Problem Solving in Mathematics Education: Approaches and Benefits
Problem Solving as Context (PSC)
For students to understand and make meaning out of the mathematics they learn in school, it should connect to the real world. Problem solving (PS) is the pedagogy that justifies teaching mathematics for real-life experience. To motivate students to realize the value of mathematics, the content connects to the real world, allowing them to gain PS experiences. PS motivates students, aiding their interest in specific areas and mathematical topics or algorithms by making use of contextual learning. When used effectively in the classroom for teaching and learning, the PS technique serves as a form of fun, often used as a reward or break from the routine way of teaching and learning.
Problem solving encourages students to practice what they learn. Perhaps the most widespread use of PS is to reinforce skills and ideas that teachers have taught directly. Teachers will recognize that when students work collaboratively to help one another and take on expert roles, their learning strengthens and refines. Teachers will consider tactics they can use to build learning communities.
With PS, teachers use mathematics as context. The stress is on finding interesting and engaging tasks or problems that help clarify a mathematical concept or procedure. To use PS as context, a teacher might teach the topic of Profit and Loss, for example, by assigning roles to students to play in the classroom. It would help students to come up with solutions to problems involving Profit and Loss, Simple Interest, and solving equations in two variables.
By providing PS context, the teacher’s goals are many. First, there is the need to create opportunities for students to make discoveries. Students could discover solutions to real-life problems involving three sets by using Venn diagrams. Second, the teacher’s aim is to help make the concepts more concrete. In groups, students could draw parallel lines and a transversal and measure all the angles to discover the relationships between corresponding angles, vertically opposite angles, alternate angles, adjacent angles, and supplementary angles. Finally, the teacher aims to offer students a rationale for learning mathematics. Students could develop a rationale for learning parabolas by using Information Communication Technology (ICT) to investigate the nature of graphs of various forms of parabolas by changing the directrix and focus.
Problem Solving as a Skill (PSS)
PS as a skill views a way to teach PS skills as a separate topic in the curriculum, rather than throughout as a means for developing understanding and basic skills. In this case, the students receive a set of general procedures for solving problems, such as drawing shapes, working backward, or making a list, and are given practice in using these procedures to solve routine problems. When teachers view PS as a collection of skills, however, they often place the skills in a hierarchy in which they expect students to first master the ability to solve routine problems before trying non-routine problems. Consequently, non-routine PS has an impact only on advanced students rather than on all students.
When defining the learning objectives of a problem-solving activity, teachers will want to be aware of the distinction between teaching PS as a separate skill and infusing PS throughout the curriculum to develop conceptual understanding as well as basic skills. For Van de Walle (2001), PS should be seen as a main teaching strategy, and he points to the importance of beginning the work from the point where students are, contrary to other ways of teaching that begin from the point where the teachers are, ignoring what the students bring with them to the classroom. He goes on to state that teaching with problems has great value, and that despite the difficulties, there are good reasons to engage in the effort.
Problem Solving as Art (PSA)
PS could take the form of an art. The art of PS is something that we learn at a very young age. It helps us through life and is something that could become a part of us. Being able to solve problems is a life skill. It is important, and we should seriously impart it to get the best results from it. Looking at PS as an art can help you to become more appreciative of it. You can begin to use PS to its full potential and really respect that PS is important. You just need to learn more about PS as a skill and an art.
In his famous book, How to Solve It, George Polya (1945) introduced the idea that PS could be taught as a practical art, like playing the piano or swimming. Polya saw PS as an act of discovery and introduced the term “modern heuristics” (the art of inquiry and discovery) to describe the abilities needed to successfully investigate new problems. He encouraged presenting mathematics not as a finished set of facts and rules, but as an experimental and inductive science. The aim of teaching PS as art is to develop students’ abilities to become skillful and enthusiastic problem solvers; to be independent thinkers who are capable of dealing with open-ended, ill-defined problems.
Constructivism in Mathematics Teaching and Learning
Confrey & Kazak (2006), reflecting over the role of constructivism in the history of mathematics education, reflected on what has been accomplished, honored the contributions of scholars around the world, and identified what remains unfinished or unexplained. In doing so, they highlighted the following:
- PS as the Root of Constructivism
- Constructivism Provides Better Understanding of Mathematics
PS as the Root of Constructivism
Confrey & Kazak (2006) located PS as one of the three roots of constructivism. Most of the traditions of mathematics education suggest that there is a need for something more than the logic of mathematics to explain, predict, and facilitate mathematics learning. Many mathematics educators have recognized that the difficulty or ease of learning could not be determined by looking at the complexity of the material under consideration. Rather, other factors, such as appropriate pedagogy, good questioning skills, motivation, perceptions, and the right attitude, would help to assist students to go through the material systematically. This would help to influence learning to improve the level of success and reduce failure. Instructional strategies like PS could have a positive impact on the SHS curriculum because topics that are considered difficult at some levels may be simple and easy to learn when imparted through PS.
Behaviorism presents teaching and learning as a set of stimulus-response connections. This is the simplest account from an external viewer’s perspective. However, a more complex psychological theory would help to capture not only the behaviors but also the experience of learning. The first tradition of constructivism is PS, which is the focus of this study. Polya (1957) had four stages (understanding, devising a plan, carrying out the plan, and looking back) and emphasized that mathematics was more than a set of formal definitions, theorems, and proofs, and acknowledged the central role of problems in generating new solutions. In the PS technique, there is always a rational sense that the problems existed independently of the solver, whereby the solver is not familiar with the problem. By learning a successful set of techniques, solutions could be more easily and effectively achieved. As a result, PS forms an acceptable extension to mathematics, not challenging in any fundamental way the effective nature of the PS technique, but only extending and enhancing it. Debates focused typically only on how much time should be devoted to it.
Constructivism Provides Better Understanding of Mathematics
From our discussions on constructivism as a root of PS, we realized that constructivism was forged, mainly, out of PS. Confrey & Kazak (2006) confirm that constructivism in relation to mathematics education has been in place over the last thirty years. The questions to ask then are: What did the theory permit us to do for all these years, and where were its resistances, in the sense of the idea of viability or fit, rather than match, over the thirty-year history? It could clearly be realized that constructivism has promoted a better understanding of mathematics, of teachers, of the processes by which students learn, construct, or discover mathematics, and has helped teachers, examination bodies, and curriculum development in mathematics education to make better decisions.
ake better decisions.