S. PHILEMON NTENZA
INVESTIGATING FORMS OF CHILDREN’S WRITING IN GRADE 7
MATHEMATICS CLASSROOMS
ABSTRACT. Recent changes in mathematics curricula, both in South Africa and elsewhere,
have begun to change the overwhelmingly symbolic nature of mathematics in schools (in
the sense of use of mathematical symbolism), promoting more use of the oral and written
language. Engaging students in ‘Writing-to-Learn’ activities in mathematics classrooms
has been identified and claimed by various mathematics education researchers as having
a positive impact on the learning of mathematics. In this paper, I report on a piece of
research, which is part of a broader study, on forms of mathematical writing and written
texts produced by learners in grade 7 (12–13-year-olds) classes in six junior high schools
in KwaZulu-Natal, in South Africa.
KEY WORDS: communication, forms of writing, junior high school, learning mathematics,
mathematical writing, writing and assessment, writing-to-learn, written text
I NTRODUCTION
A cursory glance of the South African continuous assessment policy, emanating from the National Department of Education [DoE] (1997, 2000,
2002), shows that emphasis is now being placed on formative assessment
practices which include informal assessment and other activities such as
a written assignment, investigation, journal, tutorial (DoE, 2000, my emphasis). These various initiatives and ‘new’ assessment methods have implications for mathematical writing within mathematics classrooms. There
is now, certainly, a strong ‘demand’ for more writing by students – where
they have to write more sentences in English, literally, but mixed with
mathematical symbolism – such as in journal writing, in writing a report
of a project, and so on. Hence, because South Africa is a multilingual society, there is necessarily an interplay between language and the learning of
mathematics (cf. Adler, 2001; Ntenza, 2004b; Setati, 1998), and mathematical writing in the classroom. Nevertheless, this interest in communication,
using mathematics as a tool, is also a development that is indicated in
the international literature on mathematics education research and already
appears in curriculum documents in the late 1980s and 1990s, such as the
National Curriculum for England and Wales (Department for Education
[DFE], 1995), the National Council of Teachers for Mathematics (NCTM)
“Curriculum and Evaluation Standards” in the United States (1989), and
Educational Studies in Mathematics (2006) 61: 321–345
DOI: 10.1007/s10649-006-5891-0
C
Springer 2006
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the National Statement on Mathematics for Australian Schools (Australian
Educational Council, 1990). Considering the context of major curriculum
changes in South Africa, which are noted within the Revised National
Curriculum Statement (see, for example, DoE, 2002), it is interesting to
know,
What forms of mathematical writing do students produce in South
African junior high schools?
This question will be the main focus of the present paper. In an earlier piece of work (Ntenza, 2004a) I have presented some evidence on the
cognitive benefits of students’ writing in mathematics classrooms as perceived by the teachers, and I have, but to a limited extent, made reference
to ‘forms of writing’ that are likely to be observed in South African mathematics classrooms. In this paper I intend to present new evidence and to
discuss, amongst other findings, but rather in detail, the different types or
forms of writing produced by students. I will reflect on the implications of
the lack of teaching resources on such forms of writing. I will also argue
for a more systemic approach to curriculum implementation and its related
processes, in particular, within a context where there are limited resources
in the majority of schools.
As discussed elsewhere in this article, there is a lack of research on learners’ mathematical writing, including the experience and forms of writing
that are likely to be produced by learners in South African classrooms.
Research that informs teachers, and to a large extent, contributes positively
in terms of changing teachers’ classroom practices, is badly needed in the
South African context, particularly in view of the curriculum changes that
are taking place. The overall intention of my study is to provide research
evidence that will be informative and relevant to both mathematics teachers
and curriculum experts as far as mathematical writing is concerned, and
not only for the local context in South Africa but also for the international
mathematics education community.
Research on forms of writing in the United Kingdom
Before the introduction of the National Curriculum in 1989 major research
studies of the writing taking place in different subjects in secondary schools
found little writing in mathematics classrooms. Britton et al. (1975) conducted a research study in England and Wales, and were able to collect
2122 pieces of writing from 65 secondary schools by students in the first,
third, fifth and seventh years. The researchers, in collaboration with school
teachers, drew these pieces of writing from the students’ work in all subjects
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of the curriculum where extended writing was used. They were produced
in response to teacher assigned tasks.
For purposes of analysing the pieces of writing by students, Britton
et al. described three broad function categories for “mature” writing,
that is writing expected of 10–18-year-old students at secondary schools.
These were “transactional, expressive and poetic” (p. 88) categories
of writing. Transactional writing refers to fact recording, opinion exchanges, ideas explanation and exploration, and theory construction.
Expressive writing, could be described as ‘thinking aloud on paper’
and includes diary entries, personal letters to friends or relations, addresses to public audiences (including newspaper editorials, ‘interest’
articles in specialist journals, gossip columns, autobiographies). Lastly,
poetic writing is writing which uses words selected in such a manner
as to make an arrangement or a formal pattern (Britton et al., 1975).
Using these categories to analyse the pieces of writing Britton et al.
found that they were able to independently examine writing only in
English, history, geography, science, and religious education courses. In
mathematics, the written work was too limited for it to be analysed
separately.
In Scotland, Spencer et al. (1983) did a similar survey on written work
in Scottish secondary schools applying the same categories as Britton et al.
(1975). These researchers reported that a large number of pupils did produce
some written work in mathematics during the week of their survey, but in
72% of the pupils this work consisted in copying, and only 10% of the
pupils wrote anything in their own words. The latter had only written an
average of 6.2 lines of writing, none producing extended writing over a
page in length.
Morgan (1998) mentions that recent curriculum developments in the
United Kingdom related to the introduction of investigative work into
mathematics classrooms at the General Certificate for Secondary Education (GCSE) level, have begun to focus attention on students’ writing
in two ways, that is, “firstly, . . . in supporting reflection and the development of problem solving processes. Secondly, . . . in the production of
extended writing, in many cases for the first time” (p. 22). There has been
a conscious effort, in the United Kingdom, to overcome the phenomena
observed by the earlier studies of Britton et al. (1975) and Spencer et al.
(1983).
Research on forms of writing in the United States
In the United States, Davison and Pearce (1988b, 1990) conducted a survey
to determine the amount, kinds, and uses of writing in junior high school
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mathematics classrooms. For purposes of data analysis, the researchers
argued that because of the “unique nature of mathematics” (Davison and
Pearce, 1988b, p. 10), existing categories of writing (e.g., Britton et al.,
1975) were not satisfactory and they proposed a set of categories more
appropriate for the classification of their data. These categories comprise:
direct copying and transcribing information, translating mathematical symbols into words, summarizing/interpreting, writing story problems or test
questions, creative writing, as in, e.g. a report from a mathematical project.
A detailed presentation of these categories has been included in an earlier
paper (Ntenza, 2004a).
Davison and Pearce’s research showed that only 29% of the 31 teachers
involved in the study used at most one of their writing categories, whilst
64% of the teachers used at most two categories of writing in the mathematics classroom. The results from the study also indicated that 74% of
the teachers used the most common type of writing by pupils, which was
copying and transcribing of information. Moreover, 23% of the teachers
in the study had never involved their students in writing activities except
for recording information from the board; 10% of the teachers involved
their students in writing tasks on a weekly basis; and, only 13% of the
teachers interviewed reported any creative use of the language. In terms
of whom the students’ writing was addressed, the analysis showed that
77% of the teachers had students write for the teacher almost all the time,
whilst only 26% of the teachers had students write for each other. Further, only 23% of the teachers engaged their students in some form of
pre-writing activities which included group discussion before starting to
write or the teacher demonstrating some structural format for the writing
process.
Even teachers who engaged students in writing tasks involving more
than just the copying of work from the lessons and textbooks, did so infrequently and “applied them in a way which did not increase students’
chances of success” (Davison and Pearce, 1988b, p. 13) in the mathematics classroom. The researchers were surprised by this finding; they had
expected more frequent occurrences of, for example, translation from symbols to written language and of students writing story problems, but these
activities appeared to be seldom used. Another concern raised by Davison
and Pearce (1988b) was that most of the teachers were not sure of how they
could implement writing tasks in the classroom, and they also did not see
any direct relationship between writing and mathematics. This prompted
the researchers to recommend more in-service courses and workshops for
mathematics teachers in order to improve their skills in how they could
incorporate writing activities in their classes as a means of supporting the
learning of mathematics.
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Research on forms of writing in Australia
In Australia, a study by Marks and Mousley (1990) in primary and secondary schools, aimed at describing the effect of ‘process mathematics’
on ‘mathematical literacy’. These authors also sought to identify the possible “mathematical genres” (pp. 117–118) that should be experienced in
classrooms if students were to become mathematically literate. According
to Kress (1985), “genre carries meanings about the conventional occasions
on which texts arise” (p. 20). Marks and Mousley explain ‘mathematical
genres’ as “accepted structural forms [such as proofs in Euclidean geometry, graphs, and so on] which are used to make meaning. . . . [They are]
conventionalised forms of texts, or genres” (p. 119). According to these
authors, the following mathematical genres are likely to enhance mathematical literacy among students: “Events are recounted (narrative genre),
methods described (procedural genre), the nature of individual things and
classes of things explicated (description and report genre), judgements outlined (explanatory genre), and arguments developed (expository genre)”
(p. 119).
Marks and Mousley decided to examine mathematics tasks in seven
primary and four secondary classrooms, through regular observation of
mathematics lessons, discussions and interviews with teachers. They analyzed notes of lessons used by teachers and a variety of textbooks that were
widely used in the schools. They found very limited examples of types of
writing representative of this wide range of mathematical genres. A few of
the teachers interviewed said they were giving their students opportunities
“to use language in mathematics classrooms, in order to develop and communicate their understandings in the light of their personal experiences”
(p. 123). However, these teachers – generally in the primary schools – whenever they requested students to write about their mathematics, “asked for
‘stories’ about ‘these sums’, encouraging only the recount genre – that used
in recalling a sequence of events or telling a story” (p. 123). This suggests
that, in spite of the greater influence of those who advocate writing-to-learn
activities in the United States and Australia, neither the quantity nor the
quality of writing used in mathematics classrooms has been substantial, an
observation which is also made by Morgan (1998).
A larger scale survey in Australia was done by Swinson (1992) in which
226 teachers of mathematics from 37 schools responded to a questionnaire
concerning the use of writing in classrooms. The questionnaire used in the
research study consisted, amongst other things, of a list of seven categories
of writing each with a brief description. These were “writing prompts;
journal; letter writing; summarising; essay; rewriting; any other” (p. 42),
and the teachers were requested to indicate the frequency with which they
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used each of the writing categories and examples of writing activities used
in the classroom. The findings from the survey were as follows: less than
10% of the teachers used any writing tasks frequently on a weekly basis.
Although 14% of the teachers fell in the ‘any other’ category and said they
were using writing activities on a weekly basis, a closer analysis showed
that most of these teachers gave examples that clearly pointed to students
recording and transcribing notes from the board or from a mathematics
textbook.
A few teachers from the ‘any other’ category mentioned that they used
project work or written assignments, which was given to students about five
times per term. The rest of this group seemed to indicate that they involved
their students in “reflection or the creation of knowledge” (p. 43), though
most of them used the writing activities infrequently. However, Swinson is
of the opinion that increasing the quantity of writing may not necessarily
improve the effective learning of mathematics, and suggests that teachers
select only writing activities that may support cognitive development, but
he does not give details of how such a selection may be accomplished.
Nevertheless, he concludes by saying that there is a need for research to
determine which writing activities may be the most useful in promoting
cognitive development and how they may be implemented in mathematics
classrooms.
Research in South Africa
In South Africa, no research has been done yet looking specifically at
children’s mathematical writing in the classroom. Most research studies
have tended to consider problems encountered by learners in the learning
of mathematics, and writing has not been identified as one of such problems,
although results from some of these studies might have raised extended or
independent mathematical writing as an issue that needs to be addressed.
For example, de Villiers and Njisane (1987) conducted research about
the development of geometric thinking among African high school pupils
in the KwaZulu-Natal region. They selected a sample of 4015 pupils in
grades 9–12, drawn from schools that ranged from small rural to big inner
city schools. As part of the survey a test was administered to the pupils
and it consisted, amongst other things, of questions requiring the writing
and construction of formal proofs and arguments in Euclidean geometry.
Amongst other findings, de Villiers and Njisane noted that only 20% of the
grade 12 pupils showed signs of having mastered deduction, hence proof
writing, involving more than one step.
More recently, Ndaba (1997), studying the use of language in geometry by pupils in a small number of secondary schools in KwaZulu-Natal,
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also found that pupils produced limited or no mathematical writing when
they were required to construct a mathematical argument in a geometric
proof. In one of the items given to pupils, who were working in groups,
there were at least six steps necessary to write and complete the proof.
According to Ndaba one of the aims of this item was to investigate “ . . .
whether pupils were able to group the related statements which is one of
the skills in proof writing” (p. 147). His observation was that “due to the
lack of . . . skills for proof writing not a single group managed to accomplish a proof. The groups could not [write] even one correct statement”
(p. 211).
In another survey completed by Ngcobo (1997), the purpose of which
was to explore students’ thinking by identifying the prevalent errors in
their mathematical understanding of linear equations in grade 9 (13–14year-olds) algebra, a sample of 432 students completed a test that contained
13 questions based on algebra. One of the questions in the test consisted of
two simultaneous linear equations, and the pupils were asked to translate
the given mathematical situation by writing a real-life situation in words
only. In his analysis of the results, Ngcobo observed that 59% of the pupils
omitted the question. These studies, though limited with regard to their
emphasis on learners’ writing, do bring into question both the quantity and
quality of writing that is taking place in South African mathematics classrooms, especially in the context of the ‘new’ curriculum introduced at the
beginning of 1998 in grade 1 (6-year-olds). This curriculum is popularly
known as outcomes-based education (OBE). Some of its basic and foundational principles include continuous assessment, learner-centeredness,
relevance, integration, creative thinking and flexibility. I believe, therefore,
that it is important to research current forms of writing in South African
mathematics classrooms to enable policy-makers, curriculum developers,
and teachers to make informed decisions about a successful introduction
of mathematical writing.
RESEARCH
METHODOLOGY
To find possible answers to the stated research question, I selected six
schools: three urban schools; one township school; and two rural schools.
The selection of the schools was not a random process. I was not interested
in arriving at empirical generalisations (Hammersley, 1992) from the study.
My main consideration was to choose schools that were likely to provide
rich data and information that would be useful in determining the kinds of
mathematical writing prevalent in contexts with various levels of material
and human resources (see, Ntenza, 2004a).
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For purposes of data collection, six mathematics teachers (Bridget,
Chris, Makhosi, Wendy, Xolile, and Zwide – not their real names) teaching grade 7 learners were identified in these contexts. Bridget, Chris, and
Wendy teach in the urban schools. Xolile and Zwide teach in the rural
schools. Makhosi teaches in a township school that has some facilities
such as a photocopier and an overhead projector, but the school has no
computers. The presence of a photocopying machine in her school did not,
however, have many benefits for Makhosi and the other teachers. The principal informed me that because of the large classes (cf. Kahn, 2001), an
average of 55 in each class, the school just did not have enough funds to
buy paper for use in the photocopier. Any monies available in the school are
used to buy paper to photocopy only examination papers. In most instances
learners had to pay or contribute towards buying paper if the teacher wanted
to photocopy some worksheets for the class. Obviously this was not always
possible since the majority of the learners came from poor families who
were earning very little or nothing at all.
Data collection instruments
To ensure triangulation of the data-collection methods, I decided to obtain my data through the use of a variety of data-collection instruments.
Cohen et al. (2000) describe triangulation as “the use of two or more methods of data collection in the study of some aspect of human behaviour”
(p. 112). In this phase of the research the data collection methods included
interviews of teachers, examination of students’ written work, observing
mathematics lessons, examination of lesson plans, and analysing mathematics textbooks used by teachers in the selected schools. Cohen et al.
(2000) further emphasise that triangulation methods “attempt to map out,
or explain more fully, the richness and complexity of human behaviour by
studying it from more than one standpoint . . . ” (p. 112). It was also clear
to me that to be able to collect the data I wanted I would have to visit each
of the schools at least three times. For example, because I could not take
home the students’ written work, which was in the exercise books that they
used daily to write mathematics exercises and classwork, this meant I had
to stay in the school at least half a day to examine the students’ written
work.
Teacher interviews
During the first visit I spoke at length with the principal about the school.
Besides explaining to the principal my study and its objectives, I was interested in finding out generally how the school is run, the ethos of the
school, particular problems encountered with the teaching of mathematics,
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and other subjects in general. I also wanted to get as much support as I
could from the principals, upon whom I was dependent to explain the nature of my presence to the other staff members in each of the schools I
visited. The talk with each of the principals was useful since it gave me
an overall impression of, for example, what happens in each school on
any particular school day. During these first visits to the schools I also
conducted semi-structured interviews with each of the six teachers. The
questions that I asked teachers during the interviews were mainly concerned with mathematical writing that takes place in their classrooms. The
interview schedule did not have a fixed standardised set of questions as
the interviews were based on the notion of an interview-as-conversation
(Burgess, 1984; Eisner, 1991). All the interviews were tape-recorded with
the permission of the teachers. I asked questions which were centred around
four broadly organised themes; namely, (a) amount of writing, (b) uses of
writing, (c) forms of writing that each of the teachers normally set for the
learners in their classrooms, and (d) perceived impact of outcomes-based
education on writing and assessment. All these topics, except the last one,
have been gleaned from the research literature (see, for example, Britton et
al., 1975; Davison and Pearce, 1988b, 1990; Miller, 1992; Morgan, 1998;
Shield and Swinson, 1994; Swinson, 1992; Waywood, 1992, 1994). The
last topic is rather more specific to the South African situation as it looks at
how the outcomes-based curriculum can have an impact on mathematical
writing and assessment. Nevertheless, the topic is included as a result of
research done by Clarke et al. (1993) and Waywood (1992, 1994) on the
importance of integrating writing and the mathematics curriculum. The
nature of the interviews was such that I was able to ask further questions depending on responses made by the teachers. I was also able to
probe for clarity on some issues that were raised by the teachers during the
interviews.
Examination of mathematics textbooks
During the second visit to each of the schools I examined all grade 7
mathematics textbooks used, both by the teacher and/or learners, in each
school. The main objective of the second visit to the schools was to note
the kinds of writing activities and tasks that were either or both suggested
by any of the mathematics textbooks, or set by teachers in their lesson
plans. In South Africa, the Department of Education, in each province
(and there are nine provinces in all), normally provides each school with
a list of the recommended mathematics textbooks before the end of each
academic year. Mathematics teachers in each grade in the schools select
one textbook from the recommended list of various mathematics textbooks
and then make an order for the number of learners they expect to get the
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following year in that grade. So, it is possible for different schools to use
totally different mathematics textbooks. Textbooks used by the teachers
were an important component of the study for two reasons, mainly. Firstly,
they might reveal what kinds of suggestions, if any, are made to the teachers
in terms of extended writing activities, writing tasks, projects, investigative
reports, or written assignments. Secondly, teachers depend on the mathematics textbook as a source of knowledge and as a teaching aid in mathematics classrooms. In some classrooms, particularly in rural schools where
teaching and learning resources are very limited, the mathematics textbook
is the only source of mathematical knowledge available to teachers and
learners.
Examination of lesson plans
During the third visit to each of the schools I examined lesson plans and
observed some of their mathematics lessons with the objective of noting
the kinds of writing activities that were set for those lessons. Obviously, it
was not going to be possible to note all kinds of writing activities in a single
day of classroom observation. Classroom observation was not critical in
my data collection since I was able to examine learners’ written work and
that would provide me with evidence of the amount and forms of writing
produced by learners. I requested teachers to allow me to examine their
lesson plans over a period of about two terms, approximately 22 weeks of
teaching. The teachers indeed gave me their schemes of work on a termby-term basis, schemes of work on a weekly basis, schemes of work on a
daily basis, relevant mathematics syllabus, and other artefacts. According
to Eisner (1991), documents and artefacts “provide a kind of operational
definition of what teachers value” (p. 184), therefore, they would assist
me in understanding the context within which the teachers did their work.
Consequently, by examining the lesson plans I hoped to note, for example,
if there were any intentions by teachers to give learners writing activities
in their classrooms.
Examination of learners’ written work
During the third and final visit to the school I continued working closely
with the mathematics teacher by analysing a sample of learners’ written
work, the objective being to determine the forms of writing that the students
are producing. Since the class sizes ranged from around 25–65 learners, it
was not going to be possible for me to analyse all the learners’ written work.
I asked teachers to select and give me a range of learners’ work in relation
to their mathematical performance. Hence, I was able to look at written
work from mediocre, average, and good learners. Obviously, at this stage I
depended on the teacher to make these categories about learners. One of the
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criteria the teachers used was to look at learners’ mathematical performance
in previous tests. Any learner who usually scored in a test below 40% was
classified as mediocre, 40–59% was regarded as average, and 60% and
above was taken as good. I requested exercise books containing written
work of 10 learners in each of the grade 7 classes. Some schools had more
than one grade 7 class, but since the same teacher taught them I took the
written work from one class, randomly selecting this class.
DATA
ANALYSIS
Analysis of teacher interviews
All the interviews were transcribed and then analysed. Because of a lack of
resources in some of the schools, it was generally clear from the analysis
of the interviews that the teachers had not been capacitated to the extent of
being able to apply the suggested continuous assessment policies. Hence,
the majority of the teachers still seemed to believe that ‘good’ mathematical
performance by a learner in a test or examination is a good indicator of
whether the teaching was successful and also whether that learner has
understood the topic or concepts taught (cf. Ntenza, 2004a). It seems that
the summative aspect of assessment is given much weight when it comes
to assessment of learners’ mathematical understanding and knowledge of
mathematical facts.
However, seminars, training workshops and courses have been organised, albeit in an irregular manner, for various teachers by the DoE to
cover various aspects of the new curriculum. Because of the large number
of teachers in the country, the DoE does most of its in-service training
(INSET) workshops based on the ‘cascade model’. Basically the cascade
model works according to the principle that says ‘provincial trainers’ are
trained first; they, in turn, train leader teachers from districts located in
provinces; finally, the leader teachers train groups of teachers from within
the districts. However, in most cases the leader teachers are not given
enough resources, such as adequate workshop materials and notes to enable them to run school-based or district-based workshops. Even access
to photocopying large quantities of materials becomes a problem in some
of the training and professional development workshops. Without a budget, limited support from departmental officials, limited time to do the
work, and lack of basic resources make the leader teachers’ task difficult
to do and complete. Hence, the data may be suggesting that the desired
effects of the cascade model do not reach as many teachers as intended.
If the cascade model of training does reach some teachers at grassroots
level, it is likely that what those teachers receive and get is an incomplete
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picture of what they should be actually implementing and doing in the
classrooms.
As part of the analysis of the data I will indicate some of my interpretations of the data and/or findings by giving illustrative or representative comments made by the teachers during the interviews. By illustrative
comments I mean those comments that indicate a particular position by
a particular teacher, whilst representative comments will mean a position
taken by at least more than half of the six teachers.
Amount and uses of writing
One of the initial findings from the data collected is that there seems to
be rather little and infrequent mathematical writing taking place in the six
schools. In the majority of schools there were many written pages of mathematics consisting of mathematical symbols rather than some indication of
extended mathematical writing. The following comments made by some
of the teachers I interviewed illustrate the extent to which the amount and
uses of writing are considered in their mathematics classrooms. Xolile who,
amongst other teaching duties, teaches two grade 7 mathematics classes,
one with 49 learners and the other with 52 learners, when asked whether she
would give her learners extended writing tasks, say once a week, responded
as follows: “. . . Not necessarily. Not necessarily because most of the time
if I’m leading these big classes I have to work on my timetable. Like I’m
giving them class work. I have also to allocate my time for marking that
work and to sit down with the classes I have. So with this [writing] task, it’s
only when I see fit maybe . . .”. Makhosi, Head of Department (HOD) in her
school, and who teaches four different mathematics classes each with about
55 learners, said: “. . . They do . . . writing where they’ve learned the week’s
lesson or if it’s two week’s lessons they have learned on the two week’s
lesson, [they write] how they felt about that lesson and they also point out
the problems . . .” Makhosi was actually doing journal writing with all her
learners in the four classes she taught. In fact, during the interview with
Makhosi, she informed me that after she has taught for 1 or 2 weeks, she
allows her learners to write in their journals in accordance with the format
shown in Figure 1. When I analysed the learners’ written work I found that
there was indeed evidence of some journal writing by the learners.
Journal writing is one of the new assessment methods in South Africa
suggested within the policy on continuous assessment (DoE, 1997, 2000).
It is also recommended by the outcomes-based curriculum (DoE, 2002)
currently being introduced and implemented in the schools. Makhosi mentioned that she used the same activity, as indicated in Figure 1, for all
topics in mathematics. The activity indicates that the journal writing is
supposed to take place about once in 2 weeks, though when I analysed the
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Figure 1. Journal writing activity – Makhosi.
learners’ work, the indications were that it was rather infrequent. In fact it
had been done only once in a period of 8 weeks. There was no evidence,
either from the interviews or from the learners’ written work, to suggest
that the other five teachers had tried to introduce journal writing in their
classes.
Chris, an HOD in his school, when asked about the amount of writing
that he gives, first reiterated the school’s policy on the teaching of mathematics that: “. . . the kids should have an hour and a half of mathematics
per day, four times a week. And Friday there’s no maths homework but
they have an hour lesson”. Chris was only teaching two grade 7 classes
each with about 25 learners. His opinion on the amount of writing each
day, though it was not always the case, was that: “. . . basically it should
be 30 minutes of writing time . . .”. This situation (i.e. Chris’) is exactly
the same in Wendy’s school. Incidentally, Wendy is also an HOD in her
school, but her teaching load is rather heavier as she teaches mathematics
and all other subjects in her grade 7 class. Nevertheless, for each of these
two teachers the total classroom teaching time comes to 5 and 2 hours of
homework time per week. Wendy made a similar remark: “. . . on a daily
basis, if the period is an hour, I would say written work would . . . take
half of it. And I would spend half the time either explaining or doing oral
work . . .”. From the comments made by some of the teachers, for example,
Wendy and Chris, on the amount of writing, it seems that learners spend at
least half the time allocated for teaching mathematics doing classwork. In
many instances the uses of the classwork, which consisted mainly of routine exercises, were to consolidate a topic or concept that had been taught
recently or during the other half of the teaching time. Only one teacher,
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Makhosi, gave learners a chance to write in a journal. Though this was very
minimal and inconsistent, however, the journal writing allowed learners to
write about their feelings, problems, objectives, and suggestions on some
of the mathematics topics they have learned.
Forms of writing
From the analysis of the data, in particular when I analysed the learners’
written work, I noted two forms of writing. I have referred to these as
‘symbolic writing’ and ‘mathematical writing’.
Symbolic writing. By symbolic writing I mean the mathematical symbols that learners produce as a result of doing ‘traditional’ routine exercises and sums based on multiplication, division, addition, and subtraction. The following comment by Chris is representative of the situation
that is prevalent in the six mathematics classrooms when it comes to the
kinds or forms of writing that learners are producing on a daily basis:
“Every single of these worksheets is multiplication and division. Multiplication and division and every week that’s what we emphasise because
we find when we’re teaching division or fractions the kids have no idea”.
Indeed an analysis of the learners’ written work, in the six classrooms, did
indicate lots of pages of mathematics exercises on addition, subtraction,
multiplication, division and long division, fractions, number patterns, and
percentages.
Mathematical writing. I will use here the same terminology as Davison
and Pearce (1988a, b, 1990) in describing the categories of mathematical
writing that were observed in the analysis of the data from the schools that
I worked with:
Direct use of the language. This kind of mathematical writing generally
consisted of copied and transcribed information by learners into their
exercise books from the board. An analysis of learners’ written work
shows that this consisted mainly of summaries of certain mathematical
topics, worked out examples, procedures or steps to be followed when
doing particular algorithms, rules or laws for doing some problems, and
so on. These were easily observed in the learners’ written work because
they were written, in most cases, at the end of the learners’ exercise books.
They were also clearly unmarked (the teachers normally use a red pen
to mark learners’ work) or did not show if the teachers had checked
them. For example, Bridget required that all examples, solutions, and
procedures for finding the solutions, done on the board with learners
must be copied and then marked with a red star in the margin of the
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335
learners’ exercise books. The red star next to the problem and solution
indicates to her that it is an example and she does not have to check or
mark it.
I noted that in those schools with photocopying facilities, learners
are, in most cases, given loose sheets that may contain examples, summaries, rules or procedures for doing certain problems, which they stick
onto their exercise books and which are used as reference material. The
following comment by Bridget is representative of the situation in the
more advantaged schools: “. . . I use text books myself to formulate a
workbook for them and then we’ll have various examples in the workbook, which they will then work through, either as a class, or in groups,
depending on what I want, what type of exercise it is . . .”. This practice, whilst it might have some advantages from the teacher’s perspective
such as saving on time for learners to copy and transcribe, minimises the
chances for learners to use the language in a more direct way, which is
rewriting what is on the board.
Linguistic translation. This is mathematical writing in which learners translate mathematical symbols into words. I noted that only Zwide gave
learners a few problems that would require translation from the mathematical symbols into the written language. I also noted that he encouraged his learners to create their own story problems. These story problems were exchanged amongst other learners in the class who would
then try to find their solutions. In the interview when I probed Zwide
about teaching word problems, he said that his approach was such that
if learners “. . . found it interesting then I ask them to come up with
their own problems. And I found it interesting that most of them . . .
come up with problems which are very much interesting”. When I examined learners’ written work in this particular school, I observed that
indeed they had done some work on translating mathematical symbols
into written language, and there was evidence of learners writing their
own story problems. However, as indicated earlier, only one teacher attempted these writing activities and they were not done frequently, say
once a week. This finding is similar to the one made by Davison and
Pearce (1988b) in their research. In Ngcobo’s (1997) study, 432 grade 9
learners from various schools were given a test where one of the algebra
questions required them to translate a pair of linear equations by writing
a real-life situation in words only. Ngcobo observed that only 41% of the
pupils attempted to answer the question. The study by Ngcobo showed
that most learners might not have been given much work in class, and
on a frequent basis, that would involve them in translating mathematical
symbols into written language. In fact in the case of Zwide, the writing
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activities were only attempted when he was teaching the topic on word
problems.
Summarizing and interpreting. This is mathematical writing where learners
use their own words in describing or explaining material from some text
or mathematical text produced as a result of the work the learners are
doing. The journal writing introduced by Makhosi in her classes, as
described earlier, would fall under this category. The journal writing
required learners to summarise and interpret mathematics topics that
had been taught in a number of lessons. Learners were also required to
state, amongst other things, their feelings and problems with regard to
the mathematics topics and possibly give suggestions to their problems.
As another example of mathematical writing that falls in this category I
will give an extract from a worksheet as shown below, used by Bridget.
The worksheet in Figure 2 shows two types of questions. On the one
hand, questions 2 and 3, with the usual format ‘what do you notice . . .?’
Figure 2. Example of worksheet – Bridget.
MATHEMATICAL WRITING
337
or ‘what can you say . . .?’, do indicate that learners have to interpret some
mathematical information. On the other hand, question 4, with the format
‘. . . explain what you notice . . .?’, requires an explanation about what is
observed. Worksheets with questions similar to the two types of questions
described here, though not the same, and particularly on work involving
number patterns, were quite common in those schools with photocopying
facilities. These questions do, albeit in a very limited way, encourage
learners to write about their mathematical observations of the solutions
and also allow for the interpretation of the mathematical content that the
learners are interacting with. Examination of the learners’ written work
indicated that the easiest part of this worksheet was completing those
sections requiring arithmetic calculations. I noted that it was only those
learners classified by the teachers as average or good in their general
mathematical performance in tests, who attempted to offer explanations
as required in some of the questions.
Creative use of language. This is mathematical writing in which students
explore and transmit mathematically related information such as writing
an assignment or an investigative report on a mathematics project. I found
that some teachers gave learners written assignments or investigative
reports but for reasons other than to allow learners to use language in a
creative way. For example, as far as Bridget perceived the investigative
reports were informal, optional and for enrichment activities (see Ntenza,
2004a).
The data analysis here shows that, in those schools with good teaching facilities, such as access to the Internet and a library, teachers give
students a written assignment or investigative report. Therefore, some
learners from these schools have a better chance of using the language in
a creative way than learners from rural and township schools. I say ‘some
learners’ because, for example, Bridget gave the investigative report only
to those learners who wanted to explore further the concepts that had been
taught in her classroom. Teachers in the rural schools where facilities are
limited did not attempt to set any investigative reports or written assignments. During interviews with Xolile, she commented that if learners do
not have access to a library, for example, then it is of no use giving them
an assignment or investigative report which might require access to the
library.
It is clear from the analysis of the data in the six schools that participated in the study that there is very little mathematical writing taking
place. This has implications in terms of a successful implementation
of the requirements of the new curriculum and, in particular, the introduction of writing and formative assessment practices in mathematics
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classrooms. In the next section I discuss some of the ideas from the
teachers in this respect.
Impact of outcomes-based curriculum on writing and assessment
Most of the teachers that I interviewed support the new outcomes-based
mathematics curriculum, however, they had many reservations with regard
to the filling in of forms in connection with learners’ assessment and recordkeeping (cf. DoE, 2001, p. 130). Wendy, for example, stated that the continuous assessment principles are ‘time- consuming’. More importantly, she
also felt that the formative assessment aspect, which was bringing in more
mathematical writing, meant that learners were unlikely to learn important
and basic mathematical concepts (see Ntenza, 2004a). On the other hand,
teachers in the rural schools also complained about the additional amount
of work brought in by the demands of the new curriculum, whilst their
classes remained large and with little or no supporting teaching facilities.
The findings from the study seem to indicate that most teachers are apparently unhappy about the extra work, such as filling in learner assessment
forms, brought in as a result of the introduction of the new curriculum and
its continuous assessment practices. Hence, these teachers do not see mathematical writing as a priority under such circumstances. Moreover, in most
instances the teachers have to develop and write themselves some of the
materials needed to involve learners in mathematical writing, since the majority of the textbooks that are available do not have ‘suitable’ suggestions
in this regard. In the next section I now analyse what such mathematics
textbooks may offer.
Analysis of mathematics textbooks
I examined a range of written text materials, including grade 7 mathematics textbooks (e.g., Barry and Dugmore, 1990; Bolt and Hobbs, 1998;
Cebekhulu et al., 1993; Jacobs et al., 1991; du Toit et al., 1991; Ladewig
et al., 1990; Laridon et al., 1986, 1999), that were used by the six teachers
who participated in the study. The purpose of doing this was to find out if
these textbooks gave suggestions on writing activities that could be used
by teachers in their mathematics classrooms. Firstly, I noted that different
schools were using the mathematics textbooks in different ways. In the
urban schools, where each of the learners had a copy of the governmentsupplied, recommended mathematics textbook, Bridget, Chris, and Wendy
used a variety of mathematics textbooks to formulate what they referred to
as a ‘workbook’. This is what Bridget said about a workbook she had just
done on the topic ‘Percentages’: “. . . I . . . formulated this percentages workbook. It’s a 16-page workbook where I’ve got all the different . . . examples
MATHEMATICAL WRITING
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from different textbooks. I now use about 5 or 6 textbooks and just take
examples from them and then the children work through these”.
In the rural and township schools, I noted that not all the learners had
their own copy of the government-supplied recommended mathematics
textbook for grade 7. Because of the lack of physical resources, such as
photocopiers, in these schools, teachers are not able to develop, write, and
photocopy worksheets and other written materials. For example, this is
what Zwide said when I asked how he gets photocopies of a worksheet for
his learners: “. . . there was only one task that I did – when my sister was
working in some other offices where she could get me photocopies because
the problem is the [lack] of the . . . copy machine . . . So there is very little
of that kind of work”. Even if this teacher has other mathematics materials
which he may want to distribute to his learners so that they can work individually, it seems that this would not be possible. Secondly, I noted that all
the mathematics textbooks, with the exception of the textbook by Jacobs
et al. (1991), did not have questions that would encourage mathematical
writing beyond the third level of the typology described by Davison and
Pearce (1988a, b, 1990). The textbooks analysed and the workbooks developed by some of the teachers showed, for example, many examples and
exercises that require learners to translate verbal expressions to mathematical symbols, but there were no examples or exercises showing translation
from symbols to written language or of learners writing story problems.
The mathematics textbook by Jacobs et al. (1991), in the chapter on verbal
expressions, does include a very small section – not more than one page
– on how to convert number sentences into word problems (1991, p. 79).
For most of the teachers that I interviewed, the mathematics textbook is
the most important and the only source of knowledge when it comes to
teaching mathematics. Teachers in the advantaged schools seem to be able
to use a variety of books to compile their own workbooks and to print these
individually for learners. Teachers in the under-resourced schools struggle
even to get the government-supplied, recommended mathematics textbook
supplied to all their learners. The principals in these schools informed me
that in some instances they get less textbooks that ordered for no reason
at all, whilst in other instances learners do not return school textbooks at
the end of the year let alone repay them if they are lost. Nevertheless, the
main finding from the analysis of these mathematics textbooks is that they
do not have suggestions of writing activities that may promote extended
mathematical writing. Hence, the form of writing produced by learners in
mathematics classrooms still consists largely of mathematical symbols as
it mirrors what is written in the textbook itself. Since all the teachers I
interviewed use one or more of the textbooks I analyzed, it is, therefore,
probable that what does not appear in the textbook is unlikely to show in
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their lesson plans and therefore, will also not show in their teaching of the
subject.
Analysis of lesson plans
I analysed lesson plans written by the six teachers over a period of about
two terms. I noted that Zwide, who was using the mathematics textbook
by Jacobs et al. (1991), did have some lesson plans which indicated that
he would give learners a few problems that would require translation from
the mathematical symbols into the written language. I found that in some
of the lesson plans Zwide encouraged his learners to create their own story
problems. These story problems were exchanged amongst other learners
in the class who would then try to find their solutions. Makhosi’s lesson
plans also showed how the journal writing was going to be undertaken in
her classroom. I have already discussed learners’ written work emanating from these lesson plans in an earlier section. Lesson plans by Bridget,
Chris, and Wendy had various worksheets which contained some questions
that required learners to give reasons for an answer, to offer written explanations, and to interpret mathematical information. Generally the lesson
plans that I analysed, written by the six teachers, did not indicate that much
mathematical writing was being implemented in the classrooms. There
was not much attempt by these teachers to include writing activities as
part of their assessment methods, and as suggested in the relevant policy
documents. The lesson plans showed that tests and examinations are still
regarded as the only means of determining successful learning and teaching
in the mathematics classroom.
C ONCLUDING
REMARKS
One of the claims made by Davison and Pearce (1990) is that if teachers use
writing activities frequently and systematically, for example, at least once
a week with pre-writing sessions, the performance of students improves
substantially. The analysis of the data showed that this use of writing was
not the case with the teachers who took part in the study. Therefore, in the
classrooms where some mathematical writing was observed, my main conclusion is that the amount of work written by each learner and its frequency
was lower than what has been suggested by Davison and Pearce.
Morgan (1998) observed that learners in mathematics classrooms might
write and cover pages of the exercise books without completing a single
‘ordinary English’ sentence, something that became obvious as I analysed
learners’ written work from the six schools that participated in the study.
MATHEMATICAL WRITING
341
Generally, very few learners wrote anything in their own words, and none
produced mathematical writing over half a page in length, with the exception of some learners from the class where journal writing had been
introduced. Morgan’s (1998) contention is that while symbolism and specialist vocabulary are perhaps the most obviously visible aspects of many
mathematical texts, hence provide some form of writing, especially within
the context of school mathematical texts, they are clearly inadequate to
provide a full description of the nature of mathematical texts. Morgan
(1998) further argued that it is important to move beyond the level of vocabulary and symbolism (cf. Ervynck, 1992), typical of most mathematical
writing, and consider the “grammatical structures and forms of argumentation” (p. 3) that are sometimes used and seen in mathematical texts. These
last two aspects are forms of writing that are not likely to be evident in
most children’s writing in school mathematics, although there are critical areas in the mathematics curriculum, for example, in proof writing
(and quite dominant within the Euclidean geometry that is taught in South
African schools), where learners are expected to construct mathematical
arguments.
The findings from the study show that the lack of relevant facilities in
some of the schools has a negative impact, hence, is an obstacle to teachers’
use of writing tasks in their classrooms. The issue of availability or nonavailability of resources in such schools arose in the presentation of the
data analysis as a significant issue. In particular, access to a photocopying
machine seemed to make a difference to the kinds of opportunities available
for teachers to vary the tasks presented to learners and hence to include
tasks that demand more writing. In practice, this seems to be critical, but
more importantly, I think it is a revealing finding in the context of thinking
about mathematical writing in the mathematics classroom. I believe the
South African National Department of Education can certainly lend muchneeded support, in particular, to rural schools, for example, through the
provision of basic minimum resources.
Another recommendation to the Department of Education emanating
from this study would be to provide more effective professional development courses and workshops for mathematics teachers in order to improve
their skills, such as on how they could successfully incorporate writing
activities in their classes as a means of supporting learners in the production of extended mathematical writing. There is research evidence (e.g.,
Morgan, 1998; Watson, 2001) to the effect that even experienced mathematics teachers can come up with conflicting interpretations of the mathematics from the same learners’ written work. Therefore, the suggested
workshops would assist teachers in gaining some common understanding amongst and within themselves on how to collect evidence of learn-
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ers’ understanding or knowledge of a particular mathematical topic from
a written assignment, investigative report, or research project, through
the formulation of ‘rubrics’, for example, as a guide to the assessment
practices.
An important problem already raised by researchers (e.g., Davison and
Pearce, 1988b; Ntenza, 2004) is that teachers may not perceive any relationship between writing and mathematics, or between writing and learning
of mathematics, for that matter. The findings from my own work here suggest that, to overcome some of these obstacles associated with writing in
mathematics classrooms, writing must be integrated in the whole of the
mathematics curriculum for mathematics teachers and mathematics textbook writers to change their views and practices. Given what this study
shows us about teachers’ classroom practices, their attitudes to writing and
their working environments, it seems to me that it makes sense, therefore,
and is practical, to introduce more writing as early as in the primary mathematics classroom, since at this level the children can easily develop and
learn new skills, and this may be true not only within the South African
context, but in the wider mathematics education context world-wide.
A major question that is implicitly raised by some of the findings of
this work, and is being researched within the main study, is the extent to
which writing may assist teachers in determining children’s mathematical
understanding of certain concepts and whether writing can improve the
learning of mathematics. It seems to me that one of the problems related
to doing research on writing is the ubiquitous nature of its claimed cognitive effects on learning. Nevertheless, it is hoped that the findings from
the study will contribute tentative answers to some relevant and important
questions which should be considered by anyone interested in undertaking
research in this area, such as: ‘What kind of learning, for example, takes
place when children copy and transcribe information from the board or their
textbook? Or when they must explain in writing a procedure in their words?
Or when they must complete an investigative report?’. In terms of classroom practices, both within the South African context and more broadly,
such knowledge may imply that if writing is to be used with success in
the learning of mathematics, then the structure of the mathematics classroom and the nature of the mathematics curriculum have to be radically
transformed.
A CKNOWLEDGEMENTS
The piece of research reported herein was performed with the support of
part funding from the National Research Foundation, the Canon Collins
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343
Educational Trust for Southern Africa, and the Ernest Oppenheimer
Memorial Trust. Any opinions, findings, conclusions, or recommendations
expressed are those of the author and not necessarily those of the granting
agency or agencies. I would like to express my sincere gratitude to the three
anonymous referees and the editors for their help and advice on an earlier
version of this article.
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S. PHILEMON NTENZA
University of KwaZulu-Natal, Faculty of Education
School of Science, Mathematics and Technology Education
Edgewood Campus, P.Bag X03, Ashwood, 3605
Republic of South Africa
Telephone: 27-31 260-3460
Fax: 27-31 702-0692
E-mail: [email protected] or [email protected]