Module 3.1 Theories of learning mathematics
Summary
TLDRThis session discusses different theories of learning mathematics, highlighting how understanding these theories can enhance teaching effectiveness. The speaker outlines four major learning theories: behaviorist, developmentalist, constructivist, and social theories, each influencing how mathematics is taught. The importance of aligning teaching strategies with these theories is emphasized, as well as understanding age-appropriate and individual-appropriate learning approaches. The session also touches on alternative methods like realistic mathematics, which focuses on real-world experiences and number sense before formal concepts. Educators are encouraged to reflect on their own teaching philosophy for coherence in their methods.
Takeaways
- 🧠 A strong theory helps us conceptualize learning processes, offering explanatory power and aiding in planning effective teaching strategies.
- 🔢 Mathematics education is influenced by four major learning theories: behaviorism, developmentalism (Piaget), constructivism, and social learning.
- 🛠 Behaviorist theory emphasizes reinforcement, practice, and memorization, commonly seen in the way students learn through drills like multiplication tables.
- 🏗 Developmentalist theory, influenced by Piaget, highlights different learning stages based on age, such as sensory-motor, pre-operational, and concrete operational stages.
- 🔨 Constructivist theory focuses on students creating meaning through interaction with others and engaging with concrete objects like blocks and counters.
- 👥 Social learning theory views learning as a social process, where group work, communication, and problem-solving are key to negotiating meaning.
- 👶 In early childhood education, play-based learning (guided rather than free play) is important, with influences from Reggio Emilia, Montessori, and Steiner methods.
- 📊 Age-appropriateness in learning activities is based on Piaget's predictable development stages, while individual appropriateness caters to the unique needs of each child.
- 🇳🇱 Realistic mathematics education from the Hans Freudenthal Institute emphasizes number sense and realistic, real-world problem solving, delaying formal place value instruction.
- 🎯 A coherent approach to understanding how children learn mathematics is a key factor in effective teaching, distinguishing good teachers from average ones.
Q & A
What is the main purpose of discussing different theories of learning mathematics in this session?
-The main purpose is to help conceptualize what's happening in mathematical learning, provide explanatory power, and aid in planning how to teach effectively.
What are the four dominant theories of learning mathematics mentioned in the session?
-The four dominant theories are: behaviorist theory, developmentalist theory, constructivist theory, and social theories of learning.
How does the behaviorist theory apply to learning mathematics?
-In behaviorist theory, learning mathematics is reinforced through practice and repetition. The idea is that repeated actions and rewards (like praise) help students internalize concepts, similar to Pavlov's dog experiment.
What is a 'hierarchy of learning' in mathematics, according to the session?
-A hierarchy of learning is the idea that mathematical concepts are learned in a sequential order, where understanding one concept is necessary before moving on to the next. For example, learning single-digit numbers before understanding larger numbers and fractions.
How does Piaget's developmentalist approach apply to early mathematics education?
-Piaget's developmentalist approach defines stages of cognitive development (e.g., sensory-motor, pre-operational, concrete operational) and suggests that learning activities should be appropriate for the child’s developmental stage, guiding how mathematics is taught to young children.
What does the constructivist theory emphasize in the context of mathematics education?
-Constructivist theory emphasizes that students create meaning by interacting with concrete objects and through social engagement, often using manipulatives like blocks or counters to explore mathematical concepts.
What are the main differences between behaviorist and social theories of learning?
-Behaviorist theory focuses on individual learning through reinforcement and practice, often involving memorization and drill. Social theories of learning, on the other hand, emphasize group work, problem-solving, and the importance of learning in a social context.
What is 'free play' in early childhood mathematics education, and how is it guided?
-Free play refers to activities where children are allowed to explore concepts on their own. However, teachers subtly guide these activities to ensure they are aligned with specific learning outcomes, such as skip counting or number sense.
What is 'realistic mathematics,' and how does it differ from traditional teaching methods?
-Realistic mathematics, developed by the Hans Freudenthal Institute, focuses on progressively mathematizing concepts grounded in real-world experiences. Unlike traditional methods, it delays introducing abstract concepts like place value until later, focusing first on number sense.
Why is having a coherent approach to teaching mathematics important for educators?
-A coherent approach ensures that a teacher’s methods are consistent and effective. Studies show that successful mathematics teachers have a clear understanding of how children learn mathematics, which helps them provide structured and meaningful learning experiences.
Outlines
🧠 Importance of Learning Theories in Mathematics Education
This paragraph introduces the importance of understanding learning theories in the context of teaching mathematics. It emphasizes how good theories provide explanatory power and help in effective planning. The speaker asks the audience to reflect on their own learning experiences in primary and secondary education to understand different teaching approaches. There are four dominant theories in mathematics education, which vary based on time and context, that will be explored.
📚 Behaviorist Learning Theory in Mathematics
This paragraph discusses the first dominant learning theory: behaviorism. It relates behaviorist theory to Pavlov's experiments and explains how practice and reinforcement help students internalize mathematical concepts. Hierarchies of learning are essential in mathematics, where concepts build on each other. The idea of filling learning gaps is also explained, alongside the importance of drills, memorization, and the automatic recall of multiplication facts.
🧒 Developmentalist Theory and Piaget’s Stages of Learning
The second theory discussed is developmentalism, based on Piaget’s stages of cognitive development. Each stage, from sensory motor to concrete and abstract operations, corresponds to different ages and shapes how children learn. This theory stresses the need for appropriate experiences at each stage, with specific mathematical learning tasks designed to align with the developmental phase of the child.
🔨 Constructivist and Social Learning Theories in Mathematics
Constructivist and social theories are the focus here. The constructivist theory argues that students create meaning through interaction and engagement with mathematical concepts and manipulatives. Social theories of learning emphasize that all learning occurs within a social context, contrasting the solitary behaviorist approach. The value of group work and communication in negotiating mathematical understanding is highlighted as key in these theories.
🎮 Play-Based Learning in Early Childhood Mathematics
In early childhood mathematics, play-based learning is a dominant approach, particularly for younger children. It is often structured to guide specific learning goals, despite appearing as 'free play.' The paragraph also introduces influential educators like Reggio Emilia, Steiner, and Montessori, who have shaped early childhood learning approaches. The importance of intentional design in play activities is emphasized.
👶 Age and Individual Appropriateness in Learning
This paragraph discusses the concept of appropriateness in education, both age-appropriate and individual-appropriate. Age-appropriate learning aligns with Piaget’s stages, anticipating normal developmental progress. Individual appropriateness emphasizes the uniqueness of each child and the need to tailor learning experiences to individual needs, rather than treating all children the same within an age group.
🇳🇱 Realistic Mathematics and Number Sense Development
The paragraph introduces a counter-approach known as realistic mathematics, developed in the Netherlands by the Hans Freudental Institute. This approach emphasizes grounding mathematical learning in real-life contexts and promoting number sense over formal abstract concepts like place value. It contrasts with traditional methods of introducing complex concepts too early, favoring flexibility with numbers.
📘 Synthesizing Learning Theories for Coherent Teaching
In the final paragraph, the speaker stresses the importance of having a coherent understanding of learning theories to guide effective teaching. Studies show that the best mathematics teachers have a clear, cohesive approach to how children learn. The speaker encourages the audience to engage with the readings to deepen their understanding of the theories and apply them systematically in their teaching practice.
Mindmap
Keywords
💡Behaviorist Theory
💡Hierarchy of Learning
💡Developmentalist Approach
💡Constructivist Theory
💡Social Theories of Learning
💡Play-based Learning
💡Age-Appropriate Learning
💡Individual Appropriateness
💡Realistic Mathematics Education
💡Mathematization
Highlights
Theories help conceptualize what's happening in math learning and assist in planning teaching strategies.
Mathematics learning has four dominant theories: behaviorist, developmentalist, constructivist, and social learning theories.
The behaviorist theory, drawing on Pavlov's work, emphasizes reinforcement and memorization through repeated practice.
Hierarchies of learning in mathematics involve mastering simpler concepts before advancing to more complex ones, like learning single digits before fractions.
Piaget's developmentalist theory focuses on predictable stages of learning, such as sensory motor, pre-operational, and concrete operational stages.
Constructivist theory posits that students create meaning by interacting with objects, ideas, and other learners, negotiating understanding through engagement.
Social learning theories emphasize that learning occurs in a social context, through group work, problem-solving, and communication.
Behaviorist approaches to math education are often solitary, focused on repetition and memorization.
Developmentalist approaches, especially in early childhood, involve structured play and guided activities based on predictable developmental stages.
Free play in early childhood learning should be guided by specific learning outcomes, such as skip counting, to maximize educational value.
Various early childhood educational philosophies, like Reggio Emilia, Steiner, and Montessori, contribute to the development of different approaches to learning mathematics.
Age-appropriate learning assumes that children develop along predictable sequences, while individual appropriateness focuses on tailoring teaching to each unique child's needs.
Realistic mathematics, developed in the Netherlands, emphasizes gradual understanding of math concepts through real-life experiences rather than abstract operations early on.
Effective math teachers have a coherent understanding of how children learn mathematics, distinguishing them from average teachers.
A key to successful teaching is aligning instructional methods with a consistent theoretical approach that supports children's learning development in mathematics.
Transcripts
okay so what we're doing in this session
is
working through very different theories
of learning mathematics
um often people think well what's the
point of theory actually a good
theory helps us conceptualize what's
actually going on and gives us good
explanatory power
about what's learning but also for
helping us to plan
so i want you to think back and we've
done this earlier in the course but
again think back to what your
experiences were like in your primary
years and your second year
years think about how your teacher
taught you
but also what content that you what that
teacher was actually teaching you
it's important to start to think about
how do teachers think because
how do teachers teach because it gives
us insights into their thinking
about how people learnt maths best
okay so in mathematics there's mainly
four big waves of learning theory and
these are the
the dominant theories that that um
are reflected in the different
pedagogies that teachers would use
they vary um at different points in time
um and i'll i want to come back to that
point
after we've discussed the various
theories
because different things happen for
different reasons
so if we think about what your
experiences may have been like
and most of the time when people are
asked about their teaching
their experience of maths it is that
we sat there we did lots of practice and
in the end we got it right
as if the lots of practice will help you
internalize
what it means to do operations or long
division or
volume or whatever else so the first
theory that we look at
um is what we call a behaviorist theory
uh
if you've done any psychology this is
the theory that we talk about when we
talk about pavlov's dog
so if you're doing reinforcing behaviour
it will become internalised so pavlov
pioneered this work
whereby he would ring a bell and feed
the dog the dog would salivate because
he knew he was getting food
to the point where in the end he only
had to ring the bell the dog had
associated the bell with food
and would salivate so
how does that apply to mathematics well
the idea is that
if you practice enough you will
internalize
things if you reward behaviour then
students will internalize that behavior
so with a behaviorist model there's
various what we call
hierarchies of learning and unlike any
other area of the curriculum
mathematics is
very heavily imbued with this notion of
hierarchies of learning so what's a
hierarchy a hierarchy is where
you learn one concept before you learn
the next and you have to know that one
before you know that learn the next one
so if we look at the number strand for
example
we would start teaching the single digit
numbers one to ten
then we would teach the teens then we
teach the hundreds then we teach the
thousands
and by the time the kids have got the
notion of thousands we think they're
ready now for
understanding part number so they
start to learn about fractions and
decimals at the same time they're
starting to learn big numbers
you'll often hear teachers say oh this
student is struggling with maths because
they've got gaps in their learning
what that means is they may have been
away for six months or
a few months and they haven't got the
concept of
the internal zeros in numbers like 101
or 1010 those sorts of where those zeros
are
because that in that hierarchy there's
ways that you teach that
so that's what a hierarchy of learning
is the other thing is
there's a lot of reinforcement so the
idea of the reinforcement is
that when you're learning for example
your multiplication
facts you just keep going over and over
again and you keep reinforcing to the
kids that
you know seven nines are 63 seven nines
are 63 9 7's are 63
and you're praising them for it and
eventually they'll internalize it
similarly there's a lot of drill and
practice the more you practice something
the more you will
be reinforced so the more you will learn
it
and there's a lot of memorization if you
think about the
times the the number of facts the times
times tables the multiplication facts
whatever what term you'd like to use
um students often don't know that seven
nines are 63. they'll have memorized it
but when it comes to a um being able to
articulate it if they say seven sixes
are
seven nines are 42 or they've made a
mistake in there
they actually don't know that it's wrong
and there's a big there's been a big
push away from
multiplication facts for those reasons
although
it's starting to starting to be quite a
resurgence in having
uh automaticity knowing those numbers
automatically
so that's what we call a behaviorist
approach and if you think about
your experiences in schools
my hunch is that most of your learning
would fall within what within that
paradigm
of behaviorist learning theory
what we also have in mathematics is that
um there's a particularly
important in the early years is a
developmentalist approach and this comes
a lot from the work of piaget
and versions and adaptations of piaget
piazza's
main theories of learning so these are
often
um stages of development that are quite
defined in terms of the age of the child
so when we're thinking about um
the the young the young baby the young
the young child from zero to two is
often
um in the sensory motor phase of their
life
where everything is about
physically feeling things and
internalizing what that means
so things like object permanence so if
you with a young child
if you hide an object underneath a mat
they don't know what's there so
the concept of subtraction won't
actually be understood couldn't be
understood because
when the object disappears the object
has gone it's not hidden
so as they came coming out of that stage
they start to move into
pre-operational and then into concrete
operational stages
now each of these stages and when you go
through your readings you'll actually
see how these stages are all defined
and they are quite defined in terms of
the age of the child so what we're doing
with that
um theory is looking at where the
children are coming from so the
experiences that we provide for a child
in the pre-operational stage are quite
different from
uh the concrete operations which is very
different from
the abstract thinking that we see in the
post-12 year old child
the third main theory and this has come
about largely since probably the 1970s
is constructivist and you will hear a
lot of teachers in school saying
i'm a constructivist teacher
sometimes that's an incorrect label
because they
the idea was that students create
meaning
from interacting they negotiate meaning
and they negotiate
understanding about many mathematical
concepts
through engagement with others and with
concrete objects
so often a teacher will describe
themselves as constructivists because
they're allowing the students to play
with manipulatives
and to play with various
tools and i want to say tools i mean
blocks mabs counters and so on
and using those those objects to create
some sort of meaning the final
paradigm is that what we call
social theories of learning and social
theories of learning mean that learning
is
always undertaken in a social context
you cannot
uh create meaning unless it's in a
social way
so if you think back to our first week
where we talked about
um the importance of communicating the
importance of reasoning and thinking
logically and being able to articulate
is often shaped by this notion of a
social theory of learning
now social theory of learning is in
sharp contrast to the behaviorist
approach
that you might have experienced in
school where mathematics was very much a
solitary activity
social theories are very much about
having a lot of group work a lot of
interaction a lot of problem posing a
lot of problem solving
where meaning is negotiated but with
very much within a social context
so these are our big paradigms that we
have in terms of mathematics learning
i would encourage you to undertake your
readings
around these theories so you get a
better understanding of the general
principles that underpin each one of
those approaches
in terms of the early childhood context
there are a number of
approaches that you also need to
understand and again
there's readings in your course on these
play-based learning is
a very dominant paradigm within the
early childhood setting
where again much informed by
the pre-operational stages of
development where
play is of an important thing now
there's two types of play
there's free play where it's expected
that or anticipated that just by
allowing children to have free play
that they will learn things however we
do know that
free play in and of itself is almost
hard
impossible to achieve as a teacher your
your role is really to understand what
do i want the children to learn from
engaging in these activities so your
activities that you do and the play that
you allow the child to undertake
should be very much guided by what it is
you want the children to achieve
and that the play is actually not free
play it may look
as if it's free play but it's actually
quite guided
in terms of what you want the children
to learn so if you're wanting the
children
to learn skip counting which is 2468
then you would have activities around
and play activities
that were encouraging that whole notion
of skip play
a skip counting a number of other um
significant people who have influenced
early childhood a reggio emilio
an an italian who in the post-war period
realized that there were particular ways
that we should guide
um or construct our learning
environments
the steiner school is also a very uh
important uh school of thought in terms
of how children learn as is the
montessori
and again please read your readings on
these so you get a better idea of
what are these approaches and how they
are
set up in terms of an early childhood
context and particularly around
mathematics
the other notion that we really need to
think about is of the notion of
appropriateness
and these again come back to those uh
main four main paradigms
there's two sorts of appropriateness one
is age appropriate
and that's very much informed by uh the
piagetian school of thought where there
is a predictable sequence
of development that we expect in most
normal children children who have
disabilities or children who come from
particular contexts
may not develop in that sequence but
largely
most children in the context within
which we work
are likely to develop in a particular
way provided they have
the right environment in terms of food
safety
and all of the other basic needs are
being met
age-appropriate means that the it
provides a framework
for the types of activities that we
would do so the types of activities that
we would do
for a sensory motor child is are very
different from that of a pre-operational
child as opposed to a concrete
operational child
so the activities are built around that
um what we see as
uh age-appropriate so what do we see is
appropriate for three-year-olds or
four-year-olds or five-year-olds
and so on underpinning that is though is
the assumption
that children will develop in a
predictable
normal sequence that we can anticipate
and we
arrange our activities according to that
then there's the notion of in individual
appropriateness and what this
approach um fosters is
is that every child is unique
and so rather than clumping all the
three-year-olds or the four-year-olds
together
this this approach says that there's so
much variance amongst children
that what we really need to do is look
at each child as being a unique person
and then building our learning
experiences
particularly for that child so the child
becomes the individual child becomes the
center for our planning
rather than the age of the child
however there are alternate approaches
which we don't see
a lot of evidence of in our schooling
system within australia
and you probably should understand this
theory a little more it is very much
counter to
the approaches that we have used in
australian curriculum and it's called a
realistic mathematic
realistic mathematics and it generates
comes comes from the holland
from holland in the uh
hans freudental institute it's research
based
and the idea is that we're progressively
mathematizing
concepts the approach is very realistic
it's very grounded in the real
experiences of the children
and so unlike our
our notion of teaching say place value
where we teach place value around about
um grade two grade three
uh the realistic maths recognizes that
our formal place value is a very
abstract concept
and it's actually not introduced
formally until
almost the end of primary school and
instead what is happening all the time
is there's a heavy emphasis on number
sense
so students get a sense of number before
they start
doing and when i say doing let's go back
to week one
doing mathematics and putting them in
columns and t tables and so on
and what that approach does it's
encouraging flexibility with numbers
and that becomes really important and
we'll talk a lot more about that when we
do
our number work in the curriculum so
basically what i've tried to do is
quickly give you a snapshot of the
different
theories that underpin the learning of
mathematics
it's really important for you now if you
haven't already done it to go back to
your readings
and really engage with those readings so
you get an understanding
it would also be very useful for you to
think about
what sort of theory
best underpins your work one of the
largest studies done on effective
teachers of
mathematics and it was undertaken in the
uk
showed that what marks out a good
teacher of mathematics from an average
teacher of mathematics
is that the teacher has a coherent
approach
to understanding how children learn
mathematics so
having a clear understanding in your own
head
about how you believe children learn
mathematics is really important because
then all of your work becomes quite
coherent
rather than a hodgepodge of ideas
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