Introduction are relevant to mathematics education in the present

IntroductionTheories of Learning have been widely discussed for manyyears now and research into mathematics education has also become a major fieldof research, and therefore it is only natural that research has been done intotheories of learning within mathematics education. Within this piece of work, Ihope to not only investigate the differences between the alternative theoriesbut also explain why they are relevant to mathematics education in the presentday with specific interest in classroom teaching.  There are two popular areas of theory that are commonlydebated and these are constructivist theories and socio-cultural theories. Apopular constructivism theorist was Jean Piaget and he believed that learningtends to occur within an individual rather than through interaction with otherpeople (Piaget, 1950).

This is in high contrast to that of popularsocio-cultural theorist Leo Vygotsky who believed instead that learning tendsto occur when humans interact with others (Vygotsky, 1978) in a harmoniousworking environment. Of course this is all rather broad, and that is becausewithin each theory there are different types such as within constructivism youhave naïve, radical and social constructivism.  The first person to highlight a difference between naïveconstructivism and radical constructivism was Ernst Von Glasersfeld (Thompson,1995), it is stated that naïve constructivism is when a learner would constructtheir own knowledge, In contrast to this the idea of radical constructivism iswhen you accept that naïve constructivism is not unique for the learner but canalso be applied to everyone, teachers and researchers included. Indeed, the final type of constructivism theory, socialconstructivism, is very similar to that of a socio- cultural theory and hasonly subtle differences. In essence, socio constructivism, as with all othertypes of constructivism still has the basis of knowledge being individuallyconstructed however, this occurs through engagement with others which is thenfollowed up by the learner attributing the results activated by workingtogether. On the other hand, socio-cultural theory follows an imperative focusbetween learning development through a co-operative discussion between a moreand less able group which aids the less able members learn and understand theways of thinking and behavior in the shared environment. How Can aConstructivist Theory Be Adapted to Mathematics Education?In order to answer this question, we must first answer thequestion what is mathematics and then more importantly what it is we mean bymathematics education. The answer to the first question provokes many differingresponses ranging from technical responses such as mathematics is the scienceof quantity and space (Davis & Hersh, 1980), to more general responses suchas if you ask a group of students what they perceive mathematics to be it isfairly widely accepted, certainly at introductory level mathematics, thatmathematics is black and white in the sense that it is either right or wrong.

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Although this may not be the case with the more advanced mathematics at degreelevel and beyond it is almost always the case for school mathematics across allcurriculums. Therefore, we will describe mathematics education, for the purposeof this writing, as the practice of teaching and learning of this kind ofmathematics. Within mathematics radical constructivists believe that a child’smathematical realities do not coincide with that on an adult’s mathematics (Steffe& Thompson, 2000).

In contrast social constructivists believe in radicalconstructivism individual knowers are able to construct truth that then doesnot need any support from its outside environment (Howe & Berv, 2000), thisclearly shows that social constructivists believe that there must be a correctenvironment in order to promote learning, and this would be an environmentwhich contained socially constructed attributions. Constructivism has had a huge impact on the growth ofmathematics education. It has enabled educators to appreciate that they must betentative towards the learners mathematical realities as previously educatorswould focus largely on mathematical performance. Furthermore, we spend a lot oftime when learning mathematics repeating ourselves to enable our learning. Itwas only through constructivism that this became known as repeated experiencerather than practicing. There is a difference between practice and repeatedexperience (Cooper, 1991) in that practice focuses more on the repletion ofsimilar behavior whereas repeated experience is repeating your behavior withemphasis towards repeated reasoning. Can a MathematicsClassroom Be Altered to Support a Socio-Cultural Learning Theory?In the early years of the 20th century Vygotskybegan 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 anovice interacts with an expert and this would stimulate mental functions whichhad not yet been discovered, this would imply that these functions lie in theregion between actual and potential development levels (Goos, Galibranth &Renshaw, 1999).

 Socio-cultural research on a teacher’s learning has, morerecently, drawn on two perspectives, a discourse and a practice perspective(Forman, 2003). With emphasis on mathematics education the discourseperspective describes the mathematical communications that occurs withinschools. A recent study researched how a teacher would alter their communicationbased on their perception of how the teacher and student roles would changefrom the teacher being the teller to the student as the mathematicalparticipant (Blanton, Westbrook & Carter, 2005).

This would not just happenby chance as it had always been planned by the supervisor (Blanton) in dialogueshe had with the potential teacher abut classroom communications from which shehad previously observed. What this revealed to the teacher was how studentslearn mathematics a term Blanton refers to as a “pedagogy of supervision”, sheclaims that this enables an open ZPD which can challenge a prospectiveteacher’s models of teaching in the context of actual practice (Goos, 2011). On the other hand, the practice perspective discussed byForman, combines professional activity structure and the classroom withlearning and also, identity. A framework to teach secondary mathematics whichconsisted of a situated perspective on learning was adapted (Peressini, Borko,Romagano, Knuth & Willis, 2004), that focused largely a teacher learningthrough multiple media such as teacher education courses, schools ofemployment, practicum experiences and university mathematics. It was noted thata 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 basedapproaches during the practicum they would revert to a much more traditionalapproach in their first year of full time teaching post graduation. Whilst these results may not have come of as much surpriseit is important to draw conclusions from them. It can be concluded that thechanges in style where as a result of a need to respond to the differingconstraints of the alternate scenarios, and therefore highlighting that ateachers’ knowledge will ultimately always vary depending on the context.

Theusefulness of this cannot be overstated as it allows us to confirm that acontext makes a difference to the development of mathematics teachers andindeed their professional identities.  Do Different LearningTheories Promote Different Types of Understanding?There are two different types of understanding, instrumentalunderstanding and relational understanding (Skemp, 1976). Skemp refers torelational understanding as being the traditional meaning of understanding,knowing what to do and knowing why you do that.

However, he described anothertype of understanding, instrumental understanding, he perceived this to be theless traditional, but all too common, “rules without reasons” kind ofunderstanding. At the end of Skemp’s work he Provides the reader numerousdifferent reasons as to why we have these differences in understanding withinmathematics education, ranging from the struggle to compose a test that tests aperson’s understandings, as he claims it is hard to test the differences inunderstandings, to students understanding the importance of the exams andtherefore choosing the easiest route of success, which would clearly be theinstrumental way. However, I will now attempt to provide a differentexplanation, and that is that the difference can occur through differingtheories of education. A socio-cultural theory involves a clear dialogue between anovice and an expert and thus allowing connections to be formed by the socialspace. These connections encourage an understanding, however the type ofunderstanding that is formed in this situation will depend upon the type ofunderstanding that is understood by the expert. I will now attempt to draw uponmy own education to explain. In my first few weeks of sixth form we were being taughtdifferentiation and in my class, which was deemed to be the top set we weretaught it through differentiation from first principles however in anotherstudent’s class they were taught it in a rule based way, such as you subtractone 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 thebasic principles of differentiation, however, another student from that classhad been ill that day and had therefore missed the lesson. When they returnedto school the next day the student attempted to catch up on the work that theyhad 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 anexplanation. The student was happy to oblige however they could only explain itin 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 andtherefore when the topic was discussed the novice was only able to draw on aninstrumental understanding through the connections from the shared environment.However, if this student had decided, rather than ask a peer who was in thelesson about what they missed, to look on the next page of his textbook hewould have been able to learn from the textbook, and this explaineddifferentiation from first principles and therefore they may well have beenable to develop a relational understanding. However, this relationalunderstanding would have occurred through a constructivist approach as thestudent had formed the connections within themselves. This would then provide the question, who’s understanding ofthe topic would then be better? Would it be mine or would it be the student whohad not attended the lesson but then instead read the textbook and constructedthe connections themselves? Of course we will never be able to know the actualanswer definitively, as Skemp explains it is often difficult to test betweendifferent understandings. I am inclined to say however, that I believe theunderstanding that I had which had come through a social interaction with my teacherand the connections I had formed in this environment was greater than thatwould be of a student who had worked alone and constructed an understandingindividually.

 This leadsme to the conclusion that a socio-cultural way of teaching mathematics is morepowerful as mathematics is an interactive subject, we know this through all thepeer 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 powerfulway to lead to a relational understanding however it can also be damaging whentaught in an instrumental way as it can encourage a student to purely fixate onthe rules behind a method rather than why these rules are in place. Whereas, Ibelieve if a student chose to learn from a textbook and they were given a setof rules they would be more inclined to read a further textbook to seek anexplanation as to why these rules are in place. Therefore, I believe it to bethe case that socio-cultural learning is more powerful and will lead to agreater and deeper understanding it will ultimately depend upon the quality ofdiscussion that is occurring in the environment.  Are Different Theories of Learning MorallyDifferent Also?Jo Boaler(1997) introduced the notion of differences between learning and understandingto mathematics education, “Teachers do not see any real difference between aclear transmission of knowledge and student understanding”.

She claimed thatschool classrooms put a heavy emphasis on order and control and the learning ofspecified mathematical methods. She also stated that ‘chalk and talk’ teachingwas evident in most classes. When studying at Amber Hill School what she foundwas that the reasons why their practices where disadvantaging some students wasnot down to the teacher themselves being incompetent, but instead “Teachersdid 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 theteacher’s competency but instead with the environment they created for thelearning to occur.  From thiswe can conclude that a student is much more likely to struggle in a poorlyconstructed environment as it will detract from their understanding.

Thisimplies that Boaler herself believes that mathematics is best learnt andunderstood through discussion in a good working environment. She believes thatthis discussion allows a student to form connections and therefore have a truedeep understanding. Similarly, Richard Pring (2004) refers to mathematics as”A social practice”, this clearly shows that he expects that in order formathematics to be educated in a moral manner there must be some level ofinteraction between the teacher and the learner. If we adoptthis notion of moral practice, we can conclude that it would be best of takinga socio-cultural or a social constructivist view towards mathematics educationif we wished to do so morally. This is because within these two theories thereis an interaction between teachers and learners. However, with the findings ofJo Boaler and how the difference between learning and understanding washighlighted in her work I believe that the most moral learning theory issocio-cultural theory, however when this learning theory is used in the classroomit is important, in a moral sense, to discuss the topic in enough detail inorder for the student to understand it, as it is no good adapting this theoryand then having a poor quality of discussion as this will just hinder somestudents, as we have seen with the ‘chalk and talk’ method. ConclusionMy interestin this topic has stemmed from my ever-changing opinion on what is the mostsuitable theory of learning to mathematics education.

When I was first toldabout the different theories of learning I immediately found myself drawn tosocio-cultural theory. This was down to the fact that I did not know much aboutthe different sub groups within constructivism or socio-cultural theory andtherefore my initial view was that constructivism is when someone learnsindependently and socio-culturalism is when someone learns through discussion.I therefore found myself believing a socio-cultural view as I perceived most ofmy mathematics education to have occurred within an interactive classroomenvironment, rather than alone with a textbook.  However, myopinions changed when I was introduced to social constructivism (Jaworski,2002).

I saw a new side of constructivism, a side which had discussion and asocial aspect. I was particularly interested to see if my views altered againthroughout the course of this task and unsurprisingly they did. Havingconcluded that a socio-cultural theory not only provides you with greaterunderstanding, when the quality of discussion allows for it, and also shownthat it appears to be the greatest learning theory with regards to providing amoral education, I have changed back to this view.

 However,the fickleness of my own view shows how there is not one correct theory butrather aspects from all different ones are probably best in providing the mostwholesome mathematics education.