Using and applying mathematics to different problems

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Traditionally in the United Kingdom, mathematical problems have been treated as contexts in which young learners can apply exiting knowledge. This is reflected in the use of the term 'using and applying mathematics' within the National curriculum. Problem-solving in the Netherlands is viewed somewhat differently. Problem-solving contexts are used as a starting point from which mathematical strategies and conceptual understanding are developed. The second part of this report gives ideas of teaching strategies that can be employed to promote problem solving and mathematical thinking in the developing children of the United Kingdom.

Part 1: Analysis of the progression of problem solving between the primary years from years 1 to 6

Solving problems is one of the strands in the Using and applying mathematics strand. According to the 1999 Framework for teaching mathematics, numeracy is a proficiency that requires a child to incline to and have an ability to solve problems when given different contexts. This numeracy results in children who are have the confidence to tackle mathematical problems without immediately asking their teachers and friends to help them. To become problem solvers, children need to solve problems, meaning that children have to be given the space and time to tackle mathematical problems during lessons is they are to become competent and confident problem solves. In realisation of this, problem solving for the children from primary years one to six has been embedded into mathematics teaching and learning, thereby becoming an integral part of the children's work. The renewed Primary Framework focuses on children solving problems that are set in wider ranging contexts because the children become more confident and skilled. This progression analysis highlights the increasing complexity of the mathematical problems that the children tackle as they move from one year to the next. Through years one to six Block A covers counting, partitioning and calculating. Block B covers securing number facts, understanding shape, Block C covers handling data and measures, Block D covers calculating, measuring and understanding shape and Block E covers securing number facts, relationships and calculating.

Year one

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During their first year, children are supposed to solve problems involving counting, adding, subtracting, doubling and halving in the context of numbers, measures and money. In block A of year one, children concentrate on solving problems involving counting and they extend their counting and calculation skills. The children estimate a number of objects that can be checked by counting, begin to understand place value in two-digit numbers, read and write numerals to 20 and beyond, relate addition to counting on and to combining groups and use an increasing range of vocabulary related to addition. In block B of year one, children consolidate their use of patterns and relationships to solve number problems and puzzles. In block C, children take greater responsibility for posing and answering questions. In block D, Children continue to make direct comparison of the length, weight or capacity of two objects without any counting. The children begin to use uniform non-standard units to estimate and then measure length. The children continue to work with money as well as continue to develop the concept of time by ordering the months of the year and reading time to the hour and half hour on a clock. In block E, children continue to solve practical problems involving addition or subtraction, doubling or halving and they record their solutions on a number line or in a number sentence.

Year two

During their second year, children are supposed to solve problems involving addition, subtraction, multiplication and division in contexts of numbers, measures and pound and pence. Block A does not cover any problem solving. In block B, children use their knowledge and experience of counting to learn the 2, 5 and 10 multiplication facts. The children solve one and two-step word problems involving money and measures, using all four operations. In block C, the children solve problems such as finding which soft drink is most popular with children in the class, and later make a block graph and explain what it shows to others. In block D, children continue to count in ones, twos, fives and tens. These skills come in handy in helping them to tot up a mixed set of 10p, 5p, 2p and 1p coins. The children develop the understanding of number lines to enable them read a range of scales. In block E, the children consolidate counting on from zero in steps of 2, 5 and 10 and build up times-tables, describing what they notice about numbers in the tables. They use this knowledge to predict some other numbers that would be in the count. The children understand that repeated addition can be represented using the multiplication symbol. The children use a number line to support repeated addition, recording the equal jumps on the line and writing the repeated addition statement and the matching multiplication statement. The children identify the operation(s) needed to solve a problem and explain their reasoning.

Year three

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During their third year, children are supposed to solve one-step and two-step problems involving numbers, money or measures, including time, choosing and carrying out appropriate calculations. In block A, Children solve problems involving counting, solve number puzzles and organise and explain their written responses to problems and puzzles in a systematic way. The children identify relevant information and select the appropriate operations in order to solve word problems. In block B, the children use patterns, properties and relationships between numbers to solve puzzles. In block C, the Children pose a problem and suggest systematic and appropriate approaches to collecting, organising and representing data in order to solve the problem. In block D, children add or subtract multiples of 10 or 100 and near-multiples to solve word problems and then use practical and informal written methods to solve problems involving multiplication and division. The children recognise that finding fractions of amounts involves division and find a fifth of a quantity. In block E, the children apply their skills when they solve practical measuring problems.

Year four

During their fourth year, children are supposed to solve one-step and two-step problems involving numbers, money or measures, including time; choose and carry out appropriate calculations, using calculator methods where appropriate. In block A, the children continue to derive and practise recalling multiplication and division facts to 10 - 10. The children consolidate multiplying and dividing numbers to 1000 by 10 and 100. The children develop written methods for multiplying and dividing. In block C, the children evaluate the effect of different scales on interpretation of the data. In block D, Children learn the relationships between familiar units of measurement. Practical activities help children to increase their accuracy of measurement and estimation as well as choosing appropriate instruments and units. In block E, children investigate patterns and relationships. In block E, children count in fractions along a number line from 0 to 1 and establish pairs of numbers that total 1. The children are introduced to the vocabulary of ratio and proportion

Year five

During their fifth year, children are supposed to solve one-step and two-step problems involving whole numbers and decimals and all four operations, choosing and using appropriate calculation strategies, including calculator use. In block C, children test a hypothesis by deciding what data is needed and discussing how they will collect the data. The children use ICT to help them present graphs and charts quickly, and interpret their graphs and charts to draw their conclusion. In block D, when the children measure weight, they use a range of scales. In block E, children use multiplication and division to solve problems involving ratio and proportion.

Year six

During their sixth year, children are supposed to solve multi-step problems, and problems involving fractions, decimals and percentages; choose and use appropriate calculation strategies at each stage, including calculator use. In block A, children use a calculator to explore the effect of brackets in calculations. They decide whether or not to use a calculator to solve problems. In block D, children solve practical problems by estimating and measuring using standard metric units from a range of scales. The children draw on a range of mathematics to solve problems involving estimating and measuring. The children communicate clearly how a problem was solved and explain each step and comment on the accuracy of their answer. The children explore area and perimeter of rectilinear shapes. They estimate the size of angles and use a protractor to measure acute and obtuse angles. The children describe the patterns and relationships that they discover. In block E, children solve problems in different contexts, using symbols where appropriate to explain their reasoning. The children identify and record the calculations needed, interpreting the solutions back in the original context and checking the accuracy of their answers.

Part 2: Ideas of teaching strategies to be employed to promote problem solving and mathematical thinking.

Teaching mathematics students how to solve problems is important. These students should be taught how to apply the mathematical problems to problems in everyday life. The students should be in a position to do investigational work on the mathematics problem. A problem is a task that does not provide the learner with a clear route to the solution. If the solution to a problem can be arrived at through different approaches, then that problem has some degree of openness. The term 'investigation' is used to describe such an open problem that can be solved through different solutions. An investigation is a good way to enable young learners to use and apply their abilities in mathematical knowledge. There are different levels of openness that are offered by application tasks. Exploratory problem solving is another means by which

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Application tasks exist with different levels of openness. Besides investigations, problems that have some degree of openness can be solved by exploratory problem solving. This gives the learner a chance to solve real-life problems using a mathematical approach. As a result, exploratory and investigative problem-solving offer children greater chances for developing the mathematical thinking of young learners. Word problems on the other hand are usually closed problems that have a defined solution and a standard method of calculations is applied. An example of such a problem is: How much change would I receive from a 10 pound note if I bought items costing 2.59 pounds and 3.99 pounds? Once the problem has been rewritten using symbols and numbers in a mathematical format, there is usually a standard method carrying out the resulting calculations. Word problems can still offer valuable opportunities for young learners' mathematical thinking.

A delicate balance is required between sensitive questioning which develops a child's thinking, and allowing the child time and some level of freedom to develop his own approach and strategy to problem solving. In this sense, the teacher's role is somewhat different from that when teaching other aspects of the mathematics curriculum. In such contexts, the understanding of the likely consequences of intervention and non-intervention and flexibility of approach by the teacher are critical.

Word problems can be solved by either a horizontal or vertical mathematising process (reference). The horizontal mathematising process is the easier of the two and is a strategy commonly used by children to solve word problems. Horizontal mathematising is whereby symbols are used to represent items in a mathematical word problem. Vertical mathematising is whereby the model created in vertical mathematising needs to be adapted in order for the answer to the mathematical word problem to be figured out. Askew gives two questions that are used to demonstrate the complexities surrounding word problems. The first question is: Mrs. Chang bought five video tapes that cost the same amount. If she spent 35 pounds, how much did each tape cost? The second question is: Mr. Chang bought some tapes that cost 7 ponds each. How many tapes did he buy? The first question is easier for children to solve because they can use fingers as a symbol of the number of tapes. The use of symbols supports the children's thinking within the purely mathematical context and enables them to arrive at an answer by trial-and-improvement techniques (reference).

Research findings show that children make use of a wide range of informal strategies to solve word problems (reference). However, mere use of models is not sufficient for many children to solve a word problem. This is because word problems require children to translate between the real world context and the world of mathematics and back again. Switching between the physical world and the mathematical world is difficult because there exists a mismatch between these two worlds. When the teacher is made aware of this issue it provides a way forward. In the example above, children should be asked to compare the problems in order to help them appreciate more deeply the complexities of solving such problems. Children should also be helped to categorize word problems in order to help them appreciate structural similarities and differences. Categorising problems will require children to use reasoning skills in order for them to make generalisations about solution strategies for particular classes of problems.

In the Netherlands, a different attitude to problem-solving has been adopted. This approach, known as realistic mathematics education, is based upon Freudenthal's (1968) belief that children should be given guided opportunities to reinvent mathematics through doing it. Thus, the focus of realistic mathematics education is children's mathematisation of contexts which are meaningful to them, and through this participation in the learning process, children develop mathematical understanding and strategies. Instead of using problem-solving as a vehicle for context-based application of earlier learning as a tradition in England, realistic mathematics education uses context problems as a source for the learning process. A good example of a context building up mathematical knowledge is taking the context of a city bus. The teaching starts with a real life situation where the students have to act as the driver of the city bus. The passengers are getting on and off the bus, and at each stop the students have to determine the number of passengers in the bus. Later the same is done on paper. The development of mathematical language is elicited by the need to keep track of what happened during the ride of the bus. Initially, the language is closely connected to the context, but later on it is used for describing other situations. This way, children's conceptual understanding of related strategies from within the contexts of the problem is developed from the realistic mathematics education principle.

Conclusion

A consideration of some different approaches to teaching problem-solving will inevitably lead to a consideration of the purpose of teaching problem-solving. A problem-solving approach has clear benefits for pupils in helping them to approach mathematical problems of all kinds in a more structured way. Practice in identifying the main features of a problem and rejecting redundant information, and looking for relationships and strategies in a problem-solving situation, are all transferable skills that can be used in all area of mathematics. Such transferability of skills, knowledge and understanding is however not trivial. A key challenge, therefore, is to determine how best to ensure that children learn mathematics in ways that enable them to transfer knowledge and understanding gained in one context to other contexts they encounter subsequently. The role of the teacher is important in supporting children's learning through problem-solving.