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PATTERNS LEADING TO ALGEBRA

Graham Littler & Dave Benson

Patterns are a fundamental part of mathematics and the ability of pupils/students to recognise them is important to the development of their mathematical understanding. Generally, if we see a pattern in mathematics, we look for the relationship which will describe the pattern. Patterns are found in all aspects of mathematics, arithmetic, algebra, geometry, statistics and games.

By their very nature many patterns are often associated with some spatial phenomena and this enables the pupils to visualise the patterns much more easily. It also means that the pupils must have plenty of ‘hands-on’ experiences making patterns for themselves or undertaking tasks which themselves develop patterns. In early number work children recognise the count of numbers more easily if the objects they are counting are in set patterns such as those on a die or domino.

Later on the pupil will look at sets of data to see if they can see any relationship between two parameters say, and at grades 4 and 5 be able to say in words, at least, what that relationship is without the use of symbols. This will lead naturally into the use of symbols for numbers and the start of ‘generalised arithmetic’, namely algebra. In secondary school it is also important that pupils recognise algebraic functions in terms of their graphical representation, since this gives them understanding.

To develop this work successfully, a certain classroom environment, certain teaching strategies and certain ways in which the pupils learn, are required. The teacher has to set the pupils interesting and challenging tasks so that they can use their existing mathematical knowledge to solve the task but also develop their knowledge and understanding by discovering new knowledge as they look for answers to the questions which the task raises. As they work through a task the pupils may have to find the answer to questions which are raised in their own minds or by the teacher. This means that the pupils have to take greater responsibility for their learning and instead of relying on the teacher to provide them with ‘knowledge’, that is, facts to learn and algorithms to follow, they have to find the answers for themselves. The type of task set must motivate the pupils and in turn the pupils have to show perseverance.

This style of teaching is generally known as using constructive methods. The teacher becomes a ‘facilitator’ (Littler & Koman, 2001) that is someone who helps the pupil’s learning rather appearing to be the ‘fount of all wisdom’. Some teachers find it difficult to adopt this role and to give the pupils the possibility of discovering knowledge for themselves. The tasks which follow in this unit should be capable of being attempted by less able pupils and yet enable the good pupils to reach their full potential, the difference in their solutions being the sophistication of their solutions. Many of the tasks can be extended by either the teacher or the pupils themselves asking a ‘What if…?’ question.

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Charles University Prague, The Faculty of Education
Department of Mathematics and Mathematics Education