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Closing the Gap in Education?

Redressing Marginalisation

A Study of Pedagogies for Teaching Mathematics in a Remote Australian Indigenous Community

Peter Sullivan

Monash University

Robyn Jorgensen

Griffith University

Rebecca Youdale

Community School

Introduction

The discussion in this chapter draws on a series of teaching explorations at an Indigenous community school in a remote region of Western Australia. The school was one of the sites in the Maths in the Kimberleys research project led by Robyn Jorgensen and conducted on the invitation of the Association of Independent Schools of Western Australia. The ongoing project is seeking ways to support the teaching of mathematics in small community-run schools. We see the learning of mathematics as directly connected to modernisation of communities, and that an important focus of support is to enhance the capacity of teachers to engage all students in effective mathematics learning.

The project design recognises the complexity of the educational challenges in such small communities, acknowledges those who have addressed these issues previously, and emphasises collaboration with the respective communities at each stage. This chapter outlines the context of the research, some challenges with teaching and learning mathematics for Indigenous students, the pedagogical model that we are researching, some classroom explorations that exemplify aspects of the pedagogical model, and some reflections on the opportunities and challenges with the model.

The research context

The schools serve communities that are focused on modernisation, they foster commitment to the community and there is active involvement in most schools. Most of the communities served by these schools are alcohol free, there is a mix of traditional activities such as hunting and fishing, and some access to aspects of modern living such as sporting opportunities and health care. School attendance is very good. It appears that the basic conditions for effective schools with high proportions of Indigenous students, as described by Frigo et al. (2003), are being met. In particular, Frigo et al. noted that key features of schools that supported positive outcomes were strong school leadership in partnership with local Indigenous leaders, specific actions to support regular attendance and active engagement, good teaching and Indigenous presence in the school.

The schools conduct a well-supported and highly structured program, Accelerated Literacy, which aims to set high standards for students. It revolves around allocating significant time for literacy and structured, even scripted, actions by teachers. This support for literacy is clearly a prerequisite for educational and community development, although we note that this program, given the allocated resources, has had only mixed success. While both the school leadership and the teachers are active and committed, they are inexperienced, there is a high attrition rate (although substantially less than in similar communities elsewhere) and there is limited induction to the schools, the communities and the challenges the teachers encounter.

Challenges in and approaches to teaching mathematics

Our project is exploring ways to teach the mathematics that is needed for participation in modern societies. We endorse the call by Sarra (2008) that ‘Aborigines… be afforded the capacity and freedom to engage in whatever economies in whatever part of the world they choose’. We argue that success at mathematics is a prerequisite to access to opportunities for this engagement.

Yet it seems that performance overall of Indigenous students is not preparing them for these opportunities. Lokan et al. (2001) note that, while some Indigenous students are performing at the highest levels (indeed, the recent Programme for International Student Assessment (PISA) results reported that the proportion of Indigenous students at the highest level was the international average), most are well below the overall means on most aspects of numeracy. Frigo et al. (2003) noted that, while Indigenous students performed well in assessments when commencing school, by the third year of school growth had ‘slowed considerably’ (xi) and that much of the variation in students’ scores was a function of the school. In other words, the challenges of improving access to opportunities include finding ways to improve the overall performance of Indigenous students and arresting the apparent decline in comparison with non-Indigenous students over time.

A number of studies have sought to address this decline. The theme in all of them, as articulated by Perso (2006), is that students improve when teachers recognise differences in background and learning styles, and that failing to do so can lead to teachers adopting a deficit approach, perpetuating marginalisation of these students. Frigo et al. (2003) list key elements of effective numeracy teaching – from across schools serving high proportions of Indigenous students – as teaching skills in real-life contexts, developing sound number skills, reinforcing concepts through structured activities and semi-structured play, offering low-risk opportunities to develop confidence, exploring the language of mathematics, and building on what the students know.

Several studies have attempted to explain why policies and initiatives aimed at improving Aboriginal students’ mathematics achievement have often failed (eg Baturo et al. 2004). Howard (1997) argues that the imposition of a ‘western’ curriculum has meant that ‘for many Aboriginal children… the mathematics classroom becomes an alien place characterised by tensions and conflicts about relationships and the value of what they are being taught’ (17).

There have been attempts to adapt conventional western pedagogies to Indigenous contexts. For example, the Garma Living Maths program (Perso 2006) describes an approach termed ‘two-way learning’, indicating acceptance of a mixing of western and Indigenous knowledge, and likens the meeting of these knowledge systems to the meeting of two bodies of water in a lagoon where salt and fresh water come together. A key element in this approach is the notion of not only incorporating community values into teaching approaches, but actively engaging the community in all aspects of the curriculum and pedagogies that are adopted.

An alternative approach, QuickSmart (Pegg et al. 2005), is a four-phase process for addressing the needs of low-achieving students. The approach involves initial teaching, subsequent attempts to address difficulties experienced by some students, collaborative support for teaching by a specialist and, ultimately, withdrawal from class. The approach emphasises automaticity of skills in both reading and computation, and the measures were of the extent to which the improved automaticity enhanced higher-order processing.

Our pedagogical model draws on these various approaches, and also on our understandings of mathematics learning generally. For example, we see mathematics learning as more than the development of low-level skills, and that the strategies that have been successful for learning mathematics elsewhere in the world should also be utilised in our community schools. These include creating opportunities for students to investigate mathematically rich situations, to identify patterns and seek commonalities, and to explain reasoning and justify choices. Our approach seeks to create such opportunities.

Interactive pedagogies

Our approach, termed ‘interactive pedagogies’, draws on Boaler’s (2008) extensive work and has also been informed by Burton’s research (2004) on working mathematically and productive pedagogies (Gore et al. 2004).

This approach challenges deficit models of teaching Indigenous students and seeks to promote a rich and deep mathematical learning. The approach is founded on a strong belief that all students can learn mathematics when the pedagogy is appropriate (see Grootenboer 2009; Jorgensen 2009; Sullivan 2009). The key elements of the interactive pedagogies are:

Group work: We see group work as foundational to processes of social learning. By incorporating group work with which students are familiar in out-of-school learning, it becomes possible to draw on the skills and knowledge within a group to solve problems. The roles of the group members create interdependent learning opportunities that are not possible within parallel learning.

Home language: Students are allowed to draw on their home language (in this case, Kriol) to negotiate meanings. When reporting back, they are encouraged to use standard Australian English.

High interactivity: There is a strong focus on quality interactions – within the group work and the reporting-back stage. Good questioning is critical to this approach. Teachers develop good questions to promote learning opportunities for the students as well as students learning to pose good questions to each other.

Multi-representational: Recognising the diversity among learners, we encourage use of tasks that foster, and allow for, various methods of representation that cater for the different skills and dispositions that learners bring to the task. Provided that the result is reasonable, the pathway and mode of representation is valued.

Reporting back: This is a critical part of the lesson, where students report to the class on their approach to solving the task and the responses they have developed. Ideally, students within the classroom pose questions to the reporting group so that there is quality dialogue among peers, which ultimately promotes aspects of working mathematically, such as justifying, clarifying, generalising, conjecturing and so on. The purpose is to encourage dialogue among peers that promotes rich mathematical learning.

Tasks and activities: Our focus in this chapter is on the choice of the task, for which there are three complementary elements:

  • The teachers should be clear about what they are intending to teach.
  • The lessons should build on what the students know (as distinct from what they do not know).
  • Tasks should be mathematically rich and draw on the ‘working as a mathematician’ approach, in which there should be multiple pathways and entry points for learners and multiple ways of representing thinking and learning that incorporate different learning styles and approaches.

These aspects are elaborated in the following discussion of the explorations in Rebecca’s classroom.

The classroom explorations

The project overall involved regular professional learning sessions with teachers on aspects of the interactive pedagogies, supported by occasional school visits by the research team, data collection via the video-recording of lessons by the teachers, and telephone interviews with the principals and teachers. The following discussion focuses on just one aspect of the interactive pedagogies and elaborates considerations about the selection and use of tasks and activities. It draws on observations from a set of 10 lessons, spread over three separate research visits. The lessons were planned collaboratively with Peter and taught by Rebecca.

The purpose of the observations was to examine the implementation of the interactive pedagogies in real time in a classroom. Rebecca was willing to explore all aspects of the pedagogies and these observations provide a realistic indication of what is possible. Peter observed the lessons, made video and audio recordings of key moments, gathered student work samples and interviewed Rebecca before and after the lessons. The following is a discussion of the three elements of the tasks and activities aspect of the interactive pedagogies: the importance of having clear goals; building on what students know; and mathematical richness.

The importance of having clear goals

The importance of teacher clarity is supported by Hattie and Timperley (2007), who reviewed a range of studies on the characteristics of effective classrooms. They found that feedback was among the main influences on student achievement, the key elements of which are ‘Where am I going?’, ‘How am I going?’, and ‘Where am I going to next?’ The implication is that it is best if the teacher formulates specific goals for student learning, can make decisions on expectations for performance, and has some sense of where the experiences are leading subsequently. It is therefore important that the teacher establishes clear goals, so that the many interactive classroom decisions, questions and comments are made with a clear purpose in mind.

To help make the focus of teaching plain, in each of the three sets of lessons some key ideas were extracted and elaborated at the first stage of planning. The planning was iterative and took place by email some weeks prior to the teaching. To illustrate what is meant by identifying key ideas, the following were the ideas suggested after Rebecca had proposed the topic and level (Grades 3 and 4, student ages eight or nine) that were the focus of instruction.

The first sequence of lessons was on subtraction. It was proposed that the key ideas were: stating the numbers 1 and 2 before a given number; modelling numbers in terms of their parts; mental strategies that are useful for subtraction; and connecting different representations of subtraction (see Sullivan et al. 2009a). As an indication of the success of the planning and teaching, all students were individually interviewed at the end of the lesson sequence. Nearly all Grade 4 students and most Grade 3 students were able to answer the questions: ‘I have eight biscuits and I eat three. How many do I have left?’ and ‘What is 10 take away seven?’ While these are not complex tasks, the results indicate that the lesson sequence included the weaker students effectively.

Rebecca suggested that the second sequence of lessons focus on partitioning numbers, the key stages of which were proposed to be: patterns in numbers to 10 and 100; breaking numbers into parts (eg 65 is 60 + 5 as well as 50 + 10 + 5 etc); and regrouping numbers (eg 98 + 35 = 100 + ?) (see Sullivan et al. 2009b). Students were interviewed again to obtain a sense of their learning. Nearly all of the 15 students were able to count by 10s past 100; count by fives to 90; calculate 9 + 4, where the nine objects were covered, requiring counting; state the answer to 2 + 19, and most were able to answer 27 + 10. This is evidence of learning and growth.

The third sequence was about division. The key ideas suggested were: using models to represent multiplicative situations; working multiplicatively with numbers; solving problems without using models; and moving to larger numbers.

It is argued that these represent key ideas within each of these topics, and that having a clear idea of the focus of instruction is better than merely working on a collection of loosely related activities that vaguely address the topic for instruction. This clarity is helpful for choosing tasks, for explanations, for emphasising the purpose to students, for interacting with students, for interpreting their responses and for assessing their achievements. The interview assessments were not intended to measure the ceiling of learning but the extent to which students generally had learned the desired content. In this, the lesson sequences were successful.

Building on what they know

The second element also draws on Hattie and Timperley, who argue that learning is more effective when teachers identify what the students already know, so that both the activity of the task and the feedback to students can build on their prior knowledge. The confidence that students derive from working on familiar concepts can then be used as the springboard for the subsequent challenges that teachers set that lead to real learning. Perhaps paradoxically, this is an aspect that many educators find difficult.

As part of teacher learning sessions we proposed the use of contexts with which the students are familiar. These might include linking to modern ideas such as sports, or traditional ideas such as time-marking systems or the language of directions and location.

Since the development of understanding and fluency with numbers is an essential element in education for participation in modern society, a particular challenge is to identify aspects of numbers with which students are familiar. After observing students at a school fete, Rebecca commented that the students seemed to be familiar with money. The second and third sequences were developed to build on this familiarity. The following two activities show how this was enacted.

The first activity sought to build on perceptual (as distinct from conceptual) recognition of money amounts. Various combinations of $1, $2 and $5 amounts were shown for a short time and then covered. Students first whispered their answer to the person sitting next to them and then declared their answer. The observer (Peter) noted:

The students seemed to be extraordinarily adept at doing this accurately. Given that this involves a number of key skills in combining and partitioning numbers, it created the sense of the strong foundation on which the lesson sequence could build.

This was repeated using ten 20–cent pieces first and then adding 50–cent pieces. Many of the students seemed to do this readily and it created the excitement that goes with successful completion and with challenging questions. We have video records of eight-year-old students accurately identifying money amounts as quickly as adults who have been familiar with money all their lives.

Another activity, indicating the move away from using the actual coins, was based on a well-known game, Race to $10, where students, starting at 0, in turn add $1 or $2, and the one who makes the total $10 is the winner. There is a winning strategy. This was then played as Race to $1, adding on 10 cents or 20 cents. The observer noted:

The students were able to play the game easily, and were energetically engaged. As a review of the activity, Rebecca played this for some time with individuals to see whether they would see the pattern, and recognise a winning strategy. Only one student did, and he was asked to report on this, but there was an emerging awareness in the others. Again, the success at the game, presumably derived from the familiarity with the money amounts, created an opportunity for engaging with the mathematical ideas. The money provided the springboard.

Using the money tasks, with which many students were familiar, created a sense of enthusiasm and success, and seemed to allow extension to straight number tasks, which was Rebecca’s intent in the first place. In individual interviews after the lessons, nearly all students could recognise the total of two $2 coins and two $1 coins, shown for two seconds, and over half of the students could recognise the total of three 20-cent coins and one 10-cent coin, shown for two seconds. This both confirms the initial observations that some students were fluent with these tasks and also indicates that the lessons allowed other students to attain this fluency.

Choosing rich tasks

The third element is the choice of tasks that are mathematically rich and challenging. The nature of the tasks and associated teachers’ actions are summarised in a set of recommendations for teachers (see Jorgensen and Sullivan, Chapter 4 this volume). To illustrate the nature of these tasks, the following are three examples that were used as part of the observed teaching. The first task was posed as follows:

I am thinking of two numbers. The difference between the numbers is 2. What might be the numbers?

This can be recorded symbolically as

ImageImage = 2

The point is that students can explore aspects of 2 difference and even recognise the patterns of differences that appear. We want students, for example, to be able to calculate 19 – 17 as readily as they calculate 19 – 2. This type of task gives students the opportunity to make active decisions on the numbers they use and the way they record their results. It is important to emphasise that there is more than one possible answer and that it helps if the answers are written systematically. The observer noted:

The pupils worked productively on the task, and most groups were willing and able to produce multiple solutions, some of which were systematically organised. Making choices seemed to be engaging. In this case there was a need for extended explanations of how this would work. Perhaps in the future it will be easier to pose such tasks. The students worked in groups with particular roles. Rebecca encouraged the reporter from each group to explain the process whereby the group found their particular set of answers. Again, this reflects the focus on students explaining their strategies, supported by the teacher.

The success of the groups is indicated in the diversity of responses given by the groups. Many groups developed a range of possible solutions, indicating that they had identified patterns in the solutions and laying the groundwork for knowing the answer readily to questions such as 19 – 17.

The pedagogies associated with this task are illustrative of the approach we are advocating. The task had a variety of entry levels, it could be answered in different ways, it involved group work with roles, and it allowed a detailed and focused class discussion of strategies and patterns. The concluding review allowed the teacher to highlight particular student insights.

Another task was posed, using $1, $2 and $5. In Rebecca’s words, ‘Your job is to work out as many ways as you can to make $10’. The observer noted:

This seems to be an example of the type of task that can be successful. It is complex enough to allow for multiple answers; some reasoning and problem solving is required; and it is practising a core skill toward the goal, that of ways of building to 10.

The students wrote their answers on small whiteboards. They seemed to understand the task and worked productively, with many producing multiple correct answers.

The purpose of including discussion of the third task here is slightly different. This task was presented as: ‘I have three silver coins, how much money might I have?’ It allows a similar diversity of responses to those of the previous tasks, but there was an interesting twist. The observer noted:

This again is the sort of task that should work. It has a range of answers, it prompts communication, it is challenging mathematics, and it is addressing the overall theme. It did not work as intended, with many students including $1 and $2 coins in their total, making it more complex. Yet Rebecca had explained the task well. The reason for not mixing the dollars and cents is that it makes the calculation more difficult.

Only later did it emerge that the local word for coins or change is ‘silver’ (in Australia, there are coins for $2 and $1 and these are gold), which highlights the need to consider alternative interpretations of events and language at all times. This illustrates the importance of the interactivity implied by the pedagogical model, as well as sensitivity by the teacher to the interpretations of the students.

These three tasks illustrate that the students are willing and able to engage with such number and money investigations. They are able to identify a range of possible responses and record them systematically, which presumably lays the groundwork for developing mathematical connections.

Reflecting on opportunities and challenges with the interactive pedagogies approach

The goal of the research is to investigate the challenges and opportunities afforded by this pedagogical approach. From the lesson observations overall, it can be concluded that being clear about the goals of teaching is helpful both to the teacher and to the students. In the lessons observed, the clarity meant that the activities could be thoughtfully sequenced, each activity could build on a previous experience, and the students were clear about the teacher’s goals and her expectations for them. It also appears that ‘building on what the students know’ was an effective strategy. In the sequence of activities that drew on the apparent familiarity and fluency of students with money, the class was extraordinarily energised and engaged; they participated actively and the experience seemed to lead to other learning. Rebecca was able to extend some of the students toward formulating some potentially powerful generalisations.

In these observations it also appeared that the students were both willing and able to engage with rich tasks that required decision making by them and allowed the construction of mathematical ideas. It seems useful to use such rich tasks with the students and to encourage them to create mathematical ideas based on those tasks.

In these examples, and the other lessons observed, there were many instances that would be judged outstanding teaching and learning in any school, and certainly demonstrated that students in remote schools can learn as well as their metropolitan counterparts. Three challenges emerged from these observations. The first is that care needs to be taken when making inferences about the extent of student engagement. In the sequence of activities based on the money questions the class seemed highly engaged. Yet in subsequent interviews, in the class observed, while most students were highly fluent with the money and equivalent number questions, there were two students who were not able to identify the value of any coins. This emphasises that there is a diversity of achievement within each class – and a diversity of readiness – and specific actions must be taken to accommodate this diversity. While we have not researched the potential in this, it seems that the Aboriginal Education Workers who are available in some schools could be better utilised. It also seems that there would be advantages in exploring ways of grouping students for instruction that maximise students’ opportunity to contribute meaningfully.

A second issue relates to the conduct of class reviews after rich explorations. Rebecca patiently probed the students’ thinking and invited them to explain their reasoning. Yet this was not often successful from a whole-class perspective. One example was the student who explained his strategy for winning the Race to $10 game. He gave an extended explanation and, if you knew what he was trying to say, his explanation was insightful and illustrated clear conditional thinking and argument. Yet his explanation would not have informed other listeners. There were a number of other instances where an individual gave an excellent explanation that elaborated on the desired type of thinking, but not in a way that would engage the other children. The other students were not interested in such explanations, which may be partly a function of a lack of clarity. Rebecca was energetic and committed to the approach and had worked with the class on her expectations for participation. It is suspected that specific actions will be necessary for this aspect of the approach to realise its potential. One strategy that seemed to work was for the teacher to restate the explanations given by students and to provide additional diagrammatic support for their explanations.

A further issue is the intensity of the interactivity. In many of the observations the students became tired. Noting that these classes are quite small, students are constantly under scrutiny. Clearly, in the above, there were mathematically rich and challenging experiences in which the students participated well, even beyond expectations. But it is perhaps unreasonable to expect the students to do this for the full 90 minutes of each mathematics class. It is suggested that teachers could plan some experiences that are less intensive and less interactive and these could be used to buffer shorter and more intensive parts of the lessons. These less intensive experiences could include competitive games such as card games, some aspect of physical activity combined with a mathematical experience, drawing or storytelling.

The interactive pedagogies model clearly has potential to engage students in doing significant mathematics. Our project is now seeking to elaborate on the aspects that worked well and to revise those that did not.

References

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Boaler, J. 2008. ‘Promoting “relational equity” and high mathematics achievement through an innovative mixed ability approach’. British Educational Research Journal 34 (2): 167–194.

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Sarra, C. 2008. ‘New narrative tells of brighter future together’. Australian 8 August. Accessed 20 October 2009. Available from: http://www.theaustralian.com.au/news/features/new-narrative-tells-of-brighter-future-together/story-e6frg6z6-1225759141388.

Sullivan, P. 2009. ‘Describing teacher actions after student learning from rich experiences’. In Crossing Divides, edited by Hunter, R; Bicknell, B; Burgess, T. Proceedings of the 32nd Conference of the Mathematics Education Research Group of Australasia. Vol. 1. Sydney: MERGA: 726–732.

Sullivan, P; Youdale, R; Jorgensen, R. 2009a. ‘The link between planning and teaching mathematics: An exploration in an Indigenous community school’. In Proceedings of the Biennial Conference of the Australian Association of Mathematics Teachers, edited by Kissane, B. Fremantle, Western Australia: MERGA: 247–256.

Sullivan, P; Youdale, R; Jorgensen, R. 2009b. ‘Knowing where you are going helps you know how to get there’. Australian Primary Mathematics Classroom 14 (4): 4–19.

Closing the Gap in Education?

   by Ilana Snyder and John Nieuwenhuysen