The aim of this article is to comment on the ways in which beliefs and theories of learning affect the teaching and learning of mathematics. When mathematics is viewed as a static body of knowledge, a transmission style of teaching is often employed. In contrast, a radical constructivist view of learning suggests that mathematics could be a constructive and creative endeavour. We suggest that this perspective of mathematics aligns with the principles, values and key competencies in The New Zealand Curriculum (Ministry of Education, 2007). Examples relevant to the context of primary mathematics education are considered.
Mathematics and The New Zealand Curriculum in the primary classroom
Abstract
The aim of this article is to comment on the ways in which beliefs and theories of learning affect the teaching and learning of mathematics. When mathematics is viewed as a static body of knowledge, a transmission style of teaching is often employed. In contrast, a radical constructivist view of learning suggests that mathematics could be a constructive and creative endeavour. We suggest that this perspective of mathematics aligns with the principles, values and key competencies in The New Zealand Curriculum (Ministry of Education, 2007). Examples relevant to the context of primary mathematics education are considered.
Introduction
The year 2010 heralded the mandatory implementation of The New Zealand Curriculum (NZC, Ministry of Education, 2007). This presented a timely opportunity to pause and reflect on our beliefs and teaching practices in mathematics, and to consider how these might connect with the current curriculum document. The beliefs a teacher holds about mathematics have a significant impact on the teaching and learning that occurs in the classroom setting (Grootenboer, 2008). Here we explore two beliefs that are held about mathematics, and consider how one of these resonates with the values, principles and key competencies in NZC.
One belief about mathematics is that it is a static, independently existing body of knowledge, containing truths relating to quantity, patterns, shape, space and chance (Fisher, 1990). From this absolutist perspective, mathematics exists separately from people, and students will be presented with and expected to learn about predetermined, fixed ideas. It has been claimed that teachers who primarily hold a conception of mathematics as a static body of knowledge are likely to employ a more transmission style of teaching (Dossey, 1992). In this situation children may not be given an opportunity to work as mathematicians (Davis, Sumara, & Luce-Kapler, 2000).
Another belief identified in the literature is that mathematics is a constructive, creative, experiential human endeavour (Dossey, 1992; Ernest, 1991; Mason, 2008). Ideas still remain focused on quantity, patterns, shape, space and chance, but the learner does their own exploring and discovering of ideas. The mathematics is embedded within the learner and the “doing”.
Learning theories also have an impact on the teaching and learning that occurs in the classroom (Barker, 2008a; Claxton, 1991; McCutcheon, 1995). Rather than being disconnected and remote, theories can guide and support our teaching when the time is taken to pause and reflect (Claxton, 1991). The theory of constructivism and its variants have been influential in curricula around the world since the 1970s (Barker, 2008a; Kotzee, 2010). In this paper we primarily draw on the ideas of radical constructivism (von Glasersfeld, 1984, 1987) and discourse about mathematical beliefs (Dossey, 1992) to consider the possible implications of enacting NZC in the primary mathematics classroom. This is important because forming personal perceptions of curriculum documents is an important aspect of enacting curriculum change (Barker, 2008b).
Our thinking about beliefs, learning theories and possible links to NZC has emerged from ongoing discussions between ourselves (two preservice teacher educators), analysis of NZC and a literature review. It can be valuable—and perhaps unsettling—to reflect more deeply on one’s ideas about a particular curriculum subject. Nevertheless, it has certainly been our experience that undertaking such reflection results in significant insights and can lead to changes in one’s teaching. Questions such as “What do I believe about the nature of mathematics and learning?” and “Are these beliefs evident in my teaching?” continue to help support our ongoing reflections.
Radical constructivism
Radical constructivism, as developed by the mathematics and science education theorist von Glasersfeld, holds that all communication and understanding comes from individuals reorganising their thinking in order to reconcile disruptions in the world of their personal experience (Cobb, 2011; von Glasersfeld, 1984). The importance of ongoing engagement with an experience under investigation is emphasised, and it is contended that “people’s actions are reasonable from their point of view” (Cobb, 2011, p. 159). Memory is regarded as reconstructive, and “conceptual activity is grounded in sensory-motor action” (Cobb, 2011, p. 159). Radical constructivists suggest that communication is not about “telling others” (von Glasersfeld, 1987) but rather is central to the process in which people continually adjust their actions in response to listening to and talking with others (Cobb, 2011).
Radical constructivism appears to align with an instrumental view of knowledge. Knowledge is considered to emerge “through a process of construction, or meaning making, where the students’ personal experiences, beliefs, desires and needs play integral roles in learning” (Magrini, 2010, p. 15). No claims about ultimate truths are made (von Glasersfeld, 1984, 1987). From this perspective, knowledge is regarded as always in a process of becoming (Magrini, 2010).
Mathematical beliefs and metaphors
As part of our own thinking we have found it useful to “play” with metaphors that illustrate differing beliefs about mathematics. The metaphor we explore for mathematics viewed as an unchanging, separately existing entity is suitcase mathematics. From this perspective, the role of the learner is to try to grasp hold of and hang onto the mathematical ideas. Our metaphor for a constructive, creative conception is dancing mathematics, envisaged as an open space in which dancers are exploring ideas about quantity, patterns, shape, space and chance with other dancers (learners of mathematics).
Suitcase mathematics
We have encountered suitcase mathematics in our own experiences as learners and teachers, and also in a variety of literature. It is interesting to note that this conception has its roots in the Platonist view of mathematics (Dossey, 1992). For example, Plato (427–347 BC) considered the objects of mathematics to exist independently of people (Barton, 2008; Ernest, 1991). Plato stated that mathematics was the highest expression of human thought (Barton, 2008) and that it has an appealing certainty (Sharples, 1985). From this perspective, mathematics is disconnected from everyday life and people (Ernest, 1991).
This belief is still influential. Some children, even as young as 4 years of age, have learnt that mathematics is often considered to be special and different, and that others have power and control over what mathematical ideas are to be encountered (Hughes, 1986). We believe that such beliefs could lead one to think of mathematics as separated from the self, whereby ideas are truths that are available to be picked up and put down according to circumstances. The suitcase remains separate from the individual, and the ideas are confined within set boundaries.
When mathematical ideas are thought of as existing separately from the self, the teacher’s primary role becomes that of presenter. Students learn to replicate or “do” set tasks and procedures, and they have a focus on how rather than why (Dossey, 1992). With a suitcase framework, the ideas and language used in the classroom are fixed, specific and in the process of being transmitted to the learners. The language used might include phrases such as “Yes, you are right” or “No, do it this way.”
A didactic view of the teacher as presenter of truths that need to be transmitted to passive, empty learners can be identified with an essentialist model of the curriculum (Magrini, 2010; Smith, 2011). Learners are regarded as waiting to be told essential truths held by those in authority, and instruction does not include co-operative learning, knowledge construction and meaning making (Magrini, 2010). We would contend that a suitcase view of mathematics aligns with an essentialist paradigm.
Dancing mathematics
The absolutist paradigm that positions mathematics as a collection of infallible and objective truths, separate from human affairs and values, has been dominant for over two thousand years (Ernest, 1991). This perception is now being challenged and mathematics is being affirmed as “fallible, changing, and like any other body of knowledge, the product of human inventiveness” (Ernest, 1991, p. 1). This claim of “inventiveness” seems to parallel von Glasersfeld’s (1984) contention that knowledge has to be constructed by the learner.
An alternative conception of mathematics, then, is of something that is creative, constructive, social and experiential (Ernest, 1991; Mason, 2008; Solomon, 2009). Within this conception, mathematics is thought of as being created by communication between people (Barton, 2008). These ideas imply a notion of mathematics that is not separate to one’s self, nor is it finite, but a creation that is “never finished, never completed” (Barton, 2008, p. 144). Mathematics can involve experimentation, observation, abstraction and construction (Dossey, 1992). This view of mathematics also has links to ancient Greek philosophy, this time to Aristotle, a pupil of Plato, living from 384 to 322 BC. Aristotle contended that mathematics exists in the mind and is based on “experienced reality” (Dossey, 1992, p. 40).
The metaphor used to illustrate this conception is that of dancing mathematics. We envisage the experience of dancing as involving learners creating ideas that relate to quantity, patterns, shape, space and chance. These “dances” are in constant generation and include other dancers (learners).
How mathematics is viewed has powerful educational consequences (Ernest, 1991). If mathematics is thought of as a body of truths that exist separately from people, then it would have no part to play in human affairs (Ernest, 1991). Alternatively, a conception of mathematics as constructive and creative would connect mathematics to individuals’ lives (Ernest, 1991; Renert, 2011). One’s view of mathematics therefore affects enacted pedagogy by influencing choices that are made regarding choice of activities, language used and modes of presentation.
NZC: Principles, values and key competencies
Radical constructivism and a view of mathematics as a constructive, creative endeavour give us a framework within which to consider some aspects of The New Zealand Curriculum. In NZC, principles, values and key competencies have been articulated and are expected to underpin curriculum implementation in all learning areas, including mathematics.
Some thoughts about the principles and values
Embedded within the NZC principles and values are references to equity and cultural diversity. When mathematics teaching is based on absolutist paradigms, there can be a sense of cultural alienation felt by many groups of students (Ernest, 1991; Renert, 2011). We wonder if an equitable and culturally responsive mathematics might, in part, be achieved via dancing mathematics; that is, by employing a creative and constructive pedagogy that responds to the variety of cultures that can be present in our mathematics classrooms. In New Zealand, Te Marautanga o Aotearoa suggests that mathematical problems and activities should be presented in contexts that are relevant to Māori (Ministry of Education, 2008). Similarly, contemporary mathematical contexts tailored to the needs of diverse students can be utilised (Simic-Muller, Turner, & Varley, 2009). This may entail schools learning about the variety of cultures present in their communities (Allen, Taleni, & Robertson, 2009) and embedding the mathematics of a particular community within a mathematics programme.
We suggest moving beyond learning to count in a variety of languages to exploring culturally relevant mathematical ideas. It is interesting to note that in some parts of Alaska and in the Maldives, for example, the mathematics of the indigenous culture is being incorporated into the mathematics curriculum (Barton, 2008). In New Zealand, Māori art, crafts and cultural artefacts can be aligned with mathematical topics and concepts (Ohia, 1995). For example, an investigation of the symmetries of kowhaiwhai patterns (patterns painted on the rafters in Māori meeting houses) is a rich geometrical activity.
Meeting diverse learning needs is a facet of equity. In a dancing mathematics classroom, appropriate, open-ended, investigative tasks could be incorporated. These are tasks that enable a variety of mathematical avenues to be explored and require a significant period of time to investigate. An example might be adapting the use of a question such as “Molly has two eggs, and she is given three more. How many eggs does Molly have?” to “There are five eggs in two baskets: how many eggs might be in each basket?” For a 5-year-old, such an open-ended investigative task might, depending on the individual, involve constructing a family of facts for the given number and/or exploring the relationships and commutative property of addition and subtraction. In such investigations, learners would be able to explore the particular avenues of mathematics they personally encounter.
Another open-ended investigation, this time from the strand Geometry and Measurement, could involve exploring tessellations. Rather than the children being asked to tessellate a triangle, then a square and so on, they could be asked more open-ended questions, such as, “Which shapes tessellate?”, “How do you know?”, “Have you found all of the regular shapes that tessellate?”, “Are there other shapes that tessellate?” and so on. Such open-ended investigations allow learners to construct their own theories in response to these questions and could lead in a variety of directions (e.g., regular tessellations, semi-regular tessellations, symmetry, transformations and/or the relationship between interior angles and shapes that can tessellate). Within a dancing mathematics pedagogy there would be opportunities for the teacher to support learners to explore varying aspects of the investigation. This does, however, require teachers to have an appreciation and knowledge of the multiple mathematical possibilities available for exploration.
An investigative approach can enable mathematics to be integrated with other curriculum areas, principles, values and key competencies. This can lead to an enactment of the principle of coherence, for example, which asks for “links within and across learning areas” (Ministry of Education, 2007, p. 9). With an investigation involving the geometry of patterns, there could be a link with art and design (creating a symmetrical pattern) and social studies (investigating cultural practices that are depicted in patterns).
NZC identifies innovation, inquiry and curiosity as desired values. These values are expected to be an integral part of all learning areas. Dancing mathematics could align with this position. When mathematics is encountered through open-ended investigations, there is space for learners to be innovative, inquisitive and curious. For example, it is possible to engender curiosity by using a calculator to explore number patterns (e.g., see Forrester, 2003; Huinker, 2002). In comparison, a suitcase conception of mathematics might allow a calculator to be used only as a checking (or calculating) device.
Some thoughts about the key competencies
Thinking, one of the key competencies, is described in part as “using creative, critical, and metacognitive processes to make sense of information, experiences, and ideas” (Ministry of Education, 2007, p. 12). This description of thinking aligns with the ideas encountered in radical constructivism and instrumentalism. When “thinking”, learners are constructing and making sense of their own ideas (McChesney & Cowie, 2008). We suggest these attributes align with a dancing mathematics pedagogy, whereby learners are committed to and involved in creating their own mathematics.
When mathematics is experienced as a constructive, creative enterprise there is a focus on learners’ “sense making”. Mason (2008) proposes that sense making is the most important, and possibly the most neglected, part of any pedagogical activity. In a dancing mathematics classroom it is the learners who are supported by the teacher to make sense of their own explorations. Learners do not follow particular procedures dictated by someone or something (e.g., a textbook). In a sense-making setting, learners make connections between their world and the ideas being constructed in their minds. Using equipment (e.g., decimats for learning about decimal numbers) and technology (e.g., see www.censusatschool.org.nz) has become an important way to support the making of these connections.
Another strategy to support sense making in a dancing mathematics classroom is creating a place where there is room for learners to engage in “exploratory talk”. This would mean learners critically examine and reason about their ideas (Mercer, 2000; Solomon, 2009). Time and persistence are needed to develop such exploratory talk, however. Asking process questions such as, “What are you thinking?”, “Does this remind you of another situation?” and “Would this make sense if …?” would be helpful. Learners in a dancing mathematics classroom would therefore be expected to explain and justify, carefully listen and thoughtfully respond to each other’s mathematical ideas. Such communication aligns with the radical constructivist idea of “negotiation in which people continually adjust their actions in response to their interpretations of others’ linguistic acts” (Cobb, 2011, p. 160). Communication also becomes an opportunity for learners to develop skills in relating to others, another of the five key competencies.
In addition to the communication and exploratory talk that can happen in a dancing mathematics classroom, there is also the need for silence—a spacious silence, where there is room for learners to think and reflect. Knoll (2008) suggests that “silence … is what can link doing and creating with learning … in mathematics” (p. 131). Rather than telling learners the answers, there is a need to provide times for silence, thus resisting the temptation to take over the learner’s thinking. Silences can initially be uncomfortable, but they can offer the learner an opportunity to explore their own ideas and find their own mathematical directions. Thinking spaces can help support students to see “themselves as capable learners”, an aspect of the key competency managing self (Ministry of Education, 2007, p. 12).
Another attribute of the dancing mathematics classroom is playing with mathematical ideas. It seems to be common lore that playful mathematical activity is appropriate for young children and research mathematicians, but in between these extremes learners are expected to simply do exercises and listen to teachers explain the mathematics (Barton, 2008). In the dancing mathematics classroom we advocate for play at all levels by being able to dance with both physical materials and abstract ideas to create an environment focused on playing with mathematical ideas. For example, children could be investigating their ideas about the nature of mathematics by taking photographs.
Looking back and reflecting are also vital components of learning mathematics (Mason, 2008). We suggest time be provided for learners to record, generalise and abstract their own ideas, thus offering learners opportunities to thoughtfully explain the mathematics being created. In the dancing mathematics classroom, teachers and learners could identify the specific learning outcomes at the end of a unit. Learners could identify their learning (with teacher support, if needed) after the ideas have been explored and discovered. Given our first suggestion of using open-ended investigations, each learner’s “I have learned to …” statement may be unique.
Concluding thoughts
We acknowledge that the enacted curriculum is usually different from the one intended by curriculum developers and policy makers (Smith, 2011). Taking time to consider how one’s thinking about teaching and learning connects with curriculum documents can begin to bridge the gap between intended and enacted curricula. Radical constructivism and considering beliefs about mathematics have given us a framework to examine and reflect on possible implications for enacting NZC. We contend that a view of mathematics as a constructive enterprise is congruent with an instrumentalist view of education and the intent of aspects of NZC. The work of von Glasersfeld (1984, 1987) and Cobb (2011) has been particularly useful for examining our beliefs about learning in mathematics. Metaphors have provided powerful images with which to explore and share our beliefs about mathematics, and to connect our thinking to the implementation of the principles, values and key competencies in NZC. Shall we dance?
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The authors
Merilyn Taylor and Judy Bailey are lecturers in mathematics education at the University of Waikato. Their research interests include how curriculum is enacted within mathematics contexts.
Emails: meta@waikato.ac.nz; jlbailey@waikato.ac.nz