# F. A. Khan: Mathematics and its Discontents

**TRG Poverty &
Education | Working Paper Series 5 (2015)**

**Farida
Abdulla Khan**

**Mathematics
and its Discontents**

How well does mathematics pedagogy serve the children of the poor?

Abstract

This paper describes the uniquely specific characteristics of Mathematics, the special significance of semiotic representation systems in mathematical thinking and the special demands it makes on children's cognitive processing for learning Mathematics. I argue that problems of learning in Mathematics classrooms need to be linked to and analysed within the socio-economic contexts of schooling practices and children's lives, so that the burden of failure is not attributed solely to the child and to individual capacity. The status of Mathematics within the educational system gives it tremendous power to determine children's life chances either by way of progression to higher levels of education or by a disproportionately large influence in determining admission to high status courses and professions. Ignoring the special nature of Mathematics, the out-of-school constraints that poverty imposes, and the less than adequate classrooms and schools where the poor have access to education, only serves to mask the complex ways in which learning is mediated by these factors and the serious implications this has for Mathematics learning and life chances of poor children.

## Introduction

The idea of education as a
universally positive good has not come under enough scrutiny
although critiques of this accepted view are beginning to be
taken more seriously. The fact that education and the
certification it provides is the key to securing jobs and other
economic and social advantage can hardly be contested.
Nevertheless, how educational opportunity unfolds and how and why
it fails so many children is not clearly understood. The practice
of education and of schooling is embedded within a highly complex
social, economic and political context and causeeffect relationships are difficult to trace and to
establish. This location and its dynamics influence the outcomes
of education in important ways.

An unproblematic acceptance of education as simply and always a
means of positive social change masks the ambiguities and the
processes that in fact often contribute to the very inequalities
and unjust outcomes that it is meant to overcome. A critical
tradition that highlights the ways in which this happens,
continues to elicit staunch resistance within mainstream
educational theory and practice. Althusser (1971) introduced the
concept of education as an 'ideological state apparatus' and
Bowles and Gintis (1976) made a landmark intervention with their
thesis on 'Schooling in capitalist America'. Bourdieu and
Passeron (1990) have alerted us to the forms of social and
cultural capital that reinforce class hierarchies through the
educational system and in fact promote entrenched social
inequalities; Shirley Brice Heath (1983) unveiled the middle
class culture of schools that silences other voices and
disadvantages working class children and Foucault (1991) analysed
the ways in which knowledge bestows power and the reach of social
and political control that it allows dominant power groups to
yield. These theoretical debates on education, schools and
society have initiated and allowed more complex readings of
education and inequality Ball, 2004).^{1}

Issues of access, inequality and
exclusion have been a priority in Indian governmental policy and
within the educational establishment since Independence and
continue to remain a cause for concern. Social science research
has documented the relationship between education and poverty
along with other parameters of disadvantage like caste, gender,
region and community that influence educational outcomes.
Although prior to the 1990s the focus was largely on access and
bringing the poor and the marginalised into schools, a more
critical engagement with education has been established in the
last few decades and a critical sociology of education has begun
to address these issues more substantially. Sociologists have
contributed to an understanding of institutional contexts and
'have highlighted processes of stratification, differentiation
and hierarchies of power that shape the social reality within
schools, and influence the construction of diverse "schooled”
identities' (Nambissan, 2014: 8889).^{2}

An important indicator of these hierarchies
is the differential level of academic achievement where the poor
and the socially marginalised are at a disadvantage, and
generally the least successful by any conventional measure of
academic success. The content and processes of teaching and
learning are critical to academic success and constitute a major
focus of research for understanding academic achievement. Because
education relies heavily upon the disciplines of Educational
Psychology and Child Development for explanations of learning,
academic achievement along with failure and success are most
often attributed to individual effort and ability. Unfortunately,
the relationship between sociological concerns and psychological
explanations in educational settings has not been sufficiently
examined, and linking the cognitive processes of learning in
classrooms to the larger sociological contexts of schools and
society remains amongst the least researched areas in education.
There is however a growing body of research and theory (still on
the margins of mainstream education) that explores the social
contexts of learning and cognition and the relationship between
mind and society. This has been a multi-disciplinary effort that
brings together psychology, anthropology, and philosophy, among
others, and although not directly concerned with issues of
inequality and social justice, it provides a framework and a
possibility for addressing them.^{3}

Within mainstream education, the social and class dimensions of
education are seen as the preserve of sociology and have had
little impact on the study of learning, cognition or pedagogy.
Most departments of education and teacher training institutions
have failed to engage seriously with important sociological
debates in trying to understand issues of learning and academic
achievement.

Mathematics is an important
school discipline for the study of cognition and as a 'powerful
knowledge' system, it remains a compulsory and core component of
every school syllabus. Why and what kind of Mathematics children
need to learn is not a resolved issue, and although its status
and importance has not been adequately challenged, there is some
amount of questioning and discussion around it.^{4} Notwithstanding these
controversies, however, school Mathematics has acquired a status
that has important repercussions for the academic and
subsequently the professional trajectories of students. Its role
as a mechanism of social selection is well recognized (Jorgensen
et al., 2014) as also of the fact that Mathematics '…plays a key
role in the distribution of life chances. For example, there is
widespread concern with how Mathematics acts as a "critical
filter” in depriving minority and women student's equal
opportunities in employment. Philosophical and ethical
considerations like these thus have important implications for
Mathematics and particularly for educational theory and practice'
(Ernest, 1994: 2).

In India, large numbers of
children in middle and high school are unable to cope with the
Mathematics curriculum and failure in this subject has been a
major cause of detention and subsequently for children dropping
out of school (NCERT, 2006). Within the extremely unequal and
hierarchical system of school education, Mathematics plays an
important part in determining chances of retention. It also has
the power to deny options of entry into several high-prestige
courses in higher education and failure can and does jeopardise
life chances for large numbers of children. The patterns are
predictable the more marginalised the
population, the less their chances of progress through the
educational system.

I propose in this paper to draw attention to what are usually
considered purely psychological aspects of learning and knowledge
acquisition in the Mathematics classroom and to the ways in which
this learning and transaction between teacher, pupil and text is
mediated by the socio-economic realities of children's lives as
also by the material and social contexts of their learning. I
subscribe to the argument that schools are meant to provide
access to knowledge that is not easily
available in the home or the community, and that Mathematics
classifies as what has been described as 'powerful knowledge'
that requires schools and specialised teachers to initiate and
support such learning.^{5} Although school
Mathematics is a challenge for all children, its implications for
the socio-economically disadvantaged child are critical because
the failure of learning is most often attributed to the child,
the family or the teacher whereas its relationship to structures
of poverty and privilege is overlooked. Given its importance in
determining school success, I suggest that recognising the
complexity of the content of Mathematics and its cognitive
processing along with the contexts within which learning takes
place, is vital for understanding the added disadvantage it
creates for the already disadvantaged.

I begin by describing the distinctive nature of Mathematics, the ways in which it differs from other school subjects and the specific features of its subject matter that make it so difficult for school children everywhere. I make a brief reference to the current trends in the psychology of learning and cognition which are able to explain individual differences but have not been successful with explaining group differences. I present some evidence to capture the extremely inadequate condition of the schools that are available to the poor in India and the constraints under which learning is meant to happen. This is done to emphasize how much more difficult the task of learning becomes when it is complicated with the conditions of learning and how pedagogies become ineffective when we fail to recognise the confluence of factors which goes much beyond the student, the teacher and the classroom. It attempts to explain why the difficulties inherent in the subject matter of Mathematics are so much more daunting for children in contexts of poverty and how it creates additional disadvantages for them in classrooms, in schools and in the struggle to advance within the educational system as a whole. I leave open the question of whether this can be overcome by additional resources and more intensive pedagogies, or whether the very notion of school Mathematics and its gate-keeping powers need to be challenged – therefore whether the question is a pedagogic one or a political one.

## The Specific Character of Mathematical Knowledge

The special problems that school
Mathematics poses for children both at the elementary level and
especially at the secondary level, is widely documented. The
belief that Mathematics has unique characteristics and that its
learning may require different strategies and pedagogies is
gaining more acceptance even while it remains contested. This
framework allows us to understand why mastery of a knowledge
system of this nature is particularly difficult for the average
school child. By ignoring this complexity and its implications in
the classroom, we mask the intricate ways in which disadvantage
is multiplied for children who have no resources other than the
school to draw upon.

In making the above argument, I primarily rely on the work of
Raymond Duval (1998, 2000, 2006) to elaborate this argument with
pertinent examples. Duval's work focuses on unraveling the unique
nature of mathematical knowledge and the corresponding cognitive
processes involved in its learning and its implications for
teaching and pedagogy. The choice, although somewhat arbitrary
(given that there is a growing body of research and work within
this framework), is made because of its convincing argument for
the unique nature of mathematical learning amongst the subjects
that children learn in school. To support the argument that
learning Mathematics is different and that ignoring this
difference masks the very demanding process that its learning
entails, I also briefly refer to the work of two other important
figures - Paul Ernest, a philosopher and Paul Dowling, a
sociologist. Both have engaged extensively with school
Mathematics and Mathematics education to support the link between
the complexity of Mathematics and its social implications and by
extension, especially in the case of Dowling, its capacity to
exclude. Ernest (1998, 2003) highlights the fund of unspecified
and often ignored 'tacit' knowledge required for learning
Mathematics, while Dowling (1998) describes what he terms as the
'high discursive saturation' of the subject matter of
Mathematics, its status as a discipline, and the ways in which it
creates myths that mask the exclusions that it facilitates. All
this has serious implications for Mathematics learning for the
poor, who have few resources to meet its demands and little power
to challenge its status.^{6}

Difficulties in mathematical
comprehension are commonly ascribed to the epistemological
complexity of the nature of mathematical concepts but this is
equally true of every other domain of knowledge, especially
formal knowledge systems that are expected to be mastered in the
course of schooling, whether it is the sciences or the social
sciences. Duval (2006) ascribes the difference not to the
complexity of the concepts alone but to the cognitive activity
that is required for mathematical thinking. He identifies the
three following characteristics that are unique to
Mathematics:

I*. The paramount importance of semiotic representations*:
Although all knowledge systems are dependent on semiotic
representations, mathematical thought needs semiotic
representation as an essential condition. The role of signs in
Mathematics is not to stand for some objects, as is the case in
other domains, but to *provide the capacity of substituting
some signs for others.* In Mathematics, unlike most knowledge
domains, there is NO object that can be directly perceived or
observed with instruments. Therefore access to mathematical
objects is always and *only* through signs or semiotic
representations unlike other systems where the access can be both
non-semiotic and secondarily semiotic. In Mathematics one can go
from one representation to another - the number 'ten' can be
represented by strokes or the word 'ten' or the numeral '10'; the
mathematical object that is the number itself as a concept is not
accessible in any other form and cannot be directly perceived.
Thus, for example, the phenomena of astronomy, physics,
chemistry, etc. are largely accessible by perception or by
instruments (microscopes, telescopes, measurement apparatus,
etc.) and in Biology, the semiotic sign or word 'plant' allows us
to go from the semiotic representation to the object itself, but
a sign in Mathematics never allows this shift to the actual
object, *it merely provides the capacity of substituting some
signs for others.*

II. The Cognitive paradox of
access to knowledge objects: This is a specific epistemological
situation that radically changes the cognitive use of signs. The
crucial question for learners at every stage of the curriculum
then becomes: how can they distinguish the represented object
from the semiotic representation used if they cannot get access
to the mathematical object apart from the semiotic
representations? The critical threshold for Duval, for progress
in learning and problem solving in Mathematics resides in the
ability to change from one representation system to another. The
ability to recognise that mathematical objects are only
accessible through one or another form of representation and to
identify them through multiple representations is essential to
mathematical thinking. To move from one representational medium
to another with ease is the crux of mathematical thinking and
should, according to Duval, form the essence of teaching
Mathematics in schools.

III. The large variety of semiotic representations used in
Mathematics: Mathematics also requires different semiotic
representation systems that can be used according to the task and
the problem to be solved or the question that is posed. Some
processes are easier in one system than another and some can be
carried out in one system only. Counting, for example, can be
carried out using fingers, or strokes on a paper or, as we do in
schools, using the decimal system of numbers and number names. It
is fairly obvious that counting of large quantities and
especially working with them, is done much more efficiently with
the decimal system than the use of a stroke system can
allow.

Due to the unique constraints on access to the objects of its
study, Mathematics, not surprisingly, is also the domain with the
largest range of semiotic representation systems. It uses both
systems that are common to other kinds of thinking such as
natural language, and those specific to Mathematics, such as
algebraic and formal notation. 'This emphasizes the crucial
problem of Mathematics comprehension for learners – if for any
mathematical object we can use quite different kinds of semiotic
representation how can learners recognize the same object through
semiotic representations that are produced within different
representation systems'? (Duval, 2006: 108)

Thus we use natural language to
refer to the 'three corners of a triangle' but we also represent
these as 'x, y, z', and so on. The problem then, according to
Duval, arises as much if not more, from these specific ways of
thinking as it does from the epistemological difficulties
peculiar to the introduction of new concepts. It is essential for
students to recognise the represented object through a variety of
representational systems that may be very different from each
other. For example, the shaded half of a figure (to represent the
proportion) in a visual form is a very different kind of
representation to the notation '1/2' and the cognitive distance
between the two and the effort needed to cover it is barely
understood by teachers and curriculum planners.

The case of geometry is a good case in point since it always uses
at least two systems of representation. A geometrical figure
needs to associate both discursive and visual elements, even
though at any point of time only one of these is highlighted. In
working with figures like triangles, rectangles and circles,
although the visual representation in the form of drawings of the
figures is very prominent, especially to the novice and the
child, the mathematical properties are represented in statements
like 'let ABC be a right angled triangle…'. In a classroom,
usually one kind of representation is highlighted and the other
left in the background but children are expected to go from one
to the other with ease. In actual fact, it is extremely difficult
because the common association between shapes and words and the
perceptual obviousness goes against its mathematical
implications, and intuitive interpretations are often not
mathematically appropriate. The essence of mathematical thinking
requires comprehension of the ways in which the representation
systems function. The difficulty for cognitive processing is not
only the representations but also in their transformation. Unlike
other areas of scientific knowledge, transformations of signs and
semiotic representation are at the heart of mathematical
activity.

According to Duval,
transformations are at the core of mathematical activity and
these are of two types: treatments and conversions. Treatments
are transformations that remain within the same register,
carrying out a calculation, for example, 2+2=4. The treatment of
adding two and two creates its transformation to the number four.
Conversions are the more problematic of the two and require a
change of register without changing the objects being denoted:
for example, passing from a natural language statement to a
notation using letters. Thus 'two and two is four' uses natural
language whereas 2+2=4 uses notation. A child who has not yet
learnt to read and write numerals, can count two and two to
arrive at four, but cannot be expected to understand the same
transformation when confronted with the notational form of the
problem.

Multifunctional systems pose enormous challenges for beginner
students of Mathematics and for Mathematics educators that need
to be identified before they can be addressed.

They give the illusion of being more easily accessible whereas
this is not the case and leads to a deceptive misreading of a
problem. Thus the availability of visual representation for
geometric figures and the use of natural language in explaining
appear to be an enabling process in early Math learning and has
been used as such. Although it is a good entry point for engaging
students, it needs extremely deft handling to go beyond one form
of representation to another and eventually to think beyond the
representations. The problem lies in the mistaken belief that a
geometrical figure is the mathematical object itself.

The drawing of a circle on the
pages of a book or a note-book, for example, is never the object
itself; it is always and only a semiotic representation of a
mathematical concept defined as such. But almost always students
in primary and elementary Mathematics classes, and very often
teachers, make this serious mistake that eventually inhibits
mathematical processing and thinking because it also disallows
the shift from one semiotic system to another in representing the
essential mathematical object: a circle. This flexibility of
thinking and transformation in carrying out calculations based on
the understanding of the properties of a circle is lost to some
extent when the semiotic representation (the visual drawing of a
circle) is mistaken for the mathematical object itself.
Textbooks, in an effort to make these concepts meaningful, make
precisely this mistake and therefore impede mathematical
thinking. When children 'see' the visual image of a circle as an
object and not as a representation for something else,
progressing to solving mathematical problems and thinking of
circularity and its properties mathematically is likely to pose
serious problems.

An example that illustrates the complexity of this phenomenon is
cited by Duval in a study carried out with children in middle and
primary schools in France (Duval, 2006: 118). A geometrical
problem was presented to children entering middle school which,
according to their curriculum, they are expected to solve at this
level. The problem was presented to thousands of school children
but even at the middle school level very few were able to solve
it correctly (and yet French children's scores are among the
highest in comparative studies of mathematical achievement in
international comparisons). The children's performance
illustrates how 'seeing' in geometry can actually take away from
the mathematical discourse required and how this dissociation can
actually hinder mathematical comprehension rather than facilitate
it.

Presented with a diagram (Figure 1), children were asked the following question: In the figure sketched here representing a rectangle ABCD and the circle with centre A, find the length of segment EB.

Figure
IGeometrical problem presented to
children

Children gave a number of answers but only about 920 per cent (of roughly 5,000 children) were able to give the correct answer. Figure II represents two ways of identifying the two sub-figures within the original figure representation A and representation B (Duval, 2006: pp. 117, 118).

Figure
II Two sub-figures that represent the
two different ways of viewing the problem

To find the mathematical answer
for this problem, however, students needed to look at the two
sub-figures in B and not the two subfigures in A. The sides of
the rectangle here for mathematical purposes need to be seen as
the radius of the circle and NOT as sides of a rectangle. Along
with knowledge of the properties of circles and rectangles, the
child is required to concentrate on the one-dimensional aspect of
the rectangle and its sides and although this sounds simple, it
needs considerable and conscious effort.

Our school experience and exposure to math texts intuitively
encourages the first kind of identification – a circle and a
square. Circles and squares in our classrooms are usually
presented as three dimensional and often in the shape of objects.
Although this is a good entry into understanding shapes, the
representational character of the objects needs to be abstracted
and conceptual clarity established. The multiplicity of
transformations that is needed for what looks like a fairly
simple problem is the essence of mathematical thinking. Teachers
and schools need to facilitate this if children are to become
mathematically competent and to think mathematically. Acquiring
knowledge of the systems of representations is an important first
step but it is equally important to learn to manipulate and to
use them effectively. Research in Indian classrooms provides
evidence of the fact that children in middle school, although
able to solve routine textbook problems in geometry, have fuzzy
understanding of the transformations and very little clarity
about the solutions (Kaur, 2010). These special features of
Mathematics set it apart from other school disciplines and
therefore call for serious rethinking of how the already
ambitious and accelerated school Mathematics curriculum is to be
negotiated.

Mastery of the semiotic
representational system becomes an important tool for carrying
out operations and knowing how to work with the tools that
provide solutions to a mathematical problem. The tools for
carrying out transformations are a critical component of
mathematical thinking that come from a cultural corpus of a long
intellectual history and cannot be instantly mastered or
spontaneously 'constructed'. The variety of representation
systems and the specific capacity for each to perform certain
mathematical processes poses very specific problems and needs
specific support, sustained exposure and specialised
training.

The work of Paul Ernest (1998, 2003) is useful for understanding
this analysis because it elaborates the relations between
personal and public knowledge and places Mathematics classrooms
within an epistemological perspective. He states that
Mathematics, like other disciplines, has its own very specific
disciplinary discourse that must be mastered for initiation into
the 'community' of professional mathematicians and competent
Mathematics students. Knowledge in Mathematics is never wholly
explicit; although it is often characterised as a collection of
validated propositions, which in the case of mathematical
knowledge means mainly a set of theorems with proofs. 'Most
personal knowledge,' he states, 'is not of the explicit
propositional sort, but consists of tacit knowledge of methods,
approaches and procedure, which can be applied in new situations
and to problems' (Ernest, 1998: 249). The tacit nature of this
knowledge becomes more problematic because access to the
knowledge objects is by its very nature, never direct.

He claims that standards for
proof and definition in Mathematics can never be made fully
explicit; rather, this is done in the form of exemplary problems,
solutions and proofs. Often these will be exemplified in texts
rather than in explicit statements. Mathematics therefore
includes an important body of unspecified knowledge which is
acquired largely through immersion in a mathematical community.
'Not only does it mean that mathematical knowledge extends beyond
the explicit to include a tacit dimension but that beyond the
abstract and general knowledge of the results, methods, and
language of Mathematics there is an important concrete knowledge
of Mathematics but which is also very specific and specialised.
This includes knowledge of instances and exemplars of problems,
situations, calculations, arguments and proofs, applications and
so on. Knowledge of particularities as well as generalities thus
plays an important part in Mathematics' (Ibid.: 250) This may
also explain why total immersion in seemingly mindless activities
of solving hundreds of given Mathematics problems (a common
practice for all serious students within the Indian education
system, especially those preparing for entrance exams for
institutes of technology) facilitates this mastery in some cases.
A long period of apprenticeship and deep immersion in the
language and practice of mathematical processes is an essential
requirement for learning to think mathematically and novices need
to be carefully guided and supported at every level. For children
of professional and educated parents and in most middle class
homes, academic support comes in a variety of ways – help with
homework, additional books, games, puzzles and now of course
computers, and academic support programmes that give children the
immersion into this language that is not possible within the
classroom. For those without access to all these supplementary
systems, it is an uphill struggle.

Paul Dowling (1998) makes the point that the discipline of
Mathematics is a highly specific and well-guarded field of
knowledge where entry is limited and often puzzlingly difficult.
He uses Bernstein's notion of 'restricted coding orientation',
the propensity to generate meanings that are highly context
dependent and 'elaborated coding orientation' that refers to an
inclination to context independence. This clearly resonates with
the situational/abstract thinking divide and both dichotomies,
the concrete/abstract or the local/general, have a modality of
the discursive in that they depend on language. The difference
however is that situational thinking is associated with low
discursive saturation and abstract thinking and elaborated coding
with high discursive saturation corresponding to context
dependency and context independence. Mathematics is clearly a
case of high discursive saturation, an activity which is highly
organised at the level of discourse and therefore produces
generalized utterances. Domestic and manual activities on the
other hand are examples of low discursive saturation and
therefore produce localised utterances.

He goes on to state that
practices exhibiting high discursive saturation are at highly
complex levels of discourse and exhibit comparatively complete
articulation. They are also highly organised and this makes it
possible for them to produce generalisations. Utterances of the
low discursive saturation type are characterised by implicit
regulating principles, are context specific and must be
interpreted within the context of a particular activity (Dowling,
2003). He describes Mathematics as an intellectual activity that
is both exclusive and elusive, and which maintains an elite
character in the confines of that very elite institution – the
University. He refers to Mathematics as 'a mythologizing activity
to a degree that is perhaps unparalleled on the school
curriculum' (Dowling, 1998: 2) and condemns the ascription of
what is essentially non-mathematical activity to be designated as
mathematical, which is exceedingly misleading for students and
only serves to create illusions of learning. Drawing a visual
pattern, or weaving a basket, he maintains, may contain elements
of calculation and mathematical thinking but they do NOT qualify
as Mathematics, nor will they in and of themselves initiate
children into the formal disciplinary knowledge of Mathematics.
Naming these everyday activities as Mathematics only takes away
from its abstract and complex nature and unnecessarily lulls
children into a false sense of achievement.

The use of mathematical assessment to evaluate children's
academic status while ignoring the cognitive complexity of
mathematical processing, or the severely limited nature of the
institutions within which it is transacted in conjunction with
the material conditions of children's out-of-school experiences,
only serves to hide the unequal nature of schooling and its
consequences.

## Mathematics Education and the Psychology of Learning

Notwithstanding the complex
nature of Mathematics learning and the problems that Mathematics
educators struggle with worldwide, difficulties of learning are
seldom analysed within the broader perspective of socio-economic
contexts and solutions are sought within a narrow cognitive
understanding. Presented below is a brief account of the general
trends within Psychology that have influenced Mathematics
education in India to appreciate its limitations. Although a more
serious discussion of these trends would be useful, it is beyond
the scope of this paper.

Psychologists by and large concern themselves with the child, the
teacher and the classroom to explain differential levels of
learning, and academic achievement is attributed to individual
differences. That these variations also tend consistently and
systematically to vary along lines of social and economic
disparity has been largely ignored. Issues of inequality,
discrimination and exclusion and their sociological aspects,
although recognised, are seldom taken into consideration in
explaining learning and knowledge acquisition.

Education has relied on a long tradition of behaviorist
explanations of learning that continues to hold sway in
classrooms and pedagogy despite the progressive turn to a more
child-centred approach. Piaget's genetic epistemology and notions
of the child's 'construction' of reality (Piaget, 1958; 1965)
caused an important shift in the psychology of learning and ideas
about children's thinking that seriously influenced the field of
education. As the theory became more widely known, the
educational imagination was most captured by the idea of the
child as 'active' learner and the teacher as 'facilitator'.
Piaget's primary focus of exploration was the genesis of
thinking, logical reasoning and human intelligence. The mind of
the child was the medium through which this trajectory of
'genetic epistemology' and the shift from childhood to adulthood
could be best observed. That the child is also deeply embedded in
a social, economic and political world as well as other aspects
of development was not critical to his immediate intellectual
project.

Largely as a result of this theory but also of other progressive
strands in education, efforts at providing children with
experience, practice, and freedom to explore and to construct
knowledge have gained currency in educational practice. A deeper
and nuanced understanding of how this facilitates knowledge
acquisition has not been so easily understood. In large numbers
of classrooms the world over, this shift has not been able to
yield results in terms of improving the learning of vast bodies
of knowledge that children are expected to master when they
attend school. These problems persist despite efforts to initiate
curricular reform and child-friendly pedagogies based on this
theoretical framework, as was attempted with the NCF 2005 in
India.^{7}

Within Psychology and the
cognitive sciences, mathematical thinking, as a close to perfect
example of rational thinking, has been a popular area of research
and provides rich data for testing and verifying assumptions
about human cognition. Because of its rational and abstract
nature, it is also seen as least influenced by social or cultural
factors and its learning is sought to be explained in purely
cognitive terms. All children in school are expected to learn
Mathematics and yet the difficulties they experience in doing so
remains a challenge for teachers and researchers and is a major
reason of children's alienation from learning and classroom
activity. It also, more than most subjects, marks children as
academically competent or not. Rapid advances in technology have
enhanced the prestige and status of the discipline to the extent
that nations worldwide are becoming aware of the considerable
challenge they face in the quest to improve Mathematics teaching
for their students (English, 2002)

The evidence of a strong relationship between Mathematics
achievement and socio-economic factors has inspired a corpus of
research which, according to Valero and Meaney, (2014: 977), has
a 'taken for granted' character that is accepted uncritically. It
is only in recent years that scholars in the field of Mathematics
education are beginning to unravel the complex relationship
between Mathematics classrooms, the cognition of Mathematics
learning, socio-economic factors and poverty (Zevenbergen, 2000;
Lerman, 2000).^{8} At the global level,
there is support for the hypothesis that socio-economic factors
have a significant impact on Math achievement whether it is at
the level of students, the schools or the country (Jurdak,
2014).

Research on Mathematics education
has expanded vastly and produced a rich body of work but the
persistent disadvantage in academic achievement and its
relationship to social and economic disadvantage is yet to be
adequately explained. This is an issue of concern amongst a small
but growing community that is veering Mathematics education to
focus more closely on unraveling 'more nuanced approaches for
understanding the social, political and historical constitutions
of these relationships' (Valero and Meaney, 2014:
977).^{9} This paper, as part of
this effort, is a call to look more closely at the relationship
between the cognitive processes of mathematical learning and the
constraints that limited resources impose.. Although the
relationship plays out more closely in schools and classrooms,
understanding it in a broader context brings the burden that
poverty imposes on children out into the open, and places the
responsibility for resolving it on the educational structure, the
state, educational policy and educational spending.

## State of Schools and the Contexts of Teaching and Learning:

Given that theories of learning
and cognition have not been able to resolve the persistent
problems of under achievement observed amongst the poor and the
marginalised, I suggest that exploring and understanding the ways
in which Mathematics learning is mediated by its subject matter
and the material conditions in which it takes place, are likely
to provide some answers. Introducing the material conditions of
schools and classrooms forces us to confront child poverty and
the deprivations that impact the lives of the poor. Leaving those
out of the equation of learning and academic achievement only
masks the inequalities of provision that are so deeply implicated
in the academic outputs, and on which the privileges of
successful schooling are contingent.

Concerns around school education in India became prominent in the
1980s when efforts for universalisation of education started to
gain momentum. Increased access and the Right to Education Act
(RTE), 2009 have forced issues of quality to be taken more
seriously, especially in the government school sector that is now
more or less the preserve of the poor and the deprived. The
inadequacies of the system—whether it is in terms of
availability, infrastructure, teacher education, teacher
indifference, pedagogy or curriculum and textbooks—have been
widely reported and discussed although a closer engagement with
classroom practice has traditionally remained within the domain
of Psychology. It is in the context of this systemic failure that
the 'failure' of the underprivileged child needs to be better
analysed and understood.

The implications of what and how children learn is of central
concern to the project of schooling and a long tradition of
research and theory has contributed enormously to its
understanding. Although this concern is critical for every child
in school, its importance is paramount for children who enter
schools with great expectations and aspirations of changing their
lives. The purpose of this section is to highlight the importance
of the contexts within which disciplinary knowledge is acquired
and to acknowledge the extremely impoverished environments that
are available to the poor and the marginalised. It is not my
intention to dismiss government schools or the many efforts of
teachers, students, local officials and communities that struggle
to educate more and more children in increasingly difficult
circumstances. On the contrary, given that the system still
provides education to more than half the child population in this
country, its importance and impact cannot be overlooked. Its
problems and shortcomings need highlighting precisely because it
has serious implications for the education of millions of
children mostly powerless and
disenfranchised. Unravelling the complex ways in which this
provision of schooling mediates the subject matter of Mathematics
in conditions that fail to support serious learning, allows at
least a glimpse of the burden that the poor child carries in
trying to compete and to succeed at what she is falsely led to
believe is possible within the promise of schooling.

The hierarchical and segmented
nature of the Indian school system is well documented. A
two-tiered system of schooling that divides a variety of private
schools catering to the elites and Government schools for the
middle and lower classes is slowly being eroded not only by a
slew of private schools of varying grades but also by a grading
of schools within the Government school system
itself.^{10} There is a steady
curtailing of investment in the schooling of the poor and
indirect subsidising of private schooling of the middle classes
that deprives the poor of schooling and contributes to the
reproduction of social inequalities (Abraham, 2006) and
Government schools are now left with poor girls and very poor
children (Manjrekar, 2003). 'The school(s) attended by the
children of the rich in many developing countries provide
computerised learning environments not dissimilar to those found
in best British schools. But these are not typical. More typical
are those which are struggling to provide a minimum quality of
learning against a background of diminishing resources, a rural
economy and often still a pre-literate population' (Little, 1998,
as quoted in Majumdar and Mooij, 2012).

Much of this is critically determined by public funding and the
lowest rung of Government schools, now subscribed to only by the
poorest children in the country, remain poorly provided. School
education in India has been described as the 'weakest brick in
the pillar, with unequal access, poor infrastructure, high pupil
teacher ratio, teacher absenteeism and high drop-out rate'
(Bedamatta, 2014). The failure of the Indian state to provide
universal access to quality schooling and to ensure access to all
socio-economic groups has been viewed as '... surely the more
dismal and significant failures of the development project in the
country' (Ghosh, 2011). Public spending on education in India,
despite the rhetoric surrounding it, remains below 5 per cent of
GDP, a lower ratio than many developing countries.

Access to school education in
India has increased dramatically in the last few decades and the
Net Enrolment Rate for children of the 614 years age group increased to 99.89 per cent in
201011 from 84.53 per cent in
200506. However, drop-out rates remain a
concern and retention levels for the poor are low. Educational
disadvantage includes children who have never enrolled in school
but also those who enroll but never complete even the primary
cycle as well as those who complete a few or even several years
of schooling but are unable to benefit from the schooling
process. Poverty is an overarching factor that disadvantages
children in schools and a sharply etched divide in schooling is a
stark reflection of wealth and privilege along a number of axes
such as gender, caste, region and religious community among
others (Bandyopadhyay and Govinda, 2013).

Despite several large-scale governmental efforts, there is
increasing evidence of low standards of academic achievement and
learning all over the country. Performance levels for basic
academic abilities that schools are intended to impart – reading,
writing and arithmetic are abysmal.
According to the ASER data, 2011, '… while the student enrolment
in rural India has seen a rise (96.7 per cent) in the year 2011,
a mere 53 per cent of grade 5 children in
rural India could read a grade 2 level text and 36 per cent could
solve a three digit by one digit division problem' (ASER, 2011).
In 2014, the findings show that half the children in Standard V
are barely at proficiency levels of Standard II; of all the
children enrolled in Standard V, about half cannot read even at
the Standard. II level and about half the children have not
learned mathematical skills that should have been acquired in
Standard II. Not surprisingly, the worst indicators pertain to
children from the most disadvantaged groups both socially and
economically.

The following description of a
Government Junior High School in Manipur in 2014 is not an
exceptional representation of the state of lower-end government
schools across the country:

While a small portion of the school building is made of bricks,
the rest is made of wood and mud and parts of it are broken. The
roof is made of tin, constructed years ago, and the leaks in the
roof result in the destruction of walls, furniture and the
limited teaching learning aids in the classroom. Classrooms are
dark, dingy, dusty and littered with wastepaper. Ventilation is
extremely poor with insufficient windows that are partly broken
and require urgent repairs. The veranda gathers moss and is
cracked.

"Classroom walls are cracked and have cobwebs. There are
sufficient classrooms but are partitioned by torn curtains. The
available benches, tables and chairs were in good condition in
some classrooms but damaged and insufficient in others. The table
and chairs meant for the teachers were in disrepair….

… Sadly, the blackboards were no longer usable as there were
white patches and what was written on them could hardly be seen.
The school lacks even modest aids such as globes, charts,
Mathematics and science kits, a library, children's book bank,
game materials, and even school textbooks and stationary were
inadequate. As is common in schools all over India, there are not
sufficient toilets and in the absence of safe drinking water, a
few children from the school are sent to carry about forty litres
of water to a distance of about a kilometer in the village”
(Salam, 2014).

Variations on the above are an accepted feature of schools in
urban slums, in remote areas, and all manner of schools catering
to children from marginalised economic and social backgrounds.
Thus, for example, in my own observations, a government primary
school for girls situated in a fairly well-to-do middle-class
area in the capital and attended by working-class children of
that area, shared several of the features mentioned above (lack
of furniture, broken windows, lack of water) and also had its
electric supply cut off for almost a week in the peak of summer
with temperatures soaring above 35 degrees centigrade, due to a
complicated system of payments that the government has put in
place. This, I was informed, is routine practice and, in addition
to the extremely uncomfortable physical conditions it creates,
imposes disruptions in teaching and other school activities.
Children are moved to the lower floors (for cooler rooms),
several classes are crammed into one room, with little place for
children to sit, to spread their books or to have a quiet or
comfortable working environment.

The increased access to
elementary education and the mass entry of the poor and
marginalised into the government school system has also raised
questions about teacher-pupil ratios, adequate teacher training
and teacher attitudes towards teaching and students. As the most
'expensive' part of the education system, the role and status of
teachers remains fluid and limited funding often strikes at the
expenditure associated with them. Several alternatives have been
put in place to cut costs and large scale recruitment of
para-teachers (under different names) around the country has
brought in and legitimised a cadre that are expected to have
minimum academic qualifications and no pre-service teacher
education (Sarangapani, 2011). Although this is a fraught issue
in the political economy of education and too complex to be
discussed here, it nevertheless undermines and trivialises the
practice of teaching and the professional qualities and
commitment required for school teaching. Expansion in the
recruitment of teachers lags far behind the expanding enrolments
of children and the percentage of single teacher schools
continues to be high in several states. Research also shows that
the percentage of single teacher schools is higher in blocks
populated by the most marginalised communities (Rana, 2006)

Given that first generation learners also require most support at
the primary and elementary school levels, it is interesting to
note that this sector of teacher education has been the most
neglected in India. Centralised and rigidly structured curricula,
close monitoring and increased assessments, rather than
investment in teacher education, especially in the neo-liberal
era, are seen as the panacea for improving the quality of
schooling. Focusing on 'competencies' and 'subskills' instead of
knowledge and reflection reinforces the mechanical role of the
teacher. In addition to the poor physical infrastructure of the
schools, teaching competence of the quality required, is not seen
as a priority (Sarangapani, 2011; Batra, 2013). This has serious
implications for Mathematics learning where clarity and
understanding of basic concepts is critical for more advanced
learning.

Teacher education in India has
been criticised for its superficial and largely outdated nature,
as well as its duration, curriculum and structure. Teacher
education programmes have evoked serious concern whether it is at
the level of disciplinary rigour, the inability to engage with
the complexities of the social composition of the Indian
classroom, or to engage with subject understanding of teachers
and students, among other issues. Pupil-teacher ratios (due to a
variety of factors) are a major problem and teacher attitudes
towards the poor and the under-privileged are highly problematic.
The higher class and caste backgrounds of the teachers influences
their perceptions and often an ignorance of the life conditions
of the students reinforces stereotypes about the lack of support
that children receive within their homes, apathy of parents, and
so on (Talib, 1992: 87). School teachers, although much maligned,
are also trapped within structures where they find themselves
marginalised, with little room for autonomy whether it is in
classrooms, schools or the larger educational system (Batra,
2006).^{11} Teachers function as
petty bureaucrats and in the already understaffed government
schools, their services are used for a variety of administrative
and other tasks so that little time is available for teaching and
academic activities. Teachers have low social standing, they are
not considered an intellectual cadre and there is little faith in
their capacity to think, to take academic decisions or to engage
with matters of policy (Kumar, 2008).

The diversity of children in the government school system, the
expectations of teaching to a curriculum, within a time frame,
the standardisation of systems of testing and assessment all
contribute to classrooms that have little scope for creativity,
for exploring multiple solutions to problems, working with
children individually or for meaningful interactions of a kind
needed, especially for children for whom the classroom materials
are unfamiliar. The problems of an 'overambitious curriculum'
(Pritchett and Beatty, 2012) and the 'Marks Race' (Majumdar and
Mooij, 2012), makes additional demands on learning, leaving less
room for meaningful teaching or thoughtful and calibrated
pedagogies that are consistent with the difficulty levels of the
content of learning. Majumdar and Mooij (2012) describe the
achievement model used in schools that 'interprets knowledge,
comprehension and conceptual understanding in terms of test
scores' (p. 214) and comment upon 'the low quality of education
that children receive in many rural primary schools in India, on
the heavy reliance on memorization and cramming in these schools,
on excessive examination orientation and the impact this has on
guiding the entire process of teaching and learning, curricula,
teacher involvement, parental aspirations and student activity or
passivity' (p. 216).

Schools are more focused on
completing the curriculum rather than delivering learning, the
onus of which falls on the parents and the home. According to
Majumdar and Mooij, 'The privileged have a shortcut to material
success due to a lot of cultural capital at their disposal and
middle class children squeeze through the grinding system because
of massive home support' (p.233) but 'Many from the
underprivileged sections of society, unaided by all kinds of
middle class home support, are unable to put up with the burden
of non-comprehension and hence eventually drop out of the system.
… they therefore suffer more due to the undue standardisation.
Described as 'failures', they are pushed out of the wasteful
schooling system, without gaining anything much and possibly
losing their traditional skills' (p. 234).

Given these constraints, classroom teaching is rendered
uninspiring and monotonous. Teaching to a syllabus and a
standardised examination system leaves little room for innovation
or creativity and promotes conformity and discipline. Classroom
processes consist of mechanical routine exercises such as
teachers reading out from textbooks, giving dictation to
students, children being asked to copy passages or math problems
from books, chorus repetitions and recitations, etc., the most
common activity in the classroom is rote oriented learning and
question answer sessions where answers are well defined with
little room for manoeuvre' (Majumdar and Mooij, 2012).
Mathematics classrooms have been described as 'defined by tyranny
of procedure and memorization of formulas' where, 'given the
criticality of examination performance in school life, concept
learning is replaced by procedural memory'. Partial knowledge is
not seen to be acceptable and there is little room for play in
answering questions (NCERT, 2006).

Mathematics, more than most
disciplines, relies heavily on the teachers' understanding of
Mathematics, clarity of concepts and pedagogic techniques. The
lack of interaction and communication between teachers at
different levels of schooling, and also with Mathematics
researchers and university teachers, leaves elementary teachers
ill-prepared for facing the problems that challenge them in
Mathematics classrooms. Moreover, textbook-centred pedagogy
limits initiative and curbs experimentation. Mathematics teachers
make extensive use of the chalk and talk method where problems
are presented, children are asked to work them out and the
solutions are worked out on the blackboard. Teaching aids are
rarely available or used and mechanical drill and practice
methods fill the limited time the classroom provides (Rampal and
Mahajan, 2003; Khan, 2004). The lack of conceptual knowledge of
Mathematics amongst students and often even the teachers has been
documented in schools throughout India. These methods of teaching
are least conducive to student populations who are less than
regular in attending school and have little support outside
schools to fill in the gaps.

The argument for the especially complex nature of school
Mathematics learning and its consequences is meant to demonstrate
how schooling is set up to 'fail' certain children and to
challenge schools and society on how the playing field can be
made level. Although confined to the classroom, this problems of
'failure' and 'success' needs to be addressed not only within the
classroom and the school, but at a variety of levels—within the
structures of State, education and society—to answer how it is
allowed to determine access to knowledge, to higher education, to
opportunity and ultimately to life chances. This calls for an
engagement from Sociology and the social sciences, but as
Nambissan remarks, 'the Indian school and classroom is the most
under-researched area in the Sociology of Education in India'
(2013: 83). She also notes that '... that the neglect of the
study of schools has led to a glossing over of complex processes
that mediate school experiences and influence learning of
children. This is particularly important in relation to the
education of children belonging to the most marginalized groups
in Indian society' (Ibid.)

The relationship between school
funding, child poverty and achievement in Mathematics was
meticulously analysed and perceptively discussed by Payne and
Biddle in 1999 when they reviewed large data sets and earlier
research in the area. Although their study focused on the USA,
their findings and observations are equally relevant for India
and other countries where funding of school education is
differentiated and social support systems are fragile. The
authors describe how, 'The homes of poor children provide little
access to the books, writing materials, computers, and other
supports for education that are normally present in middle class
or affluent homes in America. Poor students are also distracted
by chronic pain and disease; tend to live in communities that are
afflicted by physical decay, serious crime gangs, and drug
problems, and must face problems in their personal lives …. What
this means is that poor children have a much harder time in
school than either affluent or middle class children' and confirm
that 'poor children are also likely to attend badly funded
schools, so the raw statistics indication that their achievements
are low may also reflect the inadequacies of those schools'
(Ibid: 7). The authors conclude that 'of all aspects of home (dis)advantage that one
might study for impact on achievement, child poverty is surely a
leading candidate' (Payne and Biddle, 1999: 7)

Highlighting the highly strained conditions of schools and formal
education that the state provides and the poor avail of, should
help us become aware of inequalities and unequal systems that
allow children of the privileged classes to 'succeed' in school
(however that success may be defined) while countless others
(largely the poor, but also disadvantaged on a variety of
indicators like caste, gender, community and so on) are
absolutely overwhelmed by it. Pedagogies therefore cannot be
confined to the subject matter of a discipline and to the
proclivities of a 'child'; we need to recognise that learning and
academic achievement is deeply implicated in the material
conditions of schooling and the structural constraints of the
system and constantly mediated by it.

## Conclusions and Implications

Learning Mathematics has long
been a source of anguish for children, for teachers and for
parents, especially after the rudimentary elements of counting,
and the simple operations of addition and subtraction are
mastered. It is especially difficult for children who have
limited means, apart from the school and the classroom, to access
this extremely abstract and formal system that forms an important
barrier to be crossed in order to advance within the school
system. Children may dislike some subjects or find others
uninteresting (History as a school subject seems to be a case in
point), but given a basic competence of reading and
comprehension, a little extra effort (even if it means some rote
learning) enables them to 'pass' these subjects. The description
of mathematical thinking presented here is meant to underline its
exceptional nature and the fact that if the foundations of the
discipline, its language and its conventions are not mastered
systematically, it becomes increasingly more complex and finally
incomprehensible.

Although the benefits of schooling and a school education cannot
solely be measured in terms of academic achievement and
proficiency scores, they have become important indicators of
success and powerful assets for access to further education,
jobs, etc., and extremely important for large numbers of the less
privileged and the poor who send children to school at great cost
in the hope of securing better lives and livelihoods. The role of
those with the power to determine what counts as important
knowledge has led to suspicions about the role of knowledge in
school curricula and as a result' 'the
question of knowledge and the role of schools in its acquisition
has been neglected by both policy makers and by educational
researchers' (Young, p. 10). I have tried through this
analysis of the learning of Mathematics to emphasize that it is
important for children to acquire knowledge and to understand the
seriousness that such learning requires. The filtering capacity
of Mathematics is well known and if it is going to determine life
chances then it is the responsibility of schools to provide
academic, infrastructural and emotional support to learn
important Mathematics. It is not my brief to make every child
into a mathematician but it is only fair to accept that if the
promise of education is made to all children then some promise of
equal delivery must be assured and resources made available in an
equitable manner.

Disciplines taught in schools
have long cultural histories and accumulated knowledge bases that
children are expected to master in the course of schooling.
Learning and teaching of important and valued knowledge needs
sustained effort, time and resources and this places the
underprivileged and the poor at serious disadvantage. The
examples from Mathematics presented earlier serve to make us
aware of the fact that knowledge which counts as Mathematics is
valued as a discipline and is likely to determine entry into
higher education and into several high prestige occupations, is
not easily acquired. It requires immersion, time, effort and
imaginative teaching techniques that very few schools are able to
presently provide. This is true of all disciplines to a lesser or
greater degree and constitutes a handicap that a majority of
schools are not able to address, except a few elite and immensely
resource rich ones. To overcome the handicap, the affluent and
middle classes use various forms of advantage—whether it is
material resources or the social power that they wield—neither of
which are available to the poor.

The descriptions of schools that have been presented in this
paper illustrate the extent of deprivation that poor children
face, notwithstanding several grand schemes and policies launched
by the Government of India. Lack of funds, teachers, material
resources and very often of even classrooms or essentials like a
blackboard, make serious and engaged teaching and learning a
difficult challenge. Middle and upper class children are able to
overcome these hurdles firstly by opting for better equipped and
better managed private schools, but more importantly and less
visibly through out-of-school support systems. These children
have access to educated parents, tuitions, books, games,
educational toys and, increasingly, to technological aids to
learning. They are also much more likely to have space and time
(probably with supervision) to study after school, privileges
that poor children lack. Research evidence
suggests that children's performance, whether in government or
private schools, improves if they take tuitions and confirms the
positive role of supplemental help in raising levels of
performance on basic academic achievement in terms of reading
writing and arithmetic scores (Banerji and Wadhwa,
2013).

I also argue here that psychological theories and the notion of the child's construction of knowledge as interpreted in educational discourse have compounded this discrimination. The turn to child-centred learning although well intentioned, actually draws attention away from the importance of the systems of knowledge and their long histories of accumulated wisdom as well as the importance of a serious pedagogical commitment that is needed to master them. The normative subject of schooling conforms to the average middle class child with a rich fund of cultural capital that supports and complements the project of schooling and enables her to access the system with far less tension than the working class child for whom life and school do not seamlessly complement each other. We need to recognise the enormous burden that schooling imposes on working class children and the inadequacy of what is available for them to achieve success. Unless we search for possibilities of overcoming it, whether through increased resources, better teaching and pedagogy or through the very reconstruction of education to allow multiple routes of learning towards more fulfilling and rewarding lives, the differential benefits of education in an unequal social system are only likely to be exacerbated.

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## Acknowledgement

Grateful thanks to Geetha Nambissan, Indra Sengupta and to the Transnational Research Group for giving me the opportunity to adapt this work to a larger audience. Thanks also to Jasneet Kaur for her very helpful inputs.

Author:Farida Abdulla Khan,

Department of Educational
Studies,

Jamia Millia Islamia, New
Delhi 110 025.

Phone: +91 11 2410
6414

Email:
khan.farida@gmail.com

## Bio:

Farida Abdulla Khan is Professor in Educational Studies at the Jamia Millia Islamia, New Delhi. Her research interests include the social contexts of Mathematics education; issues of exclusion and marginalisation in education, and political conflict and education.

1 The new sociology of education in the UK and similar trends in the USA initiated more complex understandings of inequality in education and much current critical research in education acknowledges this tradition. Some prominent figures (amongst many) are Bernstein (1971) and Young and Whitty (1977) in the UK; Apple (1979), Anyon (1980) and Giroux (1983) in the USA. See volume edited by S.J. Ball (2004) for a review.

2 See Nambissan (2014) and Nambissan and Rao (2013) for a review of the changing trends of engagement with issues of disadvantage within educational research in India.

3 See Lerman (2000) for an excellent review of the social turn in Mathematics education.

4 See Khan (2012) for a more detailed discussion of this issue and its implications for the socially and economically disadvantaged child in India.

5 See Michael Young (2011) on the importance of learning, schools and 'powerful knowledge', especially for the poor.

6 See Khan (2010) for a more detailed discussion of these authors in the context of the social implications of Mathematics education in India.

7 Banerji's work, in press, supports the argument and Minocha (2013) provides some interesting data.

9 The ZDM Mathematics Education (2014) issue 46, published online, is an excellent collection of articles that represent a vibrant debate around issues of Mathematics, education, schools and society.

10 See the edited volume by Chopra and Jeffery (2005) for a number of interesting case studies that exemplify the differentiation within the system of Indian school education.

11 MHRD carried out Joint Review Mission of Teacher Education in several states across India in 201314 and the reports are available on the following website: http:/www.teindia.nic.in/rjm.aspx. These reports are a rich source of data regarding the many problems of the government school system and of teacher education across India.

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**In:**Working Papers der Transnationalen Forschungsgruppe Indien der Max Weber Stiftung "Armutsbekämpfung und Armenpolitik in Indien seit dem 19. Jahrhundert" | Working Papers of the Max Weber Foundation's Transnational Research Group India "Poverty Reduction and Policy for the Poor between the State and Private Actors: Education Policy in India since the Nineteenth Century"

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**Veröffentlicht am:**01.09.2015 14:28

**Zugriff vom:**29.01.2020 16:47