Content uploaded by Paola Valero
Author content
All content in this area was uploaded by Paola Valero on Mar 26, 2015
Content may be subject to copyright.
FROM QUESTIONS OF HOW TO QUESTIONS OF WHY
IN MATHEMATICS EDUCATION RESEARCH
Alexandre Pais, Diana Stentoft, and Paola Valero
Department of Learning, Education and Philosophy – Aalborg University
The educational sciences are generally construed around concerns of providing
research that informs practices of learning and teaching in educational institutions.
This research emphasizes questions of how to and has led to a “technification” of
educational research, as primarily concerned with providing solutions to practical
problems. In this paper we will show how mathematics education as a research field
is not an exception, by analysing how theory is understood and used in the field, to
address questions of how. We suggest that, although important, this research leave
some important areas unaddressed, namely the ones which can emerge from posing
questions of why. We argue that making this move implies rethinking and enlarging
definitions and views of mathematics education research.
INTRODUCTION
In recent decades the field of mathematics education research has opened its agenda
towards new paradigms and discourses, and it has expanded the field also to include
issues of the social, the cultural and the political. Issues of social justice (Gutstein,
2003), critical mathematics education (Skovsmose, 1994), equity (Secada, Fennema,
& Adajian, 1995), ethnomathematics (D’Ambrosio, 2002) among others, have
become influential players in a research field otherwise and continuously dominated
by research exploring psychological and cognitive aspects of students’ and teachers’
engagement with mathematics. Although we consider this move towards the socio-
political and socio-cultural a significant one, we also see a need to move the
boundaries even further. We thus suggest a move from a research agenda primarily
contained within a very specific discourse of the importance of mathematics
education, addressing primarily questions of how to improve possibilities for teaching
and learning mathematics, towards a research agenda strongly concerned with
addressing the question of why mathematics education. In making this move we see
possibilities of opening up the field to alternative discourses and ways of constructing
important understandings about the teaching and learning of mathematics in complex
social, political and economic settings. We will explore this move from questions of
how to questions of why in relation to the role of theory in mathematics education
research. We will argue that the overwhelming majority of theories constructed in the
field aim to address questions of how and, therefore, do not have the possibility of
seeing beyond a technical rationality in order to understand the whys of the
configuration of mathematics education practices in classrooms, schools and society.
Based on an analysis of recent literature addressing the role of theory in mathematics
education research, we start by pointing to the way this research is structured around
questions of how. We then analyse some recent trends in mathematics education
research (arising out of the so-called “social-turn” (Lerman, 2000)), which has
contributed to an enlargement of a field traditionally dominated by a didactical
perspective. This research has opened the field to questions broader than those strictly
concerned with providing immediate solutions for practical problems. Nevertheless,
we will argue that even research presented within the scopes of the social, cultural
and political often focuses on questions of how. We then proceed to bring in
questions of why, by exploring new discourses embedded into this simple question.
We conclude the paper with some brief comments about the implications of
transgressing the boundaries of the existing discourses shaping the field of
mathematics education research.
THEORY CONSTRUCTING RESEARCH IN MATHEMATICS EDUCATION
Theory as a key component of mathematics education research is currently on the
agenda. At ICME 11 in 2008 one of the survey teams developed a study on the notion
and role of theory in mathematics education research. This survey team had the task
of identifying, surveying, and analysing different notions and roles of ‘theory’, as
well as providing an account of the origin, nature, uses, and implications of specific
theoretical directions pertaining to different research developments in the field.
Similarly, the Second Handbook on Mathematics Teaching and Learning (Lester,
2007) contains two articles addressing issues of theory (Cobb, 2007; Silver & Herbst,
2007). In CERME there has been a working group linking, contrasting and
comparing the wide variety of theoretical approaches found in the field in order to
tackle the teaching and learning of mathematics. In 2008 the international journal
ZDM published an issue of some of the results of the CERME working group. Finally
in 2009 the theme of PME 33 was “In search for theories in mathematics education”.
These examples point to a widespread desire of the community for understanding the
role of theory in mathematics education research and a wide acknowledgement of the
variety of perspectives brought into the field through theoretical expositions. As
Silver and Herbst (2007, p. 41) state, “the moment seems propitious for a serious
examination of the role that theory plays and could play in the formulation of
problems, in the design and methods employed, and in the interpretation of findings
in education research.”
We wish to make a modest contribution to this discussion by engaging in a critical
analysis raising questions of how and why. We wish to understand in more detail how
research perspectives in general and theoretical perspectives in particular construct
and/or ignore particular discourses and, in this, our possibilities for addressing these
basic yet powerful questions.
As the “linguistic turn” in the social sciences has touched mathematics education
research (Lerman, 2000), it appears increasingly important to pay attention to the
discourses that mathematics education research constructs about itself and the
contributions and limitations of these constructions. By discourses here we
understand the ways of naming and phrasing the ideas, values and norms that emerge
from the constant and complex interactions among human beings while engaged in
social practices. Academic fields construct particular discourses about themselves
and their objects of study. Such discourses constitute systems of reason that regulate
what is possible to think and do in a given field (Popkewitz, 2004). Discourses thus
both open up possibilities and impose limitations on what we can imagine and
construct as alternatives to existing orders. Mathematics education as a field of
research is not an exception. As researchers engage in studying the field, they not
only define what is characterized as legitimate practices of mathematics education.
They also define the ways in which it is valid and legitimate to research those
practices (Valero, 2009). We have engaged elsewhere in examinating the discourses
generated in and by the field of mathematics education research, such as the idea of
mathematics education being “powerful” (Christensen, Stentoft & Valero, 2008), the
conceptions of students as mathematics learners (Valero, 2004), the concept of
learners’ identity in mathematics (Stentoft & Valero, in press b) and the concept and
view of ethnomathematics (Domite & Pais, 2009). We have also pointed to some
blind spots of some of the theoretical constructions in the field. Considering these
constructions of various discourses in the field we argue for the need to broaden the
research gaze of mathematics education research to embrace the “noises” that are
often ignored, in a search for new imaginaries for our field of study and for the
educational practices in mathematics (Stentoft & Valero, in press a).
MATHEMATICS EDUCATION RESEARCH AS A SCIENCE OF HOW
One major assumption in mathematics education research is that its main aim is to
improve students’ performance in mathematics. For example, Niss (2007, p. 1293) is
very clear when answering the question of why do we do research in mathematics
education: “We do research on the teaching and learning of mathematics because
there are far too many students of mathematics, from kindergarten to university, who
get much less out of their mathematical education than would be desirable for them
and for society.” If this is the main concern of mathematics education research, it is
not surprising that the field has grown as a space for researching in a systematic,
scientific way “the problems of practice” (Silver & Herbst, p. 45), defined as
problems relating to teaching and learning. According to Boero (in press) “this is a
rather obvious widely shared position” (p. 1). In this framework, the work of
mathematics educators is “to identify important teaching and learning problems,
considerer different existing theories and try to understand the potential and
limitations of the tools provided by these theories.” (Boero, in press, p. 1)
The above quotes demonstrate an emphasis in the field of mathematics education
research on the questions of how. How can we improve and enhance the teaching and
learning of mathematics? How can we help students to learn? These questions are
highlighted further when Cobb (2007) addresses the issue of philosophy in
mathematics education as he in a concise manner addresses assumptions engulfing
the field of research. Cobb suggests that mathematics education should be understood
as a “design science” (2007, p. 7), and provides as an example the NCTM standards.
By design science Cobb understands “the collective mission which involves
developing, testing, and revising conjectured designs for supporting envisioned
learning process” (p. 7). The ultimate goal of a science designed this way is to
“support the improvement of students’ mathematical learning” (p. 8). As part of the
pragmatic realist philosophy adopted by Cobb, attention is given to the comparison
between four significant theoretical perspectives used in mathematics education
research, namely experimental psychology, cognitive psychology, socio-cultural
theory and distributed cognition. Cobb’s discussion revolves around how these
theoretical perspectives could help improving students’ learning of mathematics. We
can research at the level of the national educational system, school or classroom,
however the goal remains the same. In Cobb’s writing, theory is understood as a tool
to give insight and understanding into learning processes with the aim of improving
them.
An alluring analogy made by Silver and Herbst (2007) between mathematics
education and medicine helps us to understand the meaning of theory as “theory for
learning”. The authors play with the analogy that mathematics education can be seen
as a science of treatment, similar to medicine: By understanding the symptoms that
characterise the difficulties of students’ mathematical learning we can propose the
proper treatment. They state: “The evolving understanding of the logic of errors has
helped support the design of better instructional treatments, in much the same way
that the evolving understanding of the logic of diseases has helped the design of
better medical treatments” (Silver & Herbst, 2007, p. 63). In this perspective, students
are seen as patients in need of treatment, and the role of mathematics education
research is to understand students’ problems and elaborate designs that direct us how
to treat those learning diseases.
This trend that focuses on learning — enhancing or remediating it—is not exclusive
to the field of mathematics education research. Philosophers of education such as
Biesta (2005) argue that over the last two decades this perspective has proliferated in
broader educational discourses where a technical language of learning has largely
dominated and almost overruled a language of education. The “learnification of
education”, in Biesta’s terms, has narrowed the possibilities to think and do education
and educational research. The disagreements about the role of school and the goals of
education that fuelled part of the educational debate during the last century1 seem to
have been overcome. We appear to have reached a consensus on the benefits of
schooling: we need to make it more effective and, therefore, we live an apparent
consensus about what concerns education. The problems with schooling and school
subjects are no longer to be political or ideological, but have become primarily
technical or didactical. In most cases, solutions to educational problems are being
reduced to the devising of better teaching and learning methods and techniques, to
improve the use of technology, to assess student’s performance, etc. Educational
1 For instance the discussions fueled by the work of John Dewey, Ivan lllich, Louis Althusser or Paulo Freire.
thinking has progressively been reduced to be a controllable, designable,
engineerable and operational framework of action for the improvement of individual
cognitive change. It is obvious that the research supporting the emergence of this type
of discourse is a research essentially concerned with questions of how.
Although the prevalence of theory as “learning theory” has allowed us to gain deeper
knowledge about the processes of teaching and learning mathematics, we suggest that
it has left important discourses faced by the educational communities in their
everyday practices unaddressed. We will argue that in order to bring these discourses
seriously into the gaze of research, we need a broader theoretical palette which allows
us to understand theory not just as “theory of learning”, but also as “theory of
education”. This leads us to propose another type of question for the research agenda,
namely the questions of why.
TOWARDS QUESTIONS OF WHY
As mentioned above, the “social turn” (Lerman, 2000) in mathematics education
brought to the field new concerns and new theories that progressively de-emphasise
cognitive psychology as the only interpretative framework and instead favour socio-
cultural theories. In this we have witnessed a move from an understanding of
children’s learning focused on the individual subject and his cognition to an
understanding that perceives learning as a product of social activity, where not only
the cognition of the subject is at stake but also his relations with other individuals and
their shared discourses.
This trend is not merely related to a displacement of the way we perceive processes
of learning. According to Lerman (2000) this trend also emerged as a result of
growing political concerns about the ways mathematics education could be linked to
reproduction of inequalities through the structures of school. Several studies in recent
years have contributed to an understanding of mathematics education in association
with issues of social exclusion according to race, gender, language, social class and
culture. Those studies have opened up a space of critique about the way mathematics
education could be contributing to systematic social exclusion of some groups
carrying particular characteristics. The critical role of mathematics education in
society is also addressed in research on ethnomathematics, particularly in studies
aiming to understand how mathematics in society conveys hegemonic discourses and
oppressive practices that promote exclusion and domination (e. g. Powell &
Frankenstein, 1997). Skovsmose (1994), analyses the way mathematics formats
reality, by creating models that end up ruling our decisions and daily lives. This
“mathematics in action” is critical since it is not neutral, but ideologically loaded,
conveying economic, military or national interests. Finally, another way of analysing
the critical role of mathematics in society is by raising the issue of power. Valero
(2004) and Skovsmose and Valero (2002) have developed a theoretical framework to
engage with the issue of power in mathematics, namely, to understand how the idea
that “mathematics empowers people” is conceived in mathematics education.
Popkewitz (2004), in his incursion into mathematics education research, applied a
Foucauldian perspective on mathematics as a school subject. He brought out the
mechanisms though which the alchemy of school mathematics constructs a set of
learning standards that are more closely related to the administration of children than
with an agenda of mathematical knowledge. This alchemy is carried out by pedagogy
(psychology and social psychology that generate knowledge about children) that
appropriates the mathematical content to transmit competences, behaviours and
attitudes (e.g., being participative, critical, having self-esteem, etc.). In this
perspective, school mathematics serves as an alibi for the appropriation of behaviours
and modes of thinking and acting that make each child governable.
Some of the research outlined above, bearing social, cultural and political
connotations, has opened up the field of mathematics education by conceiving theory
as more than “theory for learning”, and posing questions that do not imply a
“technical” response or solution but rather an intellectual and philosophical
reflection. This is research which, instead of “facilitating” the work of intervention in
the mathematics education process (particularly students and teachers), points to
potential and unexplored problems within the field, and raises more questions than
answers. This kind of research has an intention to “complicate” and to dislocate
“certainties” assumed in the field.
However, despite this invigorating openness, we argue that a significant part of
research in mathematics education labelled socio-cultural-political research shows a
tendency to understand mathematics education in a didactical sense and to aim
primarily to address questions of how: How to teach in multicultural classrooms?
How to teach for social justice? How to educate teachers for social justice? How to
integrate immigrant students in the learning of mathematics? How the socio-cultural
contexts of students influence the learning of the concepts of chance and probability?
These questions were found in the proceedings of the Mathematics Education and
Society, MES conference in Albufeira, Portugal in 2008 (Matos, Valero &
Yasukawa, 2008), and shows how even in a research environment where the
emphasis is on the political, the research persists on the question addressing the
technicalities of the field.
IMPLICATIONS OF RESEARCHING QUESTIONS OF WHY
We acknowledge the importance of raising questions of how. The research that comes
from raising such a question is one that intends to give solutions to the problems
faced by those involved in the teaching and learning of mathematics. It is what we
can call comfortable research. And all of us need some amount of comfort in our
lives. Asking questions of how opens up to discourses concerning the individuals
navigating with and in mathematics. First and foremost it invites propositions of how
students can learn, with some underlying assumption that it is important for the
student to learn mathematics. Second, it invites perspectives on teaching and the
teacher as a key player to assist in meeting the hypothesis of the importance of
mathematics education. Third, questions of how invite a broader socio-political and
socio-cultural perspective when they address issues of resources, gender, political
agendas etc. The question can in this respect hold a strong political agenda when it
asks how we distribute resources best to ensure that all receive mathematics
education. Questions of how navigate within an implicit discourse assuming and
attributing some kind of importance to mathematics education. Although potentially
political these questions do not touch upon fundamentals or put a question mark on
the nature and content of the research field itself. In other words, questions of how
take mathematics education and mathematics education research for granted and
consequently they lack a scope for what can be termed radical alternatives.
As we argued at the beginning of this paper, the ultimate goal for mathematics
education appears to be improving students’ mathematical learning. The idea
described previously of mathematics education as a therapy, a design science or a
science of how constructs education as a technological endeavour, where
mathematics education is understood as a technical engineering of students’
mathematical thinking and learning. We acknowledge the contributions that this
learnification has brought to our understanding of what happens in a mathematics
classroom at a micro-scale. Nevertheless we argue that reducing the possible meaning
of “mathematics education” to “mathematical learning” can narrow our perspectives.
And thus it becomes impossible to think and act in ways that could open spaces of
possibilities inside and outside mathematics education research. Cobb (2007) is well
aware of this. When referring to the theory that informs the researcher he mentions
that “the constraints on what is thinkable and possible are typically invisible” (2007,
p. 7). This awareness also emerges strongly in much research and it is obvious that
addressing mathematics education from the narrow perspective pointed out here,
reconfirms the fact that “if we look strictly at events as they occur in the classroom,
without consideration of the complex forces that helped to shape those learning
conditions, our understanding is only partial [and] the solutions to the problem [are]
ineffectual” (Rousseau & Tate, 2008). Very few researchers, however, have
addressed these limitations.
The MES conference appeared more than ten years ago with an intention of
broadening the research field by developing and applying new approaches, new
methodologies and new theories to the problems faced in mathematics education
research. The MES community acknowledges the need to address these problems
from cultural, social and political approaches that situate the problems in a broader
context than classrooms and schools. However, assuming a social and political
perspective of mathematics education as a research field also involves developing
research where the field itself is under critical scrutiny, and where we can formulate
questions that are not directed only towards how to develop better ways to teach and
learn mathematics (in cultural settings, for social justice, in a critical way, etc.). This
kind of research raises the question of why the theories, methods and discourses that
research constructs and is embedded into. Ultimately it raises the question of why
mathematics education, which implies an analysis about the discourses setting the
scene for its very existence.
Core questions such as the goals of mathematics education, the whys and for whom,
are political issues that should not be left unattended. The field of mathematics
education is not simply a technical field, where the teacher should improve his/her
teaching skills and where researchers should develop designs to improve teaching and
learning possibilities. To say that education is political means to bring to the field a
discussion on the construction of subjectivities through mathematics education. It
means addressing the issue of which kind of people are being formed by the learning
of mathematics, and for what and why are people to engage in the teaching and
learning of mathematics? Ultimately, we can engage in a discussion of which kind of
world is being constructed and sustained by the research in mathematics education?
Therefore, a theory of mathematics education (and not just for mathematics learning)
that places educational practices in a wider political context, where mathematics and
mathematics education are neither neutral nor intrinsically “beneficial”, makes it
possible to raise deep educational questions about the teaching and learning of
mathematics in the social, political, economic, cultural and historic contexts in which
they are immersed.
REFERENCES
Biesta, G. (2005). Against learning. Reclaiming a language for education in an age of
learning. Nordisk Pædagogik, 25(1), 54-55.
Boero, P. (in press). Autonomy and identity of mathematics education: why and how
to use external theories. To be published as part of the proceedings of the 11th
International Congress on Mathematics Education.
Cobb, P. (2007). Putting philosophy to work: coping with multiple theoretical
perspectives. In F. Lester (Ed.), Second Handbook of Research on Mathematics
and Learning (pp. 3-38), New York: Information Age.
Christensen, O. R., Stentoft, D., & Valero, P. (2008). A Landscape of Power
Distribution. In K. Nolan & E. De Freitas (Eds.), Opening the Research Text:
Critical Insights and In(ter)ventions into Mathematics Education (pp. 147-154).
New York: Springer.
D’Ambrosio, U. (2002). Etnomatemática: elo entre as tradições e a modernidade
(2ªed.). Belo Horizonte: Autêntica.
Domite, M. & Pais, A. (2009). Understanding ethnomathematics from its criticisms
and contradictions. In Proceedings of the Sixth Conference of European Research
in Mathematics Education (in press). Lyon, France.
Gutstein, E. (2003). Teaching and Learning Mathematics for Social Justice in an
Urban, Latino School. Journal for Research in Mathematics Education, 23(1), 37-
73.
Lerman, S. (2000). The social turn in mathematics education research. In J. Boaler
(Ed.), Multiple perspectives on mathematics teaching and learning (pp. 19-44).
Westport (USA): Ablex Publishing.
Lester, F. (Ed.). (2007). Second Handbook of Research on Mathematics Teaching
and Learning. Charlotte, USA: NCTM – IAP.
Matos, J. F., Valero, P. & Yasukawa, K. (eds.) (2008). Proceedings of the Fifth
International Mathematics Education and Society Conference. Albufeira, Portugal
NCTM (1992). Curriculum and evaluation standards for school mathematics.
Addenda series, grades 9-12. Reston, Virginia: Author.
Niss, M. (2007). Reflections on the state of and trends in research on mathematics
teaching and learning. In F. Lester (Ed.), Second Handbook of Research on
Mathematics and Learning (pp. 1293-1312), New York: Information Age.
Popkewitz, T. (2004). The alchemy of the mathematics curriculum: Inscriptions and
the fabrication of the child. In American Educational Research Journal, 41(1), pp.
3-34.
Powell, A. & Frankestein, M. (1997). Ethnomathematics: challenging eurocentrism
in mathematics education. State University of New York Press.
Rousseau, C., & Tate, W. F. (2008). Still separate, still unequal: Democratic access to
mathematics in U.S. schools. In L. D. English & M. G. Bartolini Bussi (Eds.),
Handbook of international research in mathematics education (2nd ed., pp. 299-
319). New York, NY: Routledge.
Secada, W., Fennema, E., & Adajian, L. (Eds.). (1995). New directions for equity in
mathematics education.Cambridge: Cambridge University.
Silver, E. & Herbst, P. (2007). Theory in mathematics education scholarship. In F.
Lester (Ed.), Second Handbook of Research on Mathematics and Learning (pp. 39-
56), New York: Information Age.
Skovsmose, O. (1994). Towards a philosophy of critical mathematics
education.Dordrecht: Kluwer.
Skovsmose, O. & Valero, P. (2002). Democratic access to powerful mathematical
ideas. En L. D. English (Ed.), Handbook of international research in mathematics
education: Directions for the 21st century (pp. 383-407). Mahwah, USA:
Lawrence Erlbaum Associates.
Stentoft, D., & Valero, P. (In press a). Fragile learning in mathematics classrooms:
How mathematics lessons are not just for learning mathematics. In M. Walshaw
(Ed.), Unpacking pedagogies. New perspectives for mathematics. Charlotte, USA:
IAP.
Stentoft, D., & Valero, P. (In press b). Identities-in-action: Exploring the fragility of
discourse and identity in learning mathematics. Nordic Studies in Mathematics
Education, 14(2).
Valero, P. (2004). Socio-political perspectives on mathematics education. In Valero,
P. & Zevenbergen, R. (Eds.). Researching the socio-political dimensions of
mathematics education: issues of power in theory and methodology (pp. 1-17).
Dordrecht: Kluwer Academic Publishers.
Valero, P. (2009). Mathematics education as a network of social practices. Invited
keynote lecture at the 6th Conference of the European Society for research in
Mathematics Education (CERME6) (forthcoming proceedings). University Joseph
Fourier, Lyon, France.
