Slow maths - a metaphor of connectedness for school mathematics

This dissertation proposes a metaphor of connectedness for school mathematics that honours the discipline of mathematics and engages students in authentic mathematical activity. It is located at the intersection of mathematics, education and philosophy, providing a fresh reading and synthesis of established ideas. It is thus an argument about what really counts in school mathematics, rather than a presentation of new empirical research. Part 1 of the thesis establishes the personal and political imperative for the thesis. It highlights the mismatch between dominant pedagogies of mathematics education and an alternative view that foregrounds students as mathematicians. I argue that the political debates that have punctuated and permeated school mathematics education over several decades are less about achievement levels or effective teaching than they are about epistemology. I examine dominant metaphors of education, arguing that they are locked into what Heidegger terms the technological enframing, casting students as products of, rather than participants in, the educational process. I describe the origins of slow food as a protest against a one-size-fits-all philosophy, and introduce the metaphor of Slow Maths as an alternative to these metaphors of education. I then re-examine the philosophical and epistemological underpinnings of mathematics education, arguing that absolutist and relativist philosophies of mathematics, rather than being binary opposites, arise from viewing mathematics from the outside and the inside, or far away and close-at-hand, respectively. Part 2 of the thesis presents extensive evidence to support three dimensions of connectedness that underpin Slow Maths: mathematical connectedness, cultural connectedness and contextual connectedness. I show that mathematics is legitimated through a knowledge mode, providing evidence that mathematics "speaks for itself". I show that it is continuously developing as a field, and describe the cultural context that promotes this development. I describe the reflexive relationship between mathematics and the world, providing evidence that mathematics both models the world and develops in response to the world. For each of these dimensions I use evidence from the discipline of mathematics itself, from the work of mathematicians and their personal narratives and from influential and contemporary mathematics education research. This thesis is unique in synthesising evidence from these three different sources in support of a philosophical position. Part 3 of the thesis moves from theory to practice. I give a practical example of a unit of work in secondary mathematics that moves beyond mechanistic solutions methods with contrived pseudo real world applications to one that has strong and rich mathematical, cultural and contextual connections. I argue for a new dimension of mathematical knowledge for teaching that I term cultural and contextual knowledge of mathematics, and provide reflections from a preservice teacher education course that affirm its value. The thesis does not purport to provide ready-made solutions. Rather it asserts the centrality of connectedness as a critical aspect of school mathematics and promotes an attitude of slowness as a way of engaging with both the discipline of mathematics and the activity of doing mathematics

Measuring Messy Mathematics: Assessing learning in a mathematical inquiry context

The pedagogy of inquiry to teach mathematics presents a seemingly messy yet busy classroom, where children are engaged in using purposeful mathematics to collaboratively generate effective and creative solutions to open-ended, ambiguous questions. Problems arise when describing student learning in inquiry settings, when assessment practices are chosen that do not align with or are not designed to capture learning in this context. This study will present an inquiry into mathematics inquiry classrooms, to analyse and interpret the relationships between three classroom elements of assessment, teaching and learning. Distinctive to this study, the researcher was also the classroom teacher which offers the reader close insight into school practices in these classrooms. The researcher aimed to understand if an alignment between these classroom elements could support teaching and learning in inquiry settings. Using design research as a methodology (Cobb et al., 2003), two primary, inquiry classrooms (Years three and six) are presented as three iterative phases of study, each using particular theoretical lenses and analytical tools. In the first phase of study, theoretical analysis of formative assessment practices was based on Dewey’s (1891) conception of thinking as a process involving abstraction, comparison and synthesis. Vygotsky’s (1978) zone of proximal development was the analytical tool in the second phase of study and was used to analyse how the classroom teacher adjusted her teaching based on feedback she received through formative assessment. The third and final phase looked closer at the mathematical learning revealed through formative assessment using the DNR framework (Harel & Koichu, 2010): a Piagetian-influenced framework. Duckworth’s (2006) belief framework was also used to consider an interrelated synthesis of findings from this study, to assist in the development and refinement of assessment practices that align with using the inquiry pedagogy to teaching mathematics. Findings from three phases in this study revealed the inadequacy of summative assessment practices in capturing and describing student learning and thinking, fostered at higher levels through inquiry. In the first phase of study, analysis of assessment completed by students as part of their everyday, classroom curriculum reflected how such assessment only requires students to perform lower-level, reproductive thinking. In contrast, formative assessment opportunities encouraged students through inquiry to conceptualise their mathematical thinking in connected and abstract ways. The second phase of study focused on teaching in one inquiry classroom and characterised the difficulties classroom teachers face as they implement inquiry into their mathematics curriculum. Analysis of inquiry teaching and learning in this phase characterised how the teacher needed to be an engineer: able to interweave student ideas as potentialities, into the scaffolding of particular learning goals. Interweaving by the teacher, of students’ connections to the mathematical topic being explored, highlighted the complexity and messiness of the inquiry classroom where frequent interactions generated feedback about students’ thinking. Analysis of student learning in the third phase of study reflected a complex journey for students which considered interactions with peers in an inquiry context. Student thinking was provoked in these interactions shifting some responsibility for learning to the student as they tried to make sense of conflicting ideas. In all phases of this study, the inquiry pedagogy supported deep and connected mathematical learning, engineered by the classroom teacher towards particular learning and assessment goals. The learning process for students, as an ongoing journey of testing and refining mathematical processes and skills, was neglected when assessment did not value these characteristics. In inquiry, when assessment of learning values the messy and personal learning journey students face, there is potential for students to continue learning beyond the constraints of narrow curriculum objectives. Further research into ‘what else’ is learned through mathematics inquiry is required, to refine the pedagogy and to make its intentions clear. This study presents potential innovations to consider for future research