Spotlight on Mathematics/Pāngarau
This page brings together a range of resources designed to help those interested in improving students’ achievement levels in mathematics in New Zealand.
Resources: What works in mathematics education?
The Iterative Best Evidence Synthesis (BES) Programme brings together research-based evidence, from New Zealand and elsewhere, to explain what does and does not work, why and how, and what makes a bigger difference for diverse learners.
BES Exemplar 1: Developing communities of mathematical inquiry
This exemplar illustrates how two teachers developed teaching practices that proved highly effective for diverse learners. The teachers took part in a professional development programme designed to help them build classroom communities of mathematical inquiry.
This BES exemplar was chosen from thousands of studies because of the big gains made for diverse learners. Māori and Pasifika students from a low decile school made 4 to 5 years of progress in just one year. The approach has been refined through many cycles of research and development and replicated.
The BES exemplar explains the kind of leadership, professional learning and teaching approaches that make such success possible. This approach also counters bullying and strengthens students' language skills. This approach has been led in New Zealand by Dr Roberta Hunter of Massey University.This exemplar has had quality assurance from practitioners and from academic experts including Professor Emerita, Courtney Cazden of Harvard University.
- Developing communities of mathematical inquiry [PDF 3.8MB]
More Resources: What works in mathematics education
Effective Pedagogy in Mathematics | Pāngarau
The synthesis provides a trustworthy overview of 660 studies of effective teaching in mathematics and pāngarau. It explains pedagogical approaches that worked in early childhood, primary and secondary settings. The focus is primary teaching.
BES Summary: Te Ako Pāngarau Whaihua | Effective Pedagogy in Mathematics
This booklet is a ten-point summary of the findings of the source best evidence synthesis that focuses of Māori-medium contexts.
This booklet is a ten-point summary of the findings of the source best evidence synthesis. Teachers and leaders have found this resource to be a useful reminder of what makes a difference.
Free online copies of this summary and others are available here. Hard copies for New Zealand educators are available from firstname.lastname@example.org This summary was commissioned by the International Academy of Education, and is also available in different languages on the IBE website.
Pedagogical Leadership in Mathematics
This case study, a US example from 'School Leadership and Student Outcomes: Best Evidence Synthesis' (pg.226-230, 2009), shows the difference that pedagogical knowledge makes to administrative decision making in an area of challenge for New Zealand mathematics teaching.
10 Key Findings for Making a Bigger Difference in Mathematics Teaching
This overview considers the findings of 103 evaluations of New Zealand mathematics education and presents 10 key findings from BES summary to help inform improvement efforts. The analysis reveals disjunctions between the new what works evidence, and policy and practice.
Early Years Mathematics Education
Chapter 3 of 'Effective Pedagogy in Mathematics | Pāngarau' focuses on Early Years Mathematics Education (p.24)
Te Aho Tukutuku Early Mathematics 2012
This resource has been designed to assist educators in early childhood education to support early mathematics learning. The disk is available and can be ordered from email@example.com
Online access coming soon.
Professional Learning and Mathematics
The 'Teacher Professional Learning and Development: Best Evidence Synthesis' (p.65, 2008) explains what did and did not work across 21 professional development initiatives to raise achievement in mathematics.
Free copies for New Zealand educators are available from firstname.lastname@example.org
Read other BES
Teaching fractions is one of our biggest challenges. For this reason several BES cases focus on fractions.
Case 5: Use mathematical tools to explore students' thinking about mathematics
This case explains how the effective use of three Numeracy Project tools (the number framework, diagnostic interview, and teaching model) enables teachers to design rich mathematical tasks for diverse (all) students. The tools support teachers to explore students' thinking about mathematics, their own expectations of students, and their beliefs about teaching mathematics.
Recent international studies show that in-school variation in mathematical achievement is greater than the variation between schools. This case describes some of the processes schools can use to build teachers' mathematical knowledge and pedagogical practices so that they can employ the tools effectively to improve mathematical outcomes for all students.
Download: BES Case 5 [PDF 368kB]
Case 9: Use everyday rhymes and waiata to develop spatial skills and awareness
Children's informal mathematical knowledge originates with their everyday activities.
This short case illustrates how a young child uses the various spatial words in a waiata to direct her actions with a poi.
See also BES Case 19 "Use fun games to build mathematical knowledge and confidence in young learners."
Download: BES Case 9 [PDF 893kB]
Case 19: Use fun games to build mathematical knowledge and confidence in young learners
Playing fun mathematical games is one of the most powerful and positive ways in which families can support mathematical learning.
This case demonstrates how adults can support children to develop deeper understanding of fractions by playing informal language games. In this case, the game was played for a few minutes several times a week. The difficulty of the tasks and problem-solving approaches varied, but all were based around the familiar context of sharing cookies amongst a number of children. This case will be useful for parents, caregivers, and educators in early childhood centres and schools.
Download: BES Case 19 [PDF 530kB]
Case 20: Use a participation framework to support students to discuss their problem-solving strategies
This case illustrates the importance of positioning students so that they are able to contribute to mathematical discussions - as problem-solvers, solution-reporters, and claim-defenders. Above all, it highlights the importance of ensuring that students who find fractions difficult are treated as competent, with legitimate contributions to make. The teacher in this case carefully supported her students by making use of their prior knowledge and providing them with multiple opportunities to learn.
Download: BES Case 20 [PDF 1.1mB]
Case 21: Develop tasks that provide high-level challenge and high-level involvement
The teacher in this case selects mathematically rich learning activities that are responsive to students' knowledge and interests, invite exploration and discussion, and develop understanding of fractions.
The case describes how a "pizza fractions" kit provided the impetus for a "fraction flags" activity when the teacher saw a student making a "flag" using pieces from the pizza fractions kit and then exploring fraction-related questions using this different representation.
Download: BES Case 21 [PDF 1.5mB]
Case 22: Support students to make their own choice of tools and representations when solving problems
This case illustrates how cognitive conflict can be a resource for learning. When an apparent contradiction surfaced during problem solving, instead of circumventing discussion by providing an explanation, the teacher in the case ensured that all students were challenged to explore their own thinking, and encouraged to use a range of tools to solve the problem. In this way, the teacher allowed the students to focus on justifying their methods, based on the logic of mathematics.
Download: BES Case 22 [PDF 709kB]
Case 23: Scaffold learning through the careful selection of tasks and problems
Teachers need to be careful that their contribution to mathematical conversations does not take from their students valuable opportunities to do some thinking.
This case describes how a secondary school teacher maintained high levels of involvement after noticing that the students were all using variations of the same incorrect strategy to tackle a particular task. The teacher introduced a different but parallel task to demonstrate the shortcomings of the strategy. Once the students saw the issue, the teacher supported them to think about another strategy.
Download: BES Case 23 [PDF 636kB]
Case 24: Expect students to be accountable for thinking through the mathematics involved in a problem
The mathematics BES emphasises the importance of basing pedagogical practices on socio-mathematical norms that collectively establish expectations about what mathematical thinking is and is not.
Through the use of contrasting examples from four different teachers, this case illustrates important differences between effective and less effective teaching. Above all, effective teaching emphasises conceptual thinking, not superficial sharing of ideas and strategies, and mathematical argumentation, not procedural description/summary.
While this case is located in grades 4-5, it is relevant to all teachers of mathematics, including secondary school teachers.
See also BES Exemplar 1 "Developing communities of mathematical inquiry"
Download: BES Case 24 [PDF 1.4mB]
Case 25: Develop a mathematical community of practice
In a mathematical community of practice, students learn to articulate their thinking and engage in exchange of ideas in an environment that is both challenging and safe. The safety is particularly important because, to develop mathematical understanding, students need to be able to get things "wrong" and learn from their "mistakes" without being embarrassed or defensive.
This case provides a window into the relationship between a teacher and two students as the teacher challenges one of the students to clarify his explanations.
See also BES Exemplar 1 "Developing communities of mathematical inquiry"
Download: BES Case 25 [PDF 722kB]
Case 26: Strengthen teacher pedagogical knowledge
Using fractions as a context, these two cases highlight the importance of teachers being able to create conceptually correct representations of mathematical ideas. Without the knowledge to do this, attempts to connect mathematics with students' lives are always going to fall short.
The examples in these cases provide a springboard for teachers to discuss their own mathematical understandings with colleagues or in a professional learning community. This process could highlight areas that need addressing.
See also BES Case 5 "Use mathematical tools to explore students' thinking about mathematics."
Download: BES Case 26 [PDF 1.0mB]
Case 29: Use pedagogical leadership to enable more equitable and effective teaching for all learners
Teaching approaches influence student identity and social outcomes as well as academic outcomes. Research shows fixed-ability grouping or streaming can have negative outcomes on all three types of outcome, especially for underachievers.
In this case, a primary principal encourages his teachers to review their beliefs about mixed-ability grouping by giving them the experience of working on a mathematical problem with a heterogeneous group of colleagues. Although the context for the case is primary mathematics, it has relevance across all levels and areas of the curriculum.
The case highlights the importance of leaders' knowledge of teaching and learning and provides a number of useful thinking tools and processes.
See also BES Exemplar 1 "Developing communities of mathematical inquiry"
Download: BES Case 29 [PDF 328kB]
Teachers and Leaders Supporting Families and Whānau with Mathematics Learning at Home
Chapter 7 of the 'School Leadership and Student Outcomes BES' (p.142) Creating Educationally Powerful Connections with Family, Whānau and Communities explains ways in which teachers and leaders can assist parents and whānau to effectively support their children's learning at home.
To learn more about this diagram go to Chapter 7 of the 'School Leadership and Student Outcomes BES'
Parents and Whānau
Your children need positive support for their mathematics learning as part of their everyday experiences. Be careful not to criticise or put too much pressure on your child in ways that have a negative effect on their learning. Playing mathematics games in ways that are fun for your child can help their learning.
Teacher Educators in New Zealand
Hard copies of many BES resources are available free of charge from email@example.com for initial and in-service teacher education courses.
"PPTA welcomes this Best Evidence Synthesis on Mathematics. It is the result of a very thorough process, inclusive of the expertise of practitioners. The final report reflects and caters to their realities, and provides some very interesting and thought-provoking reading for teachers themselves and for those involved in the pre-service and in-service education of Mathematics teachers."
Debbie Te Whaiti, (Past) President, NZPPTA, 2007.
The best evidence synthesis work shows where research and development is making a much bigger difference to improve mathematics teaching, and where there are gaps in our knowledge.
Our goal is to encourage the research community to strengthen and build on the what works evidence to support educational improvement.
More policy resources
An alternative policy approach to strengthening valued outcomes for diverse learners
- Reframing teacher professional learning [PDF 522kB]
This paper dicusses significant findings from the Teacher Professional Learning and Development Best Evidence Synthesis and their application to increasing the effectiveness of teacher professional learning that will enable improved outcomes for diverse learners:
Timperley, H., & Alton-Lee, A. (2008). Reframing teacher professional learning: An alternative policy approach to strengthening valued outcomes for diverse learners. Review of Research in Education 32, 328-369.
- (Using) evidence for educational improvement[PDF 241KB]
(Using) evidence for educational improvement provides an overview of the findings about effective professional development and highlights the potential of such evidence to inform educational improvement. The article highlights policy challenges for the systemic
use of effective professional development and illustrates what is possible in two examples of high impact research and development (R & D) that have been effective across varied contexts.
- The use of evidence to improve education and serve the public good [PDF 1.0mB]
This paper explores the challenge of mobilising research to inform ongoing improvement in valued outcomes for diverse (all) learners across school systems.
Where can I find out more?
How to support New Zealand teachers in ambitious mathematics teaching. Find out about the lead BES writer: Professor Glenda Anthony. Read Professor Glenda Anthony's Keynote presented at the International Exhibition and Forum for Education, Feb 2013, Riyadh, Saudi Arabia.
Professor Paul Cobb
Find out about the recent work of Professor Paul Cobb who was a formative quality assurer for the 'Effective Pedagogy in Mathematics/Pāngarau: BES'. Professor Cobb's work focuses on mathematics education improvement to scale across schooling systems.
Dr Roberta Hunter
Find out about recent publications of Dr Roberta Hunter whose research and development is featured in BES Exemplar 1: Developing Communities of Mathematical Inquiry.