Using and Applying Mathematics - Essay Example

of essay’s assignment is due Using and applying mathematics Introduction The first part of this report analyses the progression of problem solving for children between the primary years from years one to six. In the United Kingdom, it is customary for mathematical problems to be treated as contexts in which young learners apply existing knowledge (Brown 1998). The National curriculum reflects this fact by using the term ‘using and applying mathematics.’ Elsewhere problem-solving is viewed differently. Mathematical strategies and conceptual understandings are developed from the starting point of problem-solving contexts. A delicate balance is required between letting the child have some time and freedom to develop his own approach to strategy to problem solving, and sensitive questioning which develops the child’s thinking (Hopkins et al. 1996). Beginning with the problem-solving framework, the teacher has a very specialized and highly-involved role in the education of the students, and the recognition of the likely effect of intervention and non-intervention are critical. In this paper, teaching strategies will be presented which 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 crucial component of using and applying mathematics. According to the 1999 Framework for teaching mathematics, numeracy is a proficiency that requires a child to have an ability to solve problems when given different contexts. 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. 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; securing number facts and understanding shape in Block B; handling data and measures in Block C; calculating, measuring and understanding shape in Block D; and securing number facts, relationships and calculating in Block E (Tanner & J1s 2000). Year one In Block A, each student should be able to solve problems, recognise and utilize the number system, recognise prior experience with mathematic operations, and communicate the abstract concepts of math in a concrete, tangible form. In Block B, they name shapes and their characteristics, forming a basis for the examination of 2-D and 3-D shapes which extends through Year five. In Block C, they sort and present information in diagrams and use units of measurement and equipment- both traditional and non-traditional (The Department of Education n. d.). Year One Block Implementation BLOCK SKILL PROBLEM-SOLVING EXAMPLE A Recognizing Mental Math Ask students to raise their hand if they have ever given some of their food to a friend or chosen a cheaper toy? Iterate that these are mathematic problem-solving processes that the students already know. C Informal and Formal Basic Measurements Pick a student and ask them what their favourite animal is. Ask them how many of that animal should be able to fit in 1) a suitcase, 2) their room, and 3) a submarine. Then pull out a ruler and ask them the same three questions about the ruler. This will initiate 3-D, creative problem-solving analysis. Year two The second year begins the challenge to basic concepts. It combines the estimation of the first year with more abstractions. Two digits are linked together to make a whole new number; new numbers do not have to be whole; decimals can be rounded up. In Block A, each student should be able to determine and record ‘greater than’ or ‘less than’ and mentally calculate figures. In Block C, they should sort into diagrams using two criterion, collect, organize, interpret, and further question data, and compare units of measurement. In Block E, they should identify division as grouping and test examples (The Department of Education n. d.). Year Two Block Implementation BLOCK SKILL PROBLEM-SOLVING EXAMPLE B Shape Recognition Describe the differences between a 2-D shape and a 3-D shape. How do they appear different? How does this appearance relate to the math found in our world? Year three During their third year, children learn about solving two-step problems. In Block A, Children solve problems involving number puzzles and explain their written responses. 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 appropriate approaches to the use of data in order to solve the problem. In Block D, children solve word problems involving multiples of 10 or 100 and must realise that problems that involve finding fractions of amounts engages in the division of a quantity. In Block E, the children learn the application of their skills when they are solving practical measuring problems (The Department of Education n. d.). Year Three Block Implementation BLOCK SKILL PROBLEM-SOLVING EXAMPLE D Beginning Fractions If there is one chocolate chip cookie and Zoe eats half of the cookie, then is the remaining portion a whole number? E Solving Practical Measuring Problems If you don’t have a ruler or measuring tape, then how could you figure out how long a baseball field is along the right foul line? “You are ____ centimetres tall. How could you use that fact to figure out the length of the field?” Year four During their fourth year, children learn problem solving skills in solving two-step problems involving measurement, choose and execute appropriate calculations, and build upon the problem-solving skills of every block of Year three (The Department of Education n. d.). In Block A, the children develop written methods for multiplying and dividing. This skill set will be useful in developing other problem-solving methods, such as a scientific theory or a thesis statement for literary analysis. In Block C of Year five, data collection after the formulation of the methodology will be taught. In Block E, children investigate patterns and relationships, count in fractions, and establish pairs of numbers that total 1 and are introduced to the meaning of ratio and proportion (Haylock & Cockburn 2008). Year Four Block Implementation BLOCK SKILL PROBLEM-SOLVING EXAMPLE A Sorting You can have the students play the Guess Who? game. Ask a student to look at the people and describe one characteristic that is different in some of the people, such as red hair or glasses. E Fractions and Wholes Ask the students to select the fractions from a list which equal one whole, such as 1/4+3/4 or 5/8+6/16. Year five The fourth year emphasized proper calculations which will be sharpened through an education in mathematic formulas and rules. During their fifth year, children are engage in solving one-step and two-step problems that involve decimals and whole numbers and the four operations (The Department of Education n. d.). The children choose and use appropriate calculation strategies. In Block C, children test a hypothesis by making decisions on what data is needed and they discuss how they would collect the data. The children use information to quickly analyse graphs and charts and draw their conclusion. In Block D, the children use a range of scales to measure weight. In Block E, children solve problems that involve ratio and proportion. Year Five Block Implementation BLOCK SKILL PROBLEM-SOLVING EXAMPLE C Decision Making for Problem-Solving Students read a long word problem and identify the question being asked and what factors will be needed to answer the question. Then they proceed to identify the numbers and mathematical processes that will be used and begin their calculations. Year six During their sixth year, children learn to solve multi-step problems involving percentages, fractions and decimals- with a calculator as needed (The Department of Education n. d.). In Block A, children use a calculator to explore the effect of brackets in calculations. In Block D, children solve practical problems by estimation and measurement using standard metric units, thereby developing their formal and informal estimative skills from Years one (Block C) and three (Block D). The children learn how to clearly communicate how they solved a problem, explaining each step and commenting on the accuracy of their answer. The children learn the estimation, description, and relationship of the size of angles and then use a protractor in measuring acute and obtuse angles. In Block E, children solve problems in different contexts, use symbols where appropriate, identify and record the calculations needed, and interpret the solutions in the original context. (Bishop et. al 1997). Year Six Block Implementation BLOCK SKILL PROBLEM-SOLVING EXAMPLE A Calculators and Brackets When students put -[-3] in the calculator, they discover that positives, negatives, brackets, and parentheses have an unexpected effect on the results. D Perimeter Give the students the length and width of a fence, review the formula for perimeter, and explore the different use of operations possible to come to the same correct conclusion for a rectangle, such as 2L+2W=P or L++W+W=P. 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, but how is at the heart of success or failure. A problem is a task that does not provide the learner with a clear route to the solution (Orton 2005). The teacher provides the student with problems, the students provide the teacher with problems. There is no one answer. The term ‘investigation’ is used to describe such an open problem that can be solved through different solutions (Stewart 2008). Utilizing personal examples is reliably one of the best tools to aid in retention of concepts covered in class. Active participation is much more likely in counting or estimations exercises (typical of Year One concepts) when the result for a correct answer is consuming the object counted, such as chips or Skittles (Jordan & Levine 2009). For Year Two students, ‘greater than’ is easily remembered when the sign is depicted as an open Pac-Man mouth pointed toward the bigger number. The emphasis on the particular mathematical language used is important in all six years of instruction. Because the connotations, denotations, and other extra factors in word allocation come into play, consistency is important. However, in the first two years in particular, taking one student aside and asking them to personally explain their progress and hold-up may be the simplest course of action to determine the underlying problem. For fifth graders, it may be something as simple as reminding them that the number being divided is the quotient. All of a sudden the words on the board no longer seem to be in Latin. Students with overall mathematical learning problems will often count on their fingers longer than their classmates and still rely on other, tangible mathematical representations (Jordan et al. 2009). In such cases, early identification is key. A variety of assessment tools can be used: a verbal or written quiz, a review of their standardized test scores, a review of the teacher’s covered mathematical curriculum and/or student examples of work from the previous year. However, it is important to recall that these, too, can be inaccurate. 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 (Kirkby 1998). Research findings show that children solve word problems by making use of a wide range of informal strategies (Millett 1996). The mere use of models, however, is not adequate enough for many children who are trying solve a word problem. This is because word problems require that the children translate to and fro between the real world context and the mathematical world. 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 of Mr. And Mrs. Chang, children should be asked to compare the problems. This will help the children appreciate the complexities of solving such mathematical problems. Children should also be helped to categorize word problems in order to help them appreciate similarities and differences in the structure of mathematical problems. The children’s reasoning skills will be put into use if they were to have a simplified approach and solution strategy on how to solve particular classes of mathematical problems. A different approach to problem-solving known as Realistic Mathematics Education is used in the Netherlands (OSullivan 2005). This approach is based on the belief that children should reinvent mathematics by being given guided opportunities to tackle mathematical problems. Thus, the intention of realistic mathematics education is helping the children’s mathematise contexts which they find meaningful to them. Through the children’s participation in the learning process, they develop mathematical understanding and strategies to solving problems. In England, problem-solving is used as a means by which children can solve context-based problems. However, in the Netherlands, context problems are used by Realistic Mathematics Education as a basis for the learning process. The context of a city bus is a good example of a context building up mathematical knowledge. The children are faced with a real life situation where they act as the bus driver. Bus passengers get into and off the bus at designated bus stops, and the children count the new number of passengers in the bus. When they go back to class, the students show the findings from the bus in their books. Without the teacher having to tell the children to write down their findings while they are in the bus, the children develop a need to keep track of the number of passengers in the bus at a given time. At first, the children develop a mathematical language that is closely connected to the context. This knowledge is later used in 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 (Sellars & Lowndes 2003). Conclusion Methodological limitations aside, each classroom presents its own unique body of knowledge which a teacher can only asses through testing individual concepts. The one aspect that will always be employed is problem solving. A problem-solving approach helps pupils in approaching all kinds of mathematical problems in a more structured way. Transferable skills such as practice in looking for strategies and relationships in a problem-solving situation, and identifying the important features of a problem and ignoring the redundant information, can be used in all areas of mathematics. References Bishop, E et. al. 1997, International handbook of mathematics education, Springer. Brown, T 1998, Coordinating mathematics across the primary school, Routledge. 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Millett, A 1996, Using and applying mathematics: Innovation and change in a primary school, University of London. Orton, A 2005, Pattern in the teaching and learning of mathematics, Continuum International Publishing Group. OSullivan, L 2005, Reflective reader: Primary mathematics, Learning Matters. Sellars, E & Lowndes, S 2003, Using and applying mathematics at key stage 1: A guide to teaching problem solving and thinking skills, David Fulton. Stewart, K 2008, In the Classroom, Straightforward co Ltd. Tanner, H & J1s, S 2000, Becoming a successful teacher of mathematics, Routledge. The Department of Education n. d., The national strategies, viewed July 23 2010, . Word count: 3920 Read More