TMS3725 EXAM
PACK 2024
QUESTIONS AND
ANSWERS
FOR ASSISTANCE CONTACT
EMAIL:
, lOMoARcPSD|31863004
2
Lesson 1: The nature of mathematics at the FET level
Lesson outcomes
After reading this Lesson and engaging in the activities, you should be able to understand
1 the nature of mathematics in the FET phase
2 the discipline of mathematics
3 mathematical concepts
4 the role of the FET mathematics teacher
5 the framework and content of the CAPS document
6 the different ways of presenting knowledge and facilitating the learning of mathematics.
According to the Curriculum and Assessment Policy Statement (CAPS), mathematics is a language
that makes use of symbols and notations to describe numerical, geometric and graphical
relationships. It is a human activity that involves observing, representing and investigating patterns
and qualitative relationships in physical and social phenomena, and between mathematical objects
themselves. It helps to develop mental processes that enhance logical and critical thinking, accuracy
and problem solving, all of which will contribute to decision making. Mathematical problem solving
enables us to understand the world (physical, social and economic) around us, and, most of all, it
teaches us to think creatively (DBE 2011). Those qualities that are nurtured by mathematics include
the powers of reasoning, as well as our creativity, abstract or spatial thinking, critical thinking,
problem-solving abilities and even our communication skills.
The term ‘mathematics’ is derived from two Greek words, manthanein (which means learning) and
techne (the art or technique). Thus, mathematics refers to the art of learning as it relates to scientific
disciplines. Mathematics is a science of discovery – the discovery of relationships and the
expression of those relationships in symbolic form. It is also an intellectual game, which makes use
of puzzles, paradoxes and problem solving. It is a science of patterns and relationships.
Mathematics reveals hidden patterns that help us discover the world around us. It is a way of
thinking. It relies on logic and creativity, is pursued for a variety of practical purposes and also for
intrinsic interest. Mathematics is an art: numerous patterns are to be found in numbers and
geometric figures. It has a language of its own. Like any language, it is made up of concepts,
symbols, terminology and algorithms, and it has a syntax which is unique to it. It is based on a
certain consistent set of assumptions, and is built up from there, according to rules and logic. It is a
broad and deep discipline that continues to expand. It is a systematic, deductive science and an
Downloaded by Gabriel Musyoka ()
, lOMoARcPSD|31863004
3
experimental inductive science, involving intuitive method. It is also a science of precision and
accuracy.
Mathematics is a discipline with connections. Making those connections involves two different
processes: 1) building new mathematical ideas from learners’ prior experiences, and 2) highlighting
connections between mathematical topics. Connections between mathematical topics, between
mathematics and other subjects, and between mathematics and everyday life, all contribute to
making mathematics understandable and meaningful. According to the Department of Basic
Education (DBE 2003, p. 10), an important purpose of mathematics in the FET band/phase is the
establishment of proper connections between mathematics as a discipline and the application of
mathematics in real-world contexts. These connections are concerned with what mathematics is: 1)
its origin as a human activity, a construction, a development and a contestation that is time-
and socially dependent, 2) the processes involved in doing mathematics: problem solving,
investigating, communicating, making representations, making connections, reasoning, and
understanding the world and daily life. Mathematics is not about reasoning for its own sake: it is
concerned with reasoning, symbolising and thinking processes that are connected to activities and
problems of the social, physical and mathematical worlds, and involve human practices in all
cultures. There is thus a conceptual and social dimension to the essence and use of mathematics.
Mathematics is a highly conceptual domain, a field of knowledge consisting of concepts that are
structured in specialised ways. This means that the processes of knowing and understanding
mathematics are also specialised. According to Kilpatrick, Swafford and Findell (2001), the ability to
do mathematics well, to represent and communicate mathematics effectively, hinges on individuals
having achieved a conceptual understanding of mathematical concepts and procedures, and the
relations between those concepts and procedures. In particular, the “powerful conceptual tools” that
are made available by mathematics, enable learners to “analyse situations and arguments; make
and justify critical decisions; and take transformative action” (DBE 2007, p. 7).
Also, mathematical similarities and differences that appear in seemingly unrelated instances help
reinforce the interconnected nature of mathematics, and encourage a sense of wonder and
inquisitiveness in mathematicians and students alike. As an example, consider the many ways in
which Pascal's Triangle can be applied in combinatorial problems or in a study of the binomial
theorem. Similarly, triangular numbers appear in various visual pattern-finding settings, such as:
“Which whole numbers can be expressed as a sum of consecutive whole numbers?” If we consider,
for example, preparing students to work with connections to data and statistics, they may collect
quantitative data in a science lab or classroom, enter them as a table of values in a graphing
calculator or spreadsheet program, create a scatter plot, and establish a line or curve of best fit.
Downloaded by Gabriel Musyoka ()
, lOMoARcPSD|31863004
4
Through this process, they may be able to link the original situation to an algebraic equation. In
addition, within the domain of geometry, while ensuring that learners are mathematically literate,
they are required to work towards being able to “describe, represent and analyse shape and space
in two and three dimensions using various approaches in geometry and trigonometry in an
interrelated or connected manner” (DBE 2003, p. 10). This will enable learners to prepare for
mathematical modelling, and will show them the human creation and beauty which is mathematics.
According to the National Curriculum Statement (NCS), mathematics as a discipline is viewed as
follows:
Mathematics enables creative and logical reasoning about problems in the physical and
social world and in the context of mathematics itself. It is a distinctly human activity
practised by all cultures. Knowledge in the mathematical sciences is constructed
through the establishment of descriptive, numerical and symbolic relationships.
Mathematics is based on observing patterns; with rigorous logical thinking, this leads to
theories of abstract relations. Mathematical problem solving enables us to understand
the world and make use of that understanding in our daily lives. Mathematics is
developed and contested over time through both language and symbols by social
interaction and is thus open to change. (DBE 2003, p. 9)
Also, the National Council of Teachers in Mathematics (NCTM) (1989, p. 148) emphasises that a
suitable mathematics curriculum should include an investigation into the connections between, and
the interplay among, various mathematical topics and their applications, such that learners can
• recognise equivalent representations of the same concept;
• relate procedures in one representation to procedures in an equivalent representation;
• use and value the connections among mathematical topics; and
• use and value the connections between mathematics and other disciplines.
To be able to assist your learners in making connections in mathematics, Presmeg (2006, p. 167)
suggests first starting with an everyday practice that is meaningful to them, and seeing what
mathematical notions grow out of the linking as it is developed. Second, you might focus on the
mathematical concept you wish to teach, and then search for a starting point in the everyday
practices of your learners, to create several links with the concept as part of the chaining process.
The purpose of this module is to familiarise you with the methodological issues involved in teaching
mathematics, as proposed in the FET’s CAPS document. As a student who qualifies in this module,
Downloaded by Gabriel Musyoka ()
PACK 2024
QUESTIONS AND
ANSWERS
FOR ASSISTANCE CONTACT
EMAIL:
, lOMoARcPSD|31863004
2
Lesson 1: The nature of mathematics at the FET level
Lesson outcomes
After reading this Lesson and engaging in the activities, you should be able to understand
1 the nature of mathematics in the FET phase
2 the discipline of mathematics
3 mathematical concepts
4 the role of the FET mathematics teacher
5 the framework and content of the CAPS document
6 the different ways of presenting knowledge and facilitating the learning of mathematics.
According to the Curriculum and Assessment Policy Statement (CAPS), mathematics is a language
that makes use of symbols and notations to describe numerical, geometric and graphical
relationships. It is a human activity that involves observing, representing and investigating patterns
and qualitative relationships in physical and social phenomena, and between mathematical objects
themselves. It helps to develop mental processes that enhance logical and critical thinking, accuracy
and problem solving, all of which will contribute to decision making. Mathematical problem solving
enables us to understand the world (physical, social and economic) around us, and, most of all, it
teaches us to think creatively (DBE 2011). Those qualities that are nurtured by mathematics include
the powers of reasoning, as well as our creativity, abstract or spatial thinking, critical thinking,
problem-solving abilities and even our communication skills.
The term ‘mathematics’ is derived from two Greek words, manthanein (which means learning) and
techne (the art or technique). Thus, mathematics refers to the art of learning as it relates to scientific
disciplines. Mathematics is a science of discovery – the discovery of relationships and the
expression of those relationships in symbolic form. It is also an intellectual game, which makes use
of puzzles, paradoxes and problem solving. It is a science of patterns and relationships.
Mathematics reveals hidden patterns that help us discover the world around us. It is a way of
thinking. It relies on logic and creativity, is pursued for a variety of practical purposes and also for
intrinsic interest. Mathematics is an art: numerous patterns are to be found in numbers and
geometric figures. It has a language of its own. Like any language, it is made up of concepts,
symbols, terminology and algorithms, and it has a syntax which is unique to it. It is based on a
certain consistent set of assumptions, and is built up from there, according to rules and logic. It is a
broad and deep discipline that continues to expand. It is a systematic, deductive science and an
Downloaded by Gabriel Musyoka ()
, lOMoARcPSD|31863004
3
experimental inductive science, involving intuitive method. It is also a science of precision and
accuracy.
Mathematics is a discipline with connections. Making those connections involves two different
processes: 1) building new mathematical ideas from learners’ prior experiences, and 2) highlighting
connections between mathematical topics. Connections between mathematical topics, between
mathematics and other subjects, and between mathematics and everyday life, all contribute to
making mathematics understandable and meaningful. According to the Department of Basic
Education (DBE 2003, p. 10), an important purpose of mathematics in the FET band/phase is the
establishment of proper connections between mathematics as a discipline and the application of
mathematics in real-world contexts. These connections are concerned with what mathematics is: 1)
its origin as a human activity, a construction, a development and a contestation that is time-
and socially dependent, 2) the processes involved in doing mathematics: problem solving,
investigating, communicating, making representations, making connections, reasoning, and
understanding the world and daily life. Mathematics is not about reasoning for its own sake: it is
concerned with reasoning, symbolising and thinking processes that are connected to activities and
problems of the social, physical and mathematical worlds, and involve human practices in all
cultures. There is thus a conceptual and social dimension to the essence and use of mathematics.
Mathematics is a highly conceptual domain, a field of knowledge consisting of concepts that are
structured in specialised ways. This means that the processes of knowing and understanding
mathematics are also specialised. According to Kilpatrick, Swafford and Findell (2001), the ability to
do mathematics well, to represent and communicate mathematics effectively, hinges on individuals
having achieved a conceptual understanding of mathematical concepts and procedures, and the
relations between those concepts and procedures. In particular, the “powerful conceptual tools” that
are made available by mathematics, enable learners to “analyse situations and arguments; make
and justify critical decisions; and take transformative action” (DBE 2007, p. 7).
Also, mathematical similarities and differences that appear in seemingly unrelated instances help
reinforce the interconnected nature of mathematics, and encourage a sense of wonder and
inquisitiveness in mathematicians and students alike. As an example, consider the many ways in
which Pascal's Triangle can be applied in combinatorial problems or in a study of the binomial
theorem. Similarly, triangular numbers appear in various visual pattern-finding settings, such as:
“Which whole numbers can be expressed as a sum of consecutive whole numbers?” If we consider,
for example, preparing students to work with connections to data and statistics, they may collect
quantitative data in a science lab or classroom, enter them as a table of values in a graphing
calculator or spreadsheet program, create a scatter plot, and establish a line or curve of best fit.
Downloaded by Gabriel Musyoka ()
, lOMoARcPSD|31863004
4
Through this process, they may be able to link the original situation to an algebraic equation. In
addition, within the domain of geometry, while ensuring that learners are mathematically literate,
they are required to work towards being able to “describe, represent and analyse shape and space
in two and three dimensions using various approaches in geometry and trigonometry in an
interrelated or connected manner” (DBE 2003, p. 10). This will enable learners to prepare for
mathematical modelling, and will show them the human creation and beauty which is mathematics.
According to the National Curriculum Statement (NCS), mathematics as a discipline is viewed as
follows:
Mathematics enables creative and logical reasoning about problems in the physical and
social world and in the context of mathematics itself. It is a distinctly human activity
practised by all cultures. Knowledge in the mathematical sciences is constructed
through the establishment of descriptive, numerical and symbolic relationships.
Mathematics is based on observing patterns; with rigorous logical thinking, this leads to
theories of abstract relations. Mathematical problem solving enables us to understand
the world and make use of that understanding in our daily lives. Mathematics is
developed and contested over time through both language and symbols by social
interaction and is thus open to change. (DBE 2003, p. 9)
Also, the National Council of Teachers in Mathematics (NCTM) (1989, p. 148) emphasises that a
suitable mathematics curriculum should include an investigation into the connections between, and
the interplay among, various mathematical topics and their applications, such that learners can
• recognise equivalent representations of the same concept;
• relate procedures in one representation to procedures in an equivalent representation;
• use and value the connections among mathematical topics; and
• use and value the connections between mathematics and other disciplines.
To be able to assist your learners in making connections in mathematics, Presmeg (2006, p. 167)
suggests first starting with an everyday practice that is meaningful to them, and seeing what
mathematical notions grow out of the linking as it is developed. Second, you might focus on the
mathematical concept you wish to teach, and then search for a starting point in the everyday
practices of your learners, to create several links with the concept as part of the chaining process.
The purpose of this module is to familiarise you with the methodological issues involved in teaching
mathematics, as proposed in the FET’s CAPS document. As a student who qualifies in this module,
Downloaded by Gabriel Musyoka ()
