Young Australian Indigenous students' engagement with numeracy: Actions that assist to bridge the gap.

Warren, Elizabeth ; deVries, Eva

Grave concerns about the progress of Australian Indigenous students in both numeracy and literacy still exist. In both literacy and numeracy, Australian Indigenous students have been shown to be achieving at least two years below mainstream students by the time they reach Year 3 (Durodoye & Hildreth, 1995; Queensland Studies Authority, 2006). The latest national report on schooling in Australia (Ministerial Council on Education, Employment, Training and Youth Affairs, 2008) suggests that this trend persists. The results for Indigenous students obtained from testing in 2006 indicated that only 72% of Indigenous Queensland students were achieving at the benchmark for numeracy in Year 3. This is significantly below their achievement in both reading (88.5%) and writing (89.7%), and also significantly below the achievement of Indigenous students in five other states. The Year 5 and Year 7 numeracy results mirror the results found in Year 3. Similar trends exist in the national scores. While reading and writing scores have been gradually improving for Indigenous students since 1999, there has been little change in the numeracy results.

Unjustified criticism has been laid upon Indigenous students in the past and absenteeism, disadvantaged social background and culture have all been seen as contributing factors (Bourke & Rigby, 2000). This paradigm is seen by many as irresponsible (Cooper et al., 2004; Matthews, Howard & Perry, 2003; Sarra, 2003). Commonly, teachers or students have been blamed for the lack of outcomes in Indigenous education (Harrison, 2007). Historically, most educational efforts have aimed to assimilate Indigenous students into non-Indigenous Australian society and are based on the ideology of cultural deprivation (Prochner, 2004). New ways of adapting the conditions for learning for Australian Indigenous students are needed in order to set them up for success rather than failure in school. The literature delineates an array of successful strategies that begin to tackle these issues. Some of these are the provision of a safe, secure environment, recognition of the importance of focusing on the learning needs of the students and of Indigenous patterns of learning. With regard to literacy, another strategy includes an understanding of the benefits of an explicit teaching/learning approach and early intervention strategies to ensure the adequate acquisition of literacy skills in the early years of schooling. There were no specific strategies relating to numeracy.

While much has been written about Indigenous students' literacy learning (e.g., Gray, 2007: Luke, 2003; Nakata, 2003) there is a paucity of literature on numeracy, especially literature couched within a positive framework, indicating ways that we can assist the engagement of young Australian Indigenous students with numeracy. In addition, Thorpe et al. (2004), in a large study investigating learning experiences and teaching practices prior to Year 1, found that, over the year of their study, many students made negative progress in their understanding of basic numeracy concepts, indicating that in Queensland the early years of education are problematic with regard to numeracy outcomes for all students. There have been few published studies on the impact of early childhood education on Indigenous students (Prochner, 2004). Our longitudinal research project, Young Australian Indigenous students' Literacy and Numeracy draws on and adapts relevant mainstream research about young students' numeracy learning and endeavours to situate these findings in local settings where Indigenous cultural practices are recognised and respected. The particular focus of this paper is to investigate actions that assisted young Australian Indigenous students' engagement with numeracy.

Theoretical underpinnings

Briefly, the research base and design principles that underpinned the development of the numeracy aspect of Young Australian Indigenous students' Literacy and Numeracy project were the following:

* Mathematics ability: all children are capable of learning mathematics. Children do not have to be made ready to learn as they freely engage with informal mathematics in everyday life (Greenes, 1999).

* Role of the teachers: play is not enough to assist learning in the early years. Children learn through play but they need adult guidance to assist them to reach their full learning potential (e.g., Balfanz, Ginsburg & Greenes, 2003; Vygotsky, 1962). As compared to other cohorts of early years students, Indigenous students gain even less from attending play-based programs (Tayler, Thorpe & Bridgstock, 2006, cited in Fleer & Rabin, 2007).

* Types of activities: hands-on activity based learning best supports young Indigenous students to engage with mathematics (Cooper et al., 2006).

* Role of oral language: a focus on the language of mathematics fosters important language acquisition and assists students to acquire meta-cognitive abilities. This focus is even more relevant for students whose first language is not English (Pappas, Ginsburg & Jiang, 2003).Yet pathways for oral language experiences tend to be restricted in early childhood settings (Kennedy, Ridgway & Surman, 2006).

* Mathematics curriculum: young students are capable of dealing with a comprehensive mathematics curriculum (Greenes, Ginsburg & Balfanz, 2004).

* Indigenous students' language: Aboriginal English reflects the culture and identity of Aboriginal people (Cronin & Diezmann, 2002) and discourses of Indigenous families often do not match that of the school (Cairney, 2003). Teachers need to create a bridge for young Indigenous students between Aboriginal English and Standard Australian English as these students grapple with a new language, new concepts and the vocabulary presented for numeracy.

* Value of Western mathematics: parents of Indigenous students want their children to be bicultural and learn to live in both worlds (Partington, 1998). An understanding of Western mathematics is important for two reasons: many traditional industries are now calling for personnel with advanced skills in mathematics (Trends in International Mathematics and Science Study, 2003); and mathematics is an empowering process acting as a tool to identify power differences among socio-economic classes (Gustein, 2003).

While we acknowledge that Indigenous students enter school with their own cultural understanding of mathematics, the focus of this paper is on actions that assist them to engage with Western mathematics, an understanding that enhances their participation in a Western culture. Three key learning areas of mathematics were considered in developing the focus of this project. These were the language of mathematics, patterning and number.

Language

The use of spoken language in school and the types of interactions teachers use can either advantage or disadvantage Australian Indigenous students. Furthermore, the importance of spoken language as the foundation for all learning is often not fully recognised; many young Australian Indigenous students are not able to make a strong start in the early years of schooling as the discourses of the family often do not match that of the school (Cairney, 2003). This mismatch of home and school language has been shown to disadvantage Indigenous students' achievements in literacy and numeracy in the long term (Dickinson, McCabe & Essex, 2006; Ministerial Council on Education, Employment, Training and Youth Affairs, 2007).

It is well recognised that oral communication is dominant in the lives of Indigenous students and that their experience with print and other literacies is often limited. Patterns of classroom interactions have been shown to disadvantage some students, particularly the interaction of teacher questioning, as Indigenous students do not commonly experience this type of interaction at home or within their community (Galloway, 2003; Haig, Konisberg & Collard, 2005). Understanding and accepting Aboriginal English as a dialect of spoken English used by most Aboriginal and Torres Strait Islander people is vital and knowing that there are variations across particular communities is important (Haig, Konisberg & Collard, 2005). While Standard Australian English is the discourse of the school, teachers need to create a bridge for young Indigenous students between Aboriginal English and Standard Australian English as they grapple with a new language, new concepts and the vocabulary presented for numeracy.

In addition, language serves as a crucial window for researchers into the processes of teaching, learning and doing mathematics. All processes are conceived as socially organised, taking place not only within a social environment but also structured by that environment (Morgan, 2002). Thus studying language and its use must take into account both the immediate situation in which meanings are being exchanged (the context of the situation) and the broader culture within which the participants are embedded (the context of culture). Learning of mathematics can be viewed as an initiation into mathematical discourse. In this view, mathematical language is regarded as the carrier of pre-existing meaning and also the builder of meaning itself. The means by which linguistics is communicated in mathematics is thus crucial to the effective learning and development of mathematical thinking (Ferrari, 2004).

Number

Traditionally mathematics tends to be segregated into four stands: number, measurement, data and space. These strands are reflected in state, national and international testing regimes, with number being the most prominent. An analysis of test items indicates that up to 60% of the emphasis is on this particular strand and many of the questions in measurement and data also rely heavily on an understanding of the number system in order to be answered correctly. A deep understanding of the number system and its links to the measurement system is also seen as crucial to meeting workplace numeracy demands (National numeracy review report, 2008). In addition to its importance to mathematics, number also seems to be an area in which Australian students struggle most. The Trends in International Mathematics and Science Study results for 1994/95 and for 2002/03 revealed that the mean scores of Australian Year 4 students were significantly below the international average in number, equivalent to the international average in patterns and relationships, and above the international average in measurement, geometry and data (Thomson & Fleming, 2004). Number is also the area that Australian Indigenous students find most difficult (Queensland Studies Authority, 2006).

Early number understanding can be approached from a counting perspective, where poetic counting is converted to one-to-one counting or to a quantitative perspective. Davydov (1982) notes that number denotes a quantity and suggests that comparing the elements of sets of similar objects and applying relationships-such as equal to, greater than, and less than--gives meaning to the notion of quantity. Comparing and ordering numbers is crucial to number development. Research also suggests that these comparisons can be made independently of the ability to count (Dougherty & Zilliox, 2003), and that counting emerges from the comparison of quantities and enables discussions about comparisons. It is this perspective that underpinned this research.

Patterning

Fundamental to mathematics are the relations among patterns. Patterns are bound to one another by operators and morphisms to yield 'lasting mathematical structures' (Steen, 1988). Patterns suggest other patterns, often yielding patterns from patterns. Abstracting patterns is the basis of structural knowledge, the goal of mathematics learning in the research literature (Jonassen, Beissner & Yacci, 1993; Sfard & Linchevski, 1994). Even the very description of what it means to do mathematics can be defined in the context of patterns. The literature suggests that the ability to recognise pattern is critical to early counting and number relationships (e.g., Mulligan, Prescott & Mitchelmore, 2003; Papic, 2007). For example, students may recognise that six dots can be made from groups of three dots and from groups of two dots by focusing on the pattern of dots as much as on the groupings. Understanding repeating patterns (patterns with a discernible unit of repeat) supports students in their development of multiplicative thinking, and in particular the concept of ratio (Warren & Cooper, 2007). Papic (2007) found that students who had participated in a repeating patterning intervention in the preschool year were more advanced in counting and arithmetic skills in Year 2 than students who had not participated.

This research adopts a decolonising methodology (Smith, 2005) that aims to deal with these issues in ways that give sustained beneficial outcomes for Australian Indigenous people. As such, the research recognises the considerable capabilities of young Australian Indigenous students as they commence school and aims to assist them to participate in meaningful dialogue concerning numeracy in order to meet the challenge of improving long-term educational outcomes. In this approach, the predominant aim is to change the focus away from Indigenous people as the objects of investigation (an approach that has received extensive criticism in the literature and has been labelled as Western research) and to one of mutual benefit to the researcher and the studied Indigenous community (Bishop, 1996; Irwin, 1994). As a consequence, researchers are required to think critically about the research processes and outcomes, ensuring that Indigenous people's interests, experience and knowledge are at the heart of the research methodologies (Rigney, 1999). From a practical perspective, this meant a continual monitoring of the appropriateness of the activities for the various contexts, a watchful eye on their effectiveness in terms of Indigenous student engagement and success, and ongoing dialogue with Indigenous researchers for their critical feedback on all aspects of the research. This paper begins to tease out actions that supported young Indigenous students' engagement with numeracy concepts.

Method

The research was conducted with seven Preparatory (Prep) classrooms from four schools in north Queensland. In Queensland, Prep is a non-compulsory year of schooling that students can attend once they reach the age of five years by the end of June. Within north Queensland a considerable number of schools cater for Australian Indigenous students and students from other cultures. The sample consisted of 7 teachers and 125 students (the average age of the students was 4 years, 11 months). The design was a multi-tiered teaching quasi-experiment with the seven teachers participating in professional dialogue or learning with the researchers on four occasions throughout the school year. On each occasion, all the teachers were released from their classrooms to participate in a day of professional learning. Subsequent to these days, the researchers visited all participating classrooms to continue professional dialogue and to assist teachers in trialling resources and activities. Discussions during these visits focused on mathematics learning in the early years. The dialogue covered three areas:

* the role of mathematics language in assisting young students to engage with mathematical thinking

* representations and activities that support mathematical learning in the early years with an emphasis on the language associated with these activities * how this learning underpins higher levels of mathematical understanding.

The focus of this paper is on the actions that assisted young Indigenous students' engagement with numeracy.

During discussions with their students, teachers promoted the explicit mathematical language embedded within learning activities. They also encouraged students to communicate orally about aspects of each activity, and assisted Indigenous students to distinguish between Aboriginal English and Standard Australian English in their communication. Initially the classroom focus was not explicitly on number but how various representations worked in a numberless world and the associated mathematical language. For example, each classroom was given a large five-by-five grid and the activities involved students playing games while using their whole body. These activities gave students opportunities to answer the questions and to talk about 'What is beside you?' 'What is behind you?' 'What comes next?' 'How do you move to that position on the grid?' 'Which row is it in?' 'Which column is it in?' Students were also encouraged to act out positional worlds in their home and school environments, with teachers recording these actions digitally, and with their parents writing sentences about their actions. In the later part of the year students then mapped this language onto contexts involving numbers: for example, 'What number is beside nine?' 'What number comes after nine?' 'What number is next?' 'What numbers are between three and eight?' 'How do you move from nine to eleven?' There was also an emphasis on constructing and analysing patterns, particularly repeating patterns and mapping this understanding onto number. An example was placing clear red and blue counters on the numbers 1 to 10 in the pattern red, blue, red, blue, red, blue, and discussing the characteristics of the numbers covered by the blue counters (the even numbers).

Data-gathering techniques and procedures

All schools were a two-hour plane flight from the researchers' home town making the sites difficult to visit. The data tended to be gathered in one-week blocks, with the researchers visiting the sites five times during the year. Data collection had four components:

* pre- and post-intervention tests

* student portfolios

* classroom observations

* teacher interviews.

Teacher interviews consisted of questions pertaining to three main components:

* the usefulness of the professional learning days for classroom teaching (for example, the mathematics ideas discussed, the learning activities and concrete materials supplied for classroom use, the adaptations made to these ideas and materials within the context of the classroom)

* student learning (for example, what students had learnt from engaging with activities, what difficulties were they experiencing)

* future needs (for example, in what areas were they having difficulties, what would they like us to focus on during our classroom visits).

Both teachers and researchers were observed in the classroom as they tried new activities. Teachers were also asked to collect evidence of each student's numeracy development. The pre- and post-intervention tests and teacher interviews occurred in March and November. The data reported in this paper relates to the pre- and post-intervention tests and student portfolios.

Given the age of the students, data was gathered for the pre- and post-intervention test using a one-on-one interview. The duration of the interview was approximately 30 minutes. Due to time restrictions and the movement of some of the students from participating schools, only 48 students participated in the pre- and post-intervention numeracy interviews. Table 1 summarises data relating to the participating schools and the number of students in each school who participated in the full data collection.

All participating schools were private schools from a large regional centre in North Queensland, in which the proportion of Indigenous people is higher than in other states. Based on the number of Aboriginal and Torres Strait Islander people counted in the 2001 Census of Housing and Population, the Australian Bureau of Statistics (2001) has projected that there were 501 236 Indigenous people living in Australia in 2006, with 27.8% of this population living in cities or regional towns in Queensland.

This purposive sample comprised all Indigenous students, and a selection of Australian students and students from other cultures representative of a range of abilities, as perceived by their classroom teachers. The pre-intervention interview occurred two months after the school year commenced, and selection was based on students' cultural backgrounds and teachers' perceptions of their abilities.

Instruments were developed to ascertain students' understanding of language, patterning and number. Students' oral language ability was measured by using a commercially produced test, the Boehm Test of Basic Concepts (Boehm, 2007). The Boehm test is a commercially produced readiness test administered to grade K-2 students and evaluates mastery of basic concepts involving quantity, space and time. It consists of 50 multiple-choice items administered orally. Students are asked to point to the picture that best represents the concept. Some typical concepts tested were students' understanding of last, least, most below, between, in front of, always, some, above and first. A typical request was 'Point to the picture where the chair is in front of the desk'.

The number component was developed by the researchers and was based on the mathematics component of school entry assessment (SEA), tool designed by the New Zealand government to provide teachers with information about some of the knowledge and skills children have when they first begin school. Originally, the context used was a shop, and assessments about students' learning were made as students played a shopping game. Our adaptation-school entry number assessment (SENA)--extended the types of questions asked, the way they were asked, and modified the context and materials so that they were more suitable for Australian Indigenous students, a process referred to by Morgan (2002) as taking into account the context of the situation and the context of the culture.

The instrument consisted of three main sections: number recognition, counting, and early addition and subtraction. The total score for the test was 28. Figure 1 presents some sample questions from the interview.

The patterning component was also developed by the researchers. Students were asked to copy, continue and create repeating patterns. They were also asked to identify the repeating component in the pattern, an ability that is believed to underpin the development of an understanding of repeated addition and multiplicative thinking. The total score for the test was 11. Figure 2 presents sample questions from the interview.

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[FIGURE 2 OMITTED]

Results

Pre- and post-intervention test results

The sample tested comprised of three different groups of students. These were Australian Indigenous students (including Indigenous Australian students and Torres Strait Islander students), students from other cultures where their first language is not Australian English (including students from Japan, Palestine, the Cook Islands, the Philippines and Papua New Guinea) and others (predominantly white Australian students). These groups were named Indigenous students (n = 14, average age = 4 years 11 months), Other culture students (n = 11, average age = 4 years 11 months) and Other students (n = 23, average age = 5 years).

The results of Levene's test for homogeneity suggested that the spread of scores across all groups for the pre- and post-intervention tests is approximately equal (F = 1.37, p = .26). Thus even though the sample size is considered small, the data is well suited for the Analysis of Variance test. ANOVA was thus used to compare the results of these three groups of students on the SENA tests (pre- and post-intervention) and the Boehm tests (pre- and post-intervention). Table 2 provides a statistical summary of the results for this test.

The results indicated that the only significant difference was in the pre-intervention test results for SENA. In order to ascertain the difference between the pre-intervention test results for the three groups,Tukey pairwise comparison was performed. The results of this test indicated that the mean score of the Indigenous group was significantly different from the mean scores for the Other and Other culture groups. Figure 2 portrays the mean scores for the pre- and post-intervention tests for SENA.

[FIGURE 2 OMITTED]

To ascertain if there was any significance difference between the pre- and post-intervention test results of the three groups of students, a repeated-measures Analysis of Variance was performed. Table 3 summarises the results of this analysis.

All three groups showed a significant improvement in their scores for the post-intervention test. The values of the Eta squared indicate that difference between the pre- and post-intervention tests for all groups for all tests was very large.

In summary, at the commencement of their formal schooling (preschool) the Australian Indigenous students' results for SENA were significantly lower than the results of the Other group and Other cultures group. All groups experienced a significant increase in their results for Boehm, SENA and patterning at the end of their first year of formal schooling. At the end of their formal schooling, there was no significant difference between the results for the three groups for Boehm, SENA or patterning.

Supporting processes

This next section begins to tease out some of the teaching actions that supported Australian Indigenous students' numeracy learning. These actions were identified from classroom observations, field notes and teacher interviews.

Oral language, particularly positional language

In the initial phases, teachers focused on developing students' oral language through encouraging students to communicate about their experiences and providing a classroom environment where oral language opportunities were maximised. They particularly focused on positional language contexts, using the vocabulary commonly used in mathematics. Students were encouraged to act out words in real-world contexts and to use these words to describe situations. One classroom created a 'take-home book' where parents were encouraged to take photos of their children and write some sentences about the photos, such as 'Sam is under the water. On top of him are some floating toys.'

As indicated by the results of the Boehm test, all children had gaps in their language relating to quantity, space and time. All teachers commented that focusing on students' oral language had an impact on other areas of curriculum. As one teacher commented, 'Their descriptive language is so much more enhanced than in previous years. They use long sentences to communicate about everything.' The focus was particularly important for Australian Indigenous students and students from other cultures. These students were not only learning new concepts but also learning how to communicate more effectively in Standard Australian English with regard to these concepts. The intervention also provided a language model that demonstrated the ways in which Westerners talk about ideas and the different styles of communication.

We saw a marked difference in the style of communication between the beginning of the year and its conclusion. During the post-intervention testing in the one-on-one interactions, Australian Indigenous students made eye contact and answered our questions using mostly Standard Australian English.

Starting from numberless contexts

Teachers were asked to start the students' mathematical experiences in numberless contexts. The theory that drove this decision was that of Davydov (1975). Generalising in numberless ways before numbered contexts enables investigation of concepts and structures not traditionally discussed until students have extensive knowledge of number (Davydov, 1975). For example, as students explored five-by-five grids and empty number ladders they were asked to use their positional language to describe where they were on a grid or number track with regard to others, or they were asked to place themselves in these representations according to instructions given (for example, to stand behind Jill, to stand beside John). Both the number track (a ladder without numbers on it) and five-by-five grids are important precursors to many key representations in mathematics: the number line, Cartesian graphs and the grids commonly used in positional and location contexts. From a Davydovian perspective, the students were discovering the structure of these representations before they were even introduced to number. Teachers also presented situations where comparative language was required, comparing different groups of counters and discussing who had more, who had less, who had the most and who had the least. As evidenced by the results of the pre-intervention test for SENA, many students from a Western background are more capable of one-to-one counting than their Australian Indigenous counterparts as they enter school.

The Davydovian (1982) perspective, comparing elements of sets of similar objects and applying relationships, giving meaning to the notion of quantity, was challenging to all students. We suggest that it changed the primary focus from students using one-to-one counting for comparing groups to visually comparing groups according to their magnitude. We conjecture that it also provided Indigenous students with a context that made counting meaningful--that it was used to ascertain if their 'guess' was correct. It also provided students with a language base that allowed them to say, for example, 'Six is more than three' and 'Three is less than six'.

Mapping oral language and representations onto number contexts

As the year progressed, the students were formally introduced to counting and numbers, particularly the numbers 1 to 10. As one teacher commented: Doing the grid first allowed students to explore placing numbers on it without wondering how the grid worked. They could ask each other questions such as 'I wonder what number is next?' 'Which one is before this one?' They were able to describe numbers beyond saying 'there are three birds' and include conversations such as 'Three is just after four', 'Four is before three', 'Three is between five and one', and 'Ten is more than five'.

Directed teaching episodes

Directed teaching episodes proved particularly important in the Australian Indigenous classrooms. Teachers not only encouraged students to make different 'maths stories' using concrete materials but also to describe these stories using correct mathematical language. In their conversations with these students, the teachers deliberately distinguished how you would express these ideas in Aboriginal English and Standard Australian English: for example, explicitly making the distinction between sentences such as 'He bin boney' and 'He bin bonier' to 'He is tall' and 'He is taller'.

The use of patterning

Once students were introduced to the notion of patterning, they began to find patterns everywhere, especially in numbered contexts. As some of the Prep students placed the numbers 1 to 25 on a five-by-five grid they started discussing patterns they were seeing in the columns of the grid. For example, 2, 7, 12, 17, 22--'That is the pattern 2, 7, 2, 7, 2' and 3, 8, 13, 18, 23--'That is the pattern 3, 8, 3, 8, 3'.

Discussion and conclusions

While it is difficult to draw conclusions from the data presented in this paper, it does offer some insights and questions for further investigation. First, the research suggests that young Indigenous Australian students do not commence school with the same understanding of number concepts as either mainstream Australian students or students from other cultures. As Christie (1985, p. 11) claimed: Western notions of quantity--of more and less, of numbers, mathematics, and positivistic thinking--are not only quite irrelevant to the Aboriginal world, but contrary to it. When Aborigines see the world, they focus on the qualities and relations that are apparent, and quantities are irrelevant.

In this respect, Indigenous students are already significantly 'behind'. As number forms an important foundation for 'western mathematics', Indigenous students begin school at a disadvantage. This is not surprising, as number is not historically an important dimension of Indigenous culture and thus not necessarily a focus of child-parent interactions. We conjecture that lack of attention to number in the home environment could lead to a student being at least two years behind at the end of Year 3 (Durodoye & Hildreth, 1995).

Secondly, this research begins to delineate learning activities and classroom discourse that begin to tackle this gap. We claim that the initial focus on oral language, and in particular on positional language, allows all students to begin school on a similar footing. The results of the Boehm test suggest that all these students began school with very similar levels of understanding of the semantics and vocabulary of positional language. Thus the initial focus on this aspect of mathematical learning allowed all students to begin the compulsory years of schooling on an equal footing. The results of SENA suggest that this focus, together with an awareness of Aboriginal English and Standard Australian English, began to deal with the disadvantage identified by Cairney (2003) and the long-term disadvantage identified by Dickinson et al. (2006).

Thirdly, the initial focus on oral language also allowed teachers to focus on the pragmatics of language, how members of a speech community use language. In particular, teachers explicitly focused on how to express ideas in Aboriginal English and Standard Australian English, identifying the differences between each and ensuring that both were valued. The linguistic means of communicating mathematics is vital to all students (Ferrari, 2004), and most students experience difficulties with this aspect of mathematics. Thus this study explored how practical contexts, vocabulary and semantics impact on young Indigenous students' mathematical knowledge and skill development as they became actively involved in learning in the preschool setting.

Fourthly, situating these discussions and experimentations in a play-based context together with focused teaching and learning allowed all students to engage with quite formal mathematical language in a non-threatening environment. The students' portfolios provided evidence that they grappled with some complex ideas, expressing these ideas in their drawings and collages. Teachers' recordings of key aspects of conversations with their students indicated that these students were beginning to 'play' with mathematical language.

Finally, beginning an exploration of key aspects of number in a numberless context allowed the development of language and meaningful situations that supported the development of counting and associated language. It is conjectured that focusing on oral language development and comparing the magnitude of different quantities in numberless contexts in the initial phases of schooling allows all students to begin an exploration of number on an equal footing. Our research is beginning to show that such an approach has an impact on the keenness and willingness of Indigenous students to engage in discussions about mathematics. Its impact on their self-efficacy needs further investigation.

Acknowledgements

Many thanks to Antoinette Cole for her ongoing support and insights.

References

Australian Bureau of Statistics. (2001). Census of Population and Housing, cat. no.

Balfanz, R., Ginsburg, H., & Greenes, C. (2003). The big maths for little kids early childhood mathematics program. Teaching Children Mathematics, 264-268.

Bishop, R. (1996). Collaborative research stories: Whakawhanaungatange. Palmerston North, New Zealand: Dunmore Press.

Boehm, A. (2007). Boehm test of basic concepts--Boehm 3 (3rd edn). San Antonio, NY: Pearson.

Booker, L. (2003). Starting school: Young children and learning cultures. Buckingham, UK: Open University Press.

Bourke, P. C. J., & Rigby, K. (2000). Better practice in school attendance: Improving the school attendance of Indigenous students. Canberra, ACT: DETYA.

Cairney, T. (2003). Literacy within family life. In N. Hall, J. Larson & J. Marsh (Eds.), Handbook of early childhood literacy (pp. 85-98). Thousand Oaks, CA: Sage Publications.

Christie, M. (1985). Aboriginal perspectives on experience and learning: The role of language in Aboriginal education. Melbourne: Deakin University.

Cooper, T. J., Baturo, A., Warren, E., & Grant, E. (2006). Aim high--beat yourself: Effective mathematics teaching in a remote Indigenous community. In J. Novotna, H. Morava, M. Kratka & N. Stehlikova (Eds.), Proceedings of the 30th Conference of the International Group for the Psychology of Mathematics Education, 2, 369-376. PME: Prague

Cooper T., Baturo, A., Warren, E., & Doig, S. (2004). Young white teachers' perceptions of mathematics learning of aboriginal and non-aboriginal students in remote communities. Proceedings of the 28th Conference of the International Group for the Psychology of Mathematics Education, 1, 239-246. Bergen: Bergen University College.

Cronin, R., & Diezmann, C. (2002). Jane and Gamma go to school: Supporting young gifted Aboriginal students. Australian Journal of Early Childhood, 27(4), 12-17.

Davydov, V. V. (1975). The psychological characteristics of the 'penumbral' period of mathematics instruction. In L. P. Steffe (Ed.), Children's capacity for learning mathematics (Vol. 2, pp. 109-205). Chicago: University of Chicago.

Davydov, V. V. (1982). The psychological characteristics of the formation of elementary mathematical operations in children. In T. P. Carpenter, J. M. Moser, & T. A. Romberg (Eds.), Addition and subtraction: A cognitive perspective (pp. 224-239). Hillsdale, NJ: Lawrence Erlbaum Associates.

Dickinson, D., McCabe, A., & Essex, M. J. (2006). A window of opportunity we must open to all: The case for preschool with high-quality support for language and literacy. In D. Dickinson & B. Neuman (Eds.), Handbook of early literacy research (Vol. 2, pp. 11-28). New York: The Guilford Press.

Dougherty, B., & Zilliox, J. (2003). Voyaging from theory and practice in teaching and learning: A view from Hawaii. In N. Plateman, B. Dougherty & J. Zilliox (Eds.), Proceedings of the 27th conference of the International Group for the Psychology of Mathematics Education, 1, 31-46. College of Education: University of Hawaii.

Durodoye, B., & Hildreth, B. (1995). Learning styles and the African American student. Education, 116, 241-248.

Ferrari, P. (2004). Mathematical language and advanced mathematical learning. In M. Hones & A. Fuglestad, Proceedings of the 28th Conference of the International Group for the Psychology of Mathematics Education, 1, 239-246. Bergen: Bergen University College.

Fleer, M., & Rabin B. (2007). Constructing cultural-historical tools for supporting young children's concept formation in early literacy and numeracy. Early Years, 27(2), 103-118.

Galloway, A. (2003). Researching change and literacy development. Paper presented at the Australian Association for Research in Education Conference, Auckland, New Zealand. Retrieved April 25, 2006, from http://www.aare.edu.au/03pap/gal03580.pdf

Ginsburg, H. P., Greenes, C., & Balfanz, R. (2003). Big math for little kids. Parsippany, NJ: Dale Seymour.

Gray, B. (2007). Accelerating the literacy development of Indigenous students. Darwin: Charles Darwin University Press.

Greenes, C. (1999). Ready to learn: Developing young children's mathematical powers. In J. Copley (Ed.), Mathematics in the early years, 39-47. Reston, VA: National Council of Teachers of Mathematics.

Greenes, C., Ginsburg, H., & Balfanz, R. (2004). Big maths for little kids. Early Childhood Research Quarterly, 19(1), 159-166.

Gustein, E., (2003). Mathematics for social justice in an urban, Latino school. Journal for Research in Mathematics Education, 34(1), 37-73,

Haig, Y., Konisberg, P., & Collard, G. (2005). Teaching students who speak Aboriginal English. PEN 150. Primary English Teaching Association.

Harrison, N. (2007). Where do we look now? The future of research in Indigenous Australian Education. Australian Journal of Indigenous Education, 36, 1-5.

Hill, S. (2006). Developing early literacy: Assessment and teaching. Melbourne: Eleanor Curtain.

Irwin, K. (1994). Maori research methods and processes: An exploration. Journal for South Pacific Studies, 28, 25-43.

Jonassen, D. H., Beissner, K., & Yacci, M. (1993). Structural knowledge: Techniques for representing, conveying, and acquiring structural knowledge. Hillsdale, NJ: Erlbaum.

Kennedy, A., Ridgway, A., & Surman, L. (2006). Boundary crossing: Negotiating understandings of early literacy and numeracy. Australian Journal of Early Childhood, 31(4), 15-22.

Luke, A. (2003) Literacy and the other: A sociological approach to literacy research and policy in multilingual societies. Reading Research Quarterly, 38(1), 132-141.

Matthews, S., Howard, P., & Perry, B. (2003). Working together to enhance Australian Aboriginal students' mathematics learning. Paper presented at the Proceedings of the 26th Annual Conference of the Mathematics Education Research Group of Australasia (MERGA 26), Geelong, Victoria.

Ministerial Council on Education, Employment, Training and Youth Affairs. (2007). National report on schooling in Australia: Preliminary paper national benchmark results--reading, writing and numeracy--years 3, 5 and 7. Retrieved May 1, 2007, from http://www.mceetya.edu.au/verve/_resources/ANR2004BmrksFinal.pdf

Ministerial Council on Education, Employment, Training and Youth Affairs. (2008). 2006 national report on schooling in Australia: Preliminary paper. Retrieved February 11, 2008, from www.mceetya.edu.au/mceetya/default.asp?id=22064

Ministry of Education. (1997). School entry assessment. Wellington: New Zealand Government Press.

Morgan, C. (2002). What does social semiotics have to offer mathematical research? Paper presented at the 26th Conference of the International Group for the Psychology of Mathematics Education. Semiotics Special Interest Group. Norwich, United Kingdom.

Mulligan, J. T., Prescott, A., & Mitchelmore, M. C. (2003). Taking a closer look at visual imagery. Australian Primary Mathematics Classroom, 8(4), 23-27.

Nakata, M. (2003). Some thoughts on literacy issues in Indigenous contexts. Australian Journal of Indigenous Education, 31, 7-15.

National numeracy review report (2008). Canberra: Commonwealth of Australia.

Papic, M. (2007). Mathematical patterning in early childhood: An intervention study. Unpublished doctoral dissertation. Macquarie University, Sydney.

Pappas, S., Ginsburg, H., & Jiang, M. (2003). SES differences for young children's meta cognition in the context of mathematical problem solving. Cognitive Development, 18(3), 431-450.

Partington, G. (Ed.). (1998). Perspectives on Aboriginal and Torres Strait Islander education. Katoomba, New South Wales: Social Science Press.

Prochner, L. (2004). Early childhood education programs for Indigenous children in Canada, Australia and New Zealand. Australian Journal of Early Childhood, 29(4), 7-15.

Queensland Studies Authority (2006). Overview of statewide student performance in 2005 Queensland literacy and numeracy tests. Brisbane: Queensland Government Press.

Rigney, L. (1999). Internationalization of the indigenous anticolonial culture: Critique of research methodologies. Wicazo Sa Review, 14(2), 109-121.

Sarra, C. (2003). Young and black and deadly: Strategies for improving outcomes for Indigenous students. In Quality Teaching Series Practitioner Perspectives, Australian College of Educators. Retrieved May 12, 2006, from http://www.daretolead.edu.au/servlet/Web?s=169694&p=RA_NSW_CHERBOURG

Sfard, A., & Linchevski, L. (1994). The gains and the pitfalls of reification--the case of algebra. Educational Studies in Mathematics, 26(2-3), 191-228.

Smith, L. (2005). On tricky ground: Researching the native in the age of uncertainty. In N. Denzin and Y. Lincoln (Eds.), The Sage handbook of qualitative research, 85-108. Thousand Oaks, CA: Sage Publications.

Steen, L. (1988). The science of patterns. Science, 240, 611-616.

Tayler, C., Thorpe, K., & Bridgstock, R. (2006). Early education and care experiences as predictors of social and learning attainments in the first term of school. Poster presentation at the International Society for the Study of Behavioural Development Conference, Melbourne, Australia, 2-6 July.

Thomson, S., & Fleming, N. (2004). Summing it up: Mathematics achievement in Australian schools in TIMSS 2002. TIMSS Australia Monograph no. 6. Melbourne: ACER Press.

Thorpe, K., Tayler, C., Bridgstock. R., Grieshaber, S., Skoien, P., Dany, S., & Petriwsky, A. (2004). Preparing for school: Report on Queensland preparing for school trials. Retrieved 7 November, (2008), from education.qld.gov.au/etrf/pdf/qutreportqldpreptrial.pdf

Trends in International Mathematics and Science Study. (TIMSS) (2003). Retrieved June 12, 2008, from http://timss.bc.edu/timss2003i/released.html

Vygotsky, L. (1962). Thought and language, trans. E. Hanfmann & G. Vicar. Cambridge, MA: Massachusetts Institute of Technology. (Original work published in 1934.)

Warren, E., & Cooper, T. J. (2007). Repeating patterns and multiplicative thinking: Analysis of classroom interactions with 9 year old students that support the transition from known to the novel. Journal of Classroom Instruction, 41(2), 7-11.

Elizabeth Warren

Australian Catholic University

Eva deVries

Independent Schools Queensland

Authors

Elizabeth Warren is Professor and Associate Dean Research at the Australian Catholic University. Email: elizabeth.warren@acu.edu.au.

Eva deVries is Program Officer (Numeracy/Mathematics) for Independent Schools Queensland. Table 1 Total numbers of students and schools attended School Indigenous Number of students school? Indigenous Other Other culture A No 4 10 3 B No 0 8 6 C No 2 4 2 D Yes 8 1 0 Total 14 23 11 Table 2 Comparing the three groups' pre- and post-intervention scores for Boehm, SENA and patterning Test F value p value Oral language test (Boehm--pre-intervention test) 1.32 .276 Oral language test (Boehm--post-intervention test) 0.69 .506 School entry number assessment 6.57 * .003 (SENA--pre-intervention test) School entry number assessment 2.02 .145 (SENA--post-intervention test) Patterning (pre-intervention test) 0.7 .506 Patterning (post-intervention test) 0.21 .812 * Significant p<.005 Table 3 Results of Analysis of Variance for BOEHM, SENA and Patterning Test Group F value Eta squared Boehm oral Other [F.sub.(1, 22)] = 52.19 * .70 language assessment Indigenous [F.sub.(1, 13)] = 25.74 * .66 Other cultures [F.sub.(1, 10)] = 134.85 * .93 SENA Other [F.sub.(1, 22)] = 44.08 * .67 Indigenous [F.sub.(1, 13)] = 61.29 * .83 Other cultures [F.sub.(1, 10)] = 40.87 * .80 Patterning Other [F.sub.(1, 22)] = 57.93 * .90 Indigenous [F.sub.(1, 13)] = 54.88 * .77 Other cultures [F.sub.(1, 10)] = 27.61 * .74 * Significant p<.005