Characteristics of Effective Teaching of Mathematics

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Jonathan James Mon, Jan 10, 2022 08:49 PM UTC

Worldwide, policymakers are placing increasing demands on schools and their teachers to use effective research-informed practices. In New Zealand, a collaborative knowledge-building strategy—The Iterative Best Evidence Synthesis Program—has been implemented at the policy level. Drawing on findings from the mathematics Best Evidence Synthesis Iteration, and more recent research studies, this paper offers ten principles of effective pedagogical approaches that facilitate learning for diverse learners. In examining the links between pedagogical practices and a range of social and academic student outcomes we draw on the histories, cultures, language, and practices for the New Zealand context and comparable international contexts.

Keywords: mathematics pedagogy, effective teaching, a community of learners, tasks, discourse, teacher knowledge.

1. Journal of Mathematics Education © Education for All
December 2009, Vol. 2, No. 2, pp.147-164
Characteristics of Effective Teaching of
Mathematics: A View from the West
Glenda Anthony and Margaret Walshaw
Massey University, New Zealand
Worldwide, policy makers are placing increasing demands on schools and
their teachers to use effective research-informed practices. In New Zealand a
collaborative knowledge building strategy—The Iterative Best Evidence
Synthesis Program—has been implemented at policy level. Drawing on
findings from the mathematics Best Evidence Synthesis Iteration, and more
recent research studies, this paper offers ten principles of effective
pedagogical approaches that facilitate learning for diverse learners. In
examining the links between pedagogical practices and a range of social and
academic student outcomes we draw on the histories, cultures, language, and
practices for the New Zealand context and comparable international contexts.
Key words: mathematics pedagogy, effective teaching, community of learners,
tasks, discourse, teacher knowledge.
Mathematics, it is widely understood, plays a key role in shaping how
individuals deal with the various spheres of private, social, and civil life. Yet
today, as in the past, many students struggle with mathematics and become
disaffected as they continually confront obstacles to engagement. In order to
break this pattern it is imperative, therefore, that we understand what effective
mathematics teaching looks like. Many have looked to research to seek
evidence about what kinds of pedagogical practices contribute to desirable
student outcomes (see government funded reports by, for example, Anthony &
Walshaw, 2007; Doig, McCrae, & Rowe, 2003; Ingvarson, Beavis, Bishop,
Peck, & Ellsworth, 2004; National Mathematics Advisory Panel, 2008).
Hiebert and Grouws (2007), in their synthesis of international research, have
argued for a more detailed, richer, and coherent knowledge base to inform
policy and practice.
In a response to Hiebert and Grouws, we present findings from recent
research syntheses (Anthony & Walshaw, 2007; 2008), complemented by
evidence from recent international studies (e.g., Lester, 2007; Martin, 2007).

2. 148 Characteristics of Effective Teaching of Mathematics: A View from the West
Collectively, these reviews are closely aligned with recent mathematics
initiatives within western education systems that shift teaching and learning
away from a traditional emphasis on learning rules for manipulating symbols.
Initiatives like Principles and Standards for School Mathematics (PSSM)
(National Council of Teachers of Mathematics, 2000) focus on developing
communities of practice in which students are actively engaged with
Effective pedagogy within such communities is at the heart of this
paper. We ask: What does research tell us about the characteristics of effective
pedagogy in the west? From our investigations that have helped us answer that
question, we have developed a set of principles that underpin the kinds of
pedagogical approaches found to develop mathematical capability and
disposition within an effective learning community. The ten principles of
effective mathematics pedagogy should not be taken in isolation but
interpreted as part of a complex web of factors that can affect student learning.
They incorporate elements of practice related to the classroom community,
classroom discourse, the kinds of tasks that enhance students‘ thinking, and the
role of teacher knowledge (see Figure 1). We discuss each of these principles,
in turn, in the following sections.
Figure 1. Principles of effective pedagogy of mathematics

3. Glenda Anthony & Margaret Walshaw 149
The principles we have developed are based on recognition that
classroom teaching is a complex activity. The classroom learning community
is neither static nor linear. Rather, it is nested within an evolving network
involving the school, the wider education system, and the home and local
community. The idea that teaching sits within a nested system draws its
inspiration from the work of post-Vygotskian activity theorists such as
Davydov and Radzikhovskii (1985). The understanding of a close relationship
between social processes and conceptual development also forms the basis of
Lave and Wenger‘s (1991) well-known social practice theory, in which the
notions of ―a community of practice‖ and ―the connectedness of knowing‖ are
central features. In that theoretical framework, individual and collective
knowledge emerge and evolve within the dynamics of the spaces people share
and within which they participate.
In this paper our focus will be on the classroom as a community of
practice. Our starting point is in the understanding that teachers who foster
positive student outcomes do so through their beliefs in the rights of all
students to have access to mathematics education in a broad sense—
understanding of the big ideas of curriculum and an appreciation of their value
and application in everyday life. Additionally, we claim that effective
mathematics pedagogy:
1) acknowledges that all students, irrespective of age, can develop
positive mathematical identities and become powerful mathematical learners.
2) is based on interpersonal respect and sensitivity and is responsive to
the multiplicity of cultural heritages, thinking processes, and realities found in
everyday classrooms.
3) is focused on optimizing a range of desirable academic outcomes that
include conceptual understanding, procedural fluency, strategic competence,
and adaptive reasoning.
4) is committed to enhancing a range of social outcomes within the
mathematics classroom that will contribute to the holistic development of
students for productive citizenship.
Classroom Community
An Ethic of Care: Caring Classroom Communities that Are Focused on
Mathematics Goals Help Develop Students’ Mathematical Identities and

4. 150 Characteristics of Effective Teaching of Mathematics: A View from the West
From research studies we find that effective teachers facilitate learning
by truly caring about their students‘ engagement (Noddings, 1995). They work
at developing interrelationships that create spaces for students to develop their
mathematical and cultural identities. They have high yet realistic expectations
about enhancing students‘ capacity to think, reason, communicate, reflect upon
and critique their own practice, and they provide students opportunities to ask
why the class is doing certain things and with what effect (Watson, 2002). The
relationships that develop in the classroom become a resource for developing
students‘ mathematical competencies and identities.
Students want to learn in a ‗togetherness‘ environment (Boaler, 2008;
Ingram, 2008). Teachers can make everyone feel included by respecting and
valuing the mathematics and the cultures that students bring to the classrooms.
Ensuring that all students feel safe allows every student to get involved.
However, it is important that the caring relationships that are developed do not
encourage students to become overly dependent on their teachers. Effective
teachers promote classroom relationships that allow students to think for
themselves, to ask questions, and to take intellectual risks (Angier & Povey,
Everyday classroom routines play an important role in the development of
students‘ mathematical thinking. Effective teachers make sure that all students
are provided with opportunities to struggle with mathematics for themselves.
For example, simply inviting students to contribute a response to a
mathematical problem may not achieve anything more than cooperation from
students. Teachers need to provide students with expectations and obligations
concerning who might speak, when and in what form, and what listeners might
do (Stipek et al., 1998).
Teachers are the most important resource for developing students‘
mathematical identities (Cobb & Hodge, 2002). They influence the ways in
which student‘s think of themselves in the classroom (Walshaw, 2004). In
establishing equitable arrangements, effective teachers pay attention to the
different needs that result from different home environments, different
languages, and different capabilities and perspectives. The positive attitude
that develops raises students‘ comfort level, enlarges their knowledge base,
and gives them greater confidence in their capacity to learn and make sense of
mathematics. Confident in their own understandings, students will be more
willing to consider new ideas presented by the teacher, to consider other
students‘ ideas and assess the validity of other approaches, and to persevere in
the face of mathematical challenge.

5. Glenda Anthony & Margaret Walshaw 151
Arranging for Learning: Effective Teachers Provide Students with
Opportunities to Make Sense of Ideas both Independently and
An important role of the teacher is to provide students with working
arrangements that are responsive to their needs. All students need some time to
think and work quietly by themselves, away from the varied and sometimes
conflicting perspectives of other students (Sfard & Keiran, 2001). At other
times, partners or peers in groups can provide the context for sharing ideas and
for learning with and from others. Group or partner arrangements are useful
not only for enhancing engagement but also for exchanging and testing ideas
and generating a higher level of thinking (Ding, Li, Piccolo, & Kulm, 2007).
In supportive, small-group environments, students learn to make conjectures
and learn how to engage in mathematical argumentation and validation
(O‘Conner & Michaels, 1996). In particular, when groups are mixed in
relation to academic achievement, insights are provided at varying levels
within the group, and these insights tend to enhance overall understandings
However, teachers need to clarify expectations of participation and ensure that
roles for participants, such as listening, writing, answering, questioning, and
critically assessing, are understood and implemented (Hunter, 2008).
Whole class discussion can provide a forum for broader interpretations
and an opportunity for students to clarify their understanding. It can also assist
students in solving challenging problems when a solution is not initially
available. Teachers have an important role to play in the discussion. Focusing
attention on efficient ways of recording, they invite students to listen to and
respect one another‘s solutions and evaluate different viewpoints. In all forms
of classroom organization it is the teacher‘s task to listen, to monitor how
often students contribute, and to keep the discussion focused. When class
discussion is an integral part of an overall strategy for teaching and learning,
students provide their teachers with information about what they know and
what they need to learn.
Building on students‘ thinking: Effective teachers plan mathematics learning
experiences that allow students to build on their existing proficiencies, interest,
and experiences.
In planning for learning, effective teachers put students‘ current
knowledge and interests at the centre of their instructional decision making.
Informed by on-going assessment of students‘ competencies, including
language, reading and listening skills, ability to cope with complexity, and

6. 152 Characteristics of Effective Teaching of Mathematics: A View from the West
mathematical reasoning, teachers adjust their instruction to meet the learning
needs of their students.
With the emphasis on building on students‘ existing proficiencies,
rather than remediating weaknesses and filling gaps in students‘ knowledge,
effective teachers are able to be both responsive to their students and to the
discipline (Carpenter, Fennema, & Franke, 1996). They understand that
learners make mistakes for many reasons. Some mistakes happen because
students have not taken sufficient time or care; others are the result of
consistent, alternative interpretations of mathematical ideas that arise from
learners‘ attempts to create meaning. To help students to learn from their errors,
teachers organize discussions—with peers or the whole class—that focus
students‘ attention on the known difficulties. Asking students to share a variety
of interpretations or solution strategies enables learners to compare and re-
evaluate their ideas.
Teachers who start where students are at with their learning are also able to
design appropriate levels of challenges for their students. For low-achieving
students, teachers find ways to reduce the complexity of tasks without falling
back on repetition and busywork and without compromising the mathematical
integrity of the activity (Houssart, 2002). In order to increase the task
challenge in all classrooms, effective teachers put obstacles in the way of
solutions, remove some information, require the use of particular
representations, or ask for generalizations (Sullivan, Mousley, & Zevenbergen,
Discourse in the Classroom
Mathematical Communication: Effective Teachers Facilitate Classroom
Dialogue that Is Focused Towards Mathematical Argumentation
Teaching ways of communicating mathematically demands skilful
work on the teacher‘s part (Walshaw & Anthony, 2008). Students need to be
taught how to articulate sound mathematical explanations and how to justify
their solutions. Encouraging the use of oral, written and concrete
representations, effective teachers model the process of explaining and
justifying, guiding students into mathematical conventions. They use explicit
strategies, such as telling students how they are expected to communicate
(Hunter, 2005).
Teachers can also use the technique of revoicing (Forman & Ansell,

7. Glenda Anthony & Margaret Walshaw 153
2001), repeating, rephrasing, or expanding on student talk. Teachers use
revoicing in many ways: (i) to highlight ideas that have come directly from
students, (ii) to help the development of students‘ understandings implicit in
those ideas, (iii) to negotiate meaning with their students, and (iv) to add new
ideas, or move discussion in another direction.
When guiding students into ways of mathematical argumentation, it is
important that the classroom learning community allows for disagreements
and enables conflicts to be resolved (Chapin & O'Connor, 2007). Teachers‘
support should involve prompts for students to work more effectively together,
to give reasons for their views and to offer their ideas and opinions. Students
and teacher both need to listen to others‘ ideas and to use debate to establish
common understandings. Listening attentively to student ideas helps teachers
to determine when to step in and out of the discussion, when to press for
understanding, when to resolve competing student claims, and when to address
misunderstandings or confusion (Lobato, Clarke, & Ellis, 2005). As students‘
attention shifts from procedural rules to making sense of mathematics,
students become less preoccupied with finding the answers and more with the
thinking that leads to the answers (Fravillig, Murphy, & Fuson, 1999).
Mathematical Language: The Use of Mathematical Language Is Shaped
When the Teacher Models Appropriate Terms and Communicates Their
Meaning in a Way that Students Understand
If students are to make sense of mathematical ideas they need an
understanding of the mathematical language used in the classroom. A key task
for the teacher is to foster the use, as well as the understanding, of appropriate
mathematical terms and expressions. Conventional mathematical language
needs to be modeled and used so that, over time, it can migrate from the
teacher to the students (Runesson, 2005). Explicit language instruction and
modeling takes into account students‘ informal understandings of the
mathematical language in use. For example, words such as ―less than‖, ―more‖,
―maybe‖, and ―half‖ can have quite different meanings within a family setting.
Students can also be helped in grasping the underlying meaning through the
use of words or symbols with the same mathematical meaning, for example,
‗x‘, ‗multiply‘, and ‗times‘.
Teachers face particular challenges in multilingual classrooms. Words
such as ―absolute value‖, ―standard deviation‖, and ―very likely‖ often lack an
equivalent term in the students‘ home language. Students find the syntax of

8. 154 Characteristics of Effective Teaching of Mathematics: A View from the West
mathematical discourse difficult. Prepositions, word order, logical structures,
and conditionals are all particularly problematic for students. Students may
also be unfamiliar with the contexts in which problems have been situated.
Language (or code) switching, which involves the teacher substituting a home
language word for a mathematical word, has been shown to enhance student
understanding, especially when teachers are able to use it to capture the
specific nuances of mathematical language (Setati & Adler, 2001).
Assessment for learning: Effective teachers use a range of assessment
practices to make students‘ thinking visible and support students‘ learning.
Mathematics teachers make use of a wide range of formal and informal
assessments to monitor learning progress, to diagnose learning, and to
determine what can be done to improve learning. Within the everyday
activities of the classroom, teachers collect information about how students
learn, what they seem to know and are able to do, and what they are interested
in. This information helps teachers determine whether particular activities are
successful and informs decisions about what they should be doing to meet the
learning needs of their students (Wiliam, 2007).
Effective teachers gather information about students by watching
students as they engage in individual or group work and by talking with them.
They monitor their students‘ understanding, notice the strategies that they
prefer, and listen to the language that they use. The moment-by-moment
assessment helps them make decisions about what questions to ask next, when
to intervene in student activity, and how to answer questions. Classroom
exchanges in the form of careful questioning provide a powerful way to assess
students‘ current knowledge and ways of thinking (Steinberg, Empson, &
Carpenter, 2004). For example, questions that have a variety of solutions, or
that can be solved in more than one way, can help teachers gain insight into
students‘ mathematical thinking and reasoning.
As well as informing the teacher, assessment for learning involves
providing feedback to students. Helpful feedback explains why something is
right or wrong, and describes what to do next, or describes strategies for
improvement. Effective teachers also provide opportunities for their students
to evaluate and assess their own work. They involve students in designing test
questions, writing success criteria, writing mathematical journals, and
presenting portfolios as evidence of growth in mathematics.
Mathematical Tasks

9. Glenda Anthony & Margaret Walshaw 155
Worthwhile Tasks: Effective Teachers Understand that Selected Tasks and
Examples Influence How Students Come to View, Develop, Use, and Make
Sense of Mathematics
Tasks convey what doing mathematics is all about. By engaging in
tasks, students develop ideas about the nature of mathematics and mathematics
learning (Hodge, Zhao, Visnovska, & Cobb, 2007). Effective teachers take
care to ensure that tasks help all students to progress in their cumulative
understanding in a particular domain and engage in high-level mathematical
thinking (Henningsen & Stein, 1997).
By posing tasks and learning experiences that allow students to do
original thinking about important mathematical concepts and relationships,
teachers help learners to develop proficient ways of doing, and learning about
mathematics (Ainley, Pratt, & Hansen, 2006). Tasks should involve more than
practicing taught algorithms; they should provide opportunities for students to
struggle with important mathematical ideas. Posing tasks of an appropriate
level of mathematical challenge fosters students‘ development and use of an
increasingly sophisticated range of mathematical thinking and reasoning
activities (Watson & De Geest, 2005).
Working with open-ended and modeling tasks, in particular, provides students
with opportunities not just to apply mathematics but also to learn new
mathematics through engagement in a range of problem-solving strategies.
Essential skill development can also be part of ‗doing‘ mathematics problems.
For example, learning about perimeter and area provides opportunities to
practice multiplication and fractions computations. Modeling activities
challenge students to make sense of both the contexts and the mathematics
embedded in the tasks (English, 2006; Galbraith, Stilman, Brown, & Edwards,
2007). When working with real life complex systems, students learn that doing
mathematics involves more than simply producing right answers; applying
mathematics in everyday settings helps students learn about the value of
mathematics in society and its contributions to other disciplines.
Making Connections: Effective Teachers Support Students to Create
Connections, between Different Ways of Solving Problems, between
Mathematical Topics, and between Mathematics and Everyday
Students need to develop understandings of how a concept or skill is

10. 156 Characteristics of Effective Teaching of Mathematics: A View from the West
connected in multiple ways to other mathematical ideas (Askew, Brown,
Rhodes, Johnson, & Wiliam, 1997). Effective teachers support students to
make connections by providing them with opportunities to engage in complex
tasks and by setting expectations that they explain their thinking and solution
strategies and that they listen to the thinking of others (Anghileri, 2006).
Teachers can assist students to make connections by using carefully sequenced
examples, including examples of students‘ own solution strategies, to illustrate
key mathematical ideas (Watson & Mason, 2006). By progressively
introducing modifications that build on students‘ existing understanding,
teachers can emphasize the links between different ideas in mathematics. For
example, a teacher can introduce the idea of ‗doubling 6‘ as an alternative
strategy to ‗6 add 6‘.
Making connections across mathematical topics is important for
developing conceptual understanding. For example, the topics of fractions,
decimals, percentages, and proportions, while learning areas in their own right,
can usefully be linked through exploration of differing representations (e.g., ½
= 50%) or through problems involving everyday contexts (e.g., determining
fuel costs for a car trip).
Teachers can also help students to make connections to real
experiences. When students find they can use mathematics as a tool for
solving significant problems in their everyday lives, they begin to view the
subject as relevant and interesting.
Tools and Representations: Effective Teachers Carefully Select Tools and
Representations to Provide Support for Students’ Thinking.
Effective teachers draw on a range of representations and tools to
support learners‘ mathematical development. Tools to support and extend
mathematical reasoning and sense-making come in many forms including the
number system itself, algebraic symbolism, graphs, diagrams, models,
equations, notations, images, analogies, metaphors, stories, textbooks, and
Teachers have a critical role to play in ensuring that tools are used
effectively to support students to organize their mathematical reasoning and
support their sense-making (Blanton & Kaput, 2005). Providing students
access to multiple representations helps them to develop conceptual and
computational flexibility. Using an appropriate model, learners can think
through a problem, or test ideas. Care is needed, however, particularly when

11. Glenda Anthony & Margaret Walshaw 157
using pre-designed concrete materials (e.g., number lines, tens-frames), to
ensure that all students are able to make sense of the materials in the
mathematically intended way.
Tools are helpful in communicating ideas that are otherwise difficult to
talk about or write about. Teachers and students can use representations, such
as stories, pictures, symbols, concrete objects, and virtual manipulatives, to
assist in communicating their thinking to others. As well as ready-made tools,
effective teachers acknowledge the value of students generating and using
their own representations, but it an invented notation, or a graphical, pictorial,
tabular, or geometric representation. For example, young children frequently
create their own pictorial representations to tell stories before using the more
formal graphical tools that are fundamental to the statistics curriculum (Chick,
Pfannkuch, & Watson, 2005).
An increasing array of new technological tools is available for use in
the mathematics classroom. In addition to calculator and computer
applications, new technologies include presentation technologies (e.g., the
interactive whiteboard (Zevenbergen & Lerman, 2008), digital and mobile
technologies, and the Internet. These dynamic graphical, numerical, and visual
technological applications provide new opportunities for teachers and students
to interact, represent, and explore mathematical concepts.
Teachers must be knowledgeable decision makers in determining when
and how technology is used to support learning (Thomas & Chinnappan,
2008). Effective teachers take time to share the decision making about
technology-based approaches with their students. They require students to
monitor their own underuse or overuse of technology. With guidance from
teachers, technology can support independent inquiry and shared knowledge
Teacher Learning and Knowledge
Teacher knowledge: Effective Teachers Develop and Use Sound
Knowledge to Initiate Learning and to Act Responsively towards the
Mathematical Needs of All Their Students
How teachers organize classroom instruction is very much dependent
on what they know and believe about mathematics and on what they
understand about mathematics teaching and learning. Sound content
knowledge enables teachers to represent mathematics as a coherent and

12. 158 Characteristics of Effective Teaching of Mathematics: A View from the West
connected system (Ball & Bass, 2000). When their knowledge is robust,
teachers are able to assess their students‘ current level of mathematical
understanding. They use their knowledge to make key decisions concerning
mathematical tasks, classroom resources, talk, and actions that feed into or
arise out of the learning process.
No matter how good their teaching intentions, teachers must work out how
they can best help their students grasp core mathematical ideas (Hill, Rowan,
& Bass, 2005). In addition to having clear ideas about how they might build
students‘ procedural proficiency they need to know how to extend and
challenge students‘ thinking. To do this successfully they need substantial
pedagogical content knowledge and a grounded understanding of students as
learners. Such teachers are aware of the possibility of students‘ conceptions
and misconceptions. This knowledge informs teachers‘ on-the-spot classroom
decision making. It enables more finely tuned listening and questioning, more
focused and connected planning, and more insightful evaluation of student
The development of teacher knowledge is greatly enhanced by efforts
within the wider school community to improve teachers‘ own understandings
of mathematics and mathematics teaching and learning (Cobb & McClain,
2001; Sherin, 2002). If teachers‘ knowledge is to be enhanced, it needs the
material, systems, human and emotional support provided by professional
development initiatives. Support and resourcing can also come from the joint
efforts of other mathematics teachers within the school (Kazemi, 2008).
This paper has examined what the research says about effective
teaching of mathematics within western education systems. Current research
findings indicate that the nature of classroom mathematics teaching
significantly affects the nature and outcome of student learning. Our
conceptualization of teaching as nested within a systems network (Tower &
Davis, 2002), moves us away from prescribing pedagogical practice, towards
an understanding of pedagogical practice as occasioning students outcomes. In
this paper we have offered important insights from research about how that
occasioning might take place. Certain patterns have emerged that have
enabled us to foreground ways of doing and being that mark out an effective
pedagogical practice. Each aspect, of course, constitutes but one piece of
evidence and must be read as accounting for one variable, amongst many,
within the teaching nested system. As Hiebert and Grouws (2007) have noted,
―classrooms are filled with complex dynamics, and many factors could be

13. Glenda Anthony & Margaret Walshaw 159
responsible for increased student learning‖ (p. 371). Taking all the factors
together has allowed us to offer our ten principles as a starting point for
discussions on effective pedagogy.
Whilst the principles concern classroom pedagogical practices, we are well
aware that significant improvements in student learning outcomes will require
the efforts of many. Changes need to be negotiated and carried through in
classrooms; in mathematics teams, departments, or faculties; and in teacher
education programs. They need to be supported by resourcing. Everyone
involved in mathematics education—teachers, principals, teacher educators,
researchers, parents, specialist support services, school boards, and policy
makers, as well as students themselves—has a role to play in enhancing
students‘ mathematical proficiency. Schools, communities, and nations need to
ensure that their teachers have the knowledge, skills, resourcing, and
incentives to provide students with the very best possible learning
opportunities. In this way, every student will be able to enhance their
mathematical proficiency. In this way, too, every student has the opportunity to
enhance their view of themselves as a powerful mathematics learner.
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Glenda Anthony
Massey University, New Zealand
Email: g.j.anthony@massey.ac.nz
Margaret Walshaw
Massey University, New Zealand
Email: m.a.walshaw@massey.ac.nz

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