This workshop aims to describe the critical characteristics of expert teachers of numeracy. Naturally, most of these characteristics will be transferable to teaching other curriculum areas.
• The aim of this workshop is to describe the key
characteristics of expert teachers of numeracy.
Naturally most of these characteristics will be
transferable to teaching other areas of the
• During the course of the workshop you will be
presented with suggestions on how to facilitate
the ideas with other teachers. This will enable
you to construct your own workshop or staff
How important is good teaching?
• The first question to consider is, “How
much difference to student achievement
do teachers actually make?”
• In the next frame there is a pie chart from
research by Professor John Hattie, from the
University of Auckland. He quantifies the overall
effect on student achievement of these factors:
– Students (Personal characteristics like intelligence, co-
– Home (Expectations for success, intellectual support,
appropriate physical and emotional care)
– Teachers (Types of actions taken, expectations, effort)
– Peers (Expectations, support for each others’ efforts)
– Schools (Organisational structure, quality of resources)
Factors contributing to achievement
• Match the five sectors of the achievement pie
with the factors below:
– Schools 7%
How big is the impact of teaching?
• Hattie’s analysis of thousands of research studies
show this combined effect:
• Adrienne Alton-Lee, from the Ministry of
Education, suggests that the proportion of
student achievement due to teaching
effect varies between 15% and 60%
depending on the context. Hattie’s is a
• Under what circumstances do you think
the effect due to teaching would be high?
Is the effect uniform?
• The effect due to teaching is most significant in
situations where the learning is least supported
by the students’ everyday environment. An
obvious case is learning specialised subjects at
senior secondary and tertiary level, e.g. calculus
or Japanese as a second language.
• The teaching effect is also very high for younger
students whose home environment least
supports them educationally. Almost uniformly
there is a connection between socio-economic
status of caregivers and the achievement of
• To summarise:
Teachers make a significant difference to
student learning and they do so most in
situations where the students are needy.
Characteristics of quality teachers?
• So what is it that quality teachers do that makes the
• A good place to start is to ask your colleagues the
• You observe a teacher in action. The lesson is
sensational, the best you have seen in your career.
There is so much going on that it is hard to describe why
the lesson is so good. You decide to focus on the
• What characteristics does this teacher have that make
her an “expert”?
• Write down a few characteristics that you think an expert
• Good management is a necessary but not sufficient
• Often responses to the previous question are about
classroom management. For example, the teacher:
• Is well organised
• Ensures that students listen to instructions
• Provides appropriate work
• Develops orderly routines
• These logistical characteristics are
important but they are not enough. Hattie
distinguishes “experienced teachers”, that
is those who can manage classrooms well,
from expert teachers, that is those who
optimise their students’ learning. To be an
expert teacher you must have much more!
Clusters of quality practice
• The characteristics of expert teachers of numeracy can
be grouped in these clusters:
1. Relationships with students
2. Planning and assessment
3. Problem focus
5. Instructional responsiveness
6. Student empowerment
• This workshop will discuss each cluster.
Relationships with students
• Russell Bishop, from The University of
Waikato, lead a project called Kotahitanga
aimed at improving the achievement of
Maori student. A key finding of the project
was that students achieved best in
classrooms where the teacher related well
to them as individuals and valued their
• Bishop’s work also focused on teachers’
attribution for the achievement of Maori
students. Effective teachers had high but
realistic expectations for all their students and
conveyed their expectations to students. These
teachers believed that what they did made a
difference. Ineffective teachers attributed lack of
achievement to students’ background or
constraints imposed by “the school system.”
Planning and assessment
• A critical part of effective teaching is
mapping out anticipated learning
trajectories for students to learn a
particular idea, and providing sufficient
appropriate experiences to support this
learning. Quality teachers choose activities
for a learning purpose and are transparent
with their students about this purpose.
• Expert teachers apply a dynamic
relationship between their assessment of
students’ learning and their planning of the
next learning step. They employ a variety
of assessment techniques, particularly
their own observations, and regard any
particular assessment as formative, a
landmark on a journey rather than an
endpoint in itself. Planning is responsive to
assessment and vice versa.
• Studies into student achievement in
mathematics across countries have
compared the practice of teachers in
nations that produce high achievement.
The most astounding result has been that
lessons in particular countries have
considerable similarity. Each country
appears to have a prevalent teaching
• The teaching cultures of the high
performing nations are similar only in
the high proportion of class time
that’s students are engaged in
problem solving as opposed to
listening to teacher explanations or
• In these countries, such as Korea, The
Netherlands, Slovakia, Japan, and
Singapore, the problems presented are
carefully structured and sequenced.
Students work co-operatively or
individually on the problems, often
encountering difficulty, before the
processing of solutions collectively with
the support of the teacher.
• Expert teachers have strong pedagogical-
content knowledge (PCK). PCK is a term used
by Lee Schulman to describe what a teacher
needs to know in order to teach a topic
effectively. In numeracy, this involves knowing
the mathematical idea, how it connects to
other mathematical ideas, what contexts and
representations could be used to present it and
the cognitive obstacles and misconceptions
students commonly encounter in learning it.
• Mike Askew and Margaret Brown, from King’s
College in London, studied over 700 numeracy
lessons and concluded that the expert
teachers were those who were “connectivist”.
These teachers used their strong PCK to help
their students make connections for
themselves. Later in the workshop there are
suggestions for developing teachers’ PCK
through a workshop.
• Responsiveness implies that teachers are
prepared to alter the course of a lesson or
sequence of lessons based on the needs of
students. To do so expert teachers actively
listen to their students and mentally process
the responses. To do so expert teachers
create environments where students feel
confident to take risks, pose conjectures and
explain their ideas to others.
• Active listening to the ideas of students and
acting from these ideas is a critical aspect of
responsiveness. Paul Cobb described this as
the creation of socio-mathematical norms.
While expert teachers value all ideas from their
students, they also see their role as the
development of mathematical power. Do not
treat all ideas as equal because they are not. A
critical role of teachers is to help students
evaluate the relative strengths of ideas and
• Students’ perception of responsibility for
their own learning links strongly to high
achievement. Expert teachers develop
independence through sharing learning
outcomes with their students, requiring
students to make their own instructional
decisions, providing regular personalized
feedback and encouraging meta-cognition
(thinking about thinking).
• Expert teachers also provoke high order
thinking, such as analysing, justifying and
synthesising through the questions they
• Success for all students involves catering
for diverse needs. Quality teaching
involves careful allocation of resources,
particularly time, to maximise learning
opportunities. Expert teachers provide
additional resources for students with high
• Students learn best in situations where
they either ask questions of others or
respond to the questions of others. Expert
teachers employ a variety of instructional
groupings so students can learn from each
Improving classroom teaching
• A growing body of research into change
management in schools is highlighting the
importance of de-privatising classrooms.
Situations where teachers observe
colleagues teaching, provide feedback,
and are observed by others has shown
considerable potential to enhance
• In Victoria, Australia, Hillary Hollingsworth
has used the analysis of videoed lesson
footage as a key strategy for getting
teachers to reflect on their practice. The
dimensions of quality teaching discussed
above provide an important observational
framework for such peer observation.
An observational framework
• The first step in the process is for the teacher
observed to nominate one or two clusters that he
or she feels are an area of focus. This happens
before the observation.
• The observer then notes events that occur
during a lesson segment against the
appropriate habit/s. It is important that the
notes are factual statements about what
occurred rather than opinions or
• Use video to capture the lesson segment
so that the noted events can be replayed
• The post-observation discussion focuses
on the teacher examining the events,
discussing the rationale for the
instructional decisions made, and
considering the future implications of the
• Experience has show that observers need
practice in avoiding judgmental comments
and actively listening to the explanations
of the observed teacher. It is vital that
clear goals are set from the discussion
and both parties follow up these goals
through more observations.
A: Peer observation
• Video part of a mathematics lesson. Make sure that the tape is
no longer than 10 minutes.
• Nominate a cluster that you want them to observe.
• Play the video as they record comments.
• Role-play an observer conducting the follow-up discussion
using the video to recall events.
• Challenge the teachers to nominate a peer with whom they
will have a mutual observation arrangement. Get them to
timetable when the observations and discussions will occur.
• Schedule a staff or syndicate meeting to review the success of
Answer these questions together:
B: Pedagogical-Content Knowledge for Fractions
• To make teachers aware of the dimensions of pedagogical-content
1. What do the parts of a fraction symbol, like , mean?
a. The denominator (bottom number) means…?
b. The numerator (top number) means…?
c. The vinculum (line) means…?
2. How might a fraction like 3 relate to other mathematical ideas
a. Decimals and percentages, i.e. 0. and 66.%
b. Ratios, i.e. 2:1
c. Angles (turns), i.e. 120°
d. Measurements, i.e. 3 metres ≈ 666 millimetres
e. Division, i.e. 2÷3= 3
3. What contexts from the everyday world of
students can we use to teach fractions?
4. a. What representations (equipment,
diagrams, words, symbols) can we use in
b. What advantages/disadvantages do these
5. What misconceptions do students develop
about fractions, usually by over-generalising
what happens with whole numbers?
• The following screens list some readings
that you may find useful related to the
topic of quality teaching.
• Alton-Lee, A. (2003). Quality teaching for diverse students in schooling best
evidence synthesis. Ministry of Education:Wellington.
• Hattie, J (2002). What are the attributes of excellent teachers? In: New
Zealand Council for Educational Research Annual Conference.
• Bishop, R., Berryman, M., Richardson, C., & Tiakiwai, S. (2003).
Kotahitanga:The experiences of year 9 and 10 Maori students in
mainstream classes. Wellington:Ministry of Education.
• Stigler, J. & Hiebert, J. (1997) Understanding and Improving Classroom
Mathematics Instruction: An Overview of the TIMSS Video Study, In Raising
Australian Standards in Mathematics and Science: Insights from TIMSS,
• Clarke, D., & Hoon, S.L. (2005) Studying the Responsibility for the
Generation of Knowledge in Mathematics Classrooms in Hong Kong,
Melbourne, San Diego and Shanghai, In Chick, H. & Vincent, J. (Eds.),
Proceedings of the 29th Conference of the International Group for the
Psychology of Mathematics Education July 10-15, 2005, PME: Melbourne
• Askew, M., Brown, M., Rhodes, V., William, D. & Johnson, D. (1997).
Effective Teachers of Numeracy, King’s College, University of London:
• Shulman, L. S. (1987). Knowledge and Teaching: Foundations of the New
Reform. Harvard Educational Review, 57(1), 1-22.
• Brophy, J.,& Good, T. (1986). Teacher behaviour and student achievement.
In M.C. Wittrock (Ed.), handbook of research on teaching, 3rd ed. (pp. 328-
375). New York: MacMillan.
• Slavin, R.E. (1996). Research on co-operative learning and achievement:
What we know and what we need to know. Contemporary Educational
Psychology, 21, 43-69.