THEORIES IN MATHEMATICS EDUCATION
MATHEMATICS METHOD (GET/FET Phase)
2024
Dee Koopman
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COURSE OBJECTIVE
To introduce pre-service teachers to key theories and
philosopies in mathematics education and its implications
for teaching and learning.
DURATION
120 Minutes
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What were your experience in the maths class?
Who inspired you to do math?
How would you change the math classroom?
Which phase are your area of focus for math
teaching?
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Are you aware of how mathimatics is taught in
different schools?
How do you think you will be teaching maths?
Which are the important elements in a classroom
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OVERVIEW OF THE LESSON
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Course objective
Ideas and expectations of the course
Lesson overview
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NB of understanding philosophies and theories in mathematics education for effective teaching.
What is a theory
Different approaches tolearning theories and perspectives in mathematics education
Explanation of each theory,
Discussion of how each theory or perspective can inform instructional strategies, assessment practices, and classroom
management
Pre-service educator
– influences on pre-service educator
– characteristics of pre-service educator
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Training of educators - Gap between theory and practice
– different solution for the educator - latest research
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Application and Reflection - your own narrative - Group (10 minutes):
Group activity or discussion: Group activity - problem solving
Assignment
Summary of key points from the lesson (5 min)
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• Critical reflection
• Argumentation
• Perspectives and world view
Philosophies
• Exploration of different perspectives
Broad discipline
Concerned with fundemental questions
To understand
Philosophies and
Theories
THEORIES
• EXPLANATION OR MODEL
• CONCEPTS, ASSUMPTIONS & IMPLICATIONS
• HOW (e.g. Teaching and Learning works
• WHY something happens
• Systematic & coherent
• based on empirical studies
• makes predictions and
• tests hypothesis
• used in various fields
• math, education, science, psychology,
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HUMAN
• Human behaviour is influenced by various
factors which include
• The biological factory
• Environments circumstances and stimuli
• Social factors
• Culture
• Psychological and spiritual factors
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BACKGROUND
• Age of Reason 1600-1879 - The human being and the world replace
the focus with less focus on God as a philosophical reflection, with
human reasoning at the forefront. The philosophical approaches
created challenges for the initial scientific approach. Two distinct
bodies of knowledge (epistemology) resulted from various
philosophical questions which are:
• EMPERICISM - approach to philosophy of science that assumes
they only source of true knowledge is OBSERVATION THROUGH
THE SENSORY PERCEPTION. (Francis Bacon (1561-1626)
• RATIONALISM - propose that HUMAN REASON is the only source
of true knowledge. (Math)
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THEORIES
• What is a theory
– systematic and coherent set of facts, ideas, concepts, or
principles that explains and predicts a phenomenon or a set of
phenomena (Louw & Louw:2014)
• Key theories and philosophies
– Behaviorism, Constructivism, Socio-Cultural Theory, Realistic
Mathematics Education, Progressivism etc.
• Philosophy
– is a broader discipline that is concerned with understanding the
nature of reality, knowledge, and human experience.
– involves critical reflection, argumentation, and the exploration of
different perspectives and worldviews.
• e.g. Essentialism (teacher at the centre), progressivism,
(student at the centre) critial theory
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THEORIES COMPOSITION
• Every theory is based on (Meyer, Moore & Viljoen: 24-27)
– epistomological
• Epistemological" refers to the study or theory of knowledge,
particularly how it is acquired, justified, and applied. It comes
from the Greek words "episteme" (knowledge) and "logos"
(study)
– methodological ideas
• (deductive, inductive, qualitative, quantivative, objective
methodology
– philosophical ideas (nature and aim of maths education)
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DIFFERENT APPROACHES TO LEARNING THEORIES
• BEHAVIOURISM
– learning is a studied observation and
manipulation of stimulus-response
associations resulting in specific
behaviours
CRITICAL LEARNING THEORY
critical philosoper - Paulo Freira (Pedagogy of the
Oppressed) & Henry Giroux
Freira - traditional education systems perpetuate
social inequalities - student = passive receiver of
knowledge vs active participant
advocates a dialogic & paticipatory accproach
Giroux - critically question and challenge
dominant narratives and power structures but also
to recognize and celebrate diversity and
difference.
• COGNITIVE APPROACH
– intervening variables are necessary
components for understanding LEARNING
PROCESS
– study of thought processes
– other areas - attention theories, memory
techniques, mental imagery, problem
solving, language acquisition, decision
making
CONSTRUCTIVISM
social learning theory
central is active and passive
involvement in learning (RME)
progressive approach (RME)
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THEORY OF TEACHING
• Kerlinger(1965) has defined the terms theory of teaching:
“A theory of teaching is a set of interrelated constructs,
definitions, propositions which present a systematic view
of teaching by specifying relations among variables with
the purpose of explaining and predicting”.
Because of the dual nature of teaching and learning, the
relationship between these variables (teacher, learner,
environment, tools) is critical for mathematics education
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LEARNING AND TEACHING
• In order to ensure maximum learning, the conflation of
these theories become imperative during facilitation and
teaching of mathematics or any other subject
• The process of learning allows the educator to adjust the
teaching accordingly to the process (mathematics),
person (learner), place (environment)
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LEARNING
TEACHING
TEACHING & LEARNING THEORIES APPLICATIONS
Behaviourism
Cognitivism
Constructivism
traditional teaching approach,
where the teacher presents
mathematical concepts and
formulas, and students practice
and drill problems to reinforce
learning.
presenting mathematical
concepts in a logical and
systematic way, emphasizing the
development of students'
understanding and problemsolving skills.
providing students with handson, inquiry-based learning
experiences where they can
explore mathematical concepts
and solve authentic problems.
conditioning through repetition
and reinforcement.
Students learn through practice
and the application of
mathematical rules and
procedures.
role of the learner in actively
emphasizes the active mental
constructing their own
processes involved in learning
understanding of mathematics
mathematics,
such as how students organize through experiences and reflection.
Learning mathematics = process of
and process information, and
how they use prior knowledge to building on prior knowledge and
make sense of new mathematical making connections between
concepts.
concepts.
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LEARNING
TEACHING
TEACHING & LEARNING THEORIES APPLICATIONS
Social Constructivism
collaborative problem-solving
activities and discussions where
students work together to
construct mathematical
knowledge and understanding.
emphasizes social interactions
and cultural context in learning
mathematics.
Learning = social process that
occurs through
collaboration,discussion, and
negotiation of meaning.
Connectivism
Postmodernism
a connectivist approach might
involve recognising and valuing
involve using digital tools and
multiple perspectives and
resources to connect students
approaches to mathematical
with mathematical content and problem-solving,
to facilitate collaborative learning promoting critical reflection on
experiences.
the social and cultural contexts of
mathematics.
Focus is on technology and digital recognizing and valuing diverse
networks in learning
mathematical perspectives and
mathematics.
approaches.
Learning = distributed process
Learning = process of critically
that occurs across networks of engaging with mathematical
people and technology.
ideas and contexts.
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LEARNING
TEACHING
TEACHING & LEARNING THEORIES APPLICATIONS
Critical Theory
Multiple Intelligences Theory
critiquing traditional approaches recognizing and valuing diverse
to teaching and learning
mathematical strengths and
mathematics that may
abilities, as well as providing
perpetuate inequities, as well as opportunities for students to
advocating for more socially just engage with mathematical
and inclusive mathematics
content in ways that align with
education practices.
their individual intelligences.
promoting social justice and
recognizing and valuing diverse
equity in mathematics education. mathematical strengths and
Learning = critically examining abilities.
and challenging power structures Learning mathematics is seen as
and social inequalities.
a process of engaging with
mathematical content in ways
that align with individual
intelligences.
Ecological Systems Theory
understanding and addressing
the ecological factors that may
impact students' mathematical
learning experiences, such as
family, community, and cultural
influences.
the importance of considering
the multiple systems and
contexts that influence an
individual's mathematical
development. Learning
mathematics is seen as a process
of navigating and making
connections within these systems
and contexts.
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WHAT IS YOUR NARRATIVE / BELIEFS ON ....
APPLICATIONS OF PHILOSOPHY AND
THEORIES
A brief discussion of 5 minutes on your
belief,
It is this transcient arena that you will
problably find in the application of the
various of teaching and learning theories.
https://www.slideshare.net/diocylannrequillo/comparison-edu-philo-79947725
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A CLOSER LOOK AT BEHAVIOURIST MODEL
Human as
Individual(biological )
Individual i.r.t other people
Social Psychology
Individual i.r.t. environment
Individual i.r.t. transcendent
environment
• Originated in early 1900 - Pavlov, Skinner,
Bloom’s taxonomy
• consists of change in behaviour due to
acquisition and application of
associations between stimuli
(environment/person) & responses
(observable)
• belief any concept can be learned
given enough time and prerequisite
concepts
• concepts followed logically & learned by
rule or procedural learning
• strong drill & practice approach
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BEHAVIOURIST MODEL CONTD
Human as
Individual(biological )
Behaviour of the individual ExtraPhysically (environment )
Functioning is alsways a result of
specific environmental STIMULI
Intrapsycically
Unconscious variables in individual
psycological structure with no
intervention from the enviroment.
Freud’s ego/superego is an
internalised component
Individual i.r.t other people
Social Psychology
Social phase
Shift in thought of invidual as
social and not biological entity
only. Assumption is that indivual
can only be understood in social
context.
InterpersonalSocial environment
is a VARIABLE that influences
human functioning e.g. SOCIAL
LEARNING THEORY Social
Oriented Psycho-analytical Theory
View The individual is a social
Interaction between person and
being->>> Gestalt Psychology
environment where the
The focus on perception
ENVIRONMENT is the independent
variable. Interactional Model
Clinical Psychology Applied
The person and the environment
science and specialist field
are regarded as independent
Cognitive PsychologyIt is the study variables that has a reciprocal
of cognition & consciousness:
influence on one another.
Transactional modelPerson and
Feeling EmotionWill Information
environment are regarded as
Processing organismChange in
INTERDEPENDENT variables that
cognition = a change in behaviour cannot be defined inseparably
from the transaction.
Individual i.r.t. environment
Individual i.r.t. transcendent
environment
Environmental Psychology
The individual within their physical
environment. The person’s
relationship with their immediate
environment.
Ecosystemic Model An extention
of the interactional model with the
environment as the variable that
bring the:
Behaviourism: Values, religion,
beliefs and morality could not be
easily investigated empirically.
Psychology of ReligionPerson in
relation to TRANSCENDENT
environment which is associated
with THEOLOGY and PHILOSOPHY.
Humanistic Person Oriented
Allowances were made for religion
as part of the human psychic
functioning.
Social environmentPhysical
environment andPerson As
INTERDEPENDENT VARIABLES in
describing the person’s behaviour.
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CONSTRUCTIVISM CONT
• People construct their own realities through meaning linked to what they observe what is observed however does not only have independent meaning but should be
seen against the CO-CONSTRUCTED reality of participants about themselves,
each other, the problem/object and the world in general (Meyer, Moore &
Viljoen:472) (son in law example)
• two approaches – passive or active participation
• passive receiver - of ? - knowledge, behaviour, roles, attitude, values which is
shaped and maintained by the environment
– active participant
• dialectical - interaction between person and environment is AN ACTIVE
PROCESS
• constructivism is the learner constructing his understanding of mathematics through
resources with mathematics ahis experiences and the environment in the
CONSTRUCTION OF HIS KNOWLEDGE = LEARNER CENTERED
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STRUCTURAL THEORISTS
• originates in both behavioural and cognitive theories of psychology
• objective - focus on learners’ identification of the structures and
processes in the forming of mathematical concepts
• spiral curriculum is applicable with the development of concepts
hierarchically and at different stages which conforms to a structural
theory
PLATONIST PHILOSOPHY
• According to Lamar (2012) Platonism refers to the viewpoint that objects and
entities constructed and defined in the work of mathematics actually exist
independent of our sense preception.
https://scholarworks.calstate.edu/downloads/m900nv47k
• Various philosophies use Platonist as a tool to set their theories against.
Lamar 2012
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LEARNING THEORIES
central to discipline of educational
psychology
learning is a basic psychological
process - defined as a lasting
influence on
behaviour
knowledge
thinking
that comes through experience
however, not all learning is learned, it
is innate
Santrock 2011:217
developed to provide a common language for teachers to
discuss and exchange learning and assessment methods.
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YOUR POSITION
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Based on
past experiences - own study methods
observation of subject teachers
experiences with & from others
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New injection of new theories and methods
Professional development at universities
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Expectation is to narrow the gap between theory
and practice. (Korthagen: 2016)
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CHARACTERISTICS OF PRE-SERVICE EDUCATOR
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Limited experience
Might default to traditional methods
Exposed to newer theories in education and learning
New resources available (technical & otherwise)
New attitudes to education and learning
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GAP - TRADITIONAL THEORIES OF INSTRUCTION
MATH
• Behaviourist - teaching
– Teacher centered
– Direct instruction - no cooperative or peer learning
– Lecture style
– Drill & Practice
– Transmission fm educator to
learner (holder of information minimal engagement)
– complex task into smaller steps
– spiral curriculum
• Behaviourist - learning
– reward & punishment behaviour
– student listens and makes notes
(Higher Edu)
– memorisation & repitition
– passive learning
– includes formula learning, facts,
– pace of learning set by educator
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GAP - TRADITIONAL THEORIES OF INSTRUCTION MATH
• Procedural skills - traditional behaviourist but necessary
– numeracy skills, algorithms to solve standardised tasks, specific
mathematical knowledge
• Using the necessary “tool of delivery” for specific mathematical skills to
be acquired
– doing a series of computations for arrival at same answer
• Attention was paid to mechanical memorisation of
– definitions and statements, tables, addition, subtraction, multiplication, division,
formulas, explanations, math rules, etc
• New resources available (technical & otherwise)
• New attitudes to education and learning
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TRAINING OF MATHEMATICS EDUCATORS
• Belief system you come with to the classroom/school
– instructional belief (Lubisi, 1997)
• personal methods used to master mathematics concepts
• how you as educator were instructed
• experience gained (school, DoE, personal development)
• Objective of this programme
– impart new philosophies and theories for education and learning
– provide pedagogical tools (PCK) to influence or align with your beliefs (TH)
https://www.tandfonline.com/doi/epdf/10.1080/13540602.2016.1211523?needAccess=true
•
https://journals.co.za/doi/pdf/10.10520/EJC-c512fa4fd
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MIND THE GAP
•
Research:
– Spangenberg (2017) (UJ)
“insufficient pedagogies in teacher
education to integrate preservice
teachers theory with experiences in
teaching
– Korthagen (2016) Teacher learning
takes place at various levels and
intervals and includes the CORE
qualities of the teacher, which is
brought to the core with
REFLECTION (Deepening Reflection
by bringing the person to the
profession) – RME (Realistic Mathematics
Education)
•
https://slideplayer.com/slide/13546922/
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REFLECTION OF EDUCATOR
• Teacher learning is linked to theory - practice - self reflection (person)
• In the person, the COGNITIVE, AFFECTIVE AND MOTIVATIONAL SOURCE
of behaviour is interlinked which operates in the social environment =
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MULTIDIMENSIONAL LEARNING
REALISTIC MATHEMATICS EDUCATION (RME)
• RME was founded in the Netherlands by Wijdeveld,Goffree & Treffers. (1971)
• Students are given problem situations which they can “imagine” as realistic
situations as part of the learning process and a “reality” in the student’s mind.
• These situations serve to initiate the development of mathematical concepts, tools
and procedures as a context for later application.
• Freudental (1968,1973,1991) introduced didactical phenomenology. (concerned with
describing how a mathematical idea could emerge in a learning and teaching
process as a means to organize phenomena - jstor).
– Describe the mathematical concept, structures, ideas
– in their RELATION to the phenomena
– taking into consideration students learning processes
• objective is for students to develop mathematical tools and insights for themselves
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REALISTIC MATHEMATICS EDUCATION (RME)
• According to Freudenthal (van der Heuvel-Panhuizen: 2014)
mathematics must not be learned as a closed system but as AN
ACTIVITY OF MATHEMATIZING REALITY, and if possible, the
MATHEMATIZING OF MATHEMATICS
• He distinguishes between horisontal - vertical mathematisation
– horisontal = students problem-solve real-life situations
– vertical = reorganising within the mathematical system which
resulted in shortcuts by connecting concepts and strategies
• A balance between the horisontal and vertical mathematisation is
critical.
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RME CORE PRINCIPLES
• Treffers articulated these 6 core principles
• Activity principle
– students are active participants in the learning process
• Reality principle
– mathematics should be applied to real life problems
– mathematics should start from problem situations that are MEANINGFUL to
the students, which allows them to attach MEANING to the mathematical
constructs they developed while solving problems
• Level principle
– various levels of understanding must be passed
• from informal context related solutions
• through shortcuts and schematicization
• to acquiring insights into HOW concepts and strateies are related
– Models narrow the gap between informal context-related math and formal
maths.
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RME CORE PRINCIPLES CONTD
• Intertwinement principle
– math content domains e.g. numbers, geometry, measurement and datahandling is not isolated curriculum chapters are INTEGRATED
– students can use various mathematical tools and knowledge to solve problems
• Interactivity principle
– learning math is a social activity
– favours whole class discussions and group work
– students share strategies, insights and inventions with others thereby
strategies can be improved.
– promotes reflection which results in HIGHER LEVELS of understanding
• Guidance principle
– refers to it as the guided -re-invention of mathematics
– teachers have pro-active role in student’s learning
– programmes should shift the understanding of mathematics
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RME CONCLUDED
• The 6 principles is a long-term life-long teaching and
learning trajectory
• It is the understanding of how learners learn and how we
should deliver our teaching methodologies (pedagogies)
that unlocks the mathematics potential with every learner.
It is a partnership
Good luck with the journey
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References
• Fred Korthagen (2017) Inconvenient truths about teacher learning: towards
professional development 3.0, Teachers and Teaching, 23:4, 387-405, DOI:
• Meyer, W.F., Moore, C., Viljoen, V.G. (2015) Personology From individual to
ecosystem (4th ed) Heinemann Publishers, Johannesburg
• 10.1080/13540602.2016.1211523
• 10.1007_978-0-387-85744-2_27
• https://journals.co.za/doi/pdf/10.10520/EJC-c512fa4fd
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