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MathematicsKey features of the subjectThe StrandsThe Strands in the Upper Primary Mathematics Syllabus organise thecontent. The Upper Primary Mathematics Syllabus has five Strands. TheStrands are Number and Application, Space and Shapes, Measurement,Chance and Data and Patterns and Algebra. The five Strands are furtherorganised into a number of Sub-strands to describe specifically thedevelopment of key ideas.In each of the Strands, the content is described as learning outcomes.These outcomes identify the knowledge, skills and attitudes to be learnt ineach Upper Primary Grade.The five Strands for Mathematics are outlined below with brief explanationsof what is to be covered in each Strand.Number and ApplicationIn this Strand students learn to use all common forms of number includingfractions, decimals, percentage, indices and negative numbers. They learnto apply these to solve problems that might be encountered in everyday life.Space and ShapeIn this Strand students learn to estimate and measure length, area, volumeand angles. They learn the language required to discuss shape anddirection. They learn to locate points on a plane by way of coordinates. Theyare presented with practical applications of what they are learning.Throughout they are challenged to apply a broad range of Mathematics tosolve problems.MeasurementThis Strand concentrates on the units and practice of measuring weight,temperature and time. Students are required to record, calculate and justifythe accuracy of the measurements they make.Chance and DataThis Strand focuses on the collection, presentation and interpretation ofdata. This Strand deals with statistical information, graphs, probability andsets. It also considers methods of estimation and issues of accuracy anderror.Patterns and AlgebraThis Strand deals with patterns in packing, in number and operations. These areused to link common events to mathematical thought and the idea of abstractrepresentation of numbers and processes that is possible with algebra.5

Upper Primary Teachers GuideThe SubstrandsThe Substrands are broad topics within the Strands that allow theknowledge, processes and skills to be specific and described as learningoutcomes. The Substrands in the five Strands of the Upper PrimaryMathematics Syllabus are outlined in the table below.Strands and Substrands for Upper Primary MathematicsStrandNumber andApplicationSpace andShapeMeasurementChance andDataPatterns andAlgebra6Grade 6Grade 7Grade lsFractions and decimalsFractions and decimalsFractions and decimalsDecimals and percentageDecimals and percentageDecimals and percentageRatiosRatiosRatiosDirected numbersDirected numbersDirected AreaAreaVolume and capacityVolume and capacityVolume and nDirectionDirectionMaps and coordinatesMaps and coordinatesMaps and ccuracy and errorAccuracy and errorAccuracy and ackingAlgebraAlgebraAlgebra

MathematicsDue to the interrelated nature of two-dimensional and three-dimensionalspace, Substrands such as volume and capacity and nets and packagingneed to be programmed together. Similarly, the Substrands of shape,tessellations and angles need to be programmed together to maximiselearning opportunities and reflect the connected nature of Mathematics.Developing knowledge, skills and positiveattitudesThis Upper Primary Mathematics course is designed to enable students tosee Mathematics as an exciting, useful and creative field of study. Duringthese years, many students will solidify or build up stronger ideas aboutthemselves as learners of Mathematics—about their competence, attitude,interest and motivation. Students acquire an appreciation for, and developan understanding of, mathematical ideas if they have frequent encounterswith interesting, challenging problems.In Grades 6 -8, students should engage in mathematical activities related totheir emerging capabilities of conjecturing, that is, investigating doubtfulinformation and verifying it or coming up with solutions, abstracting andgeneralising. The content addressed in Patterns and Algebra has anobvious focus on students recognising and being able to describe generality.Problem solving in the Upper Primary Mathematics course should developthe expanding mathematical capabilities of students. This includes problemsolving that integrates such topics as probability, statistics and rationalnumber. Well-chosen problems can be particularly valuable in developing ordeepening students’ understanding of important mathematical ideas. Thisidea also provides a bridge from the known to the unknown.OutcomesThe outcomes for each of the Strands and Substrands in this Guide describewhat the students know and can do as a result of the learning experiences.They demonstrate the knowledge, skills, understanding and attitudesachieved in Mathematics at the Upper Primary level. These outcomes arenumbered with a three-digit code where each number means something. Forexample in the outcome numbered 6.1.2, the first digit, 6, means Grade 6,the second digit, 1, means Strand number 1 and the third digit, 2, meansoutcome 2 for that Strand. The list of Mathematics outcomes can be foundon pages 12-16 of the Mathematics Syllabus. They can also be found in theWorked Examples for Upper Primary Mathematics Outcomes that comes asa support document to this Teachers Guide.IndicatorsThe indicators list the kinds of things the students would be able to do, knowand understand if they are to achieve a particular outcome. These areexamples that you can use to plan your weekly and daily lessons. You candevelop other indicators depending on the needs of your students and theresources available within the school or community.7

Upper Primary Teachers GuideThis Mathematics course attempts to place Mathematics into a practical andfamiliar setting so that students have the opportunity to explore and usemathematical concepts in real life situations. You are encouraged to makeuse of this opportunity when planning and programming your lessons.Curriculum principlesThis Mathematics course is based on three fundamental learning principles: we learn best when we build new learning on what we already know, we learn well when we recognise an immediate use or need for what is tobe learned, we use many ideas and skills in a coordinated way to solve realproblems.The course continuously refers to pre-existing knowledge and setsMathematics into contexts that are familiar and of interest to the students.This contextual approach leads to real problems in interesting and familiarsettings, requiring students to participate in both problem setting andproblem solving processes.The students need to use concepts and skills from many areas ofMathematics and other sources to come up with workable solutions, as inreal life. This approach facilitates a student-based mode of learning. Bylinking new mathematical concepts to existing cultural and scholasticknowledge, the students will integrate the knowledge so that they are moreable to use it in their lives.Catering for diversityStudents’ participation in active learning is desirable. It is the teacher’s roleto therefore ensure all students in the class are given a fair opportunity todemonstrate what they have learnt.Girls and boys should be:8 given the same access to activities in and outside the classroom, encouraged to value individual differences, encouraged to have respect and understanding for others within andoutside the school community, given opportunities to develop positive self-esteem and value lifeexperiences, encouraged to participate fully, regardless of their gender, ability,language group, culture and where they live.

MathematicsTeaching and learning strategiesTeaching and learning strategiesThese are some of the many teaching and learning strategies that you canuse to teach your students Mathematics. There are many ways you canapply these strategies. You are required to use some of these strategies,along with other strategies you know will work well in the teaching andlearning of Mathematics.Experience-based learningMathematical ideas are more likely to be remembered and used if they arebased on students’ experiences. In this approach Mathematics is a doingword. Having students use three steps in most practical lessons can expandthis approach. The three steps are: predict, observe, explain.As an example of this strategy, when developing knowledge of fractions, ateacher might ask a class to predict where 1/3 of the way across theblackboard would be. Students would then be invited to make a chalk markwhere they think 1/3 of the way across is and write their initials next to themark. When enough students have made their predictions while otherstudents observe, the teacher asks, “How would we work out who is theclosest?” Students then explain how to determine 1/3 of the way across theboard and then carry out the measurements to confirm the results.Problem-based learningUsing this strategy, the teacher can set a problem or rich task for the classto solve. For example, how could we make a measuring beaker from a600 mL plastic water bottle?Steps Brainstorm students’ ideas and record them on the board. Ask related questions such as, “How could we make a cuboid with avolume of 100 cm3?” Have students carry out the investigation in groups and report back to theclass.It is important that the teacher creates a summary of what has been learntfrom solving the problem to make the learning explicit.9

Upper Primary Teachers GuideOpen-ended questionsClosed questions commonly used in Mathematics lessons only have onecorrect answer. By contrast open-ended questions can have more than oneanswer and students are encouraged to explore and come up with a varietyof answers.An example of an open-ended question is, “What could be the dimensions ofa cuboid with a volume of 240 cm3?”?One answer could be 2 cm x 12 cm x 10 cm. If a student comes up withone answer and stops, ask the class if anyone had a different answer.How many different answers are possible?You can record the various answers from the class in a table. If you want to look atall of the possible answers with whole numbers work out the prime factors of 240.Using prime factors: 240 2 x 2 x 2 x 2 x 3 x 5 we can group these tomake all of the different sets of three factors of 240. For example(2 x 2) x (2 x 2) x (3 x 5) 4 x 4 x 15.One open-ended question can provide many answers for students to findand provide quite a lot of practice at basic skills at the same time.Cooperative learningCooperative learning has students working in groups on commonproblems. The main difference between group work and cooperativelearning is that with cooperative learning all students must contribute to thegroup’s learning. A cooperative learning task might ask a group to find allof the different pentominoes: 5 connected squares, that are nets of opencubes. When groups start to work on the problem, challenge them byasking which group has found the most.Fig.1: Some examples of pentominoes10

MathematicsTeaching approachesThe teaching approach required for this course is student-centred learning.It promotes the philosophy of ‘how’ to think and not ‘what’ to think.Student-centred teaching and learning activities include practicalinvestigations and problem solving as described above. These provideopportunities for students to work cooperatively, to discuss, make decisions,plan, organise, and carry out activities, record results and report findings.Activities should also allow the children to listen to each other’s opinions,demonstrate their strategies and critically analyse results. The teaching andlearning of concepts promotes the philosophy of ‘known to unknown’, buildingon what the students know and teaching concepts using similar contexts topromote better understanding. The teaching and learning approaches must bestudent-centred and as much as possible, student-directed.Use of vernacularWhile it is recognised that English is the main language of instruction, it must also berecognised that students are still more familiar with their vernacular. You should notonly accept the use of these, but encourage their use where it will lead to betterunderstanding. Mathematics is a language in itself and different from any otherlanguages. Therefore, it is believed that the use of the students’ first languageswill help them to understand better when dealing with mathematical activities.Integration with other subjectsSome topics or teaching strategies used in the Upper Primary Mathematicscourse are also dealt with in other subject areas. These topics includemeasuring, drawing, classifying, time, collecting and presenting data,graphing, money, decimals and percentage. The skills and knowledge taughtin Mathematics are used widely in other subjects. Content from other subjectsalso provides suitable contexts in which to teach Mathematics. For example, ifstudents are studying budgeting or how to run a small business in Making aLiving, this would provide a good opportunity to introduce decimals andpercentage in Mathematics lessons.As this course is designed to be student-centred, there is a need forconsiderable flexibility in programming. The majority of Mathematics topicscan be taught in any order that suits the needs of the students. You shouldtake advantage of this flexibility to maximise the links with other subjects. Anexample of an integrated approach or links with other subjects is provided inthe Units of Work section of this document.Use of multigrade teachingThe contextual approach used for this Syllabus lends itself well to multigradeteaching as the same context can be used for all students in a multigradeclassroom. The more advanced students tackle more sophisticated workand others address similar issues at a simpler level. If you are teachingmultigrade classes, you need to plan your program of work carefully so thatstudents do not repeat the same contexts. This allows the students to progressthrough the stages described by the outcomes for each Grade. An example ofmultigrade teaching is provided in the Units of Work section of this document.11

Upper Primary Teachers GuideAssessment, recording and reportingAssessment is the ongoing process of identifying, gathering and interpretinginformation about student’s achievement of the learning outcomes.Assessment using an outcomes-based approach is criterion-referenced andprovides information on the actual learning that has taken place. Thisinformation is used to enhance further teaching and learning and to providethe basis of reports on students’ progress. To do this effectively, it isnecessary for teachers to keep accurate records of assessment information.Teacher records must describe the students’ achievement of the learningoutcomes for the purpose of: checking students’ progress, planning and programming future learning, reporting students’ achievements to parents, guardians and others, informing students about their own progress.The following information has been extracted from The National Assessmentand Reporting Policy (2003) that provides guidelines about assessment andreporting for Papua New Guinean schools.PurposeThe purpose of assessment is to improve students’ learning and is focusedon students’ demonstrations of learning outcomes. The information obtainedfrom this assessment will be used to: provide feedback to the individual learner on their progress towardsachievement of the learning outcomes, make decisions about students’ learning and to provide information toimprove teaching and learning, report students’ achievement to parents, guardians, students, teachersand others.PrinciplesAssessment and reporting must be culturally appropriate for Papua NewGuinea. For assessment to be effective, it should:12 be continuous and based on the learning outcomes, be based on a balanced approach, be valid and reliable, reflect equity principles, by being fair, sensitive and broad enough tocater for differences in gender, culture, language, religion, socioeconomic status and geographical locations, be an integral part of teaching and learning, provide opportunities for students to take responsibility for their ownlearning and to monitor their own progress.

MathematicsUpper PrimaryAssessment at Upper Primary should: be flexible and use a range of assessment methods, be continuous and show the development of knowledge, skills andunderstanding in all school subjects, use local cultural approaches to assess and report students’achievement where appropriate, be mainly internal but may include external assessment at the end ofGrade 8, result in the issue of national certificates of basic education approved bythe Board of Studies reporting academic achievement, attitudes, valuesand other relevant achievements.Responsibilities of teachersTeachers have a responsibility to: develop and implement effective assessment and reporting practiceswithin school ass

The Mathematics outcomes covered in the Syllabus, Teachers Guide and the Worked Examples for Upper Primary Mathematics Outcomes build on the aspects of Mathematics covered at the Lower Primary level. It is assumed that Mathematics will develop both academic skills for further mathematica