In this way he further melds the different theories to highlight their similarities and show how they can be considered to be positioned on a continuum of views that are not so fundamentally opposed so much as views from a different perspective. Cobb also demonstrates how the constructivist views are even spread over the continuum of theories in illustrating how the different constructivist theories place varying emphasis on the significance of societal interaction. He refers to Bauersfeld’s reference to “implicit negotiations ... outside the participant’s awareness” (Cobb, Chapter 9, Reader 2) which again helps to form a bridging of different perspectives towards that of the socio-cultural.
Cobb stresses the importance of co-ordinating these viewpoints with regard to what is taught and how it is taught. In doing so he is reflecting the framework put forward by Alexander (p9, study guide). In acknowledging that each perspective contains elements of the other and that the different viewpoints are merely positions on the continuum placing emphases in slightly different places, then the nature of curriculum content, style of teaching and assessment must accordingly reflect this range. If, for example, it is true that guided participation is in the background for constructivist theories then any constructivist educator would have to provide opportunities for guided participation within his or her learning environment.
Cobb refers to the theories of Lave and Wenger as at the extreme end of the socio-cultural perspective in that they attempt to avoid any reference to mind in the head. Whilst it seems true to think that to adopt Lave and Wenger’s position would be to negate many constructivist ideas, it is fair to say that in viewing theories of learning as a spectrum of differing perspectives, it is possible to incorporate their views into a broader theory of learning which can then act as a framework for a broader range of educational practice. As with any broad spectrum of opinion, it is easier to co-ordinate the different perspectives from the middle ground, than it is from the extremes. It is therefore reasonable that Cobb should incorporate ideas from Lave and Wenger together with those of Dewey perhaps from his essentially constructivist perspective. It would be more difficult for those on the extreme ends of socio-cultural thinking to incorporate ideas of a fundamentally symbol-processing viewpoint and vice versa.
When Cobb argues for a less pragmatic stance on the use of theories, he does so from the analysis of predominately mathematical education which can offer a quite unique window into the mind. Much mathematics and mathematical knowledge is abstract in nature which in turn can place greater importance on what is happening in the head and less on the situated nature of that thought. It is hard to consider the situated nature or social context when learning about n-dimensional space or non-Euclidian geometry. It is equally hard to assimilate a purely Vygotskyan viewpoint that “any higher mental function was external and social before it was internal” (Vygotsky cited by Cobb, Chapter 9, Reader 2) into the development of much mathematical thought, or indeed to explain how Ramanujan could develop advanced theories of numbers whilst in his Indian village without contact with any other mathematical thinkers. Indeed, it is not easy to justify any original thought from the point of view of an apprenticeship view of learning. Bredo commented that “work on situated cognition has emphasised inseparability of cognition and context” (Bredo, Chapter 2, Reader 2) which itself is in contradiction with much mathematical knowledge which currently is without real world context. In a similar way it would be hard to deny the importance of inter-subjectivity and social context in learning, either from a personal viewpoint having witnessed the manner by which my own students learn and from the strength of arguments of much educational research. Cobb argues that by adopting a less pragmatic stance, educators are able to “consider what various perspectives might have to offer relative to the problems or issues at hand” (Cobb, Chapter 9, Reader 2). In my view this is entirely consistent with Gardner’s proposal of the existence of ‘multiple intelligences’ (Chapter 7, Reader 2). In the same way that Cobb argues for educational practice to account for the entire continuum of views of learning, Gardner would argue that we need to cater for the different dimensions of intelligence. Within an activity as potentially diverse as teaching, I believe that there is plenty of room to assimilate all the theories of learning to corresponding practices within any educational establishment. Bredo characterises the symbol-processing view of the mind as a computer blindly carrying out algorithms without any real understanding of what they mean (Chapter 2, Reader 2). It is not uncommon still, anywhere in the world, to see students ‘doing’ mathematics in a similar way. It is my view that this is, at least partially, a result of a narrow view of learning. By adopting a broader perspective, one is able to view educational issues from multiple angles to better serve the multiplicity of learners that one is working with.
Part B: Observing and analysing practice.
In this part of the assignment I have opted to look at mathematics coursework, as both a learning activity and related assessment. Since 1982, when the Cockroft report highlighted the need for a greater variety of teaching methods, there has been an element of project and investigational work forced into all secondary school mathematics curriculae. Whilst essentially such work should be a valid learning activity as illustrated by many researchers and writers (for example Hiebert, Reader 2, Chapter 10), the boundaries between what is a learning activity and what is an assessment activity are frequently blurred especially when considering mathematics coursework. Currently, mathematics coursework is an integral part of GCSE assessment and as such needs to be done by all schools in the UK offering that course. Although the justification for the inclusion of coursework in that assessment is on the basis of its value as a learning activity, I am sure that my current school is no exception in its general approach to that as solely something that has to be done as part of the assessment. In my present situation the only project and investigational work undertaken are the two compulsory pieces in years 10 and 11 for GCSE together with one piece of practice coursework in year 9.
Why should it be that with most research indicating the value of such activities, that there is still so much resistance to such work and why should it be that if such work is such a valuable learning activity, then why is it that schools and departments that pay only lip-service to such activities, can continue to achieve high levels of success?
It is my view that the answers lie in the gulf that exists between the learning activities and the assessment activities. The Alexander framework indicates that views of knowledge and learning influence the methodology of teaching which in turn influences the method and content of assessment. This would certainly be a logical progression in an ideal world, but I feel that it is not always how educational practice progresses in the real world. Mathematics and mathematics education is so bound up with commonplaces that it is easy for practitioners in the subject, together with agencies involved in determining learning and assessment practices, to pursue learning and assessment activities whose worth is ‘self evident’ without the need for any real analysis of the overall aims. As a consequence, whilst research would indicate greater value in the use of problem solving and project work as a learning experience, the assessment of the subject does not effectively assess the laudable aims associated with that and consequently allows for, and even encourages, an alternative approach. For example it is common to teach number bonds in a variety of different ways in order to establish a firmer understanding of the concept of number in students. This understanding of number is then tested by questions which supposedly test that understanding. If such a question were, for example, to evaluate then it is clear that a student who calculates is likely to have a better concept of number than a student who blindly follows ‘the’ multiplication algorithm . However, in the assessment there is no advantage of showing a better understanding of number by doing the problem in the first way. Indeed it is often encouraged by teachers that students should adopt the second method as it is ‘safer’ and also lends itself to partial marks should they make a mistake in the working. It is in this way that although the stated, and intended, aims of external assessors may be to shape mathematics teaching to be more in keeping with a situated view of the mind, the assessment procedures resulting do not display the required fitness for purpose, giving rise to continued procedural mathematics teaching.
Tahta (1981) suggests that mathematical tasks have inner and outer meanings, in much the same way that Hiebert (Reader 2, Chapter 10) refers to mathematical tasks having a meta-strategic residue. Mathematics coursework provides an excellent opportunity for the students to develop problem solving strategies as well as providing a situated setting in which to solve mathematical problems. This would not be situated in the sense of a ‘real life’ problem, but more situated into the classroom culture. Hiebert (Reader 2, Chapter 10) justifies this view of situatedness in that for a student to address the inner meanings of the problem, it is not important whether that problem comes from real life or from school, but whether the student can engage that problem as his or her own.
The introduction of mathematics coursework into the assessment of the GCSE is certainly intended as “an instrument to facilitate and encourage curriculum development” (Edexcel, 2000) and it is also stated that “Coursework should be an integral part of the mathematics curriculum and not simply a bolt-on exercise aimed at satisfying new assessment criteria” (Edexcel, 2000). It is, however, apparent that this is not the end result. My current school continues to obtain exceptionally high results in spite of, and many would say because of, a non-adherence to the aims of the assessing body. It is very tempting to express the current practice of this mathematics department as driven by a symbol-processing view of the mind as the Alexander framework would indicate, but discussions with members of that department would indicate that this is not the case. It would seem that teaching methodology is often more determined by the oncoming assessment, than by the particular views of learning. Teachers are, in many respects understandably, more concerned with methodologies that ‘get the results’ than concerning themselves with the more deep rooted aims of teaching mathematics. It would therefore follow that government and external assessment agencies have a great deal of responsibility to ensure that the end assessment does provide the necessary framework to assess the students grasp of the inner meanings of the tasks and not just their ability to produce prepared responses.
I shall be examining students’ learning whilst working on the year 9 practice coursework task. The reason for choosing this particular group is that the practice coursework is meant to be more of a learning experience than the GCSE coursework assessments and so a greater degree of guidance can be offered and consequently more can be learnt about the students’ learning process through interaction between myself and the student.
The task set is that of ‘Maxi-Product’ and is shown in full in the appendix. The classes were recorded on video throughout the two week period of the coursework and transcripts of the more poignant interactions were made. The final piece of work handed in was then marked according to the criteria laid down by Edexcel (the assessing body of the GCSE). Selected students were interviewed during the activity regarding their perceptions of the work as both a learning and assessment activity. It was not possible to discuss the final assessment of the finished work with the pupils in the time available although there will be some discussion of that when it is returned to them.
Throughout the activity it would be hoped that the students develop strategies for problem solving together with recognising applications for mathematics previously discussed from a more theoretical perspective. As a consequence to treating situations problematically it is hoped that students can develop their own strategies and “adapt them later, or invent new ones, to solve new problems” (Hiebert, Reader 2, Chapter 10). The assessment of the activity is two fold. Firstly, the completed coursework is marked according to Edexcel criteria. In addition, though, there is the assessment that is ongoing throughout any learning activity by myself as the teacher. This is as a result of interactions with the students, and eavesdropping (in this case with the help of the video camera) on student interactions. It is suspected that the two forms of assessment will give some notable contrasts thereby bringing into question the validity of using mathematics coursework as a means of assessment.
Part C: Evaluating practice in relation to theoretical perspectives.
Analysing the combined evidence from my recollection of the lessons, together with the video provided some valuable, if not surprising, insights into the practice. In some respects it was a little disappointing that more pupil-pupil interaction was not clear enough on the video to be able to transcript fully, but certainly there were sufficient snippets to be able to get a good impression and there were some extracts of complete dialogue that could be understood which were of some value.
The first, most notable observation was how in the first two lessons there was an enormous amount of interaction between the pupils. This was clearly taking place whilst the students developed an understanding of the problem. In the presentation of the task, I had told them that they could, and probably should, talk about their work with each other and any other sources to get ideas, but that the final piece of work handed in should be their own. It was perfectly apparent that the students used the social interaction in small groups, and then between the groups, in order to explore ideas and establish a pattern for their work. The following transcript between a group of boys is typical of many conversations that were taking place around the room:
Stephen: If you divide it into four parts then you’ve got to divide the number by four then times it by however many times it goes into it
Some writing
Stephen: So you divide the number by however many times you’re going to split it
Isaac: and then you divide that by four. What’s three divided by four?
Reaches for calculator
and then you times that by ... how much?
We’ve done three divided by four
Stephen: Why did you do that?
So that’s twelve divided by four parts (typing into calcultor)
and that equals three
And then three times three times three times three equals eighty-one
What’s four times four times four?
David: Sixty four.
A little later
Stephen: I see a pattern emerging. All the way Dave look.
As you can plainly see (I think for the benefit of the camera!), look there
You’ve got to divide the number, yeah, by how many times you’ve got to split it which is six and then you’ve got a six (pointing) and that’s the maximum number you can get. Try and get a higher number.
By the third lesson it was evident that students had the required understanding and were then much quieter whilst they carried out their calculations and were writing up their findings. From this point on, all the individuals and groups proceeded in quiet intervals where individual work was being carried out and then periods of discussion within specific peer groups and, to a lesser extent, between groups. It was evident that most pupils would attempt to gain insight into the problem through discussion with peers and only seldom ask me for advice or help. Frequently when I was asked for help, it was more to do with the specifics of what was expected to be produced, rather than in help with understanding the problem. According to Rogoff, the interaction taking place during the two weeks of observation would be more supportive of Piaget’s view that “only when children are able to discuss problems as equals are they likely to take into account new ways of thinking” (Rogoff, Chapter 5, Reader 2). In many respects this contrasts with the Vygotskyan view that “the child is more interested in gaining from the expert partner” (Rogoff, Chapter 5, Reader 2) but this may be specific to the particular task. As it is that no particular skilled knowledge is needed for the problem, it may be just in cases like this that pupils can gain as much or more from equal status peers than from a more skilled partner.
Although it is true that most students preferred to develop their understanding from each other it was also evident that two students in the class would prefer to ask me for help and advice in preference to discussion with peers. In both cases the students did interact with other students regarding the work but would only do so from the point of view of being the more expert partner. They would discus their findings and explain how certain aspects of the problem could be approached but if they wanted to further understand a certain aspect of the problem, then they would ask me rather than discuss what could be seen as their own shortcomings. It would seem that these students lacked the self confidence to appear to their peers that they are not in control of the situation. McDermott refers to the degradation of LD pupils and discusses how the student can go to extreme measures to avoid failing at a particular task. It seems that such scenarios are not entirely restricted to pupils with learning disabilities (Chapter 1, Reader 2). The particular class I was working with was a top set and all students would be expected to achieve A or A* at GCSE in two years time.
One of these two students provided a particularly interesting learning scenario. Eleanor knew that she needed to provide some algebraic analysis of the problem to justify her results and was asking me how she should go about doing so. Once she had asked me for help, I led her to do the same as she had before with numbers and follow the same steps starting with a variable x. She seemed to understand and I left her to try to work it through. The video revealed that as soon as I left her she was looking very distressed and didn’t manage to achieve anything at all. She called me back in about ten minutes and essentially asked me exactly the same thing again. I explained again and started off the process with her and left her again. At this point she was on the verge of tears, although I hadn’t seen it at the time. This was only evident in the analysis of the video. On the third time of going to help her I sat with her whilst she worked through what I had discussed with her before and she obtained results that she could then work with. She had no difficulty at all in working through the problem, but rather her problem seemed to be a lack of confidence to embark along a route when she couldn’t see where it was leading her. She confirmed this impression in a later interview with her. I suspect that this is a common problem in the learning of mathematics at all ability levels. Mathematics is often presented in such a way as to appear as if all steps in the solution of a problem are in a logical progression. Mathematics teachers give examples which have been prepared and follow a set procedure leading always to the solution. A mathematics teacher will never embark upon an avenue in looking for a solution that comes to a dead end - if he or she did so it would probably be regarded as incompetence on the part of that teacher. A consequence is that students seldom develop the impression that the solution of mathematical problems can often involve a great deal of trial and error. Eleanor’s problem was that she couldn’t see her way through the entire problem and therefore was unprepared to start working in a direction where she was unable to see the outcome. There has been much written on the differing performance of boys and girls in mathematics, (for example see Smart, 1995), and it would seem that the possibility of boys’ greater ‘sense of adventure’ could account for this. It is certainly a need that I have identified in that I need to promote within my students a ‘suck it and see’ mentality and that students should realise that normal mathematics practice does involve investigating non-productive avenues if more worthwhile ones are to be discovered and understood.
Since working overseas where investigative, or pupil centred, learning is somewhat unusual, I have incorporated into my normal teaching a strong element of justification of learning experiences into my everyday teaching. As a consequence, students in my classes have been exposed to repeated explanations of why they are doing particular activities, with regards developing processes and skills rather than learning techniques and gaining mathematical knowledge. One student did ask me early on in the coursework what the point of it was and how she might use the results and I did explain it in such terms, emphasising that such problem solving skills were applicable to the solution of other mathematical problems as well as other real life problems. It is therefore not surprising that most of the students interviewed could identify one of the goals of the activity in terms of the development of problem solving skills. It was slightly more surprising that all the students identified the primary goal for the activity as a practice for GCSE assessment. It is thus apparent that although I, the examining board, and government agencies, would view the work as a learning activity, the students see it predominantly as an assessment activity. This is perhaps not so surprising given the overall school strategy of so little emphasis on such activities. In a similar fashion, the students perceptions of what they had learnt was predominantly in terms of preparation for the GCSE assessment. Comments such as “I have learnt what sort of things to do” or “I thought at the start it was really stupid but I learnt there was lots to do once you get going” were the norm. Only one student identified learning outcomes as the development of problem solving skills, but I suspect that his response was only a result of my earlier comments.
With respect to the appropriateness of the assessment, the students generally did not feel that it was particularly fair. One student asked me “Do I get extra marks for being the first to work out the result for negative numbers?” He referred to a result that he found that was quite significant and once he’d worked it out, the other students in the class latched onto the result. Although they each produced the working individually it was the one student who showed the initiative in considering this case. The same pupil developed other results by himself but ended up with fewer marks than some other students since his writing up of the results was not so well justified. In this way, my own ongoing judgements of how students were developing the skills needed for mathematising the problem, did not coincide with the assessment marks awarded using the GCSE coursework guidelines. Whilst I judged that the coursework provided a valuable learning experience for the students involved - all students were engaged in the problem for the full two weeks not as Hiebert noted because they had a particular interest in the particular problem, but because “the class had established a culture in which the students knew they had the freedom and responsibility to develop their own methods of solution” (Hiebert, Chapter 10, Reader 2) - it seemed to me that the activity was inappropriate as an assessment activity since the skills intended to be developed were not those which were given credit. This feeling was certainly apparent in my interviews with some students, but it is always difficult to know how much their opinions are shaped by my own when there is a spirit of openness in the classroom.
My analysis of students involved in the undertaking of coursework has not really revealed very much that is new. There is much evidence in other research of the value of coursework activities as a learning activity. What is important for government agencies and assessing bodies to evaluate is how those attributes that such activities promote can then be effectively assessed. Until that issue is successfully addressed, schools such as my present school cannot be blamed for continuing its present policy of reluctance to embrace such activities into the curriculum.
Bibliography
Reader 2. (1999) Learners, learning & assessment
Cockroft, W. H. (1982) Mathematics Counts, London: HMSO
EDEXCEL (2000) GCSE Mathematics 1385 and 1386. The assessment of Ma1 2001
Smart, T. (1995) Towards Gender Equity in Mathematics Education, edited by Gila Hanna, Kluwer Academic, Dordrecht
Tahta, D. (1981) ‘Some thoughts arising from the new Nicolet films’, Mathematics Teaching, 94, The Association of Teachers of Mathematics
Appendix:
Maxi-Product
The number 12 can be split into pairs of numbers
(12, 0) (11, 1) .....
The two numbers in each one are multiplied together and the products are:
, , ....
1. Obtain the partition of 12 into a pair of numbers which produces the maximum product.
2. Choose any numbers of your own.
In each case INVESTIGATE the split into pairs and find the partition which produces the maximum product
The number 12 could also have been split into:
1. triples (3 parts) such as
(4, 7, 1) which produces a product of
(6, 2, 4) which produces a product of
2. quadruples (4 parts) such as
(2, 5, 1, 4) which produces a product of
(3, 6, 2, 1) which produces a product of
3. Examine the splitting of any number into any number of parts. In all cases INVESTIGATE the split which produces the maximum product
4. If the number 12 is split into product pairs such as:
(4, 3) (2, 6)
INVESTIGATE the sum of the numbers
