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When teaching a topic on fractions to my year 2 pupils in the 3rd term I had almost taken it for granted that they would remember the prior information they had on reactions as we had treated it in the first term but I found out this was not always the case. This led me to discover that while the whole class knew what fractions were only a handful actually understood that fractions involved sharing objects and not just being able to color in worksheets on diagrams of halves and quarters.
This helped me discover that as a teacher I needed to be sure understanding had taken place in my class because ‘new understanding performances are built on previous understandings and new information provided by the instructional setting(Perkins, 997). I realized that while the higher ability children understood fractions as being part of a whole number of objects the lower ones saw fractions as no more than the shapes they had colored in their books. This made me realism that the concept of fractions was quite different for each individual in my class and as such I needed to discover how they were able to express their understanding.
My focus in this exercise was to investigate the ways in which I could see some sort of evidence that a concept had been understood in my classroom. Was it by verbal explanations only hen the child is able to talk about the mental representations(Perkins,1997) they have formulated in their mind or was it by when the child is able to carry out an activity based on the mental representations formed or by a combination of the two in which they would be able to use the knowledge gained flexibly ? . My concept of understanding Understanding, intelligence and knowledge are three very closely related concepts which usually occur simultaneously in education but not in all cases. Gaining knowledge is very different from understanding as far as I am concerned because if knowledge gained cannot be applied and used flexibly in real life situations then complete understanding has not taken place in the individual.
Intelligence has been defined as ‘a general ability that is found in varying degrees in all individuals. (Gardner,1993:14)’. This must mean that an individual’s level and type of intelligence will affect that individual’s ability to gain the knowledge that is required to get proper understanding of a concept. To quote David Perkins , understanding is defined as the ability to think and act flexibly with what one knows”(Perkins, 997:40). Let has also been stated that ‘….. Understanding is in essence the reshaping of old knowledge in the light of new ways of seeing things'( . Since thinking is an internal process that goes on in the human mind this meaner then that if an individual cannot interpret and act on or build on the knowledge off new concept or activity then that concept has not been fully understood. Understanding, to me, is therefore internal and mental (representational )(Perkins,1997) but should be manifested through human activity(performance)(Perkins,1997) as well.
I believe Understanding is like a small seed that is planted and grows through using it in varying degrees (flexibly) in performances until the learner becomes an expert. If understanding is internal then the ability to display the understanding of a concept will be expressed differently by different people. Since I teach in an SSL classroom I was curious to discover how an understanding of concepts was expressed and how well the pupils were able to explain and manipulate the knowledge gained to show their levels of understanding.
This is because the underlying assumption is that it is mainly through productive practice that learners internalize the system of language’ (Littleton,1984:92) and for me to encourage effective learning in my classroom the pupils must understand the language of instruction adequately so they can truly understand the concepts being learned. In this assignment my intention was to investigate the evidence of understanding in my classroom. I wanted to see how the selected pupils expressed their understanding of fractions.
I agree with David Perkins that ‘Understanding lies in the possession of the right mental structure or representation. Performances are part of this picture but simply in consequence of having the right representation (Perkins,1997:46) I wanted to see the extent to which the pupils could get the correct mental structures in place that would help them understand fractions and use this understanding in their everyday activities. I wanted to use this research as a diagnostic assessment (Black,P. ND William,D,2001) that would help me improve my teaching by identifying areas where the pupils were lacking. I also wanted to use the results I got from this investigation to help me apply differentiated teaching methods better in my lassoer . 3. Research focus For my investigation I based my research on the Social Constructivist theory of Woks. This is because according to Whisky ‘Education must be orientated not towards the yesterday of a child’s development, but towards its tomorrow'(Viscosity 1978). Take this to mean that the pupils I teach should be able to use the concept learned in the classroom in future and especially outside the classroom as this will show that they have understood the concept . The social constructivist believe that children construct new knowledge based on some prior knowledge. According to Whisky ‘any learning a child encounters in school always has a previous history since unlike other animals ‘human learning presupposes a specific social nature and a process by which children grow into the intellectual life of those around them”(Viscosity,1978:55).
This is because Wooster focuses on the link between language and societal influence on children’s learning and how children use language to show understanding of concepts and take actions or construct new knowledge based on a previous understanding they have of a concept through their social interactions. In edition, ‘much of growth starts out by turning around on our own traces and recoding in new forms, with the aid of adult tutors(teachers), what we have been doing or seeing ,then going on to new modes of organization with the new products that have been formed by these recording'(Burner, quoted by Barnes,D. 992:128) For this exercise I investigated a innumeracy class on fractions . In the Innumeracy class I investigated how the children, (between the ages of 6 and 9) understood the concepts of halves and quarters and how they were able to use this in a real life setting to share some objects in the class. Since the focus of this assignment was on evidence of understanding I wanted to see how the pupils in my class constructed new knowledge on fractions by using their prior knowledge of fractions and if they could show me they understood the concept of fractions by practically being able to share objects in halves and quarters in the classroom.
I wanted to see how well the pupils understood the concept of fractions working in groups within their Zones of Proximal Development which is the distance between the actual developmental level as determined by independent problem solving and the level of potential velveteen as determined through problem solving under adult guidance or in collaboration with more capable peers’ (Augusto, 1978:86). I wanted to see how well the two pupils selected, who I will call Jack and Oswald, would be able to display some ‘performances of understanding(Perkins,1997:42) based on the mental models(representations) they had in their mind about fractions.
I wanted to compare how well they could verbally explain what they understood about fractions with a practical demonstration of how they were able to share objects based on what they ad been taught to see if they had a proper understanding of fractions for their level. 3. 1 Data Collection Since my research question is on investigating evidence that understanding had taken place I chose to use classroom observations,an open observation(Hopkins, 2008:83) using a video camera in which I could watch the pupils carrying out a performance of understanding and then interview the pupils involved verbally to record the data on my findings.
I wanted to carry out a classroom observation in which I would be a participant only to a small extent, using what was said in espouse to my questioning as qualitative This is because I share the opinion that ‘A qualitative approach, (to research) by contrast, emphasizes meanings, experiences descriptions and so on. Raw data will be exactly what people have said (in interview or recorded conversation) or a description of what has been observed. (Colonial, 1990: 36-7 quoted by I chose video recordings over audio recording alone because Video-recordings…….. An be played and replayed, so that observations are likely to be more reliable than those made at the time. You can go back and check any point you’re not sure of, and you can ask there to check or comment on your observations(Swan,1994 :36). Another reason why I chose this research method was because it would allow the children to talk to me and explain how they had conceptualized fractions in their heads( formed mental models (Perkins,1997 :46) and how able they would be to carry out an activity on fractions(performance of understanding ) in class . realized that this would make them more comfortable since the pupils would be better able to express in words an activity they had been physically involved in. I had to observe the hillier at work using a video camera and use my observations to draw conclusions on how well they were able to understand fractions. A disadvantage of observing my pupils with a video camera was that the pupils were conscious of the camera and as such were a bit stiff and shy to speak about their activities and secondly it took a while to settle the pupils down since they were excited at the prospect of being on camera.
I preferred this research method to others like verbal reports and protocol analysis which is a ‘rigorous methodology for eliciting verbal reports of thought sequences as a valid source of data on thinking(Ericson,2002) because I was not able to organize my observation in a manner that would allow me to record my pupils thought processes as they worked since I was making the video myself . N addition I chose this research method over others like focus groups(Lewis,1992:413) which would involve conversations with a group of children because the methodology required for this would have been too complex for an inexperienced researcher like myself to use accurately. To remain within ethical standards had sought permission from the management of my school and children’s parent’s(see Appendices) to inform them about the research I was carrying out since it would entail the children’s faces being recorded even though their real names would not be mentioned in my writings. . 2 Observations The topic being discussed was fractions and how to get fractions of a number of given objects. We had treated fractions in the first term and it involved mainly being able to identify fractions of shapes and shade given fractions of shapes like halves, quarters, thirds e. T. C . This term our focus was on how to identify fractions of objects . The lesson began with a whole class game called ‘sharing leslies’ (wry. Sees. Co. UK/ resources) in which the pupils all had to share leslies into as,as, and as.
I explained the link between fractions and division because both require sharing and after this they were told to go to their seats and look for as many objects as they could find around the class which they could share into halves and quarters and arrange them in the manner they had been shared on their desks for me to see. The focus of my investigation was mainly on two pupils. One of the high ability pupils, Jack, and one of the low ability children, Oswald.
I chose them because of their learning ability roofs and their levels of spoken English. Both pupils first language is Arabic but Sacks English is much better than Oswald because his parent’s speak fluent English but Oswald English is quite poor because his parent’s do not speak any English at all. I noticed that in this class activity Jack sat still and listened during the lesson but preferred to stand up while other pupils sat on their chairs and played the game of ‘shared leslies’.
Oswald sat in his chair but did not focus on the game and was constantly moving about and rocking on his seat and he did not participate actively in the game of ‘shared leslies’. When the whole class activity was over and each child was at their desk I asked both of them if they had shared the objects in halves or in quarters and to test the level of their understanding I asked them to give me halves and quarters of the objects they had shared.
Oswald had shared objects in quarters on his desk but could not tell me whether he had shared them into halves or quarters or how many objects were in each quarter until I prompted him with more and more questions which made me see that; first of all, he had modeled the work of the other pupils in his group and secondly, he understood fractions only o the extent that fractions involved sharing but he did not understand how to interpret what had been shared.
Jack on the other hand was able to express clearly how he shared twenty colored pencils in quarters and even went as far as making a word problem out of the objects on his desk which showed that he clearly understood the concept of fractions and with this present understanding he was ready to learn further on the topic of fractions. I realized that some basic level of understanding had taken place between the two pupils because of the performance hey were able to give by practically dividing the objects into halves and quarters on their desks, but one clearly had a better understanding than the other.
I also noticed that they did not Just count out the objects but actually took time out to mentally divide the surface of their desks into twos and fours and placed the items in each section to get the correct answer. In addition, I noticed that all the children shared the objects they were given in different ways yet most of them had the same idea that halving requires dividing into two and sharing in quarters required dividing into our.
I gave the pupils worksheets to solve fractions word problems as a further test to see the levels of understanding of the concept of fractions. 3. 3 Research summary The conclusions I was able to draw from this episode on investigating evidence of understanding is that talk is very key in providing evidence of understanding in a classroom. This is because I allowed them to sit in their work groups to do this activity and I noticed they were having fun while they worked and they corrected each other when they made mistakes.
I also realized that the readiest way of working n understanding is often through talk because the flexibility of speech makes it easy for us to try out new ways of arranging what we know and easy too to change them if they seem inadequate. (Barnes,1992:125) This was seen in the way that even in my absence the pupils were able to work on their understanding of fractions by talking to their group members and being corrected by their peers when they did something wrong.
When responding to my questions, Jack used presentational talk which is ‘a final draft for display and evaluation’ he was very sure of what he wanted to say to me. Oswald on the other hand used ‘exploratory talk (Barnes,1992:126) because he had not grasped the proper understanding of fractions. I realized that for my lower ability pupils I would need to work on creating avenues for more exploratory talk to help them think and exercise their minds . 1 needed to let them use … Discovery as the paragon performance that both attains and demonstrates understanding(Perkins,1997:56). This research has also led me to discover that understanding occurs and can be evidenced at different levels for different learners and that a 100% understanding is not possible. This dad me raise the question, what is the cause for these variances in the expressions of understanding ? How is it that in the same class the way the pupils have interpreted the concept of fractions is different when they all participated in the game?
In addition it made me wonder if language alone is sufficient as evidence of understanding or must understanding be measured by a combination of both performance and language since Oswald was clearly able to share objects into quarters on his table but he was not able to use language to state clearly what he had done and what he did not understand. This also made me wonder how much of a boost or barrier language was in providing evidence that understanding has taken place since if I had Just looked at Oswald desk and not verbally questioned him about how he had arranged the objects I would have assumed he understood the topic .
I was able to realism that as a teacher I should be ‘…. Not Oust) a distributor of knowledge but a facilitator who provides an appropriate learning environment and encourages the child to formulate ideas, discover concepts and think independently(W. T. Laggard (2004)p. 2). Therefore in my class the evidence that Understanding has taken place should be in the way that the pupils are able to think out of the box and work as independently as possible to discover(construct)new knowledge for themselves.
In this case it was the knowledge that fractions is an introduction to division which they would be learning in Year 3. Another question that was raised in the course of my investigation was the place of language in the understanding process. In Oswald case I want to believe it is his level of English competency that was a barrier to his understanding because he only uses English in he classroom and had a challenge having to be taught and then express himself in English to show what he understood.
This is because the learner needs to refine his language competence in order to communicate more effectively or appropriately . (Littleton ,W. 1984 PEP) Conclusion I know that it is not possible to understand everything about something. This is why we have inactive understandings(Perkins D. 1997 pep) of concepts which is only a partial understanding but with this we can still use the concept learned flexibly so this shows some sort of understanding.
For example, in the rules of grammatical beech, but even for this we still need to have some mental models(representations) at work that help us use grammar correctly because we have been making grammatical performances for so long unconsciously that we have now become experts at it. In summary from my research I have been able to see that understanding has to be expressed both as a representation of mental models and as a performance because one without the other does not show a complete understanding of a concept.
This is because ‘Understanding lies in possession of the right mental structure or representation. Performances are part of the picture but simply in consequence of having the right representation. (Perkins,1997:46) To help the pupils in my class this research has shown me that to ensure that active learning takes place in my class I need to be sure that understanding has taken place for all the levels of learners in my class and as such I will need to improve on my differentiated teaching skills to make the pupils succeed.