Functional Elements and Problem Solving in GCSE Examinations Tony Fisher, Principal Examiner (GCSE Pilot and Functional Maths)
Objectives • To look at the assessment objectives for the new GCSE and how these link to functional elements and problem solving • To examine some examples of GCSE questions assessing functional elements and problem solving • To suggest some of the ways that teaching and learning will have to change to prepare students for the new GCSE, and introduce some activities that could help do this
Assessment objectives AO1 • Recall and use their knowledge of the prescribed content AO2 • Select and apply mathematical methods in a range of contexts AO3 • Interpret and analyse problems and generate strategies to solve them
Functional elements GCSE examinations in mathematics are required to assess functional elements Foundation 30 - 40% Higher 20 - 30%
What are functional elements? Functional elements are a subset of functional skills Functional skills are learning tools that enable people to: • Apply their knowledge and understanding to everyday life • Engage competently and confidently with others • Solve problems in both familiar and unfamiliar situations • Develop personally and professionally as positive citizens who can actively contribute to society
Functional process skills Functional skills qualifications in mathematics assess three interrelated process skills: Representing • Selecting the mathematics and information to model a situation Analysing • Processing and using mathematics Interpreting • Interpreting and communicating the results of the analysis
Functional elements 1 (a subset of functional skills) • Questions based on functional elements are set in a real life context and test one or more of the functional process skills • In GCSE examinations functional questions are categorised in terms of the assessment objectives not process skills • GCSE examinations do not lead to a functional skills qualification
Functional elements 2 Questions that assess functional elements: • Pose a purposeful problem about a realistic aspect of everyday life • Do not have to involve problem solving skills – in other words can be AO1 • Will allow students the opportunity to demonstrate mastery of at least one of the functional process skills In addition, if categorised as AO2 or AO3, questions must be written in such a way that: • Students have to select the mathematics and methods required to solve a problem
Functional or not? • At a gymnasium the notice on the rowing machine says: On this machine you lose 6 calories per minute. • On the cross trainer it says: On this machine you lose 9 calories per minute. • This question is not functional. Write the ratio of the number of calories per minute lost on the rowing machine and cross trainer in simplest form. • This question is functional. Tom trains on both the rowing machine and cross trainer. Work out a possible training plan so that Tom loses exactly 300 calories. Look at Handout 1
Modifiers of difficultyLook at Handout 1 The difficulty of a question is related to: • The technical level of the mathematics involved • The familiarity of any context in which a question is set • The complexity of the problem (e.g. the number of steps required to obtain a solution) • The amount of guidance (e.g. how the question is structured) • How obvious the mathematics involved is likely to be
Problem solving questionsLook at Handout 2 • Problem solving questions can be either functional or non-functional – they are always AO2 or AO3 • An AO3 problem solving question is one where most students need to stop and think and use one or more of these problem solving strategies • Working systematically • Working backwards • Finding an example that fits • Finding a relationship • Searching for the maths
Another problem A farmer has some sheep. 80% of the sheep have lambs. 30% of the sheep who have lambs have one lamb. 45% of the sheep who have lambs have two lambs. The rest of the sheep who have lambs have three lambs. Altogether the sheep have 546 lambs. How many sheep does the farmer have?
The challenge The present GCSE • To succeed at the current GCSE students have to able to do the maths they have been taught The new GCSE • The GCSE from 2010 involves about the same amount of doing but also a lot more thinking The challenge • To deliver learning experiences that encourage students to think mathematically in order to choose and use the maths they can do to solve problems
Maths lessons The bolt on approach (minimalist) • Will students develop problem solving skills by more of the same plus some added activities? The integrated approach (aspirational) • Or will it be more effective to integrate problem solving within teaching and learning and try to teach (or at least consolidate) some of the maths through a problem solving approach?
Getting students to think mathematically: 1 Include problem solving in every lesson Example 1 In a lesson on area and perimeter ask students to • Work out the rectangle with a perimeter of 30 cm that has the greatest area • Work out the amount of paint required to paint a skirting board Example 2 In a lesson on average and spread ask students to • Write down two sets of numbers with the same average but different spread • Compare boys and girls spelling test scores
Getting students to think mathematically: 2 Include regular problem solving lessons based on previous work Look at the example on Handout 4
Getting students to think mathematically: 3 Reinforce links For example • Teach reciprocal at the same time as teaching division by a fraction and eventually link it to negative powers and the gradients of perpendicular lines • Introduce speed as a ratio rather than an isolated rule • Introduce gradient as a ratio rather than an isolated rule • Introduce enlargement as a ratio rather than an isolated rule Look at the problem solving example on Handout 5
Getting students to think mathematically: 4 Try and introduce topics through a problem solving or a practical approach Look at the example on Handout 6 This shows how students could learn trigonometry through a (guided) problem solving approach (without mentioning SOHCAHTOA)
Getting students to think mathematically: 5 Look at the ideas on Handout 7 • How could you use these ideas in your lessons?
Summary • To succeed at the new GCSEs, students need to develop problem solving process skills • Students need to develop the confidence to think their way through problems and choose and use the mathematics they have learned to do this • One objective is to increase their chances of solving problems they might face in their future lives • To develop problem solving skills, maths lessons need to move away from their current focus on the teaching and learning of mathematical techniques • Hopefully, this seminar has given you some ideas of how you might start to meet this challenge