Encyclopedia of Mathematics Education

C 128 Cultural Diversity in Mathematics Education

communities with a stronger tradition of receiv- as participants in these practices), and (iii) social
ing immigrants, some teachers themselves have representations (the images and understandings
already had to negotiate the practices of the home that enable people to make sense of mathematical
and school culture. This complex situation may practices, such as images of learners and the
add insight into the ways that cultural differences learning process and views of mathematical
and identities come to be constructed as signifi- knowledge). These understandings emerged from
cant for the school mathematical learning. An looking at diversity from complementary perspec-
examination of studies carried out in culturally tives. One perspective focuses on the discontinu-
diverse schools in Europe reveals two views in ities between the cultural practices, and the other
the way teachers make sense of the cultural and on how discontinuity is experienced by the person
ethnic background on their students’ mathemati- as a participant in school mathematical practices.
cal learning (Abreu and Cline 2007; Gorgorio´ and This second perspective is more recent and is key
Abreu 2009). One view stresses “playing down for the development of approaches where diversity
differences” and the other “accepting differ- becomes a resource. The extent to which
ences.” The view of playing down cultural differ- approaches that stress the importance of cultural
ences draws upon representations of mathematics identities can be used as resources for change from
as a culture-free subject (that it is the same around culture-free to culturally sensitive practices in
the world). This view can also draw on a repre- mathematics education is a question for further
sentation of the child’s ability as the key determi- research. The fact that the views of cultural iden-
nant factor in their mathematical learning. The tities as mediators of school mathematical learning
universal construction of children takes priority are still marginalized can be seen as a consequence
over their ethnic and cultural backgrounds. of the dominant cultural practices and representa-
Treating everyone as equal based on their merits tions. For example, this can include practices in
is also used as a justification for not taking into teacher training, where little attention is given to
account cultural differences. The lack of recogni- preparing teachers to understand the cultural
tion of the cultural nature of mathematical prac- nature of (mathematical) learning and human
tices may restrict opportunities for students to development (see also, ▶ Immigrant Students in
openly negotiate the differences at school. This Mathematics Education). Secondly, implicit con-
way, diversity may become a problem instead of a ceptions of the social and emotional development
resource. The alternative positioning of accepting of the child at school draw on representations of
cultural differences represents a minority voice childhood which often do not take into account the
outside the consensus that mathematics is cultural diversity of current societies.
a culture-free subject and that ability is the main
factor in the mathematical learning. Cross-References

Conclusion ▶ Ethnomathematics
Diversity in mathematics education includes com- ▶ Immigrant Students in Mathematics Education
plex and multilayered phenomena that can be ▶ Situated Cognition in Mathematics Education
explored from different perspectives. Drawing on ▶ Theories of Learning Mathematics
sociocultural psychology, empirical research on
uses and learning of mathematics in different References
cultural practices offered key insights on under-
standings of cultural diversity considering (i) de Abreu G (2008) From mathematics learning out-of-
mathematical tools (the specific forms of mathe- school to multicultural classrooms: a cultural psychol-
matical knowledge associated with cultural groups ogy perspective. In: English L (ed) Handbook of inter-
and sociocultural practices), (ii) identities (the national research in mathematics education, 2nd edn.
ways differences are experienced by the students Lawrence Erlbaum, Mahwah, pp 352–383
and the impact on how they construct themselves

Cultural Influences in Mathematics Education C129

de Abreu G, Cline T (2007) Social valorization of math- Cultural Influences in Mathematics C
ematical practices: the implications for learners in Education
multicultural schools. In: Nasir N, Cobb P (eds) Diver-
sity, equity, and access to mathematical ideas. Olof Bjorg Steinthorsdottir1 and Abbe Herzig2
Teachers College Press, New York, pp 118–131 1University of Northern Iowa, Cedar Falls,
IA, USA
Asher M (2008) Ethnomathematics. In: Selin H (ed) Ency- 2University of Albany, Albany, NY, USA
clopaedia of the history of science, technology, and
medicine in non-western cultures, Springer reference. Keywords
Springer, Berlin
Achievement; Equity; Efficacy; Gender; Gender
Bishop A (1988) Mathematical enculturation: a cultural differences; Social contexts
perspective on mathematics education. Kluwer,
Dordrecht Definition

Bishop A (2002) The transition experience of immigrant For many students, mathematics is a barrier,
secondary school students: dilemmas and decisions. often having a profound effect on their further edu-
In: de Abreu G, Bishop A, Presmeg N (eds) Transitions cational and career opportunities. Some authors
between contexts of mathematical practices. Kluwer, have looked beyond features of students, curricu-
Dordrecht, pp 53–79 lum, and pedagogy to argue that the political, eco-
nomic, and social contexts of schooling also need to
Boaler J (2007) Paying the price for “sugar and spice”: be considered in this pedagogical equation (Apple
shifting the analytical lens in equity research. In: Nasir 1992; Tate 1997). Similarly, some scholars have
NS, Cobb P (eds) Improving access to mathematics: studied cultural issues affecting the study of math-
diversity and equity in the classroom. Teachers Col- ematics, including features of the societal culture in
lege, New York, pp 24–36 which education is situated and how those cultures
affect who succeeds in them (Else-Quest et al.
Cobb P, Hodge LL (2002) A relational perspective on issues 2010). In this essay, we examine cultural influences,
of cultural diversity and equity as they play out in the both within and surrounding mathematics, and their
mathematics classroom. Math Think Learn 4:249–284 effects on who succeeds in mathematics.

Cole M (1996) Cultural psychology. The Belknap Press of A Review of the Literature
Harvard University Press, Cambridge, MA
A substantial body of research has been devoted to
Crafter S, de Abreu G (2010) Constructing identities in understanding the relatively small proportion of
multicultural learning contexts. Mind Cult Act women and students of color who participate in
17(2):102–118 advanced mathematics and careers. Researchers
have explored the reasons for the differences in
De Haan M, Elbers E (2008) Diversity in the construction the achievement, attitudes, learning styles, strategy
of modes of collaboration in multiethnic classrooms. use, and persistence between girls and boys and
In: van Oers B, Wardekker W, Elbers E, van der Veer among students of different races, ethnicities,
R (eds) The transformation of learning: advances in social classes, and language proficiencies.
cultural-historical activity theory. Cambridge Univer- Although these differences have generally
sity Press, Cambridge, pp 219–241

Gorgorio´ N, de Abreu G (2009) Social representations
as mediators of practice in mathematics class-
rooms with immigrant students. Educ Stud Math
72:61–76

Gorgorio´ N, Planas N, Vilella X (2002) Immigrant chil-
dren learning mathematics in mainstream schools. In:
de Abreu G, Bishop A, Presmeg N (eds) Transitions
between contexts of mathematical practice. Kluwer,
Dordrecht, pp 23–52

Nasir NS, Cobb P (2007) Improving access to mathemat-
ics: diversity and equity in the classroom. Teachers
College Press, New York

Nunes T, Schliemann A, Carraher D (1993) Street math-
ematics and school mathematics. Cambridge Univer-
sity Press, Cambridge

Secada WG (1995) Social and critical dimensions for
equity in mathematics education. In: Secada W,
Fennema E, Adajian L (eds) New directions for equity
in mathematics education. Cambridge University
Press, New York, pp 146–164

Vygotsky L (1978) Mind in society: the development of
higher psychological processes. Harvard University
Press, Cambridge, MA

C 130 Cultural Influences in Mathematics Education

decreased, differences among groups remain, as do nature of mathematics. In an abstract context
important differences among countries (Else-Quest like the one that is common in Western mathe-
et al. 2010; Palsdottir and Sriraman 2010). Unfor- matics, a quest for certain types of understanding
tunately, the work of many researchers has had the can actually interfere with success, as when stu-
paradoxical effect of creating a discourse that dents look to understand, for example, What does
females and students of color “can’t do math” this have to do with the world? With my world?
(Fennema 2000). As a result, the identification of With my life? (Johnston 1995). Mathematics is
“who” succeeds in mathematics is too often per- often taught as a set of manipulations that lead to
ceived through the lens of a deficit model: When predetermined results or, at a more advanced level,
groups of students do not succeed or persist in as sequence of deductive proofs of clearly stated
mathematics, the reason, or so it is sometimes theorems, with little (if any) representation of the
framed in the literature, is a problem with the roles of intuition, creativity, insight, or trial and
students in groups themselves, rather than as the error, which give rise to those results and which
result of a broader social or cultural issue. give them meaning (Herzig 2002, 2010; Sriraman
and Steinthorsdottir 2007). Mathematics as it is
Building students’ sense of belongingness in commonly presented in classrooms in education is
mathematics has been proposed as a critical fea- isolated from its social and personal contexts and
ture of an equitable K-12 education (Allexsaht- applications, devoid of aesthetic considerations.
Snider and Hart 2001; Ladson-Billings 1997).
Martha Allexsaht-Snider and Laurie Hart (2001) Aside from the way mathematics is presented
defined belonging as “the extent to which each in the classroom, the way that mathematics stu-
student senses that she or he belongs as an impor- dents are perceived outside the mathematics
tant and active participant” in mathematics classroom also affects students’ involvement
(p. 97) and have argued that an important purpose and dedication to mathematics (Campbell 1995;
of schooling is to facilitate students’ sense of Damarin 2000).
belongingness and engagement with mathemat-
ics. A similar construct has been proposed at the As Noddings (1996) argued, mathematics
doctoral level, with several authors arguing educators need to find ways to make the social
belonging to or integrating into the departmental world of mathematics – its culture – more acces-
communities is important for student persistence sible to a broader range of people, and the world
(e.g., Herzig 2002, 2004a, 2010). outside of mathematics needs to change its per-
ception of those who succeed within it. Only then
There are (at least) two cultural aspects to stu- can more students, including females and people
dents’ development of a sense that they belong in of color, find a way come to feel that they truly
mathematics: (1) features of mathematics itself, as belong in some part of the mathematics world.
mathematics is presented in classrooms, and (2) the
way the broader society perceives mathematics Suzanne Damarin (2000) compared people
ability and the students who succeed in math. with mathematical ability to “marked categories”
such as women, people of color, criminals, peo-
First, mathematics is often taught in highly ple of disability, and homosexuals and identified
abstracted ways, with little or no explicit connec- these characteristics:
tion to other mathematical ideas, ideas outside of 1. Members of marked categories are ridiculed
mathematics, or the mathematical “big picture”
(Herzig 2002; Stage and Maple 1996). Some and maligned, and descriptions of marked cat-
feminist scholars have challenged the predomi- egories are used to harass, tease, and discipline
nance of abstraction in mathematics. Betty Johnston members of the larger society.
(1995) argued that abstraction in mathematics is 2. Members of marked categories are portrayed
a consequence of modern industrial society, which as incompetent in dealing with daily life.
itself is based on the idea of separating things into 3. In institutions designed to meet the needs of
manageable pieces, distinct from their context. This all, the needs of members of marked catego-
abstraction of mathematics denies the social ries are deferred to the needs of the members
of unmarked categories.

Cultural Influences in Mathematics Education C131

4. Members of marked categories are feared as “deviant” within each of the communities to C
powerful even as they are marked as powerless. which they belong.

5. Explicit or social marking serves to define In summary, researchers have made great strides
communities of the marked. in understanding why mathematics has generally
attracted to certain types of students. Rather than
6. Membership in multiple marked categories studying what is different about women and minor-
places individuals in the margins of each ities – groups that have been viewed as unsuccessful
marked community. in mathematics – studies now strive to ascertain
cultural and societal obstacles for these groups. In
7. The study of a marked category leads to the addition, the literature has shown that students are
construction and study of the complementary most engaged when in an educational environment
class of people. that fosters belonging, which can be difficult in the
mathematics field. The stereotypical views of math-
8. The unmarked category is generally larger than ematics students can make it particularly challeng-
the marked category; even when this is not the ing for women and minorities to enter the field. In
case, the marked category is not recognized as addition, the mathematically capable may not wish
the majority (Damarin 2000, pp. 72–74). to be socially or culturally marked as such due to the
Given the common perceptions of mathemat- preconceived notions many have of mathematics
students. However, by understanding the cultural
ics students as being white, male, childless, and and societal issues in the mathematics field,
socially inept, having few interests outside of researchers and educators can begin to implement
mathematics, students who explicitly do not fit policies and strategies to create more equitable
with this described group might conclude that learning environments and atmospheres.
they do not wish to fit in. Thus belonging in
mathematics might not be an entirely good Cross-References
thing, as it “marks” a student as deviant and as
socially inept. Herzig (2004b) found that some ▶ Critical Mathematics Education
female graduate students described ways that ▶ Cultural Diversity in Mathematics Education
they worked to distance themselves from some
of these common constructions of ineptness and References
social deviance, which, paradoxically, led them
to resist belonging in mathematics. Allexsaht-Snider M, Hart LE (2001) “Mathematics for
all”: how do we get there? Theory Pract 40(2):93–101
Damarin (2000) argued that membership in
the deviant category provides the “deviant” with Apple MW (1992) Do the standards go far enough?
a community with which to affiliate: Being iden- Power, policy, and practice in mathematics education.
tified and marked as mathematically able encour- J Res Math Educ 23(5):412–431
ages mathematics students to form a community
among themselves – if there are enough of them Campbell PB (1995) Redefining the ‘girl problem’ in math-
and if they have the social facility needed. Unfor- ematics. In: Secada WG, Fennema E, Adajian LB (eds)
tunately, females are members of (at least) two New directions for equity in mathematics education.
marked categories, and the double marking is not Cambridge University Press, New York, pp 225–241
merely additive: That is, females are constructed
as deviant as females separately within each Damarin SK (2000) The mathematically able as a marked
marked category in which they are placed. First, category. Gend Educ 12(1):69–85
they are marked as girls and women, but among
girls and women, their mathematical ability Else-Quest N, Janet S, Linn MC (2010) Cross-national
defines them as deviant. Second, given common patterns of gender differences in mathematics:
stereotypes of mathematics as a male domain, a meta-analysis. Psychol Bull 136(n1):103–127
mathematical women are marked among mathe-
maticians as not actually being mathematicians. Fennema E (2000) Gender and mathematics: what is
For women of color, the marking is threefold and known and what do I wish was known? Paper
even more complex, making women of color presented at the fifth annual forum of the National
Institute for Science Education, Detroit

C 132 Curriculum Resources and Textbooks in Mathematics Education

Herzig AH (2002) Where have all the students Definition of Curriculum Resources
gone? Participation of doctoral students in authentic
mathematical activity as a necessary condition for We define mathematics curriculum resources as
persistence toward the Ph.D. Educ Stud Math all the resources that are developed and used by
50(2):177–212 teachers and pupils in their interaction with math-
ematics in/for teaching and learning, inside and
Herzig AH (2004a) Becoming mathematicians: women outside the classroom. Curriculum resources
and students of color choosing and leaving doctoral would thus include the following:
mathematics. Rev Educ Res 74(2):171–214 • Text resources, such as textbooks, teacher cur-

Herzig AH (2004b) “Slaughtering this beautiful math”: ricular guidelines, websites, student sheets, and
graduate women choosing and leaving mathematics. syllabi
Gend Educ 16(3):379–395 • Other material resources, such as manipula-
tives and calculators
Herzig AH (2010) Women belonging in the social worlds • ICT-based resources, such as computer software
of graduate mathematics. Mont Math Enthus Mathematics curriculum resources, and in par-
7(2&3):177–208 ticular textbooks, are an important part of the envi-
ronment in which teachers and students work
Johnston B (1995) Mathematics: an abstracted discourse. (Haggarty and Pepin 2002). Students spend much
In: Rogers P, Kaiser G (eds) Equity in mathematics of their time in classrooms working with and
education: influences of feminism and culture. The exposed to prepared resources, such as textbooks,
Falmer Press, London, pp 226–234 worksheets, and computer software. Teachers often
rely on curriculum materials and textbooks in their
Ladson-Billings G (1997) It doesn’t add up: African day-to-day teaching, when they decide what to
American students’ mathematics achievement. J Res teach, how to teach it, and when they choose the
Math Educ 28(6):697–708 kinds of tasks, exercises, and activities to assign to
their students. In short, curriculum resources consid-
Noddings N (1996) Equity and mathematics: not a simple ered as educational artifacts are vital tools which are
issue. J Res Math Educ 27(5):609–615 of central importance for both teachers and students.
The concept of curriculum resource can also be
Palsdottir G, Sriraman B (2010) Commentary 3 on Fem- viewed in a wider sense, to include “a range of
inist pedagogy and mathematics. In: Sriraman B, human and material resources, as well as mathe-
English L (eds) Theories of mathematics education. matical, cultural and social resources” (Adler 2000,
Springer, Berlin/Heidelberg, pp 467–475 p. 210). This view would include resources such as
discussions between teachers (e.g., oral, on
Sriraman B, Steinthorsdottir O (2007) Emancipatory and a forum), knowledge and qualifications, and con-
social justice perspectives in mathematics education. textual/environmental factors (e.g., class size, time,
Interchange: Q Rev Educ 38(2):195–202 professional leadership, family support). Seen this
way, it makes the study of curriculum resources and
Stage FK, Maple SA (1996) Incompatible goals: narra- the interaction with resources a crucial ingredient of
tives of graduate women in the mathematics pipeline. teacher education and professional development.
Am Educ Res J 33(1):23–51

Tate WF (1997) Race-ethnicity, SES, gender, and
language proficiency trends in mathematics
achievement: an update. J Res Math Educ 28(6):
652–679

Curriculum Resources and
Textbooks in Mathematics Education

Birgit Pepin1 and Ghislaine Gueudet2
1Avd. for lærer- og Tolkeutdanning, Høgskolen
i Sør-Trøndelag, Trondheim, Norway
2CREAD//IUFM de Bretagne, University of
Western Brittany (UBO), Rennes Cedex, France

Keywords Curriculum Resources and Teachers’
Interaction with Resources

Curriculum resources; ICT; Internet; Profes- In this text, we particularly focus on curriculum

sional development; Teacher knowledge; Text- resources and teachers’ interactions with such

books; Use of resources resources. We present a synthesis of the

Curriculum Resources and Textbooks in Mathematics Education C133

state-of-the-art research, organized under two and in turn can be reinterpreted as contributions to C
headings: research about the resources themselves, quality studies. This issue is nevertheless particu-
about their design, and their quality (see section - larly developed in studies concerning digital
Conception, Quality, and Design of Resources) resources, as the profusion of online resources
and research about the use of resources including has created a need for quality criteria. These
the adaptation and transformation by users, in criteria have to take into account the mathemati-
particular teachers (see section The “Use” of cal content, the didactical aspects, and ergonomic
Resources). In section Evolutions and Issues for dimension (Trouche et al. 2013).
Research, we present current perspectives for
research concerning curriculum resources. The quality issue is also relevant in research
about resource design (Ruthven et al. 2009),
Conception, Quality, and Design of Resources which includes mathematical task design.
In terms of analyzing resources (and we include here Research shows that it is crucially important to
digital as well as “hard copy” resources), different provide frequent opportunities for students to
authors have pursued different lines of inquiry: engage in dynamic mathematical activity that is
1. Analyses of mathematical intentions relate to grounded in rich, worthwhile mathematical tasks.
Design-based research (Cobb et al. 2003) is par-
what mathematics is represented, the presenta- ticularly concerned about task design and quality
tion of mathematical knowledge (such as the in order to improve educational practices and
content and structure of mathematics curricu- achievement. This kind of research has clearly
lum materials, e.g., Valverde et al. 2002, or identified the involvement of research teams,
“complexity,” e.g., Schmidt et al. 1997), and where researchers and teachers work together,
also to values and beliefs implicit in curricu- as an essential ingredient for the quality of the
lum materials (e.g., Haggarty and Pepin 2002). tasks designed.
2. Analyses of pedagogical intentions of text mate-
rials address the ways in which students are It has become evident that quality and design
helped (or not) by the text. We can identify at issues are interrelated. Digital means lead to the
least three themes here: ways in which the learner development of new design modes and to new
is helped (or not) within the content of the text to possibilities of collaborative work around the
learn the materials (e.g., Van Dormolen 1986), design of resources. Research on curriculum
within the methods included in the text, or by the resources needs to address questions, such as
rhetorical voice of the text. who are the designers and in which ways does
3. Sociological analyses of texts investigate the designer/group of designers impact on the
mathematics texts, often school texts, with quality of resources? In some countries, national
respect to sociocultural factors, such as pat- “expert communities/centers” (e.g., NCTEM in
terns of social class (e.g., Dowling 1998: dif- the UK, DZLM in Germany, Enciclomedia in
ferentiation in texts between texts/exercises Mexico, Enlaces in Chile) “produce” and broad-
for “high ability” and “low ability” students). cast resources. In addition, particular communi-
4. Analyses of curriculum materials with respect ties and associations (e.g., GeoGebra community,
to different mathematical concepts are numer- Sesamath in France – see Gueudet et al. 2012)
ous (algebra, functions, geometry, etc.). These make resources available.
examine the presentation of the concept itself,
for example, the use of different representa- In the next section, we address issues involved
tions in curriculum texts. Equally, there are with the “use” of resources.
analyses of curriculum materials with respect
to different mathematical competences, such The “Use” of Resources
as “reasoning” or problem-solving. In this section, we address issues related to the
All these analyses, more or less explicitly, raise “use” of resources which include the interactions
the issue of the quality of curriculum resources between teachers and students with resources.

In terms of textbooks, large-scale studies, such
as TIMSS, recognize the importance of textbooks

C 134 Curriculum Resources and Textbooks in Mathematics Education

in teaching and learning and assert that textbooks guide (Remillard et al. 2009). Thus, it can be said
reflect, to a large extent, official curricular inten- that this kind of resources offers personal possibil-
tions and they are said to play an essential role in ities for adaptations, and teachers have always
the didactical transposition of mathematical adapted and transformed resources: selecting,
knowledge. In many countries, school textbooks changing, cutting, and rephrasing. However, the
need approval from the country’s ministry; in other main difference with digital resources, such as
countries, there is a free market for textbooks – digital textbooks, is that these adaptations are tech-
textbooks are generally seen as the “translation of nically anticipated and supported with specific
policy into practice” (Valverde et al. 2002). In technical means (Gueudet et al. 2012).
some countries (e.g., USA), textbooks have been
published with an explicit intention of influencing The two-way process, i.e., the influence of the
teacher practices, and the same holds for digital resources on the teacher and the transformation of
resources. Nevertheless, the research has also the resources by the teacher, can be described as
proven that the impact of such attempts, in terms a genesis. Gueudet, Pepin, and Trouche (2012)
of change of practice, remains limited. distinguish between resources, given to the teacher,
and documents, developed alongside such a gene-
We consider here the interactions between sis. These geneses are central in teacher profes-
students or teachers and resources from the per- sional development. They can be individual but
spective of mediated activity. This leads to con- can also involve groups of teachers working col-
sider a twofold process: on the one hand (1), the laboratively with resources. Research (e.g., Krainer
resource’s features influence the subject’s activ- and Woods 2008) suggests that these evolutions
ity and learning (for teachers, this can lead to can be supported by teacher development programs
policy choices, drawing on resources as a means which propose the design and testing of their own
for teacher education); on the other hand (2), the resources to groups of teachers.
subject shapes his/her resources, according to his/
her knowledge and beliefs. Evolutions and Issues for Research
Viewing curriculum resources as essential tools for
The features of the resources influence students’ teachers to accomplish their goals has been
learning, as well as teachers’ practices and profes- accepted for a long time. However, the vision of
sional learning. This has been evidenced by many the teacher-tool relationship (Remillard et al.
studies investigating the use of curriculum mate- 2009) has changed and needs to be explored in
rials (e.g., Remillard et al. 2009) and of ICT more depth. Moreover, considering the evolution
resources (Hoyles and Lagrange 2010) in teachers’ of resources available for teachers and students
and students’ work. (e.g., their number, nature, design mode/s), this
opens up new directions for research. It leads in
Considering the shaping of resources by particular to view the teacher as a designer of his/
teachers or pupils, the ways teachers, or students, her resources. Based on the interpretation of
use, adapt, or transform the resources depend to teaching as design, and teachers as designers,
a large extent on their knowledge and beliefs. The existing research emphasizes the vital interaction
ways students “use,” for example, a calculator is between the individuals/teachers and the tools/
said to depend on their knowledge about the calcu- resources to accomplish their goals, an accom-
lator and its affordances but also on their knowl- plishment inextricably linked to the use of cultural,
edge of the mathematics (Hoyles and Lagrange social, and physical tools. This opens the door for
2010). The same holds for textbooks (Gueudet many new avenues of researching mathematics
et al. 2012): in order to find support for solving an curriculum resources and their interaction with
exercise, some students will read the course mate- the “learner,” may it be the teacher or the student.
rials, whereas others will search for worked exam-
ples. Similarly, two teachers will use the same Studying resources for the teaching of mathe-
textbook differently. A teacher can focus on the matics requires such a stance, particularly as there
worksheets, or the provision of exercises, while have been various recent evolutions linked to the
another will consider the same book as curriculum

Curriculum Resources and Textbooks in Mathematics Education C135

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Third international handbook of mathematics educa-
tion (pp. 753–790). Springer, New York

Valverde G, Bianchi L, Wolfe R, Schmidt W, Houang
R (2002) According to the book: using TIMSS to
investigate the translation of policy into practice
through the world of textbooks. Kluwer, London

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Dordrecht, pp 141–171

D

Data Handling and Statistics parameters and testing hypotheses) (Scheaffer
Teaching and Learning 2001). Teaching and learning statistics can differ
widely across countries due to cultural, pedagog-
Dani Ben-Zvi ical, and curricular differences and the availability
University of Haifa, Israel of skilled teachers, resources, and technology.

Keywords Changing Views on Teaching Statistics
Over the Years
Statistics; Data handling; Exploratory data
analysis; Teaching and learning statistics; By the 1960s statistics began to make its way
Research on teaching and learning statistics; from being a subject taught for a narrow group
Statistical reasoning; Statistical literacy; of future scientists into the broader tertiary and
Technological tools in statistics learning school curriculum but still with a heavy reliance
on probability. In the 1970s, the reinterpretation
Definition of statistics into separate practices comprising
exploratory data analysis (EDA) and confirma-
Over the past several decades, changes in tory data analysis (CDA, inferential statistics)
perspective as to what constitute statistics and (Tukey 1977) freed certain kinds of data
how statistics should be taught have occurred, analysis from ties to probability-based models,
which resulted in new content, pedagogy and so that the analysis of data could begin to acquire
technology, and extension of teaching to school status as an independent intellectual activity. The
level. At the same time, statistics education has introduction of simple data tools, such as stem
emerged as a distinct discipline with its own and leaf plots and boxplots, paved the way for
research base, professional publications, and con- students at all levels to analyze real data interac-
ferences. There seems to be a large measure of tively without having to spend hours on the
agreement on what content to emphasize in statis- underlying theory, calculations, and complicated
tics education: exploring data (describing patterns procedures. Computers and new pedagogies
and departures from patterns), sampling and would later complete the “data revolution” in
experimentation (planning and conducting statistics education.
a study), anticipating patterns (exploring random
phenomena using probability and simulation), In the 1990s, there was an increasingly
and statistical inference (estimating population strong call for statistics education to focus more
on statistical literacy, reasoning, and thinking.
Wild and Pfannkuch (1999) provided an

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

D 138 Data Handling and Statistics Teaching and Learning

empirically based comprehensive description are uncertainty, randomness, evidence strength,
of the processes involved in the statisticians’ significance, and data production (e.g., experi-
practice of data-based inquiry from problem ment design). In the past few years, researchers
formulation to conclusions. One of the main argu- have been developing ideas of informal statistical
ments presented is that traditional approaches to reasoning in students as a way to build their
teaching statistics focus on skills, procedures, and conceptual understanding of the foundations of
computations, which do not lead students to reason more formal ideas of statistics (Garfield and
or think statistically. Ben-Zvi 2008).

These changes are implicated in a process of What Does Research Tell Us About
democratization that has broadened and diversified Teaching and Learning Statistics?
the backgrounds and motivations of those who
learn statistics at many levels with very diverse Research on teaching and learning statistics has
interests and goals. There is a growing recognition been conducted by researchers from different
that the teaching of statistics is an essential part of disciplines and focused on students at all levels.
sound education since the use of data is increas- Common faulty heuristics, biases, and misconcep-
ingly common in science, society, media, everyday tions were found in adults when they make
life, and almost any profession. judgments and decisions under uncertainty, e.g.,
the representativeness heuristic, law of small num-
A Focus on Statistical Literacy and bers, and gambler’s fallacy (Kahneman et al.
Reasoning 1982). Recognizing these persistent errors,
researchers have explored ways to help people
The goal of teaching statistics is to produce correctly use statistical reasoning, sometimes
statistically educated students who develop using specific methods to overcome or correct
statistical literacy and the ability to reason these types of problems.
statistically. Statistical literacy is the ability to
interpret, critically evaluate, and communicate Another line of inquiry has focused on how to
about statistical information and messages. develop good statistical reasoning and under-
Statistically literate behavior is predicated on the standing, as part of instruction in elementary
joint activation of five interrelated knowledge and secondary mathematics classes. These stud-
bases – literacy, statistical, mathematical, context, ies revealed many difficulties students have
and critical – together with a cluster of supporting with concepts that were believed to be fairly
dispositions and enabling beliefs (Gal 2002). elementary such as data, distribution, center,
Statistical reasoning is the way people reason and variability. The focus of these studies was
with the “big statistical ideas” and make sense to investigate how students begin to understand
of statistical information during a data-based these ideas and how their reasoning develops
activity. Statistical reasoning may involve when using carefully designed activities assisted
connecting one concept to another (e.g., center by technological tools (Shaughnessy 2007).
and spread) or may combine ideas about data and
chance. Statistical reasoning also means under- A newer line of research is the study of
standing and being able to explain statistical pro- preservice or practicing teachers’ knowledge of
cesses and being able to interpret statistical results. statistics and probability and how that understand-
ing develops in different contexts. The research
The “big ideas” of statistics that are most related to teachers’ statistical pedagogical content
important for students to understand and use are knowledge suggests that this knowledge is in
data, statistical models, distribution, center, vari- many cases weak. Many teachers do not consider
ability, comparing groups, sampling and sampling themselves well prepared to teach statistics nor
distributions, statistical inference, and covaria- face their students’ difficulties (Batanero et al.
tion. Additional important underlying concepts 2011).

Data Handling and Statistics Teaching and Learning D139

The studies that focus on teaching and learn- statistical reasoning by providing explicit atten- D
ing statistics at the college level continue to point tion to the basic ideas of statistics (such as the
out the many difficulties tertiary students have in need for data, the importance of data production,
learning, remembering, and using statistics and the omnipresence of variability); focus more on
point to some modest successes. These studies data and concepts, less on theory, and fewer
also serve to illustrate the many practical recipes; and foster active learning (Cobb 1992).
problems faced by college statistics instructors These recommendations require changes of
such as how to incorporate active or collaborative teaching statistics in content (more data analysis,
learning in a large class, whether or not to use an less probability), pedagogy (fewer lectures, more
online or “hybrid” course, or how to select one active learning), and technology (for data analy-
type of software tool as more effective than sis and simulations) (Moore 1997).
another. While teachers would like research
studies to convince them that a particular Statistics at school is usually part of the
teaching method or instructional tool leads to mathematics curriculum. New K–12 curricular
significantly improved student outcomes, that programs set ambitious goals for statistics educa-
kind of evidence is not actually available in the tion, including promoting students’ statistical lit-
research literature. However, recent classroom eracy, reasoning, and understanding (e.g., NCTM
research studies suggest some practical implica- 2000). These reform curricula weave a strand of
tions for teachers. For example, developing data handling into the traditional school mathe-
a deep understanding of statistics concepts matical strands (number and operations, geome-
is quite challenging and should not be try, algebra). Detailed guidelines for teaching
underestimated; it takes time, a well thought-out and assessing statistics at different age levels
learning trajectory, and appropriate technological complement these standards. However, school
tools, activities, and discussion questions. mathematics teachers, which are often not versed
in statistics, find it challenging to teach data han-
Teaching and Learning dling in accordance with these recommendations.

As more and more students study statistics, In order to face this challenge and promote
teachers are faced with many challenges in help- statistical reasoning, good instructional practice
ing these students succeed in learning and appre- consists of implementing inquiry or project-
ciating statistics. The main sources of students’ based learning environments that stimulate
difficulties were identified as: facing statistical students to construct meaningful knowledge.
ideas and rules that are complex, difficult, and/ Garfield and Ben-Zvi (2009) suggest several
or counterintuitive, difficulty with the underlying design principles to develop students’ statistical
mathematics, the context in many statistical prob- reasoning: focus on developing central statistical
lems may mislead the students, and being uncom- ideas rather than on presenting set of tools and
fortable with the messiness of data, the different procedures; use real and motivating data sets to
possible interpretations based on different engage students in making and testing conjec-
assumptions, and the extensive use of writing tures; use classroom activities to support the
and communication skills (Ben-Zvi and Garfield development of students’ reasoning; integrate
2004). the use of appropriate technological tools that
allow students to test their conjectures, explore
The study of statistics should provide students and analyze data, and develop their statistical
with tools and ideas to use in order to react reasoning; promote classroom discourse that
intelligently to quantitative information in the includes statistical arguments and sustained
world around them. Reflecting this need to exchanges that focus on significant statistical
improve students’ ability to reason statistically, ideas; and use assessment to learn what students
teachers of statistics are urged to emphasize know and to monitor the development of
their statistical learning, as well as to evaluate
instructional plans and progress.

D 140 Data Handling and Statistics Teaching and Learning

Technology has changed the way statisticians Cross-References
work and has therefore been changing what
and how statistics is taught. Interactive data ▶ Inquiry-Based Mathematics Education
visualizations allow for the creation of novel ▶ Mathematical Literacy
representations of data. It opens up innovative ▶ Probability Teaching and Learning
possibilities for students to make sense of data
but also place new demands on teachers to assess References
the validity of the arguments that students are
making with these representations and to facili- Batanero C, Burrill G, Reading C (2011) Teaching statis-
tate conversations in productive ways. Several tics in school mathematics: challenges for teaching and
types of technological tools are currently used teacher education (a joint ICMI/IASE study: the 18th
in statistics education to help students understand ICMI study). Springer, Dordrecht
and reason about important statistical ideas.
However, using technological tools and how to Ben-Zvi D, Garfield J (2004) The challenge of developing
avoid common pitfalls are challenging open statistical literacy, reasoning, and thinking. Springer,
issues (Biehler et al. 2013). Dordrecht

These changes in the learning goals of Biehler R, Ben-Zvi D, Bakker A, Makar K (2013)
statistics have led to a corresponding rethinking Technological advances in developing statistical rea-
of how to assess students. It is becoming more soning at the school level. In: Clements MA, Bishop A,
common to use alternative assessments such as Keitel C, Kilpatrick J, Leung F (eds) Third interna-
student projects, reports, and oral presentations tional handbook of mathematics education. Springer,
than in the past. Much attention has been paid to New York, pp 643–690
assess student learning, examine outcomes of
courses, align assessment with learning goals, Cobb GW (1992) Report of the joint ASA/MAA commit-
and alternative methods of assessment. tee on undergraduate statistics. In the American
Statistical Association 1992 proceedings of the section
For Further Research on statistical education. American Statistical Associa-
tion, Alexandria, pp 281–283
Research in statistics education has made signif-
icant progress in understanding students’ difficul- Gal I (2002) Adults’ statistical literacy: meaning, compo-
ties in learning statistics and in offering and nents, responsibilities. Int Stat Rev 70:1–25
evaluating a variety of useful instructional
strategies, learning environments, and tools. Garfield J, Ben-Zvi D (2007) How students learn
However, many challenges are still ahead of sta- statistics revisited: a current review of research on
tistics education, mostly in transforming research teaching and learning statistics. Int Stat Rev 75:
results to practice, evaluating new programs, 372–396
planning and disseminating high-quality assess-
ments, and providing attractive and effective pro- Garfield J, Ben-Zvi D (2008) Developing students’
fessional development to mathematics teachers statistical reasoning: connecting research and teaching
(Garfield and Ben-Zvi 2007). The ongoing efforts practice. Springer, New York
to reform statistics instruction and content have
the potential to both make the learning of statis- Garfield J, Ben-Zvi D (2009) Helping students develop
tics more engaging and prepare a generation of statistical reasoning: implementing a statistical reason-
future citizens that deeply understand the ratio- ing learning environment. Teach Stat 31:72–77
nale, perspective, and key ideas of statistics.
These are skills and knowledge that are crucial Kahneman D, Slovic P, Tversky A (1982) Judgment
in the current information age of data. under uncertainty: heuristics and biases. Cambridge
University Press, New York

Moore DS (1997) New pedagogy and new content: the
case of statistics. Int Stat Rev 65:123–137

NCTM (2000) Principles and standards for school mathemat-
ics. National Council of Teachers of Mathematics, Reston

Scheaffer RL (2001) Statistics education: perusing the
past, embracing the present, and charting the future.
Newsl Sect Stat Educ 7(1). http://www.amstat.org/sec-
tions/educ/newsletter/v7n1/Perusing.html

Shaughnessy JM (2007) Research on statistics learning
and reasoning. In: Lester FK (ed) The second hand-
book of research on mathematics. Information Age
Pub, Charlotte, NC, USA, pp 957–1010

Tukey J (1977) Exploratory data analysis. Addison-Wesley,
Reading

Wild CJ, Pfannkuch M (1999) Statistical thinking in
empirical enquiry. Int Stat Rev 67:223–248

Deaf Children, Special Needs, and Mathematics Learning D141

Deaf Children, Special Needs, and This serious and persistent difficulty is not uni- D
Mathematics Learning versal among children who are deaf; approxi-
mately 15 % perform at age appropriate levels.
Terezinha Nunes The successful minority indicates that deafness is
Department of Education, University of Oxford, not a direct cause of difficulty in mathematics
Oxford, UK learning (see Nunes 2004, for a discussion). This
article considers what is involved in learning math-
Keywords ematics in primary school, why deaf children may
be at a disadvantage, and how schools can support
their learning of mathematics.

Deaf children; Special needs; Mathematics Learning Mathematics in Primary School
difficulty In order to think mathematically, people need to
learn to represent quantities, relations, and space
Characteristics using culturally developed and transmitted think-
ing tools, such as oral and written number sys-
The aim of mathematics instruction in primary tems, graphs, and calculators.
school is to provide a basis for thinking mathe-
matically about the world. This is as basic Some researchers argue that numerical concepts
a skill as literacy in today’s world. Mathematical have a neurological basis that is independent
knowledge is also a means to achieve better of language learning, without which learning
employment and to enter higher education. For mathematics is extremely difficult. In view of the
all these reasons, it is of great importance that pervasiveness of deaf children’s mathematical dif-
deaf children have adequate access to mathemat- ficulties, it could be hypothesized that they have an
ical thinking, but unfortunately most deaf inadequate development of such concepts. Basic
children show a severe delay in mathematics numerical cognition has been studied in research
learning. This delay has been persistent over with young deaf children as well as adults, and the
many years. The average score in mathematics hypothesis has been discarded. Deaf children and
achievement tests for deaf children in the age adults performed at least as well as their hearing
range 8–15 in a study carried out in 1965 showed counterparts in such tasks.
that they were one standard deviation below the
average for hearing children, a result replicated The possible consequences of delays in the
about three decades later. This means that about acquisition of other language-based numerical
50 % of the deaf pupils perform similarly to concepts have also been explored. Two examples
the weakest 15 % of the hearing pupils. Later are knowledge of counting and understanding of
results continue to confirm this weak perfor- arithmetic operations.
mance. In the UK, deaf students aged 16–17
years, at the end of compulsory school, were Counting
found to have a mathematical age between Deaf children lag behind hearing children in
10 and 12.5 years. In the USA, the mathemat- learning to count, independently of whether they
ical ability of 80 % of the deaf 14-year-olds are learning to count orally or in sign (Leybaert
was described as “below basic” in problem and Van Cutsem 2002). Consequently, they
solving and knowledge of mathematical pro- perform less well than hearing children on school-
cedures. A recent systematic review confirmed entry numeracy tests, which typically include tasks
these findings (Gottardis et al. 2011) and that require counting (e.g., “show me 5 blocks”;
analyzed individual differences among deaf “tell me which number is bigger”). This delay
children. could be related to the well-established finding
that deaf people perform less well than hearing
people on serial learning tasks, in which words or
gestures must be learned in an exact sequence, just

D 142 Deaf Children, Special Needs, and Mathematics Learning

as the number string. However, they perform better are compared to same-age hearing peers. Thus, it is
if the tasks are presented differently and use spatial possible that, not knowing number words well
cues to organize the information. These findings are enough to support their mathematical reasoning,
provocative rather than conclusive. First, they raise they do not discover how to use counting to solve
the possibility that deaf children could learn to simple arithmetic problems or important ideas for
count more easily if appropriate visual and spatial their later success, such as the inverse relation
methods were used for teaching rather than serial between addition and subtraction. However,
learning instruction. Second, serial learning is not Nunes and colleagues (2008a, b) have shown that
an appropriate description of counting skills beyond relatively small amounts of teaching can effec-
a certain number (about 20 or 30 in English but this tively improve young deaf children’s performance
may differ depending on the counting system). in the mathematical reasoning and arithmetic
Research with hearing and deaf children shows tasks, with which they were struggling before the
that counting is a structured activity: for example, teaching.
errors are more likely to occur at the boundaries
between decades (e.g., . . .38, 39, 50, 51, 52. . .) than Conclusion
within decades. Therefore, in principle deaf There is little doubt that many deaf children show
children’s initial disadvantage in counting could severe and persistent difficulties in learning math-
be overcome with appropriate teaching methods ematics. Evidence suggests that there is no direct
and with support for mastery of the structure of connection between deafness and problems with
the system. However, it is possible that their initial basic number concepts that precede language.
struggle with learning to count lowers adults’ However, deaf children lag behind hearing
expectations about what they can learn in mathe- children in learning to count, whether orally or in
matics, resulting in less stimulation on mathemati- sign, and at school entry they are behind their
cal tasks, and that it also interferes with the hearing counterparts in mathematical knowledge.
children’s own discoveries in the domain of math- It is possible that falling behind in counting places
ematical reasoning. deaf children at a disadvantage from the adults’
perspective and that they end up receiving less
Early Mathematical Reasoning and Arithmetic stimulation to solve mathematical problems early
Operations on. It is also possible that their own informal
The development of mathematical reasoning starts mathematical knowledge is limited by their diffi-
before school, when children solve practical prob- culty in representing quantities explicitly with
lems using actions, which they learn to combine number words. These findings and conclusions
with counting. When most children start school (at suggest that, if parents and preschool teachers
age 5 or 6), they can already solve simple addition could find visual and spatial ways to teach
and subtraction problems by putting together or counting to deaf children, one would see positive
separating objects and counting, and some can changes in the average achievement of deaf
also solve multiplication and division problems. children in mathematics in the future.
By counting, children use explicit numerical rep-
resentation both for thinking and communicating. Cross-References
When numbers are small and the children can use
objects, deaf children do as well as hearing children ▶ Blind Students, Special Needs, and
in solving these problems, but if the numbers go Mathematics Learning
above 10 or 20, most deaf children fall behind.
When they are compared with hearing children of ▶ Concept Development in Mathematics
the same counting ability, they are just as compe- Education
tent in solving numerical tasks (Leybaert and Van
Cutsem 2002), but their disadvantage in counting is ▶ Inclusive Mathematics Classrooms
reflected in their problem-solving skills when they ▶ Language Disorders, Special Needs and

Mathematics Learning

Deductive Reasoning in Mathematics Education D143

References Mathematical and Historical-Epistemological D
Factors
Gottardis L, Nunes T, Lunt I (2011) A synthesis of 1. What is proof and what are its functions?
research on deaf and hearing children’s mathematical 2. How are proofs constructed, verified, and
achievement. Deaf Educ Int 13:131–150
accepted in the mathematics community?
Leybaert J, Van Cutsem M-N (2002) Counting in sign 3. What are some of the critical phases in the
language. J Exp Child Psychol 81:482–501
development of proof in the history of
Nunes T (2004) Teaching mathematics to deaf children. mathematics?
Whurr, London
Cognitive Factors
Nunes T, Bryant P, Burman D, Bell D, Evans D, 4. What are students’ current conceptions of
Hallett D (2008a) Deaf children’s informal knowledge
of multiplicative reasoning. J Deaf Stud Deaf Educ proof?
14:260–277 5. What are students’ difficulties with proof?
6. What accounts for these difficulties?
Nunes T, Bryant P, Burman D, Bell D, Evans D,
Hallett D et al (2008b) Deaf children’s understanding
of inverse relations. In: Marschark M, Hauser PC (eds)
Deaf cognition. Oxford University Press, Oxford,
pp 201–225

Deductive Reasoning in Instructional-Sociocultural Factors
Mathematics Education 7. Why teach proof?
8. How should proof be taught?
Guershon Harel 9. How are proofs constructed, verified, and
Department of Mathematics, University of accepted in the classroom?
California, San Diego, La Jolla, CA, USA
10. What are the critical phases in the develop-
Keywords ment of proof with the individual student and
within the classroom as a community of
Logic; Reasoning; Proof; History of proof; learners?
Proof schemes
11. What classroom environment is conducive
Definition to the development of the concept of proof
with students?
This entry examines the different facets of deduc-
tive reasoning with respect to the learning and 12. What form of interactions among the students
teaching of mathematical proof. Deductive rea- and between the students and the teacher can
soning may be defined as a formal way of reason- foster students’ conception of proof?
ing, usually top-down [from the general to the
particular] with adherence to logical consistency. 13. What mathematical activities – possibly
with the use of technology – can enhance
Characteristics of Deductive Reasoning students’ conceptions of proof?
The examinations of the learning and teaching of
proof are multifaceted. They address a broad 14. How is proof currently being taught?
range of factors: mathematical, historical-episte- 15. What do teachers need to know in order to
mological, cognitive, sociological, and instruc-
tional. Research questions involving these teach proof effectively?
factors include the following:
Theoretical Factors
16. What theoretical tools seem suitable for

investigating and advancing students’ con-
ceptions of proof?
One’s investigation of these questions is
greatly influenced by her or his philosophical
orientation to the processes of learning and teach-
ing and would reflect her or his conclusion to
questions such as the following: What bearing,
if any, does the epistemology of proof in the

D 144 Deductive Reasoning in Mathematics Education

history of mathematics have on the conceptual of real numbers, including the interpretation that
development of proof with students? What bear- the axioms, are meaningless formulas.
ing, if any, does the way mathematicians con-
struct proofs have on instructional treatments of Considerations of historical-epistemological
proof? What bearing, if any, does everyday justi- developments led to new research questions
fication and argumentation have on students’ with direct bearing on the learning and teaching
proving behaviors in mathematical contexts? of proofs. For example, to what extent and in
what ways is the nature of the content intertwined
Historical-Epistemological Developments with the nature of proving? In geometry, for
Deductive reasoning is a mode of thought com- example, does students’ ability to construct an
monly characterized as a sequence of proposi- image of a point as a dimensionless geometric
tions where one must accept any of the entity impact their ability to develop the Greek
propositions to be true if he or she has accepted conception of proof? What is the cognitive or
the truth of those that preceded it in the sequence. social mechanism by which deductive proving
This mode of thought was conceived by the can be necessitated for the students? The Greek’s
Greeks more than twentieth centuries ago and is construction of their geometric edifice seems
still dominant in the mathematics of our days. So to have been a result of their desire to
remarkable is the Greeks’ achievement that their create a consistent system that was free from
mathematics became a historical mark to which paradoxes. Would paradoxes of the same nature
other kinds of mathematics are compared. The create a similar intellectual need with students?
nature of deductive reasoning varies throughout Students encounter difficulties in moving
history (Kleiner 1991). Of particular contrast is empirical reasoning to deductive reasoning,
Greek mathematics versus modern mathematics. particularly from the Greek’s conception of
In Greek mathematics, the particular entities proof to the modern conception of proof. Exactly
under investigation are idealizations of experien- what are these difficulties? What role does the
tial spatial realities and so also are the proposi- emphasis on form rather than content in modern
tions on the relationships among these entities. mathematics (as opposed to Greek mathematics,
Logical deduction came to be central in the rea- where content is more prominent) play in this
soning process, and it alone necessitated and transition?
cemented the geometric edifice they created. In
constructing their geometry, as is depicted in Classifications of Conceptualizations of Proof
Euclid’s Elements, the Greeks had only one Harel and Sowder (1998) call these conceptuali-
model in mind – that of imageries of idealized zations proof schemes, which they classify into
physical reality. From the vantage point of mod- a system of subcategories. Their taxonomy is
ern mathematics, neither the primitive terms nor organized around three main classes of catego-
the axioms in Greek mathematics were variables, ries: the external conviction proof schemes class,
but constants referring to a single spatial model the empirical proof schemes class, and the deduc-
(Klein 1968; Wilder 1967), as is expressed in the tive proof schemes class. A partial description of
ideal world of Plato’s philosophy. In modern these classes follows.
mathematics, on the other hand, primary terms
and axioms are open to different possible External Conviction Proof Schemes
realizations. An important manifestation of Proving within the external conviction proof
this revolution is the distinction between schemes class depends either (a) on an authority
Euclid’s Elements and Hilbert’s Grundlagen. such as a teacher or a book, (b) on strictly the
The latter characterizes a structure that fits differ- appearance of the argument (e.g., proofs in geom-
ent models, that is, in an abstraction of numerous etry must have a two-column format), or (c) on
models, such as the Euclidean space, the surface symbol manipulations, with the symbols or the
of a half-sphere and the ordered pairs and triples manipulations having no potential coherent sys-
tem of referents (e.g., quantitative and spatial) in

Deductive Reasoning in Mathematics Education D145

the eyes of the student. Accordingly, the external Balacheff (1998), Bell (1976), Hersh (1993), D
conviction proof schemes class consists of three and de Villiers (1999) explicitly address these
categories: the authoritarian proof scheme cate- functions. De Villiers, who built on the work of
gory, the ritual proof scheme category, and the the others scholars mentioned here, raises two
non-referential symbolic proof scheme category. important questions about the role of proof: (a)
“What different functions does proof have within
Empirical Proof Schemes mathematics itself?” and (b) “how can these
Schemes in the empirical proof scheme class are functions be effectively utilized in the classroom
marked by their reliance on either (a) an evidence to make proof a more meaningful activity?”
from examples (sometimes just one example) of According to de Villiers, mathematical proof
direct measurements of quantities, substitutions has six not mutually exclusive roles: Verification
of specific numbers in algebraic expressions, refers to the role of proof as a means to demon-
etc., or (b) perceptions. Accordingly, this class strate the truth of an assertion according to
consists of two categories: the inductive a predetermined set of rules of logic and pre-
proof scheme category and the perceptual proof mises – the axiomatic proof scheme. Explanation
scheme category. is different from verification in that for
a mathematician it is usually insufficient to
Deductive Proof Schemes know only that a statement is true. He or she is
The deductive proof schemes class consists of likely to seek insight into why the assertion is
two subcategories, each consisting of various true. “Mathematicians routinely distinguish
proof schemes: the transformational proof proofs that merely demonstrate from proofs
scheme category and the axiomatic proof scheme which explain” (Steiner 1978, p. 135). For
category. many, the role of mathematical proofs goes
beyond achieving certainty – to show that some-
Classifications of Functions of Proof thing is true; rather, “they’re there to show. . .
In general, the empirical proof schemes and the why [an assertion] is true,” as Gleason, one of
deductive proof schemes categories correspond solvers of the solver of Hilbert’s Fifth Problem
to what Bell (1976) calls empirical justification (Yandell 2002, p. 150), points out. Two millennia
and deductive justification and Balacheff (1988) before him, Aristotle, in his Posterior Analytic,
calls pragmatic justifications and conceptual jus- asserted, “. . . We suppose ourselves to possess
tifications, respectively. Pragmatic justification unqualified scientific knowledge of a thing, as
is further divided into three categories: na¨ıve opposed to knowing it in the accidental way in
empiricism (justification by a few random exam- which the sophist knows, when we think that we
ples), crucial experiment (justification by care- know the cause on which the fact depends as the
fully selected examples), and generic example cause of the fact and of no other” (p. 4).
(justification by an example representing salient Discovery refers to the situations where through
characteristics of a whole class of cases). Con- the process of proving, new results may be dis-
ceptual justification is divided into two catego- covered. For example, one might realize that
ries: thought experiment, where the justification some of the statement conditions can be relaxed,
is disassociated from specific examples, and sym- thereby generalizing the statement to a larger
bolic calculation, where the justification is based class of cases. Or, conversely, through the prov-
solely on transformation of symbols. ing process, one might discover counterexamples
to the assertion, which, in turn, would lead to
These taxonomies are not explicit enough a refinement of the assertion by adding necessary
about many critical functions of proof within restrictions that would eliminate counterexam-
mathematics. There is a need to point to these ples. Systematization refers to the presentation
functions due to their importance in mathematics of verifications in organized forms, where each
in general and to their instructional implications result is derived sequentially from previously
in particular. The work by Hanna (1990),

D 146 Deductive Reasoning in Mathematics Education

established results, definitions, axioms, and pri- starting, simple proofs. They have difficulty
mary terms. Communication refers to the social with indirect proofs, and only a few can complete
interaction about the meaning, validity, and an indirect proof that has been started.
importance of the mathematical knowledge
offered by the proof produced. Intellectual Impact of Instruction
Challenge refers to the mental state of Students who receive more instructional time on
self-realization and fulfillment one can derive developing analytical reasoning by solving unique
from constructing a proof. problems fare noticeably better on overall test
scores. Likewise, students who have been expected
Students’ Proof Schemes to write proofs and who have had classes that
Status studies on students’ conceptualization of emphasized proof were somewhat better than
proof show the absence of the deductive proof other students. It also seems possible to establish
scheme and the pervasiveness of the empirical desirable sociomathematical norms relevant to
proof scheme among students. Students base proof, through careful instruction, often featuring
their responses on the appearances in drawings, the student role in proof-giving. There has been the
and mental pictures alone constitute the meaning concern that the ease with which technology can
of geometric terms. They justify mathematical generate a large number of examples naturally
statements by providing specific examples, not could undercut any student-felt need for deductive
able to distinguish between inductive and deduc- proof schemes. Several studies have shown that
tive arguments. Even more able students may not with careful, nontrivial planning and instruction
understand that no further examples are needed, over a period of time, progress toward deductive
once a proof has been given. Students’ preference proof schemes is possible in technology environ-
for proof is ritualistically and authoritatively ments, where such desiderata as making conjec-
based. For example, when the stated purpose tures and definitions occur.
was to get the best mark, they often felt that
more formal – e.g., algebraic – arguments might Cross-References
be preferable to their first choices. These studies
also show a lack of understanding of the ▶ Abstraction in Mathematics Education
functions of proof in mathematics, often even ▶ Argumentation in Mathematics
among students who had taken geometry ▶ Argumentation in Mathematics Education
and among students for whom the curriculum ▶ Logic in Mathematics Education
pays special attention to conjecturing and
explaining or justifying conclusions in both alge- References
bra and geometry. They believe proofs are used
only to verify facts that they already know and Balacheff N (1988) Aspects of proof in pupils’ practice of
have no sense of a purpose of proof or of its school mathematics. In: Pimm D (ed) Mathematics,
meaning. Students have difficulty understanding teachers and children. Holdder & Stoughton, London,
the role of counterexamples; many do not under- pp 216–235
stand that one counterexample is sufficient to
disprove a conjecture. Students do not see any Bell AW (1976) A study of pupils’ proof-explanations in
need to prove a mathematical proposition, espe- mathematical situations. Educ Stud Math 7:23–40
cially those they considered to be intuitively
obvious. This is the case even in a country like De Villiers MD (1999) Rethinking proof with the Geome-
Japan where the official curriculum emphasizes ter’s sketchpad. Key Curriculum Press, Emeryville
proof. They view proof as the method to examine
and verify a later particular case. Finally, the Hanna G (1990) Some pedagogical aspects of proof.
studies show that students have difficulty writing Interchange 21:6–13
valid simple proofs and constructing, or even
Harel G, Sowder L (1998) Student’s proof schemes:
results from exploratory studies. In: Schoenfeld A,
Kaput J, Dubinsky E (eds) Research in collegiate
mathematics education, vol III. American Mathemati-
cal Society, Providence, pp 234–283

22q11.2 Deletion Syndrome, Special Needs, and Mathematics Learning D147

Hersh R (1993) Proving is convincing and explaining. The majority of children will receive some D
Educ Stud Math 24:389–399 form of support at school although some
individuals experience no difficulties at all.
Klein J (1968) Greek mathematical thought and the origin Indeed a very wide level of individual differ-
of algebra (trans: Brann E). MIT Press, Cambridge, ences in attainment in individuals with 22q is
MA (Original work published 1934) noted in all studies to date.

Kleiner I (1991) Rigor and proof in mathematics: There is consistent evidence that mathemat-
a historical perspective. Math Mag 64(5):291–314 ics skills are weaker than literacy skills in the
majority children with 22q. This profile is
Steiner M (1978) Mathematical explanation. Philos Stud unusual as children with mathematics difficul-
34:135–151 ties are often reported to have comorbid reading
difficulties. Typically, performance on stan-
Yandell BH (2002) The honors class: Hilbert’s problems dardized tests of reading and spelling is within
and their solvers. A. K. Peters, Natick the normal range, but performance on mathe-
matical reasoning and arithmetic tasks is at least
22q11.2 Deletion Syndrome, Special one standard deviation below age norms in chil-
Needs, and Mathematics Learning dren with 22q. Children with 22q specifically
selected to have full scale IQ of at least 70 also
Sophie Brigstocke demonstrate this profile, thereby suggesting that
Department of Psychology, University of York, it is associated with 22q per se rather than low
Heslington, York, UK general ability.

Keywords There are very few studies examining number
skills in detail in children with 22q. De Smedt et al.
Genetic disorder; Mathematics difficulties; (2006, 2007a, b) tested children, selected to have
Cognitive impairment an IQ of more than 70, on a series of computerized
tests assessing performance in number reading
Characteristics and writing, number comparison, counting, and
single and multi-digit arithmetic. A mathematical
Chromosome 22q11.2 deletion syndrome (22q) is word-solving task was also included and reading
the most common genetic deletion syndrome ability was measured. Children were individually
with an estimated prevalence of between one in matched with typically developing children from
3,000 and 6,000 births (e.g., Kobrynski and the same class at school for gender, age, and
Sullivan 2007). It has only been detectable with parental education level. Consistent with their
100 % accuracy since 1992 using techniques such hypotheses, De Smedt et al. (2007a, b) report
as the FISH test (fluorescence in situ hybridiza- group differences on multi-digit operations
tion). Prior to identification of a single associated involving a carry, word-solving problems,
deletion, the syndrome had been given a number and speed in judging the relative value of
of different labels according to the primary med- two digits. There was no difference in reading,
ical condition, for example, velocardiofacial syn- number reading and writing, single digit addi-
drome, DiGeorge syndrome, Cayler syndrome, tion, or verbal and dot counting accuracy.
Shprintzen syndrome, and Catch 22. The difficulties with multi-digit operations
are unsurprising given the visuospatial require-
The majority of individuals with 22q ments of operations such as borrowing and
experience some degree of learning difficulty carrying. Previous researches suggest that
and generally show a marked imbalance in multi-digit arithmetic is an area of particular
performance across different subtests within IQ difficulty in children with visuospatial learning
batteries. Verbal IQ scores are usually signifi- disability as well as arithmetic difficulties
cantly higher than performance IQ scores (e.g., (Venneri et al. 2003). More research is needed
Moss et al. 1999; Wang et al. 2007).

D 148 Design Research in Mathematics Education

to further uncover the nature of the mathemat- Design Research in Mathematics
ical difficulties experienced by children with Education
22q and to aim to uncover best practice
methods for teaching number skills in 22q as Malcolm Swan
so far, certainly in the UK, no consensus has Centre for Research in Mathematics Education,
been reached. Jubilee Campus, School of Education, University
of Nottingham, Nottingham, UK
Cross-References

▶ Autism, Special Needs, and Mathematics Keywords
Learning
Engineering research; Design experiments;
▶ Deaf Children, Special Needs, and Design research
Mathematics Learning
Definition
▶ Down Syndrome, Special Needs, and
Mathematics Learning Design-based research is a formative approach to
research, in which a product or process (or “tool”)
▶ Inclusive Mathematics Classrooms is envisaged, designed, developed, and refined
▶ Language Disorders, Special Needs and through cycles of enactment, observation,
analysis, and redesign, with systematic feedback
Mathematics Learning from end users. In education, such tools
▶ Learning Difficulties, Special Needs, and might, for example, include innovative teaching
methods, materials, professional development
Mathematics Learning programs, and/or assessment tasks. Educational
theory is used to inform the design and refinement
References of the tools and is itself refined during the
research process. Its goals are to create innova-
De Smedt B, Swillen A, Devriendt K, Fryns J, Verschaffel tive tools for others to use, describe and explain
L, Ghesquiere P (2006) Mathematical disabilities in how these tools function, account for the range
young primary school children with velo-cardio-facial of implementations that occur, and develop
syndrome. Genet Couns 17:259–280 principles and theories that may guide future
designs. Ultimately, the goal is transformative;
De Smedt B, Devriendt K, Fryns JP, Vogels A, we seek to create new teaching and learning
Gewillig M, Swillen A (2007a) Intellectual abilities possibilities and study their impact on teachers,
in a large sample of children with velo-cardio- children, and other end users.
facial syndrome: an update. J Intellect Disabil Res
51:666–670 The Origins and Need for Design
Research
De Smedt B, Swillen A, Devriendt K, Fryns J, Verschaffel
L, Ghesquiere P (2007b) Mathematical disabilities in Educational research may broadly be categorized
children with velo-cardio-facial syndrome. Neuropsy- into three groups: the humanities approach, schol-
chologia 45:885–895 arly study that generates fresh insights through
critical commentary, the scientific approach
Kobrynski L, Sullivan K (2007) Velocardiofacial syndrome, that analyzes phenomena empirically to better
DiGeorge syndrome: the chromosome 22q11.2 deletion
syndromes. Lancet 370(9596):1443–1452

Moss E, Batshaw M, Solot C, Gerdes M, McDonald-
McGinn D, Driscoll D, Emanuel B, Zackai E, Wang
P (1999) Psychoeducational profile of the 22q11.2
microdeletion: a complex pattern. J Pediatr
134(2):193–198

Venneri A, Cornoldi C, Garuti M (2003) Arithmetic
difficulties in children with visuospatial learning
disability (VLD). Child Neuropsychol 9(3):
175–183

Wang P, Woodin F, Kreps-Falk R, Moss E (2007)
Research on behavioral phenotypes: velocardiofacial
syndrome (deletion 22q11.2). Dev Med Child Neurol
42:422–427

Design Research in Mathematics Education D149

understand how the world works, and the engineer- experimental psychology as a paradigm for D
ing approach that not only seeks to understand the educational research. Brown’s paper on “design
status quo but also attempts to use existing experiments” was seminal (Brown 1992). Brown
knowledge to systematically develop “high-quality recounts how her own research moved away
solutions to practical problems” (Burkhardt and from laboratory settings towards naturalistic
Schoenfeld 2003). Design research falls into this ones in which she attempted to transform class-
“engineering” category and, as such, seeks to pro- rooms from “worksites under the management
vide the tools and processes that enable the end of teachers into communities of learning.”
users of mathematics education (teachers and She vividly recounts her own struggles in
students, administrators and politicians) to tackle reconceptualizing her focus and methodology,
practical problems in authentic settings. deconstructing methodological criticisms against
it (such as the Hawthorne effect). Interestingly
Design research is an unsettled construct and Brown still saw the need for lab-based research,
the field is in its youth. It is only at the beginning both to precede and stimulate work in naturalistic
of the last two decades that we see design settings and also for the closer study of phenom-
research as an emerging paradigm for the study ena that had arisen in those settings. At about the
of learning through the systematic design of same time, Collins (1992, pp. 290–293) began
teaching strategies and tools. The beginnings of to argue for a design science in education,
this movement, at least in the USA, are usually distinguishing analytic sciences (such as physics
attributed to Brown (1992) and Collins (1992), or biology) as where research is conducted in
though in a sense, it was an idea waiting to be order to explain phenomena from design sciences
named (Schoenfeld 2004). In Europe there have (such as aeronautics or acoustics) where the goal
long been traditions of principled design-based is to determine how designed artifacts (such as
research under other guises, such as curriculum airplanes or concert halls) behave under different
development and didactical engineering (e.g., conditions. He argued strongly for the need of the
Bell 1993; Brousseau 1997; Wittmann 1995). latter in education. In mathematics education,
such designed artifacts might include, for exam-
Prior to the 1990s, much educational and psy- ple, new teaching methods, materials, profes-
chological research had relied heavily on quasi- sional development programs, assessment tasks,
experimental studies that had been developed or any combination of these.
successfully in other fields such as agriculture.
These involved experimental and control treat- Since that time, “design research” has become
ments to evaluate whether or not particular vari- more widespread and respectable in education.
ables were associated with particular outcomes. However it must be said that not all so-called
In mathematics education, for example, one “design research” studies satisfy the definition
might design a novel approach to teaching described above. Some, for example, do not sat-
a particular area of content, assign students to isfy the requirement that the designs should be
an experimental or control group, and assess theory-based and develop theory, while others do
their performance on some defined measures, not move beyond the early stages and test their
using pre- and posttesting. Though sounding designs in the hands of others not involved in the
straightforward, this practice proved highly prob- development process.
lematic (Schoenfeld 2004): the goals of education
are more complex than the mastery of specific Characterizing Design-Based Research
skills; the control of variables in naturalistic set-
tings is often impossible, undesirable, and some- There have been many attempts to characterize
times even unethical; and much of the theory is design-based research (Barab and Squire 2004;
“emergent,” only becoming apparent as one Bereiter 2002; Cobb et al. 2003; DBRC
engages in the research. 2003, p. 5; Kelly 2003; Lesh and Sriraman 2010;

In the early 1990s, a number of researchers
began to question the limitations of traditional

D 150 Design Research in Mathematics Education

Swan 2006, 2011; van den Akker et al. 2006). Rather than viewing these as negative, interfering
While design research is still in its infancy and factors, the designs and theories evolve to explain
its characterization is far from settled, most these mutations. With each cycle of the process,
researchers do seem to agree that design-based the sample size is increased and becomes more
research is: typical of the target population. From time to
time, a particular issue may arise that the
Creative and Visionary researcher wants to study closely. In such
The researcher identifies a problem in a defined a case, it is possible to go back to the small-
context and, drawing on prior research, envisions scale study of that isolated issue.
a tool that might help end users to tackle it.
A draft design is developed, possibly with the Theory-Driven
assistance of end users. For example, the The outputs of design research include develop-
researcher identifies a particular student learning ing theories about learning, interventions, and
need and uses research to design a series of les- tools. Rather than focusing on learning outcomes,
sons. The ultimate aim is to produce an effective using pre- and posttests, the research seeks to
design, an account of the theory and principles understand how designs function under different
underpinning the design, and an analysis of the conditions and in different classroom contexts.
range of ways in which the design functions in the The theories that evolve in this way are local
hands of a typical sample of the target population and humble in scope and should not be judged
of teachers and students. by their claims to “truth” but rather their claims to
be useful (Cobb et al. 2003). Theory in design
Ecologically Valid research usually focuses on an explanation of
The researcher studies and refines the design in how and why a particular design feature works
authentic settings, such as classrooms. This pre- in a particular way. It is both specific and gener-
cludes the prior manipulation of variables in the ative in that it can be used to predict ways in
study. It is important, therefore, to distinguish which future designs will function if they embody
those aspects of the design that are being studied this feature.
from those that are extraneous.

Interventionist and Iterative Some Issues and Challenges
The role of the researcher evolves as the research
proceeds. During early iterations, the design is Design research done well requires great
usually sketchy and the researcher needs to inter- skill on the part of researchers. Indeed, the
vene to make it work. With teaching materials, combination of skills required is not usually
for example, this phase may be conducted with found in individuals but in teams. A design
small samples of students. Later, as the design research team will typically involve people
evolves, the researcher holds back, in order to see with knowledge of the literature (researchers),
how the design functions in the hands of end an understanding of pedagogy (teachers), cre-
users. Early iterations are often conducted in ative “care and flair” (designers), and facility
a few favorable contexts. Early drafts of teaching with “delivering” the design (publishers IT
materials, for example, may be tested in carefully technicians).
chosen classrooms with confident teachers, in
order to gain insights into what is possible with Secondly, design research often takes a great
faithful implementation. Later iterations aim to deal longer than other forms of research. There is
study how the design functions in a wider range often a significant “entry fee” in terms of time and
of authentic contexts, with teachers who have not energy taken up with producing a prototype
been involved in the design process. Under these before any study of it can begin. This is particu-
conditions, “design mutations” invariably occur. larly true if the design involves creating
new software. Then, each cycle of design,

Design Research in Mathematics Education D151

implementation, analysis, and redesign can each Cross-References D
occupy weeks, if not months.
▶ Curriculum Resources and Textbooks in
Thirdly, design research is data rich. A Mathematics Education
mixture of qualitative and quantitative methods
is used to develop a rich description of the way ▶ Mathematics Curriculum Evaluation
the design works as well as the kinds of learning ▶ Didactic Engineering in Mathematics
outcomes that may be expected. This often results
in a proliferation of data. Brown, for example, Education
found that she “had no room to store all the data, ▶ Didactic Situations in Mathematics Education
let alone time to score it” (Brown 1992, p. 152).
Data may include lesson observations, videos References
of the designs in use, and questionnaires
and interviews with users. In early iterations, Barab S, Squire K (2004) Design-based research: putting
observation plays a dominant role. Later, how- a stake in the ground. J Learn Sci 13(1):1–14
ever, more indirect means are also needed as the
sample size grows. Reliability may be improved Bell A (1993) Principles for the design of teaching. Educ
through the use of triangulation from multiple Stud Math 24(1):5–34
data sources and repetition of analyses across
cycles of implementation and through the use of Bereiter C (2002) Design research for sustained
standardized measures. innovation. Cognit Stud Bull Jpn Cognit Sci Soc
9(3):321–327
Fourthly, design research requires discipline. It
is all too tempting to turn a “good idea” into a draft Brousseau G (1997) Theory of didactical situations in
design and then ask someone to try it out to “see mathematics (trans: Balacheff N, Cooper M, Suther-
what happens.” Good design-based research is land R, Warfield V), vol 19. Kluwer, Dordrecht
more than formative evaluation, however; it is
theory-driven. In preparation for a design-based Brown AL (1992) Design experiments: theoretical and
research study, one must try to articulate the theory methodological challenges in creating complex
and draw clear lines of connection between this interventions in classroom settings. J Learn Sci
and the design itself. This may be done by eliciting 2(2):141–178
“principles” to direct the design. The research
involves putting these principles in “harms way” Burkhardt H, Schoenfeld A (2003) Improving Educational
(Cobb et al. 2003). Then, the focus of the research Research: Toward a more useful, more influential and
needs to be articulated. For early iterations this better-funded enterprise. Educ Res 32(9):3–14
may be on the potential impact of the faithful use
of the design, while on later iterations, we may be Cobb P, Confrey J, diSessa A, Lehrer R, Schauble L
more interested in refining the design by studying (2003) Design experiments in educational research.
end users’ interpretations and mutations. Educ Res 32(1)

Finally, writing up design research is Collins A (1992) Towards a design science in education.
problematic. Most designs are too extensive to In: Scanlon E, O’Shea T (eds) New directions in edu-
be described and analyzed in traditional journal cational technology. Springer, New York, pp 15–22
articles that emphasize methods and results over
tools. Recently e-journals have begun to appear DBRC (2003) Design-based research: an emerging para-
that allow for a much clearer articulation of digm for educational inquiry. Educ Res 32(1):5–8
design-based research. These, for example,
allow extensive extracts of teaching and profes- Kelly A (2003) Theme issue: the role of design in educa-
sional development materials to be displayed, tional research. Educ Res 32(1):3–4
along with videos of the designs in use (see for
example, http://www.educationaldesigner.org). Lesh R, Sriraman B (2010) Reconceptualizing mathemat-
ics education as a design science. In: Sriraman B,
English L (eds) Theories of mathematics education:
seeking new frontiers. Springer, Berlin, pp 123–146

Schoenfeld A (2004) Design experiments. In: Elmore PB,
Camilli G, Green J (eds) Complementary methods for
research in education. American Educational Research
Association, Washington, DC

Swan M (2006) Collaborative learning in mathematics:
a challenge to our beliefs and practices. National Insti-
tute for Advanced and Continuing Education (NIACE)
for the National Research and Development Centre for
Adult Literacy and Numeracy (NRDC), London

Swan M (2011) Towards the creative teaching of mathe-
matics. In: Thomson P, Sefton-Green J (eds)
Researching creative learning: methods and issues.
Routledge, Abingdon, pp 54–67

D 152 Dialogic Teaching and Learning in Mathematics Education

van den Akker J, Graveemeijer K, McKenney S, and Refutations (Lakatos 1976). Here Lakatos
Nieveen N (eds) (2006) Educational design research. shows that a process of mathematical discovery
Routledge, London/New York is of dialogic nature, characterized by proofs and
refutations.
Wittmann E (1995) Mathematics education as a design
science. Educ Stud Math 29(4):355–374 Critical mathematics education and social
constructivism have developed dialogic teaching
Dialogic Teaching and Learning in and learning through a range of examples and
Mathematics Education studies. It has been emphasized that dialogue is
principal for establishing critical perspectives on
Ole Skovsmose mathematics and for a shared construction of
Department of Learning and Philosophy, Aalborg mathematical notions and ideas. In fact dialogic
University, Aalborg, DK, Denmark teaching and dialogic learning represents two
aspects of the same process.
Keywords
Marilyn Frankenstein (1983) has emphasized
Dialogue; Literacy; Mathemacy; Investigation; the importance of Freire’s ideas for developing
Inquiry; Inquiry cooperation model; Critical critical mathematics education, and Paul Ernest
mathematics education (1998) has opened the broader perspective of
social constructivism, also acknowledging the
Definition importance of Lakatos work.

Dialogic teaching and learning refers to certain The Inquiry Cooperation Model
qualities in the interaction between teachers and The notion of dialogue appears to be completely
students and among students. The qualities concern open. As a consequence, it becomes important to
possibilities for the students’ involvement in the try to characterize what a dialogue could mean.
educational process, for establishing enquiry pro- The Inquiry Cooperation Model as presented
cesses, and for developing critical competencies. in Alrø and Skovsmose (2002) provides such
a specification with particular references to
Characteristics mathematics.

Sources of Inspiration This model characterizes different dialogic
There are different sources of inspiration for bring- acts: Getting in contact refers to the act of tuning
ing dialogue into the mathematics classroom, and in at each other. Locating and identifying refer to
let me just refer to two rather different. forms of grasping perspectives, ideas, and argu-
ments of the other. Advocating means providing
The notion of dialogue plays a particular role arguments for a certain point of view – although
in the pedagogy of Paulo Freire. He sees dialogue not necessary one’s own. Thinking aloud means
as crucial for developing literacy, which refers to making public details of one’s thinking, for
a capacity in reading and writing the world: instance, through gestures and diagrams.
reading it, in the sense that one can interpret Reformulating refers to particular attempts in
sociopolitical phenomena, and writing it, in the grasping other ideas by rethinking, rephrasing,
sense that one becomes able to make changes. and reworking them. Challenging means
questioning certain ideas, which is an important
With explicit reference to mathematics, the way of sharpening mathematical arguments.
crucial role of dialogue can be argued with Evaluating refers to reflexive questioning, like:
allusion to Imre Lakatos’ presentation in Proof What insight might we have reached? What new
questions have we encountered?

Dialogic teaching and learning can be
characterized as a process rich of such dialogic
acts.

Didactic Contract in Mathematics Education D153

New Qualities in Teaching and Learning References D
The idea of dialogic teaching and learning is to
promote an education with new qualities. Let me Alrø H, Johnsen-Høines M (2012) Inquiry – without
refer to just a few having to do with the students’ posing questions? Math Enthus 9(3):253–270
interest, making investigations, and developing
a mathemacy. Alrø H, Skovsmose O (2002) Dialogue and learning in
mathematics education: intention, reflection, critique.
Students’ Interest. It has been emphasized Kluwer, Dordrecht
that dialogic teaching and learning includes
a sensitivity to the students’ perspectives and Ernest P (1998) Social constructivism as a philosophy of
possible interests for learning. This sensitivity mathematics. State University of New York Press,
has not only to do with the dialogic act of Albany
“getting in contact” but with all the acts
represented by the Inquiry Cooperation Frankenstein M (1983) Critical mathematics education: an
Model. A principal point of dialogic teaching application of Paulo Freire’s epistemology. J Educ
is to invite students into the learning process 164:315–339
as active learners.
Gutstein E (2006) Reading and writing the world with
Making Investigations. Dialogic teaching and mathematics: toward a pedagogy for social justice.
learning can be characterized in terms of investi- Routledge, New York
gative approaches, where both teacher and stu-
dents participate in the same inquiry process. Jaworski B (2006) Theory and practice in mathematics
Barbara Jaworski (2006) makes a particular teaching development: critical inquiry as a mode of
emphasis on establishing communities of inquiry, learning in teaching. J Math Teach Educ 9(2):
and in any such communities, dialogue plays 187–211
a defining role. Landscapes of investigations
(Skovsmose 2011) might also provide Lakatos I (1976) Proofs and refutations: the logic of
environments that facilitate dialogic teaching mathematical discovery. Cambridge University Press,
and learning. Cambridge

Similar to literacy, mathemacy refers not only Skovsmose O (2011) An invitation to critical mathematics
to a capacity in dealing with mathematical education. Sense, Rotterdam
notions and ideas but also to a capacity in
interpreting sociopolitical phenomena and acting Didactic Contract in Mathematics
in a mathematized society. Thus, mathemacy Education
combines a capacity in reading and writing
mathematics with a capacity in reading and writ- Guy Brousseau1, Bernard Sarrazy2 and
ing the world (see Gutstein 2006). Dialogue Jarmila Novotna´3
teaching and learning is in hectic develop- 1Institut Universitaire de Formation des maˆıtres
ment, both in theory and in practice. A range d’Aquitaine, Mathe´matiques, Laboratoire
of new studies and new classroom initiatives Cultures Education Societes (LACES), EA 4140,
are being developed. In particular, the very Anthropologie et diffusion des savoir, Univ.
notion of dialogue is in need of further Bordeaux – France, Bordeaux Cedex, France
development; see, for instance, Alrø and 2Departement Sciences de l’Education,
Johnsen-Høines (2012). Laboratoire Cultures Education Societes
(LACES), EA 4140, Anthropologie et diffusion
des savoir, Univ. Bordeaux – France, Bordeaux
Cedex, France
3Faculty of Education, Charles University in
Prague, Praha, Czech Republic

Cross-References Keywords

▶ Critical Mathematics Education Didactical situations; Mathematical situations;
▶ Mathematization as Social Process Didactical contract; Didactique; Milieu;
Devolution; Institutionalization

D 154 Didactic Contract in Mathematics Education

Introduction What are the respective roles of what is inex-
pressible, of what is said, of what is not said
Teachers manage didactical situations that create or cannot be said to the other in the teaching
and exploit mathematical situations where relationship?
practices are exercised and students’ mathemati-
cal knowledge is developed. The study of the Does there exist knowledge that ought not to be
didactical contract concerns the compatibility made explicit before being learned?
on this precise subject of the aspirations and The study of these questions was the origin of
requirements of the students, the teachers, the
parents, and the society. the theory of didactical situations.

Characteristics

Definition Background: Illustrative Examples
These questions arose in the course of research at
A “didactical contract” is an interpretation of the the COREM (Center for Observation and Research
commitments, the expectations, the beliefs, the on Mathematics Education, entity formed of a lab-
means, the results, and the penalties envisaged by oratory and a school establishment by the IREM of
one of the protagonists of a didactical situation the University of Bordeaux (1973–1999)) on the
(student, teacher, parents, society) for him- or possibility of assigning to mathematical situations
herself and for each of the others, a` propos of the job of managing what the teacher cannot say or
the mathematical knowledge being taught the student cannot yet understand from a text, and
(Brousseau and Otte 1989; Brousseau 1997). in the clinical observation of students failing selec-
The objective of these interpretations is to tively in mathematics:
account for the actions and reactions of the part- (a) The Case of Gae¨l. Gae¨l (8 years old) always
ners in a didactical situation.
responded in the manner of a very young
The didactical contract can be broken down child. It was not a developmental delay, but
into two parts: a contract of devolution – the rather a posture. By replacing some lessons
teacher organizes the mathematical activity (see with “games” in which he could take a chance
▶ Didactic Situations in Mathematics Education) and see the effects of his decisions and by
of the student who in response commits him- getting him to make bets – without too much
or herself to it – and a contract of institu- risk – on whether his answers were right, the
tionalization – the students propose experimenters saw his attitude changes radi-
their results and the teacher vouches for the cally and his difficulties disappear. A new
part of their results that conforms to refer- “didactical contract” with him had been
ence knowledge. constructed (Brousseau and Warfield 1998).
(b) The Age of the Captain. Researchers at the
Customary practices (Balacheff 1988), Institute for Research on the Teaching of
whether explicit or tacit, leave the hope that Mathematics (IREM of Grenoble) offered
divergences are accidental and reducible and students at age 8 the following problem:
that there exist real contracts, whether or not “On a boat there are 26 sheep and 10 goats.
they can be made explicit, that are compatible How old is the captain?” 76 of the 97 students
and satisfactory. This is not so, owing to various answered, “36 years old.”
paradoxes that became apparent in the course of This experiment produced a scandal. Some
teaching in a way that is based on mathematical accused the teachers of stupefying their students;
situations. This gave rise to many questions, others reproached the researchers for “laying stu-
among them are as follows: pid traps for the children.” In a letter to the exper-
How could students commit themselves to the imenters, G. Brousseau indicated to them that it
was a matter of an “effect of the contract” for
subject of knowledge that they have not yet
learned?

Didactic Contract in Mathematics Education D155

which neither the students nor the teachers were (a) Custom can determine pedagogical and psy- D
responsible. So the researchers asked the students: chological relationships, but not those proper
“What do you think of this problem?” The stu- to new knowledge, because new knowledge
dents responded: “It is stupid!” The researchers is a specific unexpected adventure that con-
ask: “Then why did you answer it?” The students sists of a modification and an augmentation of
answered: “Because the teacher asked for it!” The old knowledge and of its implications. Thus,
researchers ask: “And if the captain was 50 years it cannot be known in advance by the student:
old?” The students made a response: “The teacher the teacher can only commit himself to gen-
didn’t give the right numbers.” A similar experi- eral procedures, and for her part the student
ment done with established teachers produced the cannot commit herself to a project of which
same behavior: for various reasons (such as the she does not know the main part.
hope of an explanation that the teacher wanted to
hear) the subjects produce the answer least incom- (b) Paradox of devolution: the knowledge and
patible with their knowledge, even when they see will of the teacher need to become those of
very well that it is false: the obligation of answer- the student, but what the student knows
ing is stronger than that of answering correctly. or does by the will of the teacher is not
Despite these explanations, for years the initial done or decided by his own judgment. The
observation elicited strong criticisms of the work didactical contract can only succeed by being
of the teachers (Sarrazy 1996). broken: the student takes the risk of taking on
a responsibility from which he already
Didactical and Ethical Responsibility releases the teacher (a paradox similar to
The teacher has the responsibility of that of Husserl).
supporting the collective and individual activ-
ity of the students, of attesting in the end to (c) Paradox of the said and unsaid (consequence
the truth of the mathematics that has been of the preceding): it is in what the teacher
done, of confirming it or giving proofs, of does not say that the student finds what she
organizing it in the standard way, of identify- can say herself.
ing errors that have been or might be made
and passing judgment on them (without pass- (d) Paradox of the actor: the teacher must pre-
ing judgment on their authors), and of provid- tend to discover with his students knowledge
ing the students with a moderate amount of that is well known to him. The lesson is
individual help (as with the natural learning of a stage production.
a language.) Occasional individual help con-
forms to the collective process of mathemati- (e) The paradox of uncertainty: knowledge man-
cal communities. If the teacher finds himself ifests itself and is learned by the reduction of
acceding to an institutional function, he may uncertainty that it brings to a given situation.
be subject to obligations of equity and of Without uncertainty or with too much uncer-
means for which the responsibility is shared tainty, there is neither adaptation nor learn-
with the institution. Decisions made about the ing. The result is that the optimal progression
teacher and the students based on individual of normal individual or collective learning is
and isolated results are a dangerous absurdity. accompanied by a normal optimal rate of
Experts, parents, and society share the respon- errors. Artificially reducing it damages both
sibility for the effects of such decisions. individual and collective learning. It is useful
to arrange things so that it is not always the
Paradoxes of the Didactical Contract same students who are condemned to supply
The teacher wants to teach what she knows to the necessary errors.
a student who does not know it. This has many
consequences, among them are as follows: (f) As in the case of learning, excessive or pre-
mature adaptation of complex knowledge to
conditions that are too particular leads it to be
replaced by a simplified and specific knowl-
edge. This can then constitute an epistemo-
logical or didactical obstacle to its later

D 156 Didactic Contract in Mathematics Education

adaptation to new conditions. (For example, • Decomposition of the problem into inter-
division of natural numbers is associated mediate questions (decomposition of the
with a meaning, sharing, which becomes an objectives)
obstacle to understanding it in the case where
a decimal number needs to be divided by • Use of various extra-mathematical
a larger decimal number, e.g., 0.3/0.8.) rhetorical means: analogies, metaphors,
(g) The paradox of rhetoric and mathematics. To metonyms, or mnemonic minders (the
construct the students’ mathematical knowl- “Topaze effect”)
edge and its logical organization, the teacher
uses various rhetorical means, designed to cap- (c) Critical commentary on the errors, the ques-
ture their attention. The culture, pedagogical tion, the knowledge, or the material
procedures, and even mathematical discourse
(commentaries on mathematics) overflow with (d) A trial of the student and its consequences:
metaphors, analogies, metonyms, substitutions, penalties, discrimination, and individualization
word pictures, etc. The mathematical concepts In case of failure, the contract obligates the
are often constructed against these procedures
(e.g., “correlation is not causation”). The teacher to try again. The new attempt either
teacher should thus at the same time as an replaces the preceding one or criticizes and
educator teach the culture with its historical corrects it, making of it a new teaching object
mistakes and as a specialist cause the rejection (a meta-process).
of the parts that science has disqualified.
These paradoxes can only be unraveled by spe- For each of these types of response,
cific situations and processes carefully planned out there are conditions under which it is the
in the light of well-shared knowledge of mathemat- most appropriate response; thus there is no
ical and scientific didactique (Brousseau and Otte universal response.
1989; Brousseau 2005).
For example, Novotna´ and Hosˇpesova´ (2007)
Observations of Reactions of Teachers to identify and classify the behaviors whose system-
Difficulties atic repetition generates Topaze effects:
These observations and the experimental and the- 1. Explicitly, the teacher
oretical studies of the didactical contract make it
possible to understand and predict the cumulative (a) Gives the steps of the solution and trans-
effects of teachers’ decisions. forms it into the execution of a sequence of
tasks
The contract manifests itself essentially in
its ruptures. These are revealed by the reac- (b) Asks questions in a sequence that man-
tions of the students or by the interventions of dates the procedures of the solution
the teachers, and they can be classified as
follows: (c) Gives warnings about a possible error
(a) Abandonment. The teacher does not react to (d) Enumerates previous experiences or

an error made by the students (e.g., because it knowledge, pointing out analogies with
would be too complicated to explain it), or problems that have previously been
she repeats the question identically or she resolved or are obvious or well known
gives the complete solution. 2. Implicitly, he
(b) The progressive reduction or manipulation (a) Reformulates students’ propositions or his
of the students’ uncertainty, using a great own
variety of means: (b) Uses “guide” words
• Bringing in mathematical, technical, or (c) Pronounces the first syllable of words
(d) Poses new questions that orient the student
methodological information towards the solution
(e) Shows doubt about dubious initiatives
Their research confirms that the resulting
Topaze effects go unnoticed but have a high
cost. The students, apparently active, become
dependent on this aid and lose their confidence
in themselves. An error is understood to be a
transgression of the didactical contract and

Didactic Contract in Mathematics Education D157

proof itself, badly supported, becomes something The mean rate of success is a “regulated” var- D
to be learned rather than understood. iable of the system. Otherwise stated, the global
progress of all the students is less rapid if one
By using jointly the notions of milieu, of situa- requires at every stage a 100 % rate of success.
tion, and of the didactical contract, Perrin-Glorian The conception of mathematical activity as an
and Hersant (2003) were able to show in numerous adventure and a collective practice makes it pos-
examples on the one hand what the student and sible to mitigate the effects of difference in
the milieu are in charge of and thus the occasions rhythms of learning.
for learning that are their responsibility, and on
the other hand the help brought in by the teacher. It seems that today the requirements of the
different partners of teaching towards one
Predicting and Explaining Certain another are less and less compatible with each
Long-Term Effects other, perhaps because of the variety of possibil-
The uncontrolled recursive resumption of the ities, of offers, and of perspectives provided by
same type of response leads to drifting and inev- numerous ill-coordinated sciences.
itable failures. For example, for the students
studying the procedure for solving problems by The experiments on teaching rational and dec-
the same pedagogical methods, studying theo- imal numbers (Brousseau 1997) or statistics and
rems is just as costly, less sure, and less useful. probability (Brousseau et al. 2002) prove that it is
possible to organize efficient and communicable
As another example, a sequence of meta- processes with the help of didactical contracts
slippages contributed to the failure of the reform based on the nature of the knowledge to be
of “modern math”: the foundations of mathemat- acquired.
ics were interpreted by “na¨ıve” set theory, which
was itself formalized into algebra. This was met- Extensions
aphorically represented by “graphs,” which Sarrazy (1996, 1997) studied the pitfalls of these
were finally interpreted in vernacular language. meta-didactical slippages and more particularly
Each representation betrayed the preceding one those that are consequences of a teaching that
slightly and supported new conventions, and the aims at making the contractual expectations
slippages were ultimately uncontrollable. In the explicit, frequently taking the form of the teach-
absence of didactical situations and proven epis- ing of metacognitive or heuristic procedures – or
temological processes, varying the types of even of algorithms for solving problems.
response seems to be the best strategy. Complementing the work engaged in by
Schubauer-Le´oni (1986) in a psychosocial
Enforcing requirements based on the results of approach to the didactical contract, Sarrazy rad-
individuals leads to a mincing up of the objec- icalized the paradox of the consubstantial rule
tives, to the abandonment of high-level objec- (A rule does not contain in itself its conditions
tives, and to addressing the objectives by painful for use) of the contract at the intersection of the
behaviorist methods. These slow the learning and theory of situations and Wittgensteinian anthro-
lead to an individualization that slows it yet fur- pology (Wittgenstein 1953). Contrary to the psy-
ther. Each of these tends to destroy the role of chological or linguistic interpretations of the
provisional knowledge and to augment mechani- contract (such as that of “the age of the cap-
cally the time for teaching and learning without tain”), he showed how these slippages lead to
positive impact on the results. a veritable demathematization of teaching by
a displacement of the goals of the contract.
Specifying the means of teaching a subject These works also made it possible to establish
involves precise and specific protocols for per- the primacy of the role of situations and that of
formances that are known and accepted by the school cultures (Sarrazy 2002; Clanche´ and Sarrazy
population. Specifying required results for the 2002; Novotna´ and Sarrazy 2011) and family
teachers as for the students has absolutely no habits conceived as backgrounds (Searle 1979)
scientific basis. Its disastrous effects, predicted
since 1978, have been observable for 40 years.

D 158 Didactic Contract in Mathematics Education

of the didactical contract. These backgrounds Cross-References
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what Brousseau designated in 1991 “didactical Behav 18(1):1–46
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de l’enseignement des strate´gies me´tacognitives en emergence, the expression didactical engineering
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Sarrazy B, Novotna´ J (2005) Didactical contract: theoret- From its emergence as an academic field of study,
ical frame for the analysis of phenomena of teaching mathematics education has been associated with
mathematics. In: Novotna´ J (ed) Proceedings SEMT the design and experimentation of innovative
05. Univerzita Karlova v Praze, Pedagogicka´ fakulta, teaching practices, in terms of both mathematical
Prague, pp 33–45 content and pedagogy. The importance to be
attached to design was early stressed by
Schubauer-Leoni M-L (1986) Le contrat didactique: un researchers as Brousseau and Wittman, for
cadre interpre´tatif pour comprendre les savoirs instance, who very early considered that mathe-
manifeste´s par les e´le`ves en Mathe´matiques. J Eur matics education was a genuine field of research
Psychol E´ duc 1(Spe´cial 2):139–153 that should develop its own frameworks and prac-
tices and not just a field of application for other
Searle J (1979) Expression and meaning: studies in the sciences such as mathematics and psychology.
theory of speech acts. Cambridge University Press,
Cambridge The idea of didactical engineering (DE),
which emerged in French didactics in the early
Wittgenstein L (1953) Philosophical investigations. 1980s, contributed to firmly establish the place of
Blackwell, Oxford design in mathematics education research. Foun-
dational texts regarding DE such as Chevallard
Didactic Engineering in Mathematics (1982) make clear that the ambition of didactic
Education research of understanding and improving the
functioning of didactic systems where the teach-
Miche´le Artigue ing and learning of mathematics takes place
Laboratoire de Didactique Andre´ Revuz, cannot be achieved without considering these
Universite´ Paris Diderot-Paris 7, Paris, France systems in their concrete functioning, paying the
necessary attention to the different constraints
Keywords and forces acting on them. Controlled realizations
in classrooms should thus be given a prominent
Didactical engineering; Theory of didactical role in research methodologies for identifying,
situations; A priori and a posteriori analysis; producing, and reproducing didactic phenomena,
Research methodology; Classroom design; for testing didactic constructions. As a research
Development activities methodology, DE emerged with this ambition,
relying on the conceptual tools provided by the
Definition Theory of Didactical Situations (TDS), and
conversely contributing to its consolidation and
In mathematics education, there exists a tradition
of research giving a central role to the design
of teaching sessions and their experimentation
in classrooms. Didactical engineering, which
emerged in the early 1980s and continuously
developed since that time, is an important form
taken by this tradition. In the educational

D 160 Didactic Engineering in Mathematics Education

evolution (Brousseau 1997). It quickly became a To the search for fundamental situations, i.e.,
well-defined and privileged methodology in the mathematical situations encapsulating the
French didactic community, accompanying epistemological essence of the concepts
the development of research from elementary
school up to university level as evidenced in the To the characteristics of the milieu with which the
synthesis proposed at the 1989 Summer School of students will interact in order to maximize the
Didactics of Mathematics (Artigue 1990, 1992). potential it offers for autonomous action and
productive feedback
From the 1990s, DE migrated outside its
original habitat, being extended to the design of To the organization of devolution and institution-
teacher preparation and professional development alization processes by which the teacher, on the
sessions, used by didacticians from other disci- one hand, makes students accept the mathemat-
plines, for instance, physical sciences or sports, ical responsibility of solving the task and, on
and also by researchers in mathematics education the other hand, connects the knowledge they
in different countries. Simultaneously, the pro- produce to the scholarly knowledge aimed at
gressive shift of research attention towards The a priori analysis makes clear these choices
teachers increased the use of methodologies
based on naturalistic observations of classrooms, and their relation to the research hypotheses.
leading to theoretical developments and results Conjectures are made regarding the possible
that, in turn, affected DE. Moreover, design-based dynamic of the situation, students’ interaction
research perspectives emerged in other contexts, with the milieu, students’ strategies, their evolu-
independently of DE (Design-Based Research tion and their outcomes, about teacher’s neces-
Collaborative 2003). These evolutions and the sary input and role. Such conjectures regard not
resulting challenges are analyzed in Margolinas individuals but a generic and epistemic student
et al. (2011). entering the mathematical situation with some
supposed knowledge background and accepting
DE as a Research Methodology to enter the mathematical game proposed to her.
The actual realization will involve students with
As a research methodology, DE is classically their personal specificities and history, but the
structured into four different phases: preliminary goal of the a priori analysis is not to anticipate
analyses; design and a priori analysis; realization, all these personal behavior; it is to build
observation, and data collection; and a posteriori a reference with which classroom realizations
analysis and validation (Artigue 1990, 2009). will be contrasted in the a posteriori analysis.

Preliminary analyses usually include three During the phase of realization, data are
main dimensions: an epistemological analysis of collected for the analysis a posteriori. The nature
the mathematical content at stake, an analysis of of these data depends on the precise goals of
the conditions and constraints that the DE will the DE, the hypotheses tested, and the conjectures
face, and an analysis of what educational research made in the a priori analysis. The realization can
has to offer for supporting the design. lead to some adaptation of the design in itinere,
especially when the DE is of substantial size.
In the second phase, design and a priori analy- These adaptations are documented and taken
sis, research hypotheses are engaged in the pro- into account in the a posteriori analysis.
cess. Design requires a number of choices, from
global to local. They determine didactic variables, A posteriori analysis is organized in terms of
which condition the interactions between students contrast with the a priori analysis. Up to what
and knowledge, between students and between point the data collected during the realization
students and teachers, thus the opportunities that support the a priori analysis? What are the signif-
students have to learn. In line with TDS, in design, icant convergences and divergences and how to
particular importance is attached: interpret them? The hypotheses underlying the
design are put to the test in this contrast. There
are always differences between the reference pro-
vided by the a priori analysis and the contingence

Didactic Engineering in Mathematics Education D161

analyzed in the a posteriori analysis. The valida- made at reproducing Brousseau’s DE year after D
tion of the hypotheses underlying the design does year. These are only two examples among the
not, thus impose perfect match between the two many we could mention. DEs were progressively
analyses. Moreover, the validation of the research developed at all levels of schooling, covering a
hypotheses may require the collection of comple- diversity of mathematical domains and addressing
mentary data to those collected during the class- a diversity of research issues. At university level,
room, especially for appreciating the learning for instance, paradigmatic examples remain the
outcomes of the process. Statistical tools can be construction developed by Artigue and Rogalski
used, but what is essential is that validation is for the study of differential equations, combining
internal, not in terms of external comparison qualitative, algebraic, and numerical approaches to
between control and experimental groups. this topic (Artigue 1993) and that developed by
Legrand for the teaching of Riemann integral
These are the characteristics of DE as research within the theoretical framework of the scientific
methodology when associated with the concep- debate (Legrand 2001). Both were experimented
tion of a sequence of classroom sessions having with first year students and showed their resistance
a precise mathematical aim. However, as shown to students’ diversity. Constraints met at more
in Margolinas et al. (2011), this methodology has advanced levels of schooling contributed to the
been extended to other contexts such as teacher deepening of the reflection on an optimized orga-
education, more open activities such as project nization of the sharing of mathematical responsi-
work or modeling activities, and even mathemat- bilities between students and teacher in DE and to
ical activities carried out in informal settings. In the softening of the conditions and structures often
these last cases, the content of preliminary ana- imposed to design at more elementary levels. DE
lyses must be adapted; what the design ambitions was also enriched by its use in other domains
to control in terms of learning trajectories and the than mathematics and by researchers trained in
reference provided by the a priori analysis cannot other cultural traditions. A good example of it is
exactly have the same nature. provided by its use in sports, already mentioned,
and by the elaboration of DE combining the
Realizations theoretical support of TDS and that of semiotic
approaches (cf. for instance, (Falcade et al. 2007;
The first exemplars of DE research regarded ele- Maschietto 2008) using such combination for
mentary school. Paradigmatic examples are the studying the educational potential of digital
long-term designs produced by Brousseau, on the technologies). More globally, ICT has always
one hand, and by Douady, in the other hand, for been a privileged domain for DE, for exploring
extending the field of numbers from whole num- and testing the potential of new technologies,
bers to rational numbers and decimals (Brousseau and for supporting technological development as
et al. 2014; Douady 1986). The two constructions well as theoretical advances in that area.
were different, but they proved both to be success- Another interesting example is the use of DE
ful in the experimental settings where they were within the socio-epistemological framework in
tested, and they significantly contributed to the mathematics education (Farfa´n 1997; Cantoral
state of the art regarding the learning and teaching and Farfa´n 2003).
of numbers. Beyond that, they had theoretical
implications. The development of the tool-object Challenges and Perspectives
dialectics and the identification of the learning
potential offered by the organization of games DE developed as a research methodology, but
between mathematical settings by Douady are DE from the beginning had also the ambition of
intrinsically linked to her DE for the extension of providing a model for productive interaction
the number field; the idea of obsolescence of between fundamental research and action
didactic situations emerged from the attempts

D 162 Didactic Engineering in Mathematics Education

on didactic systems. DEs produced by research References
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not obey the same levels of control. In Margolinas (eds) Mathematics didactics as a scientific discipline.
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Perrin-Glorian through the idea of DE of sec-
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Courses of Study and Research (CSR) (Chevallard IREM d’Aix Marseille. Accessible at http://yves.
2006). Despite de fact that ATD and TSD chevallard.free.fr
emerged in the same culture, the visions of
design they propose today present substantial Chevallard Y (2006) Steps towards a new epistemology in
differences. Establishing productive connec- mathematics education. In: Bosch M (ed) Proceedings
tions between the two approaches without of the IVth congress of the European society
losing the coherence proper to each of them for research in mathematics education (CERME 4).
is a problem not fully solved but also Universitat Ramon Llull Editions, Barcelona,
addressed in Margolinas (2011). pp 22–30

Cross-References Design-Based Research Collaborative (2003) Design-
based research: an emerging paradigm for educational
▶ Anthropological Approaches in Mathematics enquiry. Educ Res 32(1):5–8
Education, French Perspectives
Douady R (1986) Jeux de cadres et dialectique outil-objet.
▶ Design Research in Mathematics Education Recherches en Didactique des Mathe´matiques
▶ Didactic Situations in Mathematics Education 7(2):5–32

Falcade R, Laborde C, Mariotti MA (2007) Approaching
functions: Cabri tools as instruments of semiotic
mediation. Educ Stud Math 66(3):317–334

Farfa´n R (1997) Ingenier´ıa Didactica y Matema´tica
Educativa. Un estudio de la variacio´n y el cambio.
Grupo Editorial Iberoame´rica, Me´xico

Legrand M (2001) Scientific debate in mathematics
courses. In: Holton D (ed) The teaching and learning
of mathematics at University level: an ICMI study.
Kluwer, Dordrecht, pp 127–136

Margolinas C, Abboud-Blanchard M, Bueno-Ravel L,
Douek M, Fluckiger A, Gilel P, Vandebrouck F,
Wozniak F (eds) (2011) En amont et en aval des
inge´nieries didactiques. XVe e´cole d’e´te´ de didactique
des mathe´matiques. La Pense´e Sauvage Editions,
Grenoble

Maschietto M (2008) Graphic calculators and micro-
straightness: analysis of a didactical engineering. Int
J Comput Math Learn 13(3):207–230

Didactic Situations in Mathematics Education D163

Didactic Situations in Mathematics than looking to gain credit for themselves, the D
Education students are engaged in:
– Producing “new” statements and discussing
Guy Brousseau1 and Virginia Warfield2
1Institut Universitaire de Formation des maˆıtres their validity
d’Aquitaine, Mathe´matiques, Laboratoire – Making decisions, formulating hypotheses,
Cultures Education Societes (LACES), EA 4140,
Anthropologie et diffusion des savoir, Univ. predicting and judging their consequences,
Bordeaux – France, Bordeaux Cedex, France attempting to communicate information,
2Department of Mathematics, University of producing and organizing models, argu-
Washington, Seattle, WA, USA ments and proofs, etc., adequate for certain
precise projects
Keywords – and evaluating and correcting by them-
selves the consequences of their choices
Didactical situation; A-didactical situation; It is thus not the students who are in question,
Mathematical situation; Acculturation; but some conjectures and some knowledge
Didactique (Brousseau 1997, pp. 230–235).
A “situation of institutionalization” in which
Didactical Situation the teacher:
– Takes note of the progress of the mathe-
A didactical situation in mathematics is matical situation, of the questions and
a project organized so as to cause one or some answers that have been obtained or studied
students to appropriate some piece of mathemat- from it, and of those that have emerged, and
ical reference knowledge. (The organizer and the places them within the perspective of the
student may be individuals, a population, curriculum
institutions, and so on.) – Distinguishes among the pieces of knowl-
edge (connaissances) that have appeared
Components those that have revealed themselves to be
Every didactical process is a sequence of situa- false and those that are correct, and among
tions, each pertaining to one of the following the latter those that will serve as references,
three types: presenting in that case the canonical way of
A “situation of devolution” in which the teacher formulating them
– And draws conclusions for the organization
sets the students up: of further sequences (exercises, problems,
– to accept boldly and confidently the etc.) (Brousseau 1997, pp. 235–243).

challenge of an engaging and instructive Teaching methods
mathematical situation whose instruc- Teaching methods can be distinguished first by
tions he gives in advance: conditions, the interpretation, the role, and the importance
rules, goal, and above all the criterion assigned to each of the components. Here are
for success two very different examples of this:
– and to do it without his help, on their own Example 1: In certain methods, devolution con-
responsibility (Brousseau 1997, pp. 230–235)
A “mathematical situation” that supports the sists of a prerequisite teaching of new knowl-
students in autonomous mathematical activi- edge (a lecture), followed by examples and
ties, both individual and collective, that repre- exercises, and followed by the presentation
sent those in use by mathematicians. Rather of problems whose autonomous solution by
the students constitutes the mathematical situ-
ation. Institutionalization consists of correc-
tion, evaluation, and the conclusions that the
teacher draws from them. Sometimes the

D 164 Didactic Situations in Mathematics Education

mathematical situation is considered only as geneses of fundamental mathematical concepts,
a means of verifying the individual learning in a form and by processes comparable to those
produced by the lecture. put into operation by mathematicians before the
Example 2: In other methods, devolution is final presentation of their results, in the process
reduced to the organization, presentation, and mathematical development. This idea found jus-
staging of an individual or collective mathe- tification in the work of the period: the acquisi-
matical situation aimed at provoking activities tion of language does not follow the classic
and processes like those of mathematicians: formulation of its grammar, and Piaget identified
a search for solutions or proofs but also certain mathematical structures in the genesis of
production of questions, hypotheses or conjec- logical thought in children.
tures, reformulations, definitions and study
of objects, sorting, debates, challenges, Conceiving of similar geneses, and especially
etc. Learning is the means and the product of imagining conditions capable of inducing them,
this activity. Institutionalization then consists could only arise from the competence of the
of identifying and organizing, among the cor- mathematical community. It did so through
rect pieces of knowledge produced by the stu- a gigantic effort of its researchers and of its
dents, those consistent with common usage teachers, realizing as it did so the aspirations of
and with accepted mathematical knowledge, pedagogues like Dewey, Montessori, or Freinet.
and among those the ones that are sufficiently But diffusing these conceptions more widely,
“acquired” by all of the students so that the against the traditional culture of teaching, posed
teacher and students can refer to them with yet more redoubtable problems, which have not
each other in future mathematical situations. at this point been surmounted.
The “lecture” consists of a conclusion and
of putting things in order. Exercises are Learning Mathematics by Doing It Reverses
a means of training available to the students the Classic Pedagogical Order
(Margolinas et al. 2005; Illustrative examples The teaching of mathematics is based on a text or
in Warfield 2007). some texts that express it in a canonical way (i.e.,
in the order: definitions, properties and theorems,
Origin and Necessity of the Concept of and finally proofs). The classical conception
“Didactical Mathematical Situation” consists of teaching using the texts first, so that
a student could never argue that he or she is being
The Reform of the Foundations (1907–1980) required to use a piece of knowledge that was not
The term “didactical situation” appeared in the first revealed and taught. Teaching pieces of
1960s with the meaning “mathematical situation knowledge before needing to use them gives
for teaching.” the appearance of being a “rational” method,
but it introduces a disassociation (learn with
The new mathematical concepts on which metamethods that have no relationship with the
teaching was to be rebuilt were communicated object and its use), an inversion (learn terms
by formalized texts in a symbolic language before understanding them and doing anything
unintelligible to students and/or by reformulations, with them), and finally teleological requirements:
metaphorical representations, and ambiguous the student is blamed in the course of learning
commentaries. On the other hand, they referred for not having first learned what is in fact
necessarily to examples taken from the classical the goal of the teaching that is going on. This
mathematics that they were reorganizing. The epistemological error greatly limits the field of
“fundamental” concepts were thereby postponed application, the age of learning, and the degree of
to the end of the studies. success of the classical method.

The challenge was thus to imagine conditions, Conversely, direct acculturation to specific
situations, that could induce in the students the mathematical practices that can produce these
texts brings their learning closer to that

Didactic Situations in Mathematics Education D165

of vernacular language or natural thought. But well before being able to offer teachers, in D
Everything then rests on the power of the the name of mathematicians, an aid, or some
situations to induce in the children the “process ready-to-implement solutions for teaching math-
of mathematization.” ematics, didactique must describe, understand,
and explain in a scientific manner mathematical
It would be absurd and detrimental to want to activity and its possible didactical transpositions.
exclude some method or to uniformly recom-
mend it over some other. The conditions to Didactique plays a role in the reorganization
which each is best adapted must be scientifically and transformation of mathematical knowledge.
studied and their advantages combined. For Its results are thus first addressed to the commu-
example, situations of cooperative discovery nity of mathematicians, to whom falls – for good
and collective adventures create homogeneity reasons – the responsibility towards society of the
and motivation and make it possible to acquire reference in teaching materials to the established
the classical practices by use. Exercises can help knowledge of its specialty. Didactique of mathe-
in doing well and rapidly what is worthwhile and matics requires specific concepts and methods of
has been understood (Brousseau 1992). study. It thus joins logic, computer science, epis-
temology, history of mathematics, and so on as
The Project of a Mathematical Science: one of the mathematical sciences. It takes charge
Didactique of the knowledge of everything that is specific to
The organization of these mathematical situations the discovery, the diffusion, or the appropriation
and their succession obey various reasons: mathe- of each piece of mathematical knowledge, new or
matical, epistemological, rational, empirical, ideo- not, that results from the adventures specific to it.
logical, etc. Their scientific study combines: It extends, enriches, and puts to the test the gen-
1. The (anthropological) observation and the eral contributions of classical social sciences,
which are indispensible but insufficient for
analysis (semiological) of the practices and clarifying all the facets of this teaching.
conceptions of the teachers and of the students
2. The conception, realization, and experimental Mathematical Situations
study of original mathematical situations appro-
priate to each of the pieces of mathematical Definition
knowledge aimed for (▶ Didactic engineering Every mathematical concept is the solution of at
in mathematics education) least one specific system of mathematical condi-
3. The inventory of possible choices, their tions, which itself can be interpreted by at least
modeling in the form of situations, the one situation, for example, a game, whose solu-
experimental and theoretical study of their tion (decision, message, argument) is one of the
conditions and of their properties, and the typical manifestations of the concept. A situation
creation of appropriate instruments of analysis is composed of a milieu and a project. The
(theory of didactical situations) duration of the life of a mathematical situation
The conception of these situations requires prior (the time of studying it) can vary from
and specific mathematical study of the knowledge a few seconds to several centuries for humanity
to be taught, along with that of its historical or several months for teaching.
genesis, of its epistemological properties, and
of its possible didactical geneses and their Examples
properties. But the scientific confrontation of Example 1: Children 4–5 years old. From a col-
these speculations with actual teaching is lection of thirty or so familiar objects, 5 or 6 are
fundamental. hidden in a box by a child in the morning. In the
The theory of situations, its concepts, and its afternoon, she is supposed to enumerate them to
research methods is one of the most ambitious another child, who confirms the presence or
among the numerous scientific approaches to the
phenomenon of didactique.

D 166 Didactic Situations in Mathematics Education

absence of the objects she names. The solution of He entrusts it to a milieu that is clearly stripped of
this game is the creation, enumeration, and use of teleological or pedagogical intentions [its reac-
lists. Knowing neither how to read nor how to tions depend neither on the intended goal nor on
write, the children represent the objects in their the individuals].
own way (pictograms) to distinguish them, first
individually and then collectively. The lists of The milieu of a situation is what the students
symbols represent sets; belonging or not, conjunc- exercise their actions on and what gives them
tions and disjunctions of characters are used, objective responses. The teacher thus entrusts to
corrected, understood, and formulated in vernac- the milieu the job of showing the students’ errors by
ular language (Pe´re`s 1984; Digneau 1980). their effects, without using an argument of author-
ity or revealing any intentions. The milieu may
Example 2: Children 10–11 years old. To be comprise informative texts; material objects;
certain of the number of white marbles contained in other students, cooperating or concurrent; and so
a firmly closed opaque bottle with a known number on. To this must be added the established knowl-
of marbles, some white and some black, students edge of the student as well as her memories of
invent hypothesis testing and the measure of events relevant previous events, and objective conditions,
(33 short sessions) (Brousseau et al. 2002). that may not be known to the student but that
intervene in her choices and in the effects of her
A great many researchers have imagined decisions. The cognitive variables of the situation
and studied various types of situations destined are those whose value has an influence on the issue
for all sorts of notions, for all levels of school of the situation or on the knowledge developed.
and even university. See, for example, These variables are didactical if their value can be
Bessot (2000), Laborde and Perrin-Glorian chosen by the teacher (the sex of the students may
(2005), Bloch (2003). influence the progress of a situation, but it is not a
didactical variable). The milieu can be interpreted
Types of Mathematical Knowledge, metaphorically by games that present some states
Reference Knowledge (Savoirs) that are permissible and some that are excluded,
Classical methods forbid the teacher from tolerat- rules of action, and issues of which one would be
ing without immediate correction, the manifesta- the goal sought (Warfield 2007).
tion of anything contrary to written established
mathematics. A genuine mathematical activity Examples of Milieux
necessarily gives rise to all sorts of knowledge. 1. Cabri geometry permits the student to realize,
Some is knowledge sought for – these are the
references, recognized as correct, true and in the context of geometrical objects and
known: they are professed and expected. But transformations, which of her projects are
there also necessarily appear pieces of knowledge constructible, that is, compatible with the axioms
that are ill made, ill formed, incomplete, doubtful, (Laborde et al. 1995). The projects lead the
false, or even inexpressible. They are “knowledge” students to gain knowledge of, formulate, and
in the sense of “the trace of an encounter.” Their test what the milieu permits them to glimpse.
presence, whether or not firmly nailed down, is 2. Analysis of a situation. The reader will find an
indispensible to thought. For example, a theorem example of the analysis of a didactical situation
that the student knows very well (savoir), but about (the Race to 20), of its milieu, of the strategies
whose usefulness in a situation is unsure, functions used by students, of the theorems in action that
provisionally as a simple piece of nonestablished support them, and of the didactical methods
knowledge (connaissance). that make it possible to lead them to a complete
proof and then to extend it so as to have
The teacher cannot intervene in this flow of them reinvent an algorithm: the search for
activities without blocking its functioning and the remainder of a Euclidean division, in
must therefore delegate the responsibility for Brousseau (1997, pp. 3–18). This work also
exercising a pragmatic penalty to the initiatives includes numerous other examples.
of the students that result from their knowledge.

Didactic Situations in Mathematics Education D167

The project is an objective, a final state of the information, in particular, on the milieu, the D
milieu, the response to a question, or even a same rights of refutation, and the same interest
pretext for exploration. It is what explains, in arriving at a consistent agreement (for an
justifies, or condemns after the fact the choices action on the milieu).
that have been chosen or ventured by the subject. 3. Situation of information (communication): The
transmitter and receiver cooperate on an action
The resolution is the occasion to put to trial on the milieu, in whose success they are inter-
not the student, but a way of knowing. ested and which depends on their joint action.
Neither of the two has at the same time all of
Remarks: The milieu of a situation is not the information and all of the necessary means
a natural milieu and does not turn mathematics of action. They exchange messages in order to
into a sort of experimental science. The project realize a common mathematical project.
is essential, and its goal is to establish the 4. Situation of action: A subject intervenes on the
consistency of certain statements. milieu to modify it with a determined aim. She
observes the effect of her actions and attempts
Different branches of mathematics developed to anticipate them by constructing pieces of
in different milieux: geometry in the knowledge knowledge, conscious and explainable or not.
of space, probability in the statistics of games, This situation encompasses all of the others,
algebra in arithmetic, arithmetic in the measure- but it extends beyond them by stimulating the
ment of amounts, etc. existence of inexpressible and possibly even
unconscious models of action.
In elementary teaching, knowledge of these Each of these types of situation creates distinct
milieux is neither spontaneous nor contained in typical motivations (modify a milieu, communi-
their mathematical interpretation. For example, cate some information, debate the validity of a
the knowledge that is useful for finding one’s way declaration, establish a reference) that mobilize
around a big city merits specific work that cannot and expand the corresponding repertoires
be reduced to some geometry. (implicit models of action, semiological or
linguistic repertoires, logical repertoires, mathe-
Types of Mathematical Situations matics or metamathematics, established knowl-
Characteristic of Activities, of Pieces of edge and theory) which are themselves acquired
Knowledge, and of Pieces of Mathematical according to specific different modes of learning
Learning or acculturation.
The mathematical knowledge of a student mani- The actual situations are, every one of them,
fests itself in her interactions with a milieu, as specific to a precise piece of knowledge.
a means of attaining or maintaining a desired This is the level which must be appealed to in
state. These interactions are grouped in four order to judge the relevance of the contributions
types of situations which are, in the order of of other scientific domains (pedagogy, psychol-
didactical necessity, inverse to the ordinary chro- ogy, sociology, etc.).
nological order:
1. Situation of reference: A person (student or The Processes
Different modes of composition and articulation
teacher) refers the person asking to a piece of of these elementary situations make it possible
mathematical knowledge (a proof, a theorem, to create composite situations and sequences of
a definition, etc.) that belongs to their common situations that form processes:
repertoire (Perrin-Glorian 1993). 1. Process of mathematization: A sequence
2. Situation of argumentation (of proof):
A proposer communicates to an opponent an of autonomous mathematical situations that
argument, an element of proof. He makes use are introduced by didactical interventions
for that of their common repertoire which his of the teacher and that work together towards
message tends to augment. The argument makes
reference to a milieu and a (mathematical)
project in common that gives it its meaning
and its value. The two speakers have the same

D 168 Didactic Situations in Mathematics Education

the construction of the same complex knowl- it appeared that it would be preferable to have the
edge (e.g., rational and decimal numbers students themselves produce this knowledge
(Brousseau et al. 2004, 2007, 2008, 2009)). and these texts, thanks to specific mathematical
2. Genetic situation: It introduces and without activities that best stimulated the real activity of
other external intervention generates the mathematicians.
sequence of situations that lead to the acquisi-
tion of a concept (e.g., how many white The many didactical situations realized showed
marbles [article cited]). that this project was realizable. Experiments
The didactical work of the teacher then proved it. Curricula were conceived, experimented
consists of maintaining the intensity and the with, and reproduced for all the branches of math-
relevance of the exchanges and implementing ematics and for all the basic levels of teaching
their progress and their conclusion. Examples of (3–12 years old) in an establishment conceived
process: on areas, Perrin-Glorian M.J. (1992), and for the purpose (the COREM).
on geometry, Salin M.H., Berthelot R. (1998).
Currently they cannot be developed because
Some of the Results of Research on of the complexity of knowledge necessary for the
Didactical Situation teachers to conduct them and for the public to
accept them.
The notion of didactical situation was used in many
research projects. It gave rise to numerous reflec- This research produced counterexamples to
tions and, with modifications, was expanded in most of the “universal principles,” explicit or
more work than it is possible to summarize or implicit, of classical didactics, for behaviorist
cite here: methods as well as radical constructivism. It
1. One of its first results was to establish that adap- showed, for example, that in the classical concep-
tion, errors can have no status other than that of
tation to certain conditions tends to render it being far from some norm. They are interpreted
more difficult to adapt to others and thus creates as a failure of the student and/or the teacher that
the phenomenon of didactical obstacles, then to involves their responsibility and ultimately their
show that the history of mathematics presents guilt for a failure of their will. This absurd
phenomena similar to the epistemological obsta- process generates very bad working conditions
cles detected by G. Bachelard, and finally, to for the students and for the teachers.
take advantage of this phenomenon in teaching
by use of situations presenting “jumps in Among many other results, The classical
informational complexity” (▶ Epistemological conception led to seeking out individualization
Obstacles in Mathematics Education) of teaching, but this individualization did
2. Research on situations had the goal of furnish- not improve the results, because mathematical
ing alternatives to the classical conceptions knowledge is produced by the cooperation of
that showed their limitations in the face of numerous individuals operating in the same com-
the influx of knowledge to be taught and of munity, and no isolated brain can produce the
the fundamental reorganizations necessitated exact form that history has given it. For a large
by that influx. This research showed the portion of the students, the real use of communi-
importance of the role of the unsayable in cation and mathematical debates is indispensible.
mathematical situations and of the unsaid in
the didactical relationship. The concept of situation has been the object
Rather than imagining teaching and producing and has been illustrated in a great deal of research
learning of the texts that resulted from real of different types:
mathematical activity by universal, that is 1. Empirical, so as to identify the observables of
nonspecific, nonmathematical teaching methods,
a given teaching episode and analyze them
a priori and a posteriori
2. Experimental, to conceive of either a precise
teaching project (engineering) or a teaching
design (of cognitive psychology, of sociology,
of didactique, etc.)

Didactic Situations in Mathematics Education D169

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Mathematics Education denying the existence of the process of didactic
transposition itself. It must appear that taught
Yves Chevallard1 and Marianna Bosch2 knowledge is not an invention of school. Although
1Apprentissage Didactique, Evaluation, it cannot be a reproduction of scholarly knowledge,
Formation, UMR ADEF – Unite´ Mixte de it should look like preserving its main elements. For
Recherche, Marseile, France instance, the body of knowledge taught at school
2IQS School of Management, Universitat Ramon under the label of “geometry” (or “mechanics,”
Llull, Barcelona, Spain “music,” etc.) has to appear as genuine. It is thus
important to understand the choices made in the
Keywords designation of the knowledge to be taught and the
construction of the taught knowledge to analyze
Anthropological theory of the didactic; Scholarly what is transposed and why and what mechanisms
knowledge; Knowledge to be taught; Institutional explain its final organization and to understand
transposition; Noosphere; Ecology of knowl- what aspects are omitted and will therefore not be
edge; Reference epistemological models diffused.

Definition Scope

The process of didactic transposition refers to the Besides mathematics, research on didactic
transformations an object or a body of knowledge transposition processes has been carried out in
undergoes from the moment it is produced, put into many other educational fields, such as the natural
use, selected, and designed to be taught until it is
actually taught in a given educational institution.
The notion was introduced in the field of didactics
of mathematics by Yves Chevallard (1985, 1992b).

Didactic Transposition in Mathematics Education D171

Scholarly knowledge Taught Learned/available
knowledge to be taught knowledge knowledge

Scholarly and “Noosphere” Teaching Groups of students

other institutions

Didactic Transposition in Mathematics Education, Fig. 1 Diagram of the process of didactic transposition

sciences, philosophy, music, language, technology, ways of questioning and new possibilities to D
and physical education. These investigations have modify it. The notion of didactic transposition
spread faster in the French- and Spanish-speaking is conceived, first of all, as an analytical instru-
communities (Arsac 1992; Arsac et al. 1994; Bosch ment to avoid the “illusion of transparency”
and Gasco´n 2006) than in the English-speaking concerning educational phenomena and, more
ones, although some prominent figures soon con- particularly, the nature of the knowledge
tributed to develop the first transpositive analyses involved, that is, to emancipate research from
(Kang and Kilpatrick 1992). The notion of didactic the viewpoint of the scholarly and the teaching
transposition has been generalized to institutional institutions about the knowledge involved in
transposition (Chevallard 1989, 1992a; Artaud educational processes.
1995) when knowledge is transposed from one
social institution to another. Because of social Any taught field or discipline is the product of
needs, bodies of knowledge originated and devel- an intricate process the singularity of which
oped in different “places” or institutions of society should never be underrated. As a consequence,
need to “live” in other institutions where they one should not take for granted the current,
should be transposed. They have to be transformed, observable organization of a field or discipline
deconstructed, and reconstructed in order to adapt taught at school, as if it were the only possible
to their new institutional setting. For instance, the one. Instead one should see it against the (fuzzy)
mathematical objects used by economists, geogra- set of organizations that could have existed,
phers, or musicians need to be integrated in other some of which may someday turn into reality.
practices commonly ignored by the mathemati- Considering the “scholarly knowledge” as part
cians who produced them. It is clear from the of the object of study of research in didactics
history of science that institutional transpositions – is part of this emancipatory movement of
including didactic ones – do not necessarily pro- detachment. Although school teaching has to
duce degraded versions of the initial bodies of be legitimized by external entities that guaran-
knowledge. Sometimes the transpositive work tee the pertinence and epistemological rele-
improves the organization of knowledge and vance of the knowledge taught (in a complex
makes it more understandable, structured, and process of negotiations which includes crises
accurate to the point that the knowledge originally and disagreements), researchers do not have to
transposed is itself bettered. The organization of consider these institutional perspectives as the
knowledge in fields and disciplines as it exists true or correct viewpoints or as the wrong
today is the fruit of complex and changing histor- ones; they just need to know them and inte-
ical interactional processes of institutional and grate them in the analysis of educational
didactic transpositions that are not well known yet. phenomena.

An Emancipatory Tool In some cases, the “scholarly legitimation”
of school knowledge can be questioned by the
In a field of research, new notions are not only noosphere, on behalf of its cultural relevance:
introduced to describe reality but to provide new “Is this the geometry citizens need?” Such
a conflict situation can change significantly
the conditions of teaching and learning, by
allowing a self-referential, epistemologically

D 172 Didactic Transposition in Mathematics Education

confined teaching. Moreover, there are certain basics”) pressures, canalized by the noosphere, that
teaching processes in which the scholarly cannot be presented here but that delimit the kind of
body of knowledge is created afterwards mathematical practices our students learn (or fail to
because of the need to teach a given content learn) about this body of knowledge. If we look at
that has to be organized, labeled, and recog- scholarly knowledge, the environment is different:
nized as something relevant (an illustrative negative numbers are defined as an extension of the
example is the case of accounting and its set of natural numbers N and form the ring of
corresponding body of knowledge, accoun- integers Z, without any specific discussion (http://
tancy). It is also possible that something that www.encyclopediaofmath.org/index.php/Integer).
is not even commonly recognized as a proper This has not always been the case: it is very well
body of knowledge may appear as “scholarly known that until the mid-nineteenth century, the
knowledge” for the role it assumes in a given possibility of “quantities less than zero” was still
educational process. For instance, in the denied by many scholars. Their final acceptation
teaching of sports, the scholarly knowledge, was strongly related to the needs of algebraic
albeit not academically tailored, includes that work, which explains why, for a long time, inte-
of high-level sport players, even if they are gers were called “algebraic numbers.” It also
a far cry from what we normally consider explains why the introduction of negative num-
“scholars” to be! bers was considered one of the main differences
between arithmetic and algebra. This relation-
Enlargement of the Object of Study ship to elementary algebraic work has now
completely disappeared from the scholar’s and
The second consequence of the detachment school’s conception of negative numbers,
process introduced by the notion of didactic despite the fact that some practices of calcula-
transposition is the evolution of the object of tion – for instance, those involving the product
study of didactics as a research discipline. of integers – acquire their full sense when
Besides studying students’ learning processes interpreted in this context.
and how to improve them through new teach-
ing strategies, the notion of didactic transposi- Various other analyses have brought
tion points at the object of the learning and similar results regarding how the transposition
teaching itself, the “subject matter,” as well as process has affect other different mathematical
its possible different ways of living – its contents (school algebra, linear algebra, limits
diverse ecologies – in the institutions involved of functions, proportionality, geometry, irratio-
in the transposition process. nal numbers, functions, arithmetic, statistics,
proof, modeling, etc.): more generally speak-
Let us take an example on negative numbers. ing, there is no such thing as an eternal, con-
Regarding the transpositive process, the first issue text-free notion or technique, the matter taught
is to consider what the taught knowledge is made of being always shaped by institutional forces
(what concrete activities that are proposed to the that may vary from place to place and time
students, their organization, the domain or block of to time. These investigations underline the
contents they belong to, etc.) and how official institutional relativity of knowledge and show
guidelines and noospherian discourses present to what extend most of the phenomena related
and justify these choices (the knowledge to be to the teaching and learning of mathematics
taught). Today, at most schools, negative numbers are strongly affected by constraints coming
are officially related to the measure of quantities from the different steps of the didactic trans-
with opposite directions and introduced in the con- position process. Consequently, the empirical
text of real-life situations. Where does this school unit of analysis of research in didactics
organization come from? It results from different becomes clearly enlarged, far beyond the rela-
scholar (“new mathematics”) or social (“back-to- tionships between teachers and students and
their individual characteristics.

Didactic Transposition in Mathematics Education D173

Scholarly knowledge Taught Learned/available
knowledge to be taught knowledge knowledge

Scholarly and “Noosphere” Teaching Groups of students

other institutions

D

Reference epistemological model
Research in didactics

Didactic Transposition in Mathematics Education, Fig. 2 The external position of researchers

The Need for Researchers’ Own From Didactic Transposition to the
Epistemological Models Anthropological Approach

Taking into consideration transpositive phenom- When knowledge is considered a changing
ena means moving away from the classroom and reality embodied in human practices taking
being provided with notions and elements to place in social institutions, one cannot think
describe the bodies of knowledge and practices about teaching and learning in individualistic
involved in the different institutions at different terms. The evolution of the research perspec-
moments of time. To do so, the epistemological tive towards a systematic epistemological
emancipation from scholarly and school institu- analysis of knowledge activities explicitly
tions requires researchers to create their own appears at the foundation of the anthropolog-
perspective on the different kinds of knowledge ical theory of the didactic (Chevallard 1992a,
intervening in the didactic transposition process, 2007; Winslow 2011). It is approached through
including their own way of describing knowledge the study of the conditions enabling and the con-
and cognitive practices, their own epistemology. straints hindering the production, development,
In a sense, there is no privileged reference system and diffusion of knowledge and, more generally,
from which to observe the phenomena occurring of any kind of human activity in social
in the different institutions involved in the teach- institutions.
ing process: the scholarly one, the noosphere, the
school, and the classroom. Researchers should Cross-References
build their own reference epistemological models
(Barbe´ et al. 2005) concerning the bodies of ▶ Anthropological Approaches in
knowledge involved in the reality they wish to Mathematics Education, French
approach (see Fig. 2). The term “model” is used Perspectives
to emphasize the fact that any perspective
provided by researchers (what mathematics is, ▶ Curriculum Resources and Textbooks in
what algebra is, what measuring is, what Mathematics Education
negative numbers are, etc.) always constitutes
a methodological proposal for the analysis; as ▶ Didactic Engineering in Mathematics
such, it should constantly be questioned and Education
submitted to empirical confrontation.
▶ Didactic Situations in Mathematics
Education

D 174 Didactical Phenomenology (Freudenthal)

References Didactical Phenomenology
(Freudenthal)
Arsac G (1992) The evolution of a theory in didactics: the
example of didactic transposition. In: Douady R, Marja Van den Heuvel-Panhuizen
Mercier A (eds) Research in Didactique of Freudenthal Institute for Science and
mathematics. Selected papers. La Pense´e sauvage, Mathematics Education, Faculty of Science &
Grenoble, pp 107–130 Faculty of Social and Behavioural Sciences,
Utrecht University, Utrecht, The Netherlands
Arsac G, Chevallard Y, Martinand JL, Tiberghien A,
Balacheff N (1994) La transposition didactique a` Keywords
l’e´preuve. La Pense´e sauvage, Grenoble
Phenomena in reality; Mathematical thought
Artaud M (1995) La mathe´matisation en e´conomie objects; Didactics; Realistic mathematics
comme proble`me didactique. La communaute´ education; Analyses of subject matter
des producteurs de sciences e´conomiques: une
communaute´ d’e´tude. In: Margolinas C (ed) Les de´bats What Is Meant by Didactical
de didactique des mathe´matiques. La Pense´e sauvage, Phenomenology?
Grenoble, pp 113–129
The term didactical phenomenology was coined
Barbe´ J, Bosch M, Espinoza L, Gasco´n J (2005) Didactic by Hans Freudenthal. Although his initial ideas for
restrictions on the teacher’s practice. The case of limits it date from the late 1940s, he likely first used the
of functions in Spanish High Schools. Educ Stud Math term in a German article in 1974. A few years later,
59:235–268 the term appeared in English in his book Weeding
and Sowing – Preface to a Science of Mathemat-
Bosch M, Gasco´n J (2006) Twenty-five years of the ical Education (Freudenthal 1978). Understanding
didactic transposition. ICMI Bull 58:51–64 the term requires comprehending Freudenthal’s
notion of a phenomenology of mathematics,
Chevallard Y (1985) La Transposition Didactique. Du which refers to describing mathematical concepts,
savoir savant au savoir enseigne´, 2nd edn 1991, La structures, or ideas, as thought objects (nooumena)
Pense´e sauvage, Grenoble [Spanish translation: in their relation to the phenomena (phainomena) of
Chevallard Y (1997) La transposicio´n dida´ctica. the physical, social, and mental world that can be
Del saber sabio al saber ensen˜ado. AIQUE, Buenos organized by these thought objects.
Aires]
The term didactical is used by Freudenthal in
Chevallard Y (1989) On didactic transposition theory: some the European continental tradition referring to the
introductory notes. In: Proceedings of the international way we teach students and the organization of
symposium on selected domains of research and teaching processes. This definition of didactics
development in mathematics education, Bratislava, goes back to Comenius’ (1592–1670) Didactica
pp 51–62. http://yves.chevallard.free.fr/spip/spip/IMG/ Magna (Great Didactics) that contains a well-
pdf/On_Didactic_Transposition_Theory.pdf. Accessed founded view on what and how students should
25 Oct 2012 be taught. As such, this meaning of didactics
contrasts with the Anglo-Saxon tradition in
Chevallard Y (1992a) Fundamental concepts in didactics: which it merely has a superficial meaning
perspectives provided by an anthropological approach. involving a set of instructional tricks.
In: Douady R, Mercier A (eds) Research in Didactique
of mathematics. Selected papers. La Pense´e sauvage, Combining the two terms into didactical
Grenoble, pp 131–167 phenomenology implies considering the

Chevallard Y (1992b) A theoretical approach to curricula.
J Math 13(2/3):215–230. http://yves.chevallard.free.fr/
spip/spip/IMG/pdf/A_Theoretical_Approach_to_Curric
ula.pdf. Accessed 25 Oct 2012

Chevallard Y (2007) Readjusting didactics to a changing
epistemology. Eur Educ Res J 6(2):131–134. http://
www.wwwords.co.uk/pdf/freetoview.asp?j¼eerj&
vol¼6&issue¼2&year¼2007&article¼4_Chevallard_
EERJ_6_2_web. Accessed 25 Oct 2012

Kang W, Kilpatrick J (1992) Didactic transposition in
mathematics textbooks. Learn Math 12(1):2–7

Winslow C (2011) Anthropological theory of didactic
phenomena: some examples and principles of its use
in the study of mathematics education. In: Bosch
M et al (eds) Un panorama de la TAD. An overview
of ATD. CRM Documents, vol 10. Centre de Recerca
Matema`tica, Barcelona, pp 533–551. http://www.recer
cat.net/bitstream/handle/2072/200617/Documents10.
pdf?sequence¼1. Accessed 25 Oct 2012

Didactical Phenomenology (Freudenthal) D175

phenomenology of mathematics from a didactical engagement and solving problems in a way they D
perspective. find meaningful. This attachment of meaning
to mathematical constructs students have to
Merit of a Didactical Phenomenology for develop touches on a main principle of RME.
Mathematics Education
Examples of Didactical Phenomenology
In Freudenthal’s words (1983, p. ix), a didactical
phenomenology of mathematics can “show In Weeding and Sowing, Freudenthal exemplified
the teacher the places where the learner might his idea of a didactical phenomenology by
step into the learning process of mankind.” In providing an analysis of the topic of ratio and
other words, a didactical phenomenology informs proportion. Furthermore, he announced to deal
us on how to teach mathematics, including how comprehensively with didactical phenomenology
mathematical thought objects can help organiz- in a following book. That book was Didactical
ing and structuring phenomena in reality, which phenomenology of mathematical structures
phenomena may contribute to the development of (Freudenthal 1983). In this book, he gave more
particular mathematical concepts, how students examples of didactical phenomenologies,
can come in contact with these phenomena, how including those of length, natural numbers, frac-
these phenomena beg to be organized by the tions, geometry and topology, negative numbers
mathematics intended to be taught, and how and directed magnitudes, algebraic language, and
students can be brought to higher levels of under- functions.
standing. As such, Freudenthal’s didactical phe-
nomenologies are landmarks for developing Remarkably, these examples did not just deal
teaching outlines. with connecting mathematical thought objects to
phenomena in reality to find starting points for
Relation with Realistic Mathematics learning mathematics. In fact, these examples
Education were profoundly scrutinized analyses of subject
matter in which the key concepts of a particular
By disclosing the sources of mathematics in real- mathematical topic were disclosed together with
ity, a didactical phenomenology is strongly contexts which have a model character and with
related to Realistic Mathematics Education significant landmarks in students’ learning
(RME), the domain-specific instruction theory pathways.
for mathematics, which has been developed in
the Netherlands and in which Freudenthal was The Method
heavily involved (Freudenthal 1991). In RME,
rich, realistic situations have a prominent posi- Unfortunately, in Didactical phenomenology of
tion in the learning process. These situations mathematical structures, Freudenthal did not
serve as sources for initiating the development elaborate much on how to establish these
of mathematical concepts, tools, and procedures. didactical phenomenologies. Although the book
What situations can serve as contexts for this contains a short chapter titled The method, this
development is revealed by a didactical phenom- did not reveal how to generate such phenomenol-
enology. By tracing phenomena in reality that can ogies. Nevertheless, a corner of the veil was
elicit mathematical thoughts, the students are lifted when Freudenthal (1983, p. 29) considered
given access to the sources of mathematics in the material he needed to write this book:
everyday experiences. Building on these sources
offers them an orientation basis they experience I have profited from my knowledge of mathemat-
as real and opens the possibility of personal ics, its application, and its history. I know how
mathematical ideas have come or could have
come into being. From an analysis of textbooks

D 176 Discourse Analytic Approaches in Mathematics Education

I know how didacticians judge that they can are. In Freudenthal’s view, they form the heart
support the development of such ideas in the of researching and developing mathematics
minds of learners. Finally, by observing learning education.
processes I have succeeded in understanding
a bit about the actual process of constitution of Cross-References
mathematical structures and the attainment of ▶ Realistic Mathematics Education
mathematical concepts
References
This statement and the provided examples
show how a didactical phenomenology results
from a number of analyses, each taking
a different perspective: didactical, phenomeno-
logical, epistemological, and historical-cultural.

Mathematics-Related Analyses Freudenthal H (1978) Weeding and sowing. Preface
Constituting the Didactics of to a science of mathematical education. Reidel,
Mathematics Dordrecht

These analyses have in common that they all take Freudenthal H (1983) Didactical phenomenology of
mathematics as their starting point. Didactical mathematical structures. Reidel, Dordrecht
analyses examine the nature of the mathematical
content as a basis for teaching this content. Freudenthal H (1991) Revisiting mathematics education.
By identifying the determining aspects of mathe- China lectures. Kluwer, Dordrecht
matical concepts and their relationships, knowl-
edge is gathered about didactical models that can Discourse Analytic Approaches in
help students to understand these concepts. Mathematics Education
Phenomenological analyses disclose possible
manifestations of these mathematical concepts Candia Morgan
in reality and can suggest contexts for students Institute of Education, University of London,
to meet these concepts. Epistemological analyses London, UK
focus on students’ learning processes and can
uncover how the mathematical understanding of Keywords
students in a classroom interaction may shift.
Finally, in historical-cultural analyses, we may Discourse; Discourse analysis; Foucault;
encounter current and past approaches to Identity; Language; Linguistics; Methodology;
teaching mathematics through which we can Social practice
gain a better understanding of learning mathe-
matics and how education can contribute to it. Introduction

These analyses are all included in The first challenge in addressing this topic is the
Freudenthal’s didactical phenomenology and multiplicity of ways in which the term discourse
surpass its narrow literal meaning, which would is used and defined – or not defined – within
certainly have his approval, as in Weeding and mathematics education (see Ryve 2011). It is
Sowing Freudenthal (1978, p. 116) already stated: frequently found, especially in discussions within
“[T]he name does not matter; nor is that activity the context of curriculum reform, simply to sig-
[didactical phenomenology] an invention of nify student engagement in talk in the classroom.
mine; more or less consciously it has been Without denying the value of the development of
practiced by didacticians of mathematics for such engagement, the approaches to discourse
a long time” (Freudenthal 1978, p. 116). Indeed, and discourse analysis considered in this article
the name is not essential, but these analyses

Discourse Analytic Approaches in Mathematics Education D177

all take rather more complex and theoretically its origin in the thinking of the French philoso- D
shaped views of the nature of discourse – views pher Foucault (e.g., 1972) whose work includes
that influence the focus of research and the studies of the construction of “regimes of truth”
analytic methods. An important component of about notions such as madness or sexuality.
the ways these approaches conceive of discourse Though not all discourse analytic research in
is a concern with the relationship between lan- mathematics education comes from this tradition,
guage (and other modes of communication), the it can generally be characterized as tending either
social context in which it is used, and the towards analysis of discourse, focusing on
meanings that are produced in this context communication events and the local social
(Howarth 2000). It is this concern and the funda- practices within which they arise, or towards
mental assumption that studying the way lan- analysis of Discourse, taking larger scale social
guage is used can provide insight into the practices and structures as the object of research.
activity or practice (mathematics or mathematics Of course, some approaches move between
education) in which it is used that leads the two, generating interpretation of specific
researchers to adopt discourse analytic communication events by applying knowledge
approaches. Of course, a very high proportion of of wider social practices and structures or
the data used in studies across many branches of building a picture of a significant social practice
mathematics education research is primarily lin- through analysis of local communication events.
guistic or textual: interviews, written responses to Discourse analytic approaches thus vary in two
questionnaires, classroom transcripts, written dimensions: the extent to which they make use of
texts produced by students, etc. Increasingly it detailed linguistic analysis and the extent of
has also been recognized by researchers using their focus on social practices, structures, and
a wide range of approaches that the language institutions.
produced by students or other research subjects
is not a transparent medium through which it is The adoption and development of discourse
easy to decipher an underlying truth. What distin- analytic approaches in mathematics education
guishes research that adopts a discourse analytic research largely coincided with what Lerman
approach is the assumption that the language termed the “social turn” (Lerman 2000).
is itself an inextricable part (or, for some Increased recognition of the importance of
researchers, even the whole) of the object of studying and taking account of the social nature
study. This assumption is shared with another of mathematics education practices as well as of
analytic approach, conversation analysis, and individual cognition demanded the development
some discourse analysts make use of methods of theoretical ways of conceiving of social
developed in conversation analysis. However, practices and methodological approaches to
whereas discourse analysis is generally interested studying them. Discourse analytic approaches
in characterizing the practices within which lan- provided one way of addressing this demand.
guage plays a role, conversation analysis focuses This development within the field of mathematics
primarily on how linguistic interactions them- education reflected a much wider development of
selves are organized to achieve social actions theories of discourse and discourse analytic
(see Wooffitt 2005, for an introduction to the methods within social science and the humani-
two approaches from a conversation analytic ties. As researchers have begun to draw on
perspective). theories and methods originating outside the
field of mathematics education, they have faced
Gee (1996) makes a useful distinction the challenge of ensuring that both theory and
between discourse, defined as instances of com- methods take account of the specialized nature
munication, and Discourses, the conjunctions of of mathematical communication and practices
ways of speaking, subject positions, values, and that they have the power to illuminate issues
etc. that characterize and structure particular of interest to mathematics education. Facing
social practices. The notion of Discourses has this challenge is a continuing project; notable

D 178 Discourse Analytic Approaches in Mathematics Education

contributions have come from within mathemat- participation in mathematics of students from
ics education (e.g., Morgan 1998; Sfard 2008) different social groups.
and from linguistics (e.g., O’Halloran 2005).
In this article there is no space to provide
With a few exceptions, notably the work of a detailed review of the full range of approaches
Walkerdine (1988) who used analyses of taken to discourse analysis. Instead, we provide
Discourses, including Discourses of gender and a small number of contrasting cases, exemplify-
of child-centered education, in order to under- ing the scope of discourse analytic methods
stand how differences between various social and the problems in mathematics education that
groups are constructed in mathematics education they may be used to address.
practices, early interest in discourse analytic
approaches, such as that represented in the Critical Discourse Analysis
Special Issue of Educational Studies in
Mathematics edited by Kieren et al. (2001), was Critical discourse analysis (CDA) comprises
dominated by analysis of communication events a group of analytic approaches, all of which make
(discourse), focusing on understanding class- strong analytic connections between forms of lan-
room interaction and the development of guage use, social practices, and social structures.
mathematical thinking in interaction. At a time The label “critical” indicates a concern of the
when the majority of research in mathematics researchers to make use of the knowledge achieved
education focused on the mathematical thinking through the analysis in order to enable critique and
of individuals, this application of discourse transformation of the social practices and/or struc-
analysis may be seen as an incremental manifes- tures. Research using CDA approaches thus tends
tation of the “social turn,” addressing the same to produce analyses that not only describe existing
interest in mathematical thinking but reconcep- practices but also critique the ways these practices
tualizing it as a phenomenon that is evident position students and/or teachers and the kinds of
(and, for some researchers, produced) in social mathematics and mathematical identities that are
interaction. More recently, the issues addressed valued and made possible.
by the mathematics education research commu-
nity have expanded, incorporating a wider CDA studies generally involve detailed
conceptualization of mathematics and mathemat- analyses of texts, including oral and written
ics education as social practices. Thus more texts produced and used by students and teachers
research has addressed, inter alia, identity, in the classroom but also including texts such as
power relationships, and social justice – issues the curriculum and policy documents that
that lend themselves to study using approaches structure and regulate these educational practices
that focus on Discourses. Some of this research and thus affect the interpretation of classroom
has adopted approaches that may be character- texts. Within mathematics education, probably
ized as structuralist, drawing on sociological the most widely used type of CDA is based on
accounts of social structures such as the work of the approach of Norman Fairclough (2003), using
Basil Bernstein (e.g., 2000) to describe and inter- linguistic tools drawn from systemic functional
pret discursive phenomena. Others have adopted linguistics (SFL). This approach has been used to
poststructural approaches, in which the commu- investigate specific practices such as the
nicative action itself constructs the “reality” of assessment of student reports of mathematical
which it speaks. A recent edited book entitled investigation (Morgan 1998) or the use of
Equity in Discourse for Mathematics Education “real-world problems” in an undergraduate
(Herbel-Eisenmann et al. 2012) reflects this range mathematics course (Le Roux 2008). Research
of approaches and interpretations, combining adopting a CDA approach may also use a range
detailed analyses of classroom interactions with of other methods to address textual data,
concern for how these interactions and broader including corpus analysis of large data sets (e.g.,
social practices affect the possibilities for Herbel-Eisenmann et al. 2010).