< All Reports | Print this Section | Back to Contents >

Chapter 1 - Theoretical framework

This chapter outlines the main components of the project, Training Indigenous teacher-aides and parents to support mathematics learning of Indigenous educationally disadvantaged students in junior secondary school. It describes the community in which the research was undertaken (Section 1.1), the project in terms of its aims and outcomes, its plan for meeting the aims, and the impact of the context on the plan (Section 1.2). It also states the rationale for undertaking the project (Section 1.3), provides a framework for learning and teaching mathematics (Section 1.4) and overviews the report (Section 1.5).

Research site

The research project was undertaken at Woorabinda Aboriginal Community, a rural community that is almost 2 hours' drive from any large town or city. Because only a small percentage of the Community own cars, it is, to a large extent, isolated. Woorabinda Aboriginal Community has a population of approximately 1200 people but has minimal services. For example, it has a small grocery store, a butcher shop, a delicatessen, a combined post office/bank, a hospital with no resident medical staff, and a police station that has Indigenous and non-Indigenous police.

The Community has a kindergarten, a P-7 primary school that is administered by Education Queensland from the Emerald District Office and an 8-12 high school that is administered by a council elected from the Community and is part of the Association of Independent Schools of Queensland (AISQ) Inc.

Employment levels for Indigenous adults are currently very low, necessitating a reliance on welfare, and leading to most of the problems described by Fitzgerald (2002) and the Department of Aboriginal and Torres Strait Islander Policy and Development (1999), namely, alcohol and substance abuse, teenage pregnancies, family violence, youth suicides, low health with concomitant low life expectancy, transient population, discrimination and exclusion, with a potential cost of millions of dollars to the nation.

Woorabinda's history is one of shame for the nation. It was formed by forcibly removing people from North Queensland and Taroom to live under white governance from approximately 1927-1975. This result was that at least 17 different language groups were thrust together without regard to the culture and mores of the groups. The legacy of this is still evident today in Council elections. Additional damage was done when the Government withdrew without an adequate hand-over period, leaving Indigenous people (who had been educated to Grade 4 standard) to come to terms with the white laws and government procedures. The fact that Woorabinda Aboriginal Community has developed a thriving cattle industry and has established the services required to run a small town is a credit to the people in this Indigenous community.

However, one of the most damaging outcomes of being freed from white control was that the next generation became a "lost" generation in that they tended to rely on welfare instead of finding employment whereby alcoholism and gambling became major problems. As one Elder said, "This generation didn't pass on the stories." Therefore the following generations had, to a large extent, lost their culture and respect for the Elders was diminished. The present Councils are attempting to redress this situation.

The older people at Woorabinda Aboriginal Community see education as the vehicle through which their community will thrive and prosper. Thus, they want their young people to be educated to take their place in a white society but to retain their Indigenous mores and culture. They have also indicated the need for education in social, health, law and financial issues for its adult population.

The project

The project was facilitated through the 2002 Innovative Projects Initiative which was funded by the Department of Science and Education (DEST) and undertaken collaboratively with Wadja Wadja High School, Woorabinda Aboriginal Community, the Association of Independent Schools of Queensland, Inc. (AISQ), and Queensland University of Technology (QUT).

The project's overarching aim was to improve mathematics learning outcomes for low-achieving Years 8-10 students at Wadja Wadja High School to where they could meet VET Level 1 numeration competencies (2-and 3-digit place value).

Objectives and outcomes

The objectives of the project were:

1. To develop mathematics learning activities appropriate for educationally disadvantaged Years 8-10 Indigenous students in Wadja Wadja High School.
2. To develop a training program that would enable IEWs to use the mathematics learning activities (including computer-based activities) to support the Years 8-10 students.
3. To train IEWs in the program.
4. To evaluate the effectiveness of the program in terms of the support these IEWs provided to the Years 8-10 students at Wadja Wadja High School.
5. To prepare a package of mathematics support materials and IEW training materials that could be used in other schools.

The project had initially planned to include parents as well as IEWs but it failed to attract parents and it was therefore undertaken with IEWs only.

Because of the importance of 3-digit whole numbers as the basis of understanding mathematics (and our earlier experiences in the Community had shown that the Years 8-10 students were still grappling with this understanding), the training program would strive to improve the mathematics learning outcomes of the Years 8-10 students by:

1. developing a sequence of mathematics activities, including computer-based activities, effective in improving Indigenous students' basic numeration understanding
2. developing a program (named Train a Maths Tutor Program) which comprised training and trialling cycles to support the IEWs in the use of the activities
3. developing a commercial package of materials, the Tutor Training Package, comprising Student Program, Tutor Program, Training Manual, and mathematics aids that could be used in other schools.

The specific outcomes of the project were to improve the mathematics knowledge of the Years 8-10 Wadja Wadja High School students in order to enhance their life chances, and to empower IEWs at Woorabinda through improved mathematics and mathematics tutoring knowledge and skills. The general outcomes of the project were increased knowledge of effective mathematics training programs for IEWs, a collection of effective materials for supporting the mathematics learning of these students, and a package of materials that schools could use to train IEWs to support the mathematics learning of educationally disadvantaged Indigenous junior secondary students.

Plan

The project was designed to incorporate four major stages, namely, developing mathematics activities for the Train a Maths Tutor Program, training the IEWs in the use of concrete and virtual materials and concomitant teaching questions needed to promote the mathematics outcomes embedded in the activities, trialling the activities with students, and finalising the materials into a package that could be used by other teachers. Figure 1 shows these stages and timelines.

Figure 1. Intended project stages and timelines

To facilitate transfer of knowledge from training to trialling, design and timelines were modified to amalgamate Stages 2 and 3 to be undertaken across one month only. (See Figure 2.) Whist training and trialling were still undertaken contiguously (training followed by trialling), the modified plan incorporated contingent training and trialling (trials dependent on training, next training dependent on the success of previous trialling) within the one timeframe.

Figure 2. Modified stages and timelines for the Train a Maths Tutor Program

The original plan was for Stages 2 and 3 to run for four weeks because this length of time represented the best balance between IEWs' learning needs and IEWs being absent from their school duties. However, the 4-week block set aside for training and trialling was insufficient to include computer training and trialling (see Section 1.4). For the reasons provided in Section 2.2, this component was delayed until mid-November and one week only could be allocated to it. Therefore, as shown by Figure 3, the plan was further modified so that the training was in two blocks - Block A (Concrete Materials) and Block B (Virtual Materials).

Figure 3. Final stages and timelines for the Train a Maths Tutor Program

Although the program was in two Blocks, a graduation ceremony was held at the end of Block A to celebrate the successes of the IEWs and their dedication to the first four weeks of the Train a Maths Tutor Program.

Impact of context on the research design

The Train a Maths Tutor Program was designed for IEWs and parents. Attendance of the IEWs was organised through the two schools - the Community-controlled Wadja Wadja High School and Education Queensland's Woorabinda Primary School. Attendance of parents was sought through extensive advertising at Wadja Wadja High School's open day, over the local community radio, via brochures (see Appendix B, Slide 1) at key points around the Community, and through personal contacts made by Mrs Eunice Munns (Wadja Wadja High School's Community Liaison Officer) and Ms Kylie Thompson (QUT Research Assistant). However, despite this advertisement and expressed interest of many parents prior to the training dates, no parent attended any session across the five weeks of training.

The Train a Maths Tutor Program was designed to integrate mathematics knowledge with mathematics pedagogy knowledge through activities involving concrete (real) materials and virtual (computer) materials. However, as described in Section 2.2, training was undertaken in small components of mathematics and pedagogic knowledge, followed by immediate applications of this knowledge to a trialling-with-students episode and progressed more slowly than expected. The slowing down of the training and trialling with concrete materials resulted in delaying the computer training component until November. This impacted on the trialling-with-students component of the program as attendance at the high school had diminished considerably by November which meant that computer training only could be undertaken.

Because the training and trialling episodes in Block A were planned in 4-hour daily sessions from 9 a.m. to 1 p.m. Monday to Thursday, enormous strain was put on the classrooms from which the IEWs were drawn. Furthermore, there were pre-planned dance activities (e.g., NAIDOC) in other districts that required participation of some of the IEWs. Therefore, there were times when some IEWs were unable to attend a session during Block A and thus there was not the continuity across all training and trialling sessions which the IEWs themselves would have liked.

In order to offset the first problem with respect to presence in classrooms, Block B computer training was held in the afternoons from 12 to 4 p.m.

Rationale

At the time of the project, the mathematics situation within the two schools in Woorabinda Aboriginal Community was characterised by low student performance, attendance and motivation, inadequate teacher preparation, support and experience for teaching Indigenous students, and inadequately-trained and inappropriately-used IEWs. Most teachers did not to know how to work effectively with Indigenous IEWs and tended to use them as behaviour managers rather than teaching partners. Because of the isolation and difficulties in teaching, most teachers stayed only for the time necessary (usually two years) to gain enough points within Education Queensland's Incentive Scheme to apply for another position elsewhere.

This section looks at the rationale for the project: the mathematics disadvantage of Indigenous students (Section 1.5.1) and the possible roles of the IEWs in overcoming this disadvantage (Section 1.5.2).

Indigenous students and mathematics disadvantage

Performance in diagnostic tests showed that many Indigenous students, even in Years 8-10, did not have an accurate understanding of, among other things, whole-number numeration to hundreds (two and 3-digits) and seemed to have greater difficulty than non-Indigenous children in moving from Number in terms of counting single objects to Number as groups of tens and ones and as groups of hundred, and tens and ones (Baturo, 2003). As such, they had not achieved VET Level 1 competencies and were unable to undertake VET courses that would lead to a trade, thus restricting their employment and life chances.

Indigenous students continue to be the most educationally disadvantaged group in Australia with a reducing retention rate in the post-compulsory years of schooling (Long, Frigo, & Batten, 1999). With their consistently lower levels of academic performance and higher rates of absenteeism (Bourke & Rigby, 2000; Gray, Hunter, & Schwab, 1998; South Australia Department of Education, Training and Employment {DETE}, 1999), they are poorly prepared for employment as adults and will therefore be unable to share the benefits of modern society.

As said earlier, many Indigenous high school students lack an understanding of 2- and 3-digit whole-number numeration which is required for Vocational Education and Training (VET) Level 1 (Baturo, 2003), the minimal standard for entry into TAFE traineeships and apprenticeships. To leave school without an understanding of VET Level 1 is to severely reduce academic and employment opportunities and life chances in general. However, there is a paucity of research on what Indigenous students know in mathematics, and how this knowledge can be enhanced through appropriate classroom instruction.

Table 1 provides data on Queensland Indigenous students' (ATSI) means with respect to early number and 2-digit numeration collected via the Year 2 Diagnostic Net, and is compared against the State average. Phases A, B, and C are points on a continuum and roughly correspond to Preschool, Year 1 and Year 2.

Table 1: Year 2 Number Diagnostic Net distribution and mean student performance (2000-2002)

Table 1 shows that, from Years 2000-2002: (a) only 32% of ATSI students show an understanding of 2-digit numeration (Phase C) as opposed to approximately 60% for the rest of the State; and (b) many more ATSI students (approximately 13%) fail to progress beyond a Preschool level of numeracy (Phase A), as opposed to the State (approximately 2.5%).

Therefore, this project (which focused on training IEWs to tutor Indigenous students effectively in 2- and 3-digit numeration) appeared to offer the best chance for long-term improvement in students' performance in Number and access to VET training and the increased employment that comes from this.

Role of IEWs

The IEWs at Woorabinda were similar to those described in Ashbaker and Morgan's (2001) review of literature on IEWs:

[IEWs] are typically long-term, local residents, mostly women, who work part-time for modest wages. They are often parents or grandparents of students, and therefore have a vested interest in the success of both the school and the wider community. [IEWs] often represent racial and ethnic minority groups in the community, bringing knowledge of other languages and cultures into the school. (p. 2)

Therefore, IEWs have the potential to be the key to stability in Indigenous schools and to have an important role in the mathematics learning and teaching of Indigenous students (Baturo, Cooper & Warren (2004); Clarke, 2000; DETYA, 2000; McInerney, Smyth, Lawson, & Hattam, 1999). Their familiarity with the mores of the community and the first language of the students (Aboriginal Australian English) means that IEWs should be better able to support Indigenous students to connect to the formal mathematics language required for effective teaching and learning of the subject than the short-term non-Indigenous teachers.

According to Queensland Department of Education (1984), IEWs have four main functions: general (work with one or more teachers to facilitate preparation of classes), specific roles, bilingual support (work closely with a teacher to promote understanding for children in their own language), and culture (teach children about culture). As the document emphasised, IEWs are not to be used as:

… an agent to shuttle between the two groups [Indigenous students and non-Indigenous teachers] and leave them still isolated from each other. The IEW is an agent to bring the two together with understanding. (p. 9).

It maintained that IEWs should have a job description and undertake continuous training. However, due to variations in their experience and knowledge and the schools' perceptions of their roles, the functions and treatment of IEWs at Woorabinda varied and they often had no clear role description. Education Queensland's website (2004) stated that certificate courses were offered to provide IEWs with the skills that they required, but these courses were not available in Woorabinda.

The problems that Indigenous IEWs had at Woorabinda in the way they were used in schools and in the teaching and learning of Indigenous students were similar to those described in Education Queensland (2002):

People employed through the Indigenous Education Strategic Initiatives Program (IESIP) as IEWs often report being requested, or feeling obliged, to take on roles outside their position description. Most commonly they perform duties that are more appropriately the responsibility of guidance officers, community liaison offers or community education counsellors (CECs)… Taking on these additional responsibilities, for which they have no training, can be a cause of considerable stress and there is also a feeling of being used as these other roles are generally filled by permanent officers on higher rates of pay. (p. 32)

Similar to the situations described in Education Queensland (2002), principals of the two schools at Woorabinda had difficulty finding IEWs with suitable skills. The IEWs that were employed lacked skills in literacy and numeracy and were affected by health and social issues. Again, similar to the situations described in the Education Queensland publication (2002), some of these IEWs also seemed to suffer from domestic violence, alcoholism, have a gambling addiction, or have family responsibilities that interfered with their performance at the schools and therefore interrupted the chances they would potentially be afforded to participate more effectively in classroom teaching and learning.

Some of Woorabinda's IEWs were enrolled in the Remote Area Teacher Education Program from James Cook University. In this, they were meeting a concern of WA Department of Education (2004) that there should be more Indigenous teachers who would be enabled to take a greater responsibility in schools and to be involved in guiding education policy. However, after nine years of the course's availability, no IEW in Woorabinda had graduated from the course.

Overall, the role of the IEWs in the two Woorabinda schools was problematic. Their effectiveness appeared to depend on how their role was perceived by non-Indigenous teachers, their own skills and ability levels, the amount of training they had access to, the motivation and opportunity for them to participate actively in the curriculum and how they perceived their role in the context of the school community.

Mathematics learning and teaching

This section provides a brief outline of the decimal number system used in Western society (Section 1.4.1), current theories relating to learning and teaching this system (Section 1.4.2), the role of virtual materials and computers in such learning and teaching (Section 1.4.3), and recent findings with respect to Indigenous students and mathematics learning (Section 1.4.4)

The decimal number system

The number system used throughout the Western world is referred to as the Hindu-Arabic system or the decimal number system (base 10, and therefore, 10 digits). It was developed thousands of years after other number systems such as the Babylonian or Egyptian, and hundreds of years after the Roman system. It is best taught using materials and discussion in familiar contexts.

The decimal number system has an inherent additive feature (i.e., 675 = 600 + 70 + 5) and a set of symbols to represent different values. However, what makes the decimal number system powerful is its notion of place value which is based on a multiplicative feature (i.e., 675 = 6 × 100, 7 × 10, 5 × 1) which utilises only the 10 digits (Baturo, 1997; 2000). The unit (one) may be grouped to form larger units (e.g., tens, hundreds) or it may be partitioned to form smaller units (e.g., tenths, hundredths). Thus, a value is represented by its position/place on a continuum of places.

The decimal number system features two powerful patterns, namely, the grouping of whole numbers in periods of three digits (hundreds, tens, ones of ones; hundreds, tens, ones of thousands; hundreds, tens, ones of millions; etc), and a multiplicative structure (see Figure 1) which is continuous across whole-number and decimal-number places, is bi-directional (× "shifts" numbers to the left; ÷ "shifts" numbers to the right), and is exponential - 100 (ones), 101 (tens), 102 (hundreds), and so on. Three-digit whole-number numeration (hundreds, tens, ones) encapsulates both patterns and is crucial to the development of understanding of the number system and the mathematics topics associated with this.

Figure 4. The multiplicative structure embedded in the decimal number system (Baturo, 1997)

The number system structure is manifested in the numeration processes, namely, reading, recording, representing numbers, place value (e.g., "write the number that has 3 tens, 2 hundreds and 6 ones"), counting, seriating (e.g., "count on from 612 in ones/tens/hundreds"; "what is 100/10/1 more than 358?"), comparing, ordering (e.g., "which is larger, 387 or 378?"; "arrange 482, 804, 842 in order from largest to smallest in value"), renaming (e.g., "what number has 26 tens 5 ones"; "5 tens 4 ones has the same value as 4 tens and how many ones?"), and approximating (e.g., "round 372 to the closest hundred"). Prerequisite to the numeration processes are the early number processes of sorting, classifying, patterning, rote and rational counting, digit recognition, and early grouping. (Rote counting consists of saying the number names in the correct order; rational counting consists of one-to-one correspondence; the last number named tells how many altogether; knowing what, in the set, has been counted and what has not been counted; and knowing that the counting path does not change the number in the set;) early grouping consists of making groups of 3-10 objects and developing the language − three groups of 4 and 2 ones -->3 fours 2 ones).

Learning and teaching the number system

Mathematics consists of things, relations between things, and transformations of things (Scandura, 1971). Within this paradigm of mathematics, importance lies in the relations and transformations not in the things; yet, within our research experience (Baturo, 1998; Baturo & Cooper, 2001), teachers tend to focus primarily on the "things" and neglect or downplay transformations that often give rise to patterns and therefore relationships. Mathematics learning is about the refinement, abstraction, and integration of concepts and processes, that is, to acquire structural knowledge (Sfard, 1991) of mathematics. Mathematics teaching is about facilitating this process of refinement, abstraction, and integration.

Construction of structural knowledge

The construction of structural knowledge requires teacher and learner to share an active role in the construction and in making sense of the knowledge thus constructed through discussion, reflection and validation. Thus teaching and learning become a two-way interactive process which Lampert (1986) described as "sense-making" (p.340).

Structural knowledge implies succinct yet global and integrative cognitive knowledge which facilitates access to solutions in a variety of problems. Access is facilitated because of the reduced cognitive load (succinctness) and because the integrative aspect facilitates access from a variety of perspectives. For example, the notion of fraction as "part of a whole" is succinct, integrative (the relationship between the part and the whole and the whole and its parts), and global (is applicable to fractions, per cents, probability) and is easily extended to accommodate the part/part notion of ratios.

In summary, structural knowledge of mathematics is best promoted in a social-constructivist learning environment where teachers and teacher-aides provide cognitive scaffolding (guidance) for students so that they can move from dependency upon experts or peers to being able to solve particular problems independently (Ertmer & Cennamo, 1995). In such an environment, language, along with materials, has a central role in articulating mathematical knowledge in both oral and written modes (Gibbs & Orton, 1994).

Materials, kinaesthetic activity and language

The use of external materials (e.g., real-world, concrete, pictorial) to represent a mathematical concept or process is essential in helping learners construct mental models which promote structural knowledge and connections (Baturo & Cooper, 2001; English & Halford, 1995; Halford, 1993; Payne & Rathmell, 1977). This is best promoted through posing a real-world problem of interest to the learner and then representing the problem with external materials, developing the appropriate language, and finally attaching the symbol to either the number or action. (See Figure 5.)

Figure 5. Adaptation of Payne and Rathmell's (1977) model of the components and interactions required for concept construction and integration

However, it is not enough to merely demonstrate the use of materials - students need to physically manipulate the materials to develop a kinaesthetic memory of the actions that accompany the overt representation of the concept or process. Thus students should firstly construct various mental representations (e.g., concrete, pictorial and symbolic) of numerical concepts and processes. From this foundation, they can then progress towards developing relational (integrative) understandings of abstract mathematical concepts by making connections between the various modes of representation, that is, progress towards structural knowledge. Figure 6 provides a sequence of materials which will help learners progress from concrete to abstract when learning number.

Figure 6. Sequence of representations in mathematics learning and teaching

Thus, as is described in Section 3.1, effective learning and teaching of structural understanding of mathematics is underpinned by a focus on abstract mathematics, the kinaesthetic use of materials, and appropriate language.

Role of virtual materials and computers

Until the advent of computers in classrooms, materials were limited to real-world, concrete materials and pictorial representations. Although calculators have been available for at least 20 years, they are not being used to develop structural knowledge in most classrooms today.

Concrete to pictorial material represents a sequence of abstraction because concrete materials can be physically manipulated whereas pictures cannot - the child is expected to imagine any manipulation that may have been required to transform, for example, 10 ones to 1 ten (and, conversely, 1 ten to 10 ones). Baturo & Cooper, 2001) believe that, theoretically, virtual materials (computer replications of concrete materials) are more abstract than concrete materials but less abstract than pictorial representations and therefore are able to help bridge the gap from concrete to pictorial representations and, then, to abstraction. This belief is aligned with that of Noss, Healy, and Hoyles (1997) who contend that, although students can mentally replicate the relations and transformations represented by concrete material and can abstract this mental replication to symbols and mental models, there is a gap between action and expression that is difficult to bridge.

Calculators and spreadsheets are essential for helping learners abstract the mathematics from the kinaesthetic actions. For example, knowing that counting forwards is adding so that counting forwards by ones is adding 1 (+ 1) and counting forwards by 2s, 5s, 10s translates mathematically to + 2, + 5, + 10. Conversely, counting backwards translates mathematically to subtracting.

Most activities undertaken in elementary mathematics classrooms with concrete materials involve sliding, joining, separating, grouping, ungrouping, partitioning, turning and flipping actions, all of which can be replicated with virtual materials (Baturo & Cooper, 2001), although there are nuances in differences between the materials (e.g., virtual actions are neither as overt as they are with real representations nor as covert as they are with pictorial representations). Baturo and Cooper argued that the multisensory nature of concrete materials may develop more detailed memory structures (schema), than virtual materials (e.g., tactile as well as kinaesthetic memory) but because there are fewer attributes to confuse the new learner, the less detailed/more abstract bisensory virtual materials have the potential to develop more connected mathematics understandings. Thus virtual materials are able to form a cognitive scaffold in the abstraction process of constructing mathematics schema.

Thus, mathematics teaching is enhanced if instructional activities have a dual focus, that is, they relate to mathematics and to computers. This requires considering the ways in which mathematics and computers act reciprocally with respect to knowledge and the construction of and access to knowledge. Table 2 provides some examples of the reciprocal relationship between mathematics and computers.

Table 2: Examples of the reciprocal relationship between mathematics and computers

However, the effectiveness of virtual materials is related to how they are used on the computer, particularly the relationship between the learner and the computer. The learner-computer relationship is governed by teachers' choices with regard to computer use. These choices relate to teacher beliefs as there appears to be a correlation between beliefs about mathematics and how computers are used in learning (Niederhauser & Stoddart, 1994). For example, if teachers believe that mathematics consists of facts and procedures, then they are most likely to use computers as transmitters of knowledge and their preference is for drill-and-practice software. If they believe that mathematics is a hierarchical and integrated structure and is best learnt through connecting representations, then they are more likely to use computers as just another tool in the development of mathematics knowledge. However, if teachers believe that mathematics is best learnt when students construct their own knowledge, then they are more likely to think that computers are most effective when controlled by students.

However, even if teachers agree that the computer can be just another tool in the mathematics learning process, most believe that they cannot develop their own mathematics activities. They tend to rely on finding the "right" software which is often time consuming and ultimately unsatisfactory as they lack the control to modify the activities to suit their needs (Becker, 1994). Moreover, even those teachers who have computer expertise find it difficult to translate effective mathematics teaching and learning to a computer environment. As Eraut (1994) conceded, "using an idea in one context does not enable it to be used in another contextwithout considerable further learning taking place" (p. 33).

Research also indicates that some software could actually distort learning in that its presentation may be more attractive to students than the content to be learnt so the focus of learning becomes computers and computing rather than knowledge construction (Day, 1996). Some software may also have such great computational power that processes behind concepts are hidden and understanding prevented (Doxey, 1997) and may rely too much on virtual or simulated experiences and not provide the more complete learning that comes from real experiences (Doxey, 1997).

The student-computer interaction is a particularly important issue. Sivin-Kachala, Bialo, and Langford (1997) claim that student learning is enhanced if this interaction is active and empowering (with the student in charge of the computer, either using the computer as a tool) rather than passive and controlling (with the student being directed by the computer). Successful learning tended to change significantly students' conceptions about the nature and discourse of the subject-matter being studied (Clements, 1994)

Another important issue is the link between teachers' beliefs about computer learning and computer usage. Research (e.g., Sarama, Clements, & Jacobs-Henry, 1998) has shown that if teachers believe that mathematics cannot be taught effectively with computers, then they will resist attempts to incorporate them in their classrooms. There exists, then, a need to provide mathematics computer activities that teachers feel are easy to develop, do not require specialist software, and will promote positive learning outcomes.

Thus, as described in Section 3.6, computers will be used in the Program through virtual materials using the generic Office software PowerPoint in an active and empowering way.

Indigenous students and mathematics learning

As a culturally-based discipline (Wilder, 1982), mathematics learning and teaching is an act of enculturation (Bishop, 1988). However, as present mathematics reflects white middle-class culture (Walkerdine, 1992), it conflicts with Indigenous culture. Therefore, learning mathematics can be problematic for many Indigenous students.

As argued by Matthews (2003), there is a systemic issue at the basis of Indigenous students' poor educational outcomes. By reflecting dominant society's views particularly with respect to progress and technology, education systems have devalued Indigenous culture and marginalised it as primitive, simplistic and insignificant with respect to mathematics education (Bourke, Rigby & Burden, 2000; Matthews, Howard & Perry, 2003; Sara, 2003) leading Indigenous people to believe that "they must become ‘white' to succeed" (Matthews, 2003).

To ameliorate this cultural clash, there is a move towards contextualisation, that is, using Indigenous contexts in the real world problems that begin the model-language-symbol interactions of Figure 5. Incorporating Indigenous culture and perspectives into the pedagogical approaches to mathematics education (Baturo, Norton, & Cooper, 2004; Matthews, 2003) is being seen as the way to overcome the systemic issue of Indigenous marginalisation with respect to mathematics learning (Cronin, Sarra & Yelland 2002; Jones, Kershaw & Sparrow, 1996; NSW Board of Studies, 2000).

However, the Indigenous education literature is somewhat ambivalent about the extent of the role of culture in mathematics education. Some researchers (e.g., Day, 1996; Gool & Patton, 1998; Malin, 1998) argue that successful educational performance, motivation, and attendance is primarily linked to teaching that takes account of culture as well as "hands on" projects. Thus, Indigenous students learn best when they are given learning tasks that are concrete and contextualised. Malin (1998) urges us to consider Indigenous learners as individuals who, like all learners, bring to the learning situation their own particular skills, talents, personality, knowledges and history, rather than considering them in terms of a uniform and homogenous culture. Malin's point of view has more recently been supported by Mellor and Corrigan (2004).

The ambiguity that exists in the literature regarding contextualising mathematics could stem from the different types of Indigenous communities in which prior research has been undertaken. For example, early research (e.g., Harris, 1980) was undertaken in "traditional" communities (i.e., homogeneous culture/language) communities.

There exists in Australia today, many communities of Indigenous peoples. Some are situated in rural areas and some in town/city areas. Rural communities may comprise homogenous traditional tribal groups that have one language; others may be heterogenous tribal groups (such as Woorabinda) that have been thrust together and therefore have several language groups. The homogenous groups may be thought of as "intact" cultures whereas the heterogenous groups may be thought of as "damaged" cultures. Urban (city/town) groups may be even more "damaged" as they have little opportunity to maintain their Indigenality.

With respect to learning, there is some evidence that Indigenous students "have greater sensitivity and success in dealing with visual and spatial information compared to verbal" (Barnes, 2000, p. 10), and "learn by observation and non-verbal communication" (South Australia DETE, 1999, p. 10). Indigenous students have difficulties with standard English and may not have the words to describe many mathematical ideas (Roberts, 1999) and the words they have may be ambiguous (Durkin & Shire, 1991).

Although there are some studies of Indigenous learning of mathematics in classroom settings (Nichol & Robinson, 2000), there appears to be little systematic research undertaken into how Indigenous students come to know and understand mathematical knowledge within their own culture and of how this knowledge can be developed into abstract mathematics with responsive teaching.

However, when contextualising mathematics learning, one needs to ask whose context/culture. For example, in a heterogenous culture such as Woorabinda, there is the culture of the grandparents (often the care-givers for the children) which is closest to the traditional mores of each language group, the culture of the parents which is damaged and, in many instances, disconnected from the traditional mores, and the culture of the children themselves which is more strongly associated with African-American mores (gained from television) and more inclined to revere sporting heroes and pop stars than the Elders of their Community.

Furthermore, there appears to have been no mention of the role of the effect of learning spaces in the literature on Indigenous mathematics learning. Graetz and Goliber (2002) claim that "the important role of the physical environment in shaping human social interaction" (p. 13) is ignored when providing learning spaces. This is particularly true in Australia where State Education Departments, have over the years, developed models of schools that are replicated in all communities irrespective of the nature of the learners. One can generally tell how old the school is by looking at the design of the school and classrooms. Even their newest models do not take account of Indigenous learners and their needs. As Graetz and Goliber also claim, "as students enter a classroom, they form an impression of that space and experience an emotional response" (p. 15). Thus the physical learning space may be a subtle underlying cause of truancy that is commonly reported with respect to Indigenous learners. The following quote by Fielding (2001) is cited in Skill and Young (2002).

If we see ourselves as being in the "teaching business", we will make decisions [on learning spaces] that support the … interests of teachers. If we are in the "learning business" we are more likely to respond to the needs and interests of both students and teachers …We should address a fundamental question about the environment for learners: What, in the past, was an environmental barrier to learning and what enabled learning? (p. 24)

However, teachers "inherit" the learning spaces to which they are assigned so the problem of providing learning environments that respond to learners' needs rests with the relevant education sector.

In Woorabinda, Wadja Wadja High school was once a community building which, in its history, acted as a dormitory for the children of mothers who were sent away from the Community to work on properties. This building is one level and has a large open central roofed area which flows from the front to the back and with classrooms along each side. This building gives the feeling of openness and connection to the outside. However, Woorabinda State School has grown over the years and consequently is comprised of a mix of school buildings that were standard models being built by the Queensland Education Department at the time of construction. Both schools have found that students do not graffiti buildings that have Indigenous murals.

Overview of report

Chapter 1 provided an introduction to the project, to the IEWs and to mathematics teaching and learning contexts at the start of the Train a Maths Tutor Program.

Chapter 2 describes the project's research design in terms of methodology, participants, instruments, procedure and analysis.

Chapter 3, the cornerstone of the report, describes the week-by-week activities that made up the Train a Maths Tutor Program. It provides the purposes for each week, describes the activities undertaken, discusses the IEWs' reactions to these activities, the effect of trials with students and draws inferences with regard to the efficacy of the Program and the mathematics and pedagogies that underlie it.

Chapter 4 discusses the outcomes of the study, the effect the Train a Maths Tutor Program had on the IEWs and the changes that it has appeared to have brought to the Indigenous community.

The report finishes with a synthesis of the findings of Chapters 3 and 4 into conclusions and implications.

back to top