Why can children only analyse but not decide? A reflective essay on the teaching and delivering of representation and data-handling in our curriculum.

Introduction

Defined in the English Oxford dictionary as, “A thing that represents another,” representation is the pictorial, graphical, and symbolic recording of ones data. The term representing is used since the emphasis is on children exploring their mathematical thinking through visual means. Worthington and Carruthers (2005). Explain in their work that they find children’s drawing valuable to support their work and that visual representation aids creative thinking, communication and learning. Visual representation by artists is not used just as a record but as representations of their thinking and of embryonic ideas: they fulfil many purposes that are equally relevant to our youngest mathematicians in our classrooms. Janvier, Girardon and Morland (1993) describe representation as stimuli on the senses. An idea, it would seem, that is shared in the writings of Haylock (1982) who suggests using the following model which states that through using and understanding various representations children gain a certain degree of understanding. He states that it is not an all-or-nothing state of understanding but by recording in different manners children can begin to recognise and understand pattern and make the connections needed to learn efficiently.  It is clear from this model that visual representations are important to his ethos and can help to cement mathematics, not just in data handling which is the area I chose for my study.

The problem, I believe is when children misrepresent or misconceive ideas but because they place so much emphasis on the visual these misconceptions can sometimes remain within a child’s subconscious until they learn by a method by rote. What is even more disconcerting is that a teacher can discount simple representations as mere scribbles and less emphasis is placed upon them, Matthews (2003). Considering the varied ways in which children can represent ideas which can  all relate to mathematics such as charts, tables, graphs, diagrams, models, computer graphics, and formal symbol systems it is difficult to see how simple elements can be overlooked . Pimm (1995) places great emphasis on the graphical representations that children make. He speaks of the iconic images that graphs can provide and that graphs are the drawn result of an action, a process which represents children’s understanding of the task at hand.

It is rather surprising then that Harris and Sutherland claim in their report that there has been very little research which has focused on the ways in which mathematics is presented in text books. This may be because many mathematics educators support what Voigt (1998) calls the misguided belief that the questions and the symbols of mathematics lessons have definite, clear-cut meanings. Or it may be because of a view that text books should not be used by “good” mathematics teachers.

It is with this in mind that I planned my unit focussing on the different representations a child can make whilst collecting data and using an open-ended statement to provoke independent and creative thought. I planned this solely with one statement in mind with no text-books and purely children’s own curiosity.

Planning and coaching

Carruthers and Worthington (2006) suggest that there are benefits to children visual representations as they can present a high level cognitive challenge when they are novel, creative, imaginative and are a combination of several elements.

Ensuring the coaching and teaching was valuable and extended children’s learning to include the afore mentioned cognitive challenge was important to me. As you can see from the planning found in appendix A you can clearly see the courses influence through the inclusion of Mason’s learners powers. Mason (2004) argues that if children use these powers it can help them to make sense of mathematics and the world in general and as a result children will develop mathematical independence. The planning and subsequent team teaching illustrates the need for this independence and decision making skills that ensure children understand what and why they are doing something and to make connections between other areas of maths. Delaney (2010) claims that effective teachers can help children make connections between different representations of mathematics and understand that these move between symbols, words, diagrams and objects. Delaney also states that children should be able to share their methods and value should be placed on the process and not just the end product.

I have become despondent at the teaching of data-handling where charts are provided and children simply answer questions about the data. By thinking about the learners powers and the thoughts of practitioners/researchers such as Delany it becomes clear that children need to decide the direction of the learning so that they understand why a chart or graph is chosen for a select piece of data. A view shared by Gagatsis & Elia (2004) who both state when children have no direct responsibility of the consequences of their choices and actions  they will have difficulty transferring information gained in one context to another.

It is whilst planning that I discovered my initial plan for a lesson on representation and data-handling became more of a rich task where the answer was unknown and many variables could be changed so that the manipulated data could be represented in many different ways. According to the Nrich website a rich task should step into the problem even when the solution is unclear. It should be accessible to all and children should dictate the direction of the task. Pete Griffin (2009) supports this argument and suggests that a rich task should classify mathematical objects and interpret in a range of representations. I hope that through my planning  I have interpreted this brief taking into account the opposing but justified view from Henningsen’s and Stein’s (1997) report which noted that these tasks can often be complex and are longer in duration that most mathematical, classroom activities thus leaving children more susceptible to various factors which lead to their decline and potential disengagement.

What I felt I struggled with was effectively coaching another teacher so that the ideas of the course are explained and my ideas are reflected through the teaching of the lessons. According to the Mentoring and Coaching CPD from DfES, effective mentoring is when structured dialogue takes place which articulates the existing beliefs and practises to enable accurate and effective reflection on them. I believe that my personal difficulty came when I explained that the statement for the lesson (boys have bigger hands than girls), in the planning, does not have to have a definite answer but rather the journey is more important and the way in which they choose to answer the statement. Even the DfES document stresses the need to experiment and observe by creating a learning environment that supports risk-taking.

The short period of time also meant that I struggled with gaining enough evidence base and joint teaching as I would have benefited from planning lessons on other types of chart or diagrams to compare how children developed orally and mentally with independent work.  Weston (2012), who has reported on the effectiveness of peer coaching, writes that the development process must be actively sustained for at least two terms for a large number of hours (i.e. more than 40). And it must follow cycles of trying, reflecting, and adjusting, while maintaining the focus on improved student learning – and not teacher behaviour.

Reflection

Most of my personal reflections have been taken from the jointly taught lesson from day 4 of the plan. Both the teacher (KS) and I discussed what would like as a conclusion for the lesson and then filled in the lesson reflection afterwards. Most of my further comments have been taken from these sheets found in appendix C

With reference to one of the first points made, “the children found it difficult to choose the correct chart”, see appendix C for related comment. Is this because they had very little previous experience in correctly handling data? If so, this is what I was hoping to tackle, ensuring children rely on their own independence to choose the correct manner in which the tackle the problem before them. Boaler (2009) argues that real mathematics is misrepresented in our schools and that children site maths as a list of rules and procedures which have to be remembered, which linked with the idea of rote learning but directs children away from the potential opportunity to experience real mathematics. Boaler later describes maths as the study of pattern rather than calculations and in my opinion and that of Rowland (2009), not having a known outcome gave leave for children to experience this pattern themselves and ask questions about further research, such as “What would happen if we studied adult hands?”  Rowland suggests that maths is far more than a teacher standing at the front of the classroom and telling the children what they must learn but on this occasion, a slight direction was needed to ensure all children chose the most appropriate chart even though they had worked with all of the charts, graphs and diagrams beforehand. Vygotsky draws on the notion of scaffolding to help children and according to Henningsen’s and Stein’s (1997) report, scaffolding can help children without removing or reducing the cognitive demands. However, the children did discuss how they could represent the data in a bar chart which further solidifies that point the children have the power to discuss, model and interpret their own data, given the chance to do so. AS, a boy from the class remarked that he enjoyed the activity because he had been given the chance to make his own decisions and should he be allowed to do a similar task again he would have liked to opportunity to lead it and direct its outcome, please see appendix D for relevant remarks.

The teacher remarked on the fact that there was no clear outcome and if she was to do this activity again she would have prepared the data beforehand. This is something I struggled with myself. I wanted the children to lead the way, to think for themselves which Mason (2004), agrees with, stating that children should envisage maths, change situations and make structural improvements to their work, enjoying their own joy at discovery but Boaler (2009), to some extent disagrees with this claiming that teachers must have the sufficient knowledge for their own planning to make connections beforehand and thus with their pupils. As I found no clear direction I followed the route of child discovery and although there was no definitive outcome I felt the children had still learnt many skills such as conjecturing and convincing when deciding on the most suitable chart/diagram, representing the chart/diagram in their chosen form and analysing the data (an area we did not cover during the planned/coached lesson). Again, closer inspection of the children’s evaluations in appendix D reference the other areas of learning that children were practising, such as measuring and making estimations.

The class progressed with the amended plan independently and referencing the earlier point of Weston’s (2012) that mentoring should be over a prolonged period of time, I did feel that a couple of extra sessions working with the class and the teacher may have been beneficial, rather than just the one. If I was to complete a similar task again this is something I may change.

However, the next lesson, which was led by the coached teacher, consisted of introducing and analysing data which I hoped righted the deficit of the previous lesson, See appendix E for screen captures of the lesson slides. Pictures of various groups of people were before the children and they had to compare and analyse the graphs. I later attempted the same lesson with my class of children and most could identify the connection and differences between each group. Carruthers and Worthington (2006) write that producing a range of graphs that the children can read can be purposeful, graphs and tables can provide models for discussion.

In conclusion, I would say I agree with Mason (2004) that a successful lesson incorporates all of the powers and that due to the nature of children’s own enquiring and active minds that gentle direction and independence are the way forward. Whitehead (1932) has this to say from Mason’s 2004 work, he states that children enjoy being thrown into a world of discovery and will happily search for every combination to solve a problem, every child possesses a driving force to extend their own learning.