Introduction

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.

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

mathematics.

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

individually.

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.

Conclusion

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.

However,

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.