Mathematics Pedagogy - University Of Canberra
What makesfor successfulnumeracyeducation inremote Indigenouscontexts: Anethnographic casestudy approachStories on remoteindigenousmathematicssuccessescompiled byProfessorRobyn Jorgensen2016Mathematics PedagogyBrewarrina Central SchoolBrewarrina is a town located on thebanks of the Barwon River, 787km NWof Sydney. The name of the town comesfrom the Wayilwan word, ‘burru waranha’which means ‘Acacia clumps’, ‘a nativestanding’ or ‘place where wild gooseberrygrows’. The population of Brewarrina isaround 1500 with a further 700 living onproperties surrounding the town. Thetown is located in the traditional landsof the Ngemba peoples and was oncethe tribal meeting grounds for over 5000indigenous people. Evidence of thisremains with camp sites, scar trees andfish traps still in the area.The Brewarrina Fish traps (Ngunnhu)are a very important part of the localhistory and are heritage listed. They areestimated to be more than 40,000 yearsold (they may be the oldest survivingmanmade structure in the world) and liedownstream from the weir on the bed ofthe Barwon River. The traps are made upof a collection of stone weirs and ponds,organised to create channels whichlead to a complex ‘net’. Manyyears before Europeansettlement, the Aboriginal communitieswho created these fish traps were applyinginnovative knowledge of fish migration,water ecology, physics and engineering.Once the fish were trapped within the‘net’, the Indigenous fishermen were freeto pluck the fish from the traps as theypleased. The Ngunnhu continues to bean important meeting place for the localAboriginal people to this day. Brewarrinais now a very popular area for fishingenthusiasts with fish such as Bream,Yellow Belly, Catfish and Murray Cod (thelargest officially recorded Murray Cod wascaught in the Barwon River and weighed astaggering 113kg).The first settlers arrived in the region inthe late 1830s and in 1859 the town wasdeveloped as a port for paddle steamersand was soon recognised as the furthestnavigable point on the Barwon River.Wool was shipped from Brewarrina toAdelaide on the Darling and Murray Riversfor many years. Cobb and Co. had anumber of coaches that regularly passedthrough the town which resulted in the
town thriving during the 1870s. This led to a largenumber of hotels and stores being built in the town.The NSW heritage listed Barwon Bridge was openedin 1888 so that the river could be crossed without apunt or pontoon, and this opened up the opportunityfor transporting wool to Sydney by road, rather thanjust by paddle steamer to Melbourne and Adelaide.This bridge is unusual because it is only one of twosurviving examples of the first lift bridges in NSW.The Brewarrina Ngemba Billabong has a strongIndigenous history. This was the BrewarrinaAboriginal Mission for the local Indigenous peoplefrom 1876-1967 and was the very first institution tobe established by the Aborigines Protection Board aspart of its policy to segregate Aboriginal people. LocalAboriginal people were sent to this location whentheir land was taken for grazing and this mission wasresponsible for the removal of many Aboriginal peoplefrom their homes against their will. In recognition ofthis history, the 261 ha property is now listed on theNSW State Heritage Register. The billabong wetlandsand woodlands are home to four endangered birdspecies, the brolga, blue-billed duck, freckled duckand the red-tailed, black cockatoo. They also containmany native plants such as water lilies, river redgumsand coolibah. Land management is very importantto the traditional owners and this helps to providejob opportunities as well as continuing the importanttraditions of culture and custodianship.The Brewarrina Central School is a K-12 school thatdraws its enrolment from the Brewarrina township andsurrounding settlements. A number of children travel100 to 200km by bus each day from Gongolgon andWeilmoringle. In 2015 there were 160 enrolments atthe school with 97% of the student population beingAboriginal. The school has very strong communitysupport and works closely with the local AboriginalEducation Consultative Group. The school has astrong focus on Aboriginal education and culturalawareness. The school is part of the NSW DETConnected Schools strategy. The school enjoys arelatively stable staffing profile with both teaching andsupport staff having been involved with the school forextended periods of time. Some staff members andexecutive staff have been at the school for a numberof years. The school is well resourced and has anoutstanding mathematics resource centre to supportthe approach taken at the school.Defining GrowthBrewarrina has shown growth in numeracy results inNAPLAN when compared against like schools. Theschool is aiming to continue the improvement and movestudents into a higher band in NAPLAN. The growthdescribed in this case study apply to the primary sectorof the school. There has been a very strong emphasis inthe early years (infants) of schooling with the intent thatthe strong foundations will provide students with the bestopportunities for success throughout their schooling.2012Year 3Year 5Year 7Year 920132014No score2015
Practices at theSchoolBrewarrina has adopted a number of key practicesthat are implemented across the primary yearsof school (K-6). These practices are built into allmathematics lessons in a whole-school approach. Likemany schools in this study, Brewarrina has adopteda timetable where literacy occupies the first block ofschool, numeracy the second, and the final block for allother curriculum areas. An emphasis on literacy andnumeracy is seen to be a priority for students, as thisforms the basis for a functional learner. The structureof the program is consistent throughout the whole yearso that students are familiar with the timetable andexpectations. Lessons involve group work, rotations,use of technology, games and hands-on activitiesto ensure students remain engaged and allows fordifferentiation to meet the needs of each child.Aboriginal EducatorsThe school has many long serving staff members, bothAboriginal and non-Aboriginal, making for a very stableteaching and support staff. Aboriginal Educators area key part of the school program, with some classesoperating with up to 2 Aboriginal Educators. MostAboriginal Educators have been working at the schoolfor many years and have been involved in professionaldevelopment. They know how processes work andhave much experience in working with the students.They bring considerable knowledge - school, home,cultural and community - to the classroom.Aboriginal educators are provided with release timeeach week to work with the teacher around planningand preparation. This fosters collaboration andpromotes the sharing of local knowledge to better meetthe needs of the students. Aboriginal educators play anumber of roles in the classroom including facilitator ofsmall group activities.
Lesson FormatThe school has developed a broad lesson format formathematics that all teachers in the infants/primarysector follow. The staff place great emphasis on handson, practical learning experiences with text books andworksheets only used on occasion. This approach followscurrent research and pedagogies, with staff being trainedin Targeted Early Numeracy (TEN) and Taking off withNumeracy (TOWN).ConsistencyConsistency within the lessons across the timetable, areseen to be a vital part of the planning in mathematics(and literacy) across the school. Ensuring that lessonsfollow the same format and expectations enablesthe students to know what to anticipate when theycommence their lessons, and hence be ready to engagein the lessons.Lesson StructureTeachers have flexibility within their own classrooms toorganise lessons that meet the learning needs and stylesof their students. Lessons are structured around a numberof key organisers, with teachers able to move emphasiswithin a lesson to any one or more of the lesson elementsdepending on students’ needs.Some teachers use timers (on the interactive white boards)as a visual aid for students to see how much time is left fora given activity. The lessons are designed to keep a fastpace to maximise engagement. The use of digital toolsis a feature of the lesson as well, to engage learners andpromote participation.
1. Number consolidationIn this phase of the lesson, teachers use variousstrategies to build fluency and automaticity with numberfacts. The students enjoy competition and speed withvarious games played and an emphasis placed onpersonal best rather than competing against peers.Number activities are targeted for the individual studentsso their number knowledge is commensurate with needand growth.2. Interactive white boardsAll classrooms have, and use, interactive white boards.These are used in a range of ways to stimulate, engageand motivate students.3. Hands-on, games group work rotationsAll lessons involve focussed activities in small groups.The majority of activities are hands-on and practical. Insome classes, group rotations last 15 minutes and aresupported by a teacher, Aboriginal educator or student’swork independently. The pacing is quick and someteachers use a timer (usually on the interactive whiteboard) so students can see time left in an activity. Thiskeeps the pace brisk and students on task. Groups varyfrom mixed ability to specific targeted groupings andhave intensive specialised support.4. Word problemsContextualisation of mathematics into words problems isthe final phase of a lesson. Word problems are posed,often in the group context to allow the students to workthrough a problem, using Newman’s Analysis, as acollective group. This promotes peer tutoring and fostersa deeper understanding of concepts taught.
Group WorkClass sizes are intentionally small and with atleast one Aboriginal educator in each class.Students are organised in groups, usually ofsimilar ability, so that students can supporteach other. Opportunities exist to use a mixedlevel approach which underpins a belief thatstudents can vary in their strengths, so thatat any time different students can show theirstrengths within a group.In most classes group work consists ofrotations that change every 15 minutes.The tasks in the groups are designed tobe engaging and are hands-on activities orgame-type activities.
Using Digital MediaMost lessons involve the use of some sort ofdigital media, usually interactive white boards,where the tool can be used for whole classor small group activities. It is seen to engagethe learners and promote positive learningopportunities. Teachers use a wide range ofinteractive media to enhance understanding ofnew concepts and support reluctant learners.Small ClassesBrewarrina CS has intentionally implementedsmaller class sizes to lay foundations forsuccess, not only in Primary School butthroughout their schooling. A smaller class sizeallows teachers and Aboriginal educators toknow the needs of their learners and provideexperiences to cater for varying abilitiesof individuals within a class. Smaller classsizes also provide an opportunity for positiverelationships to form and often result in a moreproductive and supportive learning environment,where individual strengths are recognised anddifficulties are addressed. Smaller classesare possible due to the additional funding thatBrewarrina Central School receives. Classesare smaller in the early years and progressivelyincrease in size as the age of the studentsincrease. This process is seen to support theneeds of students in the infant (early) years ofschooling as they transition into school, learnschool behaviours, and come to understandmany important concepts in mathematics (andliteracy).Hands-onAll classes actively promote the use of handson learning experiences. This is made possiblethrough the small group activities. Within anygroup, the students will work on similar activitiesthat are tailored for individual learners. Activitiesrange in diversity whilst using similar materialssuch as cards and dice. The degree of difficultyfor the learning opportunity is dependent on thespecific learning intention for each group andprevious experiences and concepts taught.To enable the whole school approach withhands on learning, the school has developed asubstantial resource room. The room is locatedin the library and staff are able to collect and usethe resources through the borrowing system.There are multiple copies of most resources sothat staff are able to cater for all students withinthe class.
Deconstructing Word ProblemsWith the move towards more problem-basedmathematics questions, there is a greater needfor students to be able to decode word problemsand decipher the meaning and mathematicalprocesses needed to be able to correctlyanswer questions. Based on Newman’s model,teachers use a five step process to specificallyteach students how to solve word problems. Thisincludes both the literacy and numeracy demandof the question.StepsExample1. Read the problemCan the students read the words?Students read the problem as a group or wholeclass. Words are discussed as to how they aresaid, or what they mean.Mary went to the supermarket and bought 5apples and 6 pears. How many pieces of fruitwere in her basket?2. Decipher the meaning of the questionWhat is the question asking you to do?Collectively (small groups or whole class), theclass discusses what the question is actuallyasking of themCan the students make sense of what thequestion is asking them to do?3. Identifying what the mathematicalprocesses involve in order to solve thequestion/problemSo, what do we need to do to find out theanswer? Can the students identify themathematical process required?As a group, there is discussion with regard towhat the mathematical demands are in relationto the question and how the question couldsolved. E.g. Identification that this is the additionof the two numbers in the word problem.We need to add how many apples and how manypears there are altogether.4. Undertaking the mathematicsSo, how do we do that?Doing the mathematics that is needed tocalculate the response that is required. Herethe students actually do the mathematics thatis required. (Adding the two numbers to find theresult).5 65. Constructing the correct response forthe answerSo what do we have? 11, but 11 what?What was the question asking about - finding outhow many apples were in the basket? , So theanswer needs to be ‘x’ apples.We have to work out many pieces of fruit are inthe basket.Various strategies could be used – such ascounters, fruit, counting on, etc depending onwhat the students have been doing or how theyprefer to work.Need the students to be able to interpret what the11 means – in this case, 11 pieces of fruit.
Consolidation and Fluency with NumberThere are many differences between the home,community and school in terms of numeracy practices,and the impact that this can have on the retentionand automaticity with number (facts, place value,operations, etc), mathematics lessons place a priorityon developing strength in number study.Lessons begin with number facts and are targeted foreach stage and students. Many strategies are usedas much for diversity in experiences as for catering forstudent needs. Drawing on assessments of students,the activities will appear to have a sameness to themfor the class but each activity will be different basedon student assessments. This approach keeps theclass together as a whole so that students do not feelsingled out and also meets individual students’ needs.For example, students might be making the largestnumber using a deck of cards. Differentiation occurswhen one student might be doing two digit numbers,another 4 digit numbers, while another might be usingdecimals. Similarly, students may be using dominoeswith some students counting the dots, some recordingthe dominoes on paper and counting how many andmaking an addition sentence using numbers (notdots) while another student/group might be doingmultiplication. From a class perspective, it appearsthat they are all using dominoes and unaware of thedifferentiation that has occurred.Teachers employ many resources to teach numberincluding games using cards and dice, which areparticularly popular given their familiarity with theresources. Resource packs containing dice, cards andcounters are sent home during the year to encouragelearning at home. Many other number activities areavailable through the vast array of resources in thelibrary.
Monitoring (and Celebrating)Mathematical LearningTeachers assess students every 5 weeks and thenplot data against the PLAN framework as requiredfor all EAfS schools and governed by Department ofEducation. The teachers record student performanceagainst the growth points on the continua and thenuse this data to monitor growth as well as to provideinsights into where to take learners from their currentlevels of achievement. The framework providesteachers with a solid means by which to show growthin learning. If a student is not progressing in a mannerthat would be expected, as shown through themapping process, then further support is provided.This support can be for the teacher, to help him/herdevise other activities that will scaffold learning, or itmay be additional support for the student.Depending on the ages of the students, theassessment tools will vary. In the early years ofschooling, teachers use observational methods anddocument these into their records, as well as formalindividual testing. Older students are assessed using awider array of tools including observations through tomini, focused assessments.Teachers meet weekly with the Instructional Leader(K-2) or Deputy Principal (3-6) to discuss studentprogress and strategies to move the students forwardin their mathematics learning.IN most rooms, student data is displayed on walls sothat students can see where they are on the learningcontinuum and know where they are expected to growin terms of mathematics. Students are provided with an“I can” book in which their learning is mapped againstthe continua. The books contain learning growth pointsfrom the continua expressed in language accessibleto students and families. Students mark off what theyhave learned during conferences and teachers willthen confirm success in that learning cluster. Once alllearning outcomes have been achieved within a bandor level of learning, students can be marked off ashaving achieved that level of learning. This processmakes the learning intent and success transparent tostudents and families whilst providing teachers with arecord of learning progress.
Extra Support forInfant LearnersThe program for mathematics is relatively consistentacross the K-6 school. However, it is recognised thatby stronger intervention in the early years (infant),the gap in learning between the students’ currentlevels and benchmarks could be narrowed with moretime spent on numeracy. To this end, the school hasdeveloped a program for the late afternoon (2pm)for 20 mins where the students are grouped bythe levels of achievement and focussed teachingoccurs commensurate with the levels of students’mathematics. This occurs in Terms 2, 3 and 4. Likeother lessons across the school, the focus is on theuse of hands-on activities but with very targetedlearning activities for the whole group.InstructionalLeadersAs part of the Connected Community strategy and asan Early Action for Success school, an InstructionalLeader was appointed in 2014. This role entailsworking with the teachers to develop quality programsfor the students. The role also includes buildingteacher capacity by working with teachers on planningand assessment, particularly in relation to diagnosisand then moving students into the next phase oflearning, as well as training staff in TEN. There isa K-2 Instructional Leader who works with Infantsclasses and staff, as well as Primary staff in afterschool team meetings. This role has enabled thewhole school approach to be adopted and a moveaway from text books to the hands-on approachthat was evident across the school. Teachers meetfortnightly in their stage teams (K-2 and 3-6) forplanning and professional learning. These meetingsare guided by the Instructional Leader and DeputyPrincipal and allow for a collaborative approach toteaching and learning.
Benefits for Learning and LearnersAboriginal Educators are integral to the success of teaching andlearning in the classroom. They provide a conduit between schooland community, and actively encourage students to do their best,all the time.The hands-on, activity-based approach used in mathematicslessons engage the learners and provide students withopportunities to participate in their learning. Students enjoylearning through play and discussion.With the increasing numbers of language-based problemsbeing posed in mathematics classrooms, often under theguise of being more practical or grounded in the worldbeyond schools, there is a greater need for students to beable to read and correctly interpret and respond tothese questions. This is increasingly the case inNAPLAN testing as well. The use of Newman’sAnalysis within classes enables students to haveways of working that empower them and increaseconfidence needed to tackle and solve wordproblems.Regular assessment-for-learning provides teacherswith data upon which to see success and growth,while simultaneously providing teachers witha benchmark from which to developstudents’ future learning.Advice for TeachersIt is imperative to create learning situations that meetthe needs of the learners. This can be pedagogicalas well as mathematical. Learning styles - as wellas areas of concern (such as the literacy demandof numeracy) should be built into the mathematicsprogram. Families and community members canprovide advice on the ways in which their children’slearning can be catered.Having a consistent approach across the schoolbuilds consistency and transparency for students(and community members) so that they know what toexpect when they commence mathematics lessons.This minimises lost time as students can be preparedupon entry into the mathematics lesson and ready toengage with learning.
Model for Quality LearningPrincipleA whole schoolapproach tomathematicsAboriginaleducators are anintegral part of theschoolInstructional leaderImplications for mathematics Students all take mathematicsin the second session of theday, and the approach is similaracross all classrooms.Aboriginal educators are seenas ‘teachers’ and are providedwith professional learning andsupport to be a valuable part ofthe classroom.Instructional leader appointed tosupport the infants (K-2).Focused strategies Use of hands-on, practical, games approach inthe classroom Use digital tools to engage learners Use of group work Quick pacing of lessonsAboriginal educators: work as a teaching partner supervise a small group in each rotation are provided time to work with the teacher in theplanning of mathematics lessons and their rolein the activities are provided with professional learning activitiesto help their development as teaching partners.Instructional leader: provide professional learning to teachers andaides for mathematics assist in the design, interpretation and follow onfrom student assessments provide teachers with feedback on lessons.
Key Messages – SummaryBrewarrina families support pedagogies that suit theirchildren’s preferred learning styles. Group work andhands-on activities were high on the list and these arefeatured strongly in the teaching of mathematics.A high percentage of the staff are local people (ieIndigenous) who work as qualified teachers, (includingthe assistant principal and one teacher), teachersupport staff, liaison staff and general employees inthe school. Teachers and support staff are providedwith at least 30 minutes preparation time each weekin which the class teacher can prepare for lessonsand ensure that the support staff feels supported andconfident in the activities that they will be working withthe students.Over the past few years, support staff havebeen provided with professional developmentopportunities.Being consistent and transparent with teachingmathematics is a priority. Students need to knowwhat to expect when they enter a mathematicslesson. Knowing what is going to happen allowsthem to go into a lesson ready to learn rather thanwondering what will happen.Literacy demands of numeracy need to be explicitlytaught. Providing students with a framework throughwhich they can breakdown a word problem enablesbetter access to the demands implicit in the task.This may be important for NAPLAN, but it is avaluable skill for mathematics in general.School Demographics (Brewarrina Central School)Year rangeU, K-12FTE teaching staff17.6Total enrolments160Non-teaching staff9LocationRemoteFTE non-teaching staff8ICSEA (school)655Indigenous students %97%Enrolments: Girls/Boys85/75ICSEA (distribution of students) 86% 10% 4% 0%(bottom quarter to top quarter)Teaching staff16Language background other than English 2%Student attendance rate %Contact: Robyn Jorgensen robyn.jorgensen@canberra.edu.au86%
Mathematics Pedagogy Brewarrina Central School Stories on remote indigenous mathematics successes compiled by Professor Robyn Jorgensen 2016 What makes for successful numeracy education in remote Indigenous contexts: An ethnographic case study approach Brewarrina is a town loca
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