young australian indigenous students

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  • 8/2/2019 Young Australian Indigenous Students



    4 . . . . . . . . . . . . . . . ... .. .. .. .. .. . . .. .. . . .. .. .. .. .. .. .. . . . . . . . . . . . . . . . .ENGAGEMENT WITH MATHEMATICS


    APMC 15 (1) 2010

    I n 2008, we worked with a range ofschools to trial ideas that assistedyoung Indigenous students to engagewith mathematics as they enter the schoolenvironment. Th e schools were locatedin North Queensland and the Brisbanemetropolitan area. In three of the schools,all of the students were Indigenous. Theparticipants were in the Prep and Year 1classes. The project was designed to:

    take into account pedagogy thatsupports Indigenous students'learning;

    develop learning activities that fosterdeep understandings of mathematics;and

    enhance students' engagement withmathematical learning.

    This article highlights some key pedagogicalstrategies that assisted classroom teachers toimprove Indigenous students' understandingof mathematics, particularly in the areaof number. The paper is organised underthree main sections, namely, communicatingmathematics, representing mathematics,and early number. Each section begins witha brief summary of the research whichunderpinned the project, followed byexamples of learning activities used inthe projects, an d finally a discussion ofstudents' responses.

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    Young Australian Indigenous Students: Engagement with Mathematics in the Early Years

    Communicating mathematics: Orallanguage - listening and explainingThe use of spoken language in school andthe types of interactions teachers utilise caneither advantage or disadvantage Indigenousstudents. Furthermore, the importance ofspoken language as the foundation for alllearning is often not fully recognised andmany young Indigenous students are no table to make a strong start in the earlyyears of schooling because the discourse oftheir family often does not match that ofthe school (Cairney, 2003). This mismatchof home and school language has beenshown to disadvantage Indigenous students'in terms of their achievements in literacyand numeracy in the long term (Dickinson,McCabe & Essex, 2006; MCEETIA, 2004).

    It is also well recognised that oralcommunication is dominant in the livesof Indigenous students and that theirexperiences with print and other literaciesis often limited. Patterns of classroominteractions have been shown to disadvantagesome students particularly that of teacherquestioning, because Indigenous studentsdo not commonly experience this typeof interaction at home or within theircommunity. Understanding and acceptingAboriginal English (AE) as a dialect ofspokenEnglish used by most Aboriginal and TorresStrait Islander people is vital, and knowingthat there are variations across particularcommunities is important (Haig, Konisberg& Collard, 2005). Although StandardAustralian English (SAE) is the discourse ofthe school, teachers need to create a bridgefor young Indigenous students between AEand SAE as they grapple with new language,new concepts and vocabulary presentedfor numeracy. The focus we took in theseclassrooms was, therefore, an oral approach,involving young Indigenous studentslistening to mathematical conversationsand explaining their understandingof mathematics in a supportive learningenvironment. With these young students, our

    aim was to develop the use of mathematicallanguage in a focussed play-based context.For example, describing numbers inmathematics entails relating their positionsto other numbers. This involves a veryspecific understanding of words such as"between," ~ ' n e x t to," "how far," "one morethan," "two less than." Many of the studentsinvolved in our project entered schoolwith little understanding of these types ofwords (Warren & deVries, 2009). Figure 1illustrates the type of hands-on activities usedin the classroom to assist students to beginto experiment with the use of positionallanguage as they described objects in a realworld context.

    Anyone for Breakfast?OveraIl activityStudents place the various breakfastitems in front of them. according to' thep()sitionallanguage used.Students wiU be: '.

    acting out positional languageaccording to the instructiongiven; matching theiteni's .position tolanguage.

    Language structure.o f sentences On the table, in frontoE you,place the plate: Beside your plate, putyoltl:' fork. Between your sausages, putanegg:

    Figure 1. Typical activity used for building positional language.

    APMC 15 (1) 2010 5PJn.ww

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    Warren & deVries

    As the project progressed, teachers foundthat the Indigenous students' willingness toengage in conversations about mathematicalcontexts increased. As their familiarity withthe language of mathematics increased,their "stories" about mathematical contextsalso became more complex. This increasedvocabulary also impacted on the types ofstories they shared in literacy blocks. Forexample, instead of simply saying, "The eggis on the plate," they were now sharing that,"The egg is in the middle of the plate. Besidethe egg there is a sausage, which has twochips on top of it."

    Representing mathematics:Kinaesthetic and visual strategiesRepresentations are essential to students'understanding of mathematics. They allowstudents to communicate mathematicalideas and understanding about concepts tothemselves and to others. For example, forstudents to become deeply knowledgeableabout number, they need to see numbersrepresented in a variety of ways: as a setof objects, as different lengths and areas,as bars on graphs, and as distances onnumber lines. Our approach was to use theserepresentations as we investigated numbersin a way that was hands on, visual andincorporated kinaesthetic learning (Warren,Young & deVries, 2008). Each representationbrought a new perspective to the conceptand a new set of mathematical language.Figures 2 and 3 shows two of the activitiesthat supported this approach

    Figure 2. Activity 1: Umbrellas on the number line.

    6 APMC 15 (1) 2010

    Figure 3. Activity 2: Placing eggs in cartons (ten frames).For Activity 1, students were given a

    cardboard umbrella shape with a numberon it. They were asked to "hang" theirumbrella on the skipping rope and sharehow they knew where the umbrella went.This activity supported the notion of numberas length-the length from the start of thestring to where the umbrella is placed. It alsoassisted the development of young students'understanding of proportion. The discussioncontinued to explore the idea of numbersbeing evenly spaced along the skipping rope.For Activity 2, students took turns to roll thedice and place plastic eggs in the carton. Th ecarton comprised 10 spaces in which to placethe eggs. With each turn, students were askedto create stories about the numbers of eggsin the carton. Students were encouraged toshare stories that included discussing howmany eggs were in the carton before they hadplaced their eggs, how many eggs were in thecarton after they had placed their eggs, andhow many more eggs they needed to maketen. A typical response to this activity was:"There are six eggs. We need four more tomake ten. I put two in. We now have eighteggs. We need two more."

    We also acknowledged that two keyrepresentations underpin mathematicalunderstanding and communicationthroughout all levels of mathematics; grids(graphical displays) and number lines.Unfortunately, in mathematics instruction,these have tended to be introduced tostudents in very formal contexts and taughtas an en d in themselves. It is important forstudents to represent their mathematicalideas in ways that make sense to them. Itis alsoimportant that they learn the conventionalforms of representations to facilitate their

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    Young Australian Indigenous Students: Engagement with Mathematics in the Early Years

    development of mathematical understandingand their communication with others aboutmathematical ideas.

    We introduced grids (see Figure 4) toyoung students as mats on the floor that werelarge enough for them to stand on physicallyand "be the numbers" themselves. Studentswere also encouraged to create patterns onthe grids that supported the exploration of theconventions of number charts, and create ba rgraphs. In past research (Warren & Cooper,2002) we have found that many studentsexperience difficulties with the structure ofthe hundreds board and examining diagonalpatterns (e.g., the patterns formed by themultiples of 3) on these boards. Figures 4and 5 illustrate two activities that were usedwith the grid.

    * t : , . *L ..* t : , ' t ) *t: , ' t )*t: , .* 6. 't) * :, Figure 4. Activity 3~ = = = = = = = = = : ; Patterning on the grid.

    Figure 5. Activity 4: Creating bar graphs on the floor mat.

    These activities are designed to allowyoung students to engage kinaestheticallyand visually with patterns on diagonals, thezigzag construction of the number board,

    and the construction of bar graphs beforeformalising the board in a number contextand a graphing context. Underpinning thisengagement was the continual emphasis onoral language, asking questions such as:

    What shape comes next? What shapes are between the hearts? What patterns can you see? How many more pieces of cheese are

    there than mushrooms? How many pieces of cheese and

    mushrooms are there altogether?Some typical student responses were: "There is more pepperoni than

    cheese." "The smallest one is onion." "There is the same amount of ham as

    there is mushrooms."

    Early number: Counting and subitisingstrategiesA major focus of mathematics in earlyyears' contexts is the development of anunderstanding of number. The literatureidentifies two theories ofnumber development(e.g., Gelman & Gallistel, 1978). The firststresses the role of counting. This theory isgrounded on the idea of pre-consciousnessof counting principles. In this theory, inthe preverbal stage, young students' focuson a group of items is upon gauging itsmagnitude, that is, how many objects thereare. Thus the acquisition of the first fewnumber words is achieved by mapping theword onto the magnitudes they have alreadyregistered before they can talk. Things arequantified by counting.

    Th e second relies upon the recognition ofdifference using perceptual or spatiotemporalcues - cues that are not numerical.Fundamental to this theory is the notion ofsubitising, the ability to quantifY somethingwithout really counting (either internally orexternally). Instead, things are quantified bylooking, allowing the number of objects ina small collection to be determined rapidly

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    Warren & deVries

    and accurately. The ability to subitise isno t based on preverbal counting (or evenfast counting), and is commonly limitedto no more than four objects. Research(e.g., Treacy & Frid, 2008; Willis, 2000)has shown that Indigenous students have anatural ability to subitise. The results of ourresearch indicate that this is not necessarilythe case. In fact, their ability as they enterschool is similar to non-Indigenous students.Our research also showed that the abilityto subitise improves with intervention, andit appears that no intervention results inlimited ability (Warren & deVries, in press).It is, therefore, important that teachers focusin the early years on creating activities thatassist students to learn to count and tosubitise.

    We used both of these approaches in ourclassrooms. Figures 6 and 7 show typicalsubitising activities used in the prqjectclassrooms

    Figure 6. Activity 5: Swatting flies.

    Figure 7. Activity 6: Dot concentration.

    8 APMC 15 (1) 2010

    In Activity 5, the students sat in a circlewith fly swats and the teacher put out flies andquickly called a number. The person who firstswatted the fly with that number of dots on itsback won the fly. The activity was extended toinclude questions such as, "Which numberis the one just after 3." This activity cateredfor different ability groups with the teachersmaking more use of larger numbers on theflies' backs as students' capabilities increased.Students were also encouraged to share howthey could recognise 10 dots and explain thepatterns that they could see. Activity 6 was aconcentration game with students flippingtwo cards and matching cards with the samenumbers of dots.Students simultaneously engaged withcounting activities and subitising activities,bu t it was the latter that they found mostcaptivating. Engaging with the different visualrepresentations of numbers up to 10 allowedthem not only to "guess" what the numberwas but also to talk about the numbers thatthey could see on the cards. For example,for a random dot pattern for the number 6,some typical responses were: "I can see fourdots and two dots," and "I can see three dotsand three dots."

    ConclusionPast research has suggested that success forethnically diverse students is strongly linkedto culturally responsive and empatheticteaching (Gay, 2002). Such teaching consistsof two key components, warmth and demand(Fanshawe, 1999). However, warm anddemanding "has been interpreted by manyteachers as entailing warm relationships anddemanding compliant behaviour as opposedto supporting intellectual or academic rigour"(QIECB, 2003, p. 12). Teaching Indigenousstudents entails more than an awareness oftheir culture. It requires attention to diversityin terms of both curriculum and instruction.Important to teaching Indigenous studentsis the recognition that they view learning as


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    Young Australian Indigenous Students: Engagement with Mathematics in the Early Years

    a social process. All actIVItIeS presented inthis article reflect pedagogy that supportsyoung Indigenous student learning, namely,emphasising practical experience, hands onactivities, group cooperation, and students'engagement. It also recognises that thelanguage of school is different from thelanguage of home. Added to this complexityis the introduction of the language ofmathematics.

    We are proposing that the language andrepresenta tions used to express mathematicalideas are complex. All students need theopportunity to play with this language ina supportive environment, allowing themto build their confidence and capacity touse this language to support their learning.Therefore many of the suggestions andexamples presented in this article would bebeneficial to non-Indigenous students also asthey begin their mathematical journeys.

    ReferencesCairney, T. (2003). Literacy within family life. In N.

    Hall, J. Larson & J. Marsh (Eds), Handbook ojearly childhood literacy (pp. 85-98). London: SagePublications.Dickinson, D., McCabe, A. & Essex, MJ. (2006). Awindow of opportunity we must open to all: The

    case for preschool 'With high-quality support forlanguage and literacy. In D. Dickinson & B. Neuman(Eds), Handbook oj early literacy research (Vol. 2, pp.11-28). New York: The Guilford Press.

    Fanshawe, J . P. (1999). Warmth, demandingness, andwhat else? A reassessment of what it takes to be aneffective teacher of Aboriginal and Torres StraitIsland children. Australian Journal oj IndigenousEducation, 27(2), 41-46.

    Gay, G. (2002). Preparing for culturally responsiveteaching. Journal oj Teacher Education, 53(2),106-116.Gelman, R. & Gallistel, C. R. (1978). The child'sunderstanding oj number. Cambridge, MA: HarvardUniversity Press.

    Haig, Y, Konisberg, P. & Collard, G. (2005). Teachingstudents who speak Aboriginal English. PloW 150.Primary English Teaching Association.Ministerial Council on Education, Employment,Training and Youth Affairs [MCEETYA] (2004).Preliminary paper national benchmark results - 11?ading,writing and numeracy: Yean 3, 5 and 7. Retrieved1 May 2005 from: _resources/ANR2004BrnrksFina1.pdfQueensland Indigenous Education Consultative Body[QIECB]. (2003). Position paper on schooling andteacher education. Brisbane: QIECB.Treacy, K. & Frid, S. (2008). Recognising differentstarting points in Aboriginal students' learningof number. In M. Goos, R. Brown, and K Maker(Eds), Proceedings of the 31st annual conference of theMathematics Education Research Group of Australasia(pp. 531-537). Brisbane: MERGA.Warren, E. & Cooper T (2002). Arithmetic and quasivariables: A Year 2 lesson to introduce algebra in theearly years. In B. Barton, KIrwin, M. Pfannkuchand M.Thomas (Eds.), Navigating currents andcharting directions (Proceedings of the 25th annualconference of the Mathematics Education ResearchGroup of Australasia, Vol. 2, pp. 673-681). Brisbane:MERGA.Warren, E. & deVries, E. (2009). Young AustralianIndigenous students' engagement with numeracy:Actions that assist to bridge the gap. AustralianJournal oJEducation, 53(2),159-175.Warren, E. & deVries, E. (in press). Closing the gap:Myths and truths behind subitisation. AustralianJournal ojEarly Childhood Education.Warren, E., Young,J. & deVries, E. (2008). The impactof early numeracy engagement on 4 year oldIndigenous students. Australian Journal of EarlyChildhood Education, 33(4), 2-8.

    Willis, S. (2000). Strengthening numeracy: Reducingrisk. Paper presented at the Australian Councilof Education Research conference. Improvingnumeracy learning. What does the research tell us?Brisbane: ACER.

    Elizabeth WarrenAustralian Catholic UniversityEva deVriesIndependent Schools Queensland

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