Mathematics Matters Malcolm Swan Thank Jane and Peter Effective = achieving a desired outcome What outcomes do we desire? How can we promote these? How can we assess our progress?

What promotes “effective” learning? Purposes (values) for teaching mathematics Teaching and learning principles that emerge research studies Mathematical tasks that are appropriate for each purpose Learning cultures and discourses that need to be created.

Recalling a successful lesson

A memorable, ‘successful’ lesson Age/ability range What was the task? How was the session/task introduced? How was the session/task sustained? How was the session/task concluded? What were the critical moments? What mathematics was learnt? (on plan and off plan) How was that mathematics learnt? Other memorable outcomes

Values and principles

Why teach maths? ‘Societal values’ Mathematical thinking is important for all members of a modern society as a habit of mind for its use in the workplace, business and finance; and for personal decision-making. Mathematics is fundamental to national prosperity in providing tools for understanding science, engineering, technology and economics. It is essential in public decision-making and for participation in the knowledge economy. (QCA 2007) Mathematics equips pupils with a uniquely powerful set of tools to understand and change the world. These tools include logical reasoning, problem-solving skills, and the ability to think in abstract ways. Mathematics is important in everyday life, and in many forms of employment, science and technology, medicine, the economy, the environment and development, and in public decision-making. Different cultures have contributed to the development and application of mathematics. Today, the subject transcends cultural boundaries and its importance is universally recognised. Mathematics is a creative discipline. It can stimulate moments of pleasure and wonder when a pupil solves a problem for the first time, discovers a more elegant solution to that problem, or suddenly sees hidden connections.

Why teach maths? ‘Personal values’ Mathematics equips pupils with uniquely powerful ways to describe, analyse and change the world. It can stimulate moments of pleasure and wonder for all pupils when they solve a problem for the first time, discover a more elegant solution, or notice hidden connections. Pupils who are functional in mathematics and financially capable are able to think independently in applied and abstract ways, and can reason, solve problems and assess risk. (QCA 2007) Mathematics equips pupils with a uniquely powerful set of tools to understand and change the world. These tools include logical reasoning, problem-solving skills, and the ability to think in abstract ways. Mathematics is important in everyday life, and in many forms of employment, science and technology, medicine, the economy, the environment and development, and in public decision-making. Different cultures have contributed to the development and application of mathematics. Today, the subject transcends cultural boundaries and its importance is universally recognised. Mathematics is a creative discipline. It can stimulate moments of pleasure and wonder when a pupil solves a problem for the first time, discovers a more elegant solution to that problem, or suddenly sees hidden connections.

Why teach maths? ‘Mathematical values’ Mathematics is a creative discipline. The language of mathematics is international. The subject transcends cultural boundaries and its importance is universally recognised. Mathematics has developed over time as a means of solving problems and also for its own sake. (QCA 2007) Mathematics equips pupils with a uniquely powerful set of tools to understand and change the world. These tools include logical reasoning, problem-solving skills, and the ability to think in abstract ways. Mathematics is important in everyday life, and in many forms of employment, science and technology, medicine, the economy, the environment and development, and in public decision-making. Different cultures have contributed to the development and application of mathematics. Today, the subject transcends cultural boundaries and its importance is universally recognised. Mathematics is a creative discipline. It can stimulate moments of pleasure and wonder when a pupil solves a problem for the first time, discovers a more elegant solution to that problem, or suddenly sees hidden connections.

Lacey, 2007

Why teach maths? Spiritual development, through helping pupils obtain an insight into the infinite, and through explaining the underlying mathematical principles behind some of the beautiful natural forms and patterns in the world around us Moral development, helping pupils recognise how logical reasoning can be used to consider the consequences of particular decisions and choices and helping them learn the value of mathematical truth (National Curriculum specifications, 1999) Have you ever seen a lesson where these aims have been realised? Where is the Awe and Wonder! Doubling 50 times story?

Why teach maths? Social development, through helping pupils work together productively on complex mathematical tasks and helping them see that the result is often better than any of them could achieve separately Cultural development, through helping pupils appreciate that mathematical thought contributes to the development of our culture and is becoming increasingly central to our highly technological future, and through recognising that mathematicians from many cultures have contributed to the development of modern day mathematics. (National Curriculum specifications, 1999) I think I have seen the first of these a few times now, but the second only rarely. In my own school career I was fascinated by the stories of mathematicians lives.

What content is valued? Fluency in recalling facts, performing skills Interpretations for concepts, representations Strategies for investigation and problem solving Awareness of nature of maths, learning, the ‘system’ Appreciation of and power to use mathematics in society

A range of possible outcomes An exhibition of technique An explanation of a concept A problem solution A report of a piece of research A mathematical model A plan of action A design A decision and justification ………

Tension and conflict Content coverage (Faster.. Faster) Convergence on important theorems (I want you to discover this) Challenging maths (Struggle with this hard maths) Reflection and creativity (Time to think …) Openness of investigational work (I want you to explore this) Autonomy in problem solving (Use any method you are comfortable with) First is the tension between creativity and coverage. Approaches to teaching - such as those that involve students discussing alternative approaches and creating their own examples may at first appear inefficient and rambling. Discursive approaches to learning take time and go in unpredictable directions. Transmission approaches appear to ‘cover’ content more rapidly. The pressure of coverage becomes particularly acute with older students who are approaching examinations. Inspection reports reveal this tension. On the one hand we see schools criticised for allowing insufficient time for students to explain, reason, while on the other we read that the impact of the first year of the key stage 3 mathematics framework was to encourage more rapid coverage: Use of the framework raised teachers’ expectations of students, so that they pitched work at a higher level and covered material at a faster pace (OFSTED)

Principles for teaching and learning Students learn through active participation in and reflection on social practices, internalisation and reorganisation of experience. Activate pre-existing concepts. Allow students to create multiple connections. Stimulate ‘conflict’ to promote re-interpretation, reformulation and accommodation. Devolve problems to students. Students must articulate Interpretations. ‘Production of answers’ must give way to reflective periods of ‘stillness’ for examining alternative meanings and methods. These evolved from literature on learning theories concerning the development of concepts.

Principles for teaching and learning? Teaching is more effective when it … builds on existing knowledge exposes and discusses misconceptions uses higher-order questions uses cooperative small group work encourages reasoning not ‘answer getting’ uses rich, collaborative tasks creates connections between topics uses technology in appropriate ways

OFSTED’s values… The best teaching gave a strong sense of the coherence of mathematical ideas; it focused on understanding mathematical concepts and developed critical thinking and reasoning. Careful questioning identified misconceptions and helped to resolve them, and positive use was made of incorrect answers to develop understanding and to encourage students to contribute. Students were challenged to think for themselves, encouraged to discuss problems and to work collaboratively. Effective use was made of ICT. (OFSTED, 2006)

Linking practices to values and principles

Questions for you …. What is gap between your ideal values and those currently implemented in schools, colleges and other institutions? What principles for effective teaching and learning do you think are most helpful to teachers? Which are unhelpful? Allow about 10 minutes

In groups of three… One person describes the lesson reported in the first session. The others listen and help to clarify the account (adding notes and annotations). All together, identify the values and principles that are exemplified in the account. How are these exemplified? Record this on the sheet and attach to the sample lesson. Allow about 25 minutes for this. Now switch roles….

Views of 51 delegates at first meeting

Obstacles to progress

Factors that inhibit or modify practice What are the major obstacles to progress? How do these obstacles function? What practical steps can we take to help ourselves and others to overcome these obstacles?

Most common learning strategies (GCSE classes) Statements are ranked from most to least common 1 = almost never, 2 = occasionally, 3 = half the time, 4= most of the time; 5 = almost always. Source: Swan (2005) Mean (n=779) I listen while the teacher explains. 4.28 I copy down the method from the board or textbook. 4.15 I only do questions I am told to do. 3.88 I work on my own. 3.72 I try to follow all the steps of a lesson. 3.71 I do easy problems first to increase my confidence. 3.58 I copy out questions before doing them. 3.57 I practise the same method repeatedly on many questions. 3.42 I ask the teacher questions. 3.40 I try to solve difficult problems in order to test my ability. 3.32

Least common learning strategies (GCSE classes) Statements are ranked from most to least common 1 = almost never, 2 = occasionally, 3 = half the time, 4= most of the time; 5 = almost always. Source: Swan (2005) Mean (n=779) When work is hard I don’t give up or do simple things. 3.32 I discuss my ideas in a group or with a partner. 3.25 I try to connect new ideas with things I already know. 3.20 I am silent when the teacher asks a question. 3.16 I memorise rules and properties. 3.15 I look for different ways of doing a question. 3.14 My partner asks me to explain something. 3.05 I explain while the teacher listens. 2.97 I choose which questions to do or which ideas to discuss. 2.54 I make up my own questions and methods. 2.03

Giving up on meaning… T: Let me take you back to where we don't care why we did it, I'll just tell you how. You can forget the why if you want and just remember the how. This is the way I was taught…. P: Why didn't you tell us that before? T: Because there is just a chance that you might understand why one day. P: How will this help us when we get older? T: Don't ever ask me that. Its to get a grade C at the end of the year and then you'll be sure to get a good job. The whole thing about maths is its logical thinking.

The best teaching…. focuses on: coherence of mathematical ideas; understanding concepts critical thinking and reasoning. identifying and resolving misconceptions through careful questioning. using incorrect answers constructively challenging students to think for themselves, collaborative work using ICT effectively (OFSTED, 2006)

The worst teaching…. presents mathematics as … arbitrary rules and procedures a narrow range of learning activities weak assessment / questioning, ‘teaching to the test’ (OFSTED, 2006)

Obstacles to progress… Time pressures “ It’s a gallop to the main exam.” “ Students will waste time in social chat.” Control “ How can I possibly monitor what is going on?” Views of learners “ My students cannot discuss.” “ My students are too afraid of being seen to be wrong.” Views of mathematics “ In mathematics, answers are either right or wrong – there is nothing to discuss.” “ If they understand it there is nothing to discuss. If they don’t, they are in no position to discuss anything.” Views of learning “ Mathematics is a subject where you listen and practise.” “ Mathematics is a private activity.” It is important to remember that they may never grasp certain concepts and for some learners we are talking about maintaining skills rather than making progress. These students have been doing the same thing since they were very young. They were doing “Time” when they were five years old and they are still doing “Time” now – they still haven’t grasped it. If they haven’t the ability to grasp “Time” then they haven’t got the ability to have mature mathematical discussions. These learners feel that the teacher is only there to give a method; in fact the teacher is not doing the job for which they are paid if they do not do this. Maths classes are viewed not as places for talking; they are only places for listening, writing and pondering on your own.

Obstacles to change… Objective-driven lessons “ We are told to write our objectives on the board every lesson and stick to them. If we follow students’ lines of reasoning we are told that we are going off-track.” Tangible outcomes “ If we have discussions they won’t have a neat set of notes” Form rather than substance “We have to have 3-part lessons.” “You have to have a plenary at the end of every lesson.” Assessments only test facts and skills “If it’s not on the exam we’re wasting our time if we teach it.” Lack of resources “We haven’t got the use of computers for everyone.” “I’m just too tired” It is important to remember that they may never grasp certain concepts and for some learners we are talking about maintaining skills rather than making progress. These students have been doing the same thing since they were very young. They were doing “Time” when they were five years old and they are still doing “Time” now – they still haven’t grasped it. If they haven’t the ability to grasp “Time” then they haven’t got the ability to have mature mathematical discussions. These learners feel that the teacher is only there to give a method; in fact the teacher is not doing the job for which they are paid if they do not do this. Maths classes are viewed not as places for talking; they are only places for listening, writing and pondering on your own.

Factors that inhibit or modify practice What are the major obstacles to progress? How do these obstacles function? What practical steps can we take to help ourselves and others to overcome these obstacles?

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