New Zealand Council for Educational Research - Merilyn Taylor
https://www.nzcer.org.nz/category/author/merilyn-taylor
enMathematics and The New Zealand Curriculum in the primary classroom
https://www.nzcer.org.nz/nzcerpress/curriculum-matters/articles/mathematics-and-new-zealand-curriculum-primary-classroom
<div class="field field--name-field-author-citation field--type-text field--label-hidden"><div class="field__items"><div class="field__item even">Merilyn Taylor and Judy Bailey</div></div></div><div class="field field--name-body field--type-text-with-summary field--label-above"><div class="field__label">Abstract: </div><div class="field__items"><div class="field__item even"><p>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 <em>The New Zealand Curriculum</em> (Ministry of Education, 2007). Examples relevant to the context of primary mathematics education are considered.</p></div></div></div><div class="field field--name-field-doi field--type-link-field field--label-inline clearfix"><div class="field__label">DOI: </div><div class="field__items"><div class="field__item even"><a href="https://doi.org/10.18296/cm.0125">https://doi.org/10.18296/cm.0125</a></div></div></div><div class="field field--name-field-journal-issue field--type-node-reference field--label-inline clearfix"><div class="field__label">Journal issue: </div><div class="field__items"><div class="field__item even"><a href="/nzcerpress/curriculum-matters/curriculum-matters-7-2011">Curriculum Matters 7: 2011</a></div></div></div><div class="commerce-product-field commerce-product-field-field-netsuite-internal-id field-field-netsuite-internal-id node-44865-product-field-netsuite-internal-id commerce-product-field-empty"></div><div class="commerce-product-field commerce-product-field-field-sync-failed field-field-sync-failed node-44865-product-field-sync-failed commerce-product-field-empty"></div><div class="field field--name-field-publication-full-text field--type-text-long field--label-hidden"><div class="field__items"><div class="field__item even"><p></p>
<p class="chapname">Mathematics and <i>The New Zealand Curriculum</i> in the primary classroom</p>
<p class="author">Merilyn Taylor and Judy Bailey</p>
<p class="ab">Abstract</p>
<p class="abs">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 <i>The New Zealand Curriculum</i> (Ministry of Education, 2007). Examples relevant to the context of primary mathematics education are considered.</p>
<p class="sec1">Introduction</p>
<p>The year 2010 heralded the mandatory implementation of <i>The New Zealand Curriculum</i> (<i>NZC</i>, 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 <i>NZC</i>.</p>
<p class="noindent1">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).</p>
<p class="noindent1">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”.</p>
<p class="noindent1">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 <i>NZC</i> 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).</p>
<p class="noindent1">Our thinking about beliefs, learning theories and possible links to <i>NZC</i> has emerged from ongoing discussions between ourselves (two preservice teacher educators), analysis of <i>NZC</i> 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.</p>
<p class="sec1">Radical constructivism</p>
<p>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).</p>
<p class="noindent1">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).</p>
<p class="sec1">Mathematical beliefs and metaphors</p>
<p>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 <i>suitcase mathematics</i>. 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 <i>dancing mathematics</i>, envisaged as an open space in which dancers are exploring ideas about quantity, patterns, shape, space and chance with other dancers (learners of mathematics).</p>
<p class="sec2">Suitcase mathematics</p>
<p>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).</p>
<p class="noindent1">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 <i>suitcase</i> remains separate from the individual, and the ideas are confined within set boundaries.</p>
<p class="noindent1">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 <i>how</i> rather than <i>why</i> (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.”</p>
<p class="noindent1">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.</p>
<p class="sec2">Dancing mathematics</p>
<p>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.</p>
<p class="noindent1">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).</p>
<p class="noindent1">The metaphor used to illustrate this conception is that of <i>dancing mathematics</i>. We envisage the experience of <i>dancing</i> 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).</p>
<p class="noindent1">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.</p>
<p class="sec1"><i>NZC</i>: Principles, values and key competencies</p>
<p>Radical constructivism and a view of mathematics as a constructive, creative endeavour give us a framework within which to consider some aspects of <i>The New Zealand Curriculum</i>. In <i>NZC</i>, principles, values and key competencies have been articulated and are expected to underpin curriculum implementation in all learning areas, including mathematics.</p>
<p class="sec2">Some thoughts about the principles and values</p>
<p>Embedded within the <i>NZC</i> 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, <i>Te Marautanga o Aotearoa</i> 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.</p>
<p class="noindent1">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.</p>
<p class="noindent1">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.</p>
<p class="noindent1">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.</p>
<p class="noindent1">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 <i>coherence</i>, 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).</p>
<p class="noindent1"><i>NZC</i> identifies <i>innovation</i>, <i>inquiry and curiosity</i> 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.</p>
<p class="sec2">Some thoughts about the key competencies</p>
<p><i>Thinking</i>, 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 <i>thinking</i> 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.</p>
<p class="noindent1">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 <a href="http://www.censusatschool.org.nz">www.censusatschool.org.nz</a>) has become an important way to support the making of these connections.</p>
<p class="noindent1">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 mathematic<i>s</i> 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 <i>relating to others</i>, another of the five key competencies.</p>
<p class="noindent1">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 <i>managing self</i> (Ministry of Education, 2007, p. 12).</p>
<p class="noindent1">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.</p>
<p class="noindent1">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 <i>end</i> 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.</p>
<p class="sec1">Concluding thoughts</p>
<p>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 <i>NZC</i>. 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 <i>NZC</i>. 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 <i>NZC</i>. Shall we dance?</p>
<p class="sec1">References</p>
<p class="ref">Allen, P., Taleni, L., & Robertson, J. (2009). “In order to teach you, I must know you”: The Pasifika initiative: A professional development project for teachers. <i>New Zealand Journal of Educational Studies</i>, <i>44</i>(2), 47–62.</p>
<p class="ref">Barker, M. (2008a). How do people learn?: Understanding the process. In C. McGee & D. Fraser (Eds.), <i>The professional practice of teaching</i>. Melbourne: Cengage Learning.</p>
<p class="ref">Barker, M. (2008b). <i>The New Zealand curriculum</i> and pre-service teacher education: Public document, private perceptions. <i>Curriculum Matters</i>, <i>4</i>, 7–19.</p>
<p class="ref">Barton, B. (2008). <i>The language of mathematics: Telling mathematical tales</i>. New York: Springer.</p>
<p class="ref">Claxton, G. (1991). <i>Educating the inquiring mind: The challenge for school science.</i> Hemel Hempstead, UK: Harvester Wheatsheaf.</p>
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<p class="ref">Dossey, J. (1992). The nature of mathematics: Its role and its influence. In D. A. Grouws (Ed.), <i>Handbook of research on mathematics teaching and learning</i> (pp. 39–48). New York: Maxwell Publishing.</p>
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<p class="ref">Forrester, R. (2003). In J. Way & T. Beardon (Eds.), <i>ICT and primary mathematics.</i> Maidenhead, UK: Open University Press.</p>
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<p class="ref">Hughes, M. (1986). <i>Children in number: Difficulties in learning mathematics</i>. Oxford: Basil Blackwell.</p>
<p class="ref">Huinker, D. (2002). Calculators as learning tools for young children’s exploration of number. <i>Teaching Children Mathematics</i>, <i>8</i>(6), 316–321.</p>
<p class="ref">Knoll, E. (2008). Silence! Reviving an oral tradition in mathematics education. In B. Warland (Ed.), <i>Silences in teaching and learning in higher education</i> (pp. 131–133). Hamilton, ON: Society for Teaching and Learning in Higher Education.</p>
<p class="ref">Kotzee, B. (2010). Seven posers in the constructivist classroom. <i>London Review of Education</i>, <i>8</i>(2), 177–187.</p>
<p class="ref">Magrini, J. (2010). How the conception of knowledge influences our educational practices: Toward a philosophical understanding of epistemology in education. <i>Curriculum Matters</i>, <i>6</i>, 6–27.</p>
<p class="ref">Mason, J. (2008). <i>ICMI Rome 2008: Notes towards WG2</i>. Retrieved 16 July 2008, from <a href="http://www.unige.ch/math/EnsMath/Rome2008/WG2/Papers/MASON.pdf">www.unige.ch/math/EnsMath/Rome2008/WG2/Papers/MASON.pdf</a></p>
<p class="ref">McChesney, J., & Cowie, B. (2008). Communicating, thinking, and tools: Exploring two of the key competencies. <i>Curriculum Matters</i>, <i>4</i>, 102–111.</p>
<p class="ref">McCutcheon, G. (1995). <i>Developing the curriculum: Solo and group deliberation</i>. White Plains, NY: Longman.</p>
<p class="ref">Mercer, N. (2000). <i>Words and minds: How we use language to think together</i>. London: Routledge.</p>
<p class="ref">Ministry of Education. (2007). <i>The New Zealand curriculum</i>. Wellington: Learning Media.</p>
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<p class="ref">Ohia, M. (1995). Māori and mathematics: What of the future? In B. Barton & U. Fairhall (Eds.), <i>Mathematics in Māori education: A collection of papers for the history and pedagogy of mathematics conference, Cairns, Australia, July 1995</i> (pp. 33–38)<i>.</i> Auckland: The University of Auckland.</p>
<p class="ref">Renert, M. (2011). Mathematics for life: Sustainable mathematics education. <i>For the Learning of Mathematics</i>, <i>31</i>(1), 20–26.</p>
<p class="ref">Sharples, R. (1985). <i>Plato: Meno.</i> Warminster, UK: Aris and Phillips.</p>
<p class="ref">Simic-Muller, K., Turner, E., & Varley, M. (2009). Math club problem posing. <i>Teaching Children Mathematics</i>, <i>16</i>(4), 206–213.</p>
<p class="ref">Smith, D. (2011). Neo-liberal individualism and a new essentialism: A comparison of two Australian curriculum documents. <i>Journal of Educational Administration and History</i>, <i>43</i>(1), 25–41.</p>
<p class="ref">Solomon, Y. (2009). <i>Mathematical literacy: Developing identities of inclusion. </i>Abingdon, UK: Routledge.</p>
<p class="ref">von Glasersfeld, E. (1984). An introduction to radical constructivism. In P. Watzlawick (Ed.), <i>The invented reality</i>. New York: Norton.</p>
<p class="ref">von Glasersfeld, E. (1987). Learning as a constructive activity. In C. Janvier (Ed.), <i>Problems of representation in the teaching and learning of mathematics</i> (pp. 3–17). Montreal, QC: Lawrence Erlbaum.</p>
<p class="sec1">The authors</p>
<p>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.</p>
<p class="noindent1">Emails: <a href="mailto:meta@waikato.ac.nz">meta@waikato.ac.nz</a>; <a href="mailto:jlbailey@waikato.ac.nz">jlbailey@waikato.ac.nz</a></p>
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</div></div></div>Fri, 02 Dec 2011 04:06:00 +0000joanna.morton44865 at https://www.nzcer.org.nzhttps://www.nzcer.org.nz/nzcerpress/curriculum-matters/articles/mathematics-and-new-zealand-curriculum-primary-classroom#commentsMathematics and Māori-medium education: Learners’ perspectives
https://www.nzcer.org.nz/nzcerpress/set/articles/mathematics-and-maori-medium-education-learners-perspectives
<div class="field field--name-field-author-citation field--type-text field--label-hidden"><div class="field__items"><div class="field__item even">Ngārewa Hāwera, Merilyn Taylor and Leeana Herewini</div></div></div><div class="field field--name-body field--type-text-with-summary field--label-above"><div class="field__label">Abstract: </div><div class="field__items"><div class="field__item even"><p>There is little research about the experiences of students in Māori-medium education learning mathematics. This article reveals what these students see as their teacher’s role, and the advice that they’d give to other students making the transition to secondary school.</p></div></div></div><div class="field field--name-field-doi field--type-link-field field--label-inline clearfix"><div class="field__label">DOI: </div><div class="field__items"><div class="field__item even"><a href="https://doi.org/10.18296/set.0471">https://doi.org/10.18296/set.0471</a></div></div></div><div class="field field--name-field-journal-issue field--type-node-reference field--label-inline clearfix"><div class="field__label">Journal issue: </div><div class="field__items"><div class="field__item even"><a href="/nzcerpress/set/set-2009-no-2">set 2009: no. 2</a></div></div></div><div class="commerce-product-field commerce-product-field-field-netsuite-internal-id field-field-netsuite-internal-id node-43678-product-field-netsuite-internal-id commerce-product-field-empty"></div><div class="commerce-product-field commerce-product-field-field-sync-failed field-field-sync-failed node-43678-product-field-sync-failed commerce-product-field-empty"></div><div class="field field--name-field-publication-full-text field--type-text-long field--label-hidden"><div class="field__items"><div class="field__item even"><p></p>
<div class="booksection">
<h1 class="left"><a id="a9"></a><b>Mathematics and Māori-medium education</b><br /><i>Learners’ perspectives</i></h1>
<p class="noindent1"><b>NGĀREWA HĀWERA, MERILYN TAYLOR and LEEANA HEREWINI</b></p>
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<p class="noindent"><b>KEY POINTS</b></p>
<p class="bull">•&&Students tended to perceive that their teacher’s role was to provide them with the mathematics learning they needed, rather than that the students needed to be active participants.</p>
<p class="bull">•&&Student advice for transitioning to secondary school was to work hard in Year 8, have a good grasp of basic mathematics facts and, for students transitioning to English-medium schools, get help with mathematics vocabulary.</p>
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<p class="noindent">There is little research about the experiences of students in Māori-medium education learning mathematics. This article reveals what these students see as their teacher’s role, and the advice that they’d give to other students making the transition to secondary school.</p>
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<h3 class="left3">Mā tō rourou, mā taku rourou, ka ora ai tātou</h3>
<h2 class="left">Background</h2>
<p class="noindent1"><span class="dropcap1">C</span>hildren’s perspectives are unique. The United Nations Declaration on Human Rights states explicitly that children should be given a voice on matters that affect them (New Zealand Ministry of Foreign Affairs and Trade, 1997). Children are key stakeholders in our schools, and, as such, it is important to listen to their perceptions about their experiences (Taylor, Hāwera, & Young-Loveridge, 2005). Listening to children’s voices can inform mathematics educators in ways that other stakeholders are unable to, and can enable teachers to plan classroom experiences and ways of delivery to better meet children’s needs (Hamilton, 2006).</p>
<p class="indent">Māori underachievement in mathematics has been of concern in New Zealand for many years (Crooks & Flockton, 2006). Recent government initiatives such as Te Poutama Tau have been implemented to help Māori children’s learning in Māori-medium contexts. Listening to Māori children explain their thinking in mathematics may help to reveal factors that are influential.</p>
<p class="indent">Research indicates that children in Māori-medium contexts are aware that there are a number of sources available to them for support when learning mathematics. Not surprisingly, this includes their teacher (Hāwera, Taylor, Young-Loveridge, & Sharma, 2007). When children perceive their teacher to be a facilitator or mentor rather than a transmitter of mathematics ideas, they play a more active role in their own learning (Taylor et al., 2005). Such active engagement can include children setting their own goals for mathematics. Learning goals can be closely linked to children’s achievement (Hattie, 2009). When teachers collaborate with children to set challenging learning goals, this can increase children’s engagement and participation. The monitoring and assessment of such goals can provide useful information for further learning for both the teacher and the child (Gregory, Cameron, & Davies, 2000).</p>
<p class="indent">Shared meanings and understandings should be integral to the learning process (Bishop, 2005). A mutual respect between Māori children and their teachers is required to enable reciprocal learning (ako) to occur (Macfarlane, 2004). While the teacher may assume a greater share of responsibility in the partnership, the agenda for learning experiences can be informed by the voices of the learners. Planning opportunities for children to invent their own strategies to solve mathematics problems is beneficial for the learning of mathematical ideas. Active participation between the teacher and learner can help children to make the necessary connections (Bucholz, 2004).</p>
<p class="indent">Mathematics education should aim to support people to make informed decisions regarding the events that affect them. Minority-group children can be empowered to make such decisions by knowing that they have a rich and meaningful mathematical heritage. Mathematics education should include ways to “reaffirm, and in some instances, restore the cultural dignity of children” (D’Ambrosio, 2001, p. 308). Supporting Māori children to maintain a positive view of their own racial group and cultural background would appear to be worthwhile for children’s learning (Bishop, 2005).</p>
<p class="indent">Children who move from Māori-medium to general secondary schools in Aotearoa New Zealand warrant attention. The idea that academic skills are automatically transferred across languages without appropriate intervention is flawed (Cummins, 2000). Barton (2008) argues that the learning of mathematical concepts is strongly linked to appropriate development in language. Concurrent learning of mathematics and language can optimise the quality of the discourse that children need to engage in to advance their mathematical thinking. Christensen (2003) states that this is a major challenge for teachers in Māori-medium education. Children may need to be given formal instruction in vocabulary and language (Berryman & Glynn, 2003) to support their academic development.</p>
<h2 class="left1">Method</h2>
<h3 class="left1">Participants</h3>
<p class="noindent">These data focus on the responses of 61 Years 5–8 children from four Māori-medium schools, and 10 Māori graduates from one of these settings, who were in their first year of secondary schooling (Year 9). Six of the Year 9 children attended a wharekura (secondary school in Māori medium) while the remaining four attended general secondary schools. Fifty of the total number of children had participated in Te Poutama Tau, the Māori-medium equivalent of the Numeracy Development Project. Participation varied from one to four years prior to this study. Forty-six children were female and 25 were male.</p>
<p class="center1">TABLE 1 COMPOSITION OF THE SAMPLE BY YEAR LEVEL</p>
<p class="center"><img src="/system/files/journals/set/downloads/set2009_2_052/OEBPS/images/56-a.jpg" alt="Image" /></p>
<h2 class="left1">Procedure</h2>
<p class="noindent">Schools were asked to nominate Years 5–8 children from across a range of mathematics levels. The Year 9 children were selected by being at stage seven or higher on the numeracy framework (see Numeracy Project Book 1, Ministry of Education, 2008b, for more information). Children were interviewed individually for about 30 minutes in te reo Māori or English (their choice) in a quiet place away from the classroom. They were told that the interviewer was interested in finding out about their thoughts regarding their learning of pāngarau/mathematics.</p>
<p class="indent">The Years 5–8 children were asked a collection of questions. This article analyses their responses to one. The other questions have been analysed and discussed elsewhere (see Hāwera et al., 2007). The additional question given to the Year 9 children was part of a master’s thesis study.</p>
<blockquote><p class="noindent2">The questions analysed in this paper are:</p>
<p class="noindent2">Ki ōu whakaaro, he aha ngā mahi ā tō kaiako hei āwhina i ā koe ki te ako pāngarau? [How do you think your teacher helps you to learn mathematics?]</p>
<p class="noindent2">He aha ōu tohutohu e pā ana ki te pāngarau ki tētehi tamaiti Tau 8 e haere mai ana i tētehi kura kaupapa Māori ki tēnei kura ā tērā tau? [What advice about mathematics would you give to a Year 8 child coming from a kura kaupapa Māori to this school next year?]</p>
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<h2 class="left1">Results and discussion</h2>
<h3 class="left1">Perception of the teacher’s role</h3>
<p class="noindent">All of the 71 children were asked about how they perceived their teacher helping them to learn mathematics.</p>
<p class="indent">The most common response was to refer to the teacher’s behaviour in the classroom (see Table 2). In particular, nine out of 10 of the Year 9 children did so.</p>
<p class="center1">TABLE 2 CHILDREN’S VIEWS REGARDING TEACHER’S ROLE</p>
<p class="center"><img src="/system/files/journals/set/downloads/set2009_2_052/OEBPS/images/56-b.jpg" alt="Image" /></p>
<h3 class="left1">To display a particular behaviour</h3>
<p class="noindent">About a third of the children commented on how they thought their teachers acted. The teacher was described as someone who:</p>
<blockquote><p class="noindent2">… te tuhi i runga i te papa tuhituhi. [… performs tasks like writing on the board.]</p>
<p class="noindent2">… mahi i ngā mea uaua ake mōkū. [… provides me with harder work.]</p>
<p class="noindent2">He just talks to us and explains, and explains quite a lot about, about the subject.</p>
<p class="noindent2">Mēnā kāore au i te mārama ki ētahi mahi, ka hōmai ia he tauira, he mea ngāwari ake, kia mārama ai au. [If I didn’t understand she would give an example, an easier example so I could understand.]</p>
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<h3 class="left1">To show mathematics strategies</h3>
<p class="noindent">Twenty-two of the children mentioned that the teacher helped mostly by showing them a strategy or strategies to do the mathematics. This was the most common response of the Years 5–8 children. This group of children placed great reliance on the teacher to supply them with a way or ways to do the mathematics:</p>
<blockquote><p class="noindent2">Ā, ka whakaatu mai ia i ētahi rautaki kia māmā ake te haere mo te pāngarau, … kore tahi noa iho, āhua toru, ae, ae. [He shows us some strategies so that the mathematics is easier … not just one, about three, yes, yes.]</p>
<p class="noindent2">… ka mahi ia tētahi pātai pāngarau i runga i te papatuhituhi, ana, ka pātai ia ki a mātou pēhea ka mahi tētahi rautaki mo tēnei whakautu. Ara, ka tarai mātou, ara ka tuhi ia tētahi rautaki kia mārama mātou ki tētahi rautaki rerekē mo taua pātai, ae. [… he does some mathematics questions on the board, and he asks us how would we use a strategy for this answer. We try and he writes another strategy so that we can understand a different strategy for that question.]</p>
<p class="noindent2">Tino pai, ōna mahi whakaako i ngā rautaki, i ngā rapanga ki a mātou. [He’s very good at teaching strategies and problems to us.]</p>
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<h3 class="left1">To teach mathematics knowledge</h3>
<p class="noindent">Three of the children mentioned that their teacher helped them to develop particular mathematics ideas:</p>
<blockquote><p class="noindent2">… ina kāre koe i te mohio i te rua whakarau rua, ka whakaako ia … [… if you don’t know 2 × 2, he will teach you …]</p>
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<h3 class="left1">To help with difficult mathematics</h3>
<p class="noindent">Three others mentioned that the teacher helps when the mathematics is “difficult”:</p>
<blockquote><p class="noindent2">… ka taea e ia ki te āwhina i a mātou i ētahi wā, mēnā e uaua te pātai … [… helps us sometimes when the question is difficult …]</p>
<p class="noindent2">… kia mahi mai i ngā mea māmā ki ngā mea uaua … [… does the easy to difficult ones …]</p>
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<h3 class="left1">No view offered</h3>
<p class="noindent">Eleven children did not appear to have any view about how their teacher helped them with their mathematics learning.</p>
<h3 class="left1">Discussion</h3>
<p class="noindent">Overall, these children had a view of the teacher playing a predominant part in their learning environment. They perceived that it was the teacher’s role to display a particular behaviour; for example, to “explain the work” and teach the mathematics strategies to be employed. Their responses indicated that they thought very little input was required of them. It is interesting to note that the focus on the teacher’s behaviour became more pronounced at the Year 9 level. The pedagogy that these children had been exposed to resulted in their perception that the role accorded to the teacher was one of being mainly responsible for their mathematics learning. Te Poutama Tau (Ministry of Education, 2005) and <i>Pāngarau i Roto i te Marautanga o Aotearoa</i> (Ministry of Education, 1996), however, emphasise a more reciprocal relationship for learning mathematics.</p>
<p class="indent">The way that children perceive the part that their teacher plays in their learning has implications for their participation in mathematics programmes (Taylor et al., 2005). If children perceive that the responsibility of the teacher is one of explaining mathematics ideas and demonstrating specific strategies, it may be difficult for such children to take any responsibility for developing their learning in mathematics.</p>
<p class="indent">It is of interest that 11 children did not offer a view about their teacher’s role. Reciprocity involves awareness and expectations of each other’s roles and responsibilities, without which mathematics learning may be compromised (Averill & Clarke, 2006). Perhaps mathematics learning for Māori children would be greatly enhanced if they thought their contribution to the teaching and learning process was sought and valued. Consistent expectations for such involvement must be continually explored by both parties and made explicit (Hunter, 2006). This would affirm their partnership with the teacher and foster engagement with that kaupapa (Macfarlane, 2004).</p>
<p class="indent">None of the views shared seemed to indicate a link between the teacher’s role and the children’s cultural background. The promotion of children’s identity as Māori (which is a major reason for their learning in these settings) did not feature. More explicit connections between being Māori and being a competent mathematician would be helpful. This would reaffirm the legitimacy of both factors (D’Ambrosio, 2001).</p>
<h2 class="left1">Advice about transition to secondary school</h2>
<p class="noindent">Because of the paucity of research involving the views of secondary school children who had attended Māori-medium primary schools, this group of 10 was asked an additional question. This question was about advice they would offer to children making the transition from a Māori-medium setting to learning mathematics at secondary school. Their comments were grouped broadly into three categories (see Table 3).</p>
<p class="center1">TABLE 3 YEAR 9 STUDENTS’ ADVICE TO YEAR 8 STUDENTS</p>
<p class="center"><img src="/system/files/journals/set/downloads/set2009_2_052/OEBPS/images/57-a.jpg" alt="Image" /></p>
<h3 class="left">Work hard and listen to your teacher</h3>
<p class="noindent">Three of the children thought that it was important for Year 8 pupils to listen to the teacher and work hard:</p>
<blockquote><p class="noindent2">Whakarongo ki ōna kaiako i te tau waru no te mea ka uaua ake i te tau 9. [Listen to the teacher when you are Year 8 because the work is harder in Year 9.]</p>
<p class="noindent2">Na te mea ko ngā mea ka ako i te tau 9 he tino rerekē ki ngā mea o te tau. [Because what you learn in Year 9 is very different to what you learn in Year 8.]</p>
<p class="noindent">Work hard.</p>
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<h3 class="left1">Learning</h3>
<p class="noindent">Three children commented in some way about learning:</p>
<blockquote><p class="noindent2">Keep learning.</p>
<p class="noindent2">Ka ako i ētahi mea hou. [You learn new things.]</p>
<p class="noindent2">Me mōhio pai koe ki ou mahi whakarau, whakarea, tango ērā mea katoa. [You should know all multiplication, division, subtraction and all those things.]</p>
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<h3 class="left1">Vocabulary</h3>
<p class="noindent">Two children who were now at general secondary schools made particular mention of the difficulty in learning new vocabulary:</p>
<blockquote><p class="noindent2">Vocabulary, like fractions, I didn’t know that was hautau [fractions] and algebra, all those words that are hard, like sum, that’s plus. I didn’t know that.</p>
<p class="noindent2">I would probably help, help them with the words.</p>
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<h3 class="left1">Discussion</h3>
<p class="noindent">There was no one particular idea that emerged from this small group. The responses indicate, however, that they perceived the onus was on Year 8 children to understand that learning mathematics in a secondary school was a serious enterprise and would be challenging. Year 8 children needed to know that they would be expected to behave in particular ways in order to be successful at mathematics. This advice included <i>working hard</i>, <i>listening to the teacher</i>, <i>knowing basic facts</i>. No mention was made of being prepared to share and contribute ideas and ways of making connections in mathematics.</p>
<p class="indent">Two children spoke of the difficult nature of learning mathematics vocabulary when moving from a Māori-medium setting to one where the language of instruction is English. Cummins (2000) reports that if children move to a new environment where the language of instruction is different, they may need to be supported in the transition. This supports Barton’s (2008) contention that conceptual understanding of mathematical ideas is interdependent with the development of appropriate mathematics vocabulary and discourse. Such an approach can only enhance Māori achievement in mathematics (Ministry of Education, 2006).</p>
<h2 class="left1">Conclusion</h2>
<p class="noindent">Children’s perspectives are valuable and offer insights into views about their learning. Listening to these perspectives can offer teachers insights that could inform their pedagogical practices. The data here indicate that these Māori children have strong views about the role of their teacher in their mathematics sessions at school. They expect the teacher to “provide” them with all the mathematics they need to learn and, in the main, are comfortable with this idea. However, the literature indicates that it would be more appropriate to help Māori children to understand that they should contribute more actively to their mathematics learning. They need to know that there are more mathematical strategies available to them than those the teacher introduces. Children in Māori-medium settings have the capacity, and therefore need the opportunity, to make sense of mathematics ideas in their own way. A balance between teacher direction and supporting children to take more responsibility for their learning in mathematics is required. This sharing of responsibility can also contribute to children becoming intrinsically motivated and thereby positively affect their level of academic achievement (Hattie, 2009). Increasing student participation affects the roles of both the learner and the teacher. For example, when children expect to justify their mathematical ideas, the monitoring and evaluating of the learning for both parties can be enhanced. This would be commensurate with the intent of Te āhua o ā Tātou ākonga (Ministry of Education, 2008a).</p>
<p class="indent">While the number of children interviewed here is small, research with children in Māori-medium settings, and those who transition to other secondary schools, is limited. Any voices from Māori learners in mathematics deserve consideration if we are sincere about addressing underachievement. The challenge is how to align these voices with appropriate pedagogical practice for learners in Māori-medium settings. For this to occur, research in these settings is essential, including investigating teacher perspectives regarding the teaching and learning of mathematics.</p>
<h2 class="left1">Points to ponder</h2>
<p class="noindent">It has been a privilege for us to listen to the perspectives of these children. The insights we have gained from this study lead us to suggest the following questions for teachers of children from or in Māori-medium settings:</p>
<p class="bull">•&&Does Māori culture contribute to your mathematics programme?</p>
<p class="bull">•&&What connections are explored between the mathematics and children’s own language?</p>
<p class="bull">•&&Are your children having mathematical conversations?</p>
<p class="bull">•&&Are children’s ideas being sought as a starting point for mathematical conversations and investigations?</p>
<p class="bull">•&&Do they know that they can create mathematics and ways of solving problems?</p>
<p class="bull">•&&Are they expected to prove and justify their mathematical solutions to each other?</p>
<p class="bull">•&&Do they know that they can pose mathematics problems and that this is expected?</p>
<p class="bull">•&&Do the children have opportunities to write about their thinking as well as their mathematical notation?</p>
<p class="bull">•&&Is there a public space, such as a modelling book, where children can revisit their mathematical thinking or that of their peers?</p>
<p class="bull">•&&Are your children provided with opportunities to set, monitor and assess mathematics goals?</p>
<h2 class="left1">Ngā mihi</h2>
<p class="noindent">Hei whakamutu ake tēnei wāhanga o te rangahau, ka mihi ake ki ngā whānau, ngā mātua, ngā tamariki i whakaae kia uru mai ki tēnei rangahau. Mā te mahi pēnei ka mārama pai ai te huarahi, ka hiato ngā whakatupuranga.</p>
<p class="indent">Nō reira, ngā karanga maha, ka nui te mihi.</p>
<h2 class="left1">References</h2>
<p class="hang">Averill, R., & Clarke, R. (2006). “If they don’t care, then I won’t”: The importance of caring about our students’ mathematics learning. <i><b>set</b></i>: <i>Research Information for Teachers</i>, <i>3</i>, 14–20.</p>
<p class="hang">Barton, B. (2008). <i>The language of mathematics: Telling mathematical tales</i>. New York: Springer.</p>
<p class="hang">Berryman, M., & Glynn, T. (2003). <i>Transition from Māori to English: A community approach</i>. Wellington: New Zealand Council for Educational Research.</p>
<p class="hang">Bishop, R. (2005). Pathologizing the lived experiences of the indigenous Māori people of Aotearoa/New Zealand. In C. M. Shields, R. Bishop, & A. E. Mazawi (Eds.), <i>Pathologizing practices: The impact of deficit thinking on education</i> (pp. 55–84). New York: Peter Lang.</p>
<p class="hang">Bucholz, L. (2004). The road to fluency and the license to think. <i>Teaching Children Mathematics, 10</i>(5), 362–367.</p>
<p class="hang">Christensen, I. (2003). <i>An evaluation of Te Poutama Tau</i>. Wellington: Ministry of Education.</p>
<p class="hang">Crooks, T., & Flockton, L. (2006). <i>Assessment results for students in Māori medium schools 2005</i>. Dunedin: Educational Assessment Research Unit, University of Otago.</p>
<p class="hang">Cummins, J. (2000). <i>Language, power and pedagogy: Bilingual children in the crossfire</i>. Clevedon, UK: Multilingual Matters.</p>
<p class="hang">D’Ambrosio, U. (2001). What is ethnomathematics and how can it help children in schools? <i>Teaching Children Mathematics, 7</i>(6), 308–310.</p>
<p class="hang">Gregory, K., Cameron, C., & Davies, A. (2000). <i>Self assessment and goal setting</i>. Melville, Canada: Connections Publishing.</p>
<p class="hang">Hamilton, M. (2006). Listening to student voice. <i>Curriculum Matters</i>, <i>2</i>, 128–145.</p>
<p class="hang">Hattie, J. (2009). <i>Visible learning. A synthesis over 800 meta-analyses relating to achievement</i>. New York: Routledge.</p>
<p class="hang">Hāwera, N., Taylor, M., Young-Loveridge, J., & Sharma, S. (2007). Who helps me learn mathematics, and how?: Māori children’s perspectives. In B. Annan, F. Ell, J. Fisher, J. Higgins, K. Irwin, A. Tagg, G. Thomas, T. Trinick, J. Ward, & J. Young-Loveridge, <i>Findings from the New Zealand Numeracy Development Project 2006</i> (pp. 54–66). Wellington: Ministry of Education.</p>
<p class="hang">Hunter, R. (2006). Structuring the talk towards mathematical inquiry. In P. Grootenboer, R. Zevenbergen, & M. Chinnappan (Eds.), <i>Identities, cultures and learning spaces: Proceedings of the 29th annual conference of the Mathematics Education Research Group of Australasia</i> (pp. 309–317). Canberra, ACT: Mathematics Education Research Group of Australasia.</p>
<p class="hang">Macfarlane, A. (2004). <i>Kia hiwa ra! Listen to culture—Māori students’ plea to educators</i>. Wellington: New Zealand Council for Educational Research.</p>
<p class="hang">Ministry of Education. (1996). <i>Pāngarau i roto i te marautanga o Aotearoa</i>. Wellington: Author.</p>
<p class="hang">Ministry of Education. (2005). <i>Te poutama tau: ngā rauemi tautoko</i>. Te Whanganui-a-Tara: Te Pou Taki Kōrero Whāiti.</p>
<p class="hang">Ministry of Education. (2006). <i>Māori medium student outcome overview 2001–2005. Research findings on akonga achievement in pānui, tuhituhi and pāngarau in Māori medium education</i>. Wellington: Author.</p>
<p class="hang">Ministry of Education. (2008a). <i>Te marautanga o Aotearoa</i>. Wellington: Author.</p>
<p class="hang">Ministry of Education. (2008b). <i>The number framework: Book 1</i> (Rev. ed.). Wellington: Author.</p>
<p class="hang">New Zealand Ministry of Foreign Affairs and Trade. (1997). <i>Convention on the rights of the child: Presentation of the initial report of the government of New Zealand</i>. Wellington: Author.</p>
<p class="hang">Taylor, M., Hāwera, N., & Young-Loveridge, J. (2005). Children’s view of their teacher’s role in helping them learn mathematics. In P. Clarkson, A. Downton, D. Gronn, M. Horne, A. McDonough, R. Pierce, & A. Roche (Eds.), <i>Building connections: Theory, research and practice: Proceedings of the annual conference held at RMIT, Melbourne, 7th to 9th July 2005</i> (pp. 728–734). Sydney: Mathematics Education Research Group of Australasia.</p>
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<p class="noindent"><b>NGĀREWA HĀWERA</b> and <b>MERILYN TAYLOR</b> are lecturers in mathematics education at Te Kura Toi Tangata, University of Waikato.</p>
<p class="noindent2"><b>Emails</b>: <a href="mailto:ngarewa@waikato.ac.nz">ngarewa@waikato.ac.nz</a></p>
<p class="indent2"><a href="mailto:meta@waikato.ac.nz">meta@waikato.ac.nz</a></p>
<p class="noindent1"><b>LEEANA HEREWINI</b> is an adviser for School Support Services at the University of Waikato. She has a long-standing interest in mathematics education and has always worked in bilingual or immersion settings.</p>
<p class="noindent2"><b>Email</b>: <a href="mailto:Leeana@waikato.ac.nz">Leeana@waikato.ac.nz</a></p>
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</div></div></div>Wed, 07 Sep 2011 03:05:29 +0000george.wallace43678 at https://www.nzcer.org.nzhttps://www.nzcer.org.nz/nzcerpress/set/articles/mathematics-and-maori-medium-education-learners-perspectives#comments