Introduction are relevant to mathematics education in the present


Theories of Learning have been widely discussed for many
years now and research into mathematics education has also become a major field
of research, and therefore it is only natural that research has been done into
theories of learning within mathematics education. Within this piece of work, I
hope to not only investigate the differences between the alternative theories
but also explain why they are relevant to mathematics education in the present
day with specific interest in classroom teaching.

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There are two popular areas of theory that are commonly
debated and these are constructivist theories and socio-cultural theories. A
popular constructivism theorist was Jean Piaget and he believed that learning
tends to occur within an individual rather than through interaction with other
people (Piaget, 1950). This is in high contrast to that of popular
socio-cultural theorist Leo Vygotsky who believed instead that learning tends
to occur when humans interact with others (Vygotsky, 1978) in a harmonious
working environment. Of course this is all rather broad, and that is because
within each theory there are different types such as within constructivism you
have naïve, radical and social constructivism.


The first person to highlight a difference between naïve
constructivism and radical constructivism was Ernst Von Glasersfeld (Thompson,
1995), it is stated that naïve constructivism is when a learner would construct
their own knowledge, In contrast to this the idea of radical constructivism is
when you accept that naïve constructivism is not unique for the learner but can
also be applied to everyone, teachers and researchers included.


Indeed, the final type of constructivism theory, social
constructivism, is very similar to that of a socio- cultural theory and has
only subtle differences. In essence, socio constructivism, as with all other
types of constructivism still has the basis of knowledge being individually
constructed however, this occurs through engagement with others which is then
followed up by the learner attributing the results activated by working
together. On the other hand, socio-cultural theory follows an imperative focus
between learning development through a co-operative discussion between a more
and less able group which aids the less able members learn and understand the
ways of thinking and behavior in the shared environment.


How Can a
Constructivist Theory Be Adapted to Mathematics Education?

In order to answer this question, we must first answer the
question what is mathematics and then more importantly what it is we mean by
mathematics education. The answer to the first question provokes many differing
responses ranging from technical responses such as mathematics is the science
of quantity and space (Davis & Hersh, 1980), to more general responses such
as if you ask a group of students what they perceive mathematics to be it is
fairly widely accepted, certainly at introductory level mathematics, that
mathematics is black and white in the sense that it is either right or wrong.
Although this may not be the case with the more advanced mathematics at degree
level and beyond it is almost always the case for school mathematics across all
curriculums. Therefore, we will describe mathematics education, for the purpose
of this writing, as the practice of teaching and learning of this kind of


Within mathematics radical constructivists believe that a child’s
mathematical realities do not coincide with that on an adult’s mathematics (Steffe
& Thompson, 2000). In contrast social constructivists believe in radical
constructivism individual knowers are able to construct truth that then does
not need any support from its outside environment (Howe & Berv, 2000), this
clearly shows that social constructivists believe that there must be a correct
environment in order to promote learning, and this would be an environment
which contained socially constructed attributions.


Constructivism has had a huge impact on the growth of
mathematics education. It has enabled educators to appreciate that they must be
tentative towards the learners mathematical realities as previously educators
would focus largely on mathematical performance. Furthermore, we spend a lot of
time when learning mathematics repeating ourselves to enable our learning. It
was only through constructivism that this became known as repeated experience
rather than practicing. There is a difference between practice and repeated
experience (Cooper, 1991) in that practice focuses more on the repletion of
similar behavior whereas repeated experience is repeating your behavior with
emphasis towards repeated reasoning.


Can a Mathematics
Classroom Be Altered to Support a Socio-Cultural Learning Theory?

In the early years of the 20th century Vygotsky
began to speak about a now commonly accepted concept, the Zone of Proximal Development
(ZPD). He described the ZPD as the area place which is constructed when a
novice interacts with an expert and this would stimulate mental functions which
had not yet been discovered, this would imply that these functions lie in the
region between actual and potential development levels (Goos, Galibranth &
Renshaw, 1999).


Socio-cultural research on a teacher’s learning has, more
recently, drawn on two perspectives, a discourse and a practice perspective
(Forman, 2003). With emphasis on mathematics education the discourse
perspective describes the mathematical communications that occurs within
schools. A recent study researched how a teacher would alter their communication
based on their perception of how the teacher and student roles would change
from the teacher being the teller to the student as the mathematical
participant (Blanton, Westbrook & Carter, 2005). This would not just happen
by chance as it had always been planned by the supervisor (Blanton) in dialogue
she had with the potential teacher abut classroom communications from which she
had previously observed. What this revealed to the teacher was how students
learn mathematics a term Blanton refers to as a “pedagogy of supervision”, she
claims that this enables an open ZPD which can challenge a prospective
teacher’s models of teaching in the context of actual practice (Goos, 2011).


On the other hand, the practice perspective discussed by
Forman, combines professional activity structure and the classroom with
learning and also, identity. A framework to teach secondary mathematics which
consisted of a situated perspective on learning was adapted (Peressini, Borko,
Romagano, Knuth & Willis, 2004), that focused largely a teacher learning
through multiple media such as teacher education courses, schools of
employment, practicum experiences and university mathematics. It was noted that
a teacher would vary how they taught depending on the scenario they were in;
for example, it was stated that whilst one teacher would use reform based
approaches during the practicum they would revert to a much more traditional
approach in their first year of full time teaching post graduation.


Whilst these results may not have come of as much surprise
it is important to draw conclusions from them. It can be concluded that the
changes in style where as a result of a need to respond to the differing
constraints of the alternate scenarios, and therefore highlighting that a
teachers’ knowledge will ultimately always vary depending on the context. The
usefulness of this cannot be overstated as it allows us to confirm that a
context makes a difference to the development of mathematics teachers and
indeed their professional identities.



Do Different Learning
Theories Promote Different Types of Understanding?

There are two different types of understanding, instrumental
understanding and relational understanding (Skemp, 1976). Skemp refers to
relational understanding as being the traditional meaning of understanding,
knowing what to do and knowing why you do that. However, he described another
type of understanding, instrumental understanding, he perceived this to be the
less traditional, but all too common, “rules without reasons” kind of
understanding. At the end of Skemp’s work he Provides the reader numerous
different reasons as to why we have these differences in understanding within
mathematics education, ranging from the struggle to compose a test that tests a
person’s understandings, as he claims it is hard to test the differences in
understandings, to students understanding the importance of the exams and
therefore choosing the easiest route of success, which would clearly be the
instrumental way. However, I will now attempt to provide a different
explanation, and that is that the difference can occur through differing
theories of education.


A socio-cultural theory involves a clear dialogue between a
novice and an expert and thus allowing connections to be formed by the social
space. These connections encourage an understanding, however the type of
understanding that is formed in this situation will depend upon the type of
understanding that is understood by the expert. I will now attempt to draw upon
my own education to explain.

In my first few weeks of sixth form we were being taught
differentiation and in my class, which was deemed to be the top set we were
taught it through differentiation from first principles however in another
student’s class they were taught it in a rule based way, such as you subtract
one from the power and then bring the original power to the front as a constant.
Now, I am not inferring that the teacher of that class did not understand the
basic principles of differentiation, however, another student from that class
had been ill that day and had therefore missed the lesson. When they returned
to school the next day the student attempted to catch up on the work that they
had missed out on, however they did not have a mathematics lesson that day and,
therefore they went to another student who was in their class for an
explanation. The student was happy to oblige however they could only explain it
in the way that they had been taught and that had been in the rule based way.
In this case the expert had only an instrumental understanding of the topic and
therefore when the topic was discussed the novice was only able to draw on an
instrumental understanding through the connections from the shared environment.
However, if this student had decided, rather than ask a peer who was in the
lesson about what they missed, to look on the next page of his textbook he
would have been able to learn from the textbook, and this explained
differentiation from first principles and therefore they may well have been
able to develop a relational understanding. However, this relational
understanding would have occurred through a constructivist approach as the
student had formed the connections within themselves.


This would then provide the question, who’s understanding of
the topic would then be better? Would it be mine or would it be the student who
had not attended the lesson but then instead read the textbook and constructed
the connections themselves? Of course we will never be able to know the actual
answer definitively, as Skemp explains it is often difficult to test between
different understandings. I am inclined to say however, that I believe the
understanding that I had which had come through a social interaction with my teacher
and the connections I had formed in this environment was greater than that
would be of a student who had worked alone and constructed an understanding


This leads
me to the conclusion that a socio-cultural way of teaching mathematics is more
powerful as mathematics is an interactive subject, we know this through all the
peer work that has to occur before something is deemed to be true in it.
Socio-cultural learning when taught in a relational way is the most powerful
way to lead to a relational understanding however it can also be damaging when
taught in an instrumental way as it can encourage a student to purely fixate on
the rules behind a method rather than why these rules are in place. Whereas, I
believe if a student chose to learn from a textbook and they were given a set
of rules they would be more inclined to read a further textbook to seek an
explanation as to why these rules are in place. Therefore, I believe it to be
the case that socio-cultural learning is more powerful and will lead to a
greater and deeper understanding it will ultimately depend upon the quality of
discussion that is occurring in the environment.


Are Different Theories of Learning Morally
Different Also?

Jo Boaler
(1997) introduced the notion of differences between learning and understanding
to mathematics education, “Teachers do not see any real difference between a
clear transmission of knowledge and student understanding”. She claimed that
school classrooms put a heavy emphasis on order and control and the learning of
specified mathematical methods. She also stated that ‘chalk and talk’ teaching
was evident in most classes. When studying at Amber Hill School what she found
was that the reasons why their practices where disadvantaging some students was
not down to the teacher themselves being incompetent, but instead “Teachers
did not perceive a real need to give students the opportunity to think about,
use or discuss mathematics”. This shows that the issue was not with the
teacher’s competency but instead with the environment they created for the
learning to occur.


From this
we can conclude that a student is much more likely to struggle in a poorly
constructed environment as it will detract from their understanding. This
implies that Boaler herself believes that mathematics is best learnt and
understood through discussion in a good working environment. She believes that
this discussion allows a student to form connections and therefore have a true
deep understanding. Similarly, Richard Pring (2004) refers to mathematics as
“A social practice”, this clearly shows that he expects that in order for
mathematics to be educated in a moral manner there must be some level of
interaction between the teacher and the learner.


If we adopt
this notion of moral practice, we can conclude that it would be best of taking
a socio-cultural or a social constructivist view towards mathematics education
if we wished to do so morally. This is because within these two theories there
is an interaction between teachers and learners. However, with the findings of
Jo Boaler and how the difference between learning and understanding was
highlighted in her work I believe that the most moral learning theory is
socio-cultural theory, however when this learning theory is used in the classroom
it is important, in a moral sense, to discuss the topic in enough detail in
order for the student to understand it, as it is no good adapting this theory
and then having a poor quality of discussion as this will just hinder some
students, as we have seen with the ‘chalk and talk’ method.



My interest
in this topic has stemmed from my ever-changing opinion on what is the most
suitable theory of learning to mathematics education. When I was first told
about the different theories of learning I immediately found myself drawn to
socio-cultural theory. This was down to the fact that I did not know much about
the different sub groups within constructivism or socio-cultural theory and
therefore my initial view was that constructivism is when someone learns
independently and socio-culturalism is when someone learns through discussion.
I therefore found myself believing a socio-cultural view as I perceived most of
my mathematics education to have occurred within an interactive classroom
environment, rather than alone with a textbook.


However, my
opinions changed when I was introduced to social constructivism (Jaworski,
2002). I saw a new side of constructivism, a side which had discussion and a
social aspect. I was particularly interested to see if my views altered again
throughout the course of this task and unsurprisingly they did. Having
concluded that a socio-cultural theory not only provides you with greater
understanding, when the quality of discussion allows for it, and also shown
that it appears to be the greatest learning theory with regards to providing a
moral education, I have changed back to this view.


the fickleness of my own view shows how there is not one correct theory but
rather aspects from all different ones are probably best in providing the most
wholesome mathematics education.


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