The focus of this assignment is to evaluate the use of the Cognitive Acceleration in Mathematics Education (CAME) approach to teaching mathematics in the secondary curriculum in preparation for the new exam system. I have found, through reflecting on my practice, that embedding aspects of CAME teaching continuously but gradually into lessons has had many positive effects. The alternative of delivering discrete, stand-alone, lessons has the expectation that pupils should adapt their learning to suit my teaching, or worse, discouraging the use of thinking skills. A need has therefore become apparent to further address this issue. Much research on the effectiveness of thinking skills was done in the 1990's through such initiatives as the Cognitive Acceleration in Science Education (CASE) and the CAME programmes to assess the effects that teaching thinking skills to pupils in years 7 and 8 had on pupils' exam performance.
The findings of the CAME project will be evaluated to highlight its successes and weaknesses with reference to the ideas of popular and academic theorists such as Piaget and Vygotsky. These theorists are important to seek a working definition of thinking skills however; a coherent definition has proven to be problematic, with no universally accepted definition. Opposing definitions each have differing theoretical emphases (McGuinness, 1999). For the purpose of this paper the working definition of thinking skills is that they include thinking analytically, logically and creatively to form reasoned judgments and solve problems. Popular and academic theorists have suggested generalised methods of thinking skills and how these might be implemented into the curriculum. Lastly I will reflect on my own practise as a classroom teacher in the light of the above, influenced by the ideas of Bloom, Piaget, Vygotsky and the CAME project.
The CAME Project
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The CAME Project is a project funded at King's College, University of London. Its aim is to develop cognitive skills of pupils by taking a reflective view towards their learning of mathematics (Adhami et al, 1998). The project was launched in 1993 and focussed on its delivery to pupils in years 7 and 8, a suggested 'window of opportunity' for rapid intellectual development (Adhami and Shayer, 1994), based on the fact that the next formal assessment would not take place until the end of year 9 in the form of SAT's creates this window. The CAME project developed methods for teaching mathematics which promoted 'general thinking' in the classroom and is designed to be delivered during normal mathematics lessons approximately once per fortnight (Adhami et al, 1995).
The project follows on from a previous project styled as the CASE Project which had a positive impact on cognitive development (Adey and Shayer, 1994) and the pedagogies of Jean Piaget and Lev Vygotsky who considered the cognitive and social agendas respectively within the classroom. Research titled Concepts in Secondary Mathematics and Science (CSMS Project) (1974-1979) conducted at Chelsea College found that by aged 14 only 20% of pupils where showing formal operational thinking (Hart, 1981) and criticisms in to normal instructional teaching methods also contributed to the production of the CAME project (Adhami and Shayer, 2006).
A full account of the particulars of the CAME Project and its mechanics can be found in Thinking Maths (Adhami et al, 1998). The project developed a technique of teaching mathematics designed to promote general thinking and recommends a two year programme; with specialist lessons being delivered to pupils approximately once every fortnight. Each lesson is accompanied by a comprehensive set of guidance notes for teachers and resource sheets for pupils. The structure of each lesson is broken down in to the following parts: the concrete preparation phase, whereby the theme of the lesson is fully understood by pupils. The collaborative learning phase is where pupils will work in pairs or small groups and to finish with a whole class discussion where the teacher manages the interactions between pupils in the sharing of their ideas. The emphasis of each lesson is on the sharing of ideas and thinking processes rather than specific knowledge acquisition.
CAME mathematics lessons allow pupils to develop skills in an environment where they feel that it is not always necessary to find a correct answer, if indeed a correct answer could be said to exist. I would argue, however, that pupils are socially conditioned into finding or seeking out the 'correct' answer as this is an assumption held by the majority of a population about mathematics problems. Pupils in seeking a correct answer to a specific problem can become anxious, which in turn is often where pupils develop a dislike of the subject (Richardson and Woolfolk, 1980). If the pressure of finding the correct answer is taken away, it could be argued that it becomes possible for pupils to share ideas with each other, develop collaborative thinking and bridge gaps between ideas, concepts and knowledge. Cassel and Kilshaw (2004) argue that CAME lessons develop teamwork and higher level thinking skills, proving valuable in motivating and engaging pupils. This is because the 'anti-dialogic', rote learning, teaching method usually employed by teachers as described by Friere (1970) creates no room to develop thinking skills, but rather requires that pupils receive and reproduce information, allowing pupils to later 'fit into society', rather than obtaining the skills in which they can shape their future, hence normal teaching methods are insufficient, by the fact that they are dehumanising, removing the pupil from the process and treating them as a deposit box.
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Friere (1970), creates refers to the 'anti-dialogic' model of teaching and names it as the 'Banking Model', one traditionally employed by teachers. Cassel and Kilshaw (2004) state that pupils must be conditioned into the CAME way of working as the dichotomy of different teaching styles can cause confusion. Pupils are conditioned into only operating at the lowest levels of thinking, in normal mathematics teaching, operating mainly at the 'Knowledge' and 'Understanding' stages, as described in Bloom (1956) however CAME teaching addresses much higher order thinking skills of 'Evaluation' and 'Synthesis' as described by the same author. I would argue that this confusion stems from being asked to think during CAME mathematics lessons and to reproduce during normal mathematics lessons, a conflict in styles.
The 'Banking Model' of wrote learning, is one built on the teacher narrating and the pupils collecting information. There is a set amount of information which a pupil must know as outlined by a government laid down curriculum, the curriculum being assessment driven on how much a pupil can recall. In the banking model it is the role of the teacher to give the information and the role of the pupil to receive. More often than not the pupil does not understand what they are receiving and why they are receiving it, just knowing that they must remember and restate this information in order to be considered successful (Friere, 1970). I would argue that this is problematic because without understanding the relevance pupils can become disengaged in their learning which can lead to behavioural problems. CAME addresses this by placing some of its problems in to a practical context whereby pupils can see an application of the acquired skills in real life, stimulating an interest in the subject and an appreciation of its purpose.
If a pupil is unable to fully restate the information which they may only have partially remembered then this is considered a failure. This in turn has a negative impact on the pupils' confidence and motivation in mathematics - this is why pupils tend to dislike mathematics as a subject (Middleton and Spanias, 1999). This is due to the traditional method of rote learning employed by many teachers of mathematics in which pupils must remember methods and show their working, as accepted by those in authority. In contrast, CAME allows pupils to develop their own methods under guidance, accepting the fact that pupils think differently and therefore construct arguments and approaches to problems differently.
Pupils feel comfortable talking with their peers, as they do it all day (Bell and Eaton, YYYY). CAME promotes discussion between pupils, which is a good level of criticality, therefore not a soft option. If a pupil, in discussion with another pupil, or group of pupils, has an idea which is not feasible due to insufficient or incorrect knowledge then other pupils are very quick to critique this idea. The pupil who has been 'shot down', so to speak, is very confident to ask his peers why? Whereas confronting a teacher with the question 'why?' can often be more threatening. This is supported by the findings of Bell and Eaton (YYYY) but they also warn of the frustration encountered if these discussions deviate too far from the curriculum. For this kind of group or peer work to be successful then I would argue that selective grouping of pupils needs to take place so that social hierarchy, gender or ability does not hinder cooperation or have an adverse affect on progress or confidence. Therefore I would argue that groups must contain a mix of high and low ability pupils, allowing the low ability pupils to be developed by their peers. Vygotsky (1978), supports this 'mix' and makes references to the Zone of Proximal Development (ZPD) which describes the potential development through the social interactions with more capable peers, with weaker pupils making the stronger pupils' success their own. Santrock (2004) adds to the ZPD the need for 'scaffolding' which can be removed as the pupils grows more confident and competent. I would argue that the major difficulty of establishing this mix of ability within in the classroom due to setting, where often more able pupils are taught together and likewise for the less able. This does not allow for the less able pupils to benefit from the knowledge and understanding of the more able pupils.
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CAME promotes an explorative agenda which focus on the ideas of discovering by doing, rather than assimilating information although basic information and skills must be held by the pupils in order to develop along a sensible avenue. During the small group work and even class discussions it is possible to take a pupils 'would be perceived as incorrect' idea to extend and strengthen knowledge Bell and Eaton (YYYY), as well as developing the pupils thinking skills as they begin to argue and justify their points. A well thought out CAME implementation can train pupils to think about their own thinking 'Metacognition', see for example (Simister, 2007), and challenge their own concepts and ideas, developing higher order thinking skills of synthesis and evaluation. It is recognised that fitting CAME lessons into the curricula at KS3 involves sacrificing approximately one mathematics lesson per fortnight. This is problematic because teachers come under pressure to 'cover the curriculum', which conflicts with allowing pupils to make their own learning connections (McGuinness, 1996) through sacrificing curriculum time.
It could be argued that in order for pupils to develop their thinking skills in a CAME mathematics lesson then a certain amount of baseline knowledge is required to take part due to learning not being able to take place without thinking, nor think without learning (Simister, 2007). For example, pupils cannot be expected to investigate the areas of different shapes unless they first appreciate the concept of what area is. Therefore a sensible approach has to be taken towards a solution, to find a way to deliver knowledge and skills to pupils whilst also utilising the CAME approach.
The notable influential theorists for the CAME project are Jean Piaget (1896 - 1980) and Lev Vygotsky (1896 - 1934), who through their theories have provided the world of education with descriptors, in terms of thinking, as to what pupils can do at different stages. Piaget is credited for his work in dealing with the nature of knowledge, namely, his theory of cognitive development in which he was concerned with the stages of human development (Piaget, 1983). Piaget considered that there were four stages of development; Sensorimotor, Preoperational, Concrete operational and Formal operational - each stage corresponds to an approximate age span. Each stage may be split in to sub-stages where Piaget carefully considers the level of thinking a child should have in relation to their age. The latter two stages, being Concrete operational (age 7 - 11) and Formal operational thinking which begins about age 11 and continues in to adulthood (Piaget, 1983) overlap with the earlier suggested 'window of opportunity'. These stages are of particular interest as the CAME Project was aimed at pupils in Y7 and Y8, spanning ages 11 to 13. During the concrete operational stage children are able to make appropriate logical comprehension of physical objects. They are able to sort objects by a variety of criteria, recognize logical relationships and make transitive inferences, consider multiple aspects of a problem and appreciate that physical change can be reversed. At this stage the child's egocentrism, as developed in the two earlier stages, begins to fade and children are able to perceive ideas from the point of view of others. I would argue that the degree to which this egocentrism has faded is different in each child and that this could cause problems with some children still being very egocentric, whilst others are very open to the opinions of others.
At the age of 11 children can now begin to reason and make logic of more complex situations. They are able to draw from many sources of information and hypothesize outcomes, considering problems in an abstract form. Children at this stage are able to plan the best path to follow to find solutions and apply methods of trial and improvement, learning each time how to improve their methodology. The CAME project is aimed at pupils in year 7 and 8, the age group identified by Piaget as being when pupils would be moving from the concrete operational stage to the formal operational stage, or would be in the early stages of the formal operational stage, thus the implementation of the CAME project in the target year groups reinforces and develops the skills in coincidence, that Piaget describes in his fourth stage of development.
Vygotsky (1978) considered the interactions with other children and teachers and developed through his work the Zone of Proximal Development (ZPD). The concept of the ZPD originates from Vygotskys' argument against measuring intelligence through standardised tests, arguing that it is better to measure intelligence by assessing a child's independent problem solving skills and the child's ability to solve problems with the help of a more competent person, rather than assessing them on what they know (Berk and Winsler, 1995 pp. 25-34). This difference in competencies between what a child may do alone and what they can do with help from others is the definition of the ZPD. The CAME Project, building on the concept of the ZPD, promotes the interaction between peers to solve problems under the careful management and guidance of a teacher (Adhami et al, 1998). Balaban (1995) refers to the art of scaffolding, based on the ZPD theory, and argues its effectiveness in terms of learning. I would argue that the same scaffolding concept is prevalent throughout the CAME Project, with the class teacher in effect being the engineer, removing or giving more support to groups of pupils as required, in order for them to be successful (Adhami et al, 1995). This theory is further supported by Berk and Winsler (1995, pp. 25-34) who defend the ideas of Vygotsky in that the role of education is to provide experiences which are within their ZPD, enabling children to advance their own learning.
Though both Piaget and Vygotsky are key theorists on whom the CAME Project draws inspiration (Adhami et al, 1998), both are old in terms of their writings. Vygotsky died some 76 years ago and Piaget some 30 years ago. Despite being old and the risk of their works being outdated they are still heavily referred to by authors in the field of child development. The CAME Project is just one example of this, see for example Ginsburg and Opper (1979) and Santrock (2008) who reflect on the works of Piaget and Vygotsky. Notably, some authors have built on the theories of Piaget and Vygotsky, see for instance, Tinsley and Lebak (2009) who developed Vygotskys' ZPD theory further than child development in to adulthood and have referred to the 'Zone of Reflective Capacity'. This can be constructed through interactions with others who are engaged in the same activity (Wells, 1999) and expands through positive interactions and mediations.
Lord (2007) reviewed the findings of research done between 1989 and 2005 into pupils' experiences of the national curriculum. Two points from this review is that firstly young people think about the curriculum in terms of something they have to get through in order to progress to the next stage, i.e., the progression from KS2 to KS3 where more assessment will take place. Secondly pupils' motivation dwindles at the latter stages of KS2 and continues to do so throughout the rest of their education, as pupils see little relevance between the curriculum and their own experiences. Motivation increases slightly towards the end of KS3 and again at KS4 however there is a massive drop in year 8. It could be argued that the CAME Project seeks to address this by relating mathematics to real life problems, allowing pupils to establish connections between the classroom and their own experiences, open-ended tasks provide the opportunity for pupils to not have to complete a task in order to progress to the next stage and also as a tool of motivation, driven by what Friere (1970) argued for; the dialogic practise and relevance to real life. I would argue that many GCSE exam questions avoid reality and therefore it is difficult for pupils to make connections between the curriculum, 'finding x' for instance, and the real world.
In September 2010 the new GCSE Mathematics specification will roll out across all exam boards, on the approval of Ofqual. It is evident from the new prescribed course content (Ofqual, 2009) that the new course requires a great emphasis on thinking. Pupils are to be required to use problem solving strategies, reason mathematically, make deductions and inferences and draw conclusions. As of September 2010 only 45-55% of the assessment at GCSE will rely on the pupil's ability to recall and use their knowledge of the prescribed content, leaving approximately half of the assessment based on higher order thinking skills. I would argue that this poses many problems for teachers of mathematics who have employed rote learning techniques. The pupils currently studying at KS3 who have been exposed to such traditional teaching methods have been trained to recall and restate information which is problematic due to the new specification weighting of thinking skills. It could be further argued that with the dramatic shift in assessment objectives that this is in effect an admission by the governments own qualification watchdog, Ofqual, that previous instruction and assessment methodologies were ineffective. Through its dialogic and well mediated delivery CAME develops higher order thinking in pupils, skills that will be vital for pupils to be successful in future examinations. It could be argued that pupils would do badly under the new exam specification as they have only been trained in reproducing knowledge and demonstrating basic understanding, having never been required to analyse, synthesise and evaluate. To address this potential problem a solution could be a move from the 'Banking Model' towards CAME style lessons at KS3 in order to develop the required skills for the KS4 curriculum.
My daily practise is as a secondary mathematics teacher in a mixed comprehensive high school. The setting caters for approximately 1200 pupils of compulsory school age with approximately 29% of pupils registered as having special educational needs (SEN), compared with 18% nationally (Shepherd, 2009). The school performs below the national average in terms of KS4 results and was graded in September 2009 as 'Satisfactory' by OfSTED. The school also falls under the remits of the 'National Challenge' and is under pressure to achieve a 30% pass rate with 5 A* - C GCSE grades, including English and mathematics. Therefore, I, as well as my colleagues within the mathematics department are under pressure to cover an intense curriculum, preparing pupils to be successful in the examinations.
Previous to conducting this research I would have described my own teaching as being very much in line with Friere (1970) who termed the 'Banking Model', based on 'anti-dialogic' teaching, which I would argue; appears to be the normal practise for many teachers... for instance, a prescribed course must be delivered to pupils as if they were patients in receipt of medicine so that they can attain a 'normalised' grade C. A boredom of my own 'anti-dialogic teaching' methods and a scrutiny of the new 2010 GCSE Mathematics specifications of Edexcel, OCR and AQA have led me to question my own methods in search of improving the quality of learning of the pupils whom I teach.
Given the constraints of being a classroom teacher, pressurised into achieving results I have struggled to 'fit in' CAME style lessons in to my practise. I noticed that most pupils found it difficult to communicate with each other about mathematics and appropriate concepts and they also, in the majority of cases struggled to justify their arguments logically and tended towards the answer 'I just guessed'. I decided, after many attempts of 'stand-alone' CAME style lessons that in order to improve the quality of pupils thinking then a gradual approach to developing dialogical practise, whereby pupils may interact and discuss ideas with each other, had to be taken. The purpose of which was to train to pupils to work collaboratively towards a shared goal, drawing on knowledge which the team already possessed through its composition of complex individuals and how to debate effectively so that ideas shown progression and not failure or humility. Simister (2007) argues that in order to improve the quality of thinking and learning skills then a serious whole school approach must be taken, implying that the promotion of thinking skills should not be confined to just one subject.
My methodology was to work with the idea that pupils enjoy talking to each other. I allowed pupils to discuss their work with each other, after being given the knowledge base from which to work through initially rote learnt methods. As their knowledge and understanding increased, so did dialogue between pupils. Over time I observed that pupil's confidence in discussing mathematics improved and that they could construct a more logical argument when explaining to their peers. Initially this type of activity was difficult to manage. I had to refocus pupils' discussion to the topic on many occasions. Building on this success the plenary session of each lesson was used to challenge the pupils, through discussion and collaboration to seek an answer to a problem presented to the whole class.
If a wrong turn of thought was taken then it was retraced with the whole class and pupils were asked to identify and explain at which point the thought process deviated from the correct one, leading to pupils who had not arrived at a feasible solution to say 'ah yes, I should of done...' and 'oh, I forgot how to do that but now I remember'. It was also interesting to be asked by a pupil 'Sir, I haven't done it like that but I have still wrote stuff down and got the right answer. Am I right?'. This allowed for pupils within the class to appraise the effectiveness of alternative pathways to the same solution, promoting, according to Bloom's Taxonomy of Educational Objectives (1956), pupils to demonstrate the highest order of thinking. An example of this practise was when pupils were solving linear equations; many ways to solve equations exist depending on their nature and the method which the 'solver' chooses to apply. As a preferred method of solving equations the teacher chooses to apply the 'balance method', treating the equation as if it were a set of scales, with both sides equal in size, a pupil, however, whilst observing the teachers solution finds that her method is completely different, yet she has achieved the same answer. The pupil asked the teacher 'Am I wrong?' The teacher asked the pupil to demonstrate the method which she had employed to the class; many other pupils in the class had used the same method as the demonstrating pupil but had blindly accepted the methodology of the teacher, in doing so accepting that their method was inadequate. In not telling the pupil she was wrong, and allowing her to explore and defend her own idea she was able to secure her own knowledge and understanding, appreciate that other ways exist and help her peers, whilst going through the process of metacognition. This is supported by Hancock and Jefferies (2002) who further argue that in addition to pupils being allowed to have their own conceptions in the classroom, and the necessary adjustments which the teacher must make to accommodate these new conceptions, that an open-minded culture must exist in both the classroom and the wider school in order to nurture thinking.
I have certainly seen within my middle to higher ability classes an increased enjoyment and confidence of mathematics whilst gradually implementing dialogic teaching which confirms my suspicion that traditional rote learnt mathematics teaching did not work and did not allow for pupils' thinking. Pupils are effectively teaching each other during directed class activities and in doing so are strengthening their own knowledge and understanding, allowing them to access higher order thinking. The lower ability classes which I teach, namely those with extremely low literacy skills and cognition struggled to access the 'thinking skills' type activities, specifically the pupils had great difficulty in expressing themselves to other pupils or facilitating open ended questions. As earlier suggested I believe this to be due to 'setting' in mathematics, not allowing the lower ability pupils to mix with the high ability pupils affects the ZPD and reduces the potential of the lower ability pupil (Vygotsky, 1978).
In addition to the borrowed aspects of CAME style lesson which I incorporate in to my own I set pupils, and share with them, success criteria for them to assess themselves with throughout each lesson. These criteria relate to different stages of Bloom's Taxonomy (1956) with the first criteria usually being knowledge based and the second being either understanding or applying based where the pupils are given an activity in which to show off their knowledge. The last bit of criteria asks pupils to evaluate their own methodology, and the methodologies of others incorporating suggested improvements - accessing the higher order thinking skills of synthesis and evaluation. This process is now embedded practise in every lesson and pupils are confident to critique and suggest, however, initially this was a daunting task for many, asking them to have their own opinions and ideas.
This assignment has evaluated the use of thinking skills in the CAME Project by discussing the successes and drawbacks of original research. The successes of the CAME Project are in increasing pupil motivation, confidence and promoting the development of higher order thinking skills. Major drawbacks have been the feasibility to implement CAME style, dialogic teaching methods whilst contending with the demands of the National Curriculum, and perhaps changing the perception of teachers who have always employed the traditional 'rote learnt' method of teaching. Popular theorists, such as Piaget and Vygotsky have been discussed in relation to their role in underpinning the ideas supporting the CAME Project. The need for change in traditional mathematics teaching, in light of the 2010 specification, has been addressed, identifying that this method relies on only the lower order thinking skills of remembering and understanding and leaves little if any room for the promotion of higher order thinking. I have found that the continuous and gradual implementation of CAME teaching methods and increased dialogic practise is more successful than teaching stand-alone lessons as the constant exposition reduces the dichotomy of teaching styles.