Module 3.1 Theories of learning mathematics

Summary

This 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

00:00

🧠 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.

05:01

📚 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.

10:03

🧒 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.

15:05

🔨 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.

Keywords

💡Behaviorist Theory

Behaviorist theory in education emphasizes learning through reinforcement and repetition. In the video, it is compared to Pavlov's experiment, where repeated stimuli lead to conditioned responses. This theory is applied in mathematics education by practicing concepts until they become internalized, such as memorizing multiplication tables through repetition and reinforcement.

💡Hierarchy of Learning

A hierarchy of learning refers to the structured way in which concepts are taught, where simpler concepts must be mastered before moving on to more complex ones. In mathematics, this hierarchy is evident in how children first learn single-digit numbers, then teens, hundreds, and eventually fractions and decimals, as described in the video.

💡Developmentalist Approach

The developmentalist approach, influenced by Piaget, focuses on stages of cognitive development and how children learn at different ages. The video explains how young children move from sensory-motor phases to abstract thinking, and mathematics education must align with these developmental stages.

💡Constructivist Theory

Constructivist theory suggests that learners build their own understanding through interaction and engagement with concepts. In the video, this is seen in how students negotiate meaning in mathematics through hands-on activities with manipulatives like blocks and counters, allowing them to explore mathematical ideas.

💡Social Theories of Learning

Social theories of learning emphasize the importance of social interaction in the learning process. The video contrasts this with behaviorist approaches, highlighting how group work, discussion, and collaborative problem-solving are essential for understanding mathematical concepts in a social context.

💡Play-based Learning

Play-based learning, especially in early childhood education, integrates play into structured learning activities. The video explains how free play is guided by specific learning objectives, such as skip counting, which helps children develop mathematical concepts while engaging in enjoyable activities.

💡Age-Appropriate Learning

Age-appropriate learning refers to designing educational activities based on the predictable developmental stages of children. In the video, this is linked to Piaget’s stages, where activities are tailored to what is suitable for children at different ages, such as concrete operational tasks for younger students and abstract thinking for older ones.

💡Individual Appropriateness

Individual appropriateness highlights the uniqueness of each child’s learning process, requiring tailored educational experiences. In contrast to age-appropriate methods, the video discusses how individual appropriateness focuses on each child's developmental needs, ensuring personalized approaches to their mathematics education.

💡Realistic Mathematics Education

Realistic Mathematics Education (RME) is a Dutch approach that emphasizes understanding mathematical concepts through real-world experiences rather than formal abstractions. The video describes how RME delays formal place value teaching until later, focusing on building number sense through relatable, practical experiences.

💡Mathematization

Mathematization is the process of turning real-world problems into mathematical concepts. In the context of the video, it is used in Realistic Mathematics Education, where students are encouraged to mathematize their everyday experiences, building a deeper understanding of numbers and operations before learning abstract concepts like place value.

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