CHAPTER 1
BLURRING THE BOUNDARY BETWEEN MATHEMATICS AND
THE EVERYDAY
INTRODUCTION
The first non-racial elections in South Africa, held on 27/04/1994, ushered in a
government whose political framework is grounded on principles of democracy, non-
racism and non-sexism. The ?rainbow people of the South? is a phrase popularized by
Archbishop Desmond Tutu to describe the new post-apartheid South African society. The
?rainbow? metaphor captures both the different backgrounds of South Africans and the
common non-racial values to which the nation should strive. Underlying both this
rainbow metaphor, and the new South African political framework, is an implied
intention to de-emphasize the boundaries or partitions which kept different races or
population groups of one nation apart.
It is this quest for blurring the boundaries between different categories which is mirrored
in the new South African Education curriculum, Curriculum 2005 (C2005). As stated in
one of the early policy documents emerging in the post-apartheid education:
In the past the curriculum has perpetuated race, class, gender and ethnic divisions
and has emphasized seperatedness, rather than common citizenship and
nationhood. It is therefore imperative that the curriculum be restructured to
reflect the values and principles of our new democratic society.
(DoE, 1997:1)
The substance of this statement is that an educational policy is embedded in and shaped
by a political framework of a country.
Within a ten-year period (1995 ? 2005), the new curriculum has already undergone two
major waves of curriculum change. The first wave reached its peak in 1997 when the
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first* education policy documents were publicized. It was followed, in 2000, by a second
curriculum wave characterized by the publication of the Revised National Curriculum
Statement (DoE, 2001). Both sets of documents signal clear intentions of blurring the
boundary between:
1. The mathematics and the everyday and
2. The roles of the learner and teacher
It is necessary to step back and reflect on an historical perspective regarding mathematics
and the role of the learner. Dating back to some ancient societies, mathematics was
viewed as a special subject to which access was limited to a few. Volmink, for example,
argues that the social arrangement of early civilizations were such that ?only the rich, the
powerful, the influential, had access to mathematical knowledge? (1993:123). Seleoane
also observes that Plato?s ideal society, the republic, was arranged in such a way that only
people at the top, or the philosopher kings, ?were taught thought-provoking subjects like
mathematics, astronomy and dialectics? (1988:08). Apartheid education was also
informed by intentions to reserve mathematics for a few. Dr. Verwoerd, the then Minister
of Native Affairs, stated his views on the role of education and mathematics
unambiguously:
When I have control of native education, I will reform it so that natives* will be
taught from childhood that equality with Europeans is not for them. There is no
place for him (a black child) in European society above the level of certain forms
of labour?What is the use of teaching a Bantu child mathematics when it cannot
use it in practice.
(Quoted by Khuzwayo, 2005:315)
Verwoerd?s statement was about the way in which education was to be used as a means
for preparing learners from different races for different roles in their lives. It is curious
though, that mathematics is singled out as a subject which is irrelevant for a ?Bantu?
* A number of draft publications were issued in each wave of curriculum change. I use ?first? to distinguish
this document from numerous education policy drafts produced between 1995 and 1997.
* The words Natives, Blacks and Bantu in this context are used interchangeably. Each refers to non-white
South Africans.
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child; as if to suggest that access to mathematics will enable access to a European way of
life.
This study thus takes place within a context of political transition from an apartheid-
informed education system to a new post-apartheid political context. In this chapter I
briefly introduce arguments made in relation to the way in which South African
education policy views the mathematics-everyday boundary on the one hand and the
learner-teacher role boundary on the other. I will also outline arguments, within the
mathematics education realm, in relation to the mathematics-everyday boundary. I will
suggest that throughout these debates, the learners? voice on the consequence of shifting
the boundary between the everyday and mathematics is silenced. This apparent silence is
an angle through which this particular study enters the debate. The study does not purport
to resolve debates or disputes around the mathematics-everyday boundary nor provide a
representative learners? voice on this matter. However, the study intends to further
illuminate and contribute towards these debates, informed by an interpretation of a
learners? voice.
1.1 POLICY ON SHIFTING THE MATHEMATICS-EVERYDAY AND
TEACHER-LEARNER BOUNDARIES
In the introduction above, I have separated the learner and the mathematics. However,
Bernstein cautions:
We should look at the way schools select subjects for the curriculum, the way they
teach these subjects, and the way they examine them. These things tell us about the
distribution of power in society. They also tell us about social control.
(My emphasis)
(Bernstein, 1975:85)
Bernstein suggests that in a school setting, there is a close link between what is learnt
(content like mathematics) with how is taught (access to learners). Separating the learner
and mathematics oversimplifies the complex way in which the two influence one another.
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Notwithstanding this realization; I will keep the two notions separate, at an analytical
level, in order to assist argument.
Boundary-blurring with respect to mathematics: With particular reference to contents
(like mathematics); the first policy document announced the need to collapse the
boundaries separating different school subjects. ?This most radical form [of Outcomes ?
Based Education] implies not only that we are integrating across disciplines into learning
areas but we are integrating across all learning areas in all educational activities.? (DoE,
1997:31). Along with this position was a name change of the subject Mathematics to
Mathematics Literacy, Mathematics and Mathematical Sciences (MLMMS); suggesting
that ?the boundaries have been collapsed between pure and applied mathematics, and
Statistics? (Adler, Pournara & Graven, 2000:2). The policy position and the name change
signaled a de-emphasis on the uniqueness of mathematics and accorded it (mathematics)
the same status as other subjects. This blurring of the boundary between mathematics and
other subjects was coupled with another boundary-blurring: between mathematics and the
everyday reality. In this regard, mathematics was defined as a ?human activity that deals
with patterns, problem-solving, logical thinking etc, in an attempt to understand the world
and make use of that understanding?. (DoE, 1997:2). In this way, mathematics is
construed as a subject through which reality is captured and not an isolated subject
concerned only with itself.
Owing to public feedback and following the state-commissioned C2005 Review
Committee, the boundaries between different learning contents were re-emphasised. For
mathematics, this was symbolized by a name change from MLMMS back to
Mathematics. It was also implied in a revised definition of mathematics as a distinct
subject ?Which has its own specialized language that uses symbols and notations for
describing numerical geometric and graphical relations.? (DoE, 2000:16). At a
pedagogical level though, the boundary between mathematics and the everyday remained
blurred: ?In teaching Mathematics? highlighted the policy document in an apparent
reference to mathematics teachers, ?try to incorporate contexts that can build awareness
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of human rights, and social, economic and environmental issues relevant and appropriate
to learners? realities.? (ibid).
In sum, the rigid boundary in the curriculum between mathematics and the everyday
ensured that the distinct nature of mathematics was realized; but in enabling access to this
knowledge, the boundary between mathematics and the everyday was to be blurred.
Boundary-blurring with respect to the role of the learner: Both in the first and revised
policy documents, the new curriculum views learners as active participants in knowledge
acquisition. Therefore, whilst teachers are expected to be ?professionally competent and
in touch with current developments, especially in his/her area of expertise? (DoE,
2000:06), the statement further advises that teachers should be ?open to views and
opinions held by learners which may differ with his?. This view challenges the
asymmetric relationship between learners and teachers.
By presenting an argument in favour of a learner-centred pedagogy (Adler, 1998:04), the
new curriculum has specified a new teacher-learner relationship and has therefore
redefined the learner?s role. Notwithstanding some concerns regarding the practical
aspects of this pedagogy (Brodie, 1998:166; Nkhoma, 2003); it presents the learner as an
active participant in the learning situation and not as a passive listener.
Though oversimplified, the above discussion suggests that the movement from apartheid
education policy to post-apartheid education policy was informed by a shift in political
paradigms. The development of post-apartheid education policy was, in addition, a result
of public participation. As a result, ?For the first time,? brags the Revised National
Curriculum Statement, ?decisions were made in a participatory and representative
manner? (DoE, 2001:08). What is silenced, ironically, is the learners? voice on the
possible value or significance of curriculum change. In a new curriculum that creates
space and encourages more active learner participation, one would anticipate some
aspects of the curriculum to be informed by learners? concerns. It is on the grounds of
this observation that this study emerges. In particular, I reflect on the learners?
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perspectives on the value of incorporating the everyday in mathematics. I address this
broad concern through a study guided by the key questions outlined in the section below.
1.2 KEY QUESTIONS FOR THE STUDY
Both in the title of the study and discussion so far, I have made reference to the
?everyday?. However, everydayness, as Moschkovitch (2002) and Arcavi (2002) argue, is
contextual. For example, exploring the nature of mathematical patterns is an everyday
activity for mathematicians (Moschkovitch, 2002:01). I use the term ?everyday? to refer
to learners? out-of-classroom daily experiences. The term ?everyday mathematical tasks?
refers to mathematics activities which draw in or incorporate these out-of-school aspects.
There are various factors, within a classroom context, that may influence learners?
perspectives about the role of the everyday in mathematics. One source of influence is the
type of activities incorporated in mathematics texts used by learners. In this regard,
Dowling (1998:43) asserts that pedagogic texts are not only produced for learners, they
also produce learners. Following their study on the role and use of textbooks in
improving the quality of education in schools, Crouch and Mabogoane (1997) concur that
learning materials are amongst the most important predictors of cognitive development.
The type of texts, activities and materials learners use thus shapes their views and
perspectives about the content of the subject.
Within a classroom setting, however, teachers also play a role in what learners read and
perhaps prioritize in a text. Though using the chalkboard as an example of a material
resource, Adler (1998:14) suggests that it is not that the chalkboard is good or bad, ?but
how it is used, for what and for whose benefit?. How a resource, like the mathematics
textbook is used in the classroom, determines access or lack thereof to mathematics. In
relation to this study, it is not simply the incorporation of the everyday but also its use,
which may influence learners? perspectives about its role in mathematics learning.
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The following key questions are based on the premise that both the everyday and its
treatment in the classroom shape and are shaped by the learners? perspectives. These two
aspects, the everyday and its treatment by learners are addressed respectively, by key
questions 1 and 2. Key questions 3 and 4 are motivated by the need to summon learners?
opinions on the inclusion of the everyday in mathematics. Question 5 focuses on the need
to provide as coherent account of the sense made about learners? perspectives on the
inclusion of the everyday in mathematics.
1. What type of the everyday is incorporated in mathematics texts used by learners?
This question provokes a reflection on the nature of the context(s) recruited in
activities and its relation to learners? experiences.
2. How do learners describe lessons in which the everyday is incorporated? The
particular focus of this question is on whether learners foreground the
mathematics content of the lesson and/or the everyday focus of the lesson.
3. How do learners reason and discuss mathematical activities which incorporate
everyday knowledge? This question seeks to explore the way in which learners
act and argue in class when faced with a mathematics activity which incorporates
the everyday. The question will also explore the kind of written texts produced by
learners in response to tasks that incorporate the everyday.
4. What value or significance do learners attach to the inclusion of the everyday in
mathematics? This question seeks to find out whether and in what way learners
find the incorporation of the everyday in mathematics enabling or inhibiting
access to mathematics.
5. How can learners? perspectives on the incorporation of the everyday, as gathered
in the first three key questions, be explained? This question will provoke a
generation of a coherent theoretical account of the learners? perspectives on the
inclusion of the everyday.
Debates around the relative merits and demerits of incorporating the everyday in
mathematics have occupied mathematics educators for over 20 years. What it means to
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incorporate the everyday in mathematics and its effects on the learning of mathematics
remain largely unresolved. In the following section I elaborate on some of these debates.
1.3 WHEN THE BOUNDARY BETWEEN MATHEMATICS AND THE
EVERYDAY IS BLURRED?
It is possible to discern two extreme positions regarding the effects of incorporating the
everyday in mathematics. At the one end are studies and arguments highlighting the
necessity and learning benefits of incorporating the everyday in mathematics (for
example, D?Ambrosio, 1991; Santos & Matos, 2002); at the other extreme, are studies
and arguments against (See, for example, Gellert, Jablonka & Keitel, 2001:57). These
two extremes sandwich a range of studies which present a cautious inclusion of the
everyday in mathematics. Whilst desiring the value of incorporating the everyday in
mathematics, these studies acknowledge ?that there are several conceptual and practical
difficulties in this regard that first need to be addressed? (Volmink, 1993:122). This
section elaborates and reflects on each of the three positions above.
Mathematics and its teaching should draw from the everyday
Critical mathematics education, ethnomathematics and realistic mathematics are some of
the most notable perspectives which offer inspiration and a theoretical base for the
blurring of the boundary between the everyday and mathematics. Though differently
motivated, these perspectives engage the epistemological nature of mathematics (what
mathematics is) and the pedagogic value of recruiting the everyday for teaching and
learning purposes.
Critical mathematics education: Critical mathematics education is a perspective which
argues for embedding mathematics within the socio-political context. One of the main
postulates of critical education is that education should be a platform for developing a
critical attitude among students towards a technological society (Skovsmose, 1994:338).
Mathematics has the potential to both contribute towards democracy and deny informed
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participation in democracy (Skovsmose & Valero, 2001). It is also central to
technological development which is both beneficial and destructive to mankind. In this
way, mathematics is seen as having lost its innocence. This point is stressed by
Skovsmose and Valero (2001:52), ?Mathematics cannot be assumed any more to be the
?queen of sciences?, sleeping in the limbo of neutrality, a-sociality, a-morality, and a-
politicy.?
With regard to mathematics curriculum, Skovsmose emphasizes that the problems should
relate to the fundamental social situations and conflicts and the students should be able to
realize the problems as their own. He rejects the problems that belong to ?play-realities?
with no significant purpose ??except as an illustration of mathematics as a science of
hypothetical situations? (1985:338).
Inspired by Paulo Freire and some aspects of critical mathematics education,
Mukhopadhay (1988) presented elementary mathematics teachers with an investigative
task whose context had strong social implications. Students were required to sketch an
equivalent of a popular doll, Barbie, according to real-life scale. Most students were
familiar with the Barbie dolls because its distribution in class (to these students) sparked
?giggles and laughter in spontaneous sharing of numerous personal anecdotes of Barbie?
(Mukhopadhay, 1998:156). On completion of the task, ?the students recognized how
shockingly different, even unreal and unnatural, Barbie looked compared to themselves?.
(Mukhopadhay, 1998:157). Barbie, they observed, had a pelvic area which is way too
small to bear a child. This activity served as a model for validating the usefulness of
mathematics beyond its usual abstract, context-empty existence.
Ethnomathematics: The concept of ethnomathematics draws from the realization that
mathematics is practiced among different cultural groups and that its ?identity depends
largely on focuses of interest, on motivations? (D?Ambrosio, 1991:22) which do not
belong to the realm of academic mathematics. Mathematics is thus seen as a by-product
of various cultural groupings and practices. He views the rigid boundary between
mathematics and other disciplines as largely Eurocentric. He argues, ?It is preposterously
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Eurocentric to try to identify mathematics or zoology or other disciplines as
compartmentalized pieces of knowledge in different cultures, just as it is preposterously
adult ? centric to impose on young children these structured modes of explanation and of
dealing with the world?. D?Ambrosio?s articulation views reality as the basis for
developing mathematics.
Using ethnomathematics as a theoretical base for most of his research, Paulos Gerdes
asserts that geometrical exploration constitutes the area of mathematical activity ?par
excellence? in the history of Central and Southern Africa. These activities include hair
braiding, basket weaving, tattooing etc. (Gerdes, 2001:03). The substance of Gerdes?
argument is that ethnomathematics opens up a dialogue between mathematics and various
cultural activities. As part of the Ethnomathematics Research Project at the Higher
Pedagogical Institute in Maputo, Marcos Chirenda (1993) participated in a research
project entitled a ?Circle of interest in ethnomathematics?. The study was aimed at
showing student teachers that ?it is possible to introduce mathematical concepts using
artifacts of local traditions? (1993:142). The research team led pupils* to examine
trajectory patterns of straw strips in artifacts such as hats, baskets and mats and observed
that the creation of these geo-shapes heightened the pupils? confidence. As Chirenda
argues (1993:147), ?In this way, they begin to feel that the maths they learn at school also
comes from their lives and society?. Such examples, Chirenda claims, dispel the myth
that mathematics has exclusively European roots. In a study that took place in the rural
parts of Brazil, Knijnik (1993) reflected on the value of what she terms popular
mathematics. She participated in the Landless Movement whose task was to ?produce
and market goods in the settlements? (1993:149). Working as a mathematics educator
within this context, she was exposed to what she terms popular mathematics, which is
part of cultural knowledge produced by subordinate groups and not legitimated by
dominant groups. In this project, students who knew the popular methods were the ones
who were assigned to teach them. This process allowed ?the birth of a ?synthesis-
knowledge?, which is constructed by taking popular knowledge as its starting point, and
which however, transcends it.? (1993:152).
*Grades and/or ages of pupils who participated not specified.
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Underlying both these studies is the value placed in recognizing and embracing cultural-
based practices and activities in the teaching or learning of mathematics.
Realistic mathematics: Hans Freudenthal argues in favour of a mathematics curriculum
which recruits the everyday in the teaching of mathematics. With respect to mathematics,
he calls for mathematics fraught with relations, ?I stressed the relations with lived-
through reality rather than with a dead-mock reality that has been invented with the only
purpose of serving as an example of application? (1973:79). With respect to the teaching
of mathematics, he asserts that a mathematician should never forget that mathematics is
not only meant for future mathematicians.
The Dutch school of Realistic Mathematics Education was developed along
Freudenthal?s ideas of mathematization, consisting of vertical and horizontal
mathematization. Vertical mathematization consists of ?formalizing students?
constructions and productions, moving them towards generalities of content and method
(Arcavi, 2002:21) and horizontal mathematization consists of ?moving a problem from its
context towards some form of mathematics? (ibid). Arcavi considers mathematization as
a powerful idea to bridge the gap between everyday mathematics and academic
mathematics because there is provision for students? idiosyncratic ideas to serve as
springboards towards a more formal mathematics. The success of this approach, claims
Arcavi, is backed up by ?a respectable amount of evidence? (2002:22). What is not clear
from the article, both in terms of describing the tasks or providing examples of tasks, is
whether the nature of the context (dead mock reality or lived experience) matters or not.
The crux of the arguments though, is that the learners? everyday realities have a role in
their understanding of mathematics.
Even though critical mathematics education, ethnomathematics and realistic mathematics
influenced a notable number of researchers in reflecting on the incorporation of the
everyday in mathematics; there are other studies with similar agenda, which cannot be
associated with any of these three perspectives. Boaler (1997) for example, draws from a
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range of theoretical constructs (including Bernstein?s) to relate a comparative analysis of
two schools, Amber Hill and Phoenix Park. She particularly focuses on the different ways
in which mathematics was taught at each school and the different forms of mathematical
knowledge prevalent at each. At Amber Hill, ?the individual ?contents? of mathematics
were well insulated from each other?. (1997:25). The lessons ?conformed in terms of the
explicit hierarchy which was established between teachers and students and the
disconnection of lessons from everyday realities?. (ibid). At Phoenix Park, the boundary
between mathematics and other subjects was less distinct. Mathematics ideas were
introduced as part of meaningful activities. Boaler (1997:81) concludes that:
The Phoenix Park students did not have a greater knowledge of mathematical
facts, rules and procedures, but they were more able to make use of the
knowledge they did have in different situations. Furthermore, the students at
Phoenix Park performed better than or the same as Amber Hill students in various
applied situations, conceptual questions and within more traditional questions.
She does not make much reference to the type of activities in the two schools; however,
the general implication of her study is important: Insulating mathematics may lead to
underachievement of students in applied and more traditional questions. She elaborated
this position in a more focused reflection regarding the implementation of reform-
orientated curricular. She argued that reform-orientated curricula, exemplified by among
others, the use of contextual tasks, can achieve a reduction in linguistic, ethnic and class
inequalities in schools (Boaler, 2002).
Other studies motivate for the incorporation of the everyday in mathematics on the basis
that such incorporation will lead to positive pedagogic spin-offs. Foxman and Beishuizen
(2002) suggest that the use of contextual tasks provides a useful diagnostic tool for
teachers regarding learners? difficulties in solving mathematics tasks. In their reanalysis
of mental calculation strategies by a sample of 247 11-year olds in a national survey of
schools in England, Wales and Northern Island; they observed that questions which
recruited contexts encouraged the use of more informal strategies. Learners? engagements
of these tasks, they conclude, ?invite more informal strategies and more snapshots of
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children?s inadequate use of calculation strategies, which can give the teacher more clues
on how to address them?. (2002:67). The incorporation of contextual tasks is argued for
on account of the pedagogic value they add. Similarly, though with a different focus,
Garner and Garner (2001) compared outcomes of traditional and reform calculus in terms
of students? retention of basis concepts and skills. Traditional calculus emphasized rote
memory and symbol manipulation whilst reform calculus? emphasis was on conceptual
understanding and practical application. They conclude that though the students?
retention showed no significant statistical difference, reform calculus students retained
better conceptual knowledge whilst traditional calculus students retained better
procedural knowledge. Thus, the use of reform calculus on the one hand, and traditional
calculus on the other, seem to lead to different outcomes.
The studies cited above highlight the different benefits likely to be secured from the use
of tasks characterized by a blurred boundary between the mathematics and the everyday.
There are other scholars whose argument for the incorporation of the everyday in
mathematics is accompanied by a caution. In other words, these studies highlight not only
the possible advantages of incorporating the everyday, they also emphasise the conditions
under which these tasks should be considered.
Cautious incorporation of the everyday
In this section I elaborate on studies which, similar to those cited above, are sympathetic
towards the notion of school mathematics which incorporate the everyday. However,
these studies question or caution against the pedagogic benefit of such incorporation.
These studies draw from a variety of empirical data and theoretical bases. In this section I
distinguish between studies which
1. Draw from word problems
2. Draw from school and out-of-school experiences and
3. Question the feasibility of mathematization (as previously cited).
Word problems: With specific reference to word problems, Verschaffel and De Corte
(1997) observe that there are some implicit rules and assumptions that learners need to
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understand when ?playing the game of word problems?. Non-awareness of these rules,
they argue, may lead to ?bizarre? errors and reactions. As an example of such bizarre
reactions, they make reference to some learners? responses to the following task (1997:
x): ?Pete has three apples. Ann also has some apples. Pete and Ann have 9 apples
altogether. How many apples does Ann have.?? Some of the answers provided by
learners included ?some apples?, ?a few? and ?a couple?. Their argument is that whilst
such responses are not totally incorrect, they are inappropriate for the context of word
problem-solving in school.
Wiest?s (2001) study suggests that there are certain types of word problems which elicit
creativity from children. Wiest focused on the use of fantasy contexts and real world
contexts. She observed that students expressed interest in fantasy contexts and solved
problems which incorporated this context better than tasks which incorporated real world
contexts. She suggests that fantasy contexts should be included among those used for
word problems in the teaching of mathematics since many children like fantasy. This
observation is in stark contrast with Freudenthal?s call for the use of lived through
realities. But it also highlights the different conditions under which various researchers
view the possible success of context-based mathematics tasks.
In responding to mathematical word problems, De Corte asserts that school children
believe that real world knowledge is irrelevant. In one of the studies involving items that
drew from the real world, one learner expressed this belief, ?Maths is not about things
like that [real life aspects]. It?s about getting sums right and you don?t need to know
outside things to get the sums right? (De Corte, 2000:37). This belief, according to De
Korte, is also observable amongst teachers too, a finding which supports the view that the
opinions about doing and learning mathematics of the teachers themselves ?are at least
partially responsible for the development in students of misbeliefs that have a negative
impact on the regulation of their problem-solving approach and strategies?. (ibid).
These studies do not discourage the use of word problems in mathematics classrooms,
however, they share a perspective that it should be coupled with attention to a number of
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factors. These include the type of word problems used (see Wiest above) and the benefit
of the context drawn in may be concealed to learners and some teachers.
School and out-of-school experiences: Civil and Andrade (2002) conducted a study
which focused on the development of teaching innovations to promote students? learning
of school mathematics by building on their knowledge and experience of the everyday.
The subjects for the study were Mexican?Americans whose performance in mathematics
was low. The researchers firstly tried to understand the learners? household background
which they could use as the basis for developing the mathematics. They summarise their
impressions from trying to make these connections: ?As much as we enjoy the wealth of
information that comes out of these household visits, we find ourselves constantly
wondering about the connections to the teaching of mathematics in school? (2000: 156).
The point made by Andrade and Civil is that school and out-of-school mathematics
should be seen as different discourses, therefore blurring the boundary between these two
categories may not be desirable.
That school mathematics and home mathematics, characterized by the everyday, should
be seen as two different discourses is also supported by De Abreu, Cline and Shamsi?s
observations (2002). In a study involving 24 school children (Pakistani-British and
White-British), their parents and teachers, they focused on the transitions between home
mathematics and school mathematics. They point out the ?double character? of
mathematics in these two environments. One of the conclusions they draw relates to the
importance of viewing the home and school mathematics as different practices. This
realization, they argue, enabled a mother to adjust her help at home in a way that was
supportive of her child?s success. ?A clear contrast in Kashif?s and Rachel?s parents?
accounts was their degrees of awareness of specific differences in their and their child?s
school mathematics. It was as if for Kashif?s parents there was no transition between
home and school mathematics, as if after you master a mathematical concept you could
apply it everywhere? (2002:141).
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In these two studies, the value of incorporating the everyday in mathematics is
acknowledged. However, the conclusions drawn suggest that in order to succeed or cope
with school mathematics, the everyday and mathematics should be viewed as two
different discourses consisting of different rules of engagement.
Sullivan, Zevenbergen and Mousley (2003:118) draw attention not only to the ?situation
in which the mathematics is embedded? but to the ?learning environment in which the
task is used?. They observe that the situation in which the task is embedded may be
inappropriate or irrelevant to some learners. They write:
We are not arguing that contexts should not be used; indeed we believe that
contexts have much to offer. The issue for us is that the teachers need to be fully
aware of the purpose and implications of using a particular context at a given
time, to choose a context that is relevant to both the problem content and the
children?s experience, and to have strategies for making the use of the context
clear and explicit to the students.
(2003: 118)
Implicit in their argument above, is a consideration of and attention to some form of
pedagogic practice in enabling engagement with tasks that incorporate the everyday.
In 2002, the three authors offered a more focused elaboration on the value of both the
situation in which the mathematics is embedded and the learning environment in which
the task is used. They make particular reference to a task which entailed the context of a
bank robbery. This context was viewed as abhorrent to indigenous people by indigenous
educators. A more appropriate context, they suggest, may have been a team of footballers
or netballers. ?Such a context was more likely to resonate with the students as this was an
activity central to their life experiences,? (Zevenbergen, Sullivan and Mousley, 2002:
527).
The central argument presented by Sullivan, Zevenbergen and Mousley is that two
contexts are at play. The first one is the context of the task and the second one is the
16
classroom context. Both these contexts influence how the everyday or context is engaged
in the classroom.
Feasibility of mathematization: In one of the studies previously cited, I highlighted
Arcavi?s argument wherein he viewed mathematization as one of the key concepts
contributing towards explaining the merging of the everyday and mathematics. The
studies that follow question the possibility of drawing generalizations from contexts. In
other words, they question a process which resonates with horizontal mathematization as
described by Arcavi( 2002).
Brenner invited four teachers and their junior high school students to engage a worksheet
which drew from the everyday context of Pizza. Her particular focus though, was on the
take-up by teachers of this worksheet as it made new instructional demands on them. She
observed that the invitation to use everyday mathematics incorporated into the Pizza unit
was often declined by teachers and learners. ? When this invitation was accepted,?
Claimed Brenner (2002:87) , ?we began to see the kind of reasoning that we wanted to
promote, in which students used the everyday or informal knowledge to support their
problem solving??. Shortage of time led to the use of traditional recitation. The benefit
to be derived from the everyday, the study implies, can be realized if the invitation to
consider the everyday is accepted and there is sufficient pedagogic time.. Consideration
of the everyday and subsequent movement from the everyday to mathematics is not ?a
given?.
In a study whose participants were fifth graders, Civil attempted to find out whether a
teaching innovation would enable students to advance in their learning of the prescribed
school mathematics ? in ways that are true to mathematicians? mathematics while
building on students? knowledge of and experiences of everyday mathematics?.
(2002:47). Civil distinguished between school mathematics and mathematicians?
mathematics. The former is characterized by an overreliance on paper-and-pencil
computations, prescribed algorithms and clearly defined tasks; the latter consists of ill-
defined tasks requiring persistence and collaboration with other mathematicians. She
17
observed that ?students participated when the activity was related to the everyday
mathematics but withdrew as the discussion moved to more formal mathematics?.
(2002:51). Thus, the movement from the everyday to the mathematics was not welcomed.
Moschkovitch, unlike Civil and Brenner (cited above) does not reflect on the difficulty of
moving from the everyday to mathematics. However, she illustrates that the classroom
situation shapes the sense that learners make of context-based mathematics activities. She
carried out a project which linked the use of mathematics activities as a result of which
students ?might engage during the course of their present daily lives or to future activities
in which students might engage as adults at work? (2002:95). The participants in her
study (The Antartica Project) were seventh-graders. She focused on ways in which
different classroom settings shape mathematical activities. She presented students with
expanded mathematics activities beyond traditional school mathematics problems. These
activities involved solving open-ended problems, applying mathematics to real world
problems and communicating about mathematics (2002:97). One of the observations she
made is that ? the nature of students? mathematical activity, however, depends not only
on the curricular activities used in the classroom but also on the nature of the classroom
practices, especially on the didactical contract between the teacher and students?
(2002:107).
Inclusion of the everyday is not necessary
I began this section by citing ethnomathematics, critical mathematics education and
realistic mathematics education as notable (but not only) perspectives upon which
arguments in favour of incorporating the everyday in mathematics hinge. With respect to
critical mathematics education, Skovsmose admits that the use of the everyday tends to
conceal the mathematics. It is for this reason that he argued for mathematical archaeology
(1994: 151). With regard to realistic mathematics education, I will elaborate on
arguments in this section, in which the use of ?lived-through experiences? is not regarded
as significant. With regard to ethnomathematics ; Rowlands and Carson (2002) argue that
it is only through the lens of the formal academic mathematics that the real value of
mathematics inherent in different cultures can be understood. They view mathematics as
18
a universal subject which ?transcends the civilizations of Ancient Greeks and China, the
France of Pascal and it is this universalism that has to be emphasized in the classroom,
rather than the geometrical patterns in traditional crafts, for example? (2002:98). The
literature below is organized according to arguments which suggest that
1. It is the context within which the activities (which draw from the everyday) are
engaged, that influences the quality of learning and
2. The everyday inhibits access to mathematics
Context of engagement has more influence than the everyday: Saljo and Wynhamn
(1993) challenged 332 Swedish students to determine the cost of posting a letter. The
students were provided with an official table of postage rates from the Swedish post
office. The results of their study suggest that this task is interpreted as a mathematics task
in a mathematics classroom and as a non-mathematical task in a social studies classroom
(1993:332). Therefore, it would seem that, learners in a mathematics classroom most
likely regard the tasks as mathematical in spite of these being presented in a form of
everyday tasks.
In a non-teaching context, Cooper and Dunne (2000) present a discussion which focuses
closely on the way children engage mathematical tasks that include the everyday - what
they refer to as the ?realistic? items (2000:03). They note that one child, from a middle
class family, is able to negotiate the esoteric/everyday boundary appropriately (2000:67).
This seems to suggest that this child treats some ?realistic? problems as ??merely
differently presented exemplars of standard arithmetic problems?. The inclusion of the
everyday, for this particular child does not seem to evoke the ?everyday? response.
Consequently, the inclusion of the everyday does not seem to have any effect in the way
this child engages the mathematical task. If the ?everyday nature? of a mathematical task
does not seem to influence the way the learners engage with the task; then the effect of,
and the bearing that the everyday has on the learning of mathematics does not seem clear.
In contrast, Cooper and Dunne (2000) observed that a working class boy was not able to
demonstrate his combinatorial competence by negotiating the boundaries between the
19
esoteric and the everyday in one item (2000:67). In general they noted that working class
learners are more likely to draw inappropriately from their everyday knowledge when
they respond to realistic items. In this respect, the inclusion of the everyday seems to
inhibit the acquisition of mathematics by working class learners.
The main argument flowing from studies by Cooper and Dunne (2000) and Saljo and
Wynhamn (1993) is that it is much more than the everyday in the task which determines
whether or not it (the everyday) is taken seriously. In other words, the production of a
legitimate response in relation to performance, and not so much a task itself, influences
the way learners interpret, engage and respond to the task.
The everyday inhibits access to mathematics: Floden, Buchman and Shwille (1987)
maintain that it is necessary that school mathematics remains separate from the everyday.
They suggest that the everyday restricts the students? scope of vision, it exaggerates
reliability and importance of close to home experience in the learning of mathematics and
this makes it difficult to understand the academic disciplines. This point is elaborated in a
different study whose focus was on students? arguments in explaining conceptions of
division by zero. In this particular study, Tsamir and Sheffer (2000) observed that
secondary school students justified, on the basis of concrete situations, that division by
zero results in a number. The students did not seem to notice the irrelevance of the
everyday as an explanation for this operation (division by zero). Drawing from the
context of language, Pimm (1987) explores the effect of ?borrowing? everyday English
words in creating a mathematics register. He makes particular reference to Tall?s
investigation of first-year mathematics undergraduates? interpretation of words such as
some and all. His finding is that the students regard these terms as contrastive rather than
inclusive i.e. some entails not all. Thus, the statement ?some rational numbers are real
numbers was regularly judged to be false because all rational numbers are real numbers
(Pimm, 1987:79). Even though he is not against the ?borrowing? of English words,
Pimm?s argument suggests that the use of the everyday may serve as barriers towards
mathematics.
20
In Britain, a popular text series used by students includes four sets of mathematics
textbooks; the Y (yellow) series, the R (Red) series, the B (blue) series and the G (green)
series. Paul Dowling (1998) analyses the Y series (a scheme for the highest achieving
students) and the G-series (a scheme for the lowest achieving students) of school
mathematics textbooks, SMP 11-16. From his analysis, he notes that the G-series
contains numerous examples meant to model the everyday. By contrast, the Y-series are
characterised by the language and form of the esoteric discourse in which the reader is
invited to join an international community of mathematicians. His principal argument is
that the use of G-series books serves more as a stumbling block towards an understanding
of mathematics. The users of the G-series are led towards a "different direction" to that of
community of mathematicians. Thus, rather than enabling connections and greater
mathematical meaning, these learners are effectively denied access to mathematics
learning.
The preference of authentic problems over play-realities as a platform for learning school
mathematics is also not viewed favourably by De Jager and De Jager. In defending their
use of artificial tasks for their textbook calculus chapter, De Jager and De Jager (1985):
argue that the purpose of the calculus chapter in their school textbook is to help students
to develop certain skills. They elaborate (1985:199), ? If they (the tasks) are not very
?real-life?, it will be a pity, but not a disaster, but if they are very ?real life? and develop
no skills in dealing with the problems you may meet later, that will be a disaster?. De
Jager et al value the development of mathematical skills more than the use of authentic
context. In this regard, mathematics skills are foregrounded over the use of mathematics
in lived reality and its use in acting out in social reality.
What these studies share is an epistemological approach where boundaries around
varying types of knowledge are foregrounded. The main argument presented is that the
epistemological boundary between the everyday knowledge and mathematics is
significant and cannot easily be traversed in learning.
21
The above section has provided literature whose arguments with respect to the
incorporation of the everyday in mathematics can be categorized into three: for, cautious
and against. The benefits of drawing upon the everyday in mathematics, as implied in the
arguments above, include accessibility to mathematics by making use of the familiar
contexts that learners can relate to. A contrasting but equally justifiable argument is that
drawing upon the everyday to represent or communicate mathematics ideas may delay or
deny learners? access to the body of knowledge that mathematics represents. A third
argument supports the inclusion of the everyday with a variety of cautions. Where does a
mathematics curriculum in a post-apartheid South Africa, a country in which access,
equity and meaning in education are desired goals, stand in these arguments? This
question is explored below.
In South Africa
Long before the 1994 democratic elections, concerns were expressed by some
mathematics educators over what they considered a ?very formal and highly abstract? and
?decontextualised? (Adler cited in Christie, 1991: 287) school mathematics curriculum.
One way of making the school curriculum less abstract and more meaningful was by
situating it in ?the realm of everyday experiences of people? (Volmink, 1993:123). In this
section I reflect on South African studies, considerations and arguments which relate to
the effects of a blurred mathematics-everyday boundary. The influence of Freudenthal
(realistic mathematics), Skovsmose (critical mathematics education) and D?Ambrissio
(ethnomathematics) can still be discerned, respectively, in the establishment of
REMESA1, the Ethnomathematics project at RADMASTE2 and the South African studies
with a critical mathematics education orientation ( Vithal, 2001 & Kibi, 1993). The
structure of my argument in the remainder of this chapter will still be on studies which
provide arguments for, cautiously for and against the incorporation of the everyday in
mathematics.
1 REMESA is an acronym for Realistic Mathematics Education in South Africa. It is led by Prof. Cyril
Julie and operates from the University of the Western Cape in South Africa.
2 RADMASTE is an acronym for Research and Development for Maths, Science and Technology
Education. The ethnomathematics project, led by Prof Paul Laridon, was one of its projects.
22
Arguments for: The following two studies present different versions of the everyday. In
one study the everyday is a cultural game whilst in the other it is a contrived situation. In
an study involving three Grade eight and one Grade nine learners, Mosimege (1998)
explored the use of a game, ?string figures?, with the intention of reflecting on how games
could be used for teaching mathematics. The use of this game at classroom level, he
acknowledged, needed more time ?beyond the usual one or two 30 minutes periods
allocated for mathematics lessons? (1998:283). He also observed that ?students tend to be
more engrossed in the playing than the learning process? (1998:284). These challenges
notwithstanding, Mosimege?s stance on the use of games in mathematics classes is
unambiguous: ?The fact that ultimately, after some probing, the students were able to
mention some mathematics concepts that were related to the game showed that there is a
definite use for games in mathematics lessons? (1998:284). Adendorf and Van Heerden
(2001:58) argue for the inclusion of the ?everyday situations? in the teaching of functions.
Their particular interest was in relation to the function y = ax2 + c in everyday settings.
They argue that this function can be used to describe number patterns on AIDS, the
number of AIDS victims, car accidents and netball. The basis of their argument is that the
use of the everyday situations ?could be used to change apathy within students, stimulate
interest and allow them to see a much clearer picture as to why they need to know
something about the parabola? (2001:58).
Cautious incorporation of the everyday: Mogari (2001) and Vithal (2003) engaged
learners in activities which required hands-on participation. They both caution against the
way in which such activities may spark gender issues in the classroom. Mogari?s study
involved a hands-on design of a chassis for a wire car by Grade nine pupils. The
mathematical value of this exercise was that ?constructing the chassis of a wire car can be
used to teach properties of a rectangle? (2001:205). However, he also observed that there
seemed to be a belief amongst the pupils that this activity was meant for boys. In
explaining this observation, Mogari asserts that boys perceived their female counterparts
as inferior and unable to carry out rigorous and rough assignments such as constructing
wire car artifacts. Vithal also explored what happens in a classroom when an attempt is
made to realize what may be called a social, cultural, political approach to the teaching of
23
mathematics. In her study, two Grade six learners at different schools participated in two
different projects. One class participated in a project titled the ?redesigning of an
agricultural garden? whilst the other class participated in a project titled ??fence
building?. In both activities, Vithal observed that the participation of girls was limited.
She thus cautions that summoning activities which connect with learners? lived
experiences, ? teachers have to also deal with the ?non-mathematical? aspects that are
carried in the task, content or learning material. This means that issues, such as gender,
need to be addressed in the mathematics classroom? (2003: 372).
Mogari and Vithal made their observations in different contexts, both of which involved
hands-on activities. It could be argued that gender issues arose not only due cultural
beliefs about the roles of boys and girls but because the activities themselves resonated
more with the boys? experiences and perhaps this disadvantaged or limited girls?
participation. This argument notwithstanding, the caution regarding being sensitive to
gender issues seems to be valid.
Nyabanyaba also highlights the need to be aware of instructional challenges that may
arise due to bringing in the everyday in the mathematics classroom. He explored the way
in which a context to which learners were familiar could be engaged in a mathematics
classroom. The relevant context he recruited was a ?real? soccer log table. The substance
of the task was for learners to predict, on the basis of the log standings, which soccer
team would ultimately win the league. The exercise had the potential to elicit non-
mathematical responses and arguments since soccer is the most popular sport amongst
South Africans. The teachers indicated that they would not accept answers which are
based on learners? everyday knowledge of South African soccer teams. However, ? there
was no clear indication from this study that teachers were aware of how they could assist
their students across difficulties created by the use of relevant everyday contexts in
school mathematics? (Nyabanyaba, 1999:24). Thus, Nyabanyaba concludes that for the
new curriculum (Curriculum 2005), there should be a move from merely advocating for
relevance to practically engaging teachers with the tensions of summoning the everyday.
24
Moodley?s (1992) study was more on the learners? responses to word problems. Earlier
on page 13 I elaborated on the way in which some researchers cautioned against the use
of word problems. Moodley also observed that the use of word problems may not
provoke reflection about the everyday amongst learners. Moodley (1992) shared his
experience about a learner David, who swiftly completed eight word problems. ?When
asked how he managed to read (so quickly) the whole page of word problems,? Moodley
outlines, ?he admitted he hadn?t, but had simply looked for the larger of the two numbers
in each problem and subtracted the other? (1992: 06). The substance of Moodley?s
argument is that some learners treat the everyday incorporated as a smokescreen which
they need not pay attention to. He however, suggests that relentless attempts should be
made to ?help them [learners] to learn and enjoy mathematics-through making their
experiences painless, meaningful and exciting? (Moodley, 1992: 14).
For learners and teachers, these studies suggest, summoning the everyday poses new
challenges.
Arguments against the incorporation of the everyday: Though at a theoretical level,
Muller and Taylor (1985) provide an account of limitations associated with ?border
crossing? and a ?generally educated interdisciplinary man?. Drawing on theoretical
constructs by Durkheim, Bernstein and Bourdeou, they highlight two points. Firstly they
acknowledge that for apartheid education, ? writing of syllabi and textbooks was tightly
controlled within white education departments and all interested actors outside the ruling
party were excluded from education? Secondly, they note that the post-apartheid
mathematics curriculum is guided by a hybridist model, which aims to flatten the
boundary between the everyday and the mathematics. They argue that this hybridist
model achieves a condition of ?false equality? between domains and participants and thus
will not fare better than the discriminatory elite curriculum it is designed to replace.
Taylor (1999) echoes this point again with a focus on mathematics: While admiring the
political intentions of the radicals [of wishing away the boundaries], our contention is that
the effects of this approach will be exactly the opposite of what is intended?. The crux of
Muller and Taylor (1985) and Taylor?s (1999) arguments is that far from there being
25
benefit to be derived by removing the boundaries, there are increased dangers or threats
to equity.
Murray (1992) and De Jager (1996) also argue for the maintenance of boundaries
between certain content areas (negative numbers and probability) and the everyday.
Murray cautions against the use of concrete or semi-concrete contexts such as debt to
introduce negative numbers. She observes that the need for negative numbers did not
arise out from physical context but from mathematics itself, especially in order to
?determine solutions to equations such as x + 3 = 0? (1992:287). She particularly
discourages the analogy drawn between ?film of a train going backwards is played
backwards so the train moves forward? and the operation negative times negative is
positive. De Jager (1996:127) also finds it nonsensical to use experiments for determining
the probability that ?if an event can happen in n ways and p of these are favourable, the
probability that the result will be favourable is p/n?. He explains this statement in less
abstract terms, ?If you spin a fair coin 10 times and it is heads 8 times, it means that it
was 8 times out of 10 in that case, and that is all it means.? De Jagers?s view is that
experiments are not meant to be the bases for deducing theory but to test predictions.
Therefore the everyday exercise of spinning a coin will have no influence on the already
established ratio or probability for a particular event.
In sum, the nature of studies cited in this section differed; some were based on empirical
observations whilst others were mainly theoretical. As varied as they are, a selection of
literature elaborated above suggests that the effect of the everyday on mathematics yields
no single answer. Some studies suggest the everyday will enhance the learning and
teaching of mathematics, other studies view the everyday as an obstacle whilst others
advise for the inclusion of the everyday with caution.
What is constant across all these theoretical and empirical studies, however, is that
learners? perspectives have had little influence in informing the debates on the effect of
the everyday on mathematics. Will the effect of the everyday on mathematics, from the
learners? perspectives be as varied? This study aims to describe, analyse and explain
26
South African based data through which this question may be engaged. In the next
section I outline how the chapters in this study will be organized.
1.4 CONCLUSION
In this chapter I have illustrated how the everyday has entered the curriculum at the level
of education policy development in South Africa,. I have also reflected, through the
literature reviewed, on debates around the consequences of including the everyday in
mathematics. How the incorporation of the everyday shapes both the mathematics and its
learning is a contested terrain pregnant with varied studies, findings, discussions and
viewpoints.
This study is motivated by what is silenced in both the policy discussions in South Africa
and debates within the mathematics education field cited: The learners? perspectives. It is
a study which intends to add to and illuminate the mathematics-everyday discussions
from the learners? perspective.
In the following section I highlight how the chapters in this study are organized.
1.5 ORGANISATION OF THE STUDY
The argument in this study is divided into three main sections. The first section (Chapter
1 to Chapter 3) is an orientation to the theoretical and methodological aspects of the
study. The second section (Chapter 4 ? Chapter 6) provides a detailed account of
experiences at one school, Umhlanga, and the third section (Chapter 7 to Chapter 9)
focuses attention on another school, Settlers.
In Chapter 2, I start by sketching and reflecting on a theoretical framework. I define the
theoretical framework which enables me to select and focus on particular aspects of my
experience, at each of the schools, as theoretical framework a priori. Such a framework
drew mainly from Bernstein?s code theory (1996), particularly the notions of framing and
27
classification. This framework, however, did not offer sufficient language to describe
data collected. I therefore complemented it with Dowling?s (1998) notions of describing
mathematics texts (expressive, esoteric, public and descriptive domains). I introduced
additional concepts to describe the different ways in which the everyday could be viewed
and called this extended theory a posteriori.
This study forms part of a larger study, named the Learners? Perspective Study (LPS) for
which data collection was a collaborative effort with other researchers. In Chapter 3 I
outline how I negotiate my research interest within the constraints of this larger study.
Since the empirical base for this study entails thick descriptions of classroom events and
learners? reflections on aspects of classroom events, Chapter 3 discusses the implications
and assumptions of collecting data in this way.
Chapter 4 introduces and describes one of the two schools, Umhlanga. Detailed attention
is drawn towards the broader aspects of the school: the physical structure and
organization of the school, all the nine lessons observed and the worksheets used during
these lessons. Even though it is the classroom events of some of these lessons which are
of direct interest to the study, the broader perspective highlights the ways in which these
lessons fit within the school?s events. Chapter 5 narrows down the discussion to lessons
which incorporated the everyday. Within a maze of classroom events in each of these
lessons, I pay more attention to the way in which the teacher introduces or privileges the
everyday aspects incorporated in a text and how a group of learners (different group each
day) engage the worksheets. It is in this chapter that the limitation of Bernstein?s
concepts of weak classification in relation to the incorporation of the everyday in
mathematics begins to surface. I therefore introduce and argue for the use of notions such
as authentic/inauthentic and close/far (which I will elaborate on in the next chapter). In
Chapter 6, I explore learners? views on what they thought each of the lessons entailed,
whether they welcomed the use of the everyday in mathematics and what the role of the
everyday in the lesson was. I try to explain the learners? views from drawing on the
broader context in which they experienced the everyday, including the teachers?
presentation of the lessons and views on the role of the everyday.
28
Chapters 7 to 9 parallel Chapters 4 to 6. In this way, I am able to capture and make sense
of the different ways in which learners experienced their lessons. In Chapter 7 I introduce
and describe another school, Settlers. This school is significantly different from
Umhlanga. I describe the physical structure of the school, all the fourteen lessons
observed and the worksheets used. It becomes apparent during the analysis, that the
everyday used in Umhlanga and the everyday recruited in Settlers are entirely different
from each other. The authentic/inauthentic and close/far notions introduced developed in
Chapter 4 cannot be employed unproblematically in Settlers? lessons and worksheets. I
thus complemented my theoretical constructs by appealing to Dowling?s notions of text
descriptions: esoteric, expressive, public and descriptive. Chapter 8 zooms in on lessons
in which the everyday was recruited. I focused on how the teacher balanced the
everyday-mathematics aspects and the way in which learners at this school engaged the
worksheet. Given that the everyday incorporated in the two schools differ, I try to draw
parallels on how the teachers introduce the everyday and the way in which learners
engage activities which incorporate the everyday. Chapter 9 explores Settlers learners?
reflections about the lessons, whether they welcomed the use of the everyday in
mathematics and the role of the everyday in mathematics. Similar to Chapter 6, I reflect
on and explain these experiences against the broader context in which learners
experienced the lessons.
Chapter 10 develops and makes claims on the basis experiences drawn from the two
schools. I use the word ?develop? to suggest that these claims are founded on systematic
observations, though they are themselves subject to interrogation. I draw these claims on
the basis of both methodological and theoretical aspects of the study. I also reflect on the
study as whole; outline the possible weaknesses and suggest alternative ways of
investigating the same phenomena.
29