Curriculum: theory and practice in mathematics

Ms Linda Shardlow Methodist Ladies College Kew, Victoria, Australia

Plato’s ‘ideal man’ combined inquisitiveness, theoretical grasp, sensitivity, courage, discipline and order. His educational strategy to achieve this ideal was designed around a framework that initiated individuals in a way that was practicable for the particular stage at which they were; starting first with the senses, then with the emotions, and finally, with cognitive abilities. Plato wanted the ‘harmony of the soul’ linked to the development of reason to form a ‘networked’ man.

Currently (perhaps at any stage in history), there are theories on education and, particularly, curriculum, that lead to mutually blinkered views of the roles of teachers and students; theories that end up treating both groups as objects. Debate becomes unhelpfully polarised. What curriculum can we formulate that could take educational theories and meld them into a flexible, cohesive and workable relationship between teacher and student, which changes with changing needs and addresses students’ needs in order to educate them to become people who can effectively operate in the future world? Does content in curriculum matter? Can curriculum achieve this goal by itself?

A conservative (some would say ‘traditional’) education involves a teacher/learner relationship whereby the teacher is the possessor of knowledge and the learner is the recipient of the said knowledge. The teacher is seen as the custodian of both cultural and intellectual capital and inheritance. Students enter the world of education and are initiated into an inheritance of human achievements, through the process of learning as knowledge transfer. Teaching and content is deliberate. All teaching has an element of instruction, as all knowledge consists of information to impart. The content of the curriculum consists of facts: specific, impersonal, hard, isolated, arbitrary and inert. From a curriculum point of view, the first task is to decide what information should be conveyed to the students. Secondly, the decision has to made as to how to group this information, and then in what order to teach it (the syllabus). Finally, assessment is based on how to exercise the students in the information so it can be recollected and restated.

Students are not merely seen as passive recipients of information, however. Proponents of conservative education are also interested in ‘judgement’. Judgement involves disinterested curiosity, patience, honesty, exactness, industry, concentration and doubt. Judgement, together with information, generates knowledge or the ability to do, to make or to understand and explain. Critical minds develop from knowing ‘the best that has been thought and said’ (quote attributed to Matthew Arnold). Yet, judgement, like information, is to be imparted to students via the example set by the teacher. Teaching, under this theory, consists of instructing or communicating information and imparting: communicating judgement. Assessment, of course, is about another form of judgement: the teacher’s view on how well the student has learnt the information. An inherent flaw, and perhaps complacency, with this theory is that the student is ready and willing to hear what the teacher has to say. There is no doubt as to who has the authority and who is the provider of learning – the teacher.

Others like a more balanced partnership between teacher and student – one where there is less direction and more trust of the learner and less omniscience on the part of the teacher. They like to see their approach as more student-centred – as opposed to content-based. The overarching philosophy is more humanistic and psychologically based on theories of learning. These show more interest in, and give more credence to, the ‘method’ rather than the content. Content is still there but it doesn’t have the same ritualistic and mystical symbolism as a revered icon of educational experience. They believe that a traditionalist approach, based on a content-driven curriculum, supports the following perceptions of learning for students. That:

• passive acceptance is a more desirable response to ideas than active criticism
• discovering knowledge is beyond the capabilities of students
• recall is the highest form of intellectual achievement
• the voice of authority is to be trusted and valued more than independent judgement
• feelings are irrelevant in education
• there is always a right answer to a question.

Traditionalists perhaps are more interested in the information disseminating business, where content (what the students are in schools to ‘get’; what they are supposed to learn) exists independently of, and prior to, the student and are indifferent to the media by which it is transmitted. The ‘new’ media of various technologies will reflect traditionalist paradigms in teaching and learning if they are merely added on as an additional layer to content-based curricula. It becomes just a different way of delivering the same thing. The actual form of the education received is still that of a traditionalist approach.

So, education has traditionally focused on the known; involving the passing on of a body of knowledge and transference of content. Education for the unknown has become our new focus. This doesn’t supplant education for the known but it just isn’t enough any more. Understanding is needed for the future. Mastery of a routine body of knowledge won’t assist students to operate effectively in a world of complexity and change. I believe that students need to be able to think creatively, make multiple connections and transfer that thinking between various contexts. This would involve making our curricula less inert, less narrow, less insular, less content-based, less focused on surface understandings and more focused on the acquisition of active knowledge. Assessment would also need to focus on these understandings.

Yet, teachers and secondary schools, in particular, are under increasing societal and political pressures to be accountable, and to produce results. Public perceptions of a highly competitive employment market and growing limitations on availability of places in tertiary institutions means that academic performance now almost outweighs most considerations. In the private sector especially, but perhaps not exclusively, people who buy education are insisting on value for money and the market structure of the private school creates a distinct milieu for teachers – subjecting their performance to constant appraisal in terms of the commodity the school is placing on the market. Politically, we are currently in the grip of a debate about the need for a national curriculum and performance-based pay for teachers. Both of these strands are ostensibly needed in order to secure a quality education for Australia’s young people on the world stage.

Can curricula be constructed that encourage and support the acquisition of the more active knowledge I believe students need for the emerging world and, at the same time, satisfy the demands of the societal and political pressures outlined above?

‘Our most important choice is what we choose to teach’, says David Perkins in his book, Smart Schools.

It took me a while to realise that I agree with this. My instinctive response was to change ‘what’ to ‘how’. However, concomitant with the thinking about what to teach is the thinking about the methodology employed. We can choose to devise content-based curricula or not. I am a mathematics teacher. Many mathematics syllabus documents consist of a list of exercises to cover from the chosen text. Someone must have made a choice to write it up this way. Why not create a curriculum that enables and supports thinking and understanding?

Learning is a product of thinking. Curriculum should also be a product of thinking. Can teachers articulate the reasons why certain content is to be included in the syllabus? What are the ‘big ideas’ in mathematics, for example, we want our students to understand? What questions could we ask that develop these understandings in our students? These are important constructs on which to base a curriculum. This curriculum should focus on challenge, complexity, opportunities for discussion and analysis, no set approach for solution, going beyond the straightforward and stretching students’ thinking to develop understanding.

I think there is a need for curriculum to provide a structure and, at least in mathematics, there is a need to stage content through various year levels due to the cumulative nature of the subject and the fact that a certain maturity and background knowledge are required to perform mathematics at various cognitive levels. However, within the content for these levels, syllabus documentation need not, and should not, be content-based. In order to engage students and provide them with opportunities that will enable them to operate effectively in this world of increasing complexity and rapid change, we need to build in activities and routines into classrooms that encourage and elicit quality thinking.

In designing a mathematics curriculum that addresses these needs, I would heed Julia Atkin’s advice in her paper, Enhancing Learning with Information Technology; IARTV – Incorporated Association of Registered Teachers of Victoria – Seminar Series, November 1997. She suggests the following approach. Instead of asking the question ‘What topics do we need to cover?’, we need to develop processes and approaches that result from asking ‘What are the powerful ideas and processes that we believe are important for young people to learn in mathematics?’

We should start with our focus on desired understandings, the big ideas, and then ask: ‘What information and what experiences do we engage the students in, in order for them to develop these understandings?’

The focus of assessment should then be in terms of the further question: ‘If students understand this idea, how can they demonstrate it?’

Plato said: ‘The task of the teacher is not to put knowledge where it does not exist, but rather lead the mind’s eye, so that it might see for itself’.

Human understanding requires processing in the individual’s mind. You can transfer information (rules, procedures, formulas, and so on) but understanding has to be developed. Processing for understanding cannot be left to chance – we have to provide learning opportunities within the curriculum that develop the students’ capacity to construct their own understandings and deliberately plan lessons that offer, support and develop rich and authentic thinking. This should be the primary purpose of a subject’s curriculum.

Well-documented curriculum is important because it sets the agenda, articulates the vision, and provides an enforceable structure, on which improvements to teachers' practice can be based, and referred back to. However, we have to simultaneously and continuously work on changing teachers' beliefs about teaching, pedagogy and methodology, otherwise the curriculum design will be largely ignored. Teachers ultimately determine how they will interpret any curriculum with which they work and their beliefs and ways of thinking will consequently determine the extent and success of any change. It is the teacher, and his or her pedagogical beliefs about how the learning should be shaped, that will ultimately determine whether or not learning will occur, how authentic that learning will be, and the quality of the learning that happens in their classes.

I strongly believe that a focus on learning leads to improvements in teaching and thus outcomes for our students. This said, focus needs to be continually maintained, sustained and enriched through various ways, such as curriculum documentation, engaging staff and students in reflective dialogue, modelling the sorts of behaviours that are valued and celebrating the good things that happen … even small steps. In an article by Kim Beswick, from the University of Tasmania, recently published in the Australian mathematics teacher magazine (No. 62, Volume 4, 2006), titled ‘The importance of mathematics teachers’ beliefs’, she notes ‘that what teachers believe influences their teaching yet the focus of much professional learning remains on influencing the specific practices and tools that teachers employ in their classrooms’. It is very difficult to implement changes to teachers’ practices without a concomitant shift in how they believe mathematics should be best taught and learnt. She concludes her article with the comment that: ‘Findings concerning the importance of teachers’ beliefs to the kinds of classrooms they create highlight the importance of individual teachers reflecting carefully on the beliefs they hold about the nature of mathematics and about mathematics teaching and learning’.

So, the question is this: can we, and if so, how do we, compel reflection in teachers who are not open-minded to this? Is there a bridge between where their thinking is now and what it could become with some reflection? In a recent paper (No. 161, February 2007) from the Seminar Series put out by the Centre for Strategic Education, by John Munro (‘Pedagogic capital: an essential concept for effective school leaders’),the author stresses the importance of schools having ‘pedagogic capital’ (the teaching power of the school) in order to effect changes that lead to improvements in student outcomes. He asserts that the leadership teams of schools have an important role to play in using ‘instructional leadership’ to develop a positive learning community and support an innovative, action-research based and collaborative approach to investigating issues associated with learning. The institution itself should thus take some responsibility in the development of a relevant and contextually appropriate curriculum, by providing the climate, and the means, by which its teachers can reflect on current practice.In another article titled ‘Rethinking Professional Development: Supporting reform in middle grades mathematics through the cultivation of teaching dispositions’,by Ron Ritchhart (in L. Leutzinger (ed): Mathematics in the middle, Reston, VA: NCTM – National Council of Teachers of Mathematics), Ritchhart also comments on the need for a pedagogical approach to professional development as an alternative to the ‘how to’ training of teachers. He says that we should be trying to inculcate five teaching dispositions. These are the dispositions to:

• grant students reason
• learn from students
• grapple with pedagogy
• focus on big ideas
• seek and offer collegiality.

In conclusion, teachers and students, and their schools, have mutual responsibilities and all need to reflect on their behaviours and belief systems in order to become continually evolving and developing learners. None of these groups should have a passive role when it comes to developing, implementing and sustaining a curriculum that will assist students to operate effectively in a future world of complexity and change. Like Plato’s ideal man, the creation of an appropriate curriculum needs inquisitiveness, theoretical grasp, sensitivity, courage, discipline and order. Let’s see if the ‘harmony of the soul’ can indeed be linked to the development of reason to form a ‘networked’ individual.