Contributed by:

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.

1.
Quality Teaching

2.
• 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

curriculum.

• 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

meeting programme.

characteristics of expert teachers of numeracy.

Naturally most of these characteristics will be

transferable to teaching other areas of the

curriculum.

• 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

meeting programme.

3.
How important is good teaching?

• The first question to consider is, “How

much difference to student achievement

do teachers actually make?”

• The first question to consider is, “How

much difference to student achievement

do teachers actually make?”

4.
• 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-

operation, effort)

– 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)

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-

operation, effort)

– 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)

5.
Factors contributing to achievement

• Match the five sectors of the achievement pie

with the factors below:

30%

– Students

– Home

49%

– Teachers

– Peers

– Schools 7%

7%

7%

• Match the five sectors of the achievement pie

with the factors below:

30%

– Students

– Home

49%

– Teachers

– Peers

– Schools 7%

7%

7%

6.
How big is the impact of teaching?

• Hattie’s analysis of thousands of research studies

show this combined effect:

30%

Teachers

49%

Students

7%

Schools

7%

Home 7%

Peers

• Hattie’s analysis of thousands of research studies

show this combined effect:

30%

Teachers

49%

Students

7%

Schools

7%

Home 7%

Peers

7.
• 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

combined figure.

• Under what circumstances do you think

the effect due to teaching would be high?

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

combined figure.

• Under what circumstances do you think

the effect due to teaching would be high?

8.
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

students.

• 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

students.

9.
• To summarise:

Teachers make a significant difference to

student learning and they do so most in

situations where the students are needy.

Teachers make a significant difference to

student learning and they do so most in

situations where the students are needy.

10.
Characteristics of quality teachers?

• So what is it that quality teachers do that makes the

difference?

• A good place to start is to ask your colleagues the

following question:

• 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

teacher’s behaviour.

• What characteristics does this teacher have that make

her an “expert”?

• Write down a few characteristics that you think an expert

teacher has.

• So what is it that quality teachers do that makes the

difference?

• A good place to start is to ask your colleagues the

following question:

• 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

teacher’s behaviour.

• What characteristics does this teacher have that make

her an “expert”?

• Write down a few characteristics that you think an expert

teacher has.

11.
• Good management is a necessary but not sufficient

condition

• 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

condition

• 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

12.
• 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!

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!

13.
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

4. Connections

5. Instructional responsiveness

6. Student empowerment

7. Equity

• This workshop will discuss each cluster.

• The characteristics of expert teachers of numeracy can

be grouped in these clusters:

1. Relationships with students

2. Planning and assessment

3. Problem focus

4. Connections

5. Instructional responsiveness

6. Student empowerment

7. Equity

• This workshop will discuss each cluster.

14.
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

cultural identity.

• 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

cultural identity.

15.
• 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.”

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.”

16.
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.

• 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.

17.
• 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.

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.

18.
Problem Focus

• 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

culture.

• 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

culture.

19.
• 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

practicing.

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

practicing.

20.
• 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.

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.

21.
Connections

• 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.

• 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.

22.
• 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.

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.

23.
Instructional responsiveness

• 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.

• 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.

24.
• 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

explanations.

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

explanations.

25.
Student Empowerment

• 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).

• 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).

26.
• Expert teachers also provoke high order

thinking, such as analysing, justifying and

synthesising through the questions they

ask.

thinking, such as analysing, justifying and

synthesising through the questions they

ask.

27.
Equity

• 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

needs.

• 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

needs.

28.
• 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

other.

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

other.

29.
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

classroom practice.

• 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

classroom practice.

30.
• 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.

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.

31.
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 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.

32.
• 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

interpretations.

• Use video to capture the lesson segment

so that the noted events can be replayed

several times.

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

interpretations.

• Use video to capture the lesson segment

so that the noted events can be replayed

several times.

33.
Follow-up Discussions

• 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

observations.

• 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

observations.

34.
• 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.

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.

35.
Workshop Ideas

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

the 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

the observations.

36.
Answer these questions together:

B: Pedagogical-Content Knowledge for Fractions

• To make teachers aware of the dimensions of pedagogical-content

knowledge…

2

1. What do the parts of a fraction symbol, like , mean?

3

a. The denominator (bottom number) means…?

b. The numerator (top number) means…?

c. The vinculum (line) means…?

2

2. How might a fraction like 3 relate to other mathematical ideas

like:

a. Decimals and percentages, i.e. 0. and 66.%

b. Ratios, i.e. 2:1

c. Angles (turns), i.e. 120°

2

d. Measurements, i.e. 3 metres ≈ 666 millimetres

2

e. Division, i.e. 2÷3= 3

B: Pedagogical-Content Knowledge for Fractions

• To make teachers aware of the dimensions of pedagogical-content

knowledge…

2

1. What do the parts of a fraction symbol, like , mean?

3

a. The denominator (bottom number) means…?

b. The numerator (top number) means…?

c. The vinculum (line) means…?

2

2. How might a fraction like 3 relate to other mathematical ideas

like:

a. Decimals and percentages, i.e. 0. and 66.%

b. Ratios, i.e. 2:1

c. Angles (turns), i.e. 120°

2

d. Measurements, i.e. 3 metres ≈ 666 millimetres

2

e. Division, i.e. 2÷3= 3

37.
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

teaching fractions?

b. What advantages/disadvantages do these

representations have?

5. What misconceptions do students develop

about fractions, usually by over-generalising

what happens with whole numbers?

students can we use to teach fractions?

4. a. What representations (equipment,

diagrams, words, symbols) can we use in

teaching fractions?

b. What advantages/disadvantages do these

representations have?

5. What misconceptions do students develop

about fractions, usually by over-generalising

what happens with whole numbers?

38.
Readings

• The following screens list some readings

that you may find useful related to the

topic of quality teaching.

• The following screens list some readings

that you may find useful related to the

topic of quality teaching.

39.
• 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.

NZCER:Wellington

• 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,

ACER:Melbourne

• 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

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.

NZCER:Wellington

• 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,

ACER:Melbourne

• 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

40.
• Askew, M., Brown, M., Rhodes, V., William, D. & Johnson, D. (1997).

Effective Teachers of Numeracy, King’s College, University of London:

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.

Effective Teachers of Numeracy, King’s College, University of London:

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.