Numeracy In The Middle Years Curriculum
Numeracy in theMiddle Years CurriculumA resource paper—An audit of numeracy in the SACSA Framework
AcknowledgmentsMerrilyn Goos, The University of QueenslandProf Merrilyn GoosDirector, Teaching and EducationalDevelopment InstituteThe University of QueenslandShelley Dole, The University of QueenslandDr Shelley DoleSenior Lecturer, School of EducationThe University of QueenslandVince Geiger, Australian Catholic UniversityMr Vince GeigerLecturer, School of Education, McAuley CampusAustralian Catholic UniversityDECS Curriculum ServicesPlease note:This document is an abridged version of the fullaudit. The full version includes further elaborationsfor each Learning Area and can be accessedat the DECS website for The Network—SouthAustralian Literacy and Numeracy Network at www.thenetwork.sa.edu.au . 2009 The State of South Australia, Departmentof Education and Children’s ServicesEdited by: Gunta GrovesDesigned by: She Creative Pty LtdPrinted by XXXXXXX, South Australia
ContentsIntroduction1Arts4Design and technology6English8Health and physical education10Languages12Science14Society and environment16Mathematics18Summary20Planning resource: Identifying numeracydemands in curriculum22Bibliography23
IntroductionThis audit was commissioned by the Department ofEducation and Children’s Services (DECS) to supportthe work of teachers in the Numeracy in the LearningAreas (Middle Years) Project during 2009. Its purposewas to identify the numeracy demands of the eightLearning Areas of the South Australian Curriculum,Standards and Accountability (SACSA) Frameworkand elaborate on the ways that numeracy is embeddedin each Learning Area.The SACSA Framework describes curriculum Key Ideasand Outcomes upon which learners from birth to Year12 in DECS schools, preschools and childcare settingscan expect their education to be built. These Key Ideasand Outcomes provide the basis for planning, teachingand assessing the curriculum requirements of educators’specialist or focus areas.This audit has been developed as a support for all middleyears educators as they extend their personal understanding of numeracy generally and become aware ofthe numeracy demands of their areas of specialisationor program focus. It is critical that all educators cometo understand the numeracy demands of all curriculumareas so they are able to better support improvedlearning outcomes for their students and children.The important roles of mathematics and mathematicsteachers are critical in the development of the foundationsfor numeracy and, therefore, the curriculum strands ofmathematics are used to highlight the numeracy demandsembedded in other curriculum contexts.Defining numeracyThe National numeracy review report (COAG, 2008)made the following recommendation:That all systems and schools recognise that, whilemathematics can be taught in the context ofmathematics lessons, the development of numeracyrequires experience in the use of mathematics beyondthe mathematics classroom, and hence requires anacross the curriculum commitment. Both pre- andin-service teacher education should thus recogniseand prepare all teachers as teachers of numeracy,acknowledging that this may in some cases be‘subject specific numeracy’. (p xii)Yet there is a limited research base into the numeracydemands of learning areas outside of mathematics.Numeracy is a term common in Australia and the UnitedKingdom but rarely found in America or other parts ofthe world, where expressions like quantitative literacyor mathematical literacy are used. Some definitions ofquantitative literacy focus on the ability to use quantitativetools for everyday practical purposes, while mathematicalliteracy is understood more broadly as the capacity toengage with mathematics in order to act in the worldas a constructive, concerned and reflective citizen(OECD, 2003).The 1997 Numeracy Education Strategy DevelopmentConference identified the following elements as centralto any description of numeracy: ‘numeracy involvesusing some mathematics to achieve some purposein a particular context’ (DEETYA, 1997, p 13). From thisdiscussion emerged the following description of numeracywhich it was hoped would inform future work in numeracyeducation: ‘To be numerate is to use mathematicseffectively to meet the general demands of life at home,in paid work, and for participation in community andcivic life’ (DEETYA, 1997, p 15, emphasis added).This description was later cited in the Commonwealth’snumeracy policy document Numeracy, a priority for all:Challenges for Australian schools (DETYA, 2000) asreflecting the Australian interpretation of numeracy.DRAFT Numeracy in the Middle Years Curriculum A resource paper—An audit of numeracy in the SACSA Framework1
IntroductionA numeracy modelRecently, however, Goos (2007) has argued that a broaderdescription of numeracy for new times is needed in orderto capture the rapidly evolving nature of knowledge,work, technology, social structures, and the changingcharacteristics of learners in schools. She developedthe model shown in Figure 1 to represent the dynamicand multifaceted nature of numeracy, which comprisesfour elements. A numerate person requires mathematical knowledge.This includes concepts, skills and problem-solvingstrategies, as well as the ability to use sensibleestimations.Different curriculum contexts also have distinctivenumeracy demands, so that students need to benumerate across the range of contexts in whichtheir learning takes place at school. This model is grounded in a critical orientationto numeracy since numerate people not only knowand use efficient methods, they also evaluate thereasonableness of the results obtained and are awareof uses of mathematical thinking to analyse situationsand draw conclusions.The tetrahedral model of numeracy is used as the basisfor this numeracy audit of the South Australian Curriculum,Standards and Accountability (SACSA) Framework. A numerate person also has positive dispositions—a willingness and confidence to engage with tasks,independently and in collaboration with others, andapply his/her mathematical knowledge in flexibleand adaptable ways.2CRITICAL nitiativeRiskRepresentationalPhysicalDigital Numerate practice often involves using tools. Theseinclude: physical tools, representational tools anddigital tools (technology). Because numeracy is about using mathematics to actin and on the world, people need to be numerate ina range of contexts. A numerate person can organisehis/her personal finances, personal health needs andleisure time, and think systematically about all publicissues which are often presented based on data andprojections. In addition, work-related numeracies arealways contextual.ContextsCitizenshipWorkPersonal and SocialMathematicalKnowledgeFigure 1 –A tetrahedralProblem SolvingEstimationConceptsSkillsmodel fornumeracy inthe 21st centuryThe numeracy model can be represented as a two-dimensionalshape (see Figure 1) or as a solid tetrahedron where the four facesrepresent Dispositions, Tools, Contexts and Mathematical Knowledgeand with Critical Orientation as the solid core of the tetrahedral model.DRAFT Numeracy in the Middle Years Curriculum A resource paper—An audit of numeracy in the SACSA Framework
IntroductionEvaluating the numeracydemands of Learning AreasDuring the middle years of school, students continueto consolidate the knowledge and skills acquired in theirearlier years and they need to see the relevance of whatthey are doing in the classroom to their lives beyond theschool. In addition, research has found that success isa major component in student preparedness to engagein mathematics in the middle years. So, planning fornumeracy development needs to incorporate manycontexts within mathematics, in other disciplines andin students’ lives outside school.This audit evaluates the distinctive numeracy demandsof all SACSA Framework Learning Areas in the MiddleYears Band (Years 6, 7, 8 and 9) and hence is alignedwith Standard 3 and Standard 4. Standard 5 hasalso been considered, as some Middle Years Bandstudents will be working within and towards this levelof achievement.Ways in which mathematics applies in othercurriculum contexts can be examined throughthe five strands of the mathematics Learning Areaof the SACSA Framework: exploring, analysing and modelling data measurement numberEach Learning Area has been considered in the contextof the SACSA Framework’s statement about numeracy,which is intended to be an embedded aspect of allcurriculum. The figures, which identify the numeracyemphasis in each Learning Area, use shading to indicatelow (unshaded), moderate (light shading) and high (darkshading) levels of numeracy learning demands.The audit concludes with a summary of findings acrossall Learning Areas to establish numeracy demands ofthe SACSA Framework as a whole. pattern and algebraic reasoning spatial sense and geometric reasoning.These mathematics strands have been used as a wayof organising the numeracy audit of Learning Areasother than mathematics.All Learning Areas have further examples and elaborationsin the full version of the audit of numeracy available at www.thenetwork.sa.edu.au .The SACSA Framework’s statement about how studentsdevelop their operational skills in numeracy in the LearningArea is presented first, and this is followed by a tablehighlighting the Learning Area strands that have mathematics strands embedded within the area. Numeracydemands are then evaluated by reference to each elementof the numeracy tetrahedron depicted in Figure 1.DRAFT Numeracy in the Middle Years Curriculum A resource paper—An audit of numeracy in the SACSA Framework3
ArtsCRITICAL tsCitizenshipWorkPersonal and SocialMathematicalKnowledgeProblem SolvingEstimationConceptsSkillsNumeracy evaluation in artsMathematical KnowledgeFigure 2 maps numeracy learning demands in arts onto the SACSAFramework’s Middle Years Band mathematics strands.Arts strandsSACSA numeracystatement4Learners develop and use operationalskills in numeracy to understand,analyse, critically respond to and usemathematics in different contexts.These understandings relate tomeasurement, spatial sense, patternsand algebra and data and number.This learning is evident in artswhen, for example, students designproducts using sequencing andpatterning, accurate measurementand a sense of shape, size, dimensionand perspective. Gathering, interpreting and analysing data in relation toaudience, viewer and user behaviouris another example of numeracy inarts. (DETE, 2001, Middle YearsBand, p 13)MathematicsstrandsArts practiceArts analysisand responseArtsin contextsExploring, analysingand modelling dataMeasurementNumberPattern andalgebraic reasoningSpatial sense andgeometric reasoningFigure 2 – Mathematical knowledge demands within strands of the arts Learning AreaLearning in arts makes many numeracy demands of students in termsof mathematical knowledge.In arts practice where students create arts works, students draw onelements of the mathematics strands in exploring and analysing data,and using measurement, number and spatial sense as well as patterns(which supports algebra understanding but may not promote algebraunderstanding specifically).In arts analysis and response where students analyse others’ arts works,they may conduct research that requires analysis of data and use spatialreasoning to consider elements of arts works.Through arts in contexts, students ‘understand the economic role of artsby making connections between creative, artistic endeavour and commercialuses of the arts ’ (ibid, p 29).DRAFT Numeracy in the Middle Years Curriculum A resource paper—An audit of numeracy in the SACSA Framework
Learning Area—ArtsContextsAs outlined in the SACSA Framework, the arts give voiceto thoughts, feelings and beliefs through the five main artforms: dance, drama, music, media and visual arts, and:Arts can be linked tomathematics through the studyof signs and symbolism, throughthe five main art forms: dance,drama, music, media andvisual arts.All styles of expression described by such terms astraditional, contemporary, popular, folk, commercialand fine arts are represented in arts works. Individualarts works can serve to maintain the status quo orchallenge assumptions, and critique social, cultural,economic and political practices. (ibid, p 9)The arts curriculum provides contexts for understandingsymbols and symbolism and to symbolise momentsof great importance. Arts can be linked to mathematicsthrough the study of signs and symbolism, and mathematics as semiosis has been well-considered in theliterature (see, for example, Anderson, Saenz-Ludlow,Zellweger & Cifarelli, 2003).DispositionsBecause of the extensive opportunities for developingnumeracy through arts, there is great potential to promotepositive dispositions for numeracy learning. Arts learningexperiences can be contextually rich for promotingnumeracy in an environment very different from thetraditional mathematics classroom. A major dispositionpromoted through the arts curriculum area is perseveranceand commitment, with students planning and creatingarts works—seeing arts works through from earlieststages to completion. Such a positive disposition supportsstudents as they engage in mathematical problem-solvinginvestigations. The arts Learning Area provides contextsfor developing confidence in the use of mathematics and,particularly, for developing perseverance in problem solving.ToolsStudents use a range of representational, physical anddigital tools in creating and responding to art works.In music, for example, they make notations to symboliserhythm, melody, harmony and tempo. In the visual arts,they work with materials, implements, images and mediato communicate and represent ideas in two and threedimensions. In drama, they make models of performancespaces when they study set design and constructionand consider the use of space to suit a scene in a play.They become familiar with print, film and electronic mediaas well as web-based resources such as virtual galleriesand libraries, and they use many types of digitaltechnologies to create new art forms.Critical OrientationArts has the potential to encourage students to movebeyond art works for self-expression to art works forsocial comment and critique. In this way, arts can be thevehicle through which students develop critical orientation;where they use mathematics tools to analyse facts andinformation about situations and issues. For example,banks have now introduced a range of extra feesassociated with the use of automatic teller machines(ATMs). News reports have brought this to the attentionof consumers. However, art works have the potential toensure that the injustice of this situation remains in thepublic mind. Considering this situation from a variety ofperspectives through application of numeracy (eg Howmuch will that cost the average consumer?, How muchdoes this give the banks?, How much profit did banksmake last year?) and incorporating this message withinart works has the potential to make a stronger statement.DRAFT Numeracy in the Middle Years Curriculum A resource paper—An audit of numeracy in the SACSA Framework5
Design and technologyCRITICAL tsNumeracy evaluation in designand technologyCitizenshipWorkPersonal and SocialMathematicalKnowledgeProblem SolvingEstimationConceptsSkillsMathematical KnowledgeFigure 3 maps numeracy learning demands in design and technologyonto the SACSA Framework’s Middle Years Band mathematics strands.Design and technology strandsSACSA numeracystatement6Learners develop and use operationalskills in numeracy to understand,analyse, critically respond to and usemathematics in different contexts.These understandings relate tomeasurement, spatial sense, patternsand algebra and data and number.This learning is evident in designand technology when, for example,students use spatial understandings,particularly the relationship betweenshape, structure and function toinform 2-D and 3-D design. Otherrelevant examples include studentsapplying accurate measurementand mathematical formulae whendesigning and constructing products.(DETE, 2001, Middle Years Band,pp gExploring, analysingand modelling dataMeasurementNumberPattern andalgebraic reasoningSpatial sense andgeometric reasoningFigure 3 – Mathematical knowledge demands within strands of the designand technology Learning Area‘Design and technology offers learners a rich blend of knowledge, skills,strategies and dispositions to develop their identities as individuals andto help them design shared, sustainable futures.’ (ibid, p 36)It is through the curriculum strands of designing and making that the greatestnumeracy demands are found. When designing, students consider designelements of products that they are intending to create. Through modelling,students can consider elements of their design and make modifications priorto making the product. In modelling, students construct a set of parametersfor their design and consider the impact of such parameters.When they are using mathematics to create 3-D objects from 2-D designs,they use measuring equipment with precision and consider the impact oferrors in measurement upon the final product or system.DRAFT Numeracy in the Middle Years Curriculum A resource paper—An audit of numeracy in the SACSA Framework
Learning Area—Design and technologyContextsNumeracy demands are embedded in the contexts fordesign and technology learning activities. Throughcritiquing existing designs and systems, students developnumeracy through analysing cost and architectural designand arrangement. They demonstrate numeracy whenthey analyse the positive and negative impact of productsand systems upon the environment, in terms of safetyand material selection. Through designing, studentsdemonstrate numeracy when they consider productimprovement, when they communicate their designsusing mathematics symbols, drawings and equations.When considering the cost of their designs, they usespreadsheets where mathematical formulae are utilisedto show impacts and design variations. In their designbrief, they show how their design will benefit particulargroups or users, and this analysis requires mathematicaltools and techniques. When they are engaged in making,students must adhere to timelines, which is an aspectof numeracy.DispositionsIn design and technology, students are exploring realsituations and creating new products. Because of thesubject’s link to their present and future lives, engagementin design and technology can promote positive numeracydispositions. Application of mathematics occurs in acontext, and application of mathematical tools andtechniques shows the utility of mathematics. Designand technology draws upon imagination and creativethinking. These processes are also desirable in the studyof mathematics. Promotion of these processes throughdesign and technology learning experiences encouragespositive dispositions to using mathematics in context.Through designing, studentsdemonstrate numeracy whenthey consider product improvement,when they communicate theirdesigns using mathematics symbols,drawings and equations.ToolsWhen critiquing products and systems, students drawupon mathematical tools of data analysis, number sense,estimation and mental computation. They consider form,which immerses them in shape investigation. Whendesigning, students create a design brief. They usemathematics to develop this design brief. They mayconsider a way of communicating their design brief thatinvolves scale modelling of their product. This will promoteunderstanding of proportional reasoning and scale. Theymay use a flow chart to show the stages in creation oftheir product. This relies on time understanding and timemanagement. Their design brief includes statementsabout the positive and negative impact of their product,and this requires analysis of data to justify their decisionmaking processes. In making, students use mathematicstools in context, finding out in authentic situations thingssuch as the importance of planning, of calculating andof exact measuring.Critical OrientationThe design and technology Learning Area promotescritical orientation. Through the critiquing strand, studentsconsider the impact of technologies existing in our society.As they design, they are empowered to effect changeby developing a repertoire of thinking skills and creativedispositions. They learn that there are rarely single or‘right’ answers when critiquing, designing or makingtechnology. (ibid, p 36)DRAFT Numeracy in the Middle Years Curriculum A resource paper—An audit of numeracy in the SACSA Framework7
EnglishCRITICAL tsCitizenshipWorkPersonal and SocialMathematicalKnowledgeProblem SolvingEstimationConceptsSkillsNumeracy evaluation in EnglishMathematical KnowledgeFigure 4 maps numeracy learning demands in English onto the SACSAFramework’s Middle Years Band mathematics strands.English strandsSACSA numeracystatement8Learners develop and use operationalskills in numeracy to understand,analyse, critically respond to and usemathematics in different contexts.These understandings relate tomeasurement, spatial sense, patternsand algebra and data and number.This learning is evident in Englishwhen students, for example, usegraphs and tables in oral and writtenpresentations to support an argumentor draw on understandings of spatialarrangement to understand andcritique text layout and construction.(DETE, 2001, Middle Years Band,p 66)MathematicsstrandsTexts andcontextsLanguageStrategiesExploring, analysingand modelling dataMeasurementNumberPattern andalgebraic reasoningSpatial sense andgeometric reasoningFigure 4 – Mathematical knowledge demands within strands of the EnglishLearning AreaThe English Learning Area, as described in the SACSA Framework,indicates only minimal numeracy demands and only in some aspectsof the mathematics strands.Through the strand of texts and contexts, students may interpret diagrams,turn texts into tabular form and use a variety of images to present researchfindings, which links to the mathematics strand of exploring, analysing andmodelling data. They examine past, present and future perspectives, whichprovides an avenue for developing measurement understanding in relationto time and timelines. They examine how the media influences or persuadesand explore the conventions used in advertising for this purpose. In thisway, students will develop spatial sense through exploration of signs andsymbols. In the language strand, students use spoken language to conveytheir message. Through consideration of rhythm, pace and timing, they areengaging in measurement. When they critically appraise choices of languageand modes of communication (eg camera shots), they are extending theirspatial sense and understanding of position and perspective. In the strategiesstrand, students may critically analyse statistics presented in speeches orother media and use tables to communicate or interpret. In this way, they areengaging in the mathematics strand of exploring, analysing and modelling data.DRAFT Numeracy in the Middle Years Curriculum A resource paper—An audit of numeracy in the SACSA Framework
Learning Area—EnglishNumeracy demands in the study of English are alsoevident when considering modes of communication andexpression, in a similar fashion as suggested throughlearning experiences described in arts. Additionally, in thestudy of texts and contexts in English, students analyseand explore everyday texts (eg instruction manuals,advertising brochures and campaigns, signs and signagein the built environment, maps, brochures) that may haveparticular numeracy demands.As students conduct researchfor their written texts, theyencounter numeracy demandsassociated with analysis of dataand statistics.ContextsAll three strands of the English curriculum have thepotential to provide contexts for numeracy development.The extent to which this occurs would be dependent upondecisions made by the teacher or students. Currently,the SACSA Framework for English provides few examplesof numeracy experiences, and this tends to underrepresent the numeracy demands of the Learning Area.For example, considering the publication date of a pieceof text can assist students develop understanding of timeand timelines. When students are involved in makingmagazines, greeting cards, pamphlets, arts works,promotional videos, websites, song lyrics, newspapersand advertisements, they are experiencing numeracydemands. And as students conduct research for theirwritten texts, they encounter numeracy demandsassociated with analysis of data and statistics.DispositionsIt is a sad fact that many people in our society will admitto being poor at tasks that involve mathematics. Yet, fewpeople will readily admit to having difficulties with literacy.Many students in the middle years of school have madedecisions about their capabilities in relation to Englishand mathematics and this impacts upon their personalviews of their own literacy and numeracy. ThroughEnglish learning activities having a greater focus onthe numeracy demands of this Learning Area, studentsmay have a greater opportunity to develop positivelytheir personal views about mathematics and numeracy.They will become more aware of the ways numeracyempowers and how this can be incorporated in anymedium to convey their message.ToolsIn English learning activities, students have the potentialto develop their tools for numeracy. Through texts andcontexts, language, and strategies, they can researchsituations and analyse real data. They can convert textto table form and consider how various forms of textscommunicate messages. When reading maps andinstruction manuals, students are considering pictorialtext and how this conveys messages.Critical OrientationThrough English learning activities, students developcritical literacy skills that enable them to challenge andquestion information presented to them. Using numeracycan support this critical disposition. Through analysis ofthe construction of texts, students explore informationthat is presented and consider information that is notpresented. When they are presented with text that containsstatistics and data, they can explore how they are beingpersuaded by the text-creator. Bringing numeracy tocritical analysis of texts supports development of a criticalorientation. A stronger focus on numeracy through Englishlearning activities may reduce the cultural perceptionthat being numerate is dependent upon success inschool mathematics.DRAFT Numeracy in the Middle Years Curriculum A resource paper—An audit of numeracy in the SACSA Framework9
Health and physical educationCRITICAL tsNumeracy evaluation in health andphysical educationCitizenshipWorkPersonal and SocialMathematicalKnowledgeProblem SolvingEstimationConceptsSkillsMathematical KnowledgeFigure 5 maps numeracy learning demands in health and physical educationstrands onto the SACSA Framework’s Middle Years Band mathematics strands.Health and physical education strandsSACSA numeracystatement10Learners develop and use operationalskills in numeracy to understand,analyse, critically respond to and usemathematics in different contexts.These understandings relate tomeasurement, spatial sense, patternsand algebra and data and number.This learning is evident in healthand physical education when, forexample, students use numbering,patterning and ordering in physicalactivities and when they showunderstanding of the relationshipbetween time, space and distanceto analyse physical achievement. It isalso evident when students comparespecific measuring techniques andtools for different purposes andanalyse data and statistics in relationto health issues. (DETE, 2001,Middle Years Band, pp 99–100)MathematicsstrandsPhysical activityPersonal andHealth of individualsand participation social development and communitiesExploring, analysingand modelling dataMeasurementNumberPattern andalgebraic reasoningSpatial sense andgeometric reasoningFigure 5 – Mathematical knowledge demands within strands of the health andphysical education Learning AreaThis Learning Area is rich in mathematical knowledge demands, particularlythrough measurement and the analysis and modelling of data. For example,students can measure and analyse the effects of physical activity on thebody, evaluate their food intake in relation to known health risks, and analysenutritional information on food labels.DRAFT Numeracy in the Middle Years Curriculum A resource paper—An audit of numeracy in the SACSA Framework
Learning Area—Health and physical educationContextsHealth and physical education deals with contexts of:Students can measure andanalyse the effects of physicalactivity on the body, evaluate theirfood intake in relation to knownhealth risks, and analyse nutritionalinformation on food labels. relationships, family, community and work environments It involves investigation of vocations, communityorganisations and global bodies that promote healthyactive living. This occurs through contact with the food,hospitality, health fitness, sport and recreation industries and involvement in everyday recreation and healthissues (ibid, p 96).In this Learning Area, then, it is vital to engage learnerswith the world beyond school in ways that connect withtheir personal, community and vocational interests.DispositionsThe health and physical education Learning Area aimsto promo
Printed by XXXXXXX , South Australia. Contents Introduction 1 Arts 4 Design and technology 6 English 8 Health and physical education 10 Languages 12 Science 14 Society and environment 16 Mathematics 18 Summary 20 Planning resource: Identifying numeracy 22 demands in curriculum Bibliogr
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