Mathematical Reasoning

Research by Nunes et al (2009) identified mathematical reasoning as the most important factor in a pupil's success in mathematics. Yet Ofsted's findings (Jones, 2015) show it is the least well-developed of the three national curriculum aims. Explore the reasons for this and how it could be addressed.

The 3 main aims of the national curriculum are: reasoning, fluency and application- this is the new mastery strategy which was implemented in 2012. It was influenced by high performing countries in east and southeast Asia such as: China, Singapore, Japan and North Korea (NCETM, 2014). The mastery approach places an emphasis on the depth of understanding as opposed to a breadth of knowledge (Morgan, 2015), this will give pupils the opportunity to fully understand a concept and begin reasoning mathematically before moving on. Furthermore, research suggests that this approach is having a positive effect on pupils using mastery- for example, data from the research program in 2012 (by the Programme for International Student Assessment [PISA]) suggests that by age 15 students from the high performing countries are on average up to three years ahead in maths compared to 15-year olds in England (2014, p4).

In this paper, I aim to explore one of the three key aims- reasoning. The national curriculum defines reasoning as: 'following a line of enquiry, conjecturing relationships and generalisations and developing an argument, justification or proof using mathematical language' (National Curriculum, 2013, p88). Nunes et al have advocated for reasoning being the most important factor for a pupil’s success in mathematics (p3). It is also regarded by academics as the aspect in mathematics which is at the 'core of the subject' (Haylock, 2014, p37-38). This is further highlighted by NRICH (2014), who suggest it is reasoning that helps children to use their mathematical skills, inviting them to gain a deeper conceptual understanding- making connections and draw inferences (Witt, 2014, p3). In short, reasoning is the 'glue' that helps to make mathematics make sense (NRICH, 2014).

However, teachers often struggle with teaching reasoning and how to use it to prove mathematical ideas and vice versa with students struggling to learn it (TESS India, no date, p3). This coincides with Ofsted's findings, that reasoning is the least 'well developed' of the three-national curriculum aims (Jones, 2015). It could be suggested that mathematical reasoning may be an element that cannot be taught with rules, however, it can be fostered, prompted and stimulated within pupils through various strategies. This, however, would be down to a teachers practice within the classroom. Therefore, within this paper, I would like to investigate the factors which may cause a lack of mathematical reasoning- and suggest how they may be overcome.

One factor (which is highlighted frequently by Ofsted and academics), is teaching children rules to tackle calculations without letting pupils explore or discuss- to play an active process in learning.

Atkinson (1992, p12) highlights that mathematical reasoning is 'based on understanding', therefore constantly verbalising the information, is suggested to give children a basic understanding- also known as an instrumental understanding. An instrumental understanding is when an individual learns the rules and applies them to particular circumstances (Fisher, 2005, p171). However, the issue with this is that rules are easily forgotten- thus it is seen as a shallow level of understanding which can place pressure on a learners memory. This is exampled by Cockcroft (1982), who investigated the mathematical competencies of  a group of adults. Finding that, adults (who left school with what was considered to be respectable test scores) 'appeared to have only one method of tackling a given problem' (p8). It can be inferred, from this research, that the subjects had an instrumental understanding of mathematics- as the adults were trying to replicate the 'proper method' shown to them to achieve the correct answer. However, had the adults gained a relational understanding (this is where the individual knows the reason behind the rules, therefore, they can think through and reconstruct the rules for themselves), they may have continued to try other methods- as it was reported subjects 'lacked the ability and confidence to attempt a different approach' (p8).

Therefore, it is important that we provide children with various opportunities to develop their reasoning skills- this could be done by planning an enquiry. As highlighted by Atkinson (1992, p13) 'maths with reason is rooted in action- learning through doing'. Children would need to decide how they would first tackle the problem and would then have to draw upon a range of mathematical skills to work at the enquiry (Cotton, 2013, p29). Academics and Ofsted (2012) frequently stress the importance of children playing an active rather than passive role in the classroom. I feel that enquiries- such as the ones supplied on NRICH would achieve this. A Strategy, which I would like to try in school, is placing an enquiry on a working wall, on it, children can place their work up. I believe this would help children to develop their reasoning skills as well as their problem solving as it allows pupils to view model examples of work. They can then refer to them when improving their own work, helping them to become succinct. The class is working as a problem solving team. This notion is further supported by the National Numeracy Strategy, a framework developed to raise standards in schools, advocating high quality learning is a 'two-way process in which pupils are expected to play an active part by answering questions, contributing points to discussion and explaining and demonstrating their methods to the class' (found in Myhill and Hopper, 2005, p50).

The next factor has been suggested by an article from the National Centre for Excellence in the Teaching of Mathematics (2013). They highlighted that too often teachers give children a statement and we do not give them a chance to develop their mathematical reasoning to prove it. 'Proof is a peculiarly mathematical way of reasoning' (Haylock, 2014, p43). NRICH (2015) identify that children progress in their reasoning when they go through the five stages- this starts at describing and ends at proving.

A strategy that is commonly suggested (Turner and McCullouch, 2005, Chapter 3 p9) is to allow children to choose their own resources to figure out the problem- 'manipulatives can be powerful tools to support sense making, mathematical thinking and reasoning' (NRICH, 2014). Some children may prefer to see the problem visualized others may prefer to experiment with the problem using written methods- as long as the learners are trying to make sense of the mathematical concepts. As Delaney (2001, cited in Thompson, 2010, p73) states 'there is no mathematics actually in a resource' but rather 'the mathematics is brought to the resource by those who interact with it or is developed by them as they use it to support or challenge thinking'. Children need to represent their ideas and findings, therefore using manipulatives or written down formulas relating to the problem is effective to guide the children through the process and prove their understanding (Herringer, 2013). Although, according to the research by Schoenfeld (1987), even though children may be assisted in their learning of concepts by the use of manipulative apparatus, they are often unable to apply those same concepts to problem solving situations. Therefore, it is important that teachers consistently model to use of resources to children.

My final factor, is the use of questioning. As Stripp (2015) highlights, when Chinese teachers ask for answers, they always ask follow up questions such as how? and why? Requiring children to use both inductive reasoning- making conjectures and solving the problem through convincing- and deductive reasoning- justifying and proving their conjecture (Haylock, 2014, p42-43). This can be done with both peers and teachers, challenging pupils to comprehend the maths taking place, encouraging them to discuss their conjectures and underlying thoughts, helping pupils to clarify their ideas and make mathematical connections so that they can articulate their reasoning (Ge Feng et al, 2007, p382). This was apparent in schools that took part in the China-England teacher exchange program. One child is quoted saying 'I like the Chinese lessons, I need to think very hard because I know Miss Lu will ask me to explain why and I will need to have a good answer!' (Stripp, 2015). It could be suggested that the child understands the engagement and depth required in Miss Lu's class as opposed to the other which may not require as much reasoning from the pupil.

To reflect, there are a variety of ways teachers can promote reasoning within the classroom. Fortunately, whilst exploring these factors, it became apparent that whilst I have been in schools I have observed practitioners stimulate and regularly encourage reasoning from pupils. This paper has also inspired various ways in which I hope to encourage the use of reasoning within the classroom- for example the use of a working wall. Additionally another factor, which I did not discuss but that I am aware of, is the use of mixed ability groups. In Nunes et al (2009) research, they discovered that streaming groups only slightly helped the higher ability but hindered the progress of others. Therefore, I will ensure that whilst doing investigation and enquiry work the children will be mixed ability. Overall, I believe that reasoning would be the hardest aim to teach, however, as also highlighted by Nunes at al (2009), it will help children to progress in their future mathematical development.

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