Summary

Introduction

Mathematics¹ is a powerful social entity. Arguably the most international of all curriculum subjects, mathematics plays a key role in shaping how individuals deal with the various spheres of private, social and civil life. The belief that mathematics is a key engine in the economy is widely shared by politicians and social planners, the corporate sector, parents, and the general public. Mathematical competencies and identities that make a valuable contribution to society are developed from specific beliefs and practices. Marshalling evidence about the pedagogical practices in centres/kōhanga and schools/kura that allow those competencies and identities to develop is a primary educational necessity. This best evidence synthesis describes those practices and provides a critical evidence base for effective pedagogical practice.

Mathematics in New Zealand

What do we know about mathematics in New Zealand schools and early childhood centres today?

Today, just as in past decades, many students do not succeed with mathematics; they are disaffected and continually confront obstacles to engaging with the subject. The challenge for those with an interest in mathematics education is to understand what teachers might do to break this pattern. Many of the problems associated with learning mathematics have little resemblance to those encountered in other curriculum areas. Typically the problems are domain-specific—solving them is not a straightforward matter of importing more general pedagogical cures.

If we cannot point to general education for students' lack of mathematical engagement, neither can we, today, point to exclusion practices whereby, traditionally, access to mathematics was considered the prerogative of a privileged few. In our inclusive society all students have right of access to knowledge. Precisely how teachers can enhance all students' access to powerful mathematical ideas—irrespective of socio-economic background, home language, and out-of-school affiliations—is fundamental to this best evidence synthesis.

Research has confirmed precisely what many teachers have long appreciated: that it is the classroom teacher who has a significant influence over students' learning. For example, Rowe (2004) provides evidence that when school type and the achievement and gender of students are controlled for, class/teacher effects consistently represent, on average, 59% of the residual variance in the achievements of students. Muijs and Reynolds (2001) emphasise: "All the evidence that has been generated in the school effectiveness research community shows that classrooms are far more important than schools in determining how children perform at school" (p. vii).

To a large extent, making a difference in centres and schools rests with how teachers operationalise the core dimensions of teaching. We can be sure that those core dimensions include more than the knowledge and skill that an individual teacher brings to the task. As we shall see in this synthesis, the cognitive demands of teaching, as well as the structural, organisational, management, and domain-specific choices that teachers make, are all part of the large matrix of practice. These choices include, first and foremost, the negotiation of national mathematics curriculum policy and carry over to decisions about the human, material and technological infrastructure that allow learners to achieve mathematical and social outcomes. Such infrastructural decisions involve administrators, support staff, and parents and community; they also involve the intellectual resources of curriculum materials, assessment instruments, and computational and communications technology.

Pedagogical approaches and learner outcomes

We argue throughout this synthesis for a view of pedagogy that magnifies more than what teachers know and do in centres or classrooms to support mathematical learning. And we shall look further than improved test scores. For us, 'best practice' descriptions and explanations tied to high-stakes assessment don't tell the whole story. In this synthesis, pedagogy is tied closely to interactions between people. And these interactions cannot be separated from the axes of social and material advantage or deprivation that operate to define learners. We shall see that interactions that are productive enhance not only skill and knowledge but also identity and disposition. They also add value to life and work, to the family and to the wider community of individuals (Luke, 2005).

The term 'pedagogical approaches' is taken as the unit of analysis and describes the elements of practice characterised not only by regularities but also the uncertainties of practice, both inside and beyond the centre or classroom. We link those practices to achievement outcomes as well as to a range of social and cultural outcomes, including outcomes relating to affect, behaviour, communication, and participation. In addition to what the teacher knows and does, pedagogy, so defined, takes into account the ways of knowing and thinking, language, and discursive registers made available within the physical, social, cultural, historical, and economic community of practice in which the teaching is embedded. Those characteristics extend beyond the centre or classroom to tap into the complex factors associated with family and whānau partnerships as well as those associated with institutional leadership and governance.

'Quality' or 'effective' pedagogical approaches are those that achieve their purposes. The exact nature of those purposes is, invariably, the subject of debate ─ influenced by perspectives about how things should be at a given time (Krainer, 2005). Polya (1965), for example, pressed for mathematics teachers to teach people to think: "Teaching to think means that the mathematics teachers should not merely impart information, but should try also to develop the ability of the students to use the information imparted" (p. 100). Further back in time, Ballard (1915) wrote:

"We have not yet discovered the extent to which we can trust the pupils. By adopting a general policy of mistrust, by never allowing a child to mark his own, or even another child's exercises, by making no child responsible for anybody's conduct or progress but his own, by retaining all corrective and coercive powers in the teacher's hands, we gain certain advantages; we simplify matters, we minimise the likelihood of abuse of authority, and we cultivate in the pupils the virtue of obedience. But we lose much more than we gain" (p. 19).

Times have changed. Today, as far as business and industry is concerned, the goals of mathematics relate to the intellectual capacities required for future employment and citizenship in a technologically-oriented, bicultural society. Prototypically for the general public, effective teaching is that which develops in students the skills, understandings, and numerical literacy they need for dealing confidently with the mathematics of everyday life. The current academic view in New Zealand is that the mathematics taught and learned in schools and early childhood centres should provide a foundation for working, thinking and acting like mathematicians and statisticians. In that view, "[e]ffective mathematics teaching requires understanding what students know and need to learn and then challenging and supporting them to learn it well" (National Council of Teachers of Mathematics [NCTM], 2000, p. 16).

Making a difference for all

Irrespective of their differences, the various perspectives agree that mathematics teaching should make a positive difference to the life chances of students and should enhance their participation as citizens in an information- and data-driven age (Watson, 2006). Precisely because of the "gatekeeping role that mathematics plays in students' access to educational and economic opportunities" (Cobb & Hodge, 2002, p. 249), it should assist students to develop:

• the ability to think creatively, critically, and logically;
• the ability to structure and organise;
• the ability to process information;
• an enjoyment of intellectual challenge;
• the skills to interpret and critically evaluate statistical information in a variety of contexts;
• the skills to solve problems that help them to investigate and understand the world.

Mathematical proficiency

These are the academic outcomes that exemplify mathematical proficiency. They include more than mastery of skills and concepts: they spell out the dispositions and habits of mind that underlie what mathematicians do in their work. The National Research Council (2001) has expanded on these strands further to suggest that proficient students are those who have:

• conceptual understanding: comprehension of mathematical concepts, operations, and relations;
• procedural fluency: skill in carrying out procedures flexibly, accurately, efficiently, and appropriately;
• strategic competence: the ability to formulate, represent, and solve mathematical problems;
• adaptive reasoning: ability for logical thought, reflection, explanation, and justification;
• productive disposition: habitual inclination to see mathematics as sensible, useful, and worthwhile, coupled with a belief in diligence and one's own efficacy.

These are the characteristics of an apprentice user and maker of mathematics and are appropriated by the student through effective classroom processes.² They incorporate curriculum content, classroom organisational structures, instructional and assessment strategies, and classroom discourse regarding what mathematics is, how and why it is to be learned and who can learn it. These proficiency strands, which identify the mathematical learner within communities of classroom practice and beyond, are at the core of this best evidence synthesis.

Social, affective, and participatory outcomes

We would want to add to these academic outcomes a range of other outcomes that relate to affect, behaviour, communication, and participation. Te Whāriki and The New Zealand Curriculum Framework are useful guides to identifying these social, affective and participatory outcomes, relevant to particular age groups. The outcomes include:

• a sense of cultural identity and citizenship;
• a sense of belonging (mana whenua);
• contribution (mana tangata);
• well-being (mana atua);
• communication (mana reo);
• exploration (mana aotūroa);
• whānau spirit;
• commonly held values, such as respect for others, tolerance (rangimārie), fairness, caring (aroha), diligence, non-racist behaviour, and generosity (manaakitanga); and
• preparation for democratic and global citizenship.

Diversity

Recognising mathematics pedagogy as a key lever for increasing students' post-school and citizenship opportunities involves an important shift in thinking about students' access to learning. This changed focus is able to reveal how the development of mathematical proficiency over time is characterised by an enhanced, integrated relationship between teachers' intentions and actions on the one hand and learners' learning and development on the other. Such a focus is also able to signal how persistent inequities in students' mathematics education might be addressed, and this is crucially important in the light of recent analyses of international test data. These data reveal patterns of social inequity that cannot be read simply as a recent phenomenon but confirm a trend of systemic underachievement established over past decades (see Garden & Carpenter, 1987).

That trend points to pedagogical approaches that affect learners in disproportionate ways. Findings set out by the Ministry of Education (2004) reveal that New Zealand results, compared with the results of the 32 OECD countries participating in the Programme for International Student Assessment (PISA), are widely dispersed (see also Chamberlain, 2001; Davies, 2001; McGaw, 2004). While 15-year-old New Zealand students performed significantly above the OECD average and received a placing within the second highest group of countries, this positive information is offset by the fact that a high proportion of students are at the lower levels of proficiency. Data like these, signalling low proficiency levels amongst students, provide a sobering counterpoint to claims of equitable learning opportunities for diverse students.

Issues relating to student diversity are among the most complex and challenging issues facing mathematics education today. Deficit theories have tried to explain diversity by attributing the marginal performance of particular groups to the learners themselves or their impoverished circumstances. As such, these models have blamed the learner. We do not wish to ignore the fact that a considerable number of valuable interventions have resulted from this work. We point out, however, that in organising mathematical competence around the category of learner deficiency, and measuring against a 'natural', 'neutral' benchmark, the discipline offered simplistic explanations for mathematical proficiency. It could not explain why achievement comes to some learners through a hard and painful route.

Diversity is part of the New Zealand way of life. Over the next decades, our centres and schools will cater for increasingly diverse groups of learners, and these changing demographics will require a wider understanding of diversity. Diversity, however, is "a marginalised area of research, [and] is relatively undeveloped in mathematics education" (Cobb & Hodge, 2002, p. 250). In this synthesis, diversity discounts practices that stereotype on the basis of group affiliation. Instead, diversity tries to reconcile "the identities that [students] are invited to construct in the mathematics classroom" (ibid. p. 249) with their participation in the practices of home communities, local groups and wider communities within society. Characterised in this way, diversity encompasses learner affiliations with both local and broader communities. These affiliations are revealed through ethnicity, region, gender, socio-economic status, religion, and disability as well as identifiable learning difficulties and exceptional (including special) needs.

Equity

Current efforts (e.g., Cobb & Hodge, 2002) are focused on shifting from a traditional understanding of diversity towards thinking about equity. The focus on equitable pedagogical practices in this best evidence synthesis takes issue with the ill-informed belief in New Zealand society that some, but not others, are inherently equipped to learn mathematics competently. As is emphasised in the Guidelines for Generating a Best Evidence Synthesis Iteration 2004, equity becomes a crucially important means to redress social injustices. It is not to be confused with equality. This is because equity is about interactions between contexts and people: it is not about equal outcomes and equal approaches. Neither is it about equal access to people and curriculum materials. Setting up equitable arrangements for learners requires different pedagogical strategies and paying attention to the different needs that result from different home environments, different mathematical identifications, and different perspectives (Clark, 2002).

Equity, as used in this synthesis, is defined not as a property of people but as a relation between settings and the people within those settings. Thus equity is situated rather than static and is premised on an understanding of the bicultural foundation and multicultural reality of New Zealand classrooms. Marked by fairness and justice to 'diverse realities' (Ministry of Education, 2004), it is responsive to the treaty relationship held between the Crown and Māori ─a relationship that protects te reo (Māori language) and tikanga Māori (Māori culture) and provides assurances of same educational opportunities for Māori and non-Māori. As Cobb and Hodge (2002) argue, to understand equity, the focus needs to be not only on inequitable social structures and the ideologies that prop them up but also on how such realities play out in the everyday activity within classrooms and other cultural practices.

The synthesis is also responsive to the multiple cultural heritages increasingly brought to the centre and classroom settings and to a rapidly changing demographic profile. According to the 2001 census, by 2021, Māori enrolments at centres and schools will constitute 45% of learners and by 2050, nearly 60% of all children in New Zealand will identify as either Māori or Pasifika.³ 'Pasifika' is used as an umbrella term to include the cultures and ways of thinking of the Cook Islands, Fiji, Tonga, Samoa, Tuvalu, and Niue.⁴

This synthesis comes at an important point in time for mathematics education because, although the search for the characteristics of effective pedagogy in pāngarau/mathematics is far from new, identifying and explaining effective practice that meets the needs of all learners is substantially more urgent than at any previous time. Some statistics explain why this is so. Low-decile⁵ schools tend to have a greater intake of Māori and Pasifika students. In 2001, 68% of the Pasifika school population were in decile 1, 2 or 3 schools. This compares with 9.46% of the European school population. Few students from decile 1, 2, and 3 secondary schools enrol in university courses. "Middle-class students are far more likely than working-class students to experience success at school. Five times as many students with higher professional origins obtain a university entrance level bursary or better, than those from low-skilled and nonemployed families" (Nash, 1999, p. 268). These data point to the dilemmas that teachers face as they begin with the cultural and socio-economic backgrounds of their students and try to connect them to mathematics.

A few more figures drive the point home. According to Statistics New Zealand, almost one in five of all students leave school without any formal qualifications. For Māori students, the figure is one in three; for Pasifika students, it is one in four. Despite the high achievements of many Māori (see Crooks & Flockton, 2002) and Pasifika individuals, and not in any way wanting to downplay how learner performance is being raised through Māori-medium kōhanga reo (early childhood education), kura kaupapa Māori (primary schools), and wharekura (secondary schools), the harsh reality is that average achievement, as shown in PISA and other mathematics assessments (e.g., National Education Monitoring Project), is lower for these ethnic groups.⁶

Some promising trends have been signalled in recent analyses of the 2004 Numeracy Development Project (NDP) data for 70,000 students in years 1–8 (English-medium). For all ethnic groups, achievement, as measured by the NumPA diagnostic interview, was greater than that recorded in 2003. Although the average effect size advantage for addition/subtraction was only modest (.19), the average effect sizes for the higher stages of multiplication and proportion/ratio were .4 and .43 respectively. Of particular note in the 2004 analyses was the decrease in disparities between ethnic group performances. Analysis of the 2005 achievement data in the NDP indicates that the reduction in disparities is a continuing trend (see Young-Loveridge, 2006).

To date, mathematics has been caught up in learner access to social and economic resources and hence to future wealth and power. For mathematics education, the overriding concern is to provide equitable pedagogical access to opportunities that will develop in learners a positive mathematical disposition and enhance their life chances. Pedagogical practice that acknowledges the complexity of learners and settings allows us to move away from making gross generalisations about diverse groupings of students. Given the sociopolitical realities that shape students' constantly changing out-of-school and classroom identities, the task ahead is to change patterns of underachievement that, in the past, have been connected to a range of factors. Although there is agreement about this overarching goal, there has not been shared understanding about what pedagogy might do to achieve it.

Positioning the Best Evidence Synthesis

Against the backdrop of these statistics, the task of promoting democratic access to mathematical know-how assumes formidable proportions. How do teachers work at developing empowering approaches for learners who are polarised and disempowered by their sociocultural status?What must they do to develop an understanding of the big ideas of mathematics? How do they enhance a mathematical disposition and an appreciation of the value of mathematics in life? This synthesis, despite its best intentions, does not have ready-made practices to offer. As we shall see, the dimensions and core features of effective teaching for diverse learners are multiple. One of our most important claims is that there are no hard-and-fast rules about what methods and strategies work best. We simply do not have evidence of teaching practice that could be generalised to particular kinds of learning across all settings and across all learners. As educators, we have long known that teaching differs from one centre or classroom to another.

We caution against the tradition of identifying teacher effectiveness solely through teacher uptake of curriculum reforms or through the use of test results (Koehler & Grouws, 1992). Just as we would want to believe that reformers' visions are being realised, we have long known that teachers do not always implement them in ways that were intended by curriculum designers (Millet, Brown & Askew, 2004). We also note the limitations of craft-practice approaches to teaching that highlight teacher clarity or relative time spent on lesson components. Nor, as Alton-Lee (2005) has noted, can we say with total conviction that pedagogies customised specifically for learners with special needs produce greater achievement benefit for the learners (see Lewis & Norwich, 2000). Indeed, what we shall see is that some teaching approaches and classroom arrangements produce differential results from one setting to another. Content choices factor in too. For example, figures for NCEA results reveal that atlevel 2, students taking calculus recorded the lowest 'achieved' results (40%), and at level 3, 82% of statistics students recorded the highest 'achieved' results (Ministry of Education, 2006).

One thing, however, that we have gleaned from landmark studies is a set of common, underlying pedagogical principles that appear to hold good across people and settings. It is the principles upon which teachers base their practice that tend to make a difference for diverse learners. The identification of effective practice across centre and classroom settings, based on common principles, provides a rare opportunity to offer creative solutions. We do have some promising guideposts in this work. From a series of landmark best evidence syntheses (Alton-Lee, 2003; Biddulph, Biddulph, & Biddulph, 2003; Farquhar, 2003; Mitchell & Cubey, 2003), we know that practices and conditions that are respectful of the experiences of learners can make a difference to learning. We read of teachers who have provided access to learning against all odds and have done so through their belief in the rights of all learners to have access to education in a broad sense. Their work provides evidence that the effects of social disadvantage can be halted when learners encounter curricula in the classroom. Biddulph et al. (2003) have identified families/whānau and communities as key figures in bringing about academic achievement for diverse learners. Particular family attributes and processes, and community factors, as well as centre/school, family, and community partnerships, can all make a difference.

From our own discipline, we have evidence of the sorts of factors and conditions in Māori-medium schools and classrooms that raise expectations for learners' progress. Te Poutama Tau, in its responsiveness to the goals of Māori language revitalisation and empowerment, has revealed a positive effect on student achievement (Christensen, 2004). We know too that numeracy reform efforts in the U.K. (Askew, Brown, Rhodes, Johnson, & Wiliam, 1997) and in this country (e.g., Thomas & Tagg, 2005; Young-Loveridge, 2005), have contributed to higher student performance.

Overview of chapters

What follows is a systematic and credible evidence base for pedagogical approaches that enhance both proficiency and equity for learners. It is drawn from research that explains the sorts of pedagogical approaches that lead to improved engagement and desirable outcomes for learners from diverse social groups. It represents a first-steps approach to providing insight into new definitions of effective mathematics teaching.

The synthesis is made up of eight chapters. Chapter 2 develops a theoretical and empirical framework for the BES. We provide an overview of seminal studies that pinpoint in unique ways how quality teaching might be characterised. These landmark investigations foreground the complexity of pedagogical practice and how difficult it is to come up with universal checklists of effective teaching. From these studies, a set of guiding principles is derived alongside the theoretical framing for our work in the body of the synthesis. In offering a theoretical basis for structuring the report, we explain the notion of 'communities of practice' and the terms that we use in the synthesis. A description of how evidence-based studies of quality teaching were located, and the standards that the studies had to meet to qualify for inclusion in this synthesis, is provided in an appendix.

Chapters 3, 4, 5, 6, and 7 form the backbone to this best evidence synthesis. Covering both the early years and school sectors, these chapters take a thematic approach and include vignettes that emphasise particular characteristics of quality teaching. In chapter 3, evidence-based practices that make a difference for young learners are illuminated. This chapter on quality mathematics education in the early years stresses the formative influences at work on young children and concludes with a section analysing the transition from centre to school. Chapter 4 focuses on people, relationships, and classroom environment and explores how teachers develop productive mathematical communities of learning. Tasks and tools are brought to centre stage in chapter 5. The activities that teachers choose and the sorts of mathematical enquiries that take place around those activities are clarified. Chapter 6 explores practices beyond the classroom. It offers insight into the roles that school-wide, institutional and home processes play in developing mathematical identities and capabilities. The principles unearthed in chapter 2 and the key themes developed in chapters 3–6 converge in chapter 7 around a discussion of teaching and learning fractions. Concluding thoughts about what makes a difference in centres and schools for New Zealand learners are set out in chapter 8.

What the authors are very keen to do is clarify how patterns of inequality can be countered within the mathematics classroom. These explanations are not intended to be read as prescriptions of how teachers in New Zealand centres and schools should teach mathematics. Rather, by making clear the principles and characteristics underpinning effective practice, the synthesis is intended to stimulate reflection on mathematics education within and across sectors and to generate productive critique of procedures current within the discipline. Reflection and critique will make visible a new sensibility towards the multiple dimensions of pedagogical practice.

Footnotes

1. Mathematics as used in this synthesis is inclusive of the statistics component in the curriculum.
2. By 'classroom processes' is meant the learning interactions that occur within the early learning centre and mathematics classroom.
3. Stats NZ
4. Included are the newly arrived to New Zealand, those descended from Pasifika peoples living in New Zealand for one or more generations, and those of multiple heritages.
5. A school's decile indicates the extent to which the school draws its students from low socio-economic communities. Decile 1 schools are the 10% of schools with the highest proportion of students from low socio-economic communities, whereas decile 10 schools are the 10% of schools with the lowest proportion of these students.
6. The actual difference may well be more marked. PISA cohorts represent students aged 15 years 3 months up to 16 years 2 months. 40% of Māori students leave school before the PISA upper age limit.

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