1 Problems withPatterns and NumbersAn O-level ModuleJoint Matriculation BoardShell Centre for Mathematical EducationAUTHORSThis Module has been produced by the joint efforts of many teachers working-withthe Shell Centre for Mathematical Education and the Joint Matriculation Board. Itwas developed as part of the Testing Strategic Skills programme which aimsgradually to promote a balanced range of curriculum activities through thedevelopment of new examination Module was compiled by the Shell Centre team:Alan Bell, Barbara Binns, Hugh Burkhardt, Rosemary Fraser, John Gillespie,David Pimm, Jim Ridgway, Malcolm Swan, Clare Trott,co-ordinated by Jim Ridgway and Clare Trott, and directed by Hugh responsibility for the three sections of the book was as follows:Specimen Examinations Questions:John PittsClassroom Materials:Malcolm Swanbased on the work of a group of teachers, including.
2 Barbara Binns, Mike Cannon, Alan Chisnall, Kath Cross, Tansy Hardy, GillHatch, Cath Mottram, David Wilson and the Shell Centre team, with additionalhelp fromMark Allen, Anne Baxter, Ray Crayton, Beryl Footman, David Griffiths,Frank Knowles, Mary Robinson, Glenda Taylorco-ordinated by Susie Groves and Anne Materials:Rosemary FraserThis material has been developed and tested with teachers and pupils in over 30schools, to all of whom we are indebted, with structured classroom observation bythe Shell Centre gratefully acknowledge the help we have received from:Paul Morby and the University of Birmingham Television and Film Unit in themaking of the video material,Longman Micro SoftWare, the ITMA Collaboration, the Council forEducational Technology, and the SMILE group of the Inner London EducationAuthority for the use of the microcomputer materials, and Peter Wilson and hiscolleagues at the Joint Matriculation Board, together with the staff of RichardBates Ltd.
3 In the preparation of this Module, and finally Sue Crawford andSheila Dwyer for much typing and even more patient book was designed and edited by Malcolm with Patterns and NumbersCONTENTSI ntroduction to the Module6 Specimen Examination Questions9 Classroom Materials39 Support Materials139An expanded version of the contents follows on the next page ..3 EXPANDED CONTENTSI ntroduction to the Module6 Specimen Examination Questions9 Each of these questions is accompanied by a full marking scheme illustratedwith sample "The Climbing Game"12"Skeleton Tower"18"Stepping Stones"22"Factors"28"Reverses"34 Classroom Materials39 Introduction to the Classroom Materials41 Unit AUnit A is the basic core of the Module and should be completed. It will takeabout 'Oneweek and contains a full set of worksheets and teaching Problems45Al Organising Problems46A2 Trying Different Approaches50A3 Solving a Whole Problem54 Some Solutions56 UnitBDuring Unit B, the intention is to encourage children to solve Problems withless help and to articulate and record their different approaches and Problems , "checklists" of useful strategies for pupils to use when indifficulty, full teaching notes and solutions areprovided.
4 This Unitprovidesroughly one week's "Pond Borders"72B2 "The First to 100 Game"76B3 "Sorting"80B4 "Paper Folding"884 UnitCUnitCis built around three tasks which differ in style. No printed guidance isoffered to the pupils, but the teacher has a list of strategic hints which may beoffered to those in difficulty. Solutions are provided. This Unit againoccupies roughly one " laser Wars"96C2 "Kayles"98C3 "Consecutive Sums"100 AProblem CollectionA collection of15problems and games to use as a supplement to the materialspresented in Units A, and ideas for possible extensionsare "Painted Cube, Score Draws, Cupboards, Networks, Frogs,Dots, Diagonals, The Chessboard"Games124"The Spiral Game, Nim, First One Home, Pin Them Down,The 'I-IotFat Tune' Game, Dominoes, The Treasure Hunt".Support Materials139 These materials are divided into two parts - those that arepart of this book,and those that accompany the videotape and microcomputer programs in therest of the pack.
5 Both are written under the same headings and are usabletogether or independently. They offer support to individual, or groups ofteachers who wish to develop their teaching methodology and explore widerimplications of problem solving in the Looking at lessons1422. Experiencing problem solving1443. How much support do children need?1504. How can the micro help?1565. Assessing problem solving162)Checklist for the TeacherInside back cover5 INTRODUCTION TO THE MODULEThis Module aims to develop the performance of children in tackling mathematicalproblems of a more varied, more open and less standardised kind than is normal onpresent examination papers. It emphasises a number of specific strategies which mayhelp such problem solving. These include the following:*try some simple cases*find a helpful diagram*organise systematically*make a table*spot Patterns *find a general rule*explain why the rule works*check regularlySuch skills involve bringing into the classroom a rather different balance of classroomactivities than is appropriate when teaching specific mathematical techniques; for thepupils, more independent work and more discussion in pairs or groups, or by thewhole class; for the teacher, less emphasis on detailed explanation and knowing theanswers, and more on encouragement and strategic Module is not concerned with any narrowly d~efined area of content ormathematical technique within the existing syllabus.
6 Because the strategic skills itfocuses on are demanding, it concentrates on the simpler techniques which mostpupils will have mastered ( using Numbers and discovering simple Patterns ),while giving credit to those who bring more sophisticated techniques ( algebra) tobear on the Problems . A fuller discussion of these aims and the rationale behind theModule problem solving?The Cockcroft Reportt on mathematical education said, in paragraph 243:"Mathematics teaching at all levels should include opportunities for:*exposition by the teacher;*discussion between teacher and pupils and between pupils themselves;*appropriate practical work;*consolidation and practice of fundamental skills and routines;* problem solving, including the application of mathematics to everydaysituations;*investigational work. "Many teachers would like to include more problem solving and investigational workin the mathematics curriculum.
7 Most do not because they feel under pressure toconcentrate on what is on the examination syllabus. They do not feel able to devotetMathematics Counts, HMSO to teaching mathematical strategies which are not tested in the many pupils experience only two of the six elements listed here,exposition and practice. Cockcroft's list was drawn up after a great deal ofconsultation and consideration of educational research about the way children learnmathematics. This suggested that learning best takes place if the pupil experiences avariety of mathematical activities. The Board recognises that problem solving andinvestigational work should take place as an essential part of the mathematicalcurriculum, and that this is likely to happen on a reasonable scale only if theseactivities are reflected in the assessment procedure.
8 Their importance is alreadyrecognised in the Board's current examination objectives, particularly the later onesin the list:Knowledge and abilities to be testedThe following list is intended to provide a general indication of theknowledge and abilities which the examination will be designed to Knowledge of mathematical notation, terminology, conventions andunits. The language and notation of sets together with the ideas of amapping and a function are basic to the The ability to understand information presented in verbal, graphical ortabular form, and to translate such information into The ability to recognise the mathematical methods which are suitablefor the solution of the problem under The ability to apply mathematical methods and The ability to manipulate mathematical The ability to make logical The ability to select and apply appropriate techniques to Problems inunfamiliar or novel The ability to interpret mathematical , the types of questions currently set do not adequately test these higherlevel skills.
9 This Module will allow the introduction of an appropriate question intothe examination in a way that is accessible to the teachers and pupils that the Boardserves. To ensure this, the Module offers teachers classroom material which has beencarefully developed and tested, and teacher support materials, as well as detailedinformation about the change in the examination itself. These three elements of theModule: specimen examination questions and marking schemes with pupil answers,sample classroom materials and support materials will, we hope, give a clear pictureof the intentjons. The successive trials of the Module have shown that they offer astraightforward way to realise those Examination QuestionsCONTENTSI ntroduction11 QuestionsThe Climbing Game12 Skeleton Tower18 Stepping Stones22 Factors28 Reverses3410 INTRODUCTIONThis Module introduces a somewhat new type of examination question in which themathematical processes involved, especially the choice and explanation of strategiesand discussion of results, are as important as the answers obtained.
10 This is reflectedin the marking questions set will be drawn from a wide variety of Problems which are looselylinked under the heading " Patterns and Numbers ". The range of Problems is notdefined in the conventional way by specifying a topic area or listing the mathematicaltechniques involved. Instead, the Problems will involve situations in which, startingfrom the consideration of some particular cases, a pattern has to be found and thenformulated into a general rule. These processes are important throughoutmathematics and number properties and Patterns provide a suitable field in whichthey can be should be noted that, in the examination, candidates will be given credit forexplanations of what has been attempted at each stage and for what has beendiscovered. More generally, mark schemes will be designed to give credit for:(i)showing an understanding of the problem ,(ii)organising information systematically,(iii) describing and explaining the methods used and the results obtained,(iv) formulating a generalisation or rule, in words or following sample of questions gives an indication of the variety likely to occur inthe examination.