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.
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-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)
Factors contributing to achievement • Match the five sectors of the achievement pie with the factors below: • Students • Home • Teachers • Peers • Schools
How big is the impact of teaching? • Hattie ‘s analysis of thousands of research studies show this combined effect: Teachers Students Schools Home Peers
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?
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.
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 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.
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
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 4. Connections 5. Instructional responsiveness 6. Student empowerment 7. Equity • This workshop will discuss each cluster then provide you with a tool for observing teachers in action.
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.
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.
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.
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.
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.
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.
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.
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.
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.
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).
Expert teachers also provoke high order thinking, such as analysing, justifying and synthesising through the questions they ask.
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.
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.
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.
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 “Habits of Teaching” observation form is a framework for classroom observation. It can easily be adapted for any area of the curriculum. Link to form • 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 interpretations. • Use video to capture the lesson segment so that the noted events can be replayed several times.
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.
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.
Workshop Ideas A: Peer observation • Video part of a mathematics lesson. Make sure that the tape is no longer than 10 minutes. • Introduce the “Habits of Teaching” observation form to your teachers. • 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.
Answer these questions together: B: Pedagogical-Content Knowledge for Fractions • To make teachers aware of the dimensions of pedagogical-content knowledge… • What do the parts of a fraction symbol, like , mean? • The denominator (bottom number) means…? • The numerator (top number) means…? • The vinculum (line) means…? • How might a fraction like relate to other mathematical ideas like: • Decimals and percentages, i.e. 0. and 66.% • Ratios, i.e. 2:1 • Angles (turns), i.e. 120° • Measurements, i.e. metres ≈ 666 millimetres • Division, i.e. 2÷3=
What contexts from the everyday world of students can we use to teach fractions? • a. What representations (equipment, diagrams, words, symbols) can we use in teaching fractions? b. What advantages/disadvantages do these representations have? • What misconceptions do students develop about fractions, usually by over-generalising what happens with whole numbers?
Readings • 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. 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
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.