Mathematics as a Social Construct

Pythagoras, 68, 3-14 (December 2008) 3

Mathematics as a Social Construct:
Teaching Mathematics in Context

Hayley Barnes and Elsie Venter

University of Pretoria
hayley.barnes@up.ac.za and elsie.venter@up.ac.za

Why is teaching in context an important option to consider in the teaching of
mathematics? What does it mean to teach mathematics from and in contexts? And what
are the possible challenges associated with this practice? The aim of this paper is not to
provide a comprehensive answer or solution to these questions. We attempt rather to
address these questions specifically with regard to South Africa and the theory of Realistic
Mathematics Education. In this article we consider a vignette of a more formal and
traditional mathematics lesson and then suggest possible reasons why we need to be
teaching more in context. Furthermore we discuss the application of the theory of Realistic
Mathematics Education as a potential approach to facilitate teaching in context. Finally we
present some challenges associated with this practice.

The philosophical shift that has occurred within the
domain of mathematics has brought with it a wave
of reform in mathematics education. The former
absolutist paradigm that dominated undermined the
social responsibility of mathematics in human
affairs such as value, wealth and power (Ernest,
1991). The shift has challenged the infallibility of
mathematics and acknowledged it as a product of
human inventiveness (Davis & Hersh, 1980) and a
human activity (Freudenthal, 1973), thus making it
a social construct. While the reform propagating
the teaching of mathematics as a social construct is
a positive move, along with the introduction of
mathematical literacy in many countries, how can
we effectively implement this reform? This paper
seeks to examine the area of teaching mathematics
in context (specifically in relation to the theory of
Realistic Mathematics Education) to support this
reform and the implementation thereof in school
and tertiary education.

A vignette of a traditional formal mathematics
lesson is first simulated, followed by a discussion
of mathematics as a social construct. The relevance
of teaching in context is then explored while
alluding to the theory of Realistic Mathematics
Education as a vehicle through which this can be
done. Examples of studies conducted at secondary
and tertiary level are considered, concluding with
some challenges of teaching in context.

A lesson simulation
The following vignette of a lesson will probably be
familiar to many readers:

Today’s lesson is going to look at rounding off
decimals to the nearest whole number. Let us first
revise the concept of rounding off. Remember that
rounding off helps us in estimation. Last year you
learnt to round off to the nearest ten, hundred and
thousand. You did this by looking at which ten,
hundred or thousand the number you are rounding
off is closest to. Let’s look at an example. Write
this down in your books if you have forgotten how
to round off:

Example 1: Round 63 off to the nearest ten.
On the number line 63 lies between 60 and 70. It
lies closer to 60 though so we round it off to 60, as
the nearest ten. Remember also that in this case, if
the units digit is less than the number 5, we round
down to the nearest ten.

Example 2: Round 2 499 off to:
a) the nearest ten
b) the nearest hundred
c) the nearest thousand

a) Let us start by looking at the units digit. It is
more than 5 so we round up to the nearest ten.
The answer is therefore: 2 500

Mathematics as a Social Construct: Teaching Mathematics in Context

b) Now we need to look at the tens digit because
we are rounding off to the nearest hundred. It is
also more than 5 so we round up to the next
hundred. Answer: 2 500

c) For rounding off to the nearest thousand we
look at the hundreds digit. It is less than 5 so we
round down to the nearest thousand.
Answer: 2 000

Now let us do one with decimals.

Example 3: Round 4,25 off to the nearest unit or
whole number.
4,25 lies on the number line between the
units/whole numbers 4 and 5. It is closer to 4
though. Also when rounding off the nearest whole
number, we look at the tenths digit. It is less than 5
so we round down to the nearest whole number
which is 4.

Does everyone understand? Are there any
questions?

Now let us use what we have learnt about rounding
off to help us solve some problems:

Please solve the following:

1. 17 ÷ 4 = ? Round your answer off to the

nearest whole number.

2. Mr Farmer decides to share his 17 cows evenly
between his 4 children. To avoid conflict, each
child must receive the same number of cows.
How many cows will each child get?

If you have the class work correct, please continue
with the homework which is page 23 of your
textbook, numbers 1 – 8.

In the approach applied in the lesson vignette
above, mathematics is viewed as a ready-made
system with general applicability. Consequently,
mathematics instruction is seen as a process of
breaking up formal mathematical knowledge into
learning procedures and then learning to use them
accordingly.

Let us examine another similar problem to
Example 3 above:

The problem: 17 people are trapped on a
mountain and need to be rescued by helicopter.
The helicopter can take a maximum of 4
passengers at a time, in addition to the pilot. How
many trips will the helicopter need to make?

This example illustrates where conventional
mathematics as we perform it outside of any
prescribed context, can actually support or conflict
with the answer depending on the context of the
problem. In solving the problem above, it is the
context that must take preference over the
mathematical convention of rounding off “down”
to the nearest whole number when our indicator is
below 5. But when we teach mathematics
predominantly formally and outside of context, do
our students learn to know the difference between
conventional mathematics and mathematics as a
tool operating within a social context?

Mathematics as a social construct
Formerly (prior to the 20th century) an absolutist
view of mathematical knowledge dominated the
philosophy of mathematics education. According
to Ernest (1991), this view accepts that
mathematics consists of absolute and
unchallengeable truths that can be regarded as
certain knowledge based on two types of
assumptions in terms of the actual mathematics
(axioms and definitions) and logic (axioms, rules
of inference and the formal language and its
syntax).

Early in the twentieth century a number of
antinomies and contradictions, mainly in the theory
of sets and functions, were derived in mathematics
(Kline, 1980; Kneebone, 1963; Wilder, 1965 as
cited in Ernest, 1991), which caused a crisis within
this absolutist paradigm. The certainty of
mathematics and its theorems was challenged by
the appearance of these contradictions (i.e.,
falsehoods) resulting in the development of a
number of schools in the philosophy of
mathematics. These schools aimed to account for
the nature of mathematical knowledge and to re-
establish the certainty thereof. Ernest (1991)
identifies the three major schools as being
logicism, formalism and constructivism. Without
elaborating on each school, it suffices to say that
the former absolutist paradigm that dominated
previously, undermined the social responsibility of
mathematics in human affairs such as value, wealth
and power (Ernest, 1991). The shift has challenged
the infallibility of mathematics and acknowledged
it as a product of human inventiveness (Davis &
Hersh, 1980) and a human activity (Freudenthal,
1973), thus making it a social construct. Ernest
(1991, p. 42) specifies the grounds for describing
mathematical knowledge as a social system:

Hayley Barnes & Elsie Venter

1. The basis of mathematical knowledge is
linguistic knowledge, conventions and rules,
and language is a social construction.

2. Interpersonal social processes are required to
turn an individual’s subjective mathematical
knowledge, after publication, into accepted
objective mathematical knowledge.

3. Objectivity itself will be understood to be
social.

It therefore draws on conventionalism, in accepting
that human language, rules and agreement play a
central role, including the view that mathematical
concepts develop and change. It also includes
Lakatos’ philosophical thesis that mathematical
knowledge grows through conjectures and
refutations. The above therefore constitutes the
perspective from which the following questions are
discussed in this paper:

• Why is teaching in context an important option
to consider in the teaching of mathematics?

• What does it mean to teach mathematics from
and in context?

• What are the possible challenges associated
with this practice?

The relevance of teaching in context
Within the scope of this paper, two main answers
to the first question are proposed. The first stems
directly out of the shift in mathematics education
already discussed from an absolutist paradigm to a
more social constructivist view. This shift to
emphasizing mathematics as a social construct is
certainly supported by and demonstrated in the
following definition of mathematics provided in
the Revised National Curriculum Statement
(RNCS) by the Department of Education (DoE,
2002, p. 4):

Mathematics is a human activity that involves
observing, representing and investigating
patterns and quantitative relationships in
physical and social phenomena and between
mathematical objects themselves. Through this
process, new mathematical ideas and insights
are developed. Mathematics uses its own
specialised language that involves symbols and
notations for describing numerical, geometric
and graphical relationships. Mathematical ideas
and concepts build on one another to create a
coherent structure. Mathematics is a product of
investigation by different cultures – a
purposeful activity in the context of social,
political and economic goals and constraints.

To fully embrace the extent of this definition and
to realise the goals intended by the introduction of
Mathematical Literacy as a compulsory subject, we
need to consider teaching mathematics from and in
context rather than in its absolute form for the
purpose of later applying in contexts.

The second response to the first question lies
within the results of both national as well as
international studies carried out in mathematics
education. In the systemic evaluation, South
Africa’s Grade 3 and Grade 6 learners performed
poorly on the national average. In the 1999 and
2003 results of the Trends in International
Mathematics and Science Study (TIMSS, 2003)
South Africa once again was placed last out of the
fifty countries that participated. Although a
detailed discussion and analysis of factors causing
such overall poor performance is beyond the scope
of this paper (see Howie, 2001; 2002), it can be
concluded that South African learners certainly
struggle far more than the rest of the world when
required to perform mathematics within a context.

Figures 1 shows examples of selected contextual
items from TIMSS 2003. These items were
identified (as examples not as conclusive proof)
according to their applicability to contexts from
everyday life that one might encounter. The
performance of South African Grade 8 learners on
the items in comparison to the international
average and that of selected countries are also
provided.

Teaching in and from context - The theory
of Realistic Mathematics Education
In order to examine the second question, we draw
on the work currently being done by the
Freudenthal Institute in The Netherlands. They
have been leaders in introducing and researching
teaching mathematics in and from contexts in their
theory known as Realistic Mathematics Education
(RME). Realistic Mathematics Education has its
roots in Freudenthal’s interpretation of
mathematics as a human activity (Freudenthal,
1973; Gravemeijer, 1994). To this end,
Freudenthal accentuated that the actual activity of
doing mathematics; an activity which he
propagated, should predominantly consist of
organising or mathematising subject matter taken
from reality. Learners should therefore learn
mathematics by mathematising subject matter from
real contexts and their own mathematical activity
rather than from the traditional view of presenting

Mathematics as a Social Construct: Teaching Mathematics in Context

Item 1
A garden has 14 rows. Each row has 20 plants. The gardener then plants 6 more rows with 20 plants in each
row.

How many plants are now there altogether?

Answer: _________________________

Item 2
A car has a fuel tank that holds 45  of fuel. The car consumes 8,5  of fuel for each 100 km driven. A trip
of 350 km was started with a full tank of fuel. How much fuel remained in the tank at the end of the trip?
A. 15,25 
B. 16,25 
C. 24,75 
D. 29,75 

Item 3
The graph represents the distance and time of a hike taken by Joshua and Liam.

If they both started from the same place and walked in the same direction, at what time did they meet?
A. 8:00
B. 8:30
C. 9:00
D. 10:00
E. 11:00

Performance on the items

Int. Ave RSA Tunisia Morocco Egypt Netherlands NZ USA
Item 1 61,2 17,7 48,4 38,5 36,1 88,1 76,5 78,4
Item 2 26,0 18,0 18,7 18,9 20,6 37,9 23,6 24,8
Item 3 62,4 19,3 14,5 - 47,3 81,2 72,3 79,7

Figure 1: Items from TIMSS (Source: TIMSS, 2003)

Hayley Barnes & Elsie Venter

mathematics to them as a ready-made system with
general applicability (Gravemeijer, 1994). These
real situations can include contextual problems or
mathematically authentic contexts for learners
where they experience the problem presented as
relevant and real.

The verb mathematising or its noun
mathematisation implies activities in which one
engages for the purposes of generality, certainty,
exactness and brevity (Gravemeijer, Cobb, Bowers
& Whiteneack, as cited in Rasmussen & King,
2000). Through a process of progressive
mathematisation, learners are given the opportunity
to reinvent mathematical insights, knowledge and
procedures. In doing so learners go through stages
referred to in RME as horizontal and then vertical
mathematisation (see Figure 2). Horizontal
mathematisation is when learners use their
informal strategies to describe and solve a
contextual problem and vertical mathematisation
occurs when the learners’ informal strategies lead
them to solve the problem using mathematical
language or to find a suitable algorithm (Treffers,
1987). For example, in what we would typically
refer to as a “word sum”, the process of extracting
the important information required and using an
informal strategy such as trial and error to solve
the problem, would be the horizontal
mathematising. Translating the problem into
mathematical language through using symbols and
later progressing to selecting an algorithm such as
an equation could be considered vertical
mathematisation, as it involves working with the
problem on different levels.

The traditional formal and authoritarian approach
to teaching mathematics that has dominated in
South African classrooms for a number of years
has not afforded learners many opportunities to
make use of horizontal mathematisation.

Mathematics lessons are often presented in such a
way that the learners are introduced to the
mathematical language relevant to a particular
section of work and then shown a few examples of
using the correct algorithms to solve problems
pertaining to the topic before being given an
exercise of worksheet to complete (Venter, Barnes,
Howie, & Janse van Vuuren, 2004). The exercises
or worksheets are usually intended to allow
learners to put the algorithms they have been
taught into practice and may even contain some
contextual problems that require the use of these
algorithms. According to the RME model depicted
in Figure 4, this type of approach places learners
immediately in the more formal vertical
mathematisation process. The danger in this is that
when learners have entered that process without first
having gone through a process of horizontal
mathematisation, a strong possibility exists that if
they forget the algorithms they were taught, they do
not have a strategy in place to assist them in solving
the problem. Low attainers often exhibit this lack of
strategy. This experience can be equated to
someone being shown and told what is on the other
side of a river and being expected to use what is
there for their own benefit. However, they are not
given or shown the bridge that assists one in
crossing to the other side in order to make proper
use of what is there. The horizontal mathematisation
process provides this bridge (Barnes, 2004).

When embarking on solving a contextual problem
using formal mathematical knowledge, the
following steps are usually followed. First the
problem needs to be translated from its contextual
state into mathematical terms. Available
mathematical means are then drawn on in order to
solve the problem, which then needs to be
translated back into the original context. This
process can be illustrated by the example we
encountered earlier on:

Horizontal mathematisation ( ); Vertical mathematisation ( )

Figure 2: Horizontal and vertical mathematisation Source: Adapted from Gravemeijer, 1994.

Contextual
problem

Mathematical language

Describing
Solving

Algorithm

Mathematics as a Social Construct: Teaching Mathematics in Context

The problem:
17 people are trapped on a mountain and need to
be rescued by helicopter. The helicopter can take a
maximum of four passengers at a time, in addition
to the pilot. How many trips will the helicopter
need to make?

Translation into mathematical terms:
17 people / 4 per trip

Mathematical means to solve problem:

17 ÷ 4 = 4 remainder 1 or
4
1

4 or 4.25

Translation back into original context:
Helicopter will need to make 5 trips

On the other hand, in the RME problem-centred
approach, the problem, rather than the use of a
specific mathematical tool, is the actual aim.
Instead of trying to formalise the problem into
mathematical terms, the learners are encouraged to
describe the problem in a way that makes sense to
them. This can involve using their own self-
invented symbols or pictures and identifying the
central relations in the problem situation. In this
way the problem is also simplified for the learner.
Because the symbols are meaningful for the
problem-solver, further translation and
interpretation of the problem is easier and using a
standard procedure is not mandatory. In Boxes 1
and 2 below examples of contextual problems
given to Grade 8 learners in a study using the
theory of RME is presented (Barnes, 2004). Some
responses from learners who took part in the study
are then offered as a means to demonstrate the
concept of mathematisation.

The above-mentioned study took the form of a case
study of 12 participants in an urban school in
South Africa. The learners were identified by their
mathematics teachers as low attainers in
mathematics. They were each part of a remediation
programme in mathematics that these learners took
part in while the rest of their class were enrolled
for the subject of a third language. The participants
were part of a mathematics intervention for two
lessons per week for approximately three school
terms. The intervention was designed and taught
based on the principles of Realistic Mathematics
Education. In summary the theory of RME proved
to be viable tool in teaching mathematics to low
attainers (Barnes, 2004).

Box 1
Lesson on the cat’s pills
My cat’s recent diagnosis of diabetes initiated this
problem, which served as an introductory contextual
problem in revisiting the concept of fractions. A
discussion on what diabetes is and how it occurs in
humans and cats was first embarked on with learners as
an introduction (learners had little knowledge of
diabetes and were not easy to convince that cats also get
diabetes). Learners were then presented with and asked
to solve the following problem (either in groups or
individually; they were given the choice):
Problem:
My cat needs to take two types of pills and an insulin
injection twice a day to control its diabetes. The cat takes
half a big pill in the morning and again in the evening
and a quarter of a small pill also in the morning and
again in the evening. Firstly, the vet has given me 17 big
pills and 27 small pills to start off with, how many days
will these pills last me for before I have to go back to get
more? Secondly, how many of each pill should I buy each
month so that they last me for a whole month?

About ten minutes before the end of the lesson, some
learners were asked to demonstrate and explain their
solutions to the class and a short whole class discussion
on these explanations was held before the class was
dismissed.

The solutions offered by Liya to the contextual
problem provided in Box 1 are shown in Figures 3
and 4 to exemplify this (RME) process of problem
solving. There are two parts to the solution as Liya
first tried Part A (Figure 3) and then realised from
her answers that something was not right and then
proceeded to do the solution in Figure 4. It is also
interesting to note how her Part A solution more
closely resembles the steps usually followed when
using formal mathematical knowledge in an
information processing approach.

As these learners were probably more accustomed
to using the more formal approach, and obviously
had some formal knowledge in place regarding
fractions, they initially often tried to go through the
steps of translation into mathematical terms,
searching for an adequate mathematical procedure
to solve the problem and then translating it back to
the context. In doing so though, it was noticed that
some learners did not really have a grip on which
mathematical procedure to use and even when they
chose the correct one (sometimes by chance as
they could not justify their decisions), they made
mistakes in executing them (as can be seen in Part
A of Liya’s solution).

Hayley Barnes & Elsie Venter

During the course of the study (Barnes, 2004) an
effort was continually made to encourage learners
to go through the RME approach of simplifying the
contextual problem by first representing it in their
own symbols and/or words and then further
solving and interpreting it from there. When some
of them started to do this, they found that they
could more often solve the problem, using their
informal strategies rather than formal procedures
they were unsure of. This does not mean that they
never used formal procedures or any mathematical
means but that they were expected to only use
them at a point in the problem-solving process
when they could justify the use thereof and
demonstrate an understanding of the application in
that regard.

Box 2
You decide to start making banana bread to sell in
order to earn some extra money. To start off with, you
decide to make 5 loaves of banana bread. According to

the recipe, each loaf requires 4
1
2

bananas. How many

bananas will you need to make the 5 loaves of banana
bread? Show your working out in the space provided
below.

As can be seen in Zwanela’s solution (Figure 5),
she correctly selected multiplication as her
strategy. However, when she executed the actual
multiplication, she tried to change the mixed
number into an improper fraction and in doing so

“lost” the denominator and got 9 instead of
9
2

rendering her final answer incorrect. Zwanela often
resorted immediately to vertical mathematisation
in that she searched for the “correct” formal
procedure to apply. In contrast to Zwanela’s more
formal solution, the use of horizontal
mathematisation is more evident in the solution
from Violet (Figure 6).

Gravemeijer (1994) explains that by getting
learners to solve a sequence of similar problems,
another process is induced. The problem
descriptions develop into an informal language,
which is further simplified and formalised into a
more formal mathematical language eventually. A
similar process could be experienced in terms of
the solving procedure, where solving similar kinds
of problems becomes routine and actual algorithms
take shape. Through this learning process, formal
mathematical knowledge itself can be constructed.

Figure 3: Part A of Liya’s solution to the cat pills problem

Figure 4: Part B – Liya’s second attempt at solving the cat pills problem

Mathematics as a Social Construct: Teaching Mathematics in Context

RME at tertiary level
International studies (Doorman, 2001; Streefland,
1991; Treffers, 1987; Vos, 2002), including studies
from developing countries such as Indonesia
(Armanto, 2002; Fauzan, 2002) have shown that
the RME theory is a promising direction to
improve and enhance learners’ understanding in
mathematics in primary and secondary schools. So
how can the theory of RME be employed in
tertiary education?

Kwon (2002) provides one example of how an
RME design could be used for teaching an
introductory course in differential equations at
first year university level. The teaching
experiment was conducted with a group of 43
students and data consisted of videotapes, field
notes, copies of students’ work and records of
instructional activities and decisions. Materials
implemented during the course were informed by
RME instructional heuristic and designed to assist
students to complete reinvention activities
through devising their own ways of working
through mathematical concepts (also known as
the inquiry-oriented approach). An explicit
intention of Kwon’s project was to create a
learning environment where students routinely
offered explanations of and justifications for their

explanations. A typical class period entailed
students working in groups of two to four on a task
presented by the instructor. Cycles of whole class
discussions and further group work then followed.
A continuous emphasis on reasoning and whole
class discussions resulted in key concepts
emerging.

In a later evaluation study (Rasmussen, Kwon,
Allen, Marrongelle & Burtch, 2004 as cited in
Kwon, 2005) two groups of students from four
undergraduate institutions in Korea and the USA
were compared in terms of their understanding of
central ideas and analytical methods relating to
differential equations. One group was exposed to
an inquiry-orientated approach based on the theory
of RME. The other group, however, was taught in
the traditional lecture-based manner, where a
typical class consists of a review of previous work,
presentation of new work and some time for
students to work on their own or ask for assistance
(Romberg, 2001). There was no significant
difference between the two groups on routine
problems. However, the inquiry-orientated group
did score significantly higher than the comparison
(traditional lecture-based) group on the conceptual
problems.

Figure 5: Zwanela’s solution to the banana bread problem

Figure 6: Violet’s solution to the banana bread problem

Hayley Barnes & Elsie Venter

Even after a delayed post-test a year later, Kwon
(2005) inferred that retention rates on procedural
tasks showed no significant difference, but that
retention on conceptual tasks were significantly
different between the groups. Once again the
inquiry-orientated group scored significantly higher
than the traditional group on the conceptually
oriented items. In addition, Kwon found that all the
inquiry-oriented students out-performed the
traditional group regardless of academic background
or gender. This finding has important implications
for the South African context when transition from
school level practices to that of tertiary levels is
made, especially when the quality of diverse school
experiences are considered.

As also demonstrated by Skemp’s (1971)
distinguished between relational and instructional
understanding, a chasm may exist between what
students are able to do and what they in fact
understand. Knowing what to do in a specific
situation, but not necessarily understanding why it
works, may limit the transfer of that procedure or
skill. The increasing number of procedures that
students need to commit to memory in mathematics
often results in learners in secondary school and
students at tertiary level becoming confused or
partly remembering and trying to apply procedures
they have never fully understood (Daniels &
Anghileri, 1995). Understanding on the other hand
promotes remembering and enhances transfer owing
to the reduced number of bits of knowledge that
need to be simultaneously held in the short-term
memory (Hiebert & Carpenter, 1992). The
understanding that comes from making connections,
seeing how things fit together, relating mathematics
to real situations and articulating patterns and
relationships also carries with it a satisfaction which
can further motivate students (Haylock, 1991).
Exposure to constructing mathematics at an early
level could promote the construction of mathematics
at higher levels and generating of new knowledge to
the domain. Compared to the more formal approach
of teaching mathematics we have become
accustomed to, teaching in context appears to be a
vehicle through which our would-be
mathematicians can express and develop themselves
mathematically and thus enrich the South African
community in terms of financial, economical and
scientific models for living.

Some challenges
Although a necessary and positive shift, teaching
in context does not come without its challenges. In
a heterogeneous society such as in South Africa

where we have eleven official languages, the issue
of language comes to the fore as a potential hurdle.
Research pertaining to proficiency in language as a
factor in the learning of mathematics is certainly
available and still currently being conducted
(Howie, 2002; Setati, 2002). Conclusive evidence
of the impact of the Language of Learning and
Teaching not being in the mother-tongue is not yet
available though and remains a controversial issue.
When teaching in context, it is obviously important
that the language of the problem be accessible to
the students. In our situation visual aids, such as
graphs, tables and diagrams can assist students in
this regard. Employing simple language that is not
ambiguous is also necessary.

Also relating to language is the actual selection of
the context from which the development of certain
mathematics is to evolve. The context of the
problem needs to be accessible within the
framework of the student so that they are able to
set about solving the problem, rather than getting
lost or caught up in the context. Kwon (2005) also
makes the point that the tasks need to be carefully
selected or designed as learning is not necessarily
implied through solving a sequence of problems.
This in turn calls for a high level of subject matter
knowledge and understanding of the substantive
and syntactic structures of the discipline of
mathematics (Grossman, Wilson & Shulman,
1990) for both designers and teachers.

Linking to this is the issue of material
development. As Bowie and Frith (2006)
discovered in their process of developing
mathematical literacy materials, when one attempts
to mathematise a context, it is necessary to have a
good understanding, not only of mathematics, but
also of the context. Mathematical literacy teachers
in South Africa already therefore need to
familiarise themselves with contexts such as
“voting systems, mortgages, retirement funding,
HIV/AIDS, global positioning systems and socially
responsible trade (to name but a few of the
contexts suggested in the current curriculum)”
(Bowie & Frith, 2006, p.33). The learners and
students themselves also need to be able to
understand the context and in our diverse country
we cannot, for example, assume that all learners
are familiar with the notion of a formal banking
system or the concept of risk and return on
investments. In order to successfully implement
RME, this issue of appropriate contexts will also
need to be resolved. Bowie and Frith (2006)
suggest integration of knowledge and skills across

Mathematics as a Social Construct: Teaching Mathematics in Context

subjects and terrains of practice as a possible
solution in addressing this issue. They stress that
this integration needs to be worked in at a
curriculum level in order for it to be taken
seriously. But our teachers are under enough
pressure in coping with the constant curriculum
changes. It is our opinion that we need to start
supplying teachers with materials that enable them
to teach using an RME approach.

At school level a number of learning materials
based on the theory of RME are already available
in The Netherlands, the United States and the
United Kingdom. We could consider re-
contextualising these for a South African context.
At tertiary level, not many such materials have
been developed. However, a strong developmental
research partnership between mathematicians and
mathematics educators could facilitate such a
process. Wittmann (1998 as cited in Julie, 2006)
identifies the central task of Mathematics
Education, as a field of study, as developing
learning resources for productive and meaningful
learning. He argues that teachers do not have the
time to do this and that one of the main roles of
mathematics educators is to therefore carry out the
didactical analysis of subject matter in designing
the resources. This is done through “theorisation
and thought-experimentation which leads to
hypothetical learning trajectories” (Julie, 2006:
64). These hypothesised trajectories then need to
be trialled, researched and further developed
within real classroom situations in order to
decrease the distance between the intended and
implemented curriculum (Julie, 2006). The RME
materials mentioned above have been developed in
such a manner within their respective countries
(see Gravemeijer, 1998; Gravemeijer & Cobb,
2002; Treffers & Goffree, 1985). A replication of a
similar process in South Africa would help us to
re-contextualise the materials for a South African
context as well as assist in capacity building where
classrooms become research domains and sites of
experimental implementation (Julie, 2006).

Finally, having discussed language, contexts and
material development, we cannot omit our main
challenge (and source of success) in implementing
a theory such as Realistic Mathematics Education,
namely teachers. Curriculum developers can
produce material in context to be used in
mathematics classrooms, but this does not
necessarily mean that teachers will implement
these as intended. For teaching mathematics in
context, a methodology change is required that

challenges the learners to become more independent
thinkers in order to become better problem-solvers
as well as mathematicians. The theory of RME
encourages an approach that treats each individual
student of mathematics as a mathematician with the
capacity to mathematise contexts into mathematical
problems that can be solved (Freudenthal, 1983).
Teachers who attempt to teach mathematics in
context through a more traditional approach of
giving an example and then expecting learners to
practise applying the tools to a range of contextual
problems have missed the point. It is therefore
necessary in training mathematics teachers to teach
in context, to develop the skills needed to be able to
relate mathematics and context (Brown & Schafer,
2006) in a more problem-solving or inquiry oriented
approach. An important skill that needs to be fore-
grounded in this regard is that of interpreting
mathematical concepts and skills in relation to a
context (Brown & Schafer, 2006). Having not learnt
mathematics at school or university through such an
approach, we cannot assume that our mathematics
teachers, even those proficient in mathematics, have
mastered this skill. Equipping teachers to teach in
context will therefore require extensive and
continued training. Academic institutions willing to
form partnerships with schools where mathematics
educators, mathematicians and mathematics
teachers could work together in developing,
implementing and researching an approach of
teaching in context, would go a long way to making
such a venture successful.

Conclusion
This paper has sought to examine the issue of
teaching mathematics in context as a social
construct, using the theory of Realistic
Mathematics Education. We are of the opinion that
our current mathematical practises are still not
foregrounding the shift from the absolutist
paradigm to social constructivism and adequately
empowering our learners. This has further
implications for them as students at tertiary
institutions as well as effective citizens within our
democracy. While teaching in and from context is
not without its challenges, especially within our
diverse society, it appears to hold potential as a
vehicle with which to address this problem, both at
school and tertiary level. Mathematics lecturers are
encouraged to reflect on their own practices in the
teaching of mathematics and to consider partnering
with mathematics educators (from schools and
tertiary institutions) to work on designing
classroom experiments that engender a culture of
mathematising amongst our students.

Hayley Barnes & Elsie Venter

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