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GENERATE GRAPHS AND GENERALISE THE EFFECTS OF PARAMETERS
ON GRAPHS BY MEANS OF AUTOGRAPH AND SKETCHPAD
Hennie Boshoff
Department of Mathematics
and Applied Mathematics
Nelson Mandela Metropolitan University
Dynamic mathematics computer
programs like Geometer Sketchpad (The Geometer’s
Sketchpad Resource Centre) and Autograph (Autograph, for
the Dynamic Classroom) are immensely powerful tools when
it comes to the onscreen generation of graphs. In
addition, most Mathematical software packages atones for
interactive investigations and the generalization of the
effects of parameters on graphs. For the purpose of this
paper some of the graphs listed in NCS (2007:17) will be
generated and the effects of parameter changes be
investigated by means of the Sketchpad and Autograph
packages.
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Homework
is a necessary evil in the path of learning mathematics
at school. Mathematics homework is traditionally seen
as difficult and boring. In the case of difficult
homework, “math clubs” and “math extra lessons” are
often perceived as even more difficult and more boring.
This paper describes a project where learners could get
help with their mathematics homework using MXit on their
cell phones in the afternoons after school. MXit is a
popular instant messaging system where text messages are
sent immediately between participants' cell phones and
is proprietary software of MXit Lifestyle (Pty) Ltd in
Stellenbosch. At the time of this writing, there were
over 3 million MXit users in South Africa and nearly 45%
of them were teenagers between the ages of 12 and 18. In
view of the fact that all high school learners now have
to take mathematics or mathematical literacy, MXit
offers a fun and exciting medium in which to help
learners with mathematics homework. Teachers are
welcome to refer their learners to this project or,
alternatively, teachers could easily follow the steps
taken in this project in order to set up similar systems
at their own schools.
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Failing by example: initial remarks on the constitution
of school mathematics, with special reference to the
teaching and learning of mathematics in five secondary
schools
Zain Davis
School of Education, University of Cape Town
Yusuf Johnson
Schools Development Unit,
University of Cape Town
Yusuf.Johnson@uct.ac.za
We
analyse our observations of teaching and learning of
mathematics at five secondary schools, populated by
working class students, with the goal of systematically
defining a research problem and the production of
research hypotheses. Our initial observations reveal
that the pace of teaching and learning is slow but that
students fail to learn the content adequately. We draw
on research that enables us to more productively discuss
the relation between pacing and student learning and
find that the nub of the problem centres on the absence
of the explicit operation of mathematical grounds to
support the elaboration of standard procedures. We
generate a series of hypotheses on the negative effects
of the absence of mathematical grounds on the
constitution of mathematics in the five schools and the
slowing of pacing. The hypotheses open up a series of
lines of investigation that are being pursued currently.
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CHALLENGE TO STUDENTS’ PERCEPTIONS OF LINES RELATIVE
POSITIONS IN SPATIAL GEOMETRY IN PEDAGOGICAL UNIVERSITY
– BEIRA (MOZAMBIQUE)
Paulo Diniz
Pedagogical University – Beira – Mozambique
padibene2@yahoo.com.br
In
this article I intend to share my teaching experience of
a particular concept related to Analytic geometry at
Pedagogical University in Beira, Mozambique. In 2005, I
faced a real classroom situation when I was teaching the
concept of lines’ relative position in space to my first
year mathematics students pursuing a degree in secondary
mathematics teacher education. I was surprised when in
the first test the majority of the students were unable
to respond correctly to the “easiest” task of the test.
Before the correction of the test we worked together on
the same task again, as I noticed that there were many
conceptual misunderstandings on the above mentioned
concept. This situation was very interesting to me,
particularly as I had previously considered the concept
of lines relative position as one of the easiest tasks
of the test.
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COGNITIVE ARCHITECTURE FOR ALGEBRAIC DIVISION: BASE TEN
DECOMPOSITION – AN EMULATION OF THE ‘REAL THING’
Jacques
du Plessis, Mamokgethi Setati
Marang Centre
for Maths and Science Education
University of the Witwatersrand
This paper focuses on a
didactical model that uses base ten decomposition on
naturals as cognitive architecture across the divide
that exists between arithmetic and algebraic long
division. The base ten decomposition as an algorithm for
performing division emulates the structure of algebraic
expressions and will assist in making the transition to
variable seamless. Due to page restrictions the authors
will discuss the didactical model, and reflect on key
areas of response from the grade 8 learners.
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BETWEEN UNDERSTANDING THE LANGUAGE AND UNDERSTANDING THE
MATHS? TRANSLATING FROM WRITTEN TO SYMBOLIC FORM IN A
MULTILINGUAL ALGEBRA CLASSROOM
Anthony A. ESSIEN
Marang Wits Centre for Maths
and Science Education
University of the Witwatersrand
This
paper investigates how a teacher in a multilingual
classroom supports learners who are struggling to
translate written/verbal mathematics into symbolic form.
36 multilingual grade 10 learners in one school (in
South Africa) were given a written test involving one
question and then a discussion on the solution ensued.
The results of the written test by learners, analysis of
class discussion and the interview with the teacher
reveal the complexity of discerning or situating
learners’ difficulty either as due to language
limitation or due to lack of understanding of the “math”
or both.
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A MULTIPLE CASE STUDY OF PARENTAL INVOLVEMENT WITH GRADE
8 LEARNERS OF MATHEMATICS
V. G. Govender
Eastern Cape Department of Education, PE District
This
paper focuses on a study which I conducted with selected
(mostly working class) parents of grade 8 learners of
mathematics in an urban school setting. Initially, a
survey was conducted with all the grade 8 parents at
this school. Most of the parents indicated that they
were keen on participating in a parent-assistance
programme for mathematics. Using key points from this
survey and an extensive literature review, I designed a
parent-assistance programme for mathematics. This
programme was conducted with three exclusive groups of
parents. Each group was considered to be a case study.
An in-depth interrogation of the data in
this study revealed that parents’ and children’s
perceptions of mathematics were likely to be positively
influenced. The data also suggest that children were
likely to become more confident and to improve in
mathematics. A key factor in influencing parents and
children’s perceptions of mathematics was exposure to a
wide-range of mathematical applications. This may have
also helped to develop children’s confidence in
mathematics and their improvement in the subject.
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Critique of mathematical models and applications: A
necessary component of Mathematical Literacy
Cyril Julie
School of Science and Mathematics Education, University
of the Western Cape
A
case is presented for the incorporation of the critical
engagement with mathematical modelling and applications
in the teaching of Mathematical Literacy. The different
purposes of mathematical modelling and applications are
discussed and it is demonstrated how the interrogation
of mathematical models and applications can be realised
with the different kinds models. Examples are drawn from
contemporary South African issues such as Black Economic
Empowerment (BEE) and funding for educational
institutions.
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INVESTIGATING THE USE OF LEARNERS’ HOME LANGUAGES TO
SUPPORT MATHEMATICS LEARNING
Mampho Langa,
Oprah
Winfrey Leadership Academy
Mamokgethi Setati,,
Marang Centre for Maths and Science
Education,
University of the
Witwatersrand
The paper
presents an investigation into how learners’ home
language can be used as a support for learning
mathematics. This qualitative case study was conducted
in primary school where learners were taking mathematics
in English, which is not their home language. The school
worked in collaboration with the Home Language Project
to facilitate the learning of mathematics. The study
revealed that when learners use their home languages
they interacted better and freely with their peers and
their teachers. The home language served as a reference
point for words that were ambiguous and unfamiliar to
learners. Mathematical practices such as conceptual
understanding, procedural fluency, adaptive reasoning
and strategic competence were furthermore facilitated by
the use of learners’ home language.
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LEARNING MATHEMATICS THROUGH INTEGRATION WITH CONTEXTS
THAT DRAW ON LEARNERS’ EVERYDAY EXPERIENCES BOTH IN
MATHEMATICS AND MATHEMATICAL LITERACY CLASSES: A PILOT
STUDY
Frans Machaba
Refithlile-Pele Primary School
In this paper I draw on a
pilot study to describe and explain the learning of
mathematics through integration with contexts that draw
on learners’ everyday experiences both in Mathematics
and Mathematical Literacy classes, using Bernstein codes
of recognition and realization rules and Bourdieu’s
notion of habitus. This pilot is part of a wider study
for my doctoral research which is still in progress.
The study is located within a qualitative approach,
adopting a multiple case study which will focus on two
secondary schools from contrasting backgrounds in South
Africa. For piloting and for this paper I chose only one
township school to explain and discuss the results. In
this school two classes of approximately 30 learners in
each were chosen, one Mathematical Literacy class and
one Mathematics class. The 5 learners in each class
(both Mathematics and Mathematical Literacy) were
selected based on what they have written in their
learners’ background questionnaires scripts. Thus a
total of 10 learners were chosen for working on the task
and for oral presentation. Data was collected using
task instruments and interviews. The focus is on two
issues that arose from this pilot – the problems some
learners have when negotiating the boundary between
esoteric mathematical knowledge and their everyday
knowledge and learners’ background differences in
relation to their views with mathematics and
mathematical Literacy.
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An exploratory study into the introduction of
Mathematical Literacy in Selected Cape Peninsula High
Schools
Monde
Mbekwa
University of the Western Cape
This is
an exploratory study to investigate the challenges of
implementing the new mathematical literacy curriculum in
high schools in the Western Cape. Eight schools from
previously disadvantaged areas in the Cape Peninsula of
the Western Cape Province of South Africa were studied
through class visits, interviews with teachers of
mathematical literacy, video taping of lessons and an
analysis of students’ results in the June and November
examinations in 2006. Initial results of the study show
that in these schools, the majority of teachers teaching
mathematical literacy have professional qualifications
in mathematics. There are also some teachers with other
subject specialisations who also teach the subject.
Teachers report that learners have a negative attitude
to mathematical literacy and are struggling to
understand the new subject as can be attested to by the
dismal showing in the June examinations.
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“TRYING TO DESCRIBE THE AIR” AS A METAPHOR IN
CONCEPTUALISING THE MATHEMATICS-FOR-TEACHING
Mike Mhlolo,
Marang Centre for Mathematics & Science Education,
University of the Witwatersrand – Johannesburg
Ways of presenting and
formulating concepts in Mathematics that makes the
subject comprehensible to others is at the core of
effective teaching. Current research warns that the
Mathematics that teachers need for teaching Mathematics
is not of the kind that their learners do, neither is it
just being a step ahead in content of the students learn
and neither is it some kind of a watered-down version of
formal Mathematics. It is in fact a more serious and
demanding area of Mathematical work and therefore should
be seen as tacit and as a distinct branch of
Mathematics. Because achievement in mathematics has
always been seen as symptomatic of the quality of
education and training in a country, developing this
special kind of mathematical knowledge for teaching
within a teacher has been one of the most critical
issues on the agenda for teacher professional
development endeavors the world over. However, while the
issue of teachers’ knowledge of mathematics has been a
prominent one for several decades, little progress has
been made towards a consensus on the question of ‘what
mathematics teachers need to know.’ It is like trying to
describe the air, we know it is there, we can feel it is
there but it is a very elusive concept to define or
describe!
This paper gives an analysis
of the current discourses on the dilemmas that policy
formulators and implementers face with regards the
conceptualization of this mathematics-for-teaching and
the debates for and against subject matter knowledge for
Mathematics and the Mathematics-for-teaching. The paper
will end with some suggestions for the way forward
especially for developing countries as they try to
grapple with ways of improving the teaching and learning
of Mathematics for their citizens.
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USING MULTIPLE LANGUAGES TO SUPPORT
MATHEMATICS PROFICIENCY IN A GRADE 11 MULTILINGUAL
CLASSROOM OF SECOND LANGUAGE
LEARNERS: AN ACTION RESEARCH
Terence Baron Molefe
Fons Luminis Secondary
School
This
paper provides an overview of my Master’s Research
report. The study explores the deliberate use of
multiple languages to support the development of grade
11 learners’ mathematics proficiency in a multilingual
classroom. The study is an action research aimed at I
transforming my teaching. A well-selected mathematical
task set in multiple languages was used for teaching and
multiple languages were used in a planned and proactive
manner.
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This report is based on a
doctoral thesis in research psychology, titled
“"Behavioural Correlates and Specific Attitudes in
Students Exposed to Mathematical Programmes at
Interfaculty Levels", in which the effects of short- and
long-term mathematics anxiety was probed with reference
to both short and long-term study behaviour and ultimate
performance in both algebra and geometrical
examinations. Differences in the dependent variables
were not always accompanied by corresponding differences
in performance and the procedures and results are
discussed as below.
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THE MIDMAR MILE, MIXING
CONCRETE, AND OTHER ANOMALIES IN MY MATHS LITERACY
CLASSROOM
Marc North
St. Anne’s
Diocesan College (Hilton, KZN)
In this
paper I reflect on three experiences in my Grade 10
Mathematical Literacy classroom last year in 2006 that
opened my eyes to the complexities and difficulty
involved in teaching and learning Mathematical Literacy.
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In this
paper, I describe and discuss ways in which connections
are described in the South African Mathematics National
Curriculum Statement and related documents. The
centrality of connections in the conceptualization of
mathematics and the learning outcomes and assessment
standards is identified and discussed. The notions of
representation and integration as key aspects in
understanding connections in mathematics are analysed
with respect to the National Curriculum statement.
Finally, theoretical and practical implications of
connections in the curriculum are identified.
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TEACHER’S SELECTION OF CONTEXTS IN
MATHEMATICS LEARNING SUPPORT MATERIALS FOR TEACHING THE
CONCEPT
VARIABLE
Moshe Phoshoko
Over the years mathematics has been
perceived by many people as a subject that is generally
difficult to understand and to learn. Barbeau (1990, as
cited in Picker and Berry 2001: 65) puts it more aptly
by saying: ‘Probably no area of human activity is
afflicted as mathematics with a gap between the public
perception of its nature and what its practitioners
believe it to be.’ Efforts to close this gap have
resulted in several explorations and researches on how
to make the subject accessible to the broader public
LOOKING AT HOW A GRADE SIX EDUCATOR PROMOTES CONCEPTUAL
UNDERSTANDING WHEN INTRODUCING MIXED AND IMPROPER
FRACTIONS.
Lindiwe
Tshabalala
Thuthuzekani Primary School (D2 Gauteng West)
This
research looks at how a grade six teacher promotes
conceptual understanding in learners when teaching
improper and mixed fractions. This study is informed by
notions of conceptual and procedural mathematics. The
study is also informed by constructivist theory and
looks at how the learners construct and restructure
knowledge when learning. The study was conducted in one
school. Data was collected through video-taping a lesson
and interviewing the teacher with a tape recorder. In
doing the analysis the researcher demarcated the work
into different categories. The findings indicated that
the teacher does promote conceptual understanding in
learners, but in conversions of fractions she did not.
The recommendations suggest that in order to promote
conceptual understanding, teachers should use different
representations of fractions.
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LEARNERS’ EXPERIENCES OF MATHEMATICAL LITERACY IN GRADE
10
Hamsa Venkat & Mellony
Graven
Marang Centre for Mathematics and Science Education
University of the Witwatersrand
Mathematical Literacy was introduced as a new subject in
the Further Education and Training (FET) phase in
January 2006. It is structured in the FET curriculum as
an alternative option to Mathematics, with all learners
in this phase having to take one or the other option.
This changes the previous situation in which some 40% of
all FET learners took no mathematics at all.
Our focus
in this paper is on learners’ experiences of
Mathematical Literacy (ML) in Grade 10. The data used
was collected from intensive longitudinal research in
one inner-city Johannesburg school which had three
classes taking ML in Grade 10 (alongside three
Mathematics classes).
To
date, publications in the area of ML in South Africa
have (predictably) tended to focus on more theoretical
elements such as the nature of ML as a subject
(Christiansen,
2006) (Brown & Schafer, 2006),
policy, or text-based analyses (Bowie & Frith, 2006;
Venkatakrishnan & Graven, 2006) and teacher
development issues (Brown & Schafer, 2006).
Much of this writing, including our own, has drawn
attention to potential problems, tensions and
contradictions within the policy texts, and the mixed
messages conveyed about the nature and processes
associated with ML.
Set against this backcloth
of potential problems, our weekly observations and
interactions with learners in the ML classes at the
school mentioned above, pointed from the early stages,
to positive experiences of learning. In this paper, we
begin to examine the nature of learners’ experiences
more deeply, framed as they were, in overwhelmingly
positive terms in comparison to prior experiences of
learning mathematics. Within this notion of reference
back to mathematics, our data also provided recurring
examples of vividly negative memories of mathematics
learning for the majority of Grade 10 ML learners.
Prior to
considering the nature of learners’ experiences, we
provide some background detail about the broader study,
the focal school and the three ML teachers involved, and
the data sources drawn upon within this paper.
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