CONNECTEDNESS
7.1 Reflections on curriculum
It is noteworthy that of the 45 mathematicians mentioned by name in the preceding chapters, only one, Pythagoras, receives explicit mention in the Australian Curriculum: Mathematics. Few, if any of the mathematical topics such as knot theory or chaos theory receive any more than a passing mention. None of the concepts from mathematics education research is mentioned. The writers and curriculum authorities may well argue that the curriculum is a statement of the content that should be taught and the proficiencies to be developed rather than a pedagogic prescription, and that teachers are free to use any methodologies or elaborations they feel are appropriate. In fact, this was the precise remit given to the Australian Curriculum, Assessment and Reporting Authority. Or they may argue that the document is dynamic, and that the current iteration represents a work in progress. However, I suggest that no curriculum is pedagogy-free. The absence of anything more than a passing reference to mathematicians, coupled with the almost complete absence of any mathematics developed more recently than the 18th century, illustrates the limited value placed on mathematics as a living, cultural pursuit. The absence of examples from contemporary society illustrates the limited value placed on the pivotal role played by mathematics in contemporary society. The absence of any reference to key concepts from mathematics education research shows the limited value placed on what we know about creating a classroom culture in which deep mathematical thinking is promoted and developed. Despite the avowed emphasis on mathematical proficiencies such as understanding, reasoning, fluency and problem-solving, I suggest that together these absences reinforce a view of mathematics as little more than a set of facts or skills to be learned, the teacher as a tame and complicit implementer of a fixed agenda,
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and the student as a powerless pawn in the game of school mathematics. Slow Maths seeks to challenge this.
7.1.1 A comparison of curriculum documents
Mathematics curriculum documents around the world typically contain three distinct sections. The first is normally a preamble, stating why the study of mathematics is important and listing a small number of specific goals for school mathematics. For example, the Australian Curriculum: Mathematics (Australian Curriculum and Assessment Reporting Authority [ACARA], 2013, Rationale/Aims) lists three aims for school mathematics:
• To ensure that students are confident, creative users of mathematics, able to investigate, represent and interpret situations in their personal and work lives and as active citizens;
• To ensure that students develop an increasingly sophisticated
understanding of mathematical concepts and fluency with processes, and are able to pose and solve problems and reason in Number and Algebra, Measurement and Geometry, and Statistics and Probability; and
• To ensure that students recognise connections between the areas of mathematics and other disciplines and appreciate mathematics as an accessible and enjoyable discipline to study.
Along with the stated aims of other international curriculum documents, these aims bear a remarkable similarity to the three dimensions of connectedness— mathematical, personal and contextual—articulated in Chapters 2 and 342.
The second component of all curriculum documents is a statement of the attributes that students are expected to develop through their study of
42 In fact, a careful reading of curriculum documents was a significant influence in my formulation of the three aspects of connectedness described in this thesis.
163 mathematics. For example, the Australian Curriculum: Mathematics lists four so-called proficiency strands: understanding, fluency, reasoning and problem- solving, which are an adaptation of the five strands of conceptual understanding, procedural fluency, strategic competence, adaptive reasoning and productive disposition articulated by the National Research Council (2001).
The Singapore mathematics curriculum uses a framework that includes the development of positive attitudes, meta-cognitive skills and processes concerned with reasoning, communication and connections (Figure 7.1). Perhaps significantly this framework has remained unchanged for some 20 years (Fan & Zhu, 2007).
Figure 7.1: Mathematics framework (Ministry of Education Singapore, 2006, p. 6)
The Common Core State Standards for Mathematics (Common Core State Standards Initiative, 2010) from the USA lists eight standards for mathematical practice: make sense of problems and persevere in solving them; reason
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abstractly and quantitatively; construct viable arguments and critique the reasoning of others; model with mathematics; use appropriate tools strategically; attend to precision; look for and make use of structure; look for and express regularity in repeated reasoning.
The Finland National Core Curriculum for Basic Education (2004) states:
The task of instruction in mathematics is to offer opportunities for the development of mathematical thinking, and for the learning of mathematical concepts and the most widely used problem-solving methods. The instruction is to develop the pupil’s creative and precise thinking, and guide the pupil in finding and formulating problems and in seeking solutions to them. The importance of mathematics has to be perceived broadly: it influences the pupil’s intellectual growth and advances purposeful activity and social interaction on his or her part (p. 158)
Although there are relatively minor differences between the way each of these curriculum documents describes the goals of mathematics and the processes of thinking mathematically, the intent in each is clearly the same. Each explicitly emphasises the development of mathematical thinking, problem-solving and reasoning as central goals, each refers to the importance of mathematics in understanding the world, and each, perhaps implicitly, makes reference to the value of mathematics in one’s personal and social life. Again, they bear a remarkable similarity to the three dimensions of mathematical, personal and contextual connectedness. These are undoubtedly worthy goals, yet I suggest that they are all too often subverted by what is usually the most detailed section of each document, a list of content laid out as a developmental sequence of learning.
In some cases the description of content may be relatively minimal in an attempt to emphasise depth over breadth (Common Core State Standards Initiative, 2010), yet in all cases a developmental progression is described in
165 which students are expected to follow a common sequence of learning. I suggest that the descriptions of content are based on several assumptions, each of which could be debated at length:
• Learning is hierarchical, progressing from foundational ideas to more complex knowledge;
• Learning is, if not totally predictable, at least typical relative to the age of the student;
• A common sequence of learning is appropriate for the majority of students in the majority of contexts;
• This sequence of learning is relatively stable across time and cultures; • Curriculum development process is best done centrally; and
• The teacher is the implementer of curriculum, responsible for its interpretation in a given context, but not its formulation.
These assumptions almost invariably give rise to a formal curriculum that is compartmentalised into distinct areas of content and atomised according to perceived developmental levels. For example the Australian Curriculum: Mathematics, although initially designed to focus on a limited number of big ideas at each year level in three content domains of Number and Algebra, Statistics and Probability, and Measurement and Geometry, actually contains 278 content descriptions organised into 11 distinct year levels plus an additional level for students intending to proceed to higher levels of study (Table 7.1). Each content description then has one or more elaborations that expand on the skills and concepts described in the content description by providing examples of how they can be developed or applied. Although the document organises the content descriptions into sub-strands that flow through the curriculum, it is debatable whether or not such an extensive list of content focuses on a limited number of big ideas.
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Table 7.1: Number of content descriptions in each content strand for each year level in the Australian Curriculum: Mathematics
Year level Number and
algebra Measurement and geometry Statistics and probability F 6 5 1 1 7 5 3 2 11 10 4 3 10 6 4 4 13 9 6 5 12 8 5 6 13 9 5 7 19 8 6 8 12 8 7 9 10 9 6 10 13 4 8 10A 7 6 3
Such descriptions of content have a profound impact on teachers’ enactment of a curriculum. It is hardly surprising that when confronted with a plethora of content their concerns focus almost exclusively on placement of content or resources43. Critiques of curriculum such as those in the MERGA publication Engaging the Australian Curriculum Mathematics: Perspectives from the Field (Atweh, Goos, Jorgensen, & Siemon, 2012) seem far removed from the consciousness of teachers. In our discussion of the overarching themes of the
43 See Chapter 2 in which I discuss the focus of the contributions concerning the introduction of the Australian Curriculum: Mathematics made by teachers to the Australian Association of Mathematics Teachers email list.
167 Curriculum, Bill Atweh, Donna Miller and I (2012) conclude that it is both internally inconsistent in that the valued proficiencies articulated in the aims and rationale are not always evident in the content descriptions, and externally inconsistent in that it fails to enact more general goals of schooling such as active and engaged citizenship and building the capacity for lifelong learning. (Sentence omitted)
I suggest that this is due, at least in part, to the way curriculum documents are formulated. I suggest that they are developed with a mixture of hindsight, that is using a knowledge of the structure of the discipline of mathematics that only becomes apparent after significant higher-level study of mathematics, and foresight, that is using a knowledge of how the typical student progresses through levels of understanding. Both hindsight and foresight emphasise developmental progression, hierarchical structure, stability, and a research- development-dissemination model of curriculum (Begg, 2008) that positions teachers as implementers. While hindsight and foresight have a role to play in the formulation of curriculum, I argue that there is a much more important element—insight. Insight is context-specific. It is the special quality of expert teachers that enables them to see the connectedness of a particular piece of mathematics to other areas of mathematics, the world and the student.
Hence I suggest that we need a spacious curriculum (Angier & Povey, 1999); one that gives teachers room to explore with their students the culture and traditions of mathematics, room to use mathematics to understand and critique the world, and room to appreciate mathematics as part of themselves. This does not marginalise mathematical content in the quest for what might be seen as more holistic learning as critics of programs such as New Basics (Education Queensland, 2001) have argued, but just as in slow food the process of cooking generates the need to hone, develop and practice certain skills, so the process of engaging with authentic mathematics generates opportunities for students to practice and develop fluency and understanding of key skills and concepts.
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Deep conceptual learning or authentic learning takes time—this is ‘slow learning’ (in a sense akin to ‘slow cooking’) as opposed to the more superficial fast learning aimed at improving test scores. A more extended and flexible timeframe is needed to allow learners more control over activities. More open architectures of learning can involve, for example, creative projects leading to an open exhibition for parents in the tradition of Dewey; project method, based on an agreed problem or issue; the engagement of learners in resolving a real-world problem; or storyline, a form of thematic work based on the outline of a narrative. (Wrigley et al., 2012, p. 100)
Later in this chapter I give an example of how a unit on simultaneous linear equations might be developed to emphasise mathematical, cultural and contextual connectedness.
7.1.2 An aside: mathematics and numeracy
In order to maintain a focus on what is described in curriculum documents as mathematics, I have chosen not to discuss issues concerning the relationship between numeracy, or quantitative/mathematical literacy, and mathematics. The Australian Curriculum asserts that:
[s]tudents become numerate as they develop the knowledge and skills to use mathematics confidently across all learning areas at school and in their lives more broadly. Numeracy involves students in recognising and understanding the role of mathematics in the world and having the dispositions and capacities to use mathematical knowledge and skills purposefully. (ACARA, 2013, introduction)
Steen (2001) uses the term quantitative literacy to describe “a habit of mind, an approach to problems that employs and enhances both statistics and mathematics” (p. 5). He goes on to present a variety of aspects of numeracy including elements such as confidence, cultural appreciation and making decisions with mathematics, expressions of quantitative literacy in citizenship,