Starting points and pathways in Aboriginal students

Math Ed Res J DOI 10.1007/s13394-014-0123-x O R I G I N A L A RT I C L E

Starting points and pathways in Aboriginal students’ learning of number: recognising different world views Kaye Treacy & Sandra Frid & Lorraine Jacob

Received: 21 November 2013 / Revised: 3 April 2014 / Accepted: 9 April 2014 # Mathematics Education Research Group of Australasia, Inc. 2014

Abstract This research was designed to investigate the conceptualisations and thinking strategies Indigenous Australian students use in counting tasks. Eighteen Aboriginal students, in years 1 to 11 at a remote community school, were interviewed using standard counting tasks and a ‘counting’ task that involved fetching ‘maku’ (witchetty grubs) to have enough to give a maku to each person in a picture. The tasks were developed with, and the interviews conducted by, an Aboriginal research assistant, to ensure appropriate cultural and language contexts. A main finding was that most of the students did not see the need to use counting to make equivalent sets, even though they were able to demonstrate standard counting skills. The findings highlight a need to further examine the world views, orientations and related mathematical concepts and processes that Indigenous students bring to school. Keywords Indigenous . Culturally responsive . Number . Counting . Subitising . Part-whole Frequently, a range of historical, cultural, social, demographic and curricular factors have been suggested as contributors to the underachievement of Australian Indigenous students. In the perspective in which they are framed, these factors implicitly adopt a deficit model of culture, knowledge and skills that does not provide pathways for the consideration of Australian Indigenous students’ inherent potentials as mathematics learners (Jorgensen (Zevenbergen) et al. 2010). Broader perspectives incorporating K. Treacy : S. Frid SiMERR, Perth, WA, Australia K. Treacy e-mail: [email protected] S. Frid e-mail: [email protected] L. Jacob (*) Murdoch University, Perth, Australia e-mail: [email protected]

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ethnomathematics have been utilised to create mathematics curricula deemed as more suitable for Indigenous learners than traditional Western mathematics, but these approaches also imply a deficit model of learning because they implicitly identify Western mathematics as being of higher status (Jorgensen (Zevenbergen) et al. 2010). There are perspectives however that reject the notion that the problem of underachievement lay within the students, their culture and communities. Such research has focused on Indigenous mathematics learning from the perspective of mathematics as a cultural practice in which people “seek to understand and make sense of their environment and their practices through identifying patterns that exist in organisation” (Perso 2003, p. 11). A view of mathematics as a cultural practice recognises that ‘mathematics’ is not a singular, unique entity. It is a way of interpreting and acting within one’s environment—socially, physically and spiritually—that is integrally connected to one’s world view. Although it is difficult to clearly define and detail the concept of a ‘world view’, it is in general about the “way individuals and groups see the world and how they understand that they should behave within it” (Harris 1991, p. 13). A world view can therefore be seen as linked to a person’s or group’s values as well as their actions within their daily lives. Malcolm et al. (1999) suggested that Aboriginal children develop very early, an understanding of the sophisticated complex family and social networks making up their world. Family and kinship relationships are a central component of their cultural background and the ways they interpret and function in their world, including the ways they approach ‘mathematical’ tasks. ‘Western mathematics’ has its origins in abstraction and autonomous existence of concepts and is oriented by a valuing of separation and objectivity in relation to the world. By contrast, an Indigenous world view generates a ‘mathematics’ that is “characterised by a very personal view of the universe in which humans are seen as united with nature rather than separate from it” (Harris 1991, p. 130). This mathematics is built from “ordered ways of naming and construing the relationships of natural things according to their perceived ancestral or familial linkages” (Watson and Chambers 1989, p. 31). Owens (2013) exemplifies these ideas with regard to spatial knowledge, including how places are connected to ancestral activities and maps represent relationships between people, place and objects. This way of thinking involves solving problems and explaining the physical and spiritual environment of a particular social group. Mathematical thinking, concepts and applications are carried out in everyday activities such as fishing, hunting, making decorations or artefacts (Owens 2013). Thus, it can be seen that although Western mathematics is the mathematics generally taught in schools, traditional Indigenous cultures have what are often distinctly different ways of making sense of, organising, and acting in their environments. These different world views and related social practices impact upon what is valued and used as knowledge, mathematical or otherwise. For example, as described by Perso (2003): Being able to count is highly valued by Western cultures to the point where parents boast of the ability of their pre-schoolers to count to certain numbers. By contrast, Aboriginal parents might boast of the ability of their pre-schooler to find

Starting points and pathways in Aboriginal students’ learning

their own way home from a certain distance (note that Western cultures may be appalled at a young child begin out alone without an adult presence). (p. 13) Thus, there is a need for educators to understand the mathematical concepts and processes that Aboriginal children bring to school, including the ways that language and other practices mediate their learning of both their own and Western mathematics (Meaney et al. 2008). The research study reported here arose from this need. It developed from observations from within several Western Australian schools that some Aboriginal students were able to correctly complete ‘counting’ tasks without counting. That is, their processes for completing a mathematical ‘number’ task could not be interpreted easily from a Western mathematics perspective. This apparent enigma is outlined next. Treacy (2001) created a task to find out whether students would choose to use counting to solve problems, such as for making an equivalent set. Called the Ice Cream Task, this diagnostic task has since been adopted by First Steps in Mathematics professional learning and used in schools around Western Australia and in other places across Australia (Department of Education WA 2013). The task asks students to ‘get just enough ice creams for all of the children’ displayed in a picture. The number of children in the pictures starts at six and increases to ten and then 14. The purpose of the task was to find out if students would choose to use counting to make an equivalent group, in a situation where there were no clues that counting might be an appropriate thing to do. Two very different and surprising accounts of students’ responses to this task were reported by a numeracy specialist teacher in a school in Western Australia (Stokes 2002). Those accounts of two Aboriginal students’ responses to this task are described below: Winona [eight years old] glanced at the picture (see Fig. 1) and after choosing the ice-creams very carefully … brought back six. … For the picture of fourteen (see Fig. 2) I reminded her not to forget the baby, as he would want one too, but thought she would bring back a handful as she had glanced at the picture and certainly had not had time to count them. I was amazed when she gave out fourteen ice creams. She said, ‘I didn’t count’ but could not explain how she

Fig. 1 The Ice Cream Task for six people

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Fig. 2 The Ice Cream Task for 14 scattered faces

knew how many to bring back. She had not chosen to count but ‘knew’ how many she needed. Victor [pre-primary student] looked at the picture and then went to the ice cream shop and chose his ice creams very carefully. He returned to the table with six ice creams and proceeded to hand them out saying ‘that’s one for the baby, one for the Dad, one for the sister and so on.’ In both these examples, and in others, there was no indication that the students used any form of ‘counting’, and yet they were able to complete the tasks correctly. So, how were they able to get the correct amount of ice creams? Did they have a visual image of the people and use a matching strategy to get one each? Did they use subitising? Did they use an estimation strategy? Or was it something else altogether? Further to this, the question is, why didn’t they count? Was it because they were not able to count? Thus, this research study was designed as a small-scale study to investigate these phenomena further. More specifically, the study was framed by three research questions: 1. Can some Aboriginal students complete equivalent set ‘counting’ tasks, such as the Ice Cream Task, without counting? 2. Do they have the necessary counting understanding to complete equivalent set tasks with counting if they wanted to? 3. What strategies do they use to complete equivalent set ‘counting’ tasks, such as the Ice Cream Task? In the context of these research questions, it is acknowledged that referring to the Ice Cream Task as a ‘counting’ task is a Western mathematics perspective as it seems that some Aboriginal students are able to ‘correctly’ complete the tasks without counting. In addition, in the context of this research study, it is acknowledged that “diversity is as great among Indigenous people as it is among many other groups of people” (Jorgensen (Zevenbergen) et al. 2010, p. 162). Thus, a limitation of this study is that it was conducted in one community only. Findings might be similar in other locations with other Australian Indigenous learners, but they might differ. The findings do however


have relevance with regard to the initial intent of the study to understand the mathematical concepts and processes that Aboriginal children bring to school. The study also has significance with regard to the following: (1) identification of a range of conceptualisations and thinking strategies Aboriginal children use in two different counting situations; (2) development of guidelines for teaching early number concepts and skills with Aboriginal students, in particular within ‘counting’ contexts; and (3) dissemination of information that can be used to stimulate dialogue in educational circles about Aboriginal students’ early mathematics learning and avenues for further research in this area.

Theoretical framework Number skills related to language and cultural contexts Much research has examined how specific vocabulary and language practices might impact upon the development of counting concepts and skills, particularly with regard to cultures whose languages lack a counting vocabulary (Butterworth et al. 2008), or how language structure can support or inhibit numeration and counting (Ng and Rao 2010). In the context of this study, the first of these two areas within the literature is of importance because the research findings indicate it cannot be assumed that language is an essential component of all number-related capacities. Butterworth et al. (2008) tested 45 4- to 7-year-old children from three different monolingual groups: two different Indigenous language groups in the Northern Territory of Australia and one English language group from Melbourne. The two Indigenous languages have generic number words that translate into English as the concepts of the following: singular, plural, dual plural and greater than dual plural (Warlpiri language); and singular, dual, trial and plural (Anindilyakwa language). All three groups demonstrated high levels of competence on four enumeration tasks designed to assess numerosity understanding: “memory for a number of counters, cross-modal matching of discrete sounds and counters, nonverbal exact addition and sharing play-dough disks that could be partitioned by the child” (p. 13179). Butterworth et al., concluded that number-word vocabulary is not needed for the development of enumeration concepts and that when “children learn to count, they are learning to map from their pre-existing concepts of exact numerosities onto the counting word sequence” (p. 13182). Pica et al. (2004) conducted a more extensive study of the relation between language and arithmetic by working with speakers of Munduruku, an Amazonian language with very few number words. The Munduruku language has explicit expressions for the numbers one to five, and “these expressions are long, often having as many syllables as the corresponding quantity” (p. 500). For example, ebadipdip where eba means ‘your (two) arms’ is the expression for the number four. The study found that the Manduruku: did not use their numerals in a counting sequence, nor to refer to precise quantities. They usually uttered a numeral without counting, although (if asked to do so) some of them could count very slowly and nonverbally by matching their fingers and toes to the set of dots. (p. 500)

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Further examination of the number understandings and skills of the Manduruku found that they could comprehend large numbers (sets of 20 to 80 dots) and understand the concept of relative magnitude. They could also complete approximate operations with these large numbers (addition, subtraction and comparison). The researchers noted that their findings suggest that a distinction needs to be made between exact and approximate representations of numbers and that the data provide evidence that numerical approximation is a basic competence independent of language. They also noted that their results “support the hypothesis that language plays a special role in the emergence of exact arithmetic during child development … [and] the availability of number names, in itself, may not suffice to promote a mental representation of exact number” (p. 503). Harris (1987) noted that traditional Aboriginal and Islander people have an interest in Western mathematics when the context is meaningful and practical. For example, one counts when it is necessary to count, rather than as an abstraction, a sequence of words or a count of inanimate non-meaningful objects. Further, the Western notion of precision might not be a facet of a situation unless it was needed. For example, if “in a caring extended family, you knew exactly who was sleeping in your camp, you did not need to know the precise number present” (Harris 1987, p. 36). Harris also notes that anthropologists found that “in culturally meaningful ways, Aboriginal and Islander people did use numbers with precision when the situation demanded it … [that] includes trading activities, calendar calculations, battle strategies and distributing turtle eggs” (p. 36). Malcolm et al. (1999) suggested that, in general, precision is much more central to Western society than in most Aboriginal contexts. In Aboriginal English, quantities are discussed with non-specific words rather than with specific words. In Aboriginal English, quantification is usually vague, e.g., a big mob, the biggest mob, proper big barra. (Department of Education, Western Australia 2012) In the Pitjantjatjara language, there are no comparative terms used, so words like larger, smaller, taller and tallest are not present. Jorgensen and Sullivan (2011) suggest that Pitjantjatjaraspeaking students “not only have to learn the language of … comparisons but also the deeply embedded concepts associated with such terminology” (p.52). Owens and Kaleva (2008) noted that communities in New Guinea also place different levels of importance on comparison, estimation and accuracy. She suggests that some groups use a visuospatial means of quantifying. For example, in several language groups, large numbers of objects are compared by the amount of space taken up rather than precise counting (Owens 2001). Estimation may play an important part when deciding on areas of land, houses and gardens. For example, in the Western Highlands area, the size of a sleeping room is visualised as sleeping around seven men. In addition, estimating can involve intuitive forms of proportional reasoning. For example, in the Mandang Province, people think of floor space in terms of the numbers of rows of posts. In comparing a 12-post to a 9-post house, they talk of the house as being half as big again or for a 12-post to a 6-post house as twice the size (Owens and Kaleva 2008). However, Lean collected evidence of precise counting systems for 550 languages in New Guinea (Owens 2001). In some areas, both digit-tally systems and body-part tally systems were used side by side. For example, the Kewa language group used a four-cycle counting system, where counting begins on the little finger, and so four is called ‘ki’ or ‘hand’. Twelve would be three hands, ten would be two hands and


two thumbs and seven would involve one hand and three thumbs. This group of people also used a body-part tally system with a 47-cycle. He noted that the nature of the count depended on the type of countable objects, when they are counted and which economies and exchanges were involved. (Owens 2001, p.59). Similarly, Paraides (in Owens et al. 2011) describes how one language group counted in composite units as opposed to individual objects. For example, in gathering taro tubers, two are tied, then four, then one set of six which then has the value of one. Whereas banana bunches, coconuts and wild-fowl eggs are counted as one set of four, then two, then three sets of four with 12 being the unit. Peanuts are counted in sets of ten or twos, but there is no fixed number strung together for small fish. Counting within a Western mathematics perspective Desoete et al. (2009) outline the conceptualisation and historical roots of counting. They note that children showed a sense of numbers early in their development, with infants being able to differentiate quantity in visual and audio events. Counting is then described as a key understanding and skill needed to move from a sense of quantity and numbers to a more advanced mathematics such as arithmetic operations. Key within this perspective is the concepts of procedural and conceptual counting. Procedural counting is the ability to accurately determine through a counting process the numerosity of a collection. Understanding the counting process entails a conceptual knowledge of counting and is underpinned by five counting principles as follows: stable-order principle, one-one principle, cardinality principle, order-irrelevance principle and abstraction principle (Desoete et al. 2009). It is noted here that the fifth principle as specified by Desoete at al. is not always included in outlines of counting principles (e.g., Reys et al. 2012), but is included here because it explicitly recognises counting as a mathematical process that can exist as an abstraction independent of the world. Nunes and Bryant (1996) suggest that children only really understand counting when they know what counting is for and when to use it to solve problems, in particular, when they choose counting to match sets. According to Nunes and Bryant, children initially come to understand number words and counting as a means of quantifying a single set; they then take time to generalise this understanding to the point where they can use it to compare the size of two sets or to construct equivalent sets. Nunes and Bryant suggest that children have to know not only how to count but also when it is appropriate to count. They suggest that if children do not choose to use counting to solve problems, then they have not fully understood the counting system. Munn supports this view, suggesting that children don’t really understand the function of counting until they use it spontaneously to solve problems (Munn 1998). Curriculum models of early number learning The First Steps in Mathematics Diagnostic Map for Number (Willis et al. 2004), supports a view that children do not fully understand counting unless they choose to use counting in solving problems. The map describes a series of phases students go through as they develop an increasingly abstract understanding of numbers. It suggests that students in the matching phase (3 to 6 years old) “often do not spontaneously use counting to compare

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two groups in response to questions, such as, Are there enough cups for all students?” (Willis et al. 2004, np). However, by the end of the quantifying phase (5 to 9 years old), they, “without prompting, select counting as a strategy to solve problems such as, Are there enough cups? Who has more? Will it fit?” (Willis et al. 2004, np). In this phase, students are developing a more abstract understanding of number. They “see that the significance of the number uttered at the end of the counting process is that it does not change with rearrangement of the collection or the counting strategy.” The Ice Cream Task was one of a series of tasks used to give evidence of whether students were through the matching phase and into the quantifying phase. Somewhere between the ages of five and nine, students typically respond to the task by counting the first set in order to create the equivalent set. When students are through the quantifying phase, they trust that the number at the end of the counting sequence will not change no matter how the collection is counted or arranged and therefore choose to use counting for themselves to solve equivalent set tasks. Prior to this, students typically “understand that building two collections by matching one to one leads to collections of equal size (Willis et al. 2004, np)” and are more likely to use a matching strategy to solve problems. A matching strategy would have enabled students to solve the cross-modal matching task used in the Butterworth et al. research (2008, p. 13179). The Model of Early Number Development in Fig. 3 developed by Treacy and Willis (2003) includes two pathways into number understanding. One pathway is through counting and the other is through subitising and partitioning. The counting pathway on the left of the diagram in Fig. 3 has been discussed. Attention will now focus on subitising and partitioning as shown on the right hand side of the diagram. Subitising Although defined somewhat differently by different authors, the general essence of subitising is that it is an “accurate quantitative evaluation of small sets without explicit (neither internal nor external) counting” (Benoit et al. 2004, p. 292). This concept can itself be viewed as consisting of three related concepts and skills: perceptual, perceptual-preverbal, and perceptual-verbal (Benoit et al. 2004). Perceptual subitising consists of being able to differentiate between perceptual arrays that differ by only one item. In perceptual-preverbal subitising, the focus is upon being able to recognise and differentiate between small quantities visually without counting and without having verbally naming the number in a set. Perceptual-verbal subitising involves being able to use number words or labels, without counting, to communicate perceptual determinations of quantity. Benoit notes that perceptual-verbal subitising is the traditional view of subitising. In this research, he focussed on this aspect of subitising, asking 3- to 5-yearold children to use a number word to say how many dots in familiar and unfamiliar arrangements, with sets of up to six items (Benoit et al. 2004). Research that has focused on subitising skills, as reviewed by Desoete et al. (2009), revealed a range of capacities across a range of ages, including that infants can by the age of 6 months differentiate between sets of numerosities, and young adults can


Protoquantitive comparison Same/different More/less

Can think of numbers without reference to materials

Think of numbers only in relation to amounts of materials

Counting skills

Subitizing 1,2,3 (4,5)

Counting

Part whole understanding

Use counting to get an amount

Choose to use counting to make equivalent sets.

See groups within amounts Add and subtract small amounts

See number as a representation of quantity Able to confidently break up numbers and rearrange the parts, knowing that the quantity has not changed.

Fig. 3 Children learning about number as a representation of quantity—a model

improve subitising performance through training. Other research has examined how it might be that the human brain has the capacity to subitise. There are “different theories attempting to explain number processing and subitizing” (Desoete et al. p. 58), and in particular which part of the brain is involved, but there is not a consensus on a model and explanation. Also relevant to this study is the idea of partitioning. Children who can see a collection in terms of the whole and its parts or units of units (Steffe and Cobb 1988) are using partitioning. Benz (2012) describes two processes involved in identifying the quantity of a collection. Firstly, a collection is perceived either in terms of single objects, a whole collection or the collection composed of parts. The quantity is then determined by either counting, subitising or by using a conceptual subitising process,

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that is, seeing and using the parts within the whole. The ability to decompose collections into parts is seen as an important conceptual achievement in number (Resnick 1983). According to the Treacy and Willis (2003) model above, this partwhole understanding is an extension of subitising which allows students to see groups within amounts and add and subtract small amounts. The First Steps in Mathematics Diagnostic Map for Number (Willis et al. 2004, np) states that students who are through the quantifying phase “use materials or visualise to decompose small numbers into parts empirically” (Willis et al. 2004, np). Possible strategies for solving equivalent set tasks The discussion above suggests that there are several possible strategies for solving equivalent set tasks, such as the Ice Cream Task. These include the following: & & & & & &

Using family relationships Estimation Counting Matching Subitising Partitioning

Thus, in addressing the research aims, this study was designed to examine the strategies students used in situations like the Ice Cream Task involving making equivalent sets. If they did not choose to use counting, then what strategy was used? Did they use family relationships, a matching strategy or one of the other strategies listed above?

Method Eighteen Aboriginal students, in years 1 to 11 at a remote community school in the Goldfields of Western Australia, participated in task-based interviews based on ‘counting’ tasks. The tasks involved fetching ‘maku’ (the local name for witchetty grubs) to give all the people in a picture a maku, identifying a hidden quantity when a part of a collection of maku is covered, and standard counting tasks. This paper does not report on the full findings; for focus, it reports on the maku and standard counting tasks only. An Aboriginal research assistant who spoke the same language as the students in the study (Wangkatha and Aboriginal English) was engaged to help with the task design and data gathering processes. She grew up in the community and was familiar with all of the students. The Ice Cream Task was modified to become the Maku Task, which was an attempt to make this ‘Western’ mathematics task more relevant to the students within the study. The local shop sold lolly versions of the grubs, so these were used instead of pictures of maku. Pictures of the groups of people were constructed from pictures of Aboriginal people that were found in books in the school (see Fig. 4). Specifically, the tasks in the order presented to the students were as follows:


Fig. 4 Examples of the pictures of groups of people for the Maku Task

Maku Task The student was shown pictures with 4, 6, 10 and 16 people and asked to get enough maku for all of the people in the picture. This task was designed to see whether a student would choose to use counting in a situation where it is not obvious to do so. Oral Count The younger students were asked to count from one, and the older students were asked what came after a given number, for example, 39, 59, 79, 99, 100, 109 and 199. This task looks at the extent of the student’s oral counting sequence. Get Me Task The student was asked to get a number of items (maku) and put them in a bag to take home. This task looks at whether the students can use counting when asked. The Oral Count Task asks students to count, and the Get Me Task includes a signal to count as the students were asked to get a particular number of maku. Therefore, these tasks came after the Maku Task to ensure that the students did not get any indication from the interviewer that we were interested in counting. The interviews were conducted in a combination of Wangkatha and Aboriginal English and were videotaped with permission from the students and their care-givers. The video tapes were transcribed and analysed with the assistance of the research assistant, to ensure a correct interpretation of what the students were saying. The video was reviewed to identify the strategies students used to solve the Maku Task. In the video, it was easy to see when students obviously counted the people and the maku. However, where no audible count was evident, other counting behaviours were considered, such as pointing, nodding, eye movement or sub vocalisation. Where these were not evident, the amount of time to ascertain the quantity within the picture of the people was considered. If there was no evidence of counting, and the student quickly ascertained the number of people, this was recorded as ‘No overt count’. These instances were analysed for evidence of the strategy the students used, looking for evidence of behaviours suggesting students used a matching strategy, family relationships or some other strategy.

Findings and discussion Table 1 summarises each student’s response to the Maku, Oral Count and the Get Me Tasks. This data will now be discussed in relation to the strategies and understandings the students demonstrate for each task in turn.

K. Treacy et al. Table 1 Summary of the interview data

Shaded cells show responses that are either correct or one more or one less than the required number Fictitious names have been used

Many of the students completed some or all of the items within the Maku Task without using any overt counting strategies. While it was possible to say that they were not using counting, it was not always possible to ascertain from the video footage what strategies the students used to complete the Maku Task. Whether they were using a matching strategy, family relationships or indeed other strategies, in most cases, it was not possible to say, as the students were not able to articulate enough about how they knew how many maku to collect. However, it was possible to identify if they were or were not using counting, with most students not using counting to complete the Maku Task. In comparison, when asked to get a number of items in the Get Me Task, all students, except for the youngest one, could orally count beyond 16, which was the largest number within the Maku Task. Similarly, in the Oral Count Task, all students except for the youngest could use the sequence of number names to rote count beyond 20.


Maku Task Three of the students, Katie, Doug and Philip, chose to use counting at some point in the Maku Task to work out the number of people in the picture and then used this count to collect the maku from across the room. One of these students, Katie, counted the 16 people in the picture and then collected 20 maku instead of 16. After Katie gave the people in the picture one maku each (placing four to the side), she was asked how many maku she had. She immediately said 20. She was asked how many are on the picture, and she did not know. In response to that question, Katie needed to count the maku to work out that it was 16. This suggests that she did not use the count of 16 people to work out the number of maku that she needed to collect. Katie also struggled to use the oral count when she was asked to get 17 maku. With the oral count sequence, she was able to count to 28, but was not able to continue after this number. Doug used counting for two parts of the Maku Task, but did not use counting for the largest and smallest quantities within the task. For the picture of four, he collected two items at a time and brought them back and gave them out. For the picture of 16, he simply grabbed a large handful of maku and then gave them out one at a time. When asked how many maku were in the picture, he did not know and proceeded to count. Philip was able to select the correct number of maku for the first three parts of the task without counting and then chose to use counting for the last picture of 16 people. After getting the right number for the picture of ten, the research assistant asked him several times to explain how he knew to collect ten maku. He was able to reply by saying that “I bin looked at the picture”. When asked to complete the section of the task with 16 people, he said in language to the research assistant, “Do you want me to get it right now?” to which she replied, “Yes”. He then said, “Can I count them first?” and she said, “Do what you want to do?”, so he counted the people first. It is possible that the questions that were asked of him after completing the task with ten people, one of which was, “Did you count?”, may have suggested to him that he should be doing something other than ‘just look’ to work out how many people. It is interesting to note that most of the students did not choose to count for this task. Of those that did not choose to count, most of them were able to select the exact number of maku needed each time or to select one more or one less than required. While this strategy may not be surprising for four or six maku, it is surprising for 10 or 16 maku. For 10 maku, 14 out of 18 students did not overtly count, ten were correct while three were very close selecting one more or one less than ten. For 16 maku, 14 did not overtly count. Out of those 14 students, two were correct while three were very close selecting one more or one less than 16. This suggests that for most of these students, counting was not an appropriate strategy for this type of situation. So how did the students know how many maku to select? Many of the students were not able to articulate exactly how they knew how many maku to select. When asked, many of them simply said things like “I bin looked”, suggesting that they looked at the picture and just knew how many. One student, Peter, was not able to say how he knew. After bringing back 17 maku for the 16 people in the picture, he was asked how many people in the picture, and he was not able to say. Instead, he counted the people and then said there were 16 people. This suggests that he did not even think in terms of the number of people in the picture when collecting the maku, and yet he was able to get

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just one more than the required number. He was very near the required amount, without thinking about a number. Another student, Cathy, looked at the picture of 16 people for a short time and then went to collect the maku. She showed no signs of counting, she did not touch any of the people in the picture; she did not nod her head or show any other overt sign of counting. She brought back 17 maku and placed them on the people in the picture. The research assistant asked her how many maku. She said 16. When asked to explain how she knew it was 16, she explained “when I put them in, I counted them, and I have one left”, suggesting that she had not counted the collected maku until she distributed them to the people. The research assistant asked, “Did you know how many people from the beginning?” to which Cathy shook her head and said, “When I put them in, I thought there was 17 people, but there was 16”. “Why did you think it was seventeen?” “Because there was lots”. The researcher then asked, “Did you count them to start with?” to which Cathy shook her head and said, “No, too quick”. “So how did you know to get seventeen? What were you thinking over there?” Cathy struggled to answer this question, so was asked, “Did you have a picture of the people in your head?” Cathy nodded and, glancing at the picture, said “Five men, six women and five kids”. Oral Count All of the students in this study, except Mike, were able to orally count to at least 20, which is more than the number required for the largest group of people in the Maku Task. Mike did not know what came after ten, which, when considering his age, is not surprising. Two students knew the sequence up to 100 but not after, and one student knew the sequence to 99. Interestingly, after year 4, many of the students knew the counting sequence up to 109, but did not know what came after, with some saying 200, one saying 300 and one saying 2,000. The latter student re-thought his response to this question after the task and confessed to the researcher that he knew that it wasn’t 2,000 as he was walking out the door, to which she asked, “So what is it?”, and he replied, “200 unna” (unna in this context means ‘isn’t it’). When discussing this with the students’ teachers, they said that these students had been completing three-digit addition, subtraction and multiplication, which means that all but one of these students (Tanya, who knew what came after 109) would have been completing calculations with numbers that were not in their counting sequence. Hence, they may have had a little sense of the magnitude of the numbers they were dealing with. In the students’ first language, Wangkatha, there are number words for one, two, three and four. The word for four, ‘pirni’, is used for more than four as well. Get Me Task All of the students within the group were able to use counting to get a number of maku when they were asked to. This task was the last one in the interview and the number of maku they were asked to get varied according to how they had responded on the other sections of the task. The younger students, Mike, Amanda and Alan, were asked to get less than ten items, which they succeeded in doing. Casey was one of the last students interviewed, and it was decided to try asking her to get one more than her oral count of


20, to see if she could use her count sequence in a purposeful situation. She succeeded in getting 21 maku. The rest of the students were asked to get 16 or more items to see if they could use counting in this situation, to at least the number of the largest set in the Maku Task. James made a small error when he re-counted one item twice, whereas Katie struggled to keep track of the counting sequence while collecting the requested number of items. She counted to 12 then missed out 13, said 14, 15, missed out 16 and said 17, so instead of getting 17 maku, she had 15. Katie is an older student who has a learning disability, which may explain why she had trouble recalling the counting sequence.

Conclusions and implications Most of the Aboriginal students in this study demonstrated specific counting knowledge and skills in the Oral Count and Get Me Task, yet they chose not to count in the Maku Task. Thus, although it was not always clear what strategies the students used for the Maku Task, the findings suggest that they did not see this situation as one in which counting is required. Many of the students were able to collect an appropriate number, suggesting that they were looking at the picture and using an estimation strategy to get a quantity of maku that would be ‘about right’ for the number of people. Many students described how they collected the maku by saying ‘I bin looked’, which suggests that they have used their perceptual judgement to estimate the amount required. Peter was not able to say how many maku in the 16 task, even after he had placed them onto the picture. He did not need the number of people to collect an appropriate amount of maku. It certainly seems that the students in this study were not concerned about precision or exactness, as suggested by Malcolm et al. (1999) since being ‘close’ to the exact number (e.g., one more or one less) was sufficient to complete the task. In the students’ first language, Wangkatha, there are number words for one, two, three and four. The word for four, ‘pirni’, is used for larger groups, more than four as well. (Wangkanyi Ngurra Tjurta Aboriginal Language Centre 2001/2002) This suggests that the world view of these students may not include the need to be exact. The research assistant suggested that the students were using estimation to get an amount that was ‘enough’. Estimation is a strategy that is commonly used in all cultures; however, this research suggests that it may be a preferred strategy for the students within this task. This is in contrast to non-Indigenous students completing the Ice Cream Task, who mostly choose to use counting to complete this task by the time they are 9 years of age (Willis et al. 2004). However, was it estimation or did students use a matching strategy with a mental image of family relationships in order to decide how many? This is difficult to ascertain from the video evidence. Certainly, Cathy’s strategy suggests that she saw groups of people and collected enough for each group without attending to the number within the whole collection. It is possible that other students used a similar strategy without articulating this to the researcher. There has been some research (e.g., Ginsburg 1982) into the idea of using one-to-one matching; however, most early number research has focused on using one-to-one correspondence in order to count (e.g., Baroody and Wilkins 1999; Fuson 1992; Nunes and Bryant 1996) and does not include one-to-one matching based on family relationships.

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Did they use subitising? Again, it is difficult to say. They certainly did not use perceptual-verbal subitising, as they were not able use a number word to say how many they had collected, needing to count them after they were placed on the picture. Was this perceptual or perceptual-preverbal subitising? It is difficult to say, but it is possible. However, it could also be an estimation strategy or a matching strategy based on family relationships as described above. Western mathematics has acknowledged the place of subitising in early number development, so the tendency is to view this strategy through this lens. As there is little research about alternative strategies such as estimation, matching and using family relationships, these tend to be harder to recognise. When considered in relation to the Model on Early Number Development in Fig. 3 (Treacy and Willis 2003), it seems that there may be another pathway to number understanding that is missing. This model mentions two pathways into number understanding, counting and subitising. Perhaps, a third pathway to number understanding could be added (see Fig. 5) to acknowledge the existence of other, alternative strategies such as matching, estimation or family relationships. Although it is difficult to describe the strategy the students in this study used to make the equivalent set, we can say they did not choose to count the number of people in order to work out how many maku, and they did not seem to use perceptual-verbal subitising as they did not know how many maku they collected until after laying them out. How this third pathway may connect to the existing pathways is unknown, or indeed, if it can (Fig. 5). Although it is not clear exactly what strategy the students used, what can be ascertained from the video evidence is that the majority of students were able to complete the Maku Task, getting an appropriate amount (give or take one) without counting. They were able to count to more than 16 in the other tasks, but when asked to make an equivalent set, did not see counting as an appropriate strategy. The findings of this study raise numerous issues for future research and for mathematics curriculum and teaching practices, including: &

&

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Since this study has not been able to clarify the question of whether some Aboriginal students use matching, subitising, partitioning or whether they use family relationships in situations involving making equivalent sets, more in-depth research needs to be carried out that provides appropriate contexts and opportunities for the students to talk more extensively and to disclose more of what they are thinking and doing. Additional research needs to be conducted related to the findings of this study that Aboriginal students might have a tendency towards estimation rather than exactness. The implications of these findings for teaching practices also need to be attended to, in that it cannot be assumed that Aboriginal students see a purpose for counting. The purpose of counting needs to be made explicit in activities that a teacher uses to teach counting. In this regard, there is also a need to be explicit about different cultural viewpoints, for example, comparing the value and purpose of precision, estimation or sharing in various everyday contexts. Students need to be provided with purposeful counting experiences with quantities beyond 100, to build their knowledge of the patterns in the number system and to connect quantities to this. Purposeful activities will be a challenge to identify for non-Aboriginal teachers, since what non-Aboriginal teachers might think of as purposeful might not be so for the Aboriginal students.


Protoquantitive comparison Same/different More/less

Can think of numbers without reference to materials

Think of numbers only in relation to amounts of materials

Counting skills

Counting

Subitizing 1,2,3 (4,5)

Matching, family relationships, estimation?

Part whole understanding

Use counting to get an amount

Choose to use counting to make equivalent sets.

See groups within amounts Add and subtract small amounts

See number as a representation of quantity Able to confidently break up numbers and rearrange the parts, knowing that the quantity has not changed.

Fig. 5 A third pathway added to the Model of Early Number Development

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The teachers of the students involved in the study were planning lessons based on ‘Western’ assumptions. Without support and training, they were not in the best position to recognise what their students brought with them into the classroom. Teachers need to recognise the different starting points and learning needs of all their students. This is difficult when teachers do not share the same cultural background and world view as their students and there is little available information about what they should be looking for. There is a need to value what students can do and build on it and a need to recognise and accommodate different starting points in number.

This study has significance with regard to the identification of a range of conceptualisations and thinking strategies Aboriginal students bring to school. It also

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exemplifies the kind of professional knowledge and skills that proficient teachers require to meet the Australian Institute for Teaching and School Leadership standard related to knowing what students know and about strategies for teaching Aboriginal and Torres Strait Islander students (Education Services Australia 2011). However, having the knowledge is only part of the story, having the disposition to notice strategies students’ use which may sit outside of our Western mathematics perspective is what is needed. The findings highlight a need to further examine the world views, orientations and related mathematical concepts and processes that Indigenous students bring to school. Acknowledgments The authors wish to thank the participating school and teachers and to acknowledge the contribution of the research assistant Fiona Walker. This research was funded by the National Centre for Science, Information and Communication Technology and Mathematics in Rural and Regional Australia (SiMERR).

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