Researching a Structure for Planning and Teaching Inclusive Mathematics Lessons

Inclusive mathematics teaching

Researching a structure for planning and teaching
inclusive mathematics lessons

Paper presented at the British Educational Research Association Annual Conference, Institute of Education, University of London, 5-8 September 2007

This report addresses one aspect of a project that aimed to identify strategies that teachers can use to overcome the obvious disadvantage some school students experience in learning mathematics. Currently working class and Indigenous students in Australian schools generally perform poorly in mathematics compared with their peers. This project identified factors contributing to the lack of success of such students. The focus was on ascertaining recommendations for teaching approaches that are practical and possible, while retaining a focus on student learning of mathematics. The structure for planning and teaching inclusive mathematics lessons involves using content specific open-ended tasks, planning a hypothetical trajectory of tasks, making the relevant pedagogies explicit, ensuring there is sufficient shared experience to allow meaningful discussions and review, and developing enabling prompts to support students experiencing difficulty as well as extending prompts to engage students who complete the set work.

We found that all elements of the structure are practical and contribute to lesson effectiveness, that teachers can use the structure to teach successful lessons, and they can use it to plan inclusive experiences for their classes. The most powerful finding relates to the use of enabling prompts. We identified instances of planned prompts that allowed students experiencing difficulty to engage with the core experiences of lessons to undertake alternate tasks and to subsequently complete the core tasks effectively, and so enjoy the same lessons experiences as other students.

Introduction

There are two serious challenges facing Australian mathematics educators, researchers, and teachers. First, there is a need to identify the causes of the widespread and persistent disengagement of students, particular in the middle years of schooling (e.g., Hill, Holmes-Smith, & Rowe, 1993; Russell, Mackay, & Jane, 2003). Second, while international comparisons suggest that overall performance of Australian students in mathematics is satisfactory, there are very great disparities between the high achieving and low achieving students, more so than in comparable countries (see Lokan, Ford, and Greenwood, 1997). Those most likely to perform poorly in these tests are students who come from socially and culturally disadvantaged backgrounds.

It is difficult to identify unequivocal research results that can assist teachers in addressing these two challenges in their everyday complex and multidimensional classrooms. We acknowledge the importance of factors such as classroom resources, organisation and climate, interpersonal interactions and relationships, social and cultural contexts, student motivation and sense of their futures, family expectations, and organisation of schools. Nevertheless we argue that an important component of understanding teaching and improving learning is to identify the types of tasks that prompt engagement, thinking, and the making of cognitive connections, as well as the associated teacher actions that support the use of such tasks, including addressing the needs of individual learners. The challenge for mathematics teachers is to foster mathematical learning, and the key media for pedagogical interactions between teacher and students is the tasks in which the students engage.

Assumptions about learning and classroom activity

The project is based on assumptions about knowing and learning, and about the roles of the teacher, including the need for teachers to challenge all students while offering support for students experiencing difficulty.

We draw on a socio-cultural perspective (Lerman, 2001) which extends the work of Vygotsky including his zone of proximal development (ZPD) which he described as the “distance between the actual developmental level as determined by independent problem solving and the level of potential development as determined by problem solving under adult guidance or in collaboration with more capable peers” (1978, p. 86). ZPD also provides a metaphor that defines the work of the individual or class as going beyond tasks or problems that students can solve independently, so that the students are working on challenges for which they need support. In other words, the teacher’s task is to pose to the class problems that most students are not yet able to do then to provide support for students to complete the task successfully. ZPD provides a metaphor for the support that teachers can offer to students experiencing difficulty. If the teacher poses problems that are challenges for all students, in most classes there will be some students who are not yet at the level of supported problem solving for this particular problem. We argue that adult guidance or peer collaboration might be offered to such students through adapting the task on which they are working, as distinct from, for example, grouping such students together and having them undertake quite different work, or re-teaching them using the same approach.

This notion of adapting tasks is a consistent theme in advice to teachers. For example, the Association of Teacher of Mathematics (ATM) (1988) detailed 14 specific suggestions related to interventions to support students experiencing difficulty, seven of which relate to task adaptation. Christiansen and Walther (1986) argued that, “One of the many aims of the teacher is … to differentiate according to the different needs for support but to ensure that all learners recognise that these …actions are created deliberately and with specific purposes” (p. 261). Similarly, Griffin and Case (1997) described teaching as involving knowing what individual learners understand, being aware what knowledge is within the ZPD, providing carefully constructed tasks to engage students in learning, helping learners as they construct their knowledge, and “constantly shifting or changing the ‘bridge’ to accommodate the learners’ growing knowledge” (p. 4).

These assumptions all connect to the centrality of the tasks that the teacher poses.

The Use of Open-ended Tasks

The choice of task is important. Within a socio-cultural perspective, the role of mediating tools is central in the learning process. For us, it is through and around tasks that teachers and students communicate and learn mathematical ideas, so the tasks used by the teachers and discussions around these become mediating tools. Christiansen and Walther (1986), drawing on the work of Leont’ev (1978), argued that the tasks set and the associated activity form the basis of the interaction between teaching and learning. Similarly, Brousseau (1997) proposed that, “the teacher must imagine and present to the students situations within which they can live and within which the knowledge will appear as the optimal and discoverable solution to the problems posed” (p.22). Hiebert and Wearne (1997) also proposed that, “instructional tasks and classroom discourse moderate the relationship between teaching and learning” (p. 420). In other words, the tasks used by the teachers become means for facilitating learning about particular aspects of mathematics.

Open-ended tasks have particular potential to contribute to this mathematics learning. Stein and Lane (1995) noted that student performance gains were greater with relatively open-ended tasks, when “tasks were both set up and implemented to encourage use of multiple solution strategies, multiple representation and explanations” (p. 50). Boaler (2002) provided further evidence of open-ended tasks being the key to progress when she compared the activity, operations, and achievement outcomes in two schools. The schools were chosen to represent similar socio-economic mixes of students. In one school, the teachers based their teaching on open-ended tasks and in the other traditional text-based approaches were used. After working on an “open, project based mathematics curriculum” (p. 246) in mixed ability groups, the relationship between social class and achievement was much weaker after three years, whereas the correlation between social class and achievement was still high in the school where teachers used traditional approaches. Further, the students in the school adopting open-ended approaches “attained significantly higher grades on a range of assessments, including the national examination” (p. 246). Boaler argued that her project demonstrated the “particular teaching practices that need to be considered in mathematics classrooms and the effectiveness of teachers who are committed to equity and the goals of open-ended work” (p. 254). In other words, the use of open-ended tasks proved effective in improving mathematics learning and overcoming disadvantage, but it took commitment from the teachers as well as the adoption of particular strategies.

The type of tasks used by teachers mediates the learning between the subject (student) and object (mathematics). We propose that open-ended tasks offer greater opportunities to scaffold learning opportunities for students than do closed tasks. Essentially, we assume that working on open-ended tasks can support mathematics learning by fostering operations such as investigating, creating, problematising, communicating, generalising, and coming to understand—as distinct from merely recalling—procedures. There is a substantial support for this assumption. Examples of researchers who have found that tasks or problems that have many possible solutions contribute to such learning include those working on investigations (e.g., Wiliam, 1998), those using problem fields (e.g., Pehkonen, 1997), those exploring problem posing by students (e.g., Leung, 1997), and the open approach (e.g., Nohda & Emori, 1997). It has been shown that opening up tasks can engage students in productive exploration (Christiansen & Walther, 1986), enhance motivation through increasing the students’ sense of control (Middleton, 1995), and encourage pupils to investigate, make decisions, generalise, seek patterns and connections, communicate, discuss, and identify alternatives (Sullivan, 1999). Open-ended tasks have been shown to be generally more accessible than closed examples, in that students who experience difficulty with traditional closed and abstracted questions can approach such tasks in their own ways (see Sullivan, 1999). Well-designed open-ended tasks also create opportunities for extension of mathematical operations and dimensions of thinking, since students can explore a range of options as well as considering forms of generalised response.

We use a particular form of such open-ended tasks that can be readily incorporated in conventional mathematics programs. We describe our tasks as content specific.

The nature of content specific open-ended taskscan be illustrated by some examples:

If the perimeter of a rectangle is 24 cm, what might be the area?

On squared paper, draw as many different triangles as you can with an area of six square units.

The mean height of four people in this room is 155 cm. You are one of those people. Who are the other three?

A ladder reaches 10 metres up a wall. How long might be the ladder, and what angle might it make with the wall?

What are some functions that have a turning point at (1,2)?

(For a wide range of such tasks at the primary level, see Sullivan & Lilburn, 2002)

Such tasks are content specific in that they address the type of mathematical operations that form the basis of textbooks and the conventional mathematics curriculum. The learning about specific mathematical content is at least what would be expected from completion of a typical text-book-based task, so teachers can include these as part of their teaching without jeopardising students’ performance on subsequent internal or external mathematics assessments.

These tasks are also open-ended in that there is a variety of possible operations and a variety of ways of communicating responses. Emphasis is taken off specific examples and put on to general properties, and there is a sense of open entry with relatively simple responses as well as extension possibilities.

Our earlier work on content specific open-ended questions (e.g., Sullivan, Mousley, Zevenbergen & Turner Harrison, 2003) demonstrated that the combination of such tasks with strategic questioning by teachers can lead to extension of mathematical thinking, since students explore a range of options as well as considering forms of generalised response. In each of the examples above, not only are the benefits for students in creating their own solutions, but also seeing the range of possible responses, and the patterns in those responses helps to illustrate the nature of the respective concepts.

The development of the planning and teaching model

The planning and teaching model, described below, was the product of earlier stages in our research where we sought to address the tensions in complex learning environments. For us, the learning was as much about teacher learning as student learning so a significant amount of our work was with teachers to overcome resistance to the approach and to develop a model that would successfully engage teachers as well as students. Our overall study can be described as design research that “attempts to support arguments constructed around the results of active innovation and intervention in classrooms” (Kelly, 2003, p. 3). Cobb, Confrey, diSessa, Lehrer, and Schauble (2001) argue that design experiments are appropriate for classroom explorations where the research team and teachers collaborate on determining the research focus.

Initially, though, our research identified and described aspects of classroom teaching that may act as barriers to mathematics learning for some students, and we drew on focus group advice to suggest strategies for overcoming such barriers (see Sullivan, Zevenbergen, & Mousley, 2002). For us, this provided a strong framing for our project, particularly as we were exploring the ways in which aspects of the environment mediated learning. Next, we created some partially scripted experiences taught by participating teachers and analysed by us (see Sullivan, Mousley, & Zevenbergen, 2004). This analysis allowed reconsideration of the emphasis and priority of respective teaching elements. We found that it was possible to create sets of experiences that could be taught as intended by teachers, and that many of these experiences had the effect of including most students in rich, challenging mathematical learning.

Arising from this work, we developed a model comprising five key elements of planning and teaching mathematics that can be summarised as follows.

The tasks and their sequence

As discussed above, open-ended tasks create opportunities for personal constructive activity by students directed at mathematical objects. We also consider that careful sequencing of tasks can contribute to learning. This relates closely to what Simon (1995) described as a hypothetical learning trajectory that

… provides the teacher with a rationale for choosing a particular instructional design; thus, I (as a teacher) make my design decisions based on my best guess of how learning might proceed. This can be seen in the thinking and planning that preceded my instructional interventions … as well as the spontaneous decisions that Imake in response to students’ thinking. (pp.135–136)

Simon (1995) noted that such a trajectory is made up of three components: the learning goal that determines the desired direction of teaching and learning, the activities to be undertaken by the teacher and students, and a hypothetical cognitive process, “a prediction of how the students’ thinking and understanding will evolve in the context of the learning activities” (p.136).

During our research, the use of sequenced open-ended tasks has improved students’ engagement, as evidenced by time on task, participation in discussions, and increase in successful completion of the teaching and learning activities. The use of such tasks has also had an impact on participating teachers’ notion of mathematical activity.

Enabling prompts

Teachers offer enabling prompts to allow students experiencing difficulty to engage in active experiences related to the initial goal task. These prompts can involve slightly lowering an aspect of the task demand, such as the form of representation, the size of the number, or the number of steps, so that a student experiencing difficult can proceed at that new level; and then if successful can proceed with the original task. This approach can be contrasted with the more common requirement that such students (a) listen to additional explanations; or (b) pursue goals substantially different from the rest of the class. The use of enabling prompts has generally resulted in students experiencing difficulties being able to start (or restart) work at their own level of understanding and enabled them to overcome barriers met at specific stages of the lessons.

Extending prompts

Teachers pose prompts that extend the thinking of students who complete tasks readily in ways that do not make them feel that they are merely getting more of the same. Students who complete the planned tasks quickly are posed supplementary tasks or questions that extend their thinking and activity. Extending prompts have proved effective in keeping higher-achieving students profitably engaged and supporting their development of higher-level, generalisable understandings that we associate with higher order learning.

Explicit pedagogies

Teachers make explicit for all students the usual practices, organisational routines, and modes of communication that impact on approaches to learning. These include ways of working and reasons for these, types of responses valued, views about legitimacy of knowledge produced, and responsibilities of individual learners. As Bernstein (1996) noted, through different methods of teaching and different backgrounds of experience, groups of students receive different messages about the overt and the hidden curriculum of schools. We have listed a range of particular strategies that teachers can use to make implicit pedagogies more explicit and so address aspects of possible disadvantage of particular groups (Sullivan et al., 2002). We have found that making expectations explicit enables a wide range of students to work purposefully, with teachers commenting positively about relatively low levels of teacher–student friction.

Learning community

A deliberate intention is that all students progress through learning experiences in ways that allow them to feel part of the class community and contribute to it, including being able to participate in reviews and summative class discussions about the work. To this end, we propose that all students will benefit from participation in at least some core activities that can form the basis of common discussions and shared experience, both social and mathematical, as well as a common basis for any following lessons and assessment items on the same topic. We have found that the use of tasks and prompts that support the participation of all students has resulted in classroom interactions that have a sense of learning community (Brown & Renshaw, 2006), with wide-ranging participation in leaning activities as well as group and whole-class discussions.