While there is almost a consensus that the mathematics problems appropriate for the 21st century have to be complex, unfamiliar and non routine (CUN), most of the textbooks still include only routine problems based on the application of ready-made algorithms. While students solving routine problems can rely on memorisation, solving CUN problems requires mathematical skills that include not just logic and deduction but also intuition, number sense and inference.
What is CUN (Complex, unfamiliar and non-routine problem solving)?
According to the OECD Programme for International Student Assessment (PISA), the assessment of mathematics skills for the 21st century should focus on the capacity of students to analyse, reason and communicate effectively as they pose, solve and interpret mathematical problems in a variety of situations involving quantitative, spatial, probabilistic or other mathematical concepts.
The term “problem solving” has two components, the type of problem and the knowledge and skills needed to solve the problem.
The traditional type of problems includes arithmetic computations, geometry and routine word problems. Skills needed to solve these are limited and teaching these skills consists of demonstrating the technique followed by giving some practice problems. Many students admit that memorisation is the most important skill they need to succeed in mathematics.
CUN problems differ from traditional ones in content, context and in the process needed to solve the problem. According to PISA, the context brings up the mathematical big ideas, the context relates to real life situations ranging from personal to public and scientific situations and the constructs are more complex than the traditional ones. These may include mathematical information that is not presented in an explicit form and may have multiple correct answers (depends on the basic assumptions that the solver adopts). Often students are asked to solve the given problem in different ways, to suggest creative solution processes and to reflect on and criticise their own solutions and that of others. This is not to say that routine problems are to be excluded from curriculum. They are necessary for practicing and attaining mastery and being able to respond automatically
The below examples are on the same context (buying and selling) but task 1 has multiple answers based on assumptions, task 2 (pizza task) is a more open one with some information embedded in the problem and task 3 is a routine problem.
What are the skills required to solve CUN problems?
I) Mathematical reasoning
II) Mathematical creativity, divergent thinking and posing problems
III) Mathematical communication
I) Mathematical reasoning includes proofs, logic, cause and effect, deductive thinking, inductive thinking and formal inference. NCTM, PISA, New Jersey curriculum framework etc. describes mathematical reasoning as a critical skill that enables students to make use of all other skills. With the development of mathematical reasoning, students recognise that mathematics makes sense and can be understood. They must be able to judge the accuracy of their answers. They must be able to apply reasoning in other subject areas and in daily lives. NJMCF standards summarises mathematical reasoning as “The glue that binds together all the mathematical skills.”
II) Mathematical creativity, divergent thinking and posing problems:
Torrance defined creativity as “a process of becoming sensitive to problems, identifying the difficulty, making guesses and formulating hypotheses, testing and retesting them and finally communicating the results.” Torrance identified four main components of creativity – fluency, flexibility, originality and elaboration.
Mathematical creativity involves finding or posing problems, proposing large number of problems, searching for alternative solutions, identifying innovative problems etc..
Creative thinking is a cognitive activity that results in finding solutions to a novel problem. CUN problems help students to develop critical thinking, to deal with uncertainty and decision making.
III) Mathematical communication: When students are confronted with CUN tasks, sharing ideas, discussing solutions and explaining one’s own thinking is unavoidable. While communicating, students have to be clear, convincing and precise. In a series of studies, Webb showed that although during mutual reasoning, all participants benefit from the discourse, the ones who delivers the explanations benefits more than the one who listens to it.
Mathematical communication helps in discovering common errors or misconceptions. Sometimes students make two mistakes that cancel out each other and there by attain the correct answer. Students often develop their own rules that often lead to misconceptions, eg: “multiplication always makes things bigger.” Communication by students may help teachers to identify such misconceptions.
Happy Problem Solving!!!
By Praveena K – Educational Specialist
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