Mathematical problems require decisions to be made about the mathematics needed and the strategies to be used. Problems come in all shapes and sizes from single-stage closed textbook questions to open-ended investigations. The context can be purely mathematical or can be taken from real life, including contexts from other subjects. All have their challenges but if learners only experience textbook-style questions that are based on one topic and a method that has recently been taught, they will struggle to tackle problems that require the process skills of functional mathematics.
To be able to tackle problems that are more open-ended, learners need to ask questions about the context of the problem, for example: What is this telling me? Would it make any difference if…? They should be able to sort and organize information, including deciding what is relevant and what is redundant by seeing the problem and solution as a whole rather than as a lot of small pieces. This will encourage learners to look for and spot patterns and relationships, and generalize from them where appropriate. Learners have to realize that there is not necessarily one correct way of tackling a problem and not necessarily only one correct answer to a problem. Choices have to be made but they need to be justified.
As explained in the Introductory article, helping learners to become functional in mathematics means helping them to:
• recognize situations in which mathematics can be used
• make sense of these situations
• describe the situations using mathematics
• analyse the mathematics, obtaining results and solutions
• interpret the mathematical outcomes in terms of the situation
• communicate results and conclusions.
Learners will need to experience lessons with an emphasis on activities that have sufficient scope to permit all these processes to flourish. A problem solving approach permits learners to develop all the process skills in Functional skills standards: mathematics, because these process skills are, essentially, problem solving skills. Preparing learners in functional mathematics means helping them to develop problem solving process skills.
For mathematics, this is likely to involve considerable change in the curriculum for many learners. This is because many teachers concentrate on ensuring that their learners have been introduced to all the curriculum content (thus addressing the skill that the standards call ‘analysing’) without the ‘representing’ and ‘interpreting’ skills, and many examinations reward such an approach by giving little credit for representing and interpreting.
However, in the world outside mathematics lessons, we rarely know at the time a problem is posed whether or how mathematics will help solve it and, if so, what mathematics is needed. To solve problems in the wider curriculum, in life and in the workplace it is often necessary to go through the processes outlined in the standards. Being functional in mathematics requires learners to demonstrate that they can represent situations using mathematics and interpret mathematical results in terms of the original situation. These process skills are best developed by learning to deal with substantial problems.
‘Representing’ is about being able to describe a situation mathematically. Some problems are represented by very commonplace methods, such as addition. This kind of representation can become so natural that we no longer notice ourselves deciding to use it. There are situations, however, that require more complicated representations, such as the following problem about chocolate eggs.
Mrs Newman has five children.
Three of them are girls. Two of them are boys.
The children buy chocolate eggs to give to each other.
Each girl gives each boy a red egg.
Each boy gives each girl a yellow egg.
Each girl gives each of the other girls a blue egg.
Each boy gives each of the other boys a green egg.
1. How many eggs of each colour do the children buy?
Show how you get your answer.
The children who live next door use the same rules for giving eggs.
They buy 8 red eggs, 8 yellow eggs, 2 blue eggs and 12 green eggs.
2. How many girls and how many boys live next door?
Show how you get your answer.
(Problem designed by Rita Crust and the MARS/Shell Centre team at the universities of Nottingham and Durham. Published in Developing problem solving – representing: using diagrams, tables and graphs (2005) nferNelson, ISBN 0-7087-1490-0)
The ‘Solving the eggs’ problem requires an approach to representing the information that is likely to be unfamiliar to almost all learners. Learning to deal with this kind of challenge is essential to becoming functional in mathematics.
However, even after the relationships in the problem have been clarified and represented, and the mathematical problem has been solved, the answer has still not been found. This final stage requires learners to interpret their work so that the mathematical result can be expressed in terms of numbers of eggs and numbers of girls and boys. Similarly, learners will need to become experienced in thinking about the mathematics they need to use to solve a problem. This can be illustrated by a simple example that includes three problems.
Joe buys a six-pack of cola for £3 to share among his friends. How much should he charge for each bottle? If it takes 40 minutes to bake five potatoes in the oven, how long will it take to bake one potato?
If King Henry VIII had six wives, how many wives had King Henry IV?
In current curricula, all the problems in a typical chapter on proportional reasoning will be like the first one listed; the learner does not have to choose an appropriate mathematical model. For mathematics to be functional it must include a substantial amount of modelling.
(Adapted from: Burkhardt, H., Bell, A., Pead, D. and Swan, M. (2006) Making functional mathematics happen, p. 14. Nottingham: Shell Centre Publications)
In all the examples in this section, learners are required to think for themselves. This is an experience that will stand them in good stead when they come across problems in the wider curriculum, in life or in the workplace and decide that a mathematical approach is needed.
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