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Three possible views of mathematics can be presented. The instrumentalist view of mathematics assumes the stance that mathematics is an accumulation of facts, rules and skills that need to be used as a means to an end, without there necessarily being any relation between these components. The Platonist view of mathematics sees the subject as a static but unified body of certain knowledge, in which mathematics is discovered rather than created. The problem solving view of mathematics is a dynamic, continually expanding and evolving field of human creation and invention that is in itself a cultural product. Thus mathematics is viewed as a process of enquiry, not a finished product. The results remain constantly open to revision. It is suggested that a hierarchical order exists within these three views, placing the instrumentalist view at the lowest level and the problem solving view at the highest.
Mathematics is the study of quantity, structure, space and change. Mathematicians seek out patterns, formulate new conjectures, and establish axiomatic systems by rigorous deduction from appropriately chosen axioms and definitions. Mathematics is a distinctly human activity practised by all cultures, for thousands of years. Mathematical problem solving enables us to understand the world (physical, social and economic) around us, and, most of all, to teach us to think creatively.
This corresponds well to the problem solving view of mathematics and may challenge some of our instrumentalist or Platonistic views of mathematics as a static body of knowledge of accumulated facts, rules and skills to be learnt and applied. The NCS is trying to discourage such an approach and encourage mathematics educators to dynamically and creatively involve their learners as mathematicians engaged in a process of study, understanding, reasoning, problem solving and communicating mathematically.
Below is a check list that can guide you in actively designing your lessons in an attempt to embrace the definition of mathematics from the NCS and move towards a problem solving conception of the subject. Adopting such an approach to the teaching and learning of mathematics will in turn contribute to the intended curriculum being properly implemented and attained through the quality of learners coming out of the education system.
| Practice | Example |
| Learners engage in solving contextual problems related to their lives that require them to interpret a problem and then find a suitable mathematical solution. | Learners are asked to work out which bus service is the cheapest given the fares they charge and the distance they want to travel. |
| Learners engage in solving problems of a purely mathematical nature, which require higher order thinking and application of knowledge (non-routine problems). | Learners are required to draw a graph; they have not yet been given a specific technique on how to draw (for example a parabola), but have learnt to use the table method to draw straight-line graphs. |
| Learners are given opportunities to negotiate meaning. | Learners discuss their understanding of concepts and strategies for solving problems with each other and the educator. |
| Learners are shown and required to represent situations in various but equivalent ways (mathematical modelling). | Learners represent data using a graph, a table and a formula to represent the same data. |
| Learners individually do mathematical investigations in class, guided by the educator where necessary. | Each learner is given a paper containing the mathematical problem (for instance to find the number of prime numbers less than 50) that needs to be investigated and the solution needs to be written up. Learners work independently. |
| Learners work together as a group/team to investigate or solve a mathematical problem. | A group is given the task of working together to solve a problem that requires them investigating patterns and working through data to make conjectures and find a formula for the pattern. |
| Learners do drill and practice exercises to consolidate the learning of concepts and to master various skills. | Completing an exercise requiring routine procedures. |
| Learners are given opportunities to see the interrelatedness of the mathematics and to see how the different outcomes are related and connected. | While learners work through geometry problems, they are encouraged to make use of algebra. |
| Learners are required to pose problems for their educator and peer learners. | Learners are asked to make up an algebraic word problem (for which they also know the solution) for the person sitting next to them to solve. |
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