Encyclopedia of Mathematics Education

S

Scaffolding in Mathematics collective (a group scaffolding its members
Education in a distributed way).
• The support consists of employing instruc-
Bert van Oers tional means that are supposed to help learners
Faculty of Psychology and Education, with the accomplishment of a new (mathemat-
Department Research and Theory in Education, ical) task by assisting him/her to carry out the
VU University Amsterdam, Amstelveen, required activity through providing help at
The Netherlands parts of the activity that aren’t yet indepen-
dently mastered by the learner; this is to be
Keywords distinguished from just simplifying the task by
cutting it down into a collection of isolated
Support system; Help; Zone of proximal elementary tasks.
development; Educative strategy • Scaffolding aims at providing learners help
that is contingent on the learner’s prior quali-
Definition ties and contributes to the development of
knowledge, skills, and confidence to cope
Scaffolding is generally conceived as an interac- with the full complexity of the task; as such
tional process between a person with educational scaffolding is to be distinguished from
intentions and a learner, aiming to support this straightforward instruction in correct task
learner’s learning process by giving appropriate performance.
and temporary help. Scaffolding in mathematics • As a support system scaffolding is essentially
education is the enactment of this purposive a temporary construction of external help that
interaction for the learning of mathematical is supposed to fade away in due time.
actions and problem solving strategies.
Characteristics
A number of clarifying corollary postulates are
usually added for the completion of this general Tutoring Learning
definition of scaffolding in a specific situation: The notion of educational support systems for the
• Scaffolding is an intentional support system appropriation of complex activities was first
introduced by Bruner in the 1950s in his studies
based on purposive interactions with of language development in young children.
more competent others, which can be adults In opposition to the Chomskyan explanation of
or peers; the support can be individualized language development resulting from an inherent
(one teacher scaffolding one student) or

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

S 536 Scaffolding in Mathematics Education

Language Acquisition Device (LAD), Bruner explanations and elaborations of scaffolding,
advocated a theory of language development most authors have taken this notion of the zone
that holds that parent–infant interactions of proximal development as a point of reference.
constitute a support system for children in their
attempts to accomplish communicative inten- Using a Vygotskian theoretical framework, the
tions. In Bruner’s view it is this Language work of Stone and Wertsch has contributed sig-
Acquisition Support System (LASS) that nificantly to the understanding of scaffolding.
“scaffolds” children’s language development. Stone and Wertsch (1984) have examined
scaffolding in a one-to-one remedial setting with
In a seminal article on adult tutoring in a learning-disabled child. They could show how
children’s problem solving, Bruner and his adult language directs the child to strategically
colleagues generalized the idea of learning monitor actions. Their analyses articulated the
support systems to the domain of problem solving temporary nature of the scaffold provided by
in general and explicitly coined the notion of the adult. Close observation of communicative
scaffolding as a process of tutoring children for patterns in the adult–child interactions showed a
the acquisition of new problem solving skills (see transition and progression in the source of strate-
Wood et al. 1976). They point out that scaffolding gic responsibility from adult (or other-regulated)
“consists essentially of the adult ‘controling’ actions to child (self-regulated) actions. The
those elements of the task that are initially beyond gradual reduction of the scaffolding (“fading”) is
the learner’s capacity, thus permitting him to possible through the child’s interiorization of the
concentrate upon and complete only those external support system (transforming it into
elements that are within his range of competence” “self-help”).
(Wood et al. 1976, p. 90). In the elaboration of the
scaffolding process, Wood et al. (1976, p. 98) Stone (1993) made a critical analysis of the
identify several scaffolding functions: use of the scaffolding concept as a purely instru-
1. Recruitment: scaffolding should get learners mental teaching strategy. He pointed out that
until the early 1990s most conceptions of scaf-
actively involved in relevant problem solving folding were missing an important Vygotskian
activity. dimension that has to do with the finality of
2. Reduction in degrees of freedom, i.e., keeping scaffolding for the learner. Especially the
students focused on those constituent acts that learner’s understanding of how the scaffolding
are required to reach a solution and that they and learning make sense beyond the narrow
can manage while preventing them from being achievement of a specific goal adds personal
distracted by acts that are beyond their actual sense to the cultural meaning of the actions to
competence level; these latter actions are sup- be learned through scaffolding. Stone refers
posed to be under the control of the scaffold- to this dimension with the linguistic notion of
ing tutor. “prolepsis” which can be seen here as an under-
3. Direction maintenance: The tutor has the role standing in the learner of the value of the
of keeping students in pursuit of a particular scaffolded actions in a future activity context.
objective and keeps them motivated to be Until today many applications of the scaffolding
self-responsible for the task execution. strategy are still missing this proleptic dimension
Without explicitly mentioning the Vygotskian and neglect the process of personal sense
notion of the zone of proximal development, the attachment to the scaffolded actions.
formulations used by Bruner and his colleagues
(see quote above) unequivocally refer to one of The use of scaffolding in various contexts
Vygotsky’s operationalizations of this notion has led to different educative strategies for
(see Vygotsky 1978, p. 86) as the discrepancy implementing scaffolding in classrooms with
between what a learner can do independently varying levels of explicitness as to the help given
and the learner’s performance with help (support) (for an excellent, recent, and very informative
from more knowledgeable others. In later overview and empirical testing of scaffolding
strategies, see van de Pol 2012). The most used

Scaffolding in Mathematics Education S537

scaffolding strategies with increasingly specific primary school, or enacting everyday life prac- S
help are modeling (showing the task performance), tices (going to the supermarket or calculating
giving advice (providing learners with suggestions your taxes). In a process of collaborative problem
that might help them to improve their perfor- solving (and exploratory talk, see Mercer 2000)
mance), and providing coaching in the accom- under guidance of the teacher, the teacher has to
plishment of specific actions (giving tailored take care of the contingency of the actions and
instructions for correct performance). Following solutions on all participants’ prior understandings
Stone’s critique on current scaffolding concep- but also of tailoring the scaffolding to the varying
tions, however, it is reasonable to add, as a useful needs of the students: modeling general solutions
educative strategy, embedding, which entails lur- (if necessary, when the students have problems to
ing the learner in familiar sociocultural practices find the direction of where to find the solution of
in which the new knowledge, actions, operations, the mathematical problem at hand), giving hints
and strategies to be learned are functional compo- (i.e., giving advice, if necessary, when the
nents for a full participation in that practice. This group’s problem solving seems to go astray), or
embedding in familiar sociocultural practices even stepwise coaching the execution of complex
helps students to discover the sense of both these new actions when these actions are important for
learning goals and the teacher’s scaffolding. the resolution of the problem but go beyond the
actual level of the participants’ competences. In
Attempts at employing scaffolding strategies this latter case it is important that the teacher
in mathematics education can be generated from sensitively monitors the contingency of the
the above summarized general theory of scaffold- steps in the learning process in the students.
ing, provided that it is clear what kind of mathe-
matical learning educators try to promote. If the Scaffolding in mathematics education that
formation of mathematical proficiency is reduced aims at mathematical understanding is basically
to learning to perform mathematical operations a language-based (discursive) process in which
rapidly and correctly, then scaffolding should students are collectively guided to a shared
include embedding to make clear how the mas- solution of mathematical problems and learn
tery of these operations may help students to how this contributes to their understanding of
participate autonomously in future practices. the mathematical concepts that are being
The choice for coaching on these specific actions employed. Although there is as yet a growing
in order to take care that they are mastered in body of (evidence-based) arguments for this
correct form may be an important way of scaf- discursive approach to the development of
folding the learning by repetition and practicing. mathematical thinking (see, e.g., Pimm 1995;
If, however, the focus is on learning mathematics Sfard 2008), a number of unresolved issues are
for understanding and hence on developing still waiting for elaboration:
the ability of concept-based communication • How to reconcile dialogical agreements in
and problem solving with mathematical tools, a
broader range of scaffolding strategies is needed. a group of students with the extensive body
First of all the strategy of embedding is of proofs and understandings in the wider pro-
important: helping students to connect the actions fessional mathematical community? How can
to be learned with a sociocultural practice that is a teacher scaffold the students’ processes of
recognizable and accessible for them. One may becoming a valid and reliable mathematics
think of practices like being a member of user in a variety of cultural contexts?
the mathematical community, but most of • How to scaffold the emergence of mathemat-
the time this scaffolding strategy consists in ical thinking in young children that opens
embedding the mathematical problem solving a broad and reliable basis for the development
process in cultural practices like industrial design of rich and valid mathematical thinking? How
(e.g., designing a tricycle for toddlers), or prac- can we meaningfully scaffold the process of
ticing a third-world shop in the upper grade of learning to talk, informed by mathematical
concepts? Although practical and theoretical

S 538 Semiotics in Mathematics Education

know-how is currently being expanded (see van de Pol J (2012) Scaffolding in teacher- student interac-
van Oers 2010; Fijma 2102), further ecologi- tion. Exploring, measuring, promoting and evaluating
cally valid empirical studies are needed. scaffolding. (Dissertation University of Amsterdam).
• How can teachers scaffold the process of mas- Faculteit der Maatschappij- en Gedragswetenschappen,
tery of automatization in mathematics while Amsterdam
maintaining the foundations of this process in
understanding and meaningful learning? van Oers B (2010) The emergence of mathematical think-
• How can teachers support the gradual fading of ing in the context of play. Educ Stud Math 74(1):23–37
the teacher’s scaffolding and turn this external
(interpersonal) scaffolding into a personal Vygotsky LS (1978) Mind in society. Harvard University
quality of self-scaffolding? For this it is neces- Press, Cambridge
sary to encourage the students to make and
discuss their own personal verbalizations of Wood D, Bruner JS, Ross G (1976) The role of tutoring in
the shared concepts and solutions. More study problem solving. J Child Psychol Psychiat 17:89–10
is needed into this formation of personalized
regulatory abilities on the basis of accepted Semiotics in Mathematics Education
mathematical understandings, using a combi-
nation of dialogue (interpersonal exploratory Norma Presmeg
talk) and polylogue (critical discourse with the Department of Mathematics, Illinois State
wider mathematical community). University, Maryville, TN, USA

Keywords

Cross-References Signs; Semiosis; Semiotics; Mathematical
objects; Semiotic representations; Communicat-
▶ Collaborative Learning in Mathematics ing mathematically; Decontextualization; Con-
Education textualization; Signifier; Signified; De Saussure;
Triads; Charles Sanders Peirce; Object;
▶ Inquiry-Based Mathematics Education Representamen; Interpretant; Iconic; Indexical;
▶ Zone of Proximal Development in Symbolic; Intensional interpretant; Effectual
interpretant; Communicational Interpretant;
Mathematics Education Cominterpretant; Commens; Epistemological
triangle; Semiotic bundles; Diagrammatic
References reasoning; Abduction; Onto-semiotic theoretical
model; Semiotic mediation
Fijma N (2102) Learning to communicate about
number. In: van Oers B (ed) Developmental Definitions and Background
education for young children. Springer, Dordrecht,
pp 253–270 Because mathematical objects cannot be
apprehended directly by the senses (e.g., Otte
Mercer N (2000) Words and minds. Routledge, London 2006), their ontological status requires signs such
Pimm D (1995) Symbols and meanings in school mathe- as symbols and diagrams for their communication
and learning. A sign (from ancient Greek semeion,
matics. Routledge, London meaning sign) is described by Colapietro (1993)
Sfard A (2008) Thinking as communicating. Human as “something that stands for something else”
(p. 179). Then semiosis is “a term originally used
development. The growth of discourses, and by Charles S. Peirce to designate any sign action or
mathematizing. Cambridge University Press, sign process; in general, the activity of a sign”
Cambridge, UK (p. 178). Semiotics is “the study or doctrine of
Stone CA (1993) What is missing in the metaphor of
scaffolding? In: Forman EA, Minick M, Stone CA
(eds) Contexts for learning. Sociocultural dynamics
in children’s development. Oxford University Press,
New York, pp 169–183
Stone CA, Wertsch J (1984) A social interactional analysis
of learning disabilities remediation. J Learn Disabil
17:194–199

Semiotics in Mathematics Education S539

signs; the systematic investigation of the nature, these lenses have proved useful in mathematics S
properties, and kinds of sign, especially when education.
undertaken in a self-conscious way” (p. 179).
Both Duval (2006) and Otte (2006) stressed that Ferdinand de Saussure, working in linguistics,
mathematical objects should not be confused with put forward a dyadic model of semiosis in which
their semiotic representations, although these signs a signifier (such as the word “tree”) stands for a
provide the only access to their abstract objects. signified (the concept of tree). Note that in this
Ernest (2006) suggested that there are three com- example, both the word and its concept are
ponents of semiotic systems (clearly illustrated by mental constructs, not objects accessible to the
the systems of mathematics), namely, a set of senses. Saussure’s model allows for a chaining of
signs, a set of relationships between these signs, signifiers that was used in mathematics education
and a set of rules for sign production. research by Walkerdine (1988) and Presmeg
(1998). The need to acknowledge the human
Semiotics is particularly suited to investigation subject involved in such semiosis led Presmeg
of issues in mathematics teaching and learning to the triadic model of Peirce (1992, 1998)
because it has the capacity to account for both the and to a nested chaining model that includes
general and the particular. Mathematicians and interpretation of signs (Hall 2000; Presmeg
teachers employ different symbolic practices in 2006). Charles Sanders Peirce used triads
their work, while sharing the goal of communicat- extensively in his model of semiosis. His main
ing mathematically: mathematicians aim for triad involved the components of object,
decontextualization in reporting their research representamen that stands for the object in some
whereas teachers recognize a need for contextuali- way, and interpretant, involving the meaning
zation in students’ learning of mathematical con- assigned to the object-representamen pair. An
cepts (Sa´enz-Ludlow and Presmeg 2006). Semiosis illustration used by Whitson (1997) is as follows:
is essential in both of these practices. Further, as object, it will rain; representamen, the barometer
Fried (2011) pointed out, tensions between public is falling; and interpretant, take an umbrella.
and private realms arise in a persistent way in Peirce used the term sign sometimes to designate
discussions connected with semiotics in mathemat- the representamen and at other times to refer to
ics education, reflecting “the division between stu- the whole triad. In any case, the model allows for
dents’ own inner and individual understandings of a nested chaining that may be continued indefi-
mathematical ideas and their functioning within nitely, as each interpretant in turn (and implicitly
a shared sociocultural world of mathematical thereby the whole triad) may become an object
meanings” (p. 389). that is represented by a new representamen
and interpreted. Sa´enz-Ludlow (2006) used
Semiotic Lenses and Their Uses this chaining property to illustrate the meanings
emerging in the language games of interactions
Semiotics has been a fruitful theoretical lens used in an elementary mathematics classroom,
by researchers investigating diverse issues in involving the translation of signs into new signs.
mathematics education in recent decades, as
attested by Discussion Groups held at confer- Each of the relationships comprised in the
ences of the International Group for the Psychol- Peircean triad were analyzed by him into further
ogy of Mathematics Education (PME) in 2001, triads, e.g., the relationship of the representamen
2002, 2003, and 2004 (Sa´enz-Ludlow and to its object could be iconic (like a picture),
Presmeg 2006) and at conferences with a focus indexical (pointing to it in some way, e.g.,
on semiotics in mathematics education (Radford smoke to fire), or symbolic (a conventional rela-
et al. 2011). Some theoretical formulations are tionship, e.g., the numerals to their corresponding
described briefly in this section, along with natural numbers). This model also includes the
the mention of semiotic investigations in which need for expression or communication: “Expres-
sion is a kind of representation or signification.
A sign is a third mediating between the mind

S 540 Semiotics in Mathematics Education

addressed and the object represented” (Peirce (Steinbring 2005, 2006) to show that the meaning
1992, p. 281). In an act of communication, then, of signs for individual learners is part and parcel
there are three kinds of interpretant, as follows: the of the semiotic and epistemological functions
“Intensional Interpretant, which is a determination inherent in sign interpretation.
of the mind of the utterer”; the “Effectual
Interpretant, which is a determination of the mind Elaboration and combination of constructs
of the interpreter”; and the “Communicational from semiotic theories have been necessary in
Interpretant, or say the Cominterpretant, which is research addressing the complexity of elements
a determination of that mind into which the minds involved in mathematics teaching and learning.
of utterer and interpreter have to be fused in order For instance, Arzarello introduced the construct
that any communication should take place” semiotic bundles and Arzarello and Sabena
(Peirce 1998, p. 478). The latter fused mind Peirce (2011) integrated Toulmin’s structural descrip-
designated the commens, a notion that is useful in tion of arguments; Peirce’s notions of sign, dia-
interpreting developments in the history of math- grammatic reasoning, and abduction; and
ematics through the centuries (Presmeg 2003). Habermas’s model for rational behavior. Several
The numerous triads introduced by Peirce provide research studies have used the inclusive
lenses for various larger or smaller grains of onto-semiotic theoretical model of Godino and
analysis in research in mathematics education colleagues (e.g., Santi 2011). There is also the
(Bakker 2004; Hoffman 2006). important independent branch of semiosis
known as semiotic mediation, based on the theo-
With regard to mathematical communication, retical formulations of Vygotsky, and used exten-
a different theory is provided by the social sively in research by Mariotti (e.g., Falcade et al.
semiotics of linguist Michael Halliday, as used 2007) and Bartolini Bussi (e.g., Maschietto and
in the research of Morgan (2006), who analyzed Bartolini Bussi 2009). Hoffman (2006) does not
the mathematical texts produced by secondary consider this variety of theoretical formulations
school students. Halliday emphasized “the ways of semiosis to be a problem, as long as the termi-
in which language functions in our construction nology is consistently defined and used in each
and representation of our experience and of our instance. The various research questions being
social identities and relationships” (Morgan investigated demand different tools and lenses,
2006, p. 219). A fine grain is provided in this according to the various semiotic traditions.
theory by the differentiation of context of situa-
tion, involving various kinds of specific goals, Questions for Research on Semiosis in
and context of culture, involving organizing con- Learning and Teaching Mathematics
cepts that participants hold in common, and by his
notions of field (institutional setting of an activ- Following the publication of papers from two PME
ity), tenor (relations among the participants), and discussion groups in 2001 and 2002, Sa´enz-Ludlow
mode (written and oral forms of communication). and Presmeg (2006) identified semiotic “windows
through which to explain the teaching-learning
An independent model is provided by activity while opening the gates for new avenues
Steinbring (2005), who took the position that of research in mathematics education” (p. 9) by
mathematical signs have both semiotic and epis- addressing questions such as the following:
temological functions. With regard to a particular • What exactly is entailed in the interpretation of
mathematical concept, he argued that there is
a reciprocally supported and balanced system, signs? Are signs things and/or processes? When
which he called the epistemological triangle. are signs interpreted as things and when are
The three reference points of this triangle are signs interpreted as processes by the learner?
the mathematical sign/symbol, the object/refer- • What is the role of speech and social interac-
ence context, and the mathematical concept. He tion in the interpretation of signs? What is the
provided extensive examples of interaction of role of writing in this interpretation?
learners in elementary mathematics classrooms

Semiotics in Mathematics Education S541

• Are there different levels of sign interpreta- Bakker A (2004) Design research in statistics education. S
tion? Do interpretations and the level of inter- Unpublished PhD dissertation, Utrecht University,
pretations change with respect to different Utrecht
contexts? What is the role of different contexts
in sign interpretation? Colapietro VM (1993) Glossary of semiotics. Paragon
House, New York
• Is it important for the teacher and the student
to differentiate the variety of semiotic systems Duval R (2006) A cognitive analysis of problems of com-
involved in the teaching-learning activity? prehension in a learning of mathematics. Educ Stud
Math 61:103–131
• Is there a dialectical relationship between sign
use and sign interpretation? Is there a dialectical Falcade R, Laborde C, Mariotti MA (2007) Approaching
relationship between sign interpretation and functions: Cabri tools as instruments of semiotic medi-
thinking? ation. Educ Stud Math 66:317–333

• Is it possible to involve students in creative Fried M (2011) Signs for you and signs for me: the double
acts of sign invention and sign combination to aspect of semiotic perspectives. Educ Stud Math
encapsulate the oral or written expression of 77:389–397
their conceptualizations?
Hall M (2000) Bridging the gap between everyday and
• Under what conditions do students attain the classroom mathematics: an investigation of two
ability to express themselves flexibly in the teachers’ intentional use of semiotic chains.
conventional semiotic systems of mathematics? Unpublished Ph.D. dissertation, The Florida State
University
• Can various semiotic theories be applied to
analyze data gathered using different Hoffman M (2006) What is a “semiotic perspective”, and
methodologies? what could it be? Some comments on the contributions
to this special issue. Educ Stud Math 61:279–291
• Would it be possible to have a unified semiotic
framework in mathematics education? Maschietto M, Bartolini Bussi MG (2009) Working
The latter remains an open question. However, with artefacts: gestures, drawings and speech in the
construction of the mathematical meaning of the visual
some of the potential light thrown by using semi- pyramid. Educ Stud Math 70:143–157
otic lenses in mathematics education research
has been demonstrated in investigations already Morgan C (2006) What does social semiotics have to offer
undertaken. mathematics education research? Educ Stud Math
61:219–245
Cross-References
Otte M (2006) Mathematical epistemology from a Peirc-
▶ Argumentation in Mathematics Education ean semiotic point of view. Educ Stud Math 61:11–38
▶ Concept Development in Mathematics
Peirce CS (1992) The essential Peirce, vol 1. Houser N,
Education Kloesel C (eds). Indiana University Press,
▶ Discursive Approaches to Learning Bloomington

Mathematics Peirce CS (1998) The essential Peirce, vol 2. Peirce Edi-
▶ Mathematical Language tion Project (ed). Indiana University Press,
▶ Mathematical Representations Bloomington
▶ Theories of Learning Mathematics
Presmeg NC (1998) Ethnomathematics in teacher educa-
References tion. J Math Teach Educ 1(3):317–339

Arzarello F, Sabena C (2011) Semiotic and theoretic con- Presmeg N (2003) Ancient areas: a retrospective analysis
trol in argumentation and proof activities. Educ Stud of early history of geometry in light of Peirce’s
Math 77:189–206 “commens”. Svensk Fo¨rening fo¨r Matematik
Didaktisk Forskning, Medlemsblad 8:24–34

Presmeg NC (2006) Semiotics and the “connections”
standard: significance of semiotics for teachers of
mathematics. Educ Stud Math 61:163–182

Radford L, Schubring G, Seeger F (guest eds) (2011)
Signifying and meaning-making in mathematical
thinking, teaching and learning: Semiotic perspec-
tives. Special Issue, Educ Stud Math 77(2–3)

Sa´enz-Ludlow A (2006) Classroom interpreting games
with an illustration. Educ Stud Math 61:183–218

Sa´enz-Ludlow A, Presmeg N (guest eds) (2006) Semiotic
perspectives in mathematics education. A PME Spe-
cial Issue, Educ Stud Math 61(1–2)

Santi G (2011) Objectification and semiotic function.
Educ Stud Math 77:285–311

de Saussure F (1959) Course in general linguistics.
McGraw-Hill, New York

S 542 Shape and Space – Geometry Teaching and Learning

Steinbring H (2005) The construction of new mathemati- From its very beginning, more than two and
cal knowledge in classroom interaction: an epistemo- a half thousand years ago, geometry was devel-
logical perspective. Springer, New York oped along a few main aspects:
(a) Interacting with shapes in a space. This
Steinbring H (2006) What makes a sign a mathematical
sign? An epistemological perspective on mathematical aspect arose independently in a number of
interaction. Educ Stud Math 61:133–162 early cultures as a body of practical
knowledge concerning lengths, areas, and
Walkerdine V (1988) The mastery of reason: cognitive volumes and concerning shapes’ attributes
developments and the production of rationality. and the relationships among them (the
Routledge, New York practical-intuitive aspect).
(b) Shapes, their attributes, and their changes in
Whitson JA (1997) Cognition as a semiotic process: space as fundamental ingredients for
from situated mediation to critical reflective transcen- constructing a theory (the formal logic
dence. In: Kirshner D, Whitson JA (eds) Situated approach). Elements of a formal mathemati-
cognition: social, semiotic, and psychological perspec- cal geometry emerged in the west as early as
tives. Lawrence Erlbaum Associates, Mahweh Thales (sixth century BC). By the third
century BC, this aspect of geometry was put
Shape and Space – Geometry into an axiomatic structure by Euclid
Teaching and Learning (Euclidean geometry).
(c) Shapes as basis for reflecting on visual infor-
Rina Hershkowitz mation by representing, describing, general-
Department of Science Teaching, Weizmann izing, communicating, and documenting such
Institute, Rehovot, Israel information, e.g., for better understanding
concepts, processes, and phenomena in
Keywords different areas of mathematics and science
and as a framework for realizing the contri-
Shape’s critical attributes; Euclidean geometry; bution of mathematics to domains such
Intuitive to formal; Visualization; Mathematiza- as painting, sculpture, and architecture in
tion of the reality; Deduction; Concept definition; which beauty can be generated through aes-
Concept image; Child’s representational space; thetic configurations of geometrical shapes.
Internalization; Justification; Prototypical exam- There is a classic “consensus that the first two
ple; Prototypical judgment; Dragging operation; aspects are linked because some levels of
Uncertainty conditions geometry as the science of space are needed for
learning geometry as a logical structure”
Definition and Teaching Situation (Hershkowitz et al. 1990, p. 70). These two
aspects seemed to be expressed explicitly in
Geometry (Ancient Greek: geometr´ıa; geo teaching and learning geometry in schools and
“earth,” metron “measurement”) is a mathemat- in the research work which follows it. For quite
ical area concerned with the space around us, many years the most acceptable way to teach
with the shapes in the space, their properties, geometry in K-12 was and in a sense still is
and different “patterns” and “thinking pat- hierarchical division of the themes and teaching
terns” for which they serve as trigger and approaches from intuitive (Aspect a) to formal
basis. As Freudenthal (1973) states it: “Geom- (Aspect b) along the school’s years, where the
etry can only be meaningful if it exploits the intuitive-interactive approach was the basis for
relation of geometry to the experienced elementary and preschool geometry and the for-
space. . . Geometry is one of the best opportu- mal one was left to high school. Seldom, the
nities that exist to learn how to mathematise formal approach was also used for designing
reality” (p. 407). a learning environment for high school and/or

Shape and Space – Geometry Teaching and Learning S543

universities in which learners developed an environments for developing understanding of S
understanding of geometrical structures as geometry and space,” Lehrer and Chazan (1998)
abstract systems not necessarily linked to refer- writes: “ . . .geometry and spatial visualization in
ents of a real environment, e.g., the non- school are often compressed into a caricature of
Euclidian geometries. Greek geometry, generally reserved for the
second year of high school.” Indeed in many
Approaches towards the role of visualization states in the USA, this 1-year-course in Euclid
in learning and teaching geometry and mathemat- geometry was taught in high school without any
ics as a whole (Aspect c) varied according to the geometry’s instruction before it.
observers’ eyes and interest. But, as geometry
engaged with shapes in space, which are seen, This unfortunate situation was discussed
presented, and documented visually, the role of intensively in the last few decades and as a result
visualization can’t be ignored. Two aspects of instructional and research efforts are done in
visualization which are interweaved together are order to improve it. For example, in the US
relevant to teaching and learning geometry: NCTM curriculum standards, it is claimed that
(a) Visualization as one of the ways for mathe- “the study of geometry in grades 5–8 links the
informal explorations, begun in K-4, to the more
matical thinking formalized processes studied in grades 9-12”
(b) Visualization as a representation or as “a lan- (NCTM 1989, p. 112). This intentional claim
(which unfortunately does not mention the visual
guage” by which mathematical thinking, includ- aspect) is strengthened by the hierarchical levels’
ing a visual one, might be developed, limited, structure of van Hiele’s theory (1958), which is
expressed, and communicated (Presmeg 2006) discussed in the next section.
Visual constructs are considered as a potential
support for learning other mathematical Theories Concerning Geometry
constructs, but what about geometrical con- Teaching and Learning
structs? Visualization seems to be the entrance
into geometry, the first internalization steps of the Piaget: In his developmental theories of the
learner while she/he begins to mathematize the child’s conception of space (Piaget and Inhalder
reality into geometrical constructs. There were 1967) and child’s conception of geometry (Piaget
quite many research works which were involved et al. 1960), Piaget and his colleagues describe
with visualization, but not very many that tried to the development of the child’s representational
investigate to what extent geometrical thinking is space. This is defined as the mental image of the
visual, or is interweaved with visual thinking, or real space in which the child is acting, where
affected by visual thinking? For example, when mental representation is an active reconstruction
the learner is engaged in deductive proving, what of an object at the symbolic level. Piaget in his
is the effect or the role of visual thinking if any? typical way was interested in the mental transfor-
Or, the opposite, when students are engaged in mations from the real space to the child’s represen-
a visual problem solving, what semiotic support tational space and in those attributes of real objects
they need and may have for expressing that are invariant under these transformations
their problem solving process and products? and how they develop with age. This approach
This third aspect concerning the role of visuali- is a trigger to some sub-theories. For example,
zation is the most neglected one, either because of the distinction between the concept and the concept
the lack of awareness or because of the na¨ıve image. The concept is derived from its mathemat-
assumption that the visual abilities and under- ical definition and the concept image which is the
standing are developed in a natural way and the collection of the – mental images the student has
learners do not need a special teaching. concerning the concept, or the concept as it is
The teaching and learning of geometry in reflected in the individual mind (Vinner 1983).
preschool and elementary school was neglected
in many places around the world: For example, at
the preface to their book concerning “learning

S 544 Shape and Space – Geometry Teaching and Learning

This dichotomy served as a basis for many research Freedom in Selecting Geometrical Context,
works in mathematics. Content, and Teaching/Learning Paradigms:
As a result from the above criticism, we are
van Hiele: Whereas Piagetian theory relates witnesses in the last decades to a trend of refresh-
mainly to geometry as the science of space, van ing projects in teaching and learning geometry as
Hiele’s theory combines geometry as the science a whole, but mostly at the preschool and elemen-
of space and geometry as a tool with which to tary school. These projects, which express
demonstrate mathematical structure. The theory democracy in choosing contexts, and approaches
identifies a sequence of levels of geometrical towards teaching and learning geometry,
thought from recognition and visualization up to emerged from holistic vision of what shape and
rigor (for details on the theory as a whole and on space could be, rather of what they often are in
the levels in particular, see Van-Hiele and schools (see the RME entry in this encyclopedia).
van-Hiele-Geldof 1958; Hershkowitz et al. The book edited by Lehrer and Chazan (1998) is
1990). The most relevant feature for geometry a paradigmatic example for this trend. The book
instruction and learning is van Hiele’s claim describes a variety of attractive and productive
that the development of the individual’s geomet- environments for learning about space and geom-
rical thinking, from one level to the next, is due to etry. In most of the designed learning environ-
teaching and learning experiences and does not ments, described in the book, students play active
depend much on maturity. role in constructing their own geometrical knowl-
edge. The designers’/researchers’ description of
Geometry in Preschool and student’s learning shares a collective emphasis
Elementary School on internalization, mathematization, and justifica-
tion (Hershkowitz 1998). Internalization is used in
The aspect of interaction with shape and space a Vygotskian spirit, as the transformation of exter-
(a) is the main component in elementary and nal activity into internal activity, e.g., the change
preschool geometry learning in which the clas- from “what I see?” to “how I see?” in accordance
sic main goals are constructing knowledge with the change of the observer’s position in the
about basic Euclidean geometric figures and RME curriculum (Gravemeijer 1998) and the
simple relationships among them. Around the dragging mode in dynamic geometry projects.
middle of the twentieth century, it was com- Mathematization is consistent with Freudenthal’s
mon to find a “pre-formal” course for elemen- philosophy of mathematics as human activity in
tary school, conceptualizing geometry as the which mathematizing is seen as a sort of an orga-
science of space. The focus of this course was nizing process by which elements of a context are
mostly on the identification and drawing of the transformed into mathematical objects and rela-
regular shapes, Euclidean properties of these tions. Justification is taken in a broad sense, mean-
shapes, relationships among shapes, and ing the variety of actions that students take in order
a variety of measurement activities. Since the to explain to others, as well as to themselves, what
1960s this type of course has come under they see, do, think, and why. This broad sense is
severe criticism: mainly because it lacks induc- expressed in the mathematical and cognitive free-
tive activities related to search of patterns, dom towards legitimate kinds of justifications.
because there is no implicit and explicit focus
on geometrical argumentation, and above all Some Comments on Difficulties and Relevant
because the learners are passive in constructing Research: Research has shown that common
their geometrical knowledge. It is worth to note difficulties of learning geometry at the elemen-
that teaching visual skills and visual thinking tary level emerged mostly from the unique
which is highly recommended (c) is still very mathematical structures, in which figures are
limited. represented in learning geometry: From a
mathematical point of view a geometrical

Shape and Space – Geometry Teaching and Learning S545

concept, like other mathematical concepts, is difficulties and more. By dragging elements of a S
derived from its definition which includes a drawing which was constructed geometrically on
minimal (necessary and sufficient) set of the con- the computer screen, students may provide an
cept’s critical attributes that an instance should infinite set of drawings of the same figure. This
have in order to be the concept’s example. Hence variable method of displaying a geometrical
these critical attributes may be used by students entity stresses the critical attributes, which
as a criterion to classify instances. In contrast, become the invariants of the entity under
students very often use one special example of dragging. Research indicates that students
the concept, the prototype/s as a criterion for engaged in dynamic geometry tasks are able to
classifying other examples. The prototypes are capitalize on the ambiguity of drawings in the
attained first and therefore are found in the con- learning of geometrical concepts.
cepts’ image of quite young learners. The proto-
typical example has the “longest” list of critical High-School Geometry or Shapes in
attributes (Rosch and Mervis 1975), e.g., the Space as Ingredients for Constructing
squares are prototypical example of quadrilat- a Theory
erals, and indeed they have all the quadrilaterals’
critical attributes plus their own critical attri- The two classical roles of teaching high-school
butes, like the sides’ equality. This leads to a geometry are still experiencing deductive reason-
prototypical judgment by learners and to a ing and proofs as part of human culture and
creation of biased concept images, like identify- human thinking and verifying the universality of
ing a segment as a triangle altitude, only if it is an proved geometric statements. According to
interior segment (Hershkowitz et al. 1990; Fujita extreme classical approach, experimenting,
and Jones 2007). The prototypes’ phenomenon is visualizing, measuring, inductive reasoning, and
understood better if we analyze it in the context of checking examples are not counted as valid
the typical structure of basic geometrical figures, arguments and might be that this is the reason
“the opposing directions inclusion relation- for neglecting them both in the elementary school
ships” (Hershkowitz et al. 1990), among the and in the high-school level. Geometrical proofs
sets of figures (concepts) at one direction and are considered to be on a high level of the
among their critical attributes at the opposite argumentative thinking continuum at school, and
direction. This structure explains also other the traditional high-school geometry is the
obstacles in learning geometry: e.g., the figure- essence of the secondary school geometry in
drawing obstacle (Laborde 1993, p. 49), in which many places. It starts from what can be seen with
learning difficulty emerges in situations where an the eyes, where space and shapes provide the
isolated drawing is the only representative of a environment, in which the learner gets the feeling
figure, where the figure is the geometrical con- of mathematical theory (Freudenthal 1973). At
cept as a whole. Laborde made it clear that there more advanced stage, it acquires a more abstract
is always a gap between the figure and a drawing aspect. But, even in the most abstract stage, we
which represents it, because (1) some properties still deal with some sorts of shapes and spaces,
of the drawing are irrelevant (non-critical attri- even when they can be seen with the “minds’ eye”
butes of the figure) (it becomes an obstacle when only. Nevertheless the trends of freedom towards
students try to impose these attributes as critical the meaning of justifications and towards para-
attributes on all figures’ examples), (2) the ele- digms of teaching and learning mathematics as
ments of the figure have a variability which is a whole and geometry in particular are taking
absent in a single drawing, and (3) a single draw- their way into the traditional high-school course
ing may represent various figures (Yerushalmi in geometry. Euclidian geometry is no longer
and Chazan 1990). Dynamic geometry softwares discussed in terms of “Euclid must stay” or
enable students to overcome the abovementioned

S 546 Shape and Space – Geometry Teaching and Learning

“must go,” as if it is the only representative for and understanding and less in the formal clas-
a proper argumentation on the stage. This trend is sic way in which it presented.
accelerated due to several reasons: iii. Dynamic geometry environments and
i. The difficulty of teaching proving tasks in proving: The design of dynamic geometry
learning environments raised a question
school and of understanding the role of about the place of the classical proof in the
proof: The teaching of mathematical proof curriculum, since conviction can be obtained
appears to be a failure in almost all countries quickly and relatively easily: The dragging
(Balacheff 1991; Mariotti 2006). Moreover, operation on a geometrical object enables stu-
students rarely see the point of proving. dents to apprehend a whole class of objects in
Balacheff (1991) claims that if students do which the conjectured attribute is invariant
not engage in proving processes, it is not so and hence to be convinced of its truth. The
much because they are not able to do so, but role of proof is then to provide the means to
rather they do not see any reason for it (p. 180). state the conjecture as a theorem. Dreyfus and
High-school students, even in advanced Hadas (1996) argue that students’ apprecia-
mathematics and science classes, don’t realize tion of the roles of proof can be achieved by
that a formal proof confers universal validity to activities in which the empirical investigations
a statement. A large percentage of students lead to unexpected, surprising situations. This
states that checking more examples is desirable surprise is the trigger for the question why and
(Fischbein and Kedem 1982). Many do not for the proof as an answer to this question.
distinguish between evidence and deductive
proof as a way of knowing that a geometrical Cross-References
statement is true. After a full course of
deductive geometry, most students don’t see ▶ Argumentation in Mathematics Education
the point of using deductive reasoning in geo- ▶ Concept Development in Mathematics
metrical constructions and remain still naive
empiricists whose approach to constructions Education
is an empirical guess-and-test loop (Schoenfeld ▶ Deductive Reasoning in Mathematics
1986).
ii. New thinking trends concerning the goals of Education
teaching proofs: For mathematicians, proofs ▶ Logic in Mathematics Education
play an essential role in establishing the valid- ▶ Mathematical Proof, Argumentation, and
ity of a statement and in enlightening its
meaning. In the last decades more and Reasoning
more scholars claim that the situation in ▶ Realistic Mathematics Education
school is different: Hanna (1990) suggested ▶ The van Hiele Theory
distinguishing, in school geometry, between ▶ Visualization and Learning in Mathematics
(a) proofs that only show that the theorem is
true and (b) proofs that in addition explain and Education
convince why the theorem is true. Dreyfus and
Hadas (1996) showed that when a learning References
situation is provided, in which students feel
the need in proof in order to be convinced and Balacheff N (1991) The benefits and limits of social inter-
convince others (e.g., the need to show the action: the case of mathematical proof. In: Bishop AJ
existence of hypothesis), students search for et al (eds) Mathematical knowledge: its growth
a proper proof and then a proof becomes a through teaching. Kluwer, Dordrecht, pp 175–192
meaningful mathematical tool for checking
hypothesis. In these ways the importance of Dreyfus T, Hadas N (1996) Proof as answer to the question
proof is focused in the level of its justifications why. Zentralblatt fu€r Didaktik der Matematik (ZDM)
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Fischbein E, Kedem I (1982) Proof and certitude
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matics education. Antwerp, Belgium, pp 128–131 Walters
Freudenthal H (1973) Mathematics as an educational task. Vinner S (1983) Concept definition concept image and the
Reidel, Dordrecht notion of function. Int J Math Educ Sci Technol
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quadrilaterals: towards a theoretical framing. Res obstacles with the aid of the supposer. Educ Stud
Math Educ 9:3–20 Math 21:199–219
Gravemeijer KP (1998) From a different perspective:
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Chazan D (eds) Designing learning environments for
developing understanding of geometry and space. Gilah Leder
Lawrence Earlbaum Associates, Mahwah, pp 45–66 Faculty of Education, Monash University,
Hanna G (1990) Some pedagogical aspects of proof. Inter- VIC, Australia
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Hershkowitz R (1998) Epilogue – organization and Keywords
freedom in geometry learning and teaching. In: Lehrer
R, Chazan D (eds) Designing learning environments Single-sex classroom; Single-sex school; Single-
for developing understanding of geometry and space. gender classroom; Single-gender school
Lawrence Erlbaum, Mahwah, pp 489–494
Hershkowitz R, Ben-Chaim D, Hoyles C, Lappan G, Introduction
Mitchelmore M, Vinner S (1990) Psychological
aspects of learning geometry. In: Nesher P, Kilpatrick In recent years it has become popular to replace
J (eds) Mathematics and cognition. Cambridge the word “sex” in single-sex classrooms and
University Press, Cambridge, pp 70–95 single-sex schools with the word “gender.”
Laborde C (1993) The computer as a part of learning This substitution warrants some attention for
environment: the case of geometry. In: Keitel C and “(p)recision is essential in scientific writing”
Ruthven K (eds) Learning from computers: mathemat- (American Psychological Association [APA]
ics education and technology. Springer Verlag, Nato 2010, p. 71).
ASI Series, pp 48–67
Lehrer R, Chazan D (1998) Designing learning The APA (2010) advocates the use of the term
environments for developing understanding of geome- “Gender . . .when referring to women and men as
try and space. Lawrence Erlbaum Associates, Hillsdale social groups. Sex is biological; use it when the
Mariotti MA (2006) Proof and proving in mathematics biological distinction is predominant” (p. 71).
education. In: Gutiererez A, Boero P (eds) Handbook A similar distinction is made by the World Health
of research on the psychology of mathematics Organization (WHO) (2012): “‘Sex’ refers to the
education. Sense, Rotterdam, pp 173–204 biological and physiological characteristics that
National Council of Teachers of Mathematics (1989) define men and women. ‘Gender’ . . . to the
Curriculum and evaluation standards for school socially constructed roles, behaviours, activities,
mathematics. NCTM, Reston and attributes that a given society considers
Piaget J, Inhalder B (1967) The child’s conception of appropriate for men and women.” Given that the
space. Norton, New York division of students into same-sex classrooms is
Piaget J, Inhalder B, Szeminska A (1960) The child’s invariably based on biological characteristics, it
conception of geometry. Routledge & Kegan Paul, is appropriate to retain the term sex in the heading
London of this contribution. However, explanations
Presmeg NC (2006) Research on visualization in learning for differences associated with single-sex
and teaching mathematics: emergence from psychol-
ogy. In: Gutierrez A, Boero P (eds) Handbook of
research on the psychology of mathematics education.
Sense, Rotterdam, pp 205–235
Rosch E, Mervis CB (1975) Family resemblances: studies
in the internal structure of categories. Cogn Psychol
7:605–773
Schoenfeld A (1986) On having and using geometric
knowledge. In: Hiebert J (ed) Conceptual and
procedural knowledge: the case of mathematics.
Lawrence Erlbaum Associates, Hillsdale, pp 225–264
Van-Hiele PM, van-Hiele-Geldof D (1958) A method of
initiation into geometry. In: Freudenthal H (ed) Report

S 548 Single-Sex Mathematics Classrooms

groupings are commonly linked to social expec- favor those in the single-sex school setting con-
tations, perceptions, and conventions, that is, on tinue to be replicated. Referring to England and
gender-linked differences, as defined by APA Wales, Thompson and Ungerleider (2004, p. 4)
(2010) and WHO (2012). For a more detailed pointed to the increased publicity given to school
discussion, see Leder (1992). examination results which publicize the consis-
tent and superior achievements of students grad-
Historically, more emphasis has been placed uating from single-sex private and independent
on the education of boys than of girls and origi- schools, with many of the highest scores coming
nally all-boys schools predominated. Over time, from all-girls schools. In addition to a possible
with increased expectations and demands for solution for the achievement gap, single-sex
mass education, coeducational schools were schooling is viewed in some jurisdictions as
added in many countries. In some countries, a means of balancing enrolments in subject
religious convictions have been, and are still, areas within the coeducational public system in
responsible for sex-segregated education – for which there have been extreme imbalances
example, in the predominantly Muslim countries between boys and girls.
such as Bahrain, Iran, and Saudi Arabia. In many
others, economic and political considerations, as Interpreting the finding of girls’ better perfor-
well as the increased importance attached to the mance in mathematics in single-sex schools is
education of girls, have led to the growth and problematic, however. When other factors are
ultimately dominance of coeducational schools. taken into account and in particular the fact that
In contrast, in the United States of America with single-sex schools are often independent, private
its strong history of coeducational schools, there schools which attract students from higher socio-
appears to have been a revival of single-sex economic families, it is clear that any advantages
schooling, fuelled by legislative changes. This noted cannot be attributed simplistically to the
development is hotly and continuously deplored single-sex composition of the school. Data from
and contested by many inside and outside large-scale, international mathematics tests such
educational circles (see, e.g., Brown 2011; as TIMSS and PISA illustrate unambiguously
Halpern et al. (2011, 2012)). that students’ socioeconomic background is an
important variable influencing their performance
Characteristics in mathematics: in general, the higher the level of
the socioeconomic background of students, the
Mathematics Classes in Single-Sex Schools higher their performance on the mathematics
Unlike the often short life span of single-sex component of these tests. Factors beyond
mathematics classes in coeducational schools, students’ background and system-related differ-
the single-sex grouping is maintained throughout ences in human and physical resources have also
the school life of students in single-sex schools. been shown to contribute to different achieve-
The mathematics performance and participation ment outcomes. In summary, any apparent
rates of boys attending single-sex schools have achievement advantages found in mathematics
attracted some attention, but, as noted in a report learning for girls attending a single-sex school
by the US Department of Education (2005, p. xv), cannot be attributed simplistically to one partic-
“males continue to be underrepresented in this ular school characteristic, that is, the single-sex
realm of research.” Issues related to girls’ learn- setting per se.
ing of mathematics have been a major focus of
research comparing benefits of single-sex and Longitudinal studies of the long-term impact
coeducational schools. Findings reported some of single-sex schooling are rare. Data from the
two decades ago by Leder (1992) that, when National Child Development Study “a longitudi-
differences are found, they most frequently nal study of a single cohort born in a particular
week in 1958 in Britain” (p. 314) offers one such
source (Sullivan et al. 2011). This group attended

Single-Sex Mathematics Classrooms S549

secondary school in the 1970s, at a time when background factors or school organizational S
about one-quarter of the cohort went to single-sex structures rather than the sex-segregated setting
schools – a much higher proportion than is now per se (Forgasz and Leder 2011).
the case. Sullivan et al. (2011, p. 311) report:
In most of the studies located, the focus was on
We find no net impact of single-sex schooling the shorter-term effects of the single-sex grouping.
on the chances of being employed in 2000, nor on In the few studies in which longer-term effects
the horizontal or social class segregation of mid- were examined, earlier advantages attributed to
life occupations. But we do find a positive pre- the single-sex grouping appeared to dissipate:
mium (5 %) on the wages of women (but not “The generally accepted view has been that for
men), of having attended a single-sex school. females, single-sex schooling is more advanta-
This was accounted for by the relatively good geous” (OECD 2009, p. 44). Yet nuanced explo-
performance of girls-only school students in rations of PISA data do “not uniformly support the
post-16 qualifications (including mathematics). notion that females tend to do better in a single-
sex environment” (OECD 2009, p. 45).
Mathematics Classes in Sex-Segregated
Classes in Coeducational Schools Consistent explanations for the equivocal find-
In many countries, systematic documentation of ings permeate the relevant scholarly literature:
differences in the mathematics achievement of certain groups of students (e.g., those being
boys and girls began in the early 1970s. Given harassed in a coeducational setting) were found
the important gatekeeping role or critical filter to benefit from a single-sex environment, while for
played by mathematics into further educational other groups it made no difference. Teacher strat-
and career opportunities, differences between egies, instructional materials, and the prevailing
the two groups, in favor of boys, in continued school climate, rather than the sex grouping in the
participation in advanced and post-compulsory mathematics class, were more often found to be
mathematics courses were also noted with con- critical to students’ success and perceptions of the
cern. The introduction of single-sex classes in class environment. Simplified and at times biased
coeducational schools, mostly aimed at secondary versions of these findings are regularly reported in
school students and not necessarily exclusively in the popular media and play a part in shaping the
mathematics, was among the initiatives mounted perceptions of the public and of stakeholders about
to redress the demonstrated achievement discrep- the respective benefits of single-sex and coeduca-
ancies. The move was considered to be consistent tion schooling (Forgasz and Leder 2011).
with the tenets of liberal feminism, that is, helping
females attain achievements equal to those of To conclude, many complex and interacting
males, and the apparent advantages for girls asso- factors influence the school learning environ-
ciated with the learning of mathematics in single- ment – with a single-sex classroom setting per
sex schools. se unlikely to be the most influential. Some con-
texts, including the primary years of schooling
The findings reported from single-sex mathe- and the longer-term effect of learning mathemat-
matics classes in formally coeducational schools ics in a single-sex rather than a coeducational
are largely similar to those described for other setting, have not yet received sufficient attention.
subject areas. Girls typically liked the single-sex For the present, proponents of single-sex educa-
setting and performed somewhat better academi- tion will focus on its putative benefits and critics
cally than in coeducational classes. In a number on its disadvantages.
of the studies surveyed, boys were more ambiva-
lent than girls about the single-sex setting with Cross-References
some indicating a firm preference for coeduca-
tional classes. These differences, however, ▶ Cultural Influences in Mathematics Education
could often be attributed to differences in student ▶ Equity and Access in Mathematics Education

S 550 Situated Cognition in Mathematics Education

▶ Gender in Mathematics Education Situated Cognition in Mathematics
▶ Mathematical Ability Education
▶ Socioeconomic Class in Mathematics
John Monaghan
Education School of Education, University of Leeds,
Leeds, UK
References

American Psychological Association (2010) Publication Keywords
manual of the american psychological association,
6th edn. Author, Washington Context; Knowing/knowledge; Learning; Partic-
ipation; Situation; Transfer
Brown CS (2011) Legal issues surrounding single-sex
schools in the U.S.: trends, court cases, and conflicting Introduction
laws. Sex Roles, Advance online publication. DOI
10.1007/s11199-011-0001-x “Situated cognition” is a loose term for a variety
of approaches, in education and in other fields of
Forgasz HJ, Leder GC (2011) Equity and quality inquiry, that value context. Its advocates claim
of mathematics education: research and media that how one thinks is tied to a situation. “Situa-
portrayals. In: Atweh B, Graven M, Secada W, tion” is another loose term; it may refer to a place
Valero P (eds) Mapping quality and equity in (a classroom or a laboratory), but a situation may
mathematics education. Springer, Dordrecht, also reside in relationships with people and/or
pp 205–222 artifacts, e.g., “I am with friends” and “I am at
my computer.” This entry briefly considers the
Halpern DF, Eliot L, Bigler RS, Fabes RA, history of situated approaches before looking
Hanish LD, Hyde J, Liben LS, Martin CL (2011) The - at the development of situated schools of thought
pseudoscience of single-sex schooling. Science in mathematics education. It then considers
333:1706–1707 “knowing” and, briefly, research methodologies,
implication for teaching, and critiques of situated
Halpern DF, Eliot L, Bigler RS, Fabes RA, Hanish LD, cognition.
Hyde J, Liben LS, Martin CL (2012) Response. Sci-
ence 335:166–168 Characteristics

Leder GC (1992) Mathematics and gender: changing per- History
spectives. In: Grouws DA (ed) Handbook of research Marx’s 11th thesis on Feuerbach, “Social life
on mathematics teaching and learning. Macmillan, is essentially practical. All mysteries . . . find
New York, pp 597–622 their rational solution in human practice
and in the comprehension of this practice.”
OECD [The Organisation for Economic Co-operation (Marx 1845/1968, p. 30), remains a statement
and Development] (2009) PISA equally prepared that few, if any, situated cognitivists would
for life? How 15-year-old boys and girls perform disagree with. Activity theory is an explicitly
in school. Retrieved from http://www.oecd.org/pisa/ Marxist approach used by some mathematics
pisaproducts/pisa2006/equallypreparedforlifehow education researchers which could be called
15-year-oldboysandgirlsperforminschool.htm

Sullivan A, Joshi H, Leonard D (2011) Single-sex school-
ing and labour market outcomes. Oxford Rev Educ
37(3):311–332

Thompson T, Ungerleider C (2004) Single sex schooling.
Final report. Canadian Centre for Knowledge
Mobilisation. Retrieved from http://www.cckm.ca/
pdf/SSS%20Final%20Report.pdf

U.S. Department of Education, Office of Planning, Eval-
uation and Policy Development, Policy and Program
Studies Service (2005). Single-sex versus secondary
schooling: a systematic review, Washington, D.C.,
2005. Retrieved from http://www.ed.gov/about/
offices/list/opepd/reports.html

World Health Organization (2012) What do we mean by
“sex” and “gender? Retrieved from http://www.who.
int/gender/whatisgender/en

Situated Cognition in Mathematics Education S551

“situated”; there are similarities and differences viewpoint, a comment on keeping research on S
between these two approaches, e.g., “mediation” learning within Cole’s “first psychology.”
is central to activity theory but it is not a well-
developed construct in current approaches Mathematics Education
labelled as “situated cognition” (see Kanes and In mathematics education, “situated cognition” is
Lerman 2007). often associated with studies of out-of-school
mathematics towards the end of the twentieth
Cole (1996) claims that psychology once had century, Lave (1988) and Nunes et al. (1993)
two parts, one that could and one that could not be being early and influential examples of such stud-
studied in laboratory experiments. He argues that ies. These studies presented data that people
the second part was lost in most of the twentieth- could do mathematics “better” in supermarkets
century psychology. One reason for this loss is or on the streets and argued that the mathematical
that Western social sciences in the first half of the processes carried out in out-of-school activities
twentieth century were dominated by various were radically different from those of school
forms of positivism, such as behaviorism in psy- mathematics. These studies directly challenged
chology (with a knock-on effect in education). the rationalist hegemony of academic (Western)
Positivism is a form of empiricism which posits mathematics and argued that a strong discontinu-
that we can obtain objective knowledge, a claim ity exists between school and out-of-school
that is anathema to situated cognitivists. To a mathematical practices.
behaviorist, learning concerns conditioning,
responses to stimuli, and attaching responses to According to Lave (1988), this discontinuity is
environmental stimuli. From the 1950s onwards a consequence of the fact that learning in and
JJ and EJ Gibson, in the psychology of percep- learning out of school are different social
tion, argued differently that perceptual learning practices. School mathematics is, indeed, often
was a part of an agent’s interaction with ill-suited to out-of-school practices; in some
the environment; environments afford cases the problems which arise in out-of-school
animals some actions/activities and constrain mathematics are only apparently similar to
others – a chalkboard affords the construction of school mathematics problems, but in reality
static geometric figures but an electronic white- there is a range of explicit and implicit restric-
board may afford the construction of dynamic tions which makes school methods unsuitable,
geometric figures. This can be viewed as a form and thus other methods are used (Masingila
of situated cognition (where the situation is the et al. 1996). Despite the evident discontinuity,
environment) which has influenced some some authors who do value context (situation)
research in mathematics education; see Greeno have observed an interplay between school and
(1994) for a consideration of affordances with out-of-school mathematics: Saxe (1991) found
reference to “situation theory” and mathematical evidence that school mathematics and the
reasoning. The waning of behaviorism as an mathematics of street children’s candy-selling
academic paradigm in the West, circa 1970, practice in Brazil influence each other; Magajna
however, did not immediately usher Cole’s and Monaghan (2003) found evidence that, in
second psychology. In the place of behaviorism, making sense of their practice, CAD-CAM tech-
mathematics education researchers largely nicians resorted to a form of school mathematics.
embraced cognitive models of learning such as
Piaget’s genetic epistemology and information Knowing
processing, both of which were content to capture There are many constructs associated with
data in laboratory conditions; note that this situated cognition: community of practice (CoP),
comment is not necessarily a criticism of these (legitimate peripheral) participation, boundaries,
models per se but, rather, from a situated reification, and identity. This entry does not have

S 552 Situated Cognition in Mathematics Education

space to consider these separately but they are all required in a new task is basically the same as
tied up with a central theme of knowing. knowledge acquired in a previous task”) is
a legitimate object of study for purely cognitive
The verb “knowing” rather than the noun psychologists. It is a myth to radical expositions
“knowledge” is the subheading of this section of situated cognition such as Lave (1988). View-
because situated cognitivists view this thing ing transfer as a myth can be quite upsetting for
(knowledge/knowing) as something which practical mathematics educators who might well
results from doing (participation) rather than a turn to their academic-situated colleagues and
passively acquired entity. Rather like the say “what, then, is the point in teaching?”
Gibsons’ affordances, knowing is not an Engle (2006), however, presents a situated view
absolute attribute but a relative product of of transfer as “framing” – a means of interpreting
animal-environment interaction. The “person” phenomena. Engle (2006) examines a long-term
in these interactions is equally not a laboratory science learning and teaching sequence with
subject but a whole person with goals and views regard to learner construction of content and
of themselves in relation to the CoP in which they teacher framing of the contexts of learning in
participate (hence the relevance of “identity”); terms of time, “making references to both past
such views have obvious relevance for mathe- contexts and imagined future ones . . .[to] make it
matics education studies of classroom behaviors clear to students that they are not just getting
in terms of students’ self-conceptions as “a good current tasks done, but are preparing for future
student,” “cool,” etc. But, as Kanes and Lerman learning” (456), and forms of learner participa-
(2007) point out, there are different nuances on tion. This view of transfer is far removed from the
“learning” within the situated cognition camp: a (non-situated) cognitivist view of “transfer of
view that learning is a process that may or may knowledge” and has potential for mathematics
not result from being a member of a CoP and a education, e.g., the framing of tool use in mathe-
view that learning is subordinate to social matics learning to promote intercontexuality.
processes, “learning is an integral part of gener-
ative social practice in the lived-in world” Research Methodologies
(Lave and Wenger 1991, p. 35). There is no research method specifically
associated with situated cognition although
Situated cognitivists views on knowing methods used will be primarily qualitative and,
emerged partly in exasperation with dominant possibly, mixed methods; it is hard to imagine
cognitive (non-situated) positions on knowledge: how one might research being and knowing in
mathematicized situations using only quantita-
the effect on cognitive research of “locating” prob- tive methods. Qualitative methods used in “situ-
lems in “knowledge domains” has been to separate ated research” hopefully suit the focus of the
the study of problem solving from analysis of the research. For example, it was noted above that
situations in which it occurs . . . “knowledge Kanes and Lerman (2007) point out different
domain” is a socially constructed exoticum, that nuances on “learning” within situated research
is, it lies at the intersection of the myth of decontex- and a focus on learning from being a member of
tualized understanding and professional/academic a CoP may call for discourse analysis, and a focus
specializations. (Lave 1988, p. 42) on learning as an integral part of generative social
practice in the lived-in world may call for
Contrasts between situated and cognitive ethnographic approaches.
views on knowing/knowledge have important
implication for the construct “transfer of knowl- Implication for Teaching
edge” (or “transfer” for short), which is arguably Situated cognition is an approach to understand-
the philosopher’s stone of mathematics education ing knowing and does not prescribe a teaching
research. To be fair to all, there are few serious approach. That said, reflection on situated
researchers around of any persuasion who do not
regard “transfer” as a highly problematic con-
struct. Nevertheless, “transfer” (under the right
conditions, which usually means “knowledge

Socioeconomic Class in Mathematics Education S553

cognition can be useful for teachers and teacher Greeno J (1994) Gibson’s affordances. Psychol Rev S
educators to critique their thinking about learning 101(2):336–342
and teaching, as Winbourne and Watson (1998)
do. They recognize the problems of students’ Greiffenhagen C, Sharrock W (2008) School mathematics
school mathematical experience en bloc of pro- and its everyday other? Revisiting Lave’s ‘cognition in
viding a site for students to participate in a com- practice’. Educ Stud Math 69:1–21
munity of mathematicians but provide examples
of lessons which could (and could not) be termed Kanes C, Lerman S (2007) Analysing concepts of com-
“local communities of (mathematical) practice” munity of practice. In: Watson A, Winbourne P (eds)
(p. 95), where the teachers “orchestrated” student New directions for situated cognition in mathematics
participation so that student and teacher engage- education. Springer, New York, pp 303–328
ment with mathematics, rather than simple student
behavioral compliance, was essential for the activ- Lave J (1988) Cognition in practice. Cambridge Univer-
ity in the lessons. sity Press, Cambridge

Critiques of Situated Cognition Lave J, Wenger E (1991) Situated learning: Legitimate
There is no shortage of critiques since situated peripheral participation. Cambridge University Press,
cognition has courted controversy since the pub- Cambridge
lication of Lave (1988). These include “situated
friendly” critiques such as Walkerdine (1997) Magajna Z, Monaghan J (2003) Advanced mathematical
which suggests that the regulation of individuals thinking in a technological workplace. Educ Stud
in discursive practice is not developed in Lave’s Math 52(2):101–122
work; attacks on the basic claims of situated
cognition, such as Anderson et al. (1996); and Marx K (1845/1968) Theses on Feuerbach. In: Karl Marx
questioning the existence of claims that a strong and Frederick Engels: selected works in one volume.
discontinuity exists between school and out-of- Lawrence and Wishart, London, pp 28-30
school mathematical practice (Greiffenhagen and
Sharrock 2008). Masingila J, Davidenko S, Prus-Wisniowska E (1996)
Mathematics learning and practice in and out of
Cross-References school: a framework for connecting these experiences.
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Saxe G (1991) Culture and cognitive development: studies
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▶ Mathematization as Social Process Hillsdale
▶ Theories of Learning Mathematics
Walkerdine V (1997) Redefining the subject in situated
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Anderson JR, Reder LM, Simon HA (1996) Situated perspectives. Lawrence Erlbaum, Mahwah, pp 57–70
learning and education. Educ Res 25(4):5–11
Winbourne P, Watson A (1998) Participating in
Cole M (1996) Cultural psychology: a once and future learning mathematics through shared local
discipline. Harvard University Press, Cambridge practices in classrooms. In: Watson A (ed) Situated
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nity of learners classroom. J Learn Sci 15(4):451–498
Socioeconomic Class in Mathematics
Education

Stephen Lerman
Department of Education, Centre for
Mathematics Education, London South Bank
University, London, UK

Keywords

Social class; Achievement; Equity; Assessment

S 554 Socioeconomic Class in Mathematics Education

Definition research and thus do not impact sufficiently on
mathematics education researchers, that social
Research on the relationship between social class categories of students are unquestioned, and that
and socioeconomic status and achievement in the labelling masks the fact that the populations
mathematics. with lowest achievement are the poor and ethnic
minorities.
Overview
Given the timing of the handbook in which
In many countries around the world, a correlation Secada’s chapter appears, it is to be expected that
is found between working class or low socioeco- he would have high hopes for the reform agenda
nomic status (SES) and achievement in in the USA in terms of equity. Lubienski (2000)
mathematics. Secada (1992) traces recognition found, however, that there is evidence that disad-
of the connection between social class, race, eth- vantaged students taught through the reform
nicity, and other characteristics, with achieve- pedagogy are still underachieving in national
ment in education to the United States Supreme tests. She draws on sociological theory to explain
Court case of Brown versus Board of Education why the middle classes succeed whatever reforms
in 1954. The major work on achievement in take place, though that work has been challenged
mathematics and social characteristics was (Boaler 2003) on the basis of a study of forms of
begun in the mid-to late 1980s and has been pedagogy that can claim to have been successful
a focus of a growing body of work ever since, in equity terms. Nevertheless, as a general trend,
with new theoretical perspectives developing such reproduction of advantage and disadvantage
(Valero and Zevenbergen 2004) and with needs explanation; this is addressed below.
new forums for dissemination and publication
(e.g., the Political Dimensions of Mathematics Sociopolitical Analyses
Education (PDME) conferences of the early Freire’s Marxist approach can perhaps be seen as
1990s, followed by the conferences of the Math- the earliest inspiration to researchers in
ematics Education and Society (MES) group). mathematics education in relation to raising
Here, the focus will be on social class and awareness of the idea that education is never
socioeconomic status and achievement in mathe- neutral. His Pedagogy of the Oppressed (1970)
matics. Ethnicity and race and gender in contrasted a banking concept of education,
mathematics education are separate entries in identified with the oppressor, as against a critical
the encyclopedia. pedagogy, with the goal of empowerment and
emancipation of the oppressed. His work was
Research in this area will be addressed in the based around several themes that have since
following sections: statistical evidence, sociopo- been developed in the field. Freire, taking up the
litical analyses, explanatory and analytical frame- notion that knowledge is a social construct, raised
works, and research on action or intervention. the question of whose knowledge is to be valued;
he argued for a constructivist view of learning,
Statistical Evidence not one of “banking” inert knowledge; and that
Secada (1992) provides a thorough and structured teaching is a political process and the teacher
analysis of data on achievement in the North should work dialogically, learning from their stu-
American context. While it is clear that such dents what matters to them in their lives.
analyses must be localized to be of use to
researchers, educators, and policy makers, some Research on the everyday mathematics of
of his concluding remarks are as relevant more indigenous, rural, and oppressed groups
than 20 years on. He argues that too often such (Knijnik 2000) led to and was inspired by
analyses are carried out by researchers who are ethnomathematics (D’Ambrosio 1985), a socio-
not in the mainstream of mathematics education political theory which sees academic mathemat-
ics as just one of a range of mathematical systems
used by people. D’Ambrosio (2010) argues that

Socioeconomic Class in Mathematics Education S555

we remain unconscious of how academic mathe- Critiques of the work described here come S
matics, so dominant throughout the world, can be from the poststructuralist critique of critical
used for the good of society or for domination by theory (e.g., Ellsworth 1989; Walshaw 2004)
the powerful of the powerless, the latter being the which argues that empowerment is an
most common. Valuing academic, and therefore enlightenment, universalist concept with no
school mathematics, above all, marginalizes the foundation other than ideology and from argu-
lives and values of dominated groups. ments that there is a confusion when attempting
to harness everyday practices for the purposes of
Freire’s constructivist view of learning, with teaching mathematics in school, what Dowling
elements from Vygotskian approaches too in his (2001) calls the myth of emancipation (p. 32).
insistence on the unity of cognition and affect, This latter point is developed in the following
emphasizes his rejection of the banking concept, section.
in which knowledge does not relate to what mat-
ters to the lives of underprivileged students and is Explanatory and Analytical Frameworks
inert for all students. It particularly disadvantages While the statistical evidence confirms the
the underprivileged. His critical pedagogic posi- correlation between low SES and low achieve-
tion has been taken up by many, including the ment in mathematics, and the sociopolitical
criticalmathematics group (Frankenstein 1983). perspectives argue forcefully for change,
The concatenation of the two words signals the researchers need explanatory frameworks for
particular focus of researchers in that group, why the correlation exists. It can be argued
emphasizing the development of appropriate that without such analyses, any changes being
materials that challenge hegemonic views of made in pedagogy may come from principles
the neutrality of mathematical knowledge and and values but may not make any fundamental
research studies of teaching and learning from difference.
this position. Activists such as Gutstein (2009)
have also found Freire’s work inspiring. The sociological theories of Basil Bernstein
and Pierre Bourdieu in particular, both Marxist
The dialogic view of teaching has been devel- sociologists, have been taken up by researchers in
oped by Skovsmose (1994) in particular. Empha- mathematics education to understand the causes
sizing democracy, a critique of the way that of the correlation. The ideas of these sociologists
traditional/academic mathematics formats of education have similarities and differences.
a view of the world, and the potential for equality From their Marxist origins, they both focus on
that comes with a dialogic learning process, consciousness as a product of social relations and
Skovsmose and his collaborators (e.g., Alrø and in particular relations to the means of symbolic
Skovsmose 2002) address the potential power of production.
learning and teaching that engages with what
matters in children’s lives and with how they Bourdieu (1977) introduced the notions of
can change the world. Skovsmose’s approach is habitus, cultural capital, and field. In brief, the
often referred to as a critical mathematics field provides the structuring practices which
position, with the two words separated. convey power and status. At the subjective
level, habitus is the embodiment of culture, pro-
Differences between the ethnomathematics viding the lens through which the world is
and critical mathematics education positions interpreted. The habitus of children from the
were discussed by Vithal and Skovsmose middle classes may bring with it opportunity for
(1997). Taking the case of South Africa as a power if it aligns with the expectations of
context, though emphasizing the international the school. Thus, certain forms of culture
implications, their detailed analysis of the endow the “possessor” with cultural capital that
potentialities of the two perspectives includes a can be exchanged for gains that are valued, such
concern for the empowerment of students when as success in school mathematics. Zevenbergen
an ethnomathematics approach is taken, espe- (2001) provides clear description of the analytical
cially those disadvantaged by apartheid.

S 556 Socioeconomic Class in Mathematics Education

tools Bourdieu’s framework provides and Examining gender effects of forms of assess-
uses it to analyze classroom interactions in ment in mathematics, Wiliam suggests:
a year-long ethnographic study. Gates (2004)
uses Bourdieu’s theories to examine teachers’ We are led to the conclusion that it is a third source
beliefs from a sociological rather than cognitive of difference—the definition of mathematics
perspective. In an analysis of mathematics employed in the construction of the test—that is
achievement in the context of reforms and the most important determinant of the size (and
counterreforms of the curriculum in Victoria, even the direction) of any sex differences. (Wiliam
Australia, Teese (2000) also employs Bourdieu’s 2003, p. 194)
theories.
As Lawler says, in a reexamination of one of
For Bernstein (e.g., 2000), language is an the earliest texts addressing disadvantage in
indicator of different relations to the means of mathematics (Reyes and Stanic 1988):
symbolic production, children from working-
class backgrounds exhibiting a restricted code Mathematics education does not work to realize the
and those from middle-class backgrounds living of the child, but to enact in the child partic-
exhibiting an elaborated code. Given that schools ular, culturally-defined, ways of operating and
work in an elaborated code from the first day of interacting that are deemed to be mathematical.
children’s participation in school, the manner We treat the content of mathematics as stable
in which schools reproduce advantage and structures of conventional ideas, “inert, unchang-
disadvantage, the differential distribution of ing, and unambiguous ‘things’ that children learn”
knowledge across social backgrounds becomes (Popkewitz 2004, p. 18). And although these things
obvious. Key sociological concepts of appear to make the learner more of an active
those researching who succeeds and who fails in participant by expanding the child’s role in solving
school mathematics, and why, include the problems and applying their own thinking, we
nature of knowledge discourses, the distinction simultaneously make them less active in defining
between the everyday and the “esoteric,” and its the possibilities and boundaries for their engage-
effect on students (Dowling 1998; Cooper and ment. (Lawler 2005, p. 33)
Dunne 2000); the official and unofficial fields of
pedagogic knowledge and how they are taken Lawler argues that changes over some decades
up and by whom (Morgan et al. 2002); the have not made a difference to who succeeds and
distinction between strong grammars, such as who fails. Perhaps, the challenge not addressed so
mathematical discourse, and weak grammars, far, informed by postmodern thinking, concerns
such as education, set within notions of vertical the mathematical content, not only in thinking
and horizontal knowledge structures (Lerman about what to teach but why, whether mathemat-
2010); and how forms of pedagogy can be ics should be taught to everyone, and why the
modified to improve the achievement of field is so implicated in maintaining the high
disadvantaged students (Knipping et al. 2008). status of a mathematical qualification.

Bernstein also shows how curriculum choices, Research on Action or Intervention
the recontextualization of knowledge from one The literature on interventions and radical action is
place, academic mathematics in our case, to very broad and, for the most part, does not distin-
another, school mathematics, is determined by guish between the various social characteristics of
ideology; what is deemed important for students disadvantaged groups, race, gender, social class,
to acquire is governed by beliefs and values, disability, or others. Examples can be found in the
though usually implicitly. Researchers have literature mentioned above, such as that of the crit-
taken up the issue of values in addressing how ical mathematics group, the literature of the
what currently manifests as mathematics in ethnomathematics group, or the proceedings of the
schools affects students. Mathematics Education and Society conferences.

Concluding Remarks
Localization of statistical evidence has been
mentioned above but could be seen to be vital

Socioeconomic Class in Mathematics Education S557

in all aspects of the issue of social class and Dowling P (1998) The sociology of mathematics educa- S
socioeconomic status in mathematics education. tion: mathematical myths/pedagogic texts. Falmer,
Who is disadvantaged, what the causes might be, London
what status mathematics has, whether mathemat-
ics for all is part of the values, and what kinds of Dowling P (2001) Mathematics education in late moder-
interventions might be effective are all informed nity: beyond myths and fragmentation. In: Atweh B,
by the theoretical and empirical studies described Forgasz H, Nebres B (eds) Sociocultural research on
here but are different across the world. The mathematics education: an international perspective.
research field lacks such analyses from many Lawrence Erlbaum, Mahwah, pp 19–36
parts of the world, and the complicity of main-
stream researchers in the status of mathematics in Ellsworth E (1989) Why doesn’t this feel empowering?
society may be the cause of the major focus being Working through the repressive myths of critical
on other aspects of teaching and learning. pedagogy. Harv Educ Rev 59(3):297–324

Cross-References Frankenstein M (1983) Application of Paolo Freire’s
epistemology. J Educ 165(4):315–339
▶ Cultural Diversity in Mathematics Education
▶ Equity and Access in Mathematics Education Freire P (1970) Pedagogy of the oppressed. Seabury,
▶ Indigenous Students in Mathematics New York

Education Gates P (2004) Going beyond belief systems: exploring
▶ Sociological Approaches in Mathematics a model for the social influence on mathematics
teacher beliefs. Educ Stud Math 63(3):347–369
Education
Gutstein E (2009) Possibilities and challenges in
References teaching mathematics for social justice. In: Ernest P,
Greer B, Sriraman B (eds) Critical issues in
Alrø H, Skovsmose O (2002) Dialogue and learning in mathematics education. Information Age, Charlotte,
mathematics education: Intention, reflection, critique. pp 351–374
Kluwer, Dordrecht
Knijnik G (2000) Ethnomathematics and political
Bernstein B (2000) Pedagogy, symbolic control and iden- struggles. In: Coben D, O’Donoghue J, Fitzsimmons
tity. Theory, research, critique, revised edition. G (eds) Perspectives on adults learning mathematics:
Rowman & Littlefield, New York research and practice. Kluwer, Dordrecht, pp 119–134

Boaler J (2003) Learning from teaching: exploring the Knipping C, Reid DA, Gellert U, Jablonka J (2008)
relationship between reform curriculum and equity. The emergence of disparity in performance in
J Res Math Educ 33(4):239–258 mathematics classrooms. In: Matos JF, Valero P,
Yasukawa K (eds) Proceedings of the fifth interna-
Bourdieu P (1977) Outline of a theory of practice (trans. R. tional mathematics education and society conference.
Nice). Cambridge University Press, Cambridge Centro de Investigac¸a˜o em Educac¸a˜o, Universidade de
Lisboa, Lisbon, Department of Education, Learning
Brown v. Board of Education 347 U. S. Reports (17 May and Philosophy, Aalborg University, Aalborg,
1954): 483-500 pp 320–329

Cooper B, Dunne M (2000) Assessing children’s mathe- Lawler BR (2005) Persistent iniquities: a twenty-year
matical knowledge: social class, sex and problem- perspective on “Race, Sex, Socioeconomic Status,
solving. Open University Press, Buckingham and Mathematics”. The Mathematics Educator, Mono-
graph No. 1: 29–46
D’Ambrosio U (1985) Ethnomathematics and its place in
the history and pedagogy of mathematics. For the Lerman S (2010) Theories of mathematics education: is
Learn Math 10(1):44–48 plurality a problem? In: Sriraman B, English L (eds)
Theories of mathematics education. Springer, New
D’Ambrosio U (2010) From Ea, through pythagoras, to York, pp 99–110
Avatar: different settings for mathematics. In: Pinto
MMF, Kawasaki TF (eds) Proceedings of the 34th Lubienski ST (2000) Problem solving as a means toward
Conference of the International Group for the Psychol- mathematics for all: an exploratory look through
ogy of Mathematics Education, vol 1. Belo Horizonte, a class lens. J Res Math Educ 31(4):454–482
Brazil, pp 1–20
Morgan C, Tsatsaroni A, Lerman S (2002) Mathematics
teachers’ positions and practices in discourses of
assessment. Br J Sociol Educ 23(3):443–459

Popkewitz T (2004) The alchemy of the mathematics
curriculum: inscriptions and the fabrication of the
child. Am Educ Res J 41(1):3–34

Reyes LH, Stanic GMA (1988) Race, sex, socioeconomic
status, and mathematics. J Res Math Educ 19(1):26–43

Secada WG (1992) Race, ethnicity, social class, language,
and achievement in mathematics. In: Grouws DA (ed)
Handbook of research on mathematics teaching and
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Skovsmose O (1994) Towards a philosophy of critical guides and directs research. In research on math-
mathematics education. Kluwer, Dordrecht ematics education, they have a rather short
history. They are offering vigorous and fresh
Teese R (2000) Academic success and social power: perspectives, and they have received increasing
examinations and inequality. Melbourne University attention during the last 25 years. By using
Press, Melbourne methods of empirical investigation and critical
analysis, they engage with the complex relation-
Valero P, Zevenbergen R (eds) (2004) Researching the ships between individuals, groups, knowledge,
socio-political dimensions of mathematics education: discourse, and social practice, aiming at
Issues of power in theory and methodology. Kluwer, a theoretical understanding of social processes
Dordrecht in mathematics education. These relationships
are often conceived as tensions between the
Vithal R, Skovsmose O (1997) The end of innocence: a micro level of individual agency and interaction
critique of ‘Ethnomathematics’. Educ Stud Math and the macro level of the social structure of
34(2):131–157 society. The institutions of mathematics
education and their functioning, often in terms
Walshaw M (ed) (2004) Mathematics education within the of social reproduction, are of crucial concern.
postmodern. Information Age, Greenwich
Sociological approaches in mathematics
Wiliam D (2003) Constructing difference: assessment in education refer to a field of study and a body of
mathematics education. In: Burton L (ed) Which way knowledge that are not defined by clear-cut
social justice in mathematics education? Praeger, boundaries. They use, recontextualize, and refine
Westport, pp 189–207 concepts and methods from the various branches
of sociology and their neighboring disciplines.
Zevenbergen R (2001) Mathematics, social class, and Naturally, sociology of education serves as
linguistic capital: an analysis of mathematics class- the most convenient reservoir of reference for
room interactions. In: Valero P, Zevenbergen R (eds) the sociological study of mathematics education
Researching the socio-political dimensions of (e.g., Bernstein, Bourdieu). However, studies
mathematics education: Issues of power in theory and from interpretive (interactionist (e.g., Mead)
methodology. Kluwer, Dordrecht, pp 201–215 and ethnomethodological (e.g., Garfinkel)),
phenomenological (e.g., Berger and Luckmann),
Sociological Approaches in critical (e.g., Adorno), structuralist (e.g.,
Mathematics Education Althusser), poststructuralist (e.g., Foucault) and
psychoanalytical (e.g., Lacan), political (e.g.,
Uwe Gellert Apple), feminist (e.g., Walkerdine), social
Fachbereich Erziehungswissenschaft und semiotics (e.g., Halliday), and discourse
Psychologie, Freie Universit€at Berlin, Berlin, analytical (e.g., Fairclough) perspectives have
Germany substantially contributed to our sociological
understanding of mathematics education.
Keywords
While only a few sociological studies of
Agency; Critical analysis; Control; Curriculum; mathematics education had been published
Discourse; Discrimination; Empirical investiga- before the mid-1980s, a Fifth Day Special
tion; Everyday knowledge; Feminist perspectives; Program titled Mathematics, Education, and
Identity; Ideology; Institution; Instructional Society (Keitel et al. 1989) of the 6th Interna-
mechanism; Interaction; Knowledge; Linguistic tional Congress on Mathematical Education
habitus; Mathematics textbooks; Parental partic- (ICME-6) in 1988 achieved a breakthrough,
ipation; Phenomenology; Social practice; Social quantitatively and in terms of its recognition, of
process; Parental participation; Social semiotics; research on society and institutionalized mathe-
Stratification; Structuralist perspectives matics education, conceived as the political

Topic and History

Sociological approaches in mathematics
education are those where sociological theory

Sociological Approaches in Mathematics Education S559

dimensions of mathematics education. This was While most of the presentations of sociologi- S
the start of a series of international conferences, cal research at ICME-6 still had been of
initially called Political Dimensions of Mathe- descriptive character and not systematically and
matics Education (PDME), then Mathematics explicitly based on sociological theories, they
Education and Society (MES), which served, successfully kicked off substantial advances in
and continues to serve, as the major forum for the theoretical foundation of sociological
presenting and discussing research based on approaches in mathematics education. Dowling’s
sociological approaches in mathematics educa- (1998) analysis of mathematical myths and ped-
tion (Matos et al. 2008; Gellert et al. 2010). agogic texts marks a milestone in the subsequent
development of sociological theorizing in
Issues of Research mathematics education. It examines and coordi-
nates a wide range of theoretical positions,
Sociological approaches contribute in a particular constructing a systematic and theoretically rooted
way to what Lerman (2000) has called the “social language of description for analyzing mathemat-
turn in mathematics education research.” While ics textbooks sociologically. By providing math-
much research included in this “social turn” aims ematical activities that establish positions and
at conceiving mathematical learning as more messages differentially, mathematics textbooks
social in character and as a result of action and construct a hierarchy of student voices through
interaction, sociological approaches in mathe- the distribution of the “myth of participation”
matics education investigate how mathematical (mathematics is a reservoir of use-values)
knowledge is produced, distributed, recontex- and the “myth of reference” (mathematics offers
tualized, reproduced, and evaluated by a gaze on something other than itself). Mathemat-
institutional practices. They particularly focus ical texts for high-achieving students use
on how these practices shape identities and abstraction and strategies of expansion to consis-
(re-)produce social stratifications. Another con- tently foreground generalized academic mathe-
cern is the relationships between different con- matical messages. In contrast, texts for
texts in which mathematical knowledge is low-achieving students use localizing strategies
transmitted, acquired, and assessed. As Ensor to identify the students’ voice with a public
and Galant (2005) claim, many sociological domain setting which is insulated from abstract
studies of mathematics education are, at least mathematics. The curriculum mirrors the divi-
implicitly, interested in the pedagogic forms sion of intellectual and manual labor, of class
and the mathematical knowledge supportive for distinctions, and of code orientations. However,
social justice or try to state more precisely the the ideological roots of mathematics curricula are
pathologies that impede such a development. far more hidden than overt and Dowling (1998)
Equity and access are issues that motivate some can be credited for contributing to their exposure.
sociological research in mathematics education. For many researchers, it provided an inspiring
Jablonka (2009) holds that a prevalent ingredient interpretation of the late work of the British soci-
of sociological approaches is critique, aiming at ologist of education Basil Bernstein (2000). In
uncovering ideologies, making the invisible fact, Bernstein (2000) seems to have become the
mechanisms of social functioning visible, thus most common reference in studies of mathemat-
making the unconscious conscious. Sociological ics education that take sociological approaches. It
approaches to research in mathematics education provides an ample theoretical framework with
usually draw on qualitative research methods – strong internal coherence and explicit organizing
exceptions prove the rule – which is much in principles – what Bernstein calls a strong
accord with the skepticism of the “new sociology grammar – that systematically links social struc-
of education” of the 1970s in respect of a political ture with human agency, in particular for the
arithmetic tradition. context of pedagogic discourse. The widespread
use of the concepts of the pedagogic device,

S 560 Sociological Approaches in Mathematics Education

classification and framing values, recognition scholar, and that any other position towards
and realization rules, horizontal and vertical dis- mathematics, for instance, a more critical view
course, and recontextualizing fields indicates of the nature of mathematics, is strictly discour-
a common focus and a coherent growing of aged by apparently neutral assessment practices
sociological research in mathematics education. that maximize differences between individuals
Studies of mathematics curriculum, of assess- and thus construct disparities in mathematics
ment of mathematical knowledge, of ability achievement. Ability grouping (streaming,
grouping, of pedagogic identities, and of class- setting, etc.) reinforces exclusion from the sub-
room instruction practices, which will be ject by constructing different mathematical hab-
exemplified in the next passage, are all central itus for different groups of learners (Zevenbergen
themes in sociological research in mathematics 2005). Morgan et al. (2002) report that teachers’
education. Most of the research examples link expectations and their subject position in the edu-
these central themes to each other. cation discourse are heavily influential on their
assessment practices. Consequently, teachers
Mathematics curricula can be usefully who teach in schools located in different social
described and compared in terms of the strength contexts emphasize different local assessment
of the boundaries established between everyday criteria, thus providing differential orientation
knowledge and academic mathematical knowl- towards mathematical knowledge, resulting
edge, as well as between the areas that constitute in an unequal “preparation” of students for
the school subjects. Mathematics curricula for standardized mathematics achievement tests. In
primary and for lower level secondary schools contexts of severe social discrimination and infe-
usually intend to connect school mathematical riority, what is transmitted and to be acquired is
knowledge to the local and particular of everyday often emptied of any mathematical content. The
knowledge. This aim is reflected in the high pro- tasks to be executed by students reflect a very
portion of word problems contained in the curric- weak classification between everyday and school
ulum materials for the early grades. Gellert and knowledge, and consequently, the evaluative
Jablonka (2009) discuss how students face criteria appear to be weak or absent. It appears
substantial intricacies of producing legitimate as a perverted form of recontextualization, when
text in the classroom, if and because the recontex- in socially discriminated contexts, the specialized
tualization principle of the curriculum is gener- knowledge of mathematics is subordinated to
ally not made sufficiently explicit in classroom everyday knowledge and practices.
practices. Cooper and Dunne (2000) investigate
how students with different socioeconomic class Sociological approaches to research on
backgrounds react to word and context problems. instructional practices have highlighted that
They analyzed large sets of data from the Key reform agendas often overlook the different
Stage 2 Tests for 10–11-year-old students in code orientations of groups of students. Lubienski
England. The study documents that students of (2000) argues that some instructional strategies
families where the parents do manual work have that are highly valued in current mathematics
significantly lower achievement when mathemat- education reforms disadvantage students who
ics is interwoven with context. Cooper and are characterized as of low socioeconomic status.
Dunne find that these students tend to misinter- She demonstrates how socioeconomically
pret the problems and to solve them with their advantaged students tend to profit from intensive
everyday knowledge, which means that their guided discussions in the classroom while more
mathematical competence is systematically socioeconomically disadvantaged students
underestimated in the tests. Wiliam et al. (2004) become rather confused by conflicting mathemat-
argue that for becoming successful in school ical ideas, suggesting that some characteristics of
mathematics, students need to develop a particu- discussion-intensive mathematics classrooms
lar identity, in fact that of a young mathematics might be more aligned with middle-class codes.

Sociological Approaches in Mathematics Education S561

Apparently, the linguistic habitus of socioeco- mathematics colonizes large parts of reality and S
nomically advantaged students work as cultural rearranges it. The transmission and acquisition of
capital as in school – at least at the discursive mathematical knowledge appear of direct social
level – the discursive practices are close to prac- importance when concepts of critique, democ-
tices that are common in middle-class families. racy, and Mu€ndigkeit are brought together. For
A similar effect has been observed by Brown Valero (2009), power in mathematics education
(2000) who investigates parental participation in can be conceived in terms of the structural imbal-
school mathematics. He reports that middle-class ance of knowledge control and of distributed
parents, when working together with their children positioning. The former view, which reflects
on mathematics tasks, tend to emphasize the con- a conflict theoretical stance, points to a constant
text-independent and general aspects of the tasks, struggle between structurally excluded and struc-
while working-class parents focus strictly on the turally included groups, in which the powerful
local and context-bound. Working-class children tends to win and to succeed in cushioning the
profit less than their middle-class peers from resistance on the side of the excluded. The latter
parental involvement in school mathematics. view takes power as a relational capacity of social
actors to draw on resources for self-positioning in
Inside the mathematics classroom, various situations. This definition does not only facilitate
instructional mechanisms produce a stratification analyses of how mathematics and mathematics
of achievement and success in mathematics that education is used in discourses affecting people’s
is not strictly based on the mathematical compe- lives but also opens for a self-reflective perspec-
tence of the students. These mechanisms draw on tive on how research in mathematics education is
students’ unequal competences in recognizing entangled in the distribution of power. Vithal
the rules and reading the code of mathematics (2003) investigates the role and potential power
instruction. For instance, instructional strategies of mathematics education in postapartheid South
of embedding mathematics in mundane context Africa. By coining five pairs of concepts that
and leaving implicit the relevance of that context work antagonistically and yet in cooperation
in terms of the criteria for producing legitimate with each other – freedom/structure, democracy/
text separate the students along their code authority, context/mathematics, equity/differen-
orientation. Teachers often show a well- tiation, and, pulling these four together, potenti-
distinguishable ability to maintain two different ality/actuality – the fundament for a pedagogy of
discourses at the same time, engaging some stu- conflict and dialogue is laid out. Conceiving actu-
dents in analytical mathematical arguments and ality as intrinsically conflicting, dialogue of var-
others in substantial everyday reasoning. ious forms and at many levels is suggested to
This observation is sociologically relevant since inspire and develop potentiality. Gates and
on the long run, the mathematical argument is Vistro-Yu (2003), taking on the distributions of
institutionally more highly valued. Sociological mathematical knowledge and revisiting the pro-
approaches emphasize that the diversity, or gram of Mathematics for All, describe mathemat-
heterogeneity, of groups of students is less ics as a gatekeeper to social progress and as a
a topic of concern than their positioning in filtering device. They argue for a strong role of
hierarchies of social status. the mathematics education community to avoid
and counteract the marginalization of some social
Finally, research and reflection that explicitly groups. Gender, socioeconomic class, and ethnic-
call for more attention to sociopolitical dimen- ity are discussed as examples of marginalized
sions are of fundamental importance for the voices (and at the other side of the coin, there
sociological study of mathematics education. are dominant voices); in mathematics classrooms
Here, the concept of power and its social, politi- characterized by multiple discriminations, con-
cal, and educational ramifications is fundamental. tradictions, and clashes in pedagogical practice,
Skovsmose (1994) introduces the notion of the
formatting power of mathematics to indicate that

S 562 Sociological Approaches in Mathematics Education

the marginalization tends to be reproduced and Gates P, Vistro-Yu CP (2003) Is mathematics for all? In:
exacerbated. In essence (Skovsmose and Greer Bishop AJ, Clements MA, Keitel C, Kilpatrick J,
2012), research on the sociopolitical dimensions Leung FKS (eds) Second international handbook of
of mathematics education, characterized by mathematics education. Kluwer, Dordrecht, pp 31–73
awareness of the inherently political nature of
mathematics education and by acceptance of Gellert U, Jablonka E (2009) “I am not talking about
social responsibility, is based on, and continually reality”: word problems and the intricacies of produc-
develops, the critical agency of mathematics edu- ing legitimate text. In: Verschaffel L, Greer B, Van
cation researchers. Dooren W, Mukhopadhyay S (eds) Words and worlds:
modelling verbal descriptions of situations. Sense,
Cross-References Rotterdam, pp 39–53

▶ Competency Frameworks in Mathematics Gellert U, Jablonka E, Morgan C (2010) Proceedings of
Education the sixth international Mathematics Education and
Society conference, 2nd edn. Freie Universit€at
▶ Critical Mathematics Education Berlin, Berlin. http://www.ewi-psy.fu-berlin.de/en/v/
▶ Discourse Analytic Approaches in mes6/proceedings/index.html. Accessed 2 Aug 2012

Mathematics Education Jablonka E (2009) Sociological perspectives in research in
▶ Equity and Access in Mathematics Education mathematics education. In: Kaasila R (ed)
▶ Gender in Mathematics Education Matematiikan ja luonnontieteiden: Proceedings of the
▶ Interactionist and Ethnomethodological 2008 symposium of the Finnish Mathematics and Sci-
ence Education Research Association. University of
Approaches in Mathematics Education Lapland, Rovaniemi, pp 35–67
▶ Political Perspectives in Mathematics
Keitel C, Damerow P, Bishop A, Gerdes P (eds) (1989)
Education Mathematics, education, and society. UNESCO, Paris
▶ Poststructuralist and Psychoanalytic
Lerman S (2000) The social turn in mathematics education
Approaches in Mathematics Education research. In: Boaler J (ed) Multiple perspectives in
▶ Recontextualization in Mathematics Education mathematics teaching and learning. Ablex, Westport,
▶ Socioeconomic Class in Mathematics pp 19–44

Education Lubienski ST (2000) A clash of social class cultures?
Students’ experiences in a discussion-intensive sev-
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Bernstein B (2000) Pedagogy, symbolic control and iden-
tity, 2nd (rev.) edn. Rowman & Littlefield, Lanham Matos JF, Valero P, Yakusawa K (2008) Proceedings of
the fifth international Mathematics Education and
Brown A (2000) Positioning, pedagogy and parental par- Society conference. Centro de Investigac¸a˜o em
ticipation in school mathematics: an exploration of Educac¸a˜o, Universidade de Lisboa, Lisbon. http://
implications for the public understanding of mathe- pure.ltu.se/portal/files/2376304/Proceedings_MES5.pdf.
matics. Social Epistemol 14:21–31 Accessed 2 Aug 2012

Cooper B, Dunne M (2000) Assessing children’s mathe- Morgan C, Tsatsaroni A, Lerman S (2002) Mathematics
matical knowledge: social class, sex and problem- teachers’ positions and practices in discourses of
solving. Open University, Buckingham assessment. Brit J Sociol Educ 23:445–461

Dowling P (1998) The sociology of mathematics educa- Skovsmose O (1994) Towards a philosophy of critical
tion: mathematical myths/ pedagogical texts. mathematics education. Kluwer, Dordrecht
RoutledgeFalmer, London
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perspectives, practices and possibilities. HSRC, Cape matics education? In: Ernest P, Greer B, Sriraman B
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Vithal R (2003) In search of a pedagogy of conflict and
dialogue for mathematics education. Kluwer, Dordrecht

Wiliam D, Bartholomew H, Reay D (2004) Assessment,
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years. J Curriculum Stud 37:607–619

Sociomathematical Norms in Mathematics Education S563

Sociomathematical Norms in explain one or more solution processes and that S
Mathematics Education the students attempt to understand and repeat her
reasoning on other problems. Social norms
Michelle Stephan involve participants’ expectations of each other
College of Education, Middle Secondary during discussions and can be found in
Department, The University of North Carolina at classrooms in any domain. For example,
Charlotte, Charlotte, NC, USA a student-centered science or literacy classroom
might have similar social norms above such as
Keywords explaining and justifying and understanding
students’ explanations.
Sociomath norms; Social norms; Emergent
perspective; Intellectual autonomy Sociomathematical Norms
While social norms focus on normative aspects
Definition of participation in any academic area,
sociomathematical norms, on the other hand, are
Sociomathematical norms are the normative norms that are specific to mathematical activity.
criteria by which students within classroom com- Similar to social norms, they can be found in any
munities create and justify their mathematical mathematics classroom, but they would look
work. Examples include negotiating the criteria different depending on the goals and philosophy
for what counts as a different, efficient, or sophis- of instruction. They involve the teacher and
ticated mathematical solution and the criteria for students negotiating the criteria for what counts
what counts as an acceptable mathematical as an acceptable mathematical explanation, a
explanation. different solution, an efficient solution, and a
sophisticated solution in their classroom. For
Characteristics example, a social norm for a student-centered
classroom might be that students are expected to
Social Norms explain their thinking, but what counts as an
Social norms refer to the expectations that the acceptable mathematical explanation must be
teacher and students have for one another during determined among the teacher and students. For
academic discussions. Social norms are present example, Stephan and Whitenack (2003) found
in any classroom, including science and, English, that the criteria for what counts as an acceptable
for example. However, the social norms that are mathematical explanation in one first-grade class
established within a student-centered classroom involved stating not only the procedures for
look very different from those in a traditional finding an answer but also the reasons for the
environment. Yackel and Cobb (1996) have calculations as well as what these calculations
documented at least four social norms that sup- and their results mean in terms of the problem.
port student-centered instruction: Students are The criterion necessarily changes over time as the
expected to (1) explain and justify their solutions students and teachers give and take during their
and methods, (2) attempt to make sense of others’ discussions. For instance, at the beginning of the
explanations, (3) indicate agreement or disagree- year when a first-grade teacher asked her students
ment, and (4) ask clarifying questions when the to solve the problem, Lena has 11 hearts, Dick
need arises. The social norms for more traditional has 2 hearts, how many more hearts does Lena
mathematics classes that are teacher-centered have than Dick, some students gave the answer 9
might involve expectations that the teacher while others said 11. Students felt obliged to
explain their thinking (social norm), but their
discussion simply focused on their calculations,
e.g., “I counted up 9 more to get to 11.” Students
who thought the answer was 11 argued that Lena

S 564 Sociomathematical Norms in Mathematics Education

Lena interpret how the teacher and students are
interacting and learning in a classroom:
Dick
Social Individual
Sociomathematical Norms in Mathematics Classroom social norms
Education, Fig. 1 Beliefs about own role, others’
Sociomathematical roles, and the general nature of
has 11 more than Dick so the answer is 11. Since norms mathematical activity in school
students’ explanations only drew on their calcu- Classroom
lations, the teacher attempted to initiate mathematical practices Mathematical beliefs and
a discussion about why someone might count up values
to get the answer. Because the criterion for what
counts as acceptable explanation in math class Mathematical conceptions
involved sharing their calculations, students did
not know how to explain why they counted up. As the framework shows, an individual
The teacher drew circles on the board to support forms his beliefs about his role in the class, his
students as they tried to explain why counting up mathematical beliefs, and his mathematical
was legitimate (Fig. 1). learning as he participates in and contributes to
the social and sociomathematical norms and
A student came to the board, drew a vertical classroom mathematical practices of his
line after the second “heart,” and counted by ones classroom community. Cobb and Yackel (1996)
up from Dick’s two hearts to “make them have stress that learning is both an individual and
the same amount.” In this way, the teacher, social process with neither taking primacy over
through the use of diagrams, helped the students the other.
begin to learn that an acceptable explanation
must involve their reasons for their procedures. Growth of the Concept
Many mathematics education researchers have
The criterion for what counts as an acceptable acknowledged the importance of paying close
mathematical explanation might be different attention to the establishment of certain
depending on the teaching approach used. sociomathematical norms in a variety of
For example, in more traditional classrooms, classroom settings. In fact, some argue that while
what counts as an acceptable mathematical expla- inquiry social norms are mandatory for creating
nation might involve describing only the calcula- student-centered mathematics classrooms, they
tions that one used in their procedure. In the Lena are insufficient for supporting mathematical
and Dick problem above, a traditional setting growth (Pang 2001). Pang found that teachers,
might find the following explanation acceptable, who established both strong social and
“I counted up two more from 9 on my fingers,” sociomathematical norms for inquiry instruction,
without any reference to why that method has saw more mathematical growth in their students
meaning and leads to a correct answer. than those who had only established strong inquiry
social norms. Given that sociomathematical norms
Origin of the Term focus more on the quality of the mathematical
The term sociomathematical norms was first contributions in class, Pang’s finding makes sense.
coined by Cobb and Yackel (1996) as they built
a framework for analyzing student-centered, or Mathematics education researchers have
what they called, inquiry-based classrooms. They extended Yackel and Cobb’s sociomathematical
drew on the emergent perspective, a theory norms research by analyzing the development of
that says learning occurs both cognitively these norms at the elementary (Pang 2001;
as well as in social interaction. Using the emer- Stephan and Whitenack 2003; Levenson et al.
gent perspective, they created the following 2006), middle (Akyuz 2012), high (Kaldrimidou
framework to help themselves and others et al. 2008), and college level (Rasmussen et al.
2003). Findings indicate that negotiating the
criterion for what counts as different, efficient,

Sociomathematical Norms in Mathematics Education S565

sophisticated, and an acceptable explanation in class. They also argue that the criteria for what S
inquiry settings are an important focus of the counts as acceptable should often involve more
teachers’ practice. than the procedures for solving the problem.

At the elementary level, Levenson et al. Other researchers have attempted to teach
(2006) extended the work on what counts as an sociomathematical norm development in their
acceptable explanation when they found that one professional development workshops (Shriki
teacher’s criterion for what counts as acceptable and Lavy 2005) as well as with preservice teacher
involved practically based explanations (those instruction (Dixon et al. 2009). Some articles
ground in realistic contexts), even though she detail the role of the teacher in establishing
knows that some of her students are capable of these norms (McClain 2005). Additionally,
giving more mathematically based explanations sociomathematical norms have gained attention
(those that are devoid of pictures and are more in other research fields as well with Johnson
abstract). Additionally, Pang found that teachers (2000) coining the term “sociophysics norms”
are excellent at establishing social norms that are to refer to the criteria for what counts
consistent with inquiry-based instruction, but not as inquiry-based physics discourse.
as much with sociomathematical norms. This is
a concern since Pang argues that mathematical Common Issues
discussions arise out of sociomathematical The research based upon sociomathematical
norms, not social norms. Therefore, teachers norms is growing both within the field of
must reconceptualize mathematics in their mathematics education as well as other
classrooms going beyond just expecting students disciplines. When an idea like this takes root
to explain. Stephan and Whitenack (2003) and begins to grow, oftentimes, it can change
identified a fifth sociomathematical norm, the from its original meaning. The most common
criteria for what counts as an adequate way sociomathematical norms are misinterpreted
mathematical diagram. in the literature today involves losing the fact that
they deal with the criterion for what counts as
Of the research conducted at the middle and good mathematical discourse. The fact that stu-
high school levels, most focus on documenting dents are expected to give different ideas in class
the sociomathematical norms that are established can be cast as a social norm, but the criterion for
in higher level mathematics. Kaldrimidou et al. what counts as different is negotiated within
(2008) found that the criteria for what counts as the realm of mathematics. It is the role of the
an acceptable mathematical explanation was very teacher to lead the negotiation of these criteria
procedural in a high school mathematics class and, therefore, the criterion for what counts as an
they observed while Akyuz (2012) found that it acceptable mathematical explanation depends
was more conceptual (or meaning based) in one upon the teacher’s own criterion, often influenced
middle school class founded on inquiry-based by the mathematics community.
instruction. Hershkowitz and Schwarz (1999)
documented two new sociomathematical norms Other research sometimes conflates
as they studied students who used a computer sociomathematical norms with students’ beliefs.
program to aid in their instruction. These two For example, the fact that Marcos always gives
norms involved the criteria for what counts as procedural explanations and believes that
mathematical evidence and what counts math involves calculating answers is not a
as a good hypothesis. sociomathematical norm. Rather, that is his belief
about what mathematics is (the individual side of
Keen attention to sociomathematical norms is Cobb and Yackel’s framework).
even important within college level mathematics
classrooms. Rasmussen et al. (2003) elaborate the In summary, sociomathematical norms refer
criteria for what counts as a different, elegant, to the criteria by which solutions are determined
and efficient solution as well as acceptable expla- as different, efficient, and sophisticated and
nation in an inquiry-based differential equations explanations are deemed mathematical

S 566 Stoffdidaktik in Mathematics Education

acceptable in a classroom. The teacher and her annual meeting of the American Educational Research
students create these criteria together as they Association, New Orleans
solve problems and engage is discourse with Kaldrimidou M, Sakonidis H, Tzekaki M (2008) Compar-
one another. Sociomathematical norms are pre- ative readings of the nature of the mathematical
sent in any mathematics classroom; however, the knowledge under construction in the classroom.
criteria for what counts as mathematical solutions ZDM Math Educ 40:235–245
and explanations would probably look different Levenson E, Tirosh D, Tsamir P (2006) Mathematically
from classroom to classroom, depending on how and practically-based explanations: individual
teacher- or student-centered the instruction is. preferences and socio-mathematical norms. Int J Sci
Sociomathematical norms are different from Math 4:319–344
social norms in that the former are specific to McClain K (1995) The teacher’s proactive role in
mathematics talk. Additionally, social norms supporting students’ mathematical growth.
are easier for teachers to establish in their Unpublished dissertation, Vanderbilt University,
classrooms, but mathematics grows out of Nashville
sociomathematical norms, making it extremely Pang J (2001) Challenges of reform: utility of
important for teachers to make them a clear socio-mathematical norms. Paper presented to the
focus of their teaching practice. This is one area annual meeting of the American Educational Research
that deserves more attention and research. Association, Seattle
Rasmussen C, Yackel E, King K (2003) Social and
Cross-References socio-mathematical norms in the mathematics
classroom. In: Schoen H (ed) Teaching mathematics
▶ Argumentation in Mathematics through problem solving: grades 6–12. NCTM,
▶ Learner-Centered Teaching in Mathematics Reston, pp 143–154
Shriki A, Lavy I (2005) Assimilating innovative learning/
Education teaching approaches into teacher education: why is it
▶ Manipulatives in Mathematics Education so difficult? In: Chick H, Vincent J (eds) Proceedings
▶ Mathematical Proof, Argumentation, and of the 29th conference of the international group for
the psychology of mathematics education, vol 4. PME,
Reasoning Melbourne, pp 185–192
Stephan M, Whitenack J (2003) Establishing classroom
social and socio-mathematical norms for problem
solving. In: Lester F (ed) Teaching mathematics
through problem solving: Prekindergarten-grade 6.
NCTM, Reston, pp 149–162
Yackel E, Cobb P (1996) Socio-mathematical norms,
argumentation and autonomy in mathematics. J Res
Math Educ 27:458–477

References

Akyuz D (2012) The value of debts and credits. Math Stoffdidaktik in Mathematics
Teach Middle Sch 17(6):332–338 Education

Cobb P, Yackel E (1996) Constructivist, emergent, Rudolf Straesser
and sociocultural perspectives in the context of Institut fu€r Didaktik der Mathematik,
developmental research. Educ Psychol 31:175–190 Justus–Liebig–Universit€at Giessen, Germany
Lulea, University of Technology, Sweden
Dixon J, Andreasen J, Stephan M (2009) Establishing
social and socio-mathematical norms in an undergrad- Keywords
uate mathematics content course for prospective
teachers: the role of the instructor. 6th monograph for Stoffdidaktik; Gymnasium; Ingenierie
the association of mathematics teacher educators, didactique; Elementarization; Grundvor-
scholarly practices and inquiry in the preparation of stellungen; Pedagogical content knowledge
mathematics teachers, pp 29–44

Hershkowitz R, Schwarz B (1999) The emergent
perspective in rich learning environments: some roles
of tools and activities in the construction of
socio-mathematical norms. Educ Stud Math 39:149–166

Johnson A (2000) Sociophysics norms in an innovative
physics learning environment. Paper presented to the

Stoffdidaktik in Mathematics Education S567

Definition plenary talk at a conference of German-speaking
didacticians is quite revealing: “Is there a future
An approach to mathematics education and for subject specific didactics?” (for detailed
research on teaching and learning mathematics description of this development cf. Steinbring
(i.e., didactics of mathematics), which concen- 2011, pp 44–46). Nowadays, Stoffdidaktik is
trates on the mathematical contents of the subject mainly published in journals aiming at practicing
matter to be taught, attempting to be as close as teachers of all levels of schooling in German-
possible to disciplinary mathematics. A major speaking countries (in journals such as “Der
aim is to make mathematics accessible and Mathematikunterricht MU”; URL: http://www.
understandable to the learner. friedrich-verlag.de/go/Sekundarstufe/Mathematik/
Zeitschriften/Der+Mathematikunterricht).

History Characteristics S

Stoffdidaktik or “subject matter didactics” According to Steinbring (2011, p. 45),
(translation suggested by the entry author) has Stoffdidaktik is characterized by the assumption
been a prominent approach to mathematics that mathematical knowledge – researched and
education and research into teaching and learning developed in the academic discipline – is essen-
mathematics (i.e., didactics of mathematics) in tially unchanged and absolute. “. .. it specifically
German-speaking countries (e.g., Austria, proceeds to prepare the pre-given mathematical
Germany, and parts of Switzerland). It grew out disciplinary knowledge for instruction as
of one of the two main strands of German- a mathematical content, to elementarise it and
speaking didactics of mathematics in the first to arrange it methodically.” As protagonist of
half of the twentieth century, namely, university subject matter didactics, Griesel (1974, p. 118)
studies that focused on the teaching of mathemat- has identified the following features of “didacti-
ics in “gymnasium,” the most demanding type cally oriented content analysis” as he prefers to
of school at that time in Germany. This name the approach: “The research methods of
strand was different from the strand that focused this area are identical to those of mathematics,
on teacher training for primary and the so that outsiders have sometimes gained the
majority of lower secondary schools. With pro- impression that, here, mathematics (particularly
fessors of mathematics at university interested in elementary mathematics) and not mathematics
mathematics education (like Felix Klein and education is being conducted.” In terms of
Heinrich Behnke), it had authors basically research methodology, this is a very clear and
coming from university institutions and teachers somehow very restricted preference, which – at
of gymnasium, who published in well-established least in terms of research methods – makes it
journals mainly read by mathematics difficult to distinguish Stoffdidaktik from
teachers of gymnasium (like “Zeitschrift fu€r den mathematics.
mathematischen und naturwissenschaftlichen
Unterricht ZMNU,” later “MNU” or Furthermore, Griesel continues: “The goal of
“Unterrichtsbl€atter f€ur Mathematik und ‘didactically oriented content analysis’ which
Naturwissenschaften UMN”). With the widening essentially follows mathematical methods is to
of research approaches in didactics of mathemat- give a better foundation for the formulation of
ics during the second half of the twentieth content-related learning goals and for the
century, Stoffdidaktik somehow widened its development, definition and use of a differentiated
perspective to the teaching of mathematics in all methodical set of instruments” (Griesel, 1974,
types of schools, but lost its position as one of two p. 118, both translations by Heinz Steinbring
major approaches in German-speaking didactics 2011, p. 45). The practice of “content-oriented
of mathematics. The title of Reichel’s (1995) analysis” up to the 1960s suggests that implicitly

S 568 Stoffdidaktik in Mathematics Education

Stoffdidaktik starts from the assumption that after approaches appears when the actual practice is
a decent mathematical analysis, one will find one taken into consideration: From the very begin-
and only one best way to teach a certain content ning, didactical engineering is also interested in
matter, which then should be incorporated the teacher and learner of the subject matter
into mathematics textbooks (for a critical under consideration, their preknowledge before
description of this feature of Stoffdidaktik see a teaching (experiment), and the consequences
Jahnke 1998, p. 68). after a teaching experiment. Traditional
Stoffdidaktik was not interested in the human
In the preface of a book series, which Griesel side of the teaching-learning process nor did it
himself identifies as a prototypical example of traditionally look into the consequences of a
Stoffdidaktik, Griesel (1971, p. 7) identifies six certain setup of the teaching-learning process
areas, which are important for the progress of (for a detailed comparison, see Str€aßer 1996).
didactics of mathematics. The first two are of The reason for the relative negligence of these
utmost importance, especially the first one: aspects may be the idea of the one and only best
research into the content, the methods, and the way to teach a certain subject matter, which
application of mathematics; and didactical ideas allows to forget about alternatives.
and insights, “which make it possible to attend
better, or at all, to a subject area within instruc- The notion of “pedagogical content
tion.” For him, the first area was most successful knowledge” (“PCK”), which was introduced
at that time. The other four influential factors into the debate on the professional knowledge of
are general experience, statistically based teachers by Shulman (1987), is also close to
evidence about instruction, insights into the Stoffdidaktik. With PCK as “understanding of
mathematical learning process, and the develop- how particular topics, problems, or issues are
ment-psychological and sociological conditions organized, represented, and adapted to the
(translations from Steinbring 2011, p. 45). With diverse interests and abilities of learners, and
these statements, Griesel identified some limita- presented for instruction” (Shulman, 1987, p. 8),
tions of “content-oriented analysis” using math- PCK shares a close link to subject matter knowl-
ematical methods. He even went as far as calling edge with Stoffdidaktik. In contrast,
them meaningless if the necessary follow-up Stoffdidaktik tends to be more authoritarian,
empirical investigations show that the results of looking for the best one and only mathematical
Stoffdidaktik are meaningless for the learning solution, but cares less for the personal aspects of
process of mathematics. the teaching and learning process – with
Shulman’s concept of “content knowledge”
From an international perspective, the confirming the importance of disciplinary
approach closest to Stoffdidaktik is the French mathematics for the teaching and learning of the
approach of “ingenierie didactique” – didactical subject.
engineering – especially its “a priori analysis”
part. Stoffdidaktik shares with didactical engi- Some Examples
neering a focus on disciplinary mathematics, its A rather comprehensive exemplar of
history, and its epistemology. Especially for the Stoffdidaktik is the book entitled “Mathematik
“a priori” part, didactical engineering as well as wirklich verstehen” (“Really understanding
Stoffdidaktik in its entirety heavily depend on a mathematics”) by Kirsch (1987), which covers a
detailed analysis of the content, history, and epis- major part of lower secondary mathematics (espe-
temology of the mathematical content matter cially numbers and functions with a foundation in
under analysis. If taken as a preparation to a set theory). In some sense, the title marks an impor-
teaching experiment, the a priori part of tant difference: While mathematics tends to prove
didactical engineering tends to enact the very a statement, Stoffdidaktik aims at understanding
same activities and methods, as Stoffdidaktik the statement. Vollrath (1974/2006) can be taken
would apply. A difference between these two

Stoffdidaktik in Mathematics Education S569

as the complement on equations and elementary Recent Development S
(school) algebra. Holland (1996/2007) covers Reichel’s (1995) text indicates a development with
geometry in lower secondary mathematics teach- Stoffdidaktik in the German-speaking didactics of
ing. Danckwerts and Vogel (2006) with a book on mathematics: In the last quarter of the twentieth
teaching calculus entitled “Analysis verst€andlich century, Stoffdidaktik has lost its importance as
unterrichten” (How to teach calculus understand- one of the most important and widespread research
ably) confirm the effort of Stoffdidaktik to teach approaches in the German-speaking community.
mathematics in an accessible, understandable Young researchers widened the narrow perspective
manner. of traditional Stoffdidaktik by taking into account
more aspects than disciplinary mathematics, its
The internationally best known example of history, and epistemology. In this respect, a major
Stoffdidaktik is a plenary by Kirsch at the move was the suggestion of taking into account the
ICME congress in Karlsruhe (Kirsch, 1977) beliefs, ideas, and knowledge of the learner of
entitled “Aspects of Simplification in Mathemat- mathematics. Vom Hofe (1995) was the most
ics Teaching.” The title mirrors the utmost prominent advocate of this opening up of
importance of disciplinary mathematics, which Stoffdidaktik to the learner by suggesting to care
Stoffdidaktik prepares for teaching this subject. for the “Grundvorstellungen” (i.e., basic beliefs
In order to make mathematics accessible in teach- and ideas) of the learner to link mathematics, the
ing, Kirsch suggests four activities: individual (especially: learner), and reality.
• Concentration on the mathematical heart of Grundvorstellungen are seen as a way to
better understand sense making of an individual,
the matter ways of representation that an individual develops,
• Including the “surroundings” of mathematics and her/his way of using ideas and concepts
• Recognizing and activating preexisting with respect to reality. In doing so, the concept of
Grundvorstellungen is not only meant
knowledge as a normative idea to inform curriculum
• Changing the mode of representation construction but also as a way to describe
which are often summarized under the concept the strategies and mindsets of a (potential or
“elementarization” – with a long tradition in actual) learner. Four concepts structure this
Germany (see the famous series of books by approach to didactics of mathematics, namely, the
Felix Klein Mathematics from an Advanced individual, the context, the Grundvorstellung, and
Standpoint – original title: Elementarmathematik mathematics.
vom ho¨heren Standpunkt aus).
Cross-References
For Reichel (1995), “the so-called
Stoffdidaktik was the most important part” of ▶ Cultural Influences in Mathematics Education
German didactics of Mathematics. In his “per- ▶ Curriculum Resources and Textbooks in
haps amplified understanding of that term,” he
adds a list of 15 research areas to traditional Mathematics Education
Stoffdidaktik. Besides other areas mentioned, ▶ Design Research in Mathematics Education
Stoffdidaktik should play a major role when ana- ▶ Didactic Transposition in Mathematics
lyzing the image of mathematics, in assessment
questions, in research on using computers, and on Education
language and (teaching) mathematics – to cite but ▶ History of Research in Mathematics Education
a few from Reichel’s list. This already shows that ▶ International Comparative Studies in
Reichel has a concept of Stoffdidaktik which
clearly goes further than the traditional Mathematics: An Overview
epistemology of school mathematics, content ▶ Mathematics Teacher Education Organization,
analysis, elementarization, and teaching methods
with Stoffdidaktik as a major part of research Curriculum, and Outcomes
work in didactics of mathematics.

S 570 Structure of the Observed Learning Outcome (SOLO) Model

▶ Mathematics Teachers and Curricula Structure of the Observed Learning
▶ Pedagogical Content Knowledge in Outcome (SOLO) Model

Mathematics Education John Pegg
▶ Teacher as Researcher in Mathematics Education, University of New England,
Armidale, NSW, Australia
Education

References Keywords

Danckwerts R, Vogel D (2006) Analysis Cognitive development; neo –Piagetian model;
verst€andlich unterrichten. Spektrum Akademischer, Nature of learning; Assessment framework
Heidelberg
Definition
Griesel H (1971) Die Neue Mathematik f€ur Lehrer und
Studenten (Band 1). Hermann Schroedel Verlag KG, Biggs and Collis (1982) described the Structure
Hannover of the Observed Learning Outcome (SOLO)
Model (commonly referred to as the SOLO
Griesel H (1974) U€ berlegungen zur Didaktik der Taxonomy) as a general model of intellectual devel-
Mathematik als Wissenschaft. Zentralblatt fu€r opment. SOLO had its origins in the stage develop-
Didaktik der Mathematik 6(3):115–119 ment ideas of Piaget and the information processing
concepts of the 1970s. It can be considered within
Holland G (1996/2007) Geometrie in der Sekundarstufe. the broad research framework referred to as neo-
Spektrum, Akad. Verl., Heidelberg/Berlin/Oxford, Piagetian. As such, SOLO has much in common
2nd edn. Hildesheim, Franzbecker, 3rd edn with the writings of Case (1992), Halford (1993),
and Fischer and Knight (1990) to name a few.
Jahnke T (1998) Zur Kritik und Bedeutung der
Stoffdidaktik. Math Didact 21(2):61–74 Characteristics

Kirsch A (1977) Aspects of simplification in mathematics Central to SOLO is the view that there are
teaching. In: Athen H, Kunle H (eds) Proceedings of “natural” stages in the growth of learning any com-
the third international congress on mathematical plex material or skill. Also, these stages “are similar
education. University of Karlsruhe, Karlsruhe, to, but not identical with, the developmental stages
Zentralblatt fu€r Didaktik der Mathematik, pp 98–120 in thinking described by Piaget and his co-workers”
(Biggs and Collis 1982, p. 15).
Kirsch A (1987) Mathematik wirklich verstehen – Eine
Einf€uhrung in ihre Grundbegriffe und Denkweisen. The SOLO Model has its roots in the analysis
Aulis Verlag Deubner, Ko¨ln of responses to questions posed in a variety of
subject/topic areas. The focus of analysis was on
Reichel H-C (1995) Hat die Stoffdidaktik specifying “how well” something was learned, as
Zukunft? Zentralblatt fu€r Didaktik der Mathematik a balance to the more traditional approach of
27(6):178–187 “how much” has been learned. The insight of
Biggs and Collis was that the structural organiza-
Shulman LS (1987) Knowledge and teaching: tion of knowledge was the issue that discrimi-
foundations of the new reform. Harv Educ Rev nated well-learned from poorly learned material.
57(1):1–23
For SOLO, learners actively construct their
Steinbring H (2011) Changed views on mathematical understandings by building upon earlier
knowledge in the course of didactical theory
development: independent corpus of scientific
knowledge or result of social constructions? In:
Rowland T, Ruthven K (eds) Mathematical knowledge
in teaching. Springer, Heidelberg/London/New York,
pp 43–64

Str€aßer R (1996) Stoffdidaktik und Inge´nierie didactique –
ein Vergleich. In: Kadunz G, Kautschitsch H,
Ossimitz G, Schneider E (eds) Trends und
perspektiven. Beitr€age zum 7. Internationalen K€artner
Symposium zur “Didaktik der Mathematik” in
Klagenfurt vom 26.-30.9.1994. Ho¨lder-Pichler-
Tempsky, Wien, pp 369–376

Vollrath H-J (1974) Didaktik der Algebra. Stuttgart, Klett
(later editions in 1994, 1999, 2003; from 2006 with
H-G Weigand as co-author)

vom Hofe R (1995) Grundvorstellungen mathematischer
Inhalte. Spektrum, Heidelberg/Berlin/Oxford

Structure of the Observed Learning Outcome (SOLO) Model S571

experiences and understandings. In doing this, Concrete Symbolic A person thinks through use of a
learners pass through sequential qualitatively dif- (from 6 or 7 years) symbol system such as written
ferent “stages” that represent a coherent view of language and number systems.
their world. This development is a result of pro- Formal (from 15 or 16 This is the most common mode
cesses of interaction between the learner and his years) addressed in learning in the
or her social and physical environment. upper primary and secondary
Post-formal (possibly school
Hence, understanding is viewed as an individ- at around 22 years)
ual characteristic that is both content and context A person considers more
specific (Biggs and Collis 1991). SOLO emerges abstract concepts. This can be
as a means of describing the underlying structure described as working in terms of
of an individual’s performance at a specific “principles” and “theories.”
time that is determined purely from a response. Students are no longer restricted
Describing the structure of a response is seen to a concrete referent. In its more
as a phenomenon in its own right, without neces- advanced form, it involves the
sarily representing a particular stage of intellec- development of disciplines.
tual development of the learner (Biggs and
Collis 1982). A person is able to question or
challenge the fundamental
The progressive structural complexity in structure of theories or
responses, i.e., cognitive development, is disciplines
described in two ways. First is based upon the
nature or abstractness of the task/response and is It is important to note that the ages pro-
referred to as the mode. The second is based on vided above are approximate indications of
a person’s ability to handle, with increased when a mode becomes available and is con-
sophistication, relevant cues within a mode and text dependent. There is no implication that
is referred to as the level of response. a person who is able to respond in the con-
crete symbolic mode in one context is able or
SOLO Modes would wish to respond in the same mode in
SOLO postulates that all learning occurs in one of other contexts.
five modes of functioning, and these are referred
to as Sensorimotor, Ikonic, Concrete Symbolic, Nevertheless, an implication of this descrip-
Formal, and Post-formal. The five modes of tion is that most students in primary and second-
thinking are described (briefly) below. ary school are capable of operating within the
Concrete Symbolic mode. Because of this, the
Sensorimotor (soon A person reacts to the physical Concrete Symbolic mode is considered the target S
after birth) environment. For the very young mode for instruction at school, and teaching tech-
child, it is the mode in which niques need to be adopted generally to suit
Ikonic (from 2 years) motor skills are acquired. These learners working in this mode. In the case of
play an important part in later a secondary student in certain topics, some may
life as skills associated with still respond to stimuli in the Ikonic mode, while
various sports evolve others may respond with Formal reasoning.

A person internalizes actions in Each mode has its own identity and its own
the form of images. It is in this specific idiosyncratic character. While earlier
mode that the young child acquired modes are needed to move to new
develops words and images that modes of abstraction, these earlier modes remain
can stand for objects and events. available to the individual. Within each mode,
For the adult this mode of responses become increasingly complex as the
functioning assists in the cycle of learning develops. This growth is
appreciation of art and music described in terms of levels using the same
and leads to a form of knowledge generic terms for each mode. A level refers to
referred to as intuitive a pattern of thought revealed in what a learner
says, writes, and/or does.
(continued)

S 572 Students’ Attitude in Mathematics Education

SOLO Levels and its own idiosyncratic selection and use of
Three levels form a cycle of development. These data. The main strength of the framework is in
descriptions of levels indicate an increasing its ability to offer systematic and objective qual-
sophistication in a learner’s ability to handle itative assessments of learning.
tasks associated with a mode.
References
Unistructural: The student focuses on the
domain/problem but uses only one piece of rele- Biggs J, Collis K (1982) Evaluating the quality of learn-
vant data and so may be inconsistent. ing: the SOLO taxonomy. Academic, New York

Multistructural: Two or more pieces of data Biggs J, Collis K (1991) Multimodal learning and the
are used without any relationships perceived quality of intelligent behaviour. In: Rowe H (ed) Intel-
between them. No integration occurs. Some ligence, reconceptualization and measurement.
inconsistency may be apparent. Laurence Erlbaum, New Jersey, pp 57–76

Relational: All data are now available, with Campbell K, Watson J, Collis K (1992) Volume measure-
each piece woven into an overall mosaic of rela- ment and intellectual development. J Struct Learn
tionships. The whole has become a coherent Intell Syst 11:279–298
structure. No inconsistency is present within the
known system. Case R (1992) The mind’s staircase: exploring the con-
ceptual underpinnings of children’s thought and
Each level integrates the level before it thus knowledge. Laurence Erlbaum, New Jersey
logically acquiring the elements of the prior level.
At the same time, each level forms a logical Fischer KW, Knight CC (1990) Cognitive development in
and empirically consistent structured whole. An real children: levels and variations. In: Presseisen B
important consequence is that all learner (ed) Learning and thinking styles: classroom
responses should be able to be allocated to a interaction. National Education Association,
particular level, a mixture of levels, or a mixture Washington, DC
of adjoining levels (referred to as transitional
responses). Halford GS (1993) Children’s understanding: the devel-
opment of mental models. Lawrence Erlbaum,
Research into SOLO levels since the 1990s Hillsdale
(Campbell et al. 1992; Pegg 1992) when stu-
dents’ responses were analyzed over a greater Pegg J (1992) Assessing students’ understanding at the
range of learning situations than had been under- primary and secondary level in the mathematical sci-
taken in earlier research identified more than one ences. In: Izard J, Stephens M (eds) Reshaping assess-
cycle of levels within each mode. In the case of ment practice: assessment in the mathematical
two cycles of growth identified within a mode, sciences under challenge. Australian Council of
the cycles describe a continuous pattern of devel- Educational Research, Melbourne, pp 368–385
opment with Relational level of the first cycle
linking to the Unistructural level in the second Pegg J (2003) Assessment in mathematics: a developmental
cycle. This work has resulted in a greater under- approach. In: Royer JM (ed) Advances in cognition and
standing of cognitive development (Pegg 2003). instruction. Information Age, New York, pp 227–259

Conclusion Students’ Attitude in Mathematics
SOLO is a general framework for systematically Education
assessing quality in terms of both structural and
hierarchical characteristics. The strength of Rosetta Zan and Pietro Di Martino
SOLO is the linking of the hierarchical nature of Dipartimento di Matematica, University of Pisa,
cognitive development (modes) and the cyclical Pisa, Italy
nature of learning (levels). Each mode/level of
functioning has its own integrity and structure Keywords

Affect; Beliefs; Emotions; Interpretative
approach; Students’ failure in mathematics

Students’ Attitude in Mathematics Education S573

Definition investigation are inherited from those used in S
social psychology: in particular, attitude is seen
The construct of attitude has its roots in the as “a learned predisposition or tendency on the
context of social psychology in the early part of part of an individual to respond positively or
the twentieth century. In this context, attitude is negatively to some object, situation, concept, or
considered as a state of readiness that another person” (Aiken 1970, p. 551). Recourse
exerts a dynamic influence upon an individual’s to the adverbs “positively or negatively” is very
response (Allport 1935). evident: indeed a lot of attention by researchers
is focused on the correlation between positive/
In the field of mathematics education, early negative attitude and high/low achievement.
studies about attitude towards mathematics Aiken and Dreger (1961), regarding this alleged
already appeared in 1950, but in many of these correlation between attitude and achievement,
studies the construct is used without a proper even speak of a hypothesis of the etiology of
definition. attitudes towards maths. Aiken (1970, p. 558)
claims: “obviously, the assessment of attitudes
In 1992, McLeod includes attitude among the toward mathematics would be of less concern if
three factors that identify affect (the others are attitudes were not thought to affect performance
emotions and beliefs), describing it as character- in some way.”
ized by moderate intensity and reasonable
stability. But the definition of the construct The Problematic Relationship Between
remains one of the major issues in the recent Attitude and Achievement
research on attitude: as a matter of fact, there is
no general agreement among scholars about the Until the early nineties, research into attitude
very nature of attitude. within the field of mathematics education focuses
much more on developing instruments to mea-
Therefore, in this entry, the issue of the sure attitude (in order to prove a causal correla-
definition of attitude towards mathematics (and tion between positive attitude towards maths and
also of the consequent characterization of achievement in mathematics) rather than on clar-
positive and negative attitude) is developed in ifying the object of the research.
all its complexity.
But the correlation between attitude and
The Origin of the Construct achievement that emerges from the results of
these studies is far from clear. Underlining the
Since the early studies, research into attitude has need for research into attitude, Aiken (1970)
been focused much more on the development of refers to the need of clarifying the nature of the
measuring instruments than towards the theoret- influence of attitude on achievement: he reports
ical definition of the construct, producing the results of many studies in which the correla-
methodological contributions of great impor- tion between attitude and achievement is not
tance, such as those of Thurstone and Likert. evident. Several years later, Ma and Kishor
(1997), analyzing 113 studies about attitude
As far as mathematics education is concerned, towards mathematics, confirmed that the correla-
early studies about attitude towards mathematics tion between positive attitude and achievement is
already appeared in 1950: Dutton uses Thurstone not statistically significant.
scales to measure pupils’ and teachers’ attitudes
towards arithmetic (Dutton 1951). The interest in In order to explain this “failure” in proving
the construct is justified by the vague belief that a causal correlation between positive attitude and
“something called ‘attitude’ plays a crucial role achievement, several causes have been identified:
in learning mathematics” (Neale 1969, p. 631). some related to the inappropriateness of the

In these studies, both the definition of the
construct and the methodological tools of

S 574 Students’ Attitude in Mathematics Education

instruments that had been used to assess choices (e.g., which and how many mathematics
attitude (Leder, 1985) and also achievement courses to take), there are many doubts about the
(Middleton and Spanias 1999), others that under- correlation between emotional disposition and
line the lack of theoretical clarity regarding the achievement (McLeod 1992, refers to data from
nature itself of the construct attitude (Di Martino the Second International Mathematics Study that
and Zan 2001). indicates that Japanese students had a greater
dislike for mathematics than students in other
In particular, until the early nineties, most countries, even though Japanese achievement
studies did not explicitly provide a theoretical was very high). Moreover, a positive emotional
definition of attitude and settled for operational disposition towards mathematics is important,
definitions implied by the instruments used to but not a value per se: it should be linked with
measure attitude (in other words, they implicitly an epistemologically correct view of the
define positive and negative attitude rather than discipline.
giving a characterization of attitude). Up until
that time, in mathematics education, the assess- In terms of multidimensional definition, it is
ment of attitude in mathematics is carried out more problematic to characterize the positive/
almost exclusively through the use of self-report negative dichotomy: it is different if the adjective
scales, generally Likert scales. Leder (1985) “positive” refers to emotions, beliefs, or behav-
claims that these early attempts to measure atti- iors (Zan and Di Martino 2007). The assessment
tudes are exceptionally primitive. These scales tools used in many studies try to overcome this
generally are designed to assess factors such as difficulty returning a single score (the sum of the
perspective towards liking, usefulness, and scores assigned to each item) to describe attitude,
confidence. In mathematics education a number but this is inconsistent with the assumed
of similar scales have been developed and used in multidimensional characterization of the con-
research studies, provoking the critical comment struct. Moreover, the inclusion of the behavioral
by Kulm (1980, p. 365): “researchers should not dimension in the definition of attitude exposes
believe that scales with proper names attached to research to the risk of circularity (using observed
them are the only acceptable way to measure behavior to infer attitude and thereafter
attitudes.” interpreting students’ behavior referring to the
inferred attitudes). In order to avoid such a risk,
Other studies have provided a definition of the Daskalogianni and Simpson (2000) introduce
construct that usually can be classified according a bidimensional definition of attitude that does
to one of the following two typologies: not include the behavioral component.
1. A “simple” definition of attitude which
An interesting perspective is that identified by
describes it as the positive or negative degree Kulm who moves to a more general level.
of affect associated to a certain subject. He considers the attitude construct functional to
2. A “multidimensional” definition which the researcher’s self-posed problems and for
recognizes three components of the attitude: these reason he suggests (Kulm 1980, p. 358)
affective, cognitive, and behavioral. that “it is probably not possible to offer
Both the definitions appear to be problematic: a definition of attitude toward mathematics that
first of all a gap emerges between the assumed would be suitable for all situations, and even if
definitions and the instruments used for one were agreed on, it would probably be too
measuring attitude (Leder 1985). Moreover, the general to be useful.”
characterizations of positive attitude that follow
the definitions are problematic (Di Martino and This claim is linked to an important evolution
Zan 2001). in research about attitude, bringing us to see
In the case of the simple definition, it is quite attitude as “a construct of an observer’s desire
clear that “positive attitude” means “positive” to formulate a story to account for observations,”
emotional disposition. But even if a positive emo- rather than “a quality of an individual” (Ruffell
tional disposition can be related to individual et al. 1998, p. 1).

Students’ Attitude in Mathematics Education S575

Changes of Perspective in Research into of gender differences in mathematics (Sherman S
Attitude in Mathematics Education and Fennema 1977); but he also points out the
problems that emerged in the research about atti-
In the late 80s, two important and intertwined tude (and more general affective construct),
trends strongly influenced research about attitude underlining the need for theoretical studies to
in mathematics education. better clarify the mutual relationship between
affective constructs (emotions, beliefs, and atti-
In the light of the high complexity of human tudes): “research in mathematics education
behavior, there is the gradual affirmation of the needs to develop a more coherent framework for
interpretative paradigm in the social sciences: it research on beliefs, their relationship to attitudes
leads researchers to abandon the attempt of and emotions, and their interaction with
explaining behavior through measurements or cognitive factors in mathematics learning and
general rules based on a cause-effect scheme instruction” (McLeod 1992, p. 581).
and to search for interpretative tools. Research
on attitudes towards mathematics developed, in Moreover, McLeod highlights the need to
the last 20 years, through this paradigm shift develop new observational tools and he also
from a normative-positivistic one to an interpreta- emphasizes the need for more qualitative research.
tive one (Zan et al. 2006). In line with this, the Following this, narrative tools began to assume a
theoretical construct of “attitude towards great relevance in characterizing the construct
mathematics” is no longer a predictive variable (Zan and Di Martino 2007), in observing
for specific behaviors, but a flexible and changes in individual’s attitude (Hannula 2002),
multidimensional interpretative tool, aimed at in assessing influence of cultural and environmen-
describing the interactions between affective and tal factors on attitude (Pepin 2011), and in
cognitive aspects in mathematical activity. It is establishing the relationship between attitudes
useful in supporting researchers as well as teachers and beliefs (Di Martino and Zan 2011).
in interpreting teaching/learning processes and in
designing didactical interventions. The TMA Model: A Definition of Attitude
Grounded on Students’ Narratives
Furthermore, the academic community of
mathematics educators recognized the need for In the framework described, following an
going beyond purely cognitive interpretations of interpretative approach based on the collection
failure in mathematics achievement. Schoenfeld of autobiographical narratives of students
(1987) underlines that lack at a metacognitive (more than 1800 essays with the title “Maths
level may lead students to a bad management of and me” written by students of all grade levels),
their cognitive resources and eventually to Di Martino and Zan (2010) try to identify how
failure, even if there is no lack of knowledge. students describe their relationship with mathe-
The book “Affect and mathematical problem matics. This investigation leads to a theoretical
solving” (Adams and McLeod 1989) features characterization of the construct of attitude
contributions by different scholars regarding the that takes into account students’ viewpoints
influence of affective factors in mathematical about their own experiences with mathematics,
problem solving. i.e., a definition of attitude closely related to
practice. From this study it emerges that when
This gives a new impulse to research on affect, students describe their own relationship to
and therefore on attitude, in mathematics, with mathematics, nearly all of them refer to one or
a particular interest on the characterization of the more of these three dimensions:
constructs. There is the need for a theoretical • Emotions
systematization and a first important attempt in • Vision of mathematics
this direction is done by McLeod (1992). • Perceived competence
He describes the results obtained by research
about attitude, in particular underlining the
significant results concerning the interpretation

S 576 Students’ Attitude in Mathematics Education

EMOTIONAL ▶ Metacognition
DISPOSITION ▶ Teacher Beliefs, Attitudes, and Self-Efficacy

in Mathematics Education

VISION PERCEIVED References
OF COMPETENCE
Adams V, McLeod D (1989) Affect and mathematical
MATHEMATICS problem solving: a new perspective. Springer,
New-York
Students’ Attitude in Mathematics Education,
Fig. 1 The TMA model for attitude (Di Martino and Aiken L (1970) Attitudes toward mathematics. Rev Educ
Zan 2010) Res 40:551–596

These dimensions and their mutual Aiken L, Dreger R (1961) The effect of attitudes on
relationships therefore characterize students’ rela- performance in mathematics. J Educ Psychol 52:19–24
tionship with mathematics, suggesting a three-
dimensional model for attitude (TMA) (Fig. 1): Allport G (1935) Attitudes. In: Murchinson C (ed) Hand-
book of social psychology. Clark University Press,
The multidimensionality highlighted in the Worcester, pp 798–844
model suggests the inadequacy of the positive/
negative dichotomy for attitude which referred Daskalogianni K, Simpson A (2000) Towards a definition
only to the emotional dimension. In particular the of attitude: the relationship between the affective and
model suggests considering an attitude as negative the cognitive in pre-university students. In:
when at least one of the three dimensions is neg- Nakahara T, Koyama M (eds) Proceedings of the
ative. In this way, it is possible to outline different 24th conference of the international group for
profiles of negative attitude towards mathematics. the psychology of mathematics education 3. PME,
Hiroshima, pp 217–224
Moreover, in the study a number of profiles
characterized by failure and unease emerge. Di Martino P, Zan R (2001) Attitude toward
A recurrent element is a low perceived compe- mathematics: some theoretical issues. In: van den
tence even reinforced by repeated school Heuvel-Panhuizen M (ed) Proceedings of the 25th
experience perceived as failures, often joint conference of the international group for the
with an instrumental vision of mathematics. psychology of mathematics education 3. PME,
Utrecht, pp 351–358
As Polo and Zan (2006) claim, often in
teachers’ practice the diagnosis of students’ neg- Di Martino P, Zan R (2010) ‘Me and maths’: towards
ative attitude is a sort of black box, a claim of a definition of attitude grounded on students’
surrender by the teacher rather than an accurate narratives. J Math Teach Educ 13:27–48
interpretation of the student’s behavior capable of
steering future didactical action. The identifica- Di Martino P, Zan R (2011) Attitude towards mathemat-
tion of different profiles of attitude towards ics: a bridge between beliefs and emotions. ZDM Int
mathematics can help teachers to overcome the J Math Educ 43:471–482
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an accurate diagnosis of negative attitude, struc- Dutton W (1951) Attitudes of prospective teachers toward
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dimensions, and aimed at identifying carefully
the student’s attitude profile. Hannula M (2002) Attitude toward mathematics: emotions,
expectations and values. Educ Stud Math 49:25–46
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▶ Affect in Mathematics Education Shumway R (ed) Research in mathematics education.
▶ Gender in Mathematics Education National Council of Teachers of Mathematics, Reston,
pp 356–387

Leder G (1985) Measurement of attitude to mathematics.
Learn Math 34:18–21

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between attitude toward mathematics and achievement
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S

T

Task-Based Interviews in mathematics education to gain knowledge about
Mathematics Education an individual or group of students’ existing
and developing mathematical knowledge and
Carolyn A. Maher1 and Robert Sigley2 problem-solving behaviors.
1Robert B. Davis Institute for Learning,
Graduate School of Education, Rutgers – Task-Based Interview
The State University of New Jersey, The task-based interview, a particular form of
New Brunswick, NJ, USA clinical interview, is designed so that inter-
2Rutgers – The State University of New Jersey, viewees interact not only with the interviewer
New Brunswick, NJ, USA and sometimes a small group but also with a
task environment that is carefully designed for
Keywords purposes of the interview (Goldin 2000). Hence,
a carefully constructed task is a key component
Clinical interview; Teaching experiment; of the task-based interview in mathematics
Problem solving; Task design education (Maher et al. 2011). It is intended to
elicit in subjects estimates of their existing
Definition knowledge, growth in knowledge, and also their
representations of particular mathematical ideas,
Interviews in which a subject or group of structures, and ways of reasoning.
subjects talk while working on a mathematical
task or set of tasks. In preparing a clinical task-based interview,
certain methodological considerations warrant
The Clinical Interview attention and need to be considered in protocol
Task-based interviews have their origin in clinical design. These require attention to issues of
interviews that date back to the time of Piaget, reliability, replicability, task design, and general-
who is credited with pioneering the clinical inter- izability (Goldin 2000). Some interviews are
view. In the early 1960s, the clinical interview structured, with detailed protocols determining,
was used in order to gain a deeper understanding in advance, the interviewer’s interaction and
of children’s cognitive development (e.g., Piaget questions. Other protocols are semi-structured,
1965, 1975). Task-based interviews have been allowing for modifications depending on the
used by researchers in qualitative research in judgment of the researcher. In situations where
the research is exploratory, data from the inter-
views provide a foundation for a more detailed
protocol design. In other, more open-ended

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

T 580 Task-Based Interviews in Mathematics Education

situations, a task is presented and there is minimal an 8th grade girl who has been asked to build
interaction of the researcher, except, perhaps, for a model for (a + b)3 with a set of algebra blocks.
clarification of responses or ensuring that the Stephanie, earlier in the interview, has success-
subjects understand the nature of the task. fully expanded (a + b)3 algebraically to the
expression a3 + 3a2b + 3ab2 + b3 and is challenged
Methodology by the researcher in this clip to find each of the
As subjects are engaged in a mathematical terms as it is modeled in the cube that she builds.
activity, researchers can observe their actions In this example, the researcher is assessing
and record them with audio and/or videotapes Stephanie’s ability to connect her symbolic and
for later, more detailed, analyses. The recordings, physical representations as well as observing
accompanied by transcripts, observers’ notes, how she navigates the transition from a two-
subjects’ work, or other related metadata, dimensional model of (a + b)2 to a model that
provide the data for analyses and further protocol involves three dimensions. All nine of the clips
design. Data from the interviews are then coded, from this interview are available on the Video
analyzed, and reported according to the research Mosaic Collaborative website and can be found
questions initially posed. by searching for the general title: Early algebra
ideas about binomial expansion, Stephanie’s
Techniques and Resources interview four of seven. The full title of clip 5
A variety of techniques are used in task-based is Early algebra ideas about binomial expansion,
interviews, such as thinking aloud and open- Stephanie’s interview four of seven, Clip 5 of 9:
ended prompting (Clement 2000). These can be Building (a + b)3 and identifying the pieces. The
modified and adjusted, according to the judgment link to this clip is http://hdl.rutgers.edu/1782.1/
of the researcher. rucore00000001201.Video.000065479.

Task-based interviews are used to investigate Task-Based Interviews for Assessment
subjects’ existing and developing mathematical Paper and pencil tests are limited in that they do
knowledge and ways of reasoning, how ideas are not address conceptual knowledge and the pro-
represented and elaborated, and how connections cess by which a student does mathematics and
are built to other ideas as they extend their reasons about mathematical ideas and situations.
knowledge (Maher 1998; Maher et al. 2011). Adaptations of the clinical task-based interview
Episodes of clinical, task-based interviews can have been useful in describing student knowledge
be viewed by accessing the Video Mosaic Collab- and providing insight into how mathematical
orative, VMC, website (http://www.videomosaic. solutions to tasks are built by students. By provid-
org) or Private Universe Project in Mathematics ing a structured mathematical task, researchers
(http://www.learner.org/workshops/pupmath). can gain insight into students’ mathematical
An example of a task-based interview in which thinking (Davis 1984). Also, teachers can use
the interviewee is engaged with the interviewer as task-based interviews in their classrooms to
well as the task environment that was designed by study how young children think about and learn
the researchers, see http://hdl.rutgers.edu/1782.1/ mathematics as well as to assess the mathematical
rucore00000001201.Video.000062046. The epi- knowledge of their students (Ginsburg 1977).
sode shows nine-year-old Brandon, explaining Assessments of the mathematical understanding
the notation he used to explain his reasoning. It and ways of reasoning in problem-solving situa-
also shows how the interviewer’s intervention, tions of small groups of students can also be made
asking Brandon if the solution reminded him of with open-ended task-based assessments (Maher
any other problem, prompted him, spontaneously, and Martino 1996). See http://www.learner.org/
to provide a convincing solution for an isomor- workshops/pupmath/workshops/wk2trans.html.
phic problem (Maher and Martino 1998).
A second example from the content strand of An example of a group interview facilitated by
algebra is a task-based interview of Stephanie, researchers Carolyn Maher and Regine Kiczek

Task-Based Interviews in Mathematics Education T581

with four 11th grade students who have been Significance
working on combinatorics problems as a part of There is substantial and growing evidence that
a longitudinal study of children’s mathematical clinical task-based interviews and their variations
reasoning since they were in elementary school provide important insight into subjects’ existing
(Alqahtani, 2011). In this session they were and developing knowledge, problem-solving
discussing the meaning of combinatorial notation behaviors, and ways of reasoning (Newell and
and the addition of Pascal’s identity in terms Simon 1972; Schoenfeld 1985, 2002; Ginsburg
of that notation. They were asked to write the 1997; Goldin 2000; Koichu and Harel 2007;
general form of Pascal’s identity with reference Steffe and Olive 2009; Maher et al. 2011). The
to the coefficients of the binomial expansion. interviews provide data for making students math-
Their work during the session indicates their ematical knowledge explicit. They offer insights
recognition of the isomorphism between the into the creative activity of students in
binomial expansion and the triangle and can be constructing new knowledge as they are engaged
viewed at http://videomosaic.org/viewAnalytic? in independent and collaborative problem solving.
pid¼rutgers-lib:35783.

The Teaching Experiment Cross-References T
According to Steffe and Thompson (2000),
a teaching experiment is an experimental tool ▶ Inquiry-Based Mathematics Education
that derives from Piaget’s clinical interview. In ▶ Problem Solving in Mathematics Education
this context, the interviewer and interviewee’s ▶ Questioning in Mathematics Education
actions are interdependent. However, it differs
from the clinical interview in that the interviewer References
intervenes by experimenting with inputs that
might influence the organizing or reorganizing Alqahtani M (2011) Pascal’s identity. Video annotation.
of an individual’s knowledge in that it traces Video Mosaic Collaborative. http://videomosaic.org/
growth over time. In a teaching experiment, viewAnalytic?pid¼rutgers-lib:35783
researchers create situations and ways of
interacting with students that promote modifica- Clement J (2000) Analysis of clinical interviews:
tion of existing thinking, thereby creating a focus foundation and model viability. In: Lesh R, Kelly AE
for observing the students’ constructive process. (eds) Research design in mathematics and science
There typically is continued interaction with the education. Erlbaum, Hillsdale, pp 547–589
student (or students) by the researcher who is
attentive to major restructuring of and Davis RB (1984) Learning mathematics: the cognitive
scaffolding growth in the student’s building of science approach to mathematics education. Ablex,
knowledge. In these ways, the teaching experi- Norwood
ment makes use of and extends the idea of
a clinical interview. Ginsburg, H. (1977). Children’s arithmetic: The learning
process. New York: Van Nostrand.
Yet a teaching experiment is similar to a task-
based interview in several ways. First, a problem- Ginsburg HP (1997) Entering the child’s mind: the clinical
atic situation is posed. Second, as the interviewer interview in psychological research and practice.
assesses the status of the student’s reasoning in Cambridge University Press, New York
the process of interacting with the student, new
situations are created in the attempt to better Goldin G (2000) A scientific perspective on structures,
understand the student’s thinking. Also, as in task-based interviews in mathematics education
some task-based interviews, protocols may be research. In: Lesh R, Kelly AE (eds) Research design
modified as observation of critical moments in mathematics and science education. Erlbaum,
suggests (Steffe and Thompson 2000). Hillsdale, pp 517–545

Koichu, B., & Harel, G. (2007). Triadic interaction in
clinical task-based interviews with mathematics
teachers. Educational Studies in Mathematics, 65(3),
349–365.

Maher CA (1998) Constructivism and constructivist
teaching – can they co-exist? In: Bjorkqvist O (ed)
Mathematics teaching from a constructivist point of
view. Abo Akademi, Finland, pp 29–42

T 582 Teacher as Researcher in Mathematics Education

Maher CA, Martino A (1996) Young children invent the 1980s. Cochran-Smith and Lytle (1999)
methods of proof: the gang of four. In: Nesher P, Steffe reviewed papers and books published in the
LP, Cobb P, Greer B, Goldin J (eds) Theories of United States and in England in the 1980s dis-
mathematical learning. Erlbaum, Mahwah, pp 1–21 seminating some experiences of teacher research.
The main feature of the teacher research move-
Maher CA, Martino A (1998) Brandon’s proof and iso- ment during this period seems to be an “explicit
morphism. In: Maher CA (ed) Can teachers help rejection of the authority of professional experts
children make convincing arguments? A glimpse into who produce and accumulate knowledge in
the process, vol 5. Universidade Santa Ursula, Rio de “scientific” research settings for use by others in
Janeiro, pp 77–101 (in Portuguese and English) practical settings” (1999, p. 16). Within this
movement, teachers are no longer considered as
Maher CA, Powell AB, Uptegrove E (2011) Combinator- mere consumers of knowledge produced by
ics and reasoning: representing, justifying and building experts, but as producers and mediators of knowl-
isomorphisms. Springer, New York edge, even if it is local knowledge, to be used in
a specific school or classroom. This knowledge
Newell AM, Simon H (1972) Human problem solving. aims at improving teaching practice.
Prentice-Hall, Englewood Cliffs
In mathematics education worldwide, the
Piaget J (1965) The child’s conception of number. Taylor teachers-as-researchers movement has been the
and Francis, London subject of debate within the mathematics educa-
tors’ community and of several papers presenting
Piaget J (1975) The child’s conception of the world. results of these programs or discussing certain
Littlefield Adams, Totowa aspects of teacher research (see Huillet et al.
2011). In these debates, the contention pivoted
Schoenfeld A (1985) Mathematical problem solving. around whether its outputs could be regarded as
Academic, New York research. Many research endeavors conducted by
teachers do not fill the requisites of formal
Schoenfeld A (2002) Research methods in (mathematics) research, such as systematic data collection and
education. In: English LD (ed) Handbook of analysis, as well as dissemination of the research
international research in mathematics education. results. Some researchers distinguish two forms
Lawrence Erlbaum, Mahwah, pp 435–487 of teacher research in practice: formal research,
aimed at contributing knowledge to the larger
Steffe LP, Olive J (2009) Children’s fractional knowledge. mathematics education community, and less
Springer, New York formal research, also called practical inquiry or
action research, which aims to suggest new ways
Steffe LP, Thompson PW (2000) Teaching experiment of looking at the context and possibilities for
methodology: underlying principles and essential changes in practice (Richardson 1994). A major
elements. In: Lesh R, Kelly AE (eds) Research design aim of most action research projects is the genera-
in mathematics and science education. Erlbaum, tion of knowledge among people in organizational
Hillsdale, pp 267–307 or institutional settings that is actionable – that is,
research that can be used as a basis for conscious
Teacher as Researcher in action (Crawford and Adler 1996).
Mathematics Education
The International Group for the Psychology of
Dany Huillet Mathematics Education (PME) started a working
Faculty of Sciences, University of Eduardo group called “teachers as researchers” in 1988.
Mondlane, Maputo, Mozambique This group met annually for 9 years and
published a book based on contributions from
Keywords its members (Zack et al. 1997). The book

Teacher as researcher; Teacher training; Teacher
knowledge

The term “teacher as researcher” is usually used
to indicate the involvement of teachers in educa-
tional research aiming at improving their own
practice. The teachers-as-researchers movement
emerged in England during the 1960s, in the
context of curriculum reform and extended into

Teacher Beliefs, Attitudes, and Self-Efficacy in Mathematics Education T583

comprised accounts of teachers’ different experi- References T
ences of enquiry in several countries and using
several methods which basically aimed to Benke G, Hospesova´ A, Ticha´ M (2008) The use of action
improve teaching practice. In 2003 (PME27), research in teacher education. In: Krainer K, Wood
members of a plenary panel intituled “Navigating T (eds) The international handbook of mathematics
between theory and practice. Teachers who nav- teacher education, vol 3, Participants in mathematics
igate between their research and their practice” teacher education. Sense, Rotterdam/Taipei,
shared their experience on how they connect their pp 283–307
role of teacher and researcher (Novotna´ et al.
2003). This panel was followed by a discussion Clary M (1992) Vers une formation par la recherche? In:
group called “research by teachers, research with Colomb J (ed) Recherche en didactique: contribution a`
teachers” which met at PME in 2004 and 2005, la formation des maˆıtres. Actes du colloque. INRP,
and working sessions on “teachers researching Paris, pp 237–245
with university academics” (2007–2009).
Cochran-Smith M, Lytle S (1999) The teacher
Some mathematics educators claimed that research movement: a decade later. Educ Res
teachers as researchers typically focus on their 28(7):15–25
pedagogical practice, rarely challenging the
mathematical content of their teaching Crawford K, Adler J (1996) Teachers as researchers in
(Huillet et al. 2011). They support this claim in mathematics education. In: Bishop A, Clements M,
terms of a review of several papers of the Keitel C, Kilpatrick J, Laborde C (eds) International
teachers-as-researchers movement in education. handbook of mathematics education. Kluwer,
In most of the papers reviewed, the focus is on Dortrecht, pp 1187–1205
teachers’ classroom practices. They report on
a study where teachers were not researching Huillet D, Adler J, Berger M (2011) Teachers as
their own practice but the Mathematics for researchers: placing mathematics at the centre. Educ
Teaching (MfT) limits of functions for secondary Change 15(1):17–32
school level. They suggest that teachers get
involved in research that puts mathematics at Novotna et al (2003): Proceedings of the joint meeting
the core: research on Mathematics for Teaching, of PME 27 and PME-NA 25 (Vol. 1, pp. 69–99),
with attention to both mathematical and pedagog- Honolulu, Hawai, July13-18, 2003
ical issues and their intertwining in practice.
Richardson V (1994) Conducting research on practice.
The idea of using research in teacher training Educ Res 23(5):1–10
arose long time ago. Yang (2009) contends that in
China, a school-based teaching research system Yang Y (2009) How a Chinese teacher improved class-
exists since 1952. In 1992, Clary claims that action room teaching in Teaching Research Group: a case
research can become an efficient mean of training. study on Pythagoras theorem teaching in Shanghai.
In recent years, research conducted by teachers has ZDM Math Educ 41:279–296
become an important part of some teacher educa-
tion programs (see Benke et al. 2008). Zack V, Mousley J, Breen C (eds) (1997) Developing
practice teachers’ inquiry and educational change.
Cross-References Deakin University Press, Geelong

Teacher Beliefs, Attitudes, and
Self-Efficacy in Mathematics
Education

Peter Liljedahl1 and Susan Oesterle2
1Faculty of Education, Simon Fraser University,
Burnaby, BC, Canada
2Mathematics Department, Douglas College,
New Westminster, BC, Canada

▶ Communities of Inquiry in Mathematics Keywords
Teacher Education
Beliefs; Attitudes; Self-efficacy; Affect;
▶ Reflective Practitioner in Mathematics Teaching efficacy
Education

T 584 Teacher Beliefs, Attitudes, and Self-Efficacy in Mathematics Education

Beliefs, attitudes, and self-efficacy are all aspects teachers’ espoused beliefs, enacted beliefs, actual
of the affective domain (McLeod 1992). The beliefs, and the attributed beliefs that the
affective domain can be conceptualized as an researchers assign to them.
internal representational system, comprising
emotions, attitudes, beliefs, morals, values, and Attitudes can be defined as “a disposition to
ethics (DeBellis and Goldin 2006). These are respond favourably or unfavourably to an object,
often placed on a continuum, with feelings and person, institution, or event” (Ajzen 1988, p. 4).
emotions at one end, characterized as short-lived Attitudes can be thought of as the responses that
and highly charged, and beliefs at the other end, individuals have to their belief structures. That is,
typified as more cognitive and stable in nature attitudes are the manifestations of beliefs
(Philippou and Christou 2002). In the context of (Liljedahl 2005). Negative attitudes towards
mathematics, the affective domain was introduced mathematics can interfere with teacher learning.
to explain why learners who possessed the Unfortunately, these negative attitudes can be very
cognitive resources to succeed at mathematical difficult to change in adults (Evans 2000).
tasks still failed (Di Martino and Zan 2001; see Research on the relationship between teachers’
also Affect in Mathematics Education). In the attitudes and teacher practice is rare (Philipp 2007).
context of teachers of mathematics, over the last In her cross-cultural study, Ma (1999) found that
30 years there has been a growing interest in how basic attitudes towards mathematics along with
affective factors influence classroom practice, their lack of confidence in their own abilities
specifically with reference to beliefs (Thompson affected teachers’ willingness to engage in mathe-
1992; Philipp 2007), attitudes (Ernest 1989), and matical problem solving with their students. Ernest
self-efficacy (Bandura 1997). (1988) found some indications that attitudes
towards teaching mathematics were more influen-
Philipp (2007) defines beliefs as “the lenses tial in teachers’ practice than their attitudes
through which one looks when interpreting the towards mathematics. Other desirable attitudes of
world” (p. 258). There are many different types of mathematics teachers that have been discussed in
beliefs that may influence teaching, including the literature are curiosity (Simmt et al. 2003),
but not limited to beliefs about mathematics, high motivation for success for themselves and
beliefs about the teaching of mathematics, beliefs their students (Rowan et al. 1997; Kukla-Acevedo
about the learning of mathematics, beliefs 2009), as well as appreciation for the elegance of
about students, beliefs about teachers’ own solutions and for a “good” problem (Ball 2002).
ability to do mathematics, to teach mathematics,
etc. Recognition of the power of beliefs to affect Teachers’ self-efficacy sits on the boundary
teaching has led to investigations into the beliefs between beliefs and attitudes as it also incorpo-
of preservice teachers and the role that their rates emotional factors, i.e., confidence and anxi-
experiences as mathematics students plays in ety. The research often distinguishes between, and
their initial beliefs about what it means to teach sometimes conflates, personal teaching efficacy,
mathematics (cf. Fosnot 1989; Skott 2001) and teachers’ beliefs in their own ability to teach
the role of teacher education programs to reshape effectively, and general teaching efficiency or
these beliefs (Green 1971). Research on teachers’ outcome expectancy, which relates to teachers’
beliefs is complicated by a number of factors, beliefs that teaching can make a difference
including the often blurry boundary between (Tschannen-Moran et al. 1998). Teacher self-
beliefs and knowledge (Wilson and Cooney efficacy has been found to influence teachers’
2002) and beliefs and attitudes/emotions, as attitudes and practice (Riggs and Enochs 1990),
well as challenges in finding ways to measure commitment to teaching (Coladarci 1992), and
beliefs and their impact. There is a substantial student achievement (Ashton and Webb 1986);
amount of literature on consistencies (e.g., however, research in this area is challenged by
Leatham 2006; Liljedahl 2008) and inconsis- difficulties in clearly defining and measuring self-
tencies (e.g., Hoyles 1992; Speer 2005) between efficacy and its impact (Bandura 1993). There has
also been considerable interest in the factors that

Teacher Beliefs, Attitudes, and Self-Efficacy in Mathematics Education T585

influence self-efficacy (Bandura 1997), particu- Green T (1971) The activities of teaching. McGraw-Hill, T
larly in preservice teacher education, as it has New York
been suggested that self-efficacy is most mallea-
ble early in teachers’ careers (Hoy 2004). Inter- Hoy AW (2004) The educational psychology of teacher
estingly, Swars et al. (2009) note that if teachers’ efficacy. Educ Psychol Rev 16:153–176
efficacy beliefs are connected to the traditional
teacher-centered teaching approaches, they will Hoyles C (1992) Mathematics teaching and mathematics
be in tension with the constructivist philosophies teachers: a meta-case study. Learn Math 12(3):32–44
of current reform curricula in mathematics.
So, if teacher efficacy matters at all, we need to Kukla-Acevedo S (2009) Do teacher characteristics
ensure that it is associated with “appropriate” matter? New results on the effects of teacher
pedagogical beliefs. preparation on student achievement. Econ Educ Rev
28(1):49–57
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