Abstract
The idea behind the project was to find a way of helping PPG children feel more confident when trying to understand mathematical word problems. In order to do this, I chose to teach a range of children (from Key Stage 1 and 2) on a one-to-one basis, the bar modeling strategy through a CPA approach.
Key findings: I found that most children were able to use the strategy when working with myself, but they did not tend to use the strategy instinctively when faced with a word problem. Therefore, this strategy will need to be ‘embedded’ within the school’s mathematics curriculum, in order for children to utilize it. However, the children that did not get the answer to the question correct the first time, were able to use the bar model to ‘see’ what to do and then answer correctly. This pictorial approach will need to be closely linked to the abstract and the concrete use of manipulatives.
Project focus
Since 2016, our school has been focusing on the pedagogical approach associated with Singapore mathematics (a top performing nation as seen on the Trends in International Maths and Science Study reports, from Clark, 2013) where mastery and slow in-depth learning is at its core.
In my project, I chose to look at teaching the use of rectangular bars to visually represent information given in a mathematical problem. Because of the high percentage of EAL and PPG children at our school, this approach would be beneficial as it helps children understand the ‘language of mathematics’ (Vorderman et al, 2011). I specifically wanted to know what the best approach was for teaching children how to use these bar models. As a consequence of this new knowledge, the quality of teaching and pupil learning would increase. Coupled with this, many of the teachers that I work with have had only one inset day to understand the concept.
In addition, Hertfordshire County Council has its own set group of priorities which have helped shape our school’s strategy. This includes: ‘Closing the gap between underachieving groups and all Hertfordshire children and young people’ (Hertfordshire’s Strategy for School Improvement, 2014-2017).
On a personal level, as a perpetual Key Stage 2 teacher, I was interested in working with children from Year 1 upwards, as KS1 is the best place to view the start of a child’s journey into the world of mathematics.
Also, I hope through this research I will be able to start recording these strategies and then making them available on the school webpage; this will enable parents to support their children at home. The Vorderman report specifically mentions the key influence that parents helping children with their homework can have on their learning; however, many parents are unable to help as they do not understand new strategies (Vorderman et al, 2011).
Methodology and design frame
I planned to gain some concrete data from my research. One reasons for this approach, was that you get ‘thick’ (Geertz, 1973) quantifiable data which can then be analyzed easily for generalized changes in the children’s understanding (Burton and Bartlett, 2009).
However, I also took a detached view as well, as I was trying to find out how children felt about dealing with mathematics. This is vitally important as the confidence of a mathematician links closely to their ability: as Jo Boaler (2009) says, there is no such thing as a person who cannot do math. Is this true for PPG children at my school? I also wanted to explore the children’s lives as they live in the real world competing with emotional stresses and strains (Sachs, O. 1996); especially as my sample are PPG children who may have difficult personal situations. This could though cause problems with validity as the children’s emotional states effects their performance.
I spread the sessions over six weeks, as Spindler and Spindler (1992) in Cohen et al (2007) have referred to ‘prolonged and often repetitive’ sessions being needed for impact. Also, this would improve my results reliability because only one occasion could be suspect. However, as I will have a fixed position in my interviews, reliability in that area is irrelevant (Thomas, 2013). Having a long time period is important, as Goutard (Goutard, 1968 in Ainsworth, 2007) has argued that her empirical stage can be time consuming, but necessary for greater understanding or mastery. This need for time is echoed by Askew (2012) when he says that children need time to see the link between what they are doing with manipulatives and what the teacher is doing pictorially on the paper.
In this study, the sample I decided to use was very proscriptive: I purposely used selective bias to choose children that were registered as PPG. My main variable that was changing were the children’s ages. I planned to select two children from each year group as they would give my results some reliability. However, this was not always possible.
I will need to be careful when writing the conclusion, as my own bias as a person and teacher might affect it (Thomas, 2013). However, as I am informing other teachers with the results in order to improve their practice, I think that this position is valid.
My study will be an action research project as I wish to develop my own and others’ teaching practice and as McNiff and Whitehead (2011) say, this will enable me to make a positive impact on the school’s culture as a whole. Thomas (2013) and Lewin, K. (1946) agree that this will promote positive action and change in my school in the future. In many cases, teachers that become confident in teaching will reciprocate this.
Research Methods
In order to answer the question I had set, I used a variety of different research methods in order to lead to a triangulation of data (Thomas, 2013).
The start of the project began with a questionnaire. This was useful as a starting point because it was a fast way to receive data from a large group (Burton and Bartlett, 2009). There is an element of prestige bias involved in answering a questionnaire; however, to rectify this I kept them anonymous. It gave a clear area to investigate.
I decided to use a semi-structured interview with teachers to enable me to find out how they approached the questions and also to use their help in selecting children. These interviews enable me to, ‘capture unexpected issues’ (Barbour 2005). My only issue with this approach was that as my teaching diary notes many of the teachers were aware of my role and therefore responded accordingly.
Later, I chose not to use a focus group because some children would be led by others (Puchta and Potter, 2004 in Barbour, 2008 say that some attitudes are performed: in other words, children try and play up to the crowd). Barbour (2008) also describes ‘noise’ as an issue: she means that there would too many voices at once for the author to gain a full insight. Also, as the children were from a range of abilities some might feel embarrassed solving problems in a group. Therefore, I used a 1:1 semi structured interview approach, so I had the ability to follow what the children either asked or were interested in. Ginsburg (2009) in Dunphy et al (2014) argues that 1:1 interviews give rich information about the profile of the child as a mathematical learner.
Establishing a baseline and current practice
At our school, the children are asked to show what they already know at the start of a topic. As Posner and Gertzog (1982 in Moyer & Milewicz, 2002) state, you can find out, ‘the nature and extent of an individual’s knowledge about a particular domain by identifying the relevant conceptions he or she holds and the perceived relationships among those conceptions.’ Discovering these conceptions is called a ‘cold’ task at my school and it highlights a child’s chosen methodology when solving a word problem. Then at the end of the topic children answer the same question again in order for teachers, and the children themselves, to judge their progress. I used the same approach in this investigation, as this would establish a clear baseline. Aister (2009) in Dunphy et al (2014) have said that this is useful because it gives the adult a basis on which to plan future learning activities, taking into account the specific child’s current knowledge.
Currently, bar modeling has been introduced to the school as a whole (through INSETs and staff meetings) as a method for children to use to understand problems; however, it is not in our calculation policy and many teachers are using the idea for the first time. Our school’s current calculation policy states various changing methods (e.g. a number line and the column method) across the key stages and my study showed that children are quite happy to use these.
Outcomes on learning for children
I investigated the use of Cuisenaire rods to help children develop their understanding of the bar model. I specifically focused on PPG children because of the context of my school. There was an improvement in the number of children that managed to solve the hot task compared to the cold task!
To conclude, the findings of my project reveal that a large portion of the children struggled to see the relationship between a large and a small number when using the rods. This would either stem from a poor understanding of the size of numbers, or perhaps not enough time to see the connections between the concrete experience, pictures, symbols and language that are supported by the work of Haylock and Cockburn (2003). However, the use of actual apparatus to represent the numbers helped, as argued in Nickson, M (2000), enabling the children to see the size differences themselves and therefore create their own mathematical knowledge. This is agreed by Moyer (2001: 176 in Drews, 2007) who says, manipulation of materials ‘allow learners to develop a repertoire of images that can be used in the mental manipulation of abstract concepts’. Also, as Gattegno (1963) states, they allowed the children to see the relationships between numbers, either horizontally or vertically. And finally there is a clear consensus with Piaget with his, ‘learning by doing’ (Piaget in Delaney, K. 2010).
Children need to be given the opportunity to explore manipulatives for themselves; this independence will give children the flexibility to manipulate the resource before using it to solve problems. Children need to be moved away from the concept that this resource is for the numbers 1 to 10. In every classroom there needs to be an atmosphere and culture of creativity, engagement, curiosity and safety so the children, given the opportunity to make mistakes, can thrive.
In addition, I found that even though manipulatives are able to send children on a journey towards mastery (Goutard, 1968), it is important that children do not solely rely on these resources; they need to explain their thinking clearly and be able to imagine what is happening. These children have been exposed to Cuisenaire rods before and therefore were not likely to be scared of them or want to ‘play’ with them as Moyer (2001) in Back, J. (2013) states with frustration.
If a child poses a question they should be given space to explore that idea in their own time. For example, TK (Year 2) asked in the second session, “Is blue the same as the orange rod?”; clearly she had not yet seen that the colours all represented different sizes of rods.
Children need to be exposed to the bar modeling strategy from Year 1 upwards, so they automatically try to visualize problems using bars. These efficient representations will then lead to generalizations (Clark, 2013). What is interesting, and needs to change, is that none of the teachers, in their interviews, chose to use bar modelling automatically to answer the cold task question. They chose the method that they were most comfortable and familiar with. In order to cement this idea into the school’s ethos, the concept of bar modelling will need to be added the school’s calculation policy. Bar modelling needs to be modelled by teachers at every opportunity so that children become confident and familiar with it.
Different manipulatives and representations should be used to explore bar models, as children are more likely to remember a strategy in another context if they have relational understanding (Skemp, 1976). This will also help children to see that different problems share the same mathematical structure and can be visualized in the same way (NCETM, 2013).
Outcomes and development in pedagogy for teachers
There will need to be some redevelopment in teacher’s approach to the movement between the concrete and pictorial stages of development and the strategy of bar modelling at the pictorial stage.
One particularly difficult decision is how long to use these concrete materials for. Goutard (1968) argues that there might be too great an emphasis on keeping children on concrete resources for too long in mathematics teaching. However, I disagree in this particular case, as with my findings I discovered that this period of introduction to the concrete can cause insecurity in the future to a child’s approach to a problem (NCETM, 2014).
On the other hand, I do not think that I have enough evidence to say for certain if this particular method is particularly effective for PPG children. For low achieving PPG children, more repetition with manipulatives would be helpful, but not for all. The moment where they can move on will be made clear by the child’s use of conjecture and in the way the child is convincing others (Mason, 2005 and Goutard, 1968). It is important that they can also do this when the rods are not present.
The use of manipulatives relies heavily on the questioning from the teacher: it is through skillful questioning that the child will make their own discoveries and to enable this the teacher mustn’t have ‘one way traffic’ as described by Delaney (2010). In some cases, like Mason (2005) says, the teacher needs to make informed decisions as to what parts of the model need to be stressed and others ignored.
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