

Lerman S (2000) The social turn in mathematics education Keywords
research. In: Boaler J (ed) Multiple perspectives on
mathematics teaching and learning. Ablex, Westport, Poststructuralism; Psychoanalysis
pp 19–44

P 488 Poststructuralist and Psychoanalytic Approaches in Mathematics Education

Definition the kind of being the human is. Typically, the
related ontologies are dualist in nature. They
Approaches that draw on developments within include such dichotomies as rational/irrational,
wider scholarly work that conceives of modernist objective/subjective, mind/body, cognition/
thought as limiting. affect, and universal/particular. Taken together,
these characteristically modernist beliefs about
Characteristics ontology and epistemology have informed
theories of human interaction, teaching,
Poststructuralist and psychoanalytic approaches learning, and development within mathematics
capture the shifts in scholarly thought that gained education.
currency in Western cultures during the past 50
years. Conveying a critical and self-reflective Developments within psychology and sociol-
attitude, both raise questions about the appropri- ogy that began to question these understandings
ateness of modernist thinking for understanding paved the way for a different perspective.
the contemporary social and cultural world. Since Sociology has helped seed poststructuralist
the publication of Lyotard’s The Postmodern work that aims to draw attention to the ways in
Condition (translated into English in 1984), which power works within mathematics educa-
poststructuralist and psychoanalytic thinking tion, at any level, and within any relationship, to
have provided an expression within the social constitute identities and to shape proficiencies.
sciences and humanities and, more recently, Psychology has informed a psychoanalytical
within mathematics education, for a loss of faith turn, designed to unsettle fundamental assump-
in the “grand narratives” of Western history and, tions concerning identity formations. Postmod-
in particular, enlightened modernity. A diverse ernists and psychoanalysts share some
set of initiatives in social and philosophical fundamental assumptions about the nature of the
thought, originating from the work of reality being studied, assumptions about what
Michel Foucault (e.g., 1970), Jacques Derrida constitutes knowledge of that reality, and
(e.g., 1976), Julia Kristeva (e.g., 1986), and assumptions about what are appropriate ways of
Jacques Lacan (1977), among others, helped building knowledge of that reality.
crystallize poststructuralist and psychoanalytic
ideas among researchers and scholars within Researchers in mathematics education who
mathematics education about how things might draw on this body of work have an underlying
be thought and done differently. interest in understanding, explaining, and
analyzing the practices and processes within
Poststructuralist and psychoanalytic mathematics education. Their analyses chart
approaches provide alternatives to the traditions teaching and learning, and the way in which
of psychological and sociological thought that identities and proficiencies evolve; tracking
have grounded understandings about knowledge, reflections; investigating everyday classroom
representation, and subjectivity within mathe- planning, activities, and tools; analyzing discus-
matics education. These traditions understand sions with principals, mathematics teachers,
reality as characterized by an objective structure, students, and educators; mapping out the effects
accessed through reason. More specifically, the of policy, and so forth. In the process of
traditions are based on the understanding that deconstructing taken-for-granted understand-
reason can provide an authoritative, objective, ings, they reveal how identities are constructed
true, and universal foundation of knowledge. within discourses, they demonstrate how
They also assume the transparency of language. everyday decisions are shaped by dispositions
Epistemological assumptions like these, about formed through prior events, and they provide
the relationship between the knower and the insights about the way in which language
known, are accompanied by beliefs about produces meanings and how it positions people
in relations of power. The assumptions upon
which these analyses are based enable an

Poststructuralist and Psychoanalytic Approaches in Mathematics Education P489

exploration of the lived contradictions of mathe- in the classroom and within other institutions of P
matics processes and structures. mathematics education. They do that by system-
atically constituting specific versions of the social
These analyses are developed around and natural worlds for them, all the while
a number of key organizing principles: language obscuring other possibilities from their vision.
is fragile and problematic and constitutes rather Discursivity is not simply a way of organizing
than reflects an already given reality. Meaning is what people say and do; it is also a way of
not absolute in relation to a referent, as had been organizing actual people and their systems. It
proposed by structuralism. The notion of knowing follows that “truths” about mathematics educa-
as an outcome of human consciousness and inter- tion emerge through the operation of discursive
pretation, as described by phenomenology, is also systems.
rejected. Moreover, knowing is not an outcome of
different interpretations, as claimed by hermeneu- Discursive approaches within mathematics
tics. Instead, for poststructuralist and psychoana- education draw attention to the impact of regula-
lytic scholars, reality is in a constant process of tory practices and discursive technologies on the
construction. What is warranted at one moment of constructions of teachers, students, and others. It
time may be unwarranted at another time. The reveals the contradictory realities of teachers,
claim is that because the construction process is students, policy makers, and so forth and the
ongoing, no one has access to an independent complexity and complicity of their work. Such
reality. There is no “view from nowhere,” no work emphasizes that teachers and students are
conceptual space not already implicated in that the production of the practices through which
which it seeks to interpret. There is no stable they become subjected (e.g., Hardy 2009;
unchanging world and no realm of objective Lerman 2009).
truths to which anyone has access. The notion of
a disembodied autonomous subject with agency Power in these approaches envelopes every-
to choose what kind of individual he or she might one. What the analyses reveal is that, in addition
become also comes under scrutiny. The counter- to operating at the macro-level of the school,
notion proposed is a “decentered” self – a self that power seeps through lower levels of practice
is an effect of discourse which is open to redefi- such as within teacher/student relations and
nition and which is constantly in process. school/teacher relations (see Walshaw 2010).
Even in a classroom environment that provides
Poststructuralist Approaches equitable and inclusive pedagogical arrange-
Foucault’s work is considered by many to repre- ments, poststructural approaches have shown
sent a paradigmatic example of poststructuralist that power is ever present through the classroom
thought. His work raises critical concerns about social structure, systematically creating ways of
how certain practices, and not others, become being and thinking in relation to class, gender,
intelligible and accepted, and how identities are and ethnicity and a range of other social catego-
constructed. Foucauldian analyses centered ries (see Walshaw 2001; Mendick 2006;
within mathematics educational sites explore Knijnik 2012).
lived experience, not in the sense of capturing
reality and proclaiming causes but of understand- In illuminating the impact of regulatory
ing the complex and changing processes by practices and technologies on identity and knowl-
which subjectivities and knowledge production edge production, fine-grained readings of class-
are shaped. In that sense, the focus shifts from room interaction have revealed the regulatory
examining the nature of identity and knowledge power of teachers’ discourse in providing stu-
to a focus on how identity and knowledge are dents with differential access to mathematics
discursively produced. In these analyses, “dis- (de Freitas 2010). Such readings shed light on
course” is a key concept. Discourses sketch out, how the discursive practices of teachers
for teachers, students, and others, ways of being contribute to the kind of mathematical thinking
and the kind of mathematical identities that are
possible within the classroom.

P 490 Poststructuralist and Psychoanalytic Approaches in Mathematics Education

Psychoanalytic Approaches a way that is not entirely rational nor observable.
Psychoanalytic analyses in mathematics Researchers in mathematics education who draw
education explore the question of identity. on psychoanalytic theory maintain that determi-
Lacan’s (e.g., 1977) and Zˇ izˇek’s (e.g., 1998) nations exist outside of our consciousness and, in
explanations of how identities are constructed the pedagogical relation, for example, influence
through an understanding of how others see that the way teachers develop a sense of self as
person have been influential in revealing that teacher and influence their interactions in the
teachers, students, and others are not masters of classroom. The identities teachers have of them-
their own thoughts, speech, or actions. Zˇ izˇek’s selves are, in a very real sense, “comprised,”
psychoanalytic position is that the self is not made in and through the activities, desires,
a center of coherent experience: “there are no interests, and investments of others. Understand-
identities as such. There are just identifications ings like these invite unknowingness, fluidity,
with particular ways of making sense of the world and becoming, which, in turn, have the effect of
that shape that person’s sense of his self and his producing different knowledge.
actions” (Brown and McNamara 2011, p. 26).
A person’s identifications are not reducible to Emancipatory Possibilities
the identities that the person constructs of Although both poststructuralist and psychoana-
himself. Rather, the self is performed within the lytic theorists question the modernist concept of
ambivalent yet simultaneous relationship of enlightenment, in reconceptualizing emancipa-
subjection/agency. tion away from individualist sensibilities, they
highlight possibilities for where and in what
Psychoanalytic observations of identity for- ways mathematics educational practices might
mation are likely to reveal how identities develop be changed (see Radford 2012). In addition to
through discourses and networks of power that uncovering terrains of struggle, poststructuralist
shift continually in a very unstable fashion, and psychoanalytic analyses foster democratic
changing as alliances are formed and reformed. provision, enabling a vision of critical-ethical
When identities are formed in a very mobile teaching where different material and political
space, what emerge are fragmented selves, layers conditions might prevail. What is clarified in
of self-understandings, and multiple positionings these approaches is that discourses are not
within given contexts and time (see Hanley entirely closed systems but are vehicles for
2010). This psychoanalytic idea is fundamental reflecting on where mathematics education is
to understanding that teachers and students today, how it has come to be this way, and the
(among others) negotiate their way through lay- consequences of conventional thought and
ered meanings and contesting perceptions of actions. Importantly, such analyses are
what a “good” teacher or student looks like. To a political resource for transforming the
complete a negotiation, there is a level at which processes and structures that currently deny
the teacher or student invests, or otherwise, teachers, students, policy makers, and others the
in a discursive position made available achievement of their ethical goals within
(see Bibby 2009). mathematics education.

A teacher’s, for example, investments within Cross-References
one discourse rather than another is explained
through the notion of affect and, more especially, ▶ Psychological Approaches in Mathematics
through the notions of obligation and reciprocity. Education
Affect, in the psychoanalytic analysis, is not
a derivative aspect but a constitutive quality of ▶ Sociological Approaches in Mathematics
classroom life (see Walshaw and Brown 2012). It Education
is not an interior experience, but rather, it
operates through processes that are historical in

Probability Teaching and Learning P491

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Different Meanings of Probability
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object or operation (rectangles, division, etc.) is
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Age, Charlotte, pp 129–151 per se, different philosophical theories and the
emerging conceptions of probability still persist,
Derrida J (1976) Of grammatology. The John Hopkins among which the most relevant for teaching are
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axiomatic or formal conceptions (Batanero et al.
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events, an assumption which is reasonable in
Knijnik G (2012) Differentially positioned language such games as throwing dice. In the classical
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Laplace in 1814 in his Philosophical essay on
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Columbia University, New York the number of favorable cases to a particular
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P 492 Probability Teaching and Learning

In his endeavor to extend the scope of measure theory, Kolmogorov, who corroborated
probability to insurance and life-table problems, the frequentist view, derived in 1933 an
Jacob Bernoulli justified to assign probabilities to axiomatic. This axiomatic was accepted by the
events through a frequentist estimate by elaborat- different probability schools because with some
ing the Law of Large Numbers. In the frequentist compromise the mathematics of probability
approach sustained later by von Mises or Renyi, (no matter the classical, frequentist or subjectivist
probability is defined as the hypothetical number view) may be encoded by Kolmogorov’s theory;
towards which the relative frequency tends. the interpretation would differ according to the
Such a convergence had been observed in many school one adheres to. However, the discussion
natural phenomena so that the frequentist about the meanings of probability and the long
approach extended the range of applications history of paradoxes is still alive in intuitions of
enormously. A practical drawback of this people who often conflict with the mathematical
conception is that we never get the exact value rules of probability (Borovcnik et al. 1991).
of probability; its estimation varies from one
repetition of the experiments (called sample) to Probability in the School Curriculum
another. Moreover, this approach is not appropri-
ate if it is not possible to repeat the experiment Students are surrounded by uncertainty in
under exactly the same conditions. economic, meteorological, biological, and
political settings and in their social activities
While in the classical and in the frequentist such as games or sports. The ubiquity of random-
approaches probability is an “objective” value we ness implies the student’s need to understand
assign to each event, the Bayes’s theorem, random phenomena in order to make adequate
published in 1763, proved that the probability decisions when confronted with uncertainty; this
for a hypothetical event or cause could be revised need has been recognized by educational
in light of new available data. Following this authorities by including probability in the
interpretation, some mathematicians like Keynes, curricula from primary education to high school
Ramsey, or de Finetti considered probability as and at university level.
a personal degree of belief that depends on
a person’s knowledge or experience. Bayes’ the- The philosophical controversy about the
orem shows that an initial (prior) distribution meaning of probability has also influenced teach-
about an unknown probability changes by rela- ing (Henry 1997). Before 1970, the classical view
tive frequencies into a posterior distribution. of probability based on combinatorial calculus
Consequently, from data one can derive an inter- dominated the school curriculum, an approach
val so that the unknown probability lies within its that was difficult, since students have problems
boundaries with a predefined (high) probability. to find the adequate combinatorial operations to
This is another proof that relative frequencies solve probability problems. In the “modern
converge and justifies using data to estimate mathematics” era, probability was used to
unknown probabilities. However, the status of illustrate the axiomatic method; however this
the prior distribution in this approach was approach was more suitable to justify theories
criticized as subjective, even if the impact of the than to solve problems. Both approaches hide
prior diminishes by objective data, and de the multitude of applications since the
Finetti proposed a system of axioms to justify equiprobability assumption is restricted to
this view in 1937. games of chance. Consistently, many school
teachers considered probability as a subsidiary
Despite the fierce discussion on the founda- part of mathematics, and either they taught it in
tions, progress of probability in all sciences and this style or they left it out of class. Moreover,
sectors of life was enormous. Throughout the students hardly were able to apply probability in
twentieth century, different mathematicians out-of-school contexts.
tried to formalize the mathematical theory of
probability. Following Borel’s work on set and

Probability Teaching and Learning P493

With increasing importance of statistics at the current curricula in spite of its increasing use P
school and progress of technology with easy in applications and that it may help to overcome
access to simulation, today there is a growing many paradoxes, especially those linked to
interest in an experimental introduction of conditional probabilities (Borovcnik 2011).
probability as a limit of stabilized frequencies
(frequentist approach). We also observe a shift When organizing the teaching of probability,
in the way probability is taught from a formula- there is moreover a need to decide what content to
based approach to a modern experiential include at different educational levels. Heitele
introduction where the emphasis is on probabilis- (1975) suggested a list of fundamental probabi-
tic experience. Students (even young children) listic concepts, which can be studied at various
are encouraged to perform random experiments degrees of formalization, each of which increases
or simulations, formulate questions or predictions in cognitive and linguistic complexity as one
about the tendency of outcomes in a series proceeds through school to university. These
of these experiments, collect and analyze data concepts played a key role in the history and
to test their conjectures, and justify their form the base for the modern theory of probabil-
conclusions on the basis of these data. This ity while at the same time people frequently hold
approach tries to show the students that incorrect intuitions about their meaning or their
probability is inseparable from statistics, and application in absence of instruction. The list of
vice versa, as it is recognized in the curriculum. fundamental concepts include the ideas of
random experiment and sample space, addition
Simulation and experiments can help and multiplication rules, independence and con-
students face their probability misconceptions ditional probability, random variable and
by extending their experience with randomness. distribution, combinations and permutations,
It is important, however, to clarify the distinction convergence, sampling, and simulation.
between ideally repeated situations and one-off
decisions, which are also frequent or perceived as All these ideas appear along the curriculum,
such by people. By exaggerating simulation and although, of course, with different levels of for-
a frequentist interpretation in teaching, students malization. In primary school, an intuitive idea of
may be confused about their differences or return probability and the ability to compute simple
to private conceptions in their decision making. probabilities by applying the Laplace rule or via
the estimation from relative frequencies using
Moreover, a pure experimental approach is a simple notation seems sufficient. By the end
not sufficient in teaching probability. Though of high school, students are expected to discrim-
simulation is vital to improve students’ probabi- inate random and deterministic experiments, use
listic intuitions and in materialize probabilistic combinatorial counting principles to describe the
problems, it does not provide the key about how sample space and compute the associate proba-
and why the problems are solved. This justifica- bilities in simple and compound experiments,
tion depends on the hypotheses and on the theo- understand conditional probability and indepen-
retical probability model on which the computer dence, compute and interpret the expected value
simulation is built, so that a genuine knowledge of discrete random variables, understand how to
of probability can only be achieved through the draw inferences about a population from random
study of some probability theory. However, the samples, and use simulations to acquire an
acquisition of such formal knowledge by students intuitive meaning of convergence.
should be gradual and supported by experience
with random experiments, given the complemen- It is believed today that in order to become
tary nature of the classic and frequentist a probability literate citizen, a student should
approaches to probability. It is also important to understand the use of probability in decision
amend these objective views with the subjectivist making (e.g., stock market or medical diagnosis)
perspective of probability which is closer to how or in sampling and voting. In scientific or profes-
people think, but is hardly taken into account in sional work, or at university, a more complex
meaning of probability including knowledge of

P 494 Probability Teaching and Learning

main probability distributions and even the simultaneously the global regularity and the
central limit theorem seems appropriate. particular variability of each randomly
generated distribution. However, later research
Intuitions and Misconceptions contradicted some of their results; Green’s
(1989) investigation with 2,930 children
For teaching, it is important to take into account indicates that the percentage of students recog-
informal ideas that people relate to chance and nizing random distributions decreases with age.
probability before instruction. These ideas appear
in children who acquire experience of random- Moreover, research in Psychology has shown
ness when playing chance games or by observing that adults tend to make erroneous judgments
natural phenomena such as the weather. They use in their decisions in out-of-school settings even if
qualitative notions (probable, unlikely, feasible, they are experienced in probability. The
etc.) to express their degrees of belief in the well-known studies by Kahneman and his collab-
occurrence of random events in these settings; orators (see Kahneman et al. 1982) identify that
however their ideas are too imprecise. Young people violate normative rules behind scientific
children may not see stable properties in random inference and use specific heuristics to simplify
generators such as dice or marbles in urns and the uncertain decision situation. According to
believe that such generators have a mind of their them, such heuristics reduce the complexity of
own or are controlled by outside forces. these probability tasks and are in general useful;
however, under specific circumstances, heuristics
Although older children may realize the need cause systematic errors and are resistant to change.
of assigning numbers (probabilities) to events to
compare their likelihood, probabilistic reasoning For example, in the representativeness
rarely develops spontaneously without instruc- heuristics, people estimate the likelihood of an
tion (Fischbein 1975), and intuitions are often event taking only into account how well it repre-
found to be wrong even in adults. For example, sents some aspects of the parent population
the mathematical result that a run of four consec- neglecting any other information available, no
utive heads in coin tossing has no influence on the matter how relevant it is for the particular
probability that the following toss will result in decision. People following this reasoning might
heads seems counterintuitive. This belief maybe believe that small samples should reflect the
due to the confusion between hypotheses and population distribution and consistently rely too
data: when we deal with coin tossing, we usually much on them. In case of discrepancies between
assume that the experiment is performed sample and population, they might even predict
independently. In spite of the run of four heads next outcomes to reestablish the alleged similar-
observed, the model still is used and, then, the ity. Other people do not understand the purpose
probability for the next outcome remains half of probabilistic methods, where it is not possible
for heads; however intuitively these data to predict an outcome with certainty but the
prompt people to abandon the assumption of behavior of the whole distribution, contrary to
independence and use the pattern of past data to what some people expect intuitively. A detailed
predict the next outcome. survey of students’ intuitions, strategies,
and learning at different ages may be found in
Piaget and Inhelder (1951) investigated the different chapters of Jones (2005) and
children’s understanding of chance and probabil- in Jones et al. (2007).
ity and described stages in the development of
probabilistic reasoning. They predicted a mature Another fact complicates the teaching of
comprehension of probability at the formal probability (Borovcnik and Peard 1996): whereas
operational stage (around 15 years of age), in other branches of mathematics counterintuitive
which comprises that adolescents understand the results are encountered only at higher levels
law of large numbers – the principle that explains of abstraction, in probability counterintuitive
results abound even with basic concepts such
as independence or conditional probability.

Probability Teaching and Learning P495

Furthermore, while in logical reasoning – the experiment such as flipping a coin, a child obtains P
usual method in mathematics – a proposition is different results each time the experiment is
true or false, a proposition about a random event performed, and the experiment cannot be
would only be true or false after the experiment reversed. Therefore, it is harder for children to
has been performed; beforehand we only can understand (and acknowledge) the structure
consider the probability of possible results. This behind the experiments, which may explain why
explains that some probability theorems (e.g., the they do not always develop correct probability
central limit theorem) are expressed in terms of conceptions without instruction.
probability.
Our previous discussion also suggests several
Challenges in Teaching Probability important questions to be considered in future
research: How should we take advantage of the
The preceding philosophical and psychological multifaceted nature of probability in organizing
debate suggests that teachers require a specific instruction? How to conduct children to gradually
preparation to assure their competence to teach view probability as an a priori degree of
probability. Unfortunately, even if prospective uncertainty, as the value to which relative fre-
teachers have a major in mathematics, they quencies tend in random experiments repeated
usually studied only probability theory and under the same conditions, and as a personal
consistently lack experience in designing degree of belief, where “subjectivist” does not
investigations or simulations (Stohl 2005). mean arbitrariness, but use of expert knowledge?
They may be unfamiliar with different mean- How to make older students realize that proba-
ings of probability or with frequent misconcep- bility should be viewed as a mathematical model,
tions in their students. Research in statistics and not a property of real objects? And finally,
education has shown that textbooks lack to how best to educate teachers to become compe-
provide sufficient support to teachers: they pre- tent in the teaching of probability?
sent an all too narrow view of concepts; appli-
cations are restricted to games of chance; even Cross-References
definitions are occasionally incorrect or
incomplete. ▶ Data Handling and Statistics Teaching
and Learning
Moreover, teachers need training in pedagogy
related to teaching probability as general References
principles valid for other areas of mathematics
are not appropriate (Batanero et al. 2004). For Batanero C, Godino JD, Roa R (2004) Training teachers to
example, in arithmetic or geometry elementary teach probability. J Stat Educ 12 [Online]. http://
operations can be reversed and reversibility can wwwamstatorg/publications/jse/
be represented by concrete materials: when
joining a group of three marbles with another Batanero C, Henry M, Parzysz B (2005) The nature of
group of four, a child always obtains the same chance and probability. In: Jones G (ed) Exploring
result (seven marbles); if separating the second probability in school: challenges for teaching and
set from the total, the child always returns to the learning. Springer, New York, pp 15–37
original set provided that the marbles are seen as
equivalent (and there is hardly a dispute on such Borovcnik M (2011) Strengthening the role of probability
an abstraction). These experiences are vital to within statistics curricula. In: Batanero C, Burrill G,
help children progressively abstract the structure Reading C (eds) Teaching statistics in school mathe-
behind the concrete situation, since they remain matics – challenges for teaching and teacher educa-
closely linked to concrete situations in their tion. A joint ICMI/IASE study. Springer, New York,
mathematical thinking. However, with a random pp 71–83

Borovcnik M, Peard R (1996) Probability. In: Bishop A,
Clements K, Keitel C, Kilpatrick J, Laborde C (eds)
International handbook of mathematics education.
Kluwer, Dordrecht, pp 239–288

P 496 Problem Solving in Mathematics Education

Borovcnik M, Bentz HJ, Kapadia R (1991) A probabilistic Introduction
perspective. In: Kapadia R, Borovcnik M (eds) Chance
encounters: probability in education. Kluwer, The core and essence of a problem solving
Dordrecht, pp 27–73 approach to learn mathematics is summarized in
the following quotation: Problem solving is a
Fischbein E (1975) The intuitive source of probability lifetime activity. Experiences in problem solving
thinking in children. Reidel, Dordrecht are always at hand. All other activities are subor-
dinate. Thus, the teaching of problem solving
Green DR (1989) Schools students’ understanding of should be continuous. Discussion of problems,
randomness. In: Morris R (ed) Studies in mathematics proposed solutions, methods of attacking
education, vol 7, The teaching of statistics. UNESCO, problems, etc. should be considered at all times
Paris, pp 27–39 (Krulik and Rudnick 1993, p. 9).

Hacking I (1975) The emergence of probability Characteristics
Cambridge. Cambridge University Press, Cambridge,
MA What Does Mathematical Problem Solving
Involve?
Heitele D (1975) An epistemological view on fundamen- Mathematical problem solving is a research
tal stochastic ideas. Educ Stud Math 6:187–205 and practice domain in mathematics education
that fosters an inquisitive approach to develop
Henry M (1997) L’enseignement des statistiques et des and comprehend mathematical knowledge
probabilite´s [Teaching of statistics and probability]. Santos-Trigo 2007. As a research domain, the
In: Legrand P (ed) Profession enseignant: Les maths problem-solving agenda includes analyzing
en colle`ge et en lyce´e. Hachette-E´ ducation, Paris, cognitive, social, and affective components
pp 254–273 that influence and shape the learners’ develop-
ment of problem-solving proficiency. As an
Jones GA (ed) (2005) Exploring probability in schools instructional approach, the agenda includes the
challenges for teaching and learning. Springer, design and implementation of curriculum pro-
New York posals and corresponding materials that
enhance problem-solving activities. Key ele-
Jones G, Langrall C, Mooney E (2007) Research in prob- ments in both the research and practice
ability: responding to classroom realities. In: Lester endeavors are the characterization of problems
F (ed) Second handbook of research on mathematics and what the problem-solving processes entail.
teaching and learning. Information Age Publishing and Often, a list of routine and nonroutine problems
NCTM, Greenwich is chosen as a means to elicit and develop
students’ problem-solving competencies. Also,
Kahneman D, Slovic P, Tversky A (eds) (1982) Judge- the same mathematical contents to be learned
ment under uncertainty: Heuristics and biases. and textbooks problems are seen as opportuni-
Cambridge University Press, New York ties for learners to engage in problem-solving
activities. These activities involve making
Piaget J, Inhelder B (1951) La gene´se de l’ide´e de hasard sense of concepts or problem statements;
chez l’enfant [The origin of the idea of chance in looking for different ways to represent, explore,
children]. Presses Universitaires de France, Paris and solve the tasks; extending the tasks’ initial
domain; and developing a proper language to
Stohl H (2005) Probability in teacher education and communicate and discuss results. The ways to
development. In: Jones G (ed) Exploring probability organize and implement problem-solving activ-
in schools: challenges for teaching and learning. ities might take different routes depending on
Springer, New York, pp 345–366

Problem Solving in Mathematics
Education

Manuel Santos-Trigo
Center for Research and Advanced Studies,
Department of Mathematics Education,
Cinvestav-IPN, San Pedro Zacateno,
Mexico D.F., Mexico

Keywords

Problem solving; Inquiring approach; Modeling
activities; Representations; Curriculum; Digital
technology

Problem Solving in Mathematics Education P497

the instructor’s aims, educational level, and Research programs structured around problem P
students’ background. solving have made significant contributions to the
understanding of the complexity involved in
In university and graduate levels, the Moore developing the students’ deep comprehension of
method, a variant of problem-solving approach, mathematics ideas, in using research results in the
might involve the selection of a list of theorems design and structure of curricular frameworks,
and course problems that students are asked to and in directing mathematical school practices.
unpack, explain, and prove within a learning
community that foster the members’ participation The cumulative findings in problem solving
including the instructor as a moderator (Halmos provide useful information on how the field has
1994). Other problem-solving approaches rely on evolved during the last 40 years in terms of
promoting scaffolding activities to gradually themes, research designs, curriculum proposals,
guide students’ construction of problem-solving and mathematical instruction. Of course, there
abilities. Instructional strategies involve fostering are traces of mathematical problem-solving
and valuing students’ small group participation, activities throughout the history of human civili-
plenary group discussions, the instructor presen- zation that have contributed to the development
tations through modeling problem-solving of the problem-solving agendas. For example, the
behaviors, and the students’ constant mathemat- same year that Polya published his How to Solve
ical reflection. Lesh and Zawojewski (2007) It book, Hadamard published Essay on the Psy-
identify modeling activities as essential for chology of Invention in the Mathematics Field.
students to develop knowledge and Hadamard asked 100 physicists/mathematicians
problem-solving experiences. They contend that how they performed their work and identified
in modeling processes, interactive cycles a four-step model that describes their problem-
represent opportunities for learners to constantly solving experiences: preparation, incubation,
reflect on, revise, and refine tasks’ models. Thus, illumination, and verification. However, in the
the multiplicity of interpretations of problem 1970s, the discussion of previous problem-
solving has become part of the identity of solving developments, including Krutetskii’s
the field. (1976) work on the study of mathematical
abilities of gifted children, became an important
Problem-Solving Developments: issue in the mathematics education agenda.
Frameworks, Focus, and Current Themes
The most salient feature of the problem-solving In the following sections, the goal is to provide
research agenda is that the themes, questions, and an account of the main problem-solving
research methods have changed perceptibly and developments that appeared during the last 40
significantly through time. Shifts in research years. This account includes some examples of
themes are intimately related to shifts in research problem-solving directions in specific countries,
designs and methodologies (Lester and Kehle identifies current issues, and possible future
2003). Early problem-solving research relied on directions in the field.
quantitative methods and statistical hypothesis
testing designs; later, approaches were, and Focus on Problem-Solving Activities and
continue to be, based mostly on qualitative Mathematicians’ Work
methodologies. In addition, the development It is recognized that mathematical problems and
and systematic use of digital technologies not their solutions are a key ingredient in the making
only has offered new paths to represent and and development of the discipline. What does the
explore mathematical situations (through the process of formulating and solving mathematical
use of dynamic models); also the students’ problems entail? The discussion of this type of
appropriation of these tools becomes an impor- question, within the mathematics community,
tant issue in the research and instructional agenda provided valuable information to characterize
(Hoyles and Lagrange 2010). problem-solving processes and to think of the
students’ construction of mathematical knowledge

P 498 Problem Solving in Mathematics Education

in terms of problem-solving activities. In mathe- Specifically, it was important to analyze in detail
matics education, the mathematicians’ work and how students could gradually develop a way of
developments in disciplines as psychology became thinking consistent with mathematical practice in
relevant to relate problem-solving activities and terms of problem-solving activities (Schoenfeld
the students’ learning of mathematics. Schoenfeld 1992).
(1985) suggests that open critiques (Kline 1973) to
the new math and the back-to-basic reforms in the During this period, problem-solving perspec-
USA were important to focus on problem-solving tives appeared explicitly in curriculum discus-
activities as a way for students to learn mathemat- sions, and in mathematical instruction, the
ics. Polya (1945) reflected on his own experience Polya’s model was a predominant approach to
as a mathematician to write about the process guide teaching strategies. The need to do research
involved and ways to be successful in problem- to support, structure, and implement problem-
solving activities. (Polya used retrospection solving activities became crucial, and the
(looking back at events that already have taken development of qualitative methods was relevant
place) and introspection (self-examination of to complement and extend previous quantitative
one’s conscious thought and feelings) methods to problem-solving analysis.
write about problem solving and ways to teach it.)
He proposed a general framework that describes The Importance of Problem-Solving Frameworks
four problem-solving stages (understanding the and the Design of Curriculum Proposals
problem, devising a plan, carrying out the plan, The use of qualitative tools provided a means
looking back). He also discussed the role and to analyze and discuss both features of mathe-
importance of using heuristic methods in students’ matical thinking and the process involved in
construction of mathematical knowledge. His problem-solving approaches. Schoenfeld (1985)
ideas not only shaped initial research programs in implemented a research program that focused on
problem solving but also appeared in curriculum analyzing students’ development of mathemati-
proposals and teaching scenarios (Krulik and Reys cal ways of thinking consistent with current
1980). mathematics practices. A key issue in his
program was to characterize what it means to
Polya’s ideas were found in curriculum think mathematically and to document how
materials and in the ways the development of students become successful or develop profi-
mathematical instruction was organized and ciency in solving mathematical tasks. He used a
structured. The use of heuristic methods was set of nonroutine tasks to engage first year uni-
deemed relevant, and instruction or teaching versity students in problem-solving activities. As
activities were organized and centered on the a result, Schoenfeld proposed a framework to
teacher who was in charge of modeling explain and document students’ problem-solving
problem-solving behaviors for the students. behaviors in terms of four dimensions: the
use of basic mathematical resources, the use of
Research studies included quantitative designs cognitive or heuristic strategies, the use of
to document and contrast groups or classes of metacognitive or self-monitoring strategies, and
students’ problem-solving behaviors exposed to students’ beliefs about mathematics and problem
differential approaches. Research results indi- solving.
cated that the identification of problem-solving
strategies and the process of modeling their use This framework has been used extensively not
in instruction was not sufficient for students only to document the extent to which problem
to foster their comprehension of mathematical solvers succeed in their problem-solving attempts
knowledge and problem-solving approaches. but also to organize and foster students’ develop-
This recognition allowed the mathematics education ment of problem-solving experiences in the class-
community to reflect on ways to characterize rooms. Schoenfeld (1992) also shed light on the
and explain the students’ development of mathemat- strengths and limitations associated with the use
ical thinking and problem-solving approaches. of Polya’s heuristics. Schoenfeld (1992, p. 353)

Problem Solving in Mathematics Education P499

pointed out that “Polya’s characterization did Regional or country mathematics education P
not provide the amount of detail that would traditions also play a significant role in shaping
enable people who were not already familiar and pursuing a problem-solving agenda. Artigue
with the strategies to be able to implement and Houdement (2007) summarized the use of
them.” In this period, the shift from using quanti- problem solving in mathematics education, and
tative to qualitative methods appeared in research they pointed out that in France problem solving is
studies, and students’ interactions were valued conceptualized through the lens of two influential
and promoted in mathematical instruction. The and prominent theoretical and practical
importance of students’ previous knowledge frameworks in didactic research: the theory of
was recognized when engaging themselves in didactic situations (TDS) and the anthropological
problem-solving discussions; students became theory of didactics (ATD).
the center of instruction that valued their active
participation as a part of a learning community. In the Netherlands, the problem-solving
In addition, the NCTM (1989) launched a curric- approach is associated with the theory of Realis-
ulum framework structured around problem- tic Mathematics that pays special attention to the
solving approaches. This framework was updated process involved in modeling the real-world sit-
in 2000 (NCTM 2000). This is the curriculum uations. They also recognized a strong connec-
proposal that best promotes the students develop- tion between mathematics as an educational
ment of mathematical experiences based on prob- subject and problem solving as defined by the
lem-solving approaches. Recently, the Common PISA program Doorman et al. 2007.
Core State Mathematics Standards (CCSMS)
(2010) also identified problem solving as one of Cai and Nie (2007) pointed out that problem-
the standard processes to develop students’ math- solving activities in Chinese mathematics
ematical proficiency. Through all grades, students education have a long history and are viewed as
are encouraged to engage in problem-solving a goal to achieve and as an instructional approach
practices that involve making sense of problems, supported more on experience than a cognitive
and persevere in solving them, to look for and analysis. In the classroom, teachers stress
express regularity in repeated reasoning, to use problem-solving situations that involve discus-
appropriate tools strategically, etc. sion: one problem multiple solutions, multiple
problems one solution, and one problem multiple
Problem-solving frameworks offer valuable changes. “The purpose of teaching problem
information regarding the main aspects that solving in the classroom is to develop students’
influence the development of problem-solving problem solving skills, help them acquire ways of
competencies. In addition, they provided basis thinking, form habits of persistence, and build
to analyze in deep the role of metacognition and their confidence in dealing with unfamiliar
beliefs systems in learners’ comprehension of situations” (Cai and Nie 2007, p. 471).
mathematics.
Recently, Schoenfeld (2011) updated his 1985
Regional Problem-Solving Developments and the problem-solving framework to explain how and
Use of Digital Tools why problem solvers make decisions that shape
Developments in mathematical problem solving and guide their problem-solving behaviors. He
have gone hand in hand with development and proposes three constructs to explain in detail
discussions in mathematics education. For what problem solvers do on a moment-by-moment
example, a situated cognition perspective links basis while engaging in a problem-solving pro-
the learning process to problem-solving activities cess: the problem solver’s resources, goals, and
within specific contexts, and a community of orientations. He suggests that these constructs
practices perspective emphasizes student’s social offer teachers, and problem solvers in other
interactions as a way to make sense and work on domains, tools for reflecting on their practicing
mathematical problems. decisions. In his book, he uses the framework to
analyze and predict the behaviors of mathematics
and science teachers and a medical doctor.

P 500 Problem Solving in Mathematics Education

The development and use of digital technol- instructional approaches. However, it is not
ogy opened up new paths for problem-solving clear how teachers implement and assess their
approaches. For example, basis, frameworks, students’ development of problem-solving com-
and instructional approaches that emerged from petences. In this context, teachers, together with
analyzing students’ problem-solving experiences researchers, need to be engaged in problem-
centered on the use of paper and pencil should be solving experiences where all have an opportu-
reexamined in accordance to what the use of nity to discuss and design problem-solving
the tools brings in to play. That is, they need to activities and ways to implement and evaluate
be adjusted or extended to incorporate and them in actual classroom settings. In addition,
document ways in which the use of digital there are different paths for students to develop
technology fosters new methods of representing mathematical thinking, and the use of tools
and reasoning about problem situations (Hoyles shapes the ways they think of, represent, and
and Lagrange 2010). The systematic use of explore mathematical tasks or problems. Then,
technology not only enhances what teachers and theoretical frameworks used to explain learners’
students do with the use of paper and pencil but construction of mathematical knowledge need to
also extends and opens new routes and ways of capture or take into account the different ways of
reasoning for students and teachers to develop reasoning that students might develop as a result
mathematics knowledge (Santos Trigo and of using a set of tools during the learning
Reyes-Rodriguez 2011). Thus, emerging reason- experiences. As a consequence, there is a need
ing associated with the use of the tools needs to be to develop or adjust current problem-solving
characterized and made explicit in curriculum frameworks to account not only the students’
and conceptual frameworks in order for teachers processes of appropriation of the tools but also
to incorporate it and to foster its development in the need to characterize the ways of reasoning,
teaching practices. In terms of curriculum including the use of new heuristics, for example,
materials and instruction, the use of several dig- dragging in dynamic representations, with which
ital technologies could transform the rigid and students construct learning as a result of using
often static nature of the content presentation digital tools in problem-solving approaches.
into a dynamic and flexible format where learners In addition, it is important to develop methodo-
can access to several tools (dynamic software, logical tools to observe, analyze, and evaluate
online encyclopedias, widgets, videos, etc.) group’s problem-solving behaviors that involve
while dealing with mathematical tasks. the use of digital technology.

The advent and use of computational technol- Cross-References
ogy in society and education influence and shape
the academic problem-solving agenda. The ▶ Affect in Mathematics Education
learners’ tools appropriation to use them in ▶ Collaborative Learning in Mathematics
problem-solving activities involves extending
previous frameworks and to develop different Education
methods to explain mathematical processes that ▶ Communities of Practice in Mathematics
are now enhanced with the use of those tools.
Education
Directions for Future Research ▶ Competency Frameworks in Mathematics
In retrospective, research in problem solving has
generated not only interesting ideas and useful Education
results to frame and discuss paths for students ▶ Critical Thinking in Mathematics Education
to develop mathematical knowledge and ▶ Inquiry-Based Mathematics Education
problem-solving proficiency; it has also ▶ Learning Practices in Digital Environments
generated ways to incorporate this approach into ▶ Mathematical Ability
the design of curriculum proposals and ▶ Mathematical Modelling and Applications in

Education

Professional Learning Communities in Mathematics Education P501

▶ Mathematical Representations NCTM (1989) Curriculum and evaluation standards for P
▶ Metacognition school mathematics. NCTM, Reston
▶ Problem Solving in Mathematics Education
▶ Realistic Mathematics Education NCTM (2000) Principles and standards for school
▶ Scaffolding in Mathematics Education mathematics. National Council of Teachers of
▶ Task-based Interviews in Mathematics Mathematics, Reston

Education Polya G (1945) How to solve it. Princeton University
▶ Technology and Curricula in Mathematics Press, Princeton

Education Santos Trigo M, Reyes-Rodriguez A (2011) Teachers’ use
▶ Visualization and Learning in Mathematics of computational tools to construct and explore
dynamic mathematical models. Int J Math Educ Sci
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Halmos PR (1994) What is teaching. Am Math Mon Karin Brodie
101(9):848–854 School of Education, University of the
Witwatersrand, Johannesburg, South Africa
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education and technology –rethinking the terrain. The Characteristics
17th ICMI study. Springer, New York
During the past decade, professional learning
Kline M (1973) Why Johnny can’t add: the failure of the communities have drawn the attention of educa-
new math. St. Martin’s Press, New York tionists interested in school leadership, school
learning, and teacher development. Professional
Krulik S, Reys RE (eds) (1980) Yearbook of the national learning communities aim to establish school
council of teachers of mathematics, Problem solving in cultures, which are conducive to ongoing
school mathematics. NCTM, Reston learning and development, of students, teachers,
and schools as organizations (Stoll et al. 2006).
Krulik S, Rudnick JA (1993) Reasoning and problem Professional learning communities refer to
solving. A handbook for elementary school teachers. groups of teachers collaborating to inquire into
Allyn and Bacon, Boston their teaching practices and their students’ learn-
ing with the aim of improving both. In order to
Krutestkii VA (1976) The psychology of mathematical improve practice and learning, professional
abilities in school children. University of Chicago learning communities interrogate their current
Press, Chicago practices and explore alternatives in order to

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Lester F, Kehle PE (2003) From problem solving to model-
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tives on mathematics problem solving, learning and
teaching. Lawrence Erlbaum, Mahwah, pp 501–518

P 502 Professional Learning Communities in Mathematics Education

refresh and re-invigorate practice (McLaughlin The idea is that teachers who work together learn
and Talbert 2008). Exploring alternatives is together, making for longer-term sustainability of
particularly important in mathematics education new practices and promoting community-
where a key goal of teacher development is to generated shifts in practice, which are likely to
support teachers’ orientations towards under- provide learners with more coherent experiences
standing and engaging students’ mathematical across the subject or school (Horn 2005;
thinking in order to develop conceptual under- McLaughlin and Talbert 2008). Professional
standings of mathematics among students. learning communities support teachers to
“coalesce around a shared vision of what counts
A key principle underlying professional learn- for high-quality teaching and learning and begin
ing communities is that if schools are to be intel- to take collective responsibility for the
lectually engaging places, all members of the students they teach” (Louis and Marks 1998,
school community should be intellectually p. 535). Ultimately, a school-wide culture of col-
engaged in learning on an ongoing basis (Curry laboration can be promoted, although working
2008). Professional learning communities are across subject disciplines can distract from
“fundamentally about learning – learning for a focus on subject knowledge (Curry 2008).
pupils as well as learning for teachers, learning Networked learning communities, where
for leaders, and learning for schools” (Katz and professional learning communities come together
Earl 2010, p. 28). Successful learning communi- across schools in networks, provide further
ties are those that challenge their members to support and sustainability for individual commu-
reconsider taken-for-granted assumptions in nities and improved teacher practices (Katz and
order to generate change, for example, challeng- Earl 2010).
ing the notion that working through procedures
automatically promotes conceptual understand- There is differing terminology for learning
ings of mathematics. At the same time, not all communities, which illuminate subtle but
current practices are problematic, and successful important differences in how communities are
professional learning communities integrate the constituted. These include “communities of
best of current practice with ideas for new practice,” “communities of enquiry,” and “criti-
practices. cal friends groups.” The key emphasis in the
notion of professional learning is that it signals
A number of characteristics of successful the focus of the community and the learning as
professional learning communities have been both data-informed and knowledge-based.
identified: they create productive relationships
through care, trust, and challenge; they de- Data-Informed and Knowledge-Based
privatize practice and ease the isolation often Enquiry
experienced by teachers; they foster collabora- Professional learning communities can be
tion, interdependence, and collective responsibil- established within or across subjects, and in
ity for teacher and student learning; and they each case the communities would choose
engage in rigorous, systematic enquiry on different focuses to work on. Working within
a challenging and intellectually engaging focus. mathematics suggests that the focus would be
Professional learning communities in mathemat- on knowledge of and intellectual engagement
ics education focus on supporting teachers to with mathematics and the teaching and learning
develop their own mathematical knowledge and of mathematics. Effective communities focus
their mathematical knowledge for teaching, par- on addressing student needs through a focus on
ticularly in relation to student thinking (Brodie student achievement and student work, joint
2011; Curry 2008; Jaworski 2008; Katz et al. lesson and curriculum planning, and joint
2009; Little 1990). observations and reflection on practice, through
watching actual classroom lessons or videotaped
The notion of collective learning in
professional learning communities is important.

Professional Learning Communities in Mathematics Education P503

recordings of classroom practice. Mathematics practice suggests that only research-based evi- P
learning communities support teachers to focus dence is good enough to inform teacher profes-
on learner thinking through examples of learners sional development. Data-informed professional
solving rich problems (Borko et al. 2008; development suggests that teachers themselves,
Whitcomb et al. 2009) or through teachers’ with some expert guidance, can and should
analyzing learner errors (Brodie 2011). interpret data that is available to them and
integrate research knowledge with their local
In many cases data comes from national tests, circumstances).
and teachers work together to understand the data
that the tests present and to think about ways to Leadership
improve their practice that the data suggests. Leadership in professional learning communities
Working with data as a mechanism to improve is central, particularly in helping to bring together
test scores can be seen as a regulatory practice, data from practice and the findings of research.
with external accountability to school managers Leaders can be school-based or external, for
and education department officials. Proponents of example, district officials or teacher-educators
teacher-empowered professional learning from universities. For long-term sustainability,
communities argue strongly that the goal of such there should be leadership within the school, or
data analysis must be to inform teachers’ conver- within a cluster of schools.
sations in the communities, as a form of internal
accountability to knowledge and learning (Earl Two key roles have been established as impor-
and Katz 2006). Data can also include teachers’ tant for leaders in professional learning commu-
own tests, interviews with learners, learners’ work, nities. The first is promoting a culture of inquiry
and classroom observations or videotapes. and mutual respect, trust, and care, where
teachers are able to work together to understand
The professional focus of professional challenges in their schools more deeply and sup-
learning communities requires that the learning port each other in the specific challenges that they
in these communities be supported by face as teachers. The second is to support teachers
a knowledge base as well as by data. As teachers to focus on their students’ knowledge and subse-
engage with data, their emerging ideas are quently their own knowledge and teaching prac-
brought into contact with more general findings tices. The second role is crucial in supporting
from research. Jackson and Temperley (2008) professional learning communities where sub-
argue for a model where practitioner knowledge ject-specific depth is the goal, depth in learning
of the subject, learners, and the local context and knowledge for both teachers and learners.
meets public knowledge, which is knowledge
from research and best practice. The interaction It is important for leaders in professional
between data from classrooms and wider public learning communities to also be learners and to
knowledge is central in creating professional be able to admit their own weaknesses (Brodie
knowledge, for two reasons. First, without 2011; Katz, et al. 2009). At the same time, it is
outside ideas coming into the communities’ con- important for leaders to have and present exper-
versations, they can become solipsistic and tise, which helps the community to move for-
self-preserving and may continue to maintain ward. In mathematics, leaders need to recognize
the status quo rather than invigorate practice. opportunities for developing mathematical
Second, data and knowledge work together to knowledge and knowledge of learning and teach-
promote internal accountability, to the learners ing mathematics among teachers, for example,
and teachers and to support the creation of new what counts as appropriate mathematical expla-
professional knowledge, which is research-based, nations, representations, and justifications and
locally relevant, and collectively generated. how these can be communicated with learners.
(Data-informed practice is different from Other functions for leaders in professional learn-
evidence-based practice. Evidence-based ing communities are developing teachers’

P 504 Professional Learning Communities in Mathematics Education

capacities to analyze classroom data; supporting References
teachers to observe and interpret data rather than
evaluate and judge practice; supporting teachers Borko H, Jacobs J, Eiteljorg E, Pittman M (2008) Video as
to choose appropriate problem of practices to a tool for fostering productive discussions in
work on, once the data has been interpreted; and mathematics professional development. Teach Teach
helping teachers to work on improving their prac- Educ 24:417–436
tice and monitoring their own and progress in
doing this, as well as their learners’ progress Boudett KP, Steele JL (2007) Data wise in action:
(Boudett and Steele 2007). So leadership in pro- stories of schools using data to improve teaching
fessional learning communities is a highly spe- and learning. Harvard University Press, Cambridge,
cialized task. MA

Impact and Research Brodie K (2011) Teacher learning in professional learning
There is a growing body of research that shows communities. In: Venkat H, Essien A (eds) Proceed-
that professional learning communities do pro- ings of the 17th national congress of the association for
mote improved teacher practices and improved mathematics education of south Africa (AMESA).
student achievement (Stoll et al. 2006). However, AMESA, Johannesburg, pp 25–36
the evidence is mixed depending on which
aspects of learning different studies choose to Curry M (2008) Critical friends groups: the possibilities
focus on. Research into professional learning and limitations embedded in teacher professional com-
communities invariably must confront how to munities aimed at instructional improvement and
recognize and describe learning, both in the con- school reform. Teach Coll Rec 110(4):733–774
versations of the community and in classrooms. It
is well known from situated theory that learning Earl L, Katz S (2006) How networked learning communi-
does not travel untransformed between sites, ties work. Centre for strategic education, seminar
rather it is recontextualized and transformed as series paper no 155
it travels from classrooms to communities and
back again. Horn IS (2005) Learning on the job: a situated account of
teacher learning in high school mathematics depart-
A second issue that research into professional ments. Cognit Instr 23(2):207–236
learning communities must confront is the rela-
tionship between group and individual learning. Jackson D, Temperley J (2008) From professional
While the focus of the community is on group learning community to networked learning commu-
learning and interdependence, ultimately each nity. In: Stoll L, Louis KS (eds) Professional learning
person contributes in particular ways to the com- communities: divergence, depth and dilemmas.
munity and brings particular expertise, and dif- Open University Press and McGraw Hill Education,
ferent people will learn and grow in different Maidenhead, pp 45–62
ways. Kazemi and Hubbard (2008) suggest
a situated framework for research into how the Jaworski B (2008) Building and sustaining enquiry com-
individual and the group coevolve in mathemat- munities in mthematics teaching development:
ics professional learning communities. Group teachers and didacticians in collaboration. In:
and individual trajectories can be examined in Krainer K, Wood T (eds) Participants in mathematics
relationship to each other, through a focus on teacher education: individuals, teams, communities
particular practices and artifacts of practice and networks. Sense, Rotterdam, pp 309–330
discussed by the community. How particular
practices travel from the classroom into the com- Katz S, Earl L (2010) Learning about networked learning
munity and back again can be traced through communities. Sch Eff Sch Improv 21(1):27–51
linking what happens in the community to what
happens in teachers’ classrooms. Katz S, Earl L, Ben Jaafar S (2009) Building and
connecting learning communities: the power of
networks for school improvement. Corwin, Thousand
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design and study of professional development.
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Professional learning communities: divergence, depth
and dilemmas. Open University Press and McGraw
Hill Education, Maidenhead, pp 151–165

Psychological Approaches in Mathematics Education P505

Stoll L, Bolam R, McMahon A, Wallace M, Thomas between mathematical practices in different soci-
S (2006) Professional learning communities: a review eties. The reasons for these ties are profound, and
of the literature. J Educ Change 7:221–258 beyond the very different approaches adopted,
mathematics represents a domain through which
Whitcomb J, Borko H, Liston D (2009) Growing talent: human cognition, cognitive development, or
promising professional development models and human development can be studied. We focus
practices. J Teach Educ 60(3):207–212 here on some psychological approaches adopted
in mathematics education. Although these
Psychological Approaches in approaches have come out at different times,
Mathematics Education approaches were not merely replaced and each
of them is still vibrant in the community of
Baruch B. Schwarz researchers in mathematics education.
The Hebrew University, Jerusalem, Israel

Keywords The Constructivist Approach P

Behaviorism; Collaborative learning; Construc- Our review of psychological approaches in math-
tivism; Concept development; Teaching experi- ematics is not exhaustive. We mention the
ments; Design research; Technology-Based approaches that contribute to our understanding
learning environments; Abstraction; Socio-math- of learning and teaching processes and that can
ematical norms help in what we consider as their improvement.
For this reason, we overlooked behavioristic
Characteristics approaches. We will begin with constructivism –
a learning theory with a very long history that can
Cognitive psychology, developmental psychol- be traced to John Dewey. The simple and general
ogy, and educational psychology are general idea according to which learning occurs when
fields of research for which mathematics educa- humans actively engage in tasks has been under-
tion naturally seems one among many domains of stood very differently by different psychologists.
application. However, the history of these For some, constructivism means discovery-based
domains of research and of the development of teaching techniques, while for others, it means
research in mathematics education is much more self-directedness and creativity. Wertsch (1998)
complex, and not at all hierarchical. For example, adopts a social version of constructivism –
in their monumental Human Problem Solving socioculturalism – to encourage the learner to
(Newell and Simon 1972), Newell and Simon arrive at his or her version of the truth, influenced
acknowledged that many of their ideas (which by his or her background, culture, or embedded
became among the fundamentals of Cognitive worldview. Historical developments and symbol
Psychology) were largely inspired from George systems, such as language, logic, and mathemat-
Po´lya’s How to Solve It (Po´lya 1945). Another ical systems, are inherited by the learner as
prestigious link is of course Piaget’s Episte´ a member of a particular culture, and these are
mologie Ge´ne´tique – his theory of human learned throughout the learner’s life. The fuzzi-
development: the theory was based on memora- ness and generality of the definition of construc-
ble experiments in which Piaget designed tivism led to inconsistent results. It also led to the
conservation tasks in which mathematical entities memorable “math wars” controversy in the
were focused on (number, quantity, length, pro- United States that followed the implementation
portions, etc.). Also Cole’s Cultural Psychology of constructivist-inspired curricula in schools
(1996) is largely based on the comparison with textbooks based on new standards. In spite
of many shortcomings, the constructivist
approach had the merit to lead scientists to

P 506 Psychological Approaches in Mathematics Education

consider the educational implications of the the- identified for scientific concepts and then in
ories of human development of Piaget and mathematics (e.g., the acquisition of the concept
Vygotsky in particular in mathematics education of fraction requires radical changes in the
(von Glasersfeld 1989; Cobb and Bauersfeld preexisting concept of natural number, Hartnett
1995). and Gelman 1998). Misconceptions were thought
to develop when new information is simply added
The Piagetian Approach: Research on to the incompatible knowledge base, producing
Conceptions and Conceptual Change synthetic models, like the belief that fractions are
always smaller than the unit. Learning tasks, in
The impact of Piaget’s theory of human develop- which students were faced with a cognitive
ment had and still has an immense impact on conflict, were expected to replace their miscon-
research on mathematics education. Many ceptions by the current accepted conception.
researchers adapted the Piagetian stages of Researchers in mathematics education continue
cognitive growth to describe learning in school studying the discordances and conflicts between
mathematics. Collis’ research on formal opera- many advanced mathematical concepts and na¨ıve
tions and his notion of closure (Collis 1975) are mathematics. Intuitive beliefs may be the cause of
examples of this adaptation. With the multi-base students’ systematic errors (Fischbein 1987;
blocks (also known as Dienes blocks), Dienes Stavy and Tirosh 2000; Verschaffel and De
(1971) was also inspired by Piaget’s general Corte 1993). Incompatibility between prior
idea that knowledge and abilities are organized knowledge and incoming information is one
around experience to sow the seeds of contempo- source of students’ difficulties in understanding
rary uses of manipulative materials in mathemat- algebra (Kieran 1992), fractions (Hartnett and
ics instruction to teach structures to young Gelman 1998), and rational numbers (Merenluoto
students. and Lehtinen 2002). The conceptual change
approach is still vivid because of its instructional
Since the 1970s researchers in science implications that help to identify concepts in
education realized that students bring to learning mathematics that are going to cause students
tasks alternative frameworks or misconceptions great difficulty, to predict and explain students’
that are robust and difficult to extinguish. The systematic errors, to understand how counterintu-
idea of misconception echoed Piagetian ideas itive mathematical concepts emerge, to find the
according to which children consistently elabo- appropriate bridging analogies, and more gener-
rate understandings of reality that do not fit ally, to develop students as intentional learners
scientific standards. Researchers in mathematics with metacognitive skills required to overcome
education adopted these ideas in terms of tacit the barriers imposed by their prior knowledge
models (Fischbein 1989) or of students’ concept (Schoenfeld 2002). However, harsh critiques
images (Tall and Vinner 1981). These frame- pointed out that cognitive conflict is not an effec-
works were seen as theories to be replaced by tive instructional strategy and that instruction that
the accepted, correct scientific views. Bringing “confronts misconceptions with a view to
these insights into the playground of learning and replacing them is misguided and unlikely to
development was a natural step achieved through succeed” (Smith et al. 1993, p. 153). As a
the idea of conceptual change. This idea is used to consequence, misconceptions research in
characterize the kind of learning required when mathematics education was abandoned in the
new information comes in conflict with the early 1990s. Rather, researchers began studying
learners’ prior knowledge usually acquired on the knowledge acquisition process in greater
the basis of everyday experiences. It is claimed detail or as stated by Smith et al. (1993) to focus
that then a major reorganization of prior on “detailed descriptions of the evolution of
knowledge is required – a conceptual change. knowledge systems” (p. 154) over long periods
The phenomenon of conceptual change was first of time.

Psychological Approaches in Mathematics Education P507

Departing from Piaget: From Research plane – of the interactions between teacher and P
on Concept Formation to Teaching students. Vygotsky’s theory of human develop-
Experiments ment was a natural source of inspiration for
researchers in mathematics education in this con-
The fine-grained description of knowledge text. A series of seminal studies on street mathe-
systems in mathematics education was initiated matics (e.g., Nunes et al. 1993) on the ways
as an effort to adapt his theory to mathematics unschooled children used mathematical practices
education (Skemp 1971). Theories of learning in showed the situational character of mathematical
mathematics were elaborated, among them activity. Rogoff’s (1990) integration of Piagetian
the theory of conceptual fields (Vergnaud 1983), and Vygotskian theories to see in guided partici-
the notion of tool-object dialectic (Douady 1984, pation a central tenet of human development
1986), and theories of process-object duality of fitted these developments in research in mathe-
mathematical conceptions (Sfard 1991; matics education. Rogoff considered learning
Dubinsky 1991). Van Hiele’s theory of develop- and development as changes of practice. For
ment of geometric thinking (Van Hiele 2004) her, learning is mutual as the more knowledge-
seems at a first glance to fit Piaget’s view of able (the teacher) as well as students learn to
development with its clear stages. However, it attune their actions to each other. Cobb and col-
clearly departed from Piaget’s theory in the leagues took the mathematics classroom in its
sense that changes result from teaching rather complexity as the natural context for learning
than from independent construction on the part mathematics (Cobb et al. 2001; Yackel and
of the learner. The method of the teaching Cobb 1996). He introduced the fundamental
experiment was introduced to map trajectories notion of social and socio-mathematical norms
in the development of students’ mathematical to point at constructs that result from the
conceptions. Steffe et al. (2000) produced fine- recurring enactment of practices in classrooms
grained models of students’ evolving conceptions (an embryonic version of this notion had already
that included particular types of interactions with been elaborated by Bauersfeld (1988)). Cobb and
a teacher and other students. It showed that learn- colleagues showed that those norms are funda-
ing to think mathematically is all but a linear mental for studying individual and group
process, but that what can be seen as mistakes or learning: learning as a change of practice entails
confusions may be essential in the learning pro- identifying the establishment of various norms.
cess. Moreover, “misconceptions” often resist Vygotsky’s intersubjectivity as the necessary
teacher’s efforts, but they eventually are neces- condition for maintaining communication was
sary building blocks in the learning of concep- replaced by Cobb and colleagues by taken-as-
tions. In the same vein, Schwarz et al. (2009) shared beliefs. Cobb also considered the mathe-
elaborated the RBC model of abstraction in con- matical practices of the classrooms (standards of
text to identify the building blocks of mathemat- mathematical argumentation, ways to reasoning
ical abstraction which are often incomplete or with tools and symbols) as other general collec-
flawed. Such studies invite considering alternative tive constructs to be taken into account to trace
approaches to understand the development of learning. Norms are constructed in the mini-
mathematical thinking. The RBC model takes culture of the classroom in which researchers
into account the impressive development of socio- are not only observers but actively participate in
cultural approaches in mathematics education. the establishment of this mini-culture. Cobb
adopts here a new theoretical approach in the
Sociocultural Approaches Learning Sciences – Design Research (Collins
et al. 2004). This interesting approach led to
Descriptions of students learning in teaching many studies in mathematics education, but also
experiments stressed the importance of the social raised the tough issue of generalizability of
design experiments.

P 508 Psychological Approaches in Mathematics Education

Although according to Cobb and his and argumentative practices, or teacher’s
followers, learning is highly situational, facilitation of group work. A good example of
knowledge that emerges in the classroom is dialogic teaching enacted in mathematics class-
presented in a decontextualized form that fits (or rooms is Accountable Talk (Michaels et al. 2009).
not) accepted mathematical constructs. The writ- Dialogic Teaching raises harsh psychological
ings of influential thinkers challenged this view. issues as in contrast with sociocultural
In L’arche´ologie du savoir, Michel Foucault approaches for which adult guidance directs
(1969) convincingly traced the senses given to emergent learning, dialogism involves symmetric
ideas such as “madness” along the history relations.
through the analysis of texts. Instead of identify-
ing knowledge as a static entity, he forcefully Numerous technological tools have been
claimed that human knowledge should be viewed designed by CSCL (Computer-Supported
as “a kind of discourse” – a special form of Collaborative Learning) scientists to facilitate
multimodal communication. Leading mathemat- (un-)guided collaborative work for learning math-
ics education researchers adopted this perspec- ematics. These new tools enable new discourse
tive (Lerman 2001; Kieran et al. 2002). In her practices with different synchronies and enriched
theory of commognition, Sfard (2008) viewed blended multimodalities (oral, chat, computer-
discourse as what changes in the process of learn- mediated actions, gestures). Virtual Math Teams
ing, and not the internal mental state of an indi- (Stahl 2012) is a representative project which
vidual learner. From this perspective, studying integrates powerful dynamic mathematics
mathematics learning means exploring pro- applications such as GeoGebra in a multiuser plat-
cesses of discourse development. The method- form for (un)guided group work on math prob-
ology of the theory of commognition relies on lems, so that small groups of students can share
meticulous procedures of data collecting and their mathematical explorations and co-construct
analysis. The methods of analysis are adapta- geometric figures online. In a recent book, Trans-
tions of techniques developed by applied lin- lating Euclid, Stahl (2013) convincingly shows
guists or by discursively oriented social how collaborating students can reinvent Euclid-
scientists. The discourse of the more knowl- ean geometry with minimal guidance and suitable
edgeable other is for Sfard indispensable, not CSCL tools. The possibilities opened by new
only as an ancillary help for the discovering technologies challenge the tenets of sociocultural
student but as a discourse to which he or she psychology: the fact that students can collaborate
should persist to participate, in spite of the fact during long periods without adult guidance chal-
its nature is incommensurable with the nature lenges neo-Vygotskian approaches for which
of his or her own discourse. Sfard’s theory and adult guidance is central for development. To
Cobb’s theory, which stemmed from research in what extent can it be said that the designed tools
mathematics education, have become influential embody adult discourse? In spite of the fact that
in the Learning Sciences in general. the teacher is often absent, new forms of partici-
pation of the teacher fit dialogism (e.g., modera-
Open Issues tion as caring but minimally intrusive guidance).
The psychological perspective that fit changes in
Leading modern thinkers such as Bakhtin have participation and the role of multiple artifacts in
headed towards dialogism, a philosophy based on these changes is an extension of the Activity The-
dialogue as a symmetric and ethical relation ory, the theory of Expansive Learning (Engestro¨m
between agents. This philosophical development 1987) to the learning of organizations rather than
has yielded new pedagogies that belong to what is the learning of individuals. The mechanisms of
called Dialogic Teaching, and new practices, for the emergent learning of the group are still mys-
example, (un-)guided small group collaborative terious, though. It seems then, that, again, mathe-
matics education pushes psychology of learning
to unconquered lands.

Psychological Approaches in Mathematics Education P509

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Q

Quasi-empirical Reasoning (Lakatos) the claims on the methodology of mathematics,
related to explaining how it is that mathematical
Bharath Sriraman1 and Nicholas Mousoulides2 knowledge qualifies for superlative epistemolog-
1Department of Mathematical Sciences, The ical qualities such as certainty, indubitability, and
University of Montana, Missoula, MT, USA infallibility.
2University of Nicosia, Nicosia, Cyprus
Lakatos’ attempted to illustrate the fallibility
Keywords of mathematics. Written as a fictionalized
classroom dialogue, Lakatos’ book (1976)
Lakatos; Maverick traditions in the philosophy of presented an innovative, captivating, and power-
mathematics; Fallibilism; Proofs and refutations; ful context for a reconstructed historical debate
Proof; Philosophy of mathematics and proof of the Descartes-Euler theorem for
polyhedral, as a generic example of the
Definition development of mathematical knowledge.
Lakatos appealed to the history of the theorem,
This entry examines Lakatos’ assertion that the by embedding what he had discovered in his
nature of mathematical knowledge is quasi- dissertation (dissertation topic suggested to him
empirical, in attempting to describe the growth by George Polya). The Descartes-Euler theorem
of mathematical knowledge and its implications asserts that for a polyhedron p we have V – E +
for mathematics education. F ¼ 2, where V, E, and F are, respectively, the
number of vertices, edges, and faces of p. He
Characteristics showed how Descartes-Euler’s theorem and the
concepts involved in it evolved through proofs,
The Hungarian philosopher Imre Lakatos (1976) counterexamples, and proofs modified in light
considered mathematics to be a quasi-empirical of the counterexamples, thereby illustrating the
science in his famous book “Proofs and Refuta- fallibility of mathematics.
tions: The Logic of Mathematical Discovery.”
The book, popularized within the mathematics The core of Lakatos’ philosophy of mathemat-
community by Reuben Hersh (1978) after this ics is that mathematical theorems are defeasible
paper “Introducing Imre Lakatos” (Hersh 1978), and subject to refutations not unlike claims
might also be considered as Lakatos’ response to in empirical sciences. Lakatos (1976, 1978)
attempted to establish an analogy between
Popper’s (1962) conjectures and refutations in
science and the logic of attempts at deductive
proofs and refutations in mathematics and to
describe the rational growth of mathematical

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

Q 512 Quasi-empirical Reasoning (Lakatos)

knowledge (p. 5). In extending Popper’s studies the implications of this view as a means
(1958, 1962) critical philosophy of science to of developing a model of mathematical inquiry,
mathematics, Lakatos claims that mathematical in attempting to relate this epistemological
theorems are not irrefutably true statements, but framework to actual classroom situations. During
conjectures, since we cannot know that a theorem the last 30 years, a significant number of philos-
will not be refuted. While in science, bold ophers and mathematics educators alike have
conjectures can be a starting point of the growth appropriated his ideas in Proofs and Refutations
of knowledge, in mathematics presenting and inferred great meanings for the classroom
tentative proofs can be the starting point of the practices of both teachers and students
growth of knowledge, even if they contain (Sierpinksa and Lerman 1996).
hidden assumptions or lemmas that have not
been proved yet. Lakatos’ (1976, 1978) gave substantial impe-
tus to developments in the sociology of mathe-
Lakatos’ approach in the philosophy of matical knowledge. Lakatos’ work can well serve
mathematics resulted in the argumentation that as a basis for a social constructivist philosophy of
mathematics, like the sciences, is a quasi- mathematics, which in turn can be used to
empirical theory. Such theories have their develop a theory of learning, such as constructiv-
“crucial truth-value injection” at the bottom. ism. A social constructivist perspective clearly
The logical flow in quasi-empirical theories is prefer the “Lakatosian” conception of mathemat-
not the transmission of truth, but rather the ical certainty as being subject to revision over
retransmission of falsity. The term quasi- time to put forth a fallible and non-Platonist
empirical describes the nature of the truth-value viewpoint about mathematics (Ernest 1991).
transmission in a particular deductive system,
like mathematics, not whether the system is Ernest (1991) claimed that the fallibilist phi-
empirical. Lakatos argues that “from special losophy and social construction of mathematics
theorems at the bottom (“basic statements”) up presented by Lakatos not only had educational
towards the set of axioms . . . a quasi-empirical implications but that Lakatos was even aware of
theory – at best -[can claim] to be well – corrob- these implications (p. 208). Various examples
orated, but always conjectural” (pp. 33–34). He propose a classroom discourse that conveys the
further explains that informal, quasi-empirical, thought-experimental view of mathematics as
mathematics does not grow through a that of continual conjecture-proof-refutation that
monotonous increase in the number of indubita- involves rich mathematizing experiences for stu-
bly established theorems but through the inces- dents. Ernest argued that school mathematics
sant improvement of guesses by speculation and should take on the socially constructed nature
criticism and by the logic of proofs and refuta- presented by Lakatos and also that teacher and
tions. His opinion that mathematics is conjectural students should engage in ways identical to those
is in contrast to the view that mathematics is in his dialogue, specifically posing and solving
Euclidean in nature. According to Lakatos the problems, articulating and confronting assump-
efforts of Russell and Hilbert to Euclideanize tions, and participating in genuine discussion
mathematics failed: “the Grande Logiques cannot (p. 208). In line with Ernest’s recommendations,
be proved true or even consistent; they can only Agassi (1980) identified that mathematics
be proved false or even inconsistence” (p. 15). education could be benefited by a Lakatos’
method of inspired teaching. Agassi proposed a
Although Lakatos described his work as Lakatosian method for the classroom, which had
a study of “mathematical methodology,” much “the merit of taking the student from where he
writing since then has used it as a font of sugges- stands and using his interruptions of the lecture as
tion concerning mathematics education, includ- a chief vehicle of his progress, rather than
ing school mathematics education. Researchers worrying about the teacher’s progress” (p. 30).
(e.g., Sriraman 2006) claim that Lakatos adopts Likewise, Fawcett (1938) attempted to conduct
the philosophical position of fallibilism and a classroom situation like the one presented in

Questioning in Mathematics Education Q513

Lakatos’ book. In a 2-year teaching experiment References Q
that highlighted the role of argumentation in
choosing definitions and axioms, the students Agassi J (1980) On mathematics education: the
in Fawcett’s study created suitable definitions, Lakatosian revolution. Learn Math 1(1):27–31
choose relevant axioms when necessary, and
created a Euclidean geometry system by Ernest P (1991) The philosophy of mathematics
using the available mathematics of Euclid’s education. Falmer Press, Bristol
time period.
Fawcett HP (1938) The nature of proof.
How would mathematics teaching and Thirteenth yearbook of the NCTM. Bureau of
learning have changed in a Lakatosian perspec- Publications, Teachers College, Columbia University,
tive? If a quasi-empirical view is taken, students New York
no longer need to ignore their common sense,
their experiences. Students’ explorations can Hersh R (1978) Introducing Imre Lakatos. Mathematical
become a central aspect of teaching. The didactic Intelligencer 1(3):148–151.
possibilities of Lakatos’ thought experiment
abound but not much is present in the mathemat- Lakatos I (1976) Proofs and refutations: the logic of
ics education literature in terms of teaching mathematical discovery. Cambridge University Press,
experiments that try to replicate the “ideal” class- Cambridge
room conceptualized by Lakatos. Sriraman
(2006) suggests the use of combinatorial Lakatos I (1978) The methodology of scientific
problems involving the use of sophisticated research programmes: philosophical papers, vol 1–2.
counting strategies with high school students to Cambridge University Press, Cambridge
explore the Lakatosian possibilities of furthering
mathematical discourse. Further with the advent Larvor B (1998) Lakatos: an introduction. Routledge,
of technology for mathematics learning which London/New York
support students’ explorations of visual represen-
tations, students’ creation of mathematical state- Popper K (1958) The logic of scientific discovery. Basic
ments based on exploration becomes a feasible Books, New York
and legitimate classroom activity.
Popper K (1962) Conjectures and refutations. Basic
Lakatos’ work is situated within the philoso- Books, New York
phy of science and clearly not intended for nor
advocates a didactic position on the mathematics Sierpinska A & Lerman S (1996) Epistemologies of math-
education, but it has implications for teaching and ematics and mathematics education. In Bishop AJ,
learning of mathematics (Sriraman 2006). The Clements MA, Keital C, Kilpatrick J, Laborde C
legacy of Lakatos is not restricted to counterex- (eds) International Handbook of Mathematics Educa-
amples and fallibility (Larvor 1998), but rather tion, Springer, pp 827–876.
implies for a program based on sensitivity to the
history of mathematics, an appreciation for Sriraman B (2006) An ode to Imre Lakatos: quasi-thought
the dynamics of its concepts and standards, its experiments to bridge the ideal and actual
relation with other fields, and on the central role mathematics classrooms. Interchange Q Rev Educ
students might play in developing mathematical 37(1–2):151–178
concepts.
Questioning in Mathematics
Education

John Mason
University of Oxford and The Open University,
Oxford, UK

Keywords

Questioning; Asking; Telling; Listening-to;
Listening-for; Prompts; Focusing; Attention;
Being mathematical

Cross-References

▶ Argumentation in Mathematics John Mason has retired
▶ Argumentation in Mathematics Education

Q 514 Questioning in Mathematics Education

Definition in order to build up to complexity and that
students have to have their hand held as they
Questioning means here the use of questions and negotiate these steps.
other prompts offered to students so as to help
them get unstuck or to direct their attention If a teacher is frightened that students will not
in a potentially useful way so that they make be able to address a problem, what they offer their
mathematical progress. students will reinforce lack of challenge and
hence lack of resilience and resourcefulness
Introduction (Claxton 2002). Their students are likely to
develop the view that they will always be given
On the face of it, being taught mathematics simple tasks, and so with little or no experience of
consists mainly of responding to mathematical how unfamiliar challenges can be tackled, they
questions posed either by a text or a teacher. Sup- are likely to balk when asked an unusual or chal-
port for how to respond comes from worked exam- lenging question. Worse, they may be reinforced
ples and exposition in the text and from questions in believing that their intelligence is bounded and
and exposition by the teacher. But whether some- so try to stay away from failure, which means
thing said or written is actually a genuine question refusing even relatively simple challenges
or a question masquerading as instruction is not (Dweck 2000). One of the biggest obstacles to
always easy to discern. Furthermore, student student success is the assumptions made by the
responses to apparent questions may themselves teacher about the capabilities of their students;
be questions rather than answers. another is the impression formed by students
from their teachers, parents, and the institution
Mathematical Perspectives of what they are capable of achieving. Even when
James Stigler and James Hiebert (1999) observed a challenging task is used from a textbook, there
that whereas in American (and indeed in most is evidence that when students get stuck it is
English speaking) mathematics classrooms, a natural tendency for the teacher to “dumb
students are asked to obtain the answers to math- down” the question, essentially engaging in
ematical questions, in Japan it is more usual to be mathematical funnelling so that their students
asked “in how many different ways can you find can succeed (Stein et al. 1996).
the answer?” Asking a complex and thought
provoking question to initiate work on a topic By contrast, students who experience chal-
makes assumptions about student competence lenges, whose teacher strives to be mathematical
and engagement. This pedagogical stance with and in front of the students so that they are
can be described as “deep end” or “complexity- exposed to “what to do when you are stuck”
oriented,” as opposed to “shallow end” or “sim- (Mason et al. 1982/2010), are more likely to
plicity-oriented” teaching. The two approaches develop resilience and resourcefulness and to be
are based on entirely different assumptions reinforced in the belief that they can succeed if
about students as human beings. The first sees they try hard enough and cleverly enough. Stu-
people as having demonstrated the powers neces- dents, whose teacher strives to be mathematical
sary to tackle complexity, to make sense of math- with and in front of their students, are likely to
ematics, and as willing to persevere in the use gain insight into mathematics as a constructive
those powers when challenged, frequently “fold- and creative enterprise. Students who see their
ing back” (Pirie and Kieren 1994) in a spiral of teacher sometimes getting stuck, and then
frequent returns to the same ideas in increasingly unstuck, and who experience being stuck but
complex ways (Bruner 1966). The second is then are encouraged to become aware of how
based on a “staircase” theory that learning they managed to get unstuck again are more
proceeds in careful, simple but inexorable steps likely to learn “what to do when you get stuck”
(Mason et al. 1982/2010) and to develop resil-
ience and resourcefulness. They are likely to
learn to “know what to do when they don’t

Questioning in Mathematics Education Q515

know what to do” (Claxton 2002). Students waiting until a question is asked that can be Q
whose teacher challenges them appropriately answered. Unfortunately the sequence of ques-
but significantly are likely to develop flexibility tions is entirely ephemeral, and no learning has
and creativity in their thinking. taken place. John Holt (1964, p. 24) describes
such an incident beautifully, and Heinrich
Asking as Telling Bauersfeld (1988, p. 36) called this pedagogic
Many apparent questions are actually rhetorical: trap “funnelling,” because the questions funnel
simply placing an interrogative voice tone at the student attention more and more narrowly,
end of an utterance does not guarantee that becoming simpler and simpler (see also Wood
a question is being asked. For example, “what 1998). An alternative strategy is to exit from the
do we do with our rulers?” is actually drawing interaction by acknowledging being caught in
attention to inappropriate behavior and is not “guess what is in my mind” and telling students
a genuine question (Ainley 1987). It is intended what “came to mind” or more extremely, taking a
to focus attention on the behavior, and it is telling different approach or abandoning the issue
the student(s) to change their behavior. altogether in order to return at a later date.

A great deal of spontaneous classroom Asking as Enquiring
questioning is actually “telling” masquerading Some questions are genuine, in the sense that the
as “asking.” In the flow of the classroom, the person asking does not know the answer and is
teacher has something come to mind and then presumably seeking that answer. For example,
asks a question which is intended to direct or drawing attention to the status of an utterance
focus student attention on what has come to with a question like “Is that a conjecture, or a
mind. The question interrupts and structures fact or what?” or frequently asking students
students’ attention. Students may experience the “How do you know . . .?”, students can respond
question as genuine, and try to respond, but to genuine enquiry, and the teacher can be
usually students experience the question as genuinely interested in the students’ response.
a shift of attention into an instruction to “guess The difference between “asking as telling” and
what is in my mind,” while the teacher expects “asking as asking” lies in a distinction made by
students to be “attending the way the teacher is Brent Davis (1996) between listening-for an
(now) attending.” Often it is only a student’s expected response and listening-to what students
“inappropriate” or unexpected reply to a question are saying (and watching what students are doing).
that provokes an awareness that there is an
expected, even an intended, answer in the Listening-to what students are saying and doing
teacher’s mind. These “telling” questions can be rather than listening-for what you expect is a form
very subtle, but almost always plunge students of “teaching by listening” (Davis 1996) which
into “guess what the teacher wants to hear,” sounds paradoxical at first but is certainly possible,
which may not advance their learning. by setting up tasks (asking questions) that encour-
age students to reveal their thinking. A good way
When you find yourself having asked such to get students to reveal their thinking is to ask
a question, you can either keep going or bail them “how do you know?” when they make an
out. If you keep going, you are likely to find assertion, for it reinforces the awareness that math-
yourself asking another even more focusing ematics is about making and justifying conjec-
question leading to a sequence of ever more pre- tures. Another good way is to ask students “will
cise and focused questions until eventually the that always be the case” when they make an actual
student can answer without any effort. Although or an implied generality and to ask “when else
the teacher is following a train of increasing might that be the case” to jolt them out of the
particularity or detail, the student is experiencing particular into widening their scope of generality.
a sequence of interventions. Even though the
teacher has followed a train of thought, the Another good way to get students to reveal
student has no access to that thinking, simply their thinking is to ask them to construct

Q 516 Questioning in Mathematics Education

mathematical problems for themselves. For objects, care must be taken not to confuse
example, asking students to construct a problem “absence of evidence” from “evidence of
(like the ones in a set of exercises, say) can be very absence”: just because a student does not vary
revealing about the scope of generality that they something that can be varied, or change some-
perceive in those exercises. Variants include “a thing in a particular way, does not mean that they
really simple example of a problem of this type,” did not think of it, only that they did not reveal it.
“a complicated example,” or even “a general
example” and can be augmented with “an exam- It is important to be clear here that “teaching
ple that will challenge other students” or “an by listening” is only one form of pedagogic
example that shows you know how to tackle strategy and is unlikely to succeed as the sole
problems like these.” Not only do these reveal mode of interaction. Even Socrates asked
dimensions of possible variation (Watson and questions and made the occasional observation!
Mason 2005) of which the student is aware, but
it is also a good study technique to pose and then Intention and Effect
solve your own problems. Furthermore it is much It is evident that all classroom questions (and
more engaging to work on problems you have many outside the classroom) are interventions in
posed than on well-worked-over problems in a the flow of students’ mentation and as such have
standard text. Students can respond with two aspects: the intention, which is to focus or
a degree of self-challenge that they feel comfort- direct attention, to re-orient perspective, and the
able with, and even if they do not completely effect, which is either to re-orient student
successfully solve the problems they pose, they behavior or to reveal an as-yet unknown answer.
are learning something. Even a genuine question is an intervention, an
interruption. Even asking a question when the
Another good way to get students to reveal student is immersed in being stuck and “not
their accessible example space (Watson and thinking about anything” except being stuck is
Mason 2005) is to get them to construct examples an interruption in the student’s state. Too many
of mathematical objects meeting various con- interventions, too frequent intervention, too
straints. By carefully choosing the constraints so intrusive an intervention may result in students
as to force students to think beyond the first (usu- coming to depend on the teacher rather than
ally rather simple) example that comes to mind developing resilience and resourcefulness. What
enriches their example space while revealing the constitutes too much, too many, or too intrusive is
dimensions they are aware of that can be changed, a delicate matter which cannot be automated or
and even something about the range of permissi- even taught: it is a judgement that comes from
ble change in those dimensions. For example, experience, both as a learner and as a teacher.
asking students to write down three pairs of num-
bers that differ by two often reveals a preference Classroom Ethos
for whole numbers, even when students are told The sociocultural-mathematical norms of
they will not be asked to do anything with those a classroom (the classroom rubric) have
numbers. The same “construction” can be used a significant affect on what is possible in the
with a pair of fractions, a pair of numbers whose way of asking questions and getting thoughtful
logarithms differ by 2, a pair of trig functions that responses (Yackel and Cobb 1996). In
look different, a pair of integrals, and so on. As an a classroom in which mathematical questions
example of increasing constraints, asking for a are asked which have simple answers that are
decimal number between 2 and 3, and without either correct or incorrect, students can become
using the digit 5, and with at least one digit a 7 dependent on the teacher asking appropriate
is highly revealing about students’ appreciation of questions. Uncertainty as to the correctness of
how decimals are constructed. an answer is likely to lead to increasing reticence
in answering, of fear of being wrong and looking
When listening-to students justifying conjec- foolish. This in turn can lead to an instrumental
tures, constructing problems or constructing

Questioning in Mathematics Education Q517

and intelligence-testing view of mathematics. By likely to come to depend on the teacher asking Q
contrast, in a classroom in which everything said that question. The question remains associated
(by students and the teacher) is taken to be with the classroom rather than being internalized
a conjecture that needs to be tested and justified, by students. By contrast, if the teacher begins by
students can be encouraged to try to articulate using a question type repeatedly and effectively
what they do understand, certain that they will and then gradually makes their prompts less and
be helped to modify their conjectures without less explicit, students’ attention can be directed to
being ridiculed. In a conjecturing atmosphere the types of questions that the teacher is asking
those who are certain hold back or ask helpful and eventually to students spontaneously asking
questions, while those who are uncertain try themselves the question. For example, asking
to articulate their uncertainty. questions like “what question am I going to ask
you?” or “What did you do yesterday when you
Questions as Typifying What Mathematics were stuck?” provides a metacognitive shift, an
Is About impetus for students to become aware of what
Since students’ experience of mathematics is they have been asked rather than remaining
dominated by the questions they are asked, their immersed in their task and simply responding to
impression of what mathematics is about, of what the question (Bauersfeld 1995). Meanwhile the
the mathematics enterprise is about, is likely to be teacher can begin introducing a different question
formed by the nature and content of the questions or prompt.
they are asked. Anne Watson and John Mason
(1998; see also a primary version Jeffcoat et al. The term “scaffolding” (Wood et al. 1976) is
2004) built on a collection of mathematically often used to refer to the temporary support that
structured question types developed by Zygfryd a teacher can provide for students, in which the
Dyrslag (1984) to provide a wide-ranging teacher acts as “consciousness for two” (Bruner
collection of questions that draw attention to 1986). This notion applies both to “horizontal
mathematical ways of thinking. They used a list mathematization” in which students are
of verbs of mathematics including prompted to become aware of other situations in
Exemplifying, Specializing, Completing, Deleting, which their thinking could be used (“utility” in
the sense of Ainley and Pratt 2002) and to
Correcting, Comparing, Sorting, Organizing, “vertical mathematization” in which students
Changing, Varying, Reversing, Altering, Gen- are prompted to become aware of what they
eralizing, Conjecturing, Explaining, Justifying, have been doing as instances of some more
Verifying, Convincing, and Refuting general or “abstract” action (Treffers 1987).
and types of mathematical statements including Shifting between levels of thinking is not entirely
Definitions, Facts, Properties, Theorems, Exam- natural for many or even most students. It is
ples, Counterexamples, Techniques, Instruc- a major role for teachers of mathematics.
tions, Conjectures, Problems, Representation,
Notation, Symbolization, Explanations, Justi- However, teacher interventions, whether as
fications, Proofs, Reasoning, Links, Relation- reminders or as re-orientations of attention are
ships, and Connections likely to go unnoticed, because the student is
to generate a grid of mathematically fruitful and immersed in the action. In order that students
pedagogically effective questions which are become aware of the questions they are asked,
founded in the mathematical practices of experts. the prompts they are given that serve to redirect
their attention usefully, it is usually necessary for
Internalizing Questions the teacher to engage in what Brown et al. (1989)
If students are always asked the same question or called fading or, in other words, to use increas-
type of question whenever they get stuck, when- ingly indirect, even metacognitive prompts, so
ever a new topic is being presented, or whenever that eventually the students internalize the
a topic is being reviewed, then most students are prompts for themselves (Love and Mason
1992). Learning and independence can really

Q 518 Questioning in Mathematics Education

only be said to have been achieved when students What is worthy of further investigation are
spontaneously question themselves and each questions of the following form:
other. A good example can be found in Brown What is it about a situation that brings certain
and Coles (2000) regarding the question “what is
the same and what is different about . . .”. questions or prompts to my mind? How
might this inform ways of working with
Students Asking “Good Questions” students so that they begin to come to mind
It must be every competent teacher’s dream that for the students?
students will ask “good” mathematical questions What is it about a situation that could bring
and a potential nightmare to be asked a lot of certain useful questions or prompts to mind?
questions beyond the teacher’s competence. What blocks or deters me from asking certain
Every teacher needs strategies to deal with the types of questions?
unexpected and difficult or challenging What is it about some questions and prompts that
question. Displacement and deferral strategies attracts teachers to try to use them, while other
include inviting students to record their conjec- questions and prompts do not?
ture for discussion later, having a public place
reserved for current conjectures and questions, Cross-References
and seeking assistance from colleagues in the
same or other institutions or on the web. But ▶ Affect in Mathematics Education
such questions are unlikely to come out unless ▶ Curriculum Resources and Textbooks in
students are being encouraged to pose questions
and to make conjectures. The best way to Mathematics Education
stimulate genuine mathematical questions from ▶ Hypothetical Learning Trajectories in
students is to ask genuine questions oneself, to be
seen to be enquiring, to have strategies (special- Mathematics Education
izing and generalizing, representing, and ▶ Inquiry-Based Mathematics Education
transforming) to use, and to be satisfied to leave ▶ Manipulatives in Mathematics Education
an enquiry as a conjecture for later (even much ▶ Mathematical Representations
later) consideration. This is what is meant by ▶ Noticing of Mathematics Teachers
“being mathematical with and in front of ▶ Problem Solving in Mathematics Education
students,” and it is the best way to offer students ▶ Sociomathematical Norms in Mathematics
experience of the thrill and pleasure of thinking
mathematically. Education
▶ Zone of Proximal Development in

Mathematics Education

Further Investigation References
It might be tempting to research questions of the
form “which form of questioning is the most Ainley J (1987) Telling questions. Math Teach 118:24–26
effective?” or “Which order of questions is most Ainley J, Pratt D (2002) Purpose and utility in pedagogic
effective?”, but in mathematics education, any
assertion of a generality has counterexamples task design. In: Cockburn A, Nardi E (eds) Proceed-
(Tahta, personal communication, 1990). There ings of the 26th annual conference of the international
is no universality, because so much of what group for the psychology of mathematics education,
happens depends on the rapport and relationship vol 2. PME, Norwich, pp 17–24
between teacher and students and between Bauersfeld H (1988) Interaction, construction, and
teacher and mathematics (Kang and Kilpatrick knowledge – alternative perspectives for mathematics
1992; Handa 2011). education. In: Grouws DA, Cooney TJ (eds)
Perspectives on research on effective mathematics
teaching: research agenda for mathematics education,
vol 1. NCTM and Lawrence Erlbaum Associates,
Reston, pp 27–46

Questioning in Mathematics Education Q519

Bauersfeld H (1995) “Language games” in the mathemat- Love E, Mason J (1992) Teaching mathematics: action
ics classroom: their function and their effects. In: and awareness. Open University, Milton Keynes
Cobb P, Bauersfeld H (eds) The emergence of mathe-
matical meaning: interaction in classroom cultures. Mason J, Burton L, Stacey K (1982/2010) Thinking
Lawrence Erlbaum Associates, Hillsdale, pp 271–291 mathematically. Addison Wesley, London

Brown L, Coles A (2000) Same/different: a ‘natural’ way Pirie S, Kieren T (1994) Growth in mathematical
of learning mathematics. In: Nakahara T, Koyama M understanding: how can we characterise it and
(eds) Proceedings of the 24th conference of the how can we represent it? Educ Stud Math
international group for the psychology of mathematics 26(2–3):165–190
education. Hiroshima, Japan, pp 2-153–2-160
Stein M, Grover B, Henningsen M (1996) Building
Brown S, Collins A, Duguid P (1989) Situated cognition student capacity for mathematical thinking and
and the culture of learning. Educ Res 18(1):32–41 reasoning: an analysis of mathematical tasks used in
reform classrooms. Am Educ Res J 33(2):455–488
Bruner J (1966) Towards a theory of instruction. Harvard
University Press, Cambridge Stigler J, Hiebert J (1999) The teaching gap: best ideas
from the world’s teachers for improving education in
Bruner J (1986) Actual minds, possible worlds. Harvard the classroom. Free Press, New York
University Press, Cambridge
Treffers A (1987) Three dimensions, a model of goal and
Claxton G (2002) Building learning power: helping young theory description in mathematics education. Reidel,
people become better learners. TLO, Bristol Dordrecht

Davis B (1996) Teaching mathematics: towards a sound Watson A, Mason J (1998) Questions and prompts for
alternative. Ablex, New York mathematical thinking. Association of Teachers of
Mathematics, Derby
Dweck C (2000) Self-theories: their role in motivation,
personality and development. Psychology Press, Watson A, Mason J (2005) Mathematics as a constructive
Philadelphia activity: learners generating examples. Erlbaum,
Mahwah
Dyrszlag Z (1984) Sposoby Kontroli Rozumienia Pojec
Matematycznych. Oswiata i Wychowanie 9B:42–43 Wood T (1998) Funneling or focusing? Alternative
patterns of communication in mathematics class. In:
Handa Y (2011) What does understanding mathematics Steinbring H, Bartolini-Bussi MG, Sierpinska A (eds)
mean for teachers? Relationship as a metaphor for Language and communication in the mathematics
knowing, Studies in curriculum theory series. classroom. National Council of Teachers of Mathe-
Routledge, London matics, Reston, pp 167–178

Holt J (1964) How children fail. Penguin, Harmondsworth Wood D, Bruner J, Ross G (1976) The role of tutoring in
Jeffcoat M, Jones M, Mansergh J, Mason J, Sewell H, problem solving. J Child Psychol 17:89–100

Watson A (2004) Primary questions and prompts. Yackel E, Cobb P (1996) Sociomathematical norms,
ATM, Derby argumentation, and autonomy in mathematics. J Res
Kang W, Kilpatrick J (1992) Didactic transposition in Math Educ 27:458–477
mathematics textbooks. Learn Math 12(1):2–7

Q

R

Realistic Mathematics Education Although “realistic” situations in the meaning
of “real-world” situations are important in RME,
Marja Van den Heuvel-Panhuizen1 and “realistic” has a broader connotation here.
Paul Drijvers2 It means students are offered problem situations
1Freudenthal Institute for Science and which they can imagine. This interpretation of
Mathematics Education, Faculty of Science & “realistic” traces back to the Dutch expression
Faculty of Social and Behavioural Sciences, “zich REALISEren,” meaning “to imagine.”
Utrecht University, Utrecht, The Netherlands It is this emphasis on making something real in
2Freudenthal Institute, Utrecht University, your mind that gave RME its name. Therefore, in
Utrecht, The Netherlands RME, problems presented to students can come
from the real world but also from the fantasy
Keywords world of fairy tales, or the formal world of
mathematics, as long as the problems are
Domain-specific teaching theory; Realistic experientially real in the student’s mind.
contexts; Mathematics as a human activity;
Mathematization The Onset of RME

What is Realistic Mathematics The initial start of RME was the founding in 1968
Education? of the Wiskobas (“mathematics in primary
school”) project initiated by Edu Wijdeveld and
Realistic Mathematics Education – hereafter Fred Goffree and joined not long after by Adri
abbreviated as RME – is a domain-specific Treffers. In fact, these three mathematics
instruction theory for mathematics, which has didacticians created the basis for RME. In 1971,
been developed in the Netherlands. Characteristic when the Wiskobas project became part of the
of RME is that rich, “realistic” situations are given newly established IOWO Institute, with Hans
a prominent position in the learning process. Freudenthal as its first director and in 1973
These situations serve as a source for initiating when the IOWO was expanded with the Wiskivon
the development of mathematical concepts, tools, project for secondary mathematics education; this
and procedures and as a context in which students basis received a decisive impulse to reform the
can in a later stage apply their mathematical prevailing approach to mathematics education.
knowledge, which then gradually has become
more formal and general and less context specific. In the 1960s, mathematics education in the
Netherlands was dominated by a mechanistic
teaching approach; mathematics was taught

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

R 522 Realistic Mathematics Education

directly at a formal level, in an atomized manner, which scientifically structured curricula were
and the mathematical content was derived from used and students were confronted with ready-
the structure of mathematics as a scientific disci- made mathematics as an “anti-didactic inver-
pline. Students learned procedures step by step sion.” Instead, rather than being receivers of
with the teacher demonstrating how to solve ready-made mathematics, students should be
problems. This led to inflexible and reproduc- active participants in the educational process,
tion-based knowledge. As an alternative for this developing mathematical tools and insights by
mechanistic approach, the “New Math” move- themselves. Freudenthal considered mathematics
ment deemed to flood the Netherlands. Although as a human activity. Therefore, according to him,
Freudenthal was a strong proponent of the mathematics should not be learned as a closed
modernization of mathematics education, it was system but rather as an activity of mathematizing
his merit that Dutch mathematics education was reality and if possible even that of mathematizing
not affected by the formal approach of the mathematics.
New Math movement and that RME could be
developed. Later, Freudenthal (1991) took over Treffers’
(1987a) distinction of horizontal and vertical
Freudenthal’s Guiding Ideas About mathematization. In horizontal mathematization,
Mathematics and Mathematics the students use mathematical tools to organize
Education and solve problems situated in real-life situations.
It involves going from the world of life into that of
Hans Freudenthal (1905–1990) was a symbols. Vertical mathematization refers to the
mathematician born in Germany who in 1946 process of reorganization within the mathematical
became a professor of pure and applied system resulting in shortcuts by using connections
mathematics and the foundations of mathematics between concepts and strategies. It concerns mov-
at Utrecht University in the Netherlands. As a ing within the abstract world of symbols. The two
mathematician he made substantial contributions forms of mathematization are closely related
to the domains of geometry and topology. and are considered of equal value. Just stressing
RME’s “real-world” perspective too much may
Later in his career, Freudenthal (1968, 1973, lead to neglecting vertical mathematization.
1991) became interested in mathematics educa-
tion and argued for teaching mathematics that is The Core Teaching Principles of RME
relevant for students and carrying out thought
experiments to investigate how students can be RME is undeniably a product of its time and
offered opportunities for guided re-invention of cannot be isolated from the worldwide reform
mathematics. movement in mathematics education that
occurred in the last decades. Therefore, RME
In addition to empirical sources such as text- has much in common with current approaches to
books, discussions with teachers, and observa- mathematics education in other countries. Never-
tions of children, Freudenthal (1983) introduced theless, RME involves a number of core princi-
the method of the didactical phenomenology. By ples for teaching mathematics which are
describing mathematical concepts, structures, inalienably connected to RME. Most of these
and ideas in their relation to the phenomena for core teaching principles were articulated origi-
which they were created, while taking into nally by Treffers (1978) but were reformulated
account students’ learning process, he came to over the years, including by Treffers himself.
theoretical reflections on the constitution of men-
tal mathematical objects and contributed in this In total six principles can be distinguished:
way to the development of the RME theory. • The activity principle means that in RME stu-

Freudenthal (1973) characterized the then dents are treated as active participants in the
dominant approach to mathematics education in learning process. It also emphasizes that

Realistic Mathematics Education R523

mathematics is best learned by doing geometry, measurement, and data handling R
mathematics, which is strongly reflected in are not considered as isolated curriculum
Freudenthal’s interpretation of mathematics chapters but as heavily integrated. Students
as a human activity, as well as in Freudenthal’s are offered rich problems in which they can
and Treffers’ idea of mathematization. use various mathematical tools and knowl-
• The reality principle can be recognized in edge. This principle also applies within
RME in two ways. First, it expresses the domains. For example, within the domain of
importance that is attached to the goal of number sense, mental arithmetic, estimation,
mathematics education including students’ and algorithms are taught in close connection
ability to apply mathematics in solving to each other.
“real-life” problems. Second, it means that • The interactivity principle of RME signifies
mathematics education should start from that learning mathematics is not only an
problem situations that are meaningful to individual activity but also a social activity.
students, which offers them opportunities to Therefore, RME favors whole-class discus-
attach meaning to the mathematical constructs sions and group work which offer students
they develop while solving problems. Rather opportunities to share their strategies and
than beginning with teaching abstractions inventions with others. In this way students
or definitions to be applied later, in RME, can get ideas for improving their strategies.
teaching starts with problems in rich contexts Moreover, interaction evokes reflection,
that require mathematical organization or, which enables students to reach a higher
in other words, can be mathematized and level of understanding.
put students on the track of informal context- • The guidance principle refers to Freudenthal’s
related solution strategies as a first step in idea of “guided re-invention” of mathematics.
the learning process. It implies that in RME teachers should have
• The level principle underlines that learning a proactive role in students’ learning and that
mathematics means students pass various levels educational programs should contain scenar-
of understanding: from informal context-related ios which have the potential to work as a lever
solutions, through creating various levels of to reach shifts in students’ understanding. To
shortcuts and schematizations, to acquiring realize this, the teaching and the programs
insight into how concepts and strategies are should be based on coherent long-term teach-
related. Models are important for bridging the ing-learning trajectories.
gap between the informal, context-related
mathematics and the more formal mathematics. Various Local Instruction Theories
To fulfill this bridging function, models have
to shift – what Streefland (1985, 1993, 1996) Based on these general core teaching principles, a
called – from a “model of” a particular number of local instruction theories and paradig-
situation to a “model for” all kinds of other, matic teaching sequences focusing on specific
but equivalent, situations (see also Gravemeijer mathematical topics have been developed over
1994; Van den Heuvel-Panhuizen 2003). time. Without being exhaustive some of these
local theories are mentioned here. For example,
Particularly for teaching operating with Van den Brink (1989) worked out new approaches
numbers, this level principle is reflected in to addition and subtraction up to 20. Streefland
the didactical method of “progressive schema- (1991) developed a prototype for teaching
tization” as it was suggested by Treffers fractions intertwined with ratios and proportions.
(1987b) and in which transparent whole- De Lange (1987) designed a new approach to
number methods of calculation gradually teaching matrices and discrete calculus. In the
evolve into digit-based algorithms. last decade, the development of local instruction
• The intertwinement principle means mathe-
matical content domains such as number,

R 524 Realistic Mathematics Education

theories was mostly integrated with the use of For example, Kindt (2010) showed how
digital technology as investigated by Drijvers practicing algebraic skills can go beyond repeti-
(2003) with respect to promoting students’ under- tion and be thought provoking. Goddijn et al.
standing of algebraic concepts and operations. (2004) provided rich resources for realistic
Similarly, Bakker (2004) and Doorman (2005) geometry education, in which application and
used dynamic computer software to contribute proof go hand in hand.
to an empirically grounded instruction theory
for early statistics education and for differential Worldwide, RME is also influential.
calculus in connection with kinematics, For example, the RME-based textbook series
respectively. “Mathematics in Context” Wisconsin Center
for Education Research & Freudenthal Institute
The basis for arriving at these local instruction (2006) has a considerable market share in
theories was formed by design research, as the USA. A second example is the RME-based
elaborated by Gravemeijer (1994), involving a “Pendidikan Matematika Realistik Indonesia” in
theory-guided cyclic process of thought Indonesia (Sembiring et al. 2008).
experiments, designing a teaching sequence, and
testing it in a teaching experiment, followed by a A Long-Term and Ongoing Process of
retrospective analysis which can lead to Development
necessary adjustments of the design.
Although it is now some 40 years from the incep-
Last but not least, RME also led to new tion of the development of RME as a domain-
approaches to assessment in mathematics specific instruction theory, RME can still be seen
education (De Lange 1987, 1995; Van den as work in progress. It is never considered a fixed
Heuvel-Panhuizen 1996). and finished theory of mathematics education.
Moreover, it is also not a unified approach to
Implementation and Impact mathematics education. That means that through
the years different emphasis was put on different
In the Netherlands, RME had and still has a con- aspects of this approach and that people who were
siderable impact on mathematics education. In the involved in the development of RME – mostly
1980s, the market share of primary education text- researchers and developers of mathematics
books with a traditional, mechanistic approach education and mathematics educators from
was 95 % and the textbooks with a reform-oriented within or outside the Freudenthal Institute – put
approach – based on the idea of learning mathe- various accents in RME. This diversity, however,
matics in context to encourage insight and under- was never seen as a barrier for the development of
standing – had a market share of only 5 %. In RME but rather as stimulating reflection and
2004, reform-oriented textbooks reached a 100 % revision and so supporting the maturation of
market share and mechanistic ones disappeared. the RME theory. This also applies to the
The implementation of RME was guided by the current debate in the Netherlands (see Van den
RME-based curriculum documents including Heuvel-Panhuizen 2010) which voices the
the so-called Proeve publications by Treffers and return to the mechanistic approach of four
his colleagues, which were published from decades back. Of course, going back in time is
the late 1980s, and the TAL teaching-learning not a “realistic” option, but this debate has
trajectories for primary school mathematics, made the proponents of RME more alert to
which have been developed from the late 1990s keep deep understanding and basic skills more
(Van den Heuvel-Panhuizen 2008; Van den in balance in future developments of RME and
Heuvel-Panhuizen and Buys 2008). to enhance the methodological robustness of
the research that accompanies the development
A similar development can be seen in second- of RME.
ary education, where the RME approach also
influenced textbook series to a large extent.

Recontextualization in Mathematics Education R525

Cross-References Treffers A (1987b) Integrated column arithmetic R
according to progressive schematisation. Educ Stud
▶ Didactical Phenomenology (Freudenthal) Math 18:125–145

References Van den Brink FJ (1989) Realistisch rekenonderwijs aan
jonge kinderen [Realistic mathematics education for
Bakker A (2004) Design research in statistics education: young children]. OW&OC, Universiteit Utrecht,
on symbolizing and computer tools. CD-Be`ta Press, Utrecht
Utrecht
Van den Heuvel-Panhuizen M (1996) Assessment
De Lange J (1987) Mathematics, insight and meaning. and realistic mathematics education. CD-ß
OW & OC, Utrecht University, Utrecht Press/Freudenthal Institute, Utrecht University,
Utrecht
De Lange J (1995) Assessment: no change without prob-
lems. In: Romberg TA (ed) Reform in school mathe- Van den Heuvel-Panhuizen M (2003) The didactical use
matics. SUNY Press, Albany, pp 87–172 of models in realistic mathematics education: an
example from a longitudinal trajectory on percentage.
Doorman LM (2005) Modelling motion: from trace graphs Educ Stud Math 54(1):9–35
to instantaneous change. CD-Be`ta Press, Utrecht
Van den Heuvel-Panhuizen M (ed) (2008) Children learn
Drijvers P (2003) Learning algebra in a computer mathematics. A learning-teaching trajectory with
algebra environment. Design research on the under- intermediate attainment targets for calculation with
standing of the concept of parameter. CD-Be`ta Press, whole numbers in primary school. Sense Publishers,
Utrecht Rotterdam/Tapei

Freudenthal H (1968) Why to teach mathematics so as to Van den Heuvel-Panhuizen M (2010) Reform under
be useful. Educ Stud Math 1:3–8 attack – forty years of working on better mathematics
education thrown on the scrapheap? no way! In:
Freudenthal H (1973) Mathematics as an educational task. Sparrow L, Kissane B, Hurst C (eds) Shaping the
Reidel Publishing, Dordrecht future of mathematics education: proceedings of the
33rd annual conference of the Mathematics Education
Freudenthal H (1983) Didactical phenomenology of math- Research Group of Australasia. MERGA, Fremantle,
ematical structures. Reidel Publishing, Dordrecht pp 1–25

Freudenthal H (1991) Revisiting mathematics education. Van den Heuvel-Panhuizen M, Buys K (eds)
China lectures. Kluwer, Dordrecht (2008) Young children learn measurement and
geometry. Sense Publishers, Rotterdam/Taipei
Goddijn A, Kindt M, Reuter W, Dullens D (2004)
Geometry with applications and proofs. Freudenthal Wisconsin Center for Education Research & Freudenthal
Institute, Utrecht Institute (ed) (2006) Mathematics in context.
Encyclopaedia Britannica, Chicago
Gravemeijer KPE (1994) Developing realistic mathemat-
ics education. CD-ß Press/Freudenthal Institute, Recontextualization in Mathematics
Utrecht Education

Kindt M (2010) Positive algebra. Freudenthal Institute, Paul Dowling
Utrecht Institute of Education, Department of Culture,
Communication and Media, University
Sembiring RK, Hadi S, Dolk M (2008) Reforming math- of London, London, UK
ematics learning in Indonesian classrooms through
RME. ZDM Int J Math Educ 40(6):927–939 Keywords

Streefland L (1985) Wiskunde als activiteit en de realiteit Anthropological theory of didactics;
als bron. Nieuwe Wiskrant 5(1):60–67 Classification; Didactic transposition; Discursive
saturation; Domains of action; Emergence;
Streefland L (1991) Fractions in realistic mathematics Framing; Institutionalisation; Noosphere;
education. A paradigm of developmental research. Pedagogic device; Recontextualisation; Social
Kluwer, Dordrecht activity method; Sociology; Strategic action

Streefland L (1993) The design of a mathematics course.
A theoretical reflection. Educ Stud Math
25(1–2):109–135

Streefland L (1996) Learning from history for teaching in
the future. Regular lecture held at the ICME-8 in
Sevilla, Spain; in 2003 posthumously. Educ Stud
Math 54:37–62

Treffers A (1978) Wiskobas doelgericht [Wiskobas goal-
directed]. IOWO, Utrecht

Treffers A (1987a) Three dimensions. A model of
goal and theory description in mathematics
instruction – the Wiskobas project. D. Reidel Publish-
ing, Dordrecht

R 526 Recontextualization in Mathematics Education

Characteristics into “knowledge actually taught.” Even this
knowledge is not necessarily equivalent to the
Recontextualization refers to the contention that knowledge acquired by the student, which is the
texts and practices are transformed as they are product of a further transposition. The precise
moved between contexts of their reading or enact- nature of the transposition at each stage is a func-
ment. This simple claim has profound implications tion of the nature of the knowledge (scholarly, to
for mathematics education and for education gen- be taught, actually taught) being recontextualized
erally. There are three major theories in the gen- and of historical, cultural, and pedagogic
eral field of educational studies that directly and specificities. TDT – which has been developed
explicitly concern recontextualization: the Theory in terms of conceptual complexity as the
of Didactic Transposition (later the Anthropolog- Anthropological Theory of Didactics
ical Theory of Didactics) of Yves Chevallard, (ATD, Chevallard 1992) – invites researchers to
Basil Bernstein’s pedagogic device, and Paul investigate the precise processes whereby the
Dowling’s Social Activity Method. These are all recontextualizations have been achieved in
complex theories, so their presentation here of particular locations and in respect of particular
necessity entails substantial simplification. regions of the curriculum, so revealing the
conditions and constraints on the teaching of
The Theory of Didactic Transposition (TDT) mathematics in these contexts. This has been
proposes, essentially, that constituting a practice attempted in, for example, the topics of calculus
as something to be taught will always involve a (Bergsten et al. 2010), statistics (Wozniak 2007),
transformation of the practice. This is a general and the limits of functions (Barbe´ et al. 2005).
claim that can be applied to any practice and any
form of teaching, but Chevallard’s (1985, 1989) Bernstein describes the “pedagogic device” as
work and that of many of those who have worked “the condition for culture, its productions,
with the TDT is most centrally concerned with reproductions and the modalities of their interre-
the teaching of mathematics in formal schooling lations” (1990; see also Bernstein 2000). It is a
(primary, secondary, or higher education phases). central feature of a highly complex theory that
The work of the didactic transposition is carried was developed over a period of some 40 years, so
out, firstly, by agents of what Chevallard referred its representation here is of necessity radically
to as the noosphere and involves the production simplified. Whereas Chevallard’s theory is
of curricula in the form of policy documents, concerned with the epistemological and cultural
syllabuses, textbooks, examinations, and so constraints on didactics, Bernstein’s interest lies
forth constituting the “knowledge to be taught.” in the manner in which societies are reproduced
The first task in this work is the construction of a and transformed. Pedagogy and, in
body of source knowledge as the referent practice particular, transmission occur in all sociocultural
of the “knowledge to be taught.” In the case of institutions, although much of the work inspired
school mathematics, this source or “scholarly by Bernstein has focused on formal schooling. An
knowledge” has been produced by mathemati- important exception to this is his early dialogue
cians over a very long historical period and in with the anthropologist, Mary Douglas
diverse contexts. In its totality, then, it is not a (see Douglas 1996/1970), which contributed to
practice that is currently enacted by mathemati- Douglas’s cultural theory and Bernstein’s funda-
cians, but is compiled in the noosphere. The next mental concepts, classification (regulation
task is the constitution of the “knowledge to be between contexts) and framing (regulation within
taught” from this “scholarly knowledge,” and it is a context). The pedagogic device regulates what is
the former that is presented to teachers as the transmitted to whom, when, and how and consists
curriculum. There is a further move, however, as of three sets of rules, hierarchically organized:
the teacher in the classroom must, through inter- distribution, recontextualization, and evaluation.
pretation and the production and management of Recontextualization rules, in particular, regulate
lessons, transpose the “knowledge to be taught” the delocation of discourses from the fields of

Recontextualization in Mathematics Education R527

their production – the production of physics Dowling’s (2009, 2013) Social Activity R
discourse in the university, for example – and Method (SAM) presents a sociological organiza-
their relocation as pedagogic discourse. This tional language that takes seriously lessons from
is achieved by the embedding of these constructionism and poststructuralism. As is the
instructional discourses in regulatory discourses case with Chevallard’s TDT, Dowling’s work
involving principles of selection, sequencing, and began with an interest in mathematics education
pacing. Recontextualization is achieved by agents (see Dowling 1994, 1995, 1996, 1998) but is
in the official recontextualizing field – policy more fundamentally sociological, giving a degree
makers and administrators – and the pedagogic of priority to social relations over cultural
recontextualizing field (teacher educators, the practices. For Dowling, the sociocultural is
authors of textbooks, and so forth) that together characterized by social action that is directed at
might be taken to coincide with Chevallard’s noo- the formation, maintenance, and destabilizing of
sphere in terms of membership. Superficially, alliances and oppositions. These alliances and
there might seem to be similarities between oppositions, however, are emergent upon
Bernstein’s and Chevallard’s theories. A crucial the totality of social action rather than being the
distinction, however, is that recontextualization deliberate outcomes of individual actions.
for Bernstein, but not for Chevallard, is always Alliances are visible in terms of regularities of
governed by distribution. This entails that peda- practice that give the appearance of regulating
gogic discourse is always structured by the social who can do, say, think what, though, as emergent
dimensions of class, gender, and race. Bernstein’s outcomes, they might be thought of, metaphori-
is a sociological theory, while Chevallard’s might cally, as advisory rather than determinant.
reasonably be described (in English) as an Another feature of Dowling’s theory is that it
educational theory. Through the sociological con- has a fractal quality, which is to say, the same
cept, relative autonomy, Bernstein also allows for language can be applied at any level of analysis
the possibility of the transformation of culture and the language is also capable of describing
and, ultimately, of society. A further distinction itself. School mathematics is an example of what
lies in that Bernstein describes pedagogic might be taken to exhibit a regularity of practice
discourse in terms of his fundamental categories, including the institutionalization of expression
classification, and framing, which enables (signifiers) and content (signifieds) in texts. The
a description of form but not of content. Further strength of institutionalization varies, however,
resources for the description of the form of dis- between strong and weak, giving rise to the
courses are available in Bernstein’s (2000) work scheme of domains of practice in Fig. 1, which
on horizontal and vertical discourses and on constitutes part of the structure of all contexts,
knowledge structures where he describes mathe- which is to say, of all alliances. Human agents
matics as a vertical discourse having horizontal might be described as seeing the world in terms of
knowledge structure and a strong grammar. In this the scheme in Fig. 1 or, more precisely, from
description he seems to be making no epistemo- the perspective of the esoteric domain. Where
logical distinction between mathematics in its the particular context is school mathematics, the
field of production, on the one hand, and school agent may cast a gaze beyond school mathematics
mathematics, on the other. onto, for example, domestic practices such as

Content (signifieds)

Expression (signifiers) I+ I−

Recontextualization in I+ esoteric domain descriptive domain
Mathematics Education,
Fig. 1 Domains of action I− expressive domain public domain
(Source: Dowling 2009)
I+/− represents strong / weak institutionalisation.

R 528 Recontextualization in Mathematics Education

Recontextualization in Practice
Mathematics Education,
Fig. 2 Modes of Representation DS− DS+
recontextualization DS− improvising de-principling
(Source: Dowling 2013) DS+ re-principling

rationalising

shopping. The deployment of principles of recog- for example, has directed an elaborated version of
nition and realization that are specific to school the scheme at literary studies.
mathematics will result in the recontextualization
of domestic shopping as mathematized shopping. Another category from SAM is discursive sat-
This contributes to the public domain of school uration, which refers to the extent to which a
mathematics, which thereby appears to be about practice makes its principles linguistically avail-
something other than mathematics. This contrasts able. To the extent that an activity or part of
with esoteric domain text that is unambiguously an activity can be described as high or low
about mathematics, the descriptive domain – the discursive saturation (DS+ or DSÀ), then another
domain of mathematical modelling – that appears scheme is generated that describes modes of
to be about something other than mathematics but recontextualization. This scheme is shown in
that is presented in the language of mathematics, Fig. 2. If school mathematics can generally
and the expressive domain (the domain of be described as DS+ and domestic shopping as
pedagogic metaphors) that appears to be about DSÀ, then the recontextualizing of domestic
mathematics but that is presented in the language shopping as school mathematics public domain –
of other practices (an equation is a balance, and so the representation of shopping by mathematics –
forth). This scheme enables the description of can be described as rationalizing and the
complex mathematical texts and settings in recontextualizing of, say, banking by school
terms of the distribution of the different domains mathematics as re-principling.
of mathematical practice to different categories of
student (e.g., in terms of social class). It can also These three theories of recontextualization –
reveal distinctions between modes of pedagogy those of Chevallard, Bernstein, and Dowling –
that take different trajectories around the scheme. offer different possibilities to researchers,
It should be emphasized that public domain and practitioners in mathematics education and
shopping is not the same thing as domestic shop- themselves draw on different theoretical and
ping; the recontextualization of practice always disciplinary antecedents. They are, however, not
entails a transformation as is illustrated by in competition as much as being complementary.
Brantlinger (2011) in respect of critical All three present languages that can be and have
mathematics education. The gaze of mathematics been deployed far more widely than mathematics
education is described (Dowling 2010) as fetching education, though Chevallard’s and Dowling’s
practices from other activities and recontex- theories certainly have their roots in this field of
tualizing them as mathematical practice. This is, research. Naturally, all three theories have
in a sense, a didactic necessity in the production of undergone more or less transformative action in
apprentices to mathematics who must, initially, be respect of their recontextualization for the
addressed in a language that is familiar to them. purposes of this entry.
A danger, however, lies in the pushing of the
results back out of mathematics as the result no Cross-References
longer has ecological validity. The scheme in
Fig. 1 is reproduced in all activities that can be ▶ Anthropological Approaches in Mathematics
recognized as exhibiting regularity of practice and Education, French Perspectives
at all levels within any such practice. Chung (2011),
▶ Calculus Teaching and Learning
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▶ Curriculum Resources and Textbooks in Douglas M (1996/1970) Natural symbols: explorations in R
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▶ Didactic Transposition in Mathematics Dowling PC (1994) Discursive saturation and school
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Education Dowling PC (1995) A language for the sociological
▶ Mathematical Knowledge for Teaching description of pedagogic texts with particular
▶ Mathematical Language reference to the secondary school mathematics scheme
▶ Mathematics Curriculum Evaluation SMP 11–16. Collect Orig Resour Educ 19(2): no
▶ Metaphors in Mathematics Education journal page numbers
▶ Semiotics in Mathematics Education
▶ Socioeconomic Class in Mathematics Dowling PC (1996) A sociological analysis of school
mathematics texts. Educ Stud Math 31:389–415
Education
▶ Sociological Approaches in Mathematics Dowling PC (1998) The sociology of mathematics
education: mathematical myths/pedagogic texts.
Education Falmer Press, London

References Dowling PC (2009) Sociology as method: departures from
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Chevallard Y (1985) La transposition didactique du savoir Mathematics Education
savant au savoir enseigne´. Editions Pense´e Sauvage,
Grenoble Barbara Jaworski
Loughborough University, Loughborough,
Chevallard Y (1989) On didactic transposition theory: Leicestershire, UK
some introductory notes. In: The proceedings of
the international symposium on selected domains Reflective practice is a commonly used term in
of research and development in mathematics educa- mathematics education, often without careful
tion, Bratislava, 3–7 August 1988, pp 51–62. http:// definition, implying a contemplative reviewing
yves.chevallard.free.fr/spip/spip/IMG/pdf/On_Didactic_ of learning and/or teaching in mathematics in
Transposition_Theory.pdf. Accessed 26 Aug 2012 order to approve, evaluate, or improve practice.
A feedback loop is often suggested in which
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perspectives provided by an anthropological approach. initiation of practice providing possibilities for
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of mathematics. La Pense´e Sauvage, Grenoble,
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R 530 Reflective Practitioner in Mathematics Education

improved practice. More precise definitions often reflect observe
draw on Dewey, who wrote: plan act

Active, persistent and careful consideration of any reflect observe
belief or supposed form of knowledge in the light plan act
of the grounds that support it and the further con-
clusions to which it tends constitutes reflective reflect observe
thought (1933, p. 9) plan act

. . . reflective thinking, in distinction to other Reflective Practitioner in Mathematics Education,
operations to which we apply the name of thought, Fig. 1 Action-reflection cycle (McNiff (1988), pp 44,
involves (1) a state of doubt, hesitation, perplexity, Fig 3.7)
mental difficulty, in which thinking originates, and
(2) an act of searching, hunting, inquiring, to find of action in the process of learning, and relates
material that will resolve the doubt and dispose of doing and learning through a reflective process.
the perplexity (p. 12).
Our knowing is ordinarily tacit, implicit in our
. . . Demand for the solution of a perplexity is patterns of action and in our feel for the stuff with
the steadying and guiding factor in the entire pro- which we are dealing. It seems right to say that
cess of reflection. (p. 14) our knowing is in our action (1983, p. 49).

Rather than a perspective just of contempla- Scho¨n refers to knowing-in-action as “the
tive thought, Dewey emphasizes the important sorts of know-how we reveal in our intelligent
element of action in reflection and the goal of action – publicly observable, physical perfor-
an action outcome. This has led to a linking of mances like riding a bicycle and private opera-
reflective practice with so-called action research tions like instant analysis of a balance sheet”
which is research conducted by practitioners into (1997, p. 25). He claims a subtle distinction
aspects of (their own) professional practice. between knowing-in-action and reflection-in-
Stephen Kemmis a leading proponent of action action. The latter he links to moments of surprise
research spoke of reflection as “meta-thinking,” in action: “We may reflect on action, thinking
thinking about thinking. He wrote: back on what we have done in order to discover
how our knowing-in-action may have contributed
We do not pause to reflect in a vacuum. We pause to an unexpected outcome” (p. 26). “Alterna-
to reflect because some issue arises which demands tively,” he says, “we may reflect in the midst of
that we stop and take stock or consider before we action without interrupting it . . . our thinking
act. . . . We are inclined to see reflection as some- serves to reshape what we are doing while we
thing quiet and personal. My argument here is that are doing it” (p. 26). Scho¨n distinguishes reflec-
reflection is action-oriented, social and political. Its tion-on-action and reflection-in-action. The first
product is praxis (informed committed action) the involves looking back on an action and reviewing
most eloquent and socially significant form of its provenance and outcomes with the possibility
human action. (Kemmis 1985, p. 141) then of modifying future action; the second is
especially powerful, allowing the person acting
Kemmis conceptualized action research with to recognize a moment in the action, possibly
reference to a critically reflective spiral in action with surprise, and to act, there and then,
research of plan, act and observe, and reflect
(Kemmis and McTaggart 1981; Carr and
Kemmis 1986), and other scholars have adapted
this subsequently (e.g., McNiff 1988) (Fig. 1).

More recent scholars relate ideas of reflection,
seminally, to the work of Donald Scho¨n who has
written about the reflective practitioner in pro-
fessions generally and in education particularly
(Scho¨n 1983, 1987). Scho¨n relates reflection to
knowing and describes knowing-in-action and
reflection-in-action. With reference to Dewey,
he writes about learning by doing, the importance

Reflective Practitioner in Mathematics Education R531

differently. John Mason has taken up this idea in practice in reflective cycles. However, rather than R
his discipline of noticing: we notice, in the the theorized systematicity of action research
moment, something of which we are aware, pos- (e.g., McNiff 1988), Jaworski described the cyclic
sibly have reflected on in the past and our notic- process of growth of knowledge for these teachers
ing afford us the opportunity to act differently, to as evolutionary, as “lurching” from time to time,
modify our actions in the process of acting opportunity to opportunity, as teachers grappled
(Mason 2002). with the heavy demands of being a teacher and
sought nevertheless to reflect on and in their prac-
Michael Eraut (1995) has criticized Scho¨n’s tice. As Eraut suggested, the nature of teaching in
theory of reflection-in-action where it applies to classrooms is demanding and complex for the
teachers in classrooms. He points out that Scho¨n teacher, as is the ongoing life in a school and the
presents little empirical evidence of reflection-in- range of tasks a teacher is required to undertake.
action, especially where teaching is concerned. Teachers’ reflection on their practice, evidenced
The word action itself has different meanings by reports at project meetings and observations of
for different professions. In teaching, action usu- teacher educator researchers, led to noticing in the
ally refers to action in the classroom where moment in classrooms, reflection-in-action, and
teachers operate under pressure. Eraut argues concomitant changes in action resulting from such
that time constraints in teaching limit the scope noticing.
for reflection-in-action. He argues that there is too
little time for considered reflection as part of the A question that arises in considering reflective
teaching act, especially where teachers are practice in mathematics education concerns what
responding to or interacting with students. difference it makes (to reflective practice) that it
Where a teacher is walking around a classroom is being used in relation to mathematics and to the
of children quietly working on their own, learning and teaching of mathematics. Although
reflection-in-action is more possible but already in the mathematics education literature there are
begins to resemble time out of action. Thus Eraut many references to the reflection of practitioners,
suggests that, in teaching, most reflection is there is a singular lack of relating reflective
reflection-on-action, or reflection-for-action. He practice directly to mathematics. We see writings
suggests that Scho¨n is primarily concerned with by mathematics educators referring, for example,
reflection-for-action, reflection whose purpose is to mathematics teachers who are reflective prac-
to affect action in current practice. titioners, reflecting on their practice of teaching
mathematics; however, the mathematics is rarely
In mathematics education research into addressed per se. We read about specific
teaching practices in mathematics classrooms, approaches to teaching mathematics and to
Jaworski (1998) has worked with the theoretical engagement in reflective practice, for example,
ideas of Scho¨n, Mason, and Eraut to characterize the identification of “critical incidents,” or the
observed mathematics teaching and the thinking, use of a “lesson study approach.” To a great
action, and development of the observed teachers. extent, the same kinds of practices and issues
The research was undertaken as part of a project, might be reported if the writers were talking
the Mathematics Teachers’ Enquiry (MTE) Pro- about science or history teaching. There is also
ject, in which participating teachers engaged in a dearth of research in which mathematics stu-
forms of action research into their own teaching. dents are seen as reflective practitioners.
Jaworski claims that the three prepositions
highlighted in the above discussion, on, in, and References
for, “all pertain to the thinking of teachers at
different points in their research” (p. 9) and pro- Carr W, Kemmis S (1986) Becoming critical: education,
vides examples from observations of teaching and knowledge and action research. Routledge Falmer,
conversations with teachers. To some degree, all London
the teachers observed engaged in action research
in the sense that they explored aspects of their own Dewey J (1933) How we think. D.C. Heath, London

R 532 Rural and Remote Mathematics Education

Eraut M (1995) Scho¨n shock: a case for reframing reflection- live in rural local units, and (3) they will not
in-action? Teach Teach Theory Pract 1(1):9–22 contain an urban center of over 200,000 people
(OECD 2010a).
Jaworski B (1998) Mathematics teacher research: process,
practice and the development of teaching. J Math Developing regions around the world, in
Teach Educ 1(1):3–31 particular Africa and Asia, are still mostly rural.
However, by 2030 these regions will join the
Kemmis S (1985) Action research and the politics of reflec- developed world in having a mostly urban
tion. In: Boud D, Keogh R, Walker D (eds) Reflection: population. Although the developed world has
turning experience into learning. Kogan Page, London been predominantly urban since the early 1950s,
some countries have a relative high proportion of
Kemmis S, McTaggart R (1981) The action research the population outside major cities (e.g., Australia,
planner. Deakin University, Geelong 34 %; Canada, 19 %) (Australian Bureau of
Statistics [ABS] 2012; Statistics Canada 2008).
Mason J (2002) Researching your own classroom practice. Social indicators show that people living in rural
Routledge Falmer, London areas have less access to a high quality of life
than do those living in urban areas, based on
McNiff J (1988) Action research: principles and practice. factors such as employment, education, health,
Macmillan, London and leisure (UN 2011). To some degree, research
in this area has been considered from a deficit
Scho¨n DA (1983) The reflective practitioner. Temple perspective, often perceived as backward, attached
Smith, London to tradition, and anti-modern (Howley et al. 2010).

Scho¨n DA (1987) Educating the reflective practitioner. Differences in Student Performance
Jossey-Bass, Oxford Students in large urban areas tend to outperform
students in rural schools by the equivalent of
Rural and Remote Mathematics more than one year of education (OECD 2012).
Education Severe poverty, often exacerbated in rural areas
due to a lack of employment, education opportu-
Tom Lowrie nities, and infrastructure, manifests the situation
Faculty of Education, Charles Sturt University, (Adler et al. 2009). Although socioeconomic
Wagga Wagga, NSW, Australia background accounts for part of the difference,
performance difference remain even when
Keywords socioeconomic background is removed as a
factor (OECD 2012). In other situations, severe
Rural mathematics education; Remote mathe- environmental conditions, including drought and
matics education; Distance education flood, heighten the challenging nature of educa-
tional opportunities in rural areas (Lowrie 2007).
Definition(s) Differences in students’ success in mathematics
are often correlated with the size of their commu-
Definitions of rural and remote mathematics con- nity, along with its degree of remoteness (Atweh
texts differ considerably from country to country et al. 2012). Rural, and especially remote, com-
and region to region – nevertheless most defini- munities face challenges of high staff turnover,
tions consider geographical position, population reduced professional learning opportunities, and
density, and distance from the nearest urban area. difficulty in accessing quality learning opportu-
The Organisation for Economic Co-operation nities for students (Lyons et al. 2006). The capac-
and Development (OECD) classifies regions ity to attract teachers with strong mathematics
within its member countries into three groups pedagogical content knowledge – already a chal-
based on population density – predominantly lenge in many countries – is heightened in rural
urban, intermediate, or predominantly rural.
A region is considered rural if it meets three
methodology criteria: (1) “local units” within a
region are rural if they have a population density
of less than 150 inhabitants per square kilometer,
(2) more than 50 % of the population in the region

Rural and Remote Mathematics Education R533

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