Why study theories of learning mathematics?
To able to plan how to teach
mathematics effectively, there need to be some understanding of how student
learn mathematics. A major uk review of effective teachers of numeracy (Askew
et al, 1997) has shown that one of the key factors in develpoing high quality
practices and outcomes in numeracy learning is the role of teory. The authors
argue that the “teacher” beliefs and understandings of the mathematics and
pedagogical purposes behind particular classroom practices seems to be more
important than the forms of practices themselves (p.3) it is not the ways in
which teachers use particular practices in classroom (e.g whole group work,
mental math, direct teaching, or other method) but the beliefs they hold
towards their teaching approaches that are critical. The role of theory in
underpinning practice is an essential element of quality teaching in
mathematics.
By having an idea of how student
learn, teachers are better able to plan for and anticipate in particular ways
and to create learning environment to facilitate better learning. Three
significant classes of theories have heavily influenced our understanding of
how students come to learn and understand mathematics :
- Cognitive
theories that focus on student thinking
- Sociocultural
theories that seek to understand cognition within a social context and
- Social
(or socially critical) theories
Cognitive Theories ?
·
The influence of piaget
The work of jean
piaget 1972 has had some of the most significant impact on mathematics
education. His work on two aspects of how ideas were formed spanned a
considerable part of the 20th century. His early and late periods were
dominated by the active construction of meaning, where he proposed that, trough
the win processes of acommodation and assimilation, schema were constructed.
These ideas are foundational to the significant impact of contructivism in
mathematics education. His middle period was dominated ny stage theories, where
he tried to develop a theory of how young students went through particular
stage in their thinking patterns. Piaget’s writing on childern’s cognitive and
affective development were extremely influential in education, particularly
from the 1950s onward. His stage theory of cognitive development, for example
strongly influenced early childhood and primary education in the 1960s and
1970s. In more years, the influence of this theory has declined because of the
view that it tends to highlight what childern cannot do, rather than what they
can do. As well, stage theory, in the forms in which it has been translated
(from french), interpreted and applied, is criticesed on the basis that it
highlight notions of readiness. In doing so, it can hold back particular
aspects of instruction, at the expense of teaching particular content.
One important
focus of piaget’s work, highly relevant to our contrast, is the development of
logico-mathematical knowledge. In particular,
he made major contribution to the understanding of the developmrnt of
number concepts in young children as well as the development of concept
relating to logic, time, space, and geometry and movement and speed. Since the
early 1980’s piaget’s theories have constituted one of the main bases for development
of constructivist theories in education.
·
Contructivism
Contructivism is
a term that has been used in education and educational psychology with increasing frequency since
the late 1970s .Today , any serious discussion of learning theory related to
mathematics , science or literacy , for example , would include a detailed
discussion of constructivism. As
outlined by Cobb (2000) ; A rangeof psychological theories about learning and
understanding falls under the heading of constructivism. The common element that ties together this family of theories is
the assumption that people actively build or construct their knowledge of the world and of each
other (p.277).
Cobb describes how
constructivists reject notions of
stimulus-respon theory (behaviourist learning theory) and how remembering is
more than direct retrieval , as thought processes are in operation. The
individual mind of the learner is central to constructivism.
The impact of piaget’s work in
contemporary mathematics is obvious in the way in which constructivism ,
and its numerous versions , has been
taken up by teacher and curriculum writers.
There are number of different forms of constructivism , but underpinning
all versions are three premises :
1. Rather
than being passively received . knowledge is actively constructed by students.
2. Mathematical
knowledge is created by students as they
reflect on their physical and mental actions. By observing relationships ,
identifying patterns and making abstractions and generalisations , students come
to integrate new knowledge into their existing mathematical schemas.
3. Learning
mathematics is a social process where , though dialogue and interaction ,
students come to construct more refined mathematical knowledge. Through
engaging in the physical and social aspects of mathematics , students come to
construct more robust understandings of mathematical consepts and
processes through processes of negotiation , explanation and justification.
Constructivism recognises that mathematics must make
sense to students if they are to retain and laern mathematics. For students to
develop appropriate knowledge they must be provided with rich learning
experiences so that their constructed meanings and understandings are in
keeping with the of mathematics.
·
The importance of dialogue and argumentation
Within the
constructivist paradigm, the role of language and dialogue is central to
fostering learning environments. Providing appropriately organised esperiences
where students can talk with their peers allows them to explore ideas in
language and concepts that is similiar to their own. This enables higher
achieving students to practise their control of language and lower achieving
students to hear ideas being modelled in a language that is more likely to be
in a genre that they can access. For example , when talking about the
properties of three-dimensional shapes, the formal language of edges , faces
and vertices may be introduced. The high achieving students may find this
language useful as they have been confused
by the use of the term ‘side’ –does
it refer to the face or an edge?
The appropriate language aids their learning. Other students may be
grappling with notions of three-dimensional shapes. Having students explain
their ideas to their peers often supports both sets of learners.
·
Constructivism in the mathematics classroom
Within a
constructivist classroom , the teacher
acknowledges that students will have constructed a range of understandings from
any given interaction on the basis that they have entered the context from a
range of different perspectives and experiences . a contructivist perspective recognises that
it is not possible to assume that the teaching of a concept relates to the
development of the ideas proposed by the teacher and that , indeed ,there will
be a multiplicity of understandings constructed by the range of students in the
classroom. A constructivist teacher realises that having taught something does
not mean that students have learnt exactly what was envisaged by the teacher.
It is important for the teacher to use a range of tools and techniques to
assess what the students have constructed. By identifying what the students
have constructed, the teacher is then able to identify constructions that are
akin to what has been the objective of the lesson along with being able to identify
misconceptions. It is the misconceptions
that allow the teacher to access what the students have come to construct and
thus develop teaching strategies that will move the students into more
appropriate constructions.
Sosiocultural Theories : The
influence of Vigotsky
Lev Vygotsky is regarded as the
founder of sociocultural theory , or what can be described as the
sociohistorical approach in psychology (e.g. Cole ,1966;Moll,1990). Vygotsky’s
work , which is embodied in literature on sociocultural theories of learning
mathematics , has gained increasing importance in theorising how students learn
mathematics. Vygotsky saw that students internalised complex ideas
(Daniels,1990), but he extended the general constructivist approach by arguing
that the internalisation of knowledge could be better achieved when students
were guided by good , analytic questions posed by the teacher.
The
expert teacher is central to Vygotskian theory . the teacher’s role is to identify
the student’s current mode of representation and then, through the use of good
discourse, questioning or learning situations, provoke the students to move
forward in her/his thinking. The recognotions of a students representation or
thinking was seen as her/his zone of proximal development and the teachers
actions for supporting learning was described as scaffolding. When working in
the zone of proximal development particular attention is paid to the language
being used since the language of the student influences how she/he will
interpret and build understanding (Bell and woo, 1998) within a Vygotskian
approach it is seen to be important that teachers use and build considerable
language and communication opportunities within the classroom environment in
order to build mathematical understandings.
·
Scaffolding
Good teaching
involves teachers knowing their students current thinking about mathematical
concepts and then knowing how to move the students towards more complex ,
complete and robust constructions through the use of organised learning
activities and environments. Good questions are important in facilitating
learning. Typically , good questions are those that foster deeper levels of
learning as opposed to recall.
Socially Critical Theories
Sociological theories, and
particularly critical sociology, have gained increasing importance in
mathematics. These theories shift the focus of learning mathematics away from
the individual to a more macro level of analysis. In part, this interest has
stemmed from the consistent poor performance of students who come from
particular backgrounds. It is now recognised internationally that particular
students are more at risk of not performing well. Aside from students with
learning disabilities , these are indigenous students of almost all nationals,
students from working class (or low socioeconomic status) families ; students
who live in remote or rural areas , and students whose first language is not
english (in english –speaking countries ). When gender is considered in concert
with these variables, differences are exacerbated (walkerdine,1988,1989). It
has long been recognised that girls have been particularly disadvantaged in the
study of mathematics due to genered practices in teaching and assessment
(Fennema and Meyer,1989; Leder, 1992). However , its is now recognised that
this applies not to girls per se, but to girls from particular social and
cultural backgrounds. When considering the ways in which the practices of
mathematics work to exclude girls, it becomes important to recognise that some
girls ( namely middle class girls ) are more likely to be successful than their
peers from working class backgrounds (both girls and boys) , thus gender is not
the sole variable , but must be considered in concert with other variables.
Rather than assume that success is due to some innate ‘mathematical ability’ ,
socially critical theories have the practices of school mathematics as the
focus of attention.
Socially
critical theories explore the practices of mathematics education to see how
they are implicated in the reproduction of inequities and in so doing,
challenge such practices to change. Assessment , mathematical language (
Zevenbergen,2000,2001) and classroom talk, textbooks (Dowling, 1998) and
ability grouping (Boaler,1997) are some of the areas that have been critically
examined in terms of the way in which social, cultural, linguistic and gender
differences are reproduced through mathematics education. These studies have
illustrated the very subtle ways in which school mathematics contributes to ,
and legitimates , the failure of particular groups of students. This book
devotes a full chapter to the study of equity from this prespective.
Mathematics
is one of the most important subjects in the school curriculum and it serves a
particular role (among others) as a social filter. The work of Lamb (1997) show
that succes in school mathematics is the best predictor for success in life.
People who come to believe that they are not good at mathematics tend to accept
their position in life. Thus it is important for all students to succeed in
school mathematics – regardless of background , gender or language. By knowing
how practices are implicated in the construction of differences , teachers can
change their practice in order to produce more equitable classrooms and
outcomes.
New Times : New Learnings
Current educational theory has
moved to a stronger recognition of how society impacts on learning. Postmodern
theorising has drawn considerable attention to how current times are very
different from old times ( the Industrial Age ), largely due to the use of
technology . As early as 30 years ago , educationalists were commenting on how
television was influencing attention rates and that young students had much
shorter attention spans than in the past. Similarly , the advent of television
meant that reading books was being replaced by viewing television and there was
seen to be a decline in reading skills and motivations. More recently , the use
of computers and other computer technologies have been seen to have
considerable influence over students ‘ thinking and behaviours, so much so that terms such as ‘cyberkids’ ,
‘technoliteracy’ and numerous others have become part of the educational
discourse ( Luke , 2000).
The
world that schools are preparing students for is fundamentally different from
that of even a few years ago. The term ‘New Times’ has been coined to represent
the very different social , economic , political and educational times of the
society in which young people now live (Gee,2002). Many of the new terms in
educational theory are prefaced with the word ‘New’ to represent this thinking.
Students who have grown up in a technological age are intimidated by technology
and henceits insertion into mathematics
is an important change.
·
Technology and new learning
In terms of
learning , New Times and New Learnings embrace the use of technology so that
the tedium often associated with working calculations (‘doing sums’) can be
replaced with technologies to support the development of mathematival thinking.
This change is fundamental in preparing students for life in New Times.
Technology in this sense includes calculators as well as computers so that ways
of thinking-such as the development of algebraic thinking –can be very well
supported by the use of technology (Asp and McCrae,2000)
New learning in an old
curriculum has been challenged by the work of Stacey and Groves (1996). In the
old curriculum , students in the first year or two of schooling would only work
with numbers to 20 . Through the use of calculators , young students number
sense can now be developed to four or more digit numbers (Groves,1995) . Thus,
rather than technology replacing skills, it can be used enhance mathematical
thinking.
Within this framework,
consideration must also be given to what is seen as mathematics. Often this is
framed within basic skills where the wider society bemoans young graduates who
are unable to calculate. Within New Times , the emphasis becomes somewhat
different due to the saturation of technology. Thirty years ago, shops could
only enter the value of the goods in the cash register-calculation were done on
paper, and change given using a counting –on method. Today’s society is much
richer in technology-some stores have registers that scan items, so no data
need be manually entered , others require the assistant only to press the item
of purchase (e.g.McDonald’s) . Most registers also calculate the amount of
refund (change) required. Thus old basics have been superseded by new
basic-sales assistants need to be able to estimate , to problem solve , to
validate, in order to evaluate whether items have been scanned or not and thus
verify the validity of the total amount or the change given ( in case a wrong
amount is entered for amount tendered) in other words, they need some of the
old basics, but technology has brought with it a range of new skills. These
sklills are different from, but not exclusive of, the attributes that would be
sought in mathematics.
Within a New Learning framework,
the teaching of mathematics involves a much higher emphasis on the use and
integration of learning technologies (such as computers, calculators , graphic
calculators) . Since most western students have been exposed to such
technologies, they are more likely to think and respond to these forms of
teaching.
Furthermore , there is a renewed
emphasis on the mathematics curriculum reflecting new forms of mathematics that
reflect the needs of the world beyond. New Times reflect the supersaturation of
information, so students need to be more ‘literate’ in terms of analysing,
interpreting and being able to critique the text to which they are exposed.
This often means being able to use and apply theire mathematical thinking and
analysis to texts containing data-such as graphs, measures of central tendency
and so on. They not only need the skills to construct or calculate such
measures, but within New Times , they also need to be able to interpret more
carefully such information. This demands a refocusing of the mathematics
curriculum into such areas.
Theory into practice
The value of a good theory is its
capacity to enable teachers to develop good practice that supports and enhances
learning. Teachers need to have a strong theoretical basis to their work. By
understanding how students learn, teachers are able to organise the learning in
ways that will enhance the capacity for laerning. Rather than advocate one
theory as being superior to another, it is more appropriate to consider what is
being learnt. When considering how to teach the memorisation of the
multiplication facts, behaviourism may be a better option than constructivism
since the aim is to remember rather than to understand. When understanding is
to be achieved , constructivism may offer more potential. When making links
with the world beyond schools, New Learnings may offer better frameworks for
planning. When trying to understand why some students have more difficulty in
passing mathematics than others, socially critical theories are useful for
examining the practices of mathematics.
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