WORKING MATHEMATICALLY

Students use mathematical thinking processes and skills in interpreting and dealing with mathematical and non-mathematical situations. In particular, they:

1. Call on a repertoire of general problem solving techniques, appropriate technology and personal and collaborative management strategies when working mathematically.

Students draw on a range of general strategies when dealing with mathematical problems to which they have no readily available method of solution. These include such things as: act it out; guess, check and improve; look for patterns; draw a picture or make a model; solve a simpler version of the problem first; identify and attempt subtasks; generate and systematically list possibilities; and eliminate possibilities. They make thoughtful use of technology to enhance their mathematical work and have a range of management strategies to help them get started and keep going in individual and collaborative problem solving. They know that working independently on a problem can enable them to get a firm grasp of its features and bring their own unique perspective to its solution. But they also recognise the value of working with others, cooperating to pool ideas and welcoming, and dealing constructively with, conflicting perspectives and views.

2. Choose mathematical ideas and tools to fit the constraints in a practical situation, interpret and make sense of the results within the context and evaluate the appropriateness of the methods used.

Students recognise when mathematics may assist in dealing with a practical problem. They choose mathematical ideas, procedures and technology which suit the physical, social or ethical constraints in a situation, considering the assumptions they need to make in order to use the mathematics involved. Thus, they know that for most people multiplying the distance they can run in a minute by 60 won’t give a good estimate of how far they can run in an hour, but multiplying the water lost from a dripping tap in a minute by 60 will give a good estimate of the water lost in an hour.

They consider the levels of precision and accuracy needed, make appropriate use of the results of using technology and express results in ways suited to the context, perhaps by rounding to a sensible number: for example, asked how many buses are needed to transport 397 students if each bus can hold 54, they say 8 rather than the 7.3518518 which appears on their calculator or the 7 to which the calculator answer rounds.

They also judge the appropriateness of the methods used. Thus, given a problem of six hungry children and only three apples, they respond to a suggestion that the children draw ‘lots’ for the apples by querying whether having an equal chance of getting an apple makes this a ‘fair’ or ‘good’ solution and ask, ‘Is there a better way?’

3.Investigate, generalise and reason about patterns in number, space and data, explaining and justifying conclusions reached.

Students observe regularities and differences and describe them mathematically. By identifying common features in mathematical situations, they make generalisations about numbers, space and data. Thus, they may observe that every time they combine 3 things with 9 things, they get 12 things and make the generalisation that 3 add 9 is always 12. They know that many patterns may be observed in the one situation, and generate and investigate a number of different conjectures about it. Students understand that a mathematical generalisation must be true always rather than mostly and that one exception invalidates it. They attempt to confirm or refute their own and others’ generalisations and prepare arguments to convince themselves and others that a generalisation must hold in every case and not only for all the cases tried. Thus, in investigating quadrilaterals they may note that every four-sided figure which they try tessellates. This leads them to conjecture that all quadrilaterals tessellate and to search for a general argument which will convince both themselves and their peers of this. They write (and speak) mathematics clearly and precisely, expressing and explaining their generalisations verbally and with standard algebraic conventions.