Problem solving exposes children to opportunities to apply, reinforce and develop their knowledge and skills of concepts and principles which can be applied to solve problems. As defined by Thompson (2003), problem solving is a central component of mathematical reasoning which allows children to see the relevance mathematics has to their everyday lives. In order to understand mathematics one must not only develop knowledge on the terms, concepts and principles used but must also develop an understanding of the different ways to reason and justify their ideas (Haylock 2014). Bernard (2014) defines reasoning as the critical skill that enables pupils to deepen their critical and analytical thinking and make sense of mathematics. The element of investigation is also crucial for mathematical learning as it allows children to explore different ways of solving a problem and promotes reflective thinking. As defined by Erlina (2010: 1) mathematical investigation ‘refers to the sustained exploration of a mathematical situation. It distinguishes itself from problem solving because it is open-ended.’ This rationale critically analyzes the potential problem solving, reasoning and investigation can have for mathematical learning and highlights effective approaches for teaching.
NATIONAL CURRICULUM- PROBLEM SOLVING ENVIRONMENT
Problem solving is the skill of identifying, understanding and solving problems by applying the correct skills systematically (Pennant 2014). It is an ongoing central part of the curriculum which impacts teaching and learning. The National Curriculum (2014: 3) highlights three main aims which teacher need to ensure their pupils achieve. These relate to the fluency in the fundamentals of mathematics, the ability to reason mathematically by ‘following a line of enquiry, conjecturing relationships and generalisations, and developing an argument, justification or proof using mathematical language’ and the ability to solve problems. All three aims have a significant effect on the teaching and learning of problem solving. As McClure (2013) explains, a strong problem solving approach within the classroom will impact the quality of learning that takes place. It will ensure pupils develop their mathematical thinking and build their confidence to persevere when confronted with a problem. Therefore implementing a teaching and learning environment which offers high quality problem solving connections is essential for the development of critical thinking. As Pennant (2013) states, a strong problem solving environment will have a positive impact on teaching and learning as it will encourage pupils to grow as independent mathematicians.
THEORY POLYA & BRUNER (BENEFITS & DRAWBACKS)
Developing an explicit understanding of how to problem solve can have vast benefits to a child’s everyday life as it will enable them to tackle any type of problem they encounter (Witt 2014). Polya’s (1945) problem solving approach of learning provides an explanation of what problem solving should entail. For Polya, problem solving should be related to real problems which allow children to explore and develop their mathematical reasoning. He refers to four stages which should be used when tackling a problem which includes the following: understanding the problem, devising a plan, carrying out this plan and looking back to ensure the answer has been obtained. In a previous school experience, children implemented Polya’s theory when solving problems. They were encouraged to independently identify the problem related to money, choose a strategy to solve it and used it. This approach allowed children to interpret the question, make their own choices as to how they were going to work it out and enabled them to apply their knowledge to lead towards the solution. Therefore, Polya’s explanation of problem solving can be of major importance to the teacher as it can be very effective to develop children’s higher order thinking skills, fluency when selecting the appropriate mathematical strategies and applying it, and deepen their understanding of how mathematics works. In contrast, Bruner (1957) believes that Polya’s approach is not accessible to novice mathematicians. Bruner highlights how such a complex process can have a negative effect on novices’ confidence and motivation to tackle more challenging problems. Bruner presents the idea of scaffolding learning and providing an organised, gradual approach which will ensure progression.
Adding on, problem solving within a classroom environment must be purposeful and relational in order to be beneficial for pupils. Teachers can enable this to happen by providing opportunities for children to learn through investigative and problem solving approaches. As Pennant (2013) highlights, creating an environment where investigative thinking is valued, mistakes are seen as useful and all suggestions are valued, is fundamental to develop confident problem solvers. Providing children with a range of structured, well-planned problem solving, investigation and reasoning activities, which are presented in context, will have a positive effect on a child’s ability to reason logically and deductively and will greatly enhance their mathematical critical thinking. In a previous school experience, an activity called magic squares was used to improve children’s ability to reason mathematically. It consisted of making each row, column and diagonal add up to the magic total. This activity was differentiated to ensure it was accessible to all children; as a result, it developed children’s understanding of the many possible ways to work out the problem and enabled them to reflect on the process of finding the solution. Burkharat (1981) proposed a range of taxonomy problems which children should be exposed to children to help better their understanding of the process of mathematising in the real world. Some of these included action problems, believable problems, and educational problems. As Barmby et al (2014) highlight, exposing children to a range of different mathematical problems and making the problem accessible to them is crucial to enrich a child’s conceptual understanding. Giving them the opportunity to tackle problems which require them to make connections across mathematical ideas and use a greater range of prior skills will also help children to develop problem solving strategies in mathematics which can be applied in other subject areas (National Curriculum 2014). However, there can be disadvantages associated with the support children receive and the engagement of pupils. As Briggs (2014) points out, the ‘skills of persistence trial and error and risk-taking’ appears to lack when children are faced with problem solving activities. Therefore, less interest and confidence is demonstrated when having to undertake such activities.
EFFECTIVE TEACHING APPROACHES (COLLABORATION AND DISCUSSION)
Effective teaching approaches are essential to encourage meaningful mathematical learning. Pratts (2006) highlights two key teaching approaches which are fundamental to enable children to learn and flourish. These include collaboration and discussion, and questioning. Collaboration and discussion is one of the best ways to get pupils interest and engaged with problem solving. Watkins (2005) defines collaborative learning as a teaching approach which involves a joint effort by learners to achieve an end result. Providing plenty of opportunities where children feel confident to share their ideas and understanding of problem solving through promoting group work is vital to increase motivation for learning. Research carried out by Topping and Ehly (1998) associates increased collaboration within the classroom environment with increased motivation to tackle problem solving activities. Therefore, the application of collaboration and discussion within learning is imperative to develop children’s mathematical knowledge. As Rogers (2004) states, collaborative learning is critical to developing children’s high-level thinking skills, build their self-esteem and encourage diverse understanding.
QUESTIONING
Questioning is another teaching approach which, when used correctly, can be extremely effective to develop a child’s mathematical reasoning skills. Morgan (), director of NCETM, highlights the fact that the use of formative questions can be vital to ‘uncover a pupil’s reasoning behind the answer’. Morgan emphasises the importance of focusing on the wrong answers to enable children to explore concepts in greater depth. In contrast, the use of questioning can also have a negative effect. Gershon (2013) points out the consequence closed-ended questions can have on a child’s mathematical development. Gershon highlights the fact that these types of questions do not give children the chance to think about their own responses but offer possible answers from which to choose. Questions can also be misinterpreted by children and therefore can affect their confidence when tackling problems.
Conclusion
In conclusion, problem solving skills have a vital role in allowing children to cope with everyday life. The effective implementation of problem solving within a classroom environment is critical to developing a child’s mathematical reasoning skills. Therefore, choosing appropriate teaching strategies which enable purposeful problem solving to be a central part of learning is fundamental for children to develop valuable and transferable problem solving skills. As highlighted throughout this essay, problem solving is a crucial skill children develop when they are provided with opportunities to explore concepts and principles, experiment and apply knowledge, and reflect on mathematical thinking.