Improving learning in mathematics

1. The average company doesn’t invest enough in skills.That’s why they’re average!.(Bob Putnam, Chairman Ford UK) Improving learning in mathematics

2. Every School a Good School

3. Better Mathematics

4. Improving learning in mathematics Programme Aim To enhance the learning experiences of all pupils by promoting quality teaching of mathematics

5. Improving learning in mathematics Learning Intentions By the end of the two-day programme participants will be better able to: • Challenge pupils understanding through skilful questioning • Use an appropriate variety of teaching activities and learning strategies • Encourage pupils to think and talk about how they learn mathematics and what they have learnt • Contribute to departmental planning and the dissemination of good practice within and across schools. Every School a Good School A strategy for raising achievement in literacy and numeracy. Better Maths

6. Improving learning in mathematics Session 1 Beliefs about learning and teaching Learning Intentions This session is intended to help us to reflect on our current assumptions, beliefs and teaching practices

7. Improving learning in mathematics Beliefs about learning and teachingWorkshop 1a • With your partner discuss the 6 statements in your envelope. • At your table share your thoughts on all of the statements and as a group decide on which ones reflect good practice in a mathematics classroom. • Use post-its to record the statements you disagree with on the bar chart.

8. Improving learning in mathematics Beliefs about learning and teaching Workshop1b • Discuss those statements you believe reflect good practice in a mathematics classroom. • Choose one statement which you think : • is well addressed in your classroom • you would need work towards in the future.

9. Engage learners in discussing and explaining ideas, Challenging and teaching one another, Creating and solving each other's questions and working collaboratively to share methods and results. Improving learning in mathematics To help learners to adopt more active approaches towards learning

10. 2 + 2 = 5 Traditional, 'transmission' approaches involve simplifying ideas and methods by explaining them to learners one step at a time. In contrast, this model emphasises the interconnected nature of mathematics, and it is 'challenging' in that it seeks to confront common conceptual difficulties head on. Improving learning in mathematics To develop more 'connected' and 'challenging' teaching methods.

11. Improving learning in mathematics The types of activity Classifying mathematical objects Evaluating mathematical statements Creating problems Analysing reasoning and solutions Interpreting multiple representations

12. Improving learning in mathematics Personal reflection

13. Improving learning in mathematics Please be back in 30 minutes

14. Improving learning in mathematics Session 2 Effective Questioning The answer to my question is 48! What is the question?

17. Improving learning in mathematics Bowland Charitable Trust • What different types of questions are there? • What different purposes do your questions serve? • Which type of question do you use most frequently? Record your comments on the worksheet provided

18. Why Do We Ask Questions? To manage and organise pupils’ behaviour To find out what pupils know To stimulate interest in a new topic To focus on an issue or topic Improving learning in mathematics • To structure a task for • maximum learning • To identify, diagnose • difficulties or blocks to • learning • To stimulate pupils to ask • questions • To give pupils opportunity • to assimilate, reflect and • learn through discussion

19. What Is Effective Questioning? Questions are planned and related to session objectives. Questions are mainly open. Teacher allows ‘wait time’. Both right and wrong answers are followed up. Questions are carefully graded in difficulty. Teacher encourages learners to explain and justify answers. Teacher allows collaboration before answering. All participate e.g. using mini-whiteboards. Learners ask questions too. Improving learning in mathematics

20. Promoting Pupil Questioning Model questioning for pupils. Provide opportunities for pupils to practise their skills. Plan time for pupils’ questions and for dealing with them effectively. Improving learning in mathematics

21. Different types of questions Creating examples and special cases Evaluating and correcting Comparing and organising Modifying and changing. Generalising and conjecturing Explaining and justifying Improving learning in mathematics

22. Creating examples and special cases Show me an example of: a number between and ; a hexagon with two reflex angles; a shape with an area of 12 square units and a perimeter of 16 units; a quadratic equation with a minimum at (2,1); a set of 5 numbers with a range of 6…and a mean of 10…and a median of 9 Improving learning in mathematics

23. Evaluating and correcting What is wrong with these statements? How can you correct them? When you multiply by 10, you add a nought. + = Squaring makes bigger. If you double the radius you double the area. An increase of x% followed by a decrease of x% leaves the amount unchanged. Every equation has a solution. Improving learning in mathematics

24. Comparing and organising What is the same and what is different about these objects? Improving learning in mathematics • Square, trapezium, parallelogram. • An expression and an equation. • (a + b)2 and a2 + b2 • Y = 3x and y = 3x +1 as examples of straight lines. • 2x + 3 = 4x + 6; 2x + 3 = 2x + 4; 2x + 3 = x + 4

25. 1, 2, 3, 4, 5, 6, 7, 8, 9,10 , , , , , Comparing and organising Improving learning in mathematics How can you divide each of these sets of objects into 2 sets? • y = x2 - 6x + 8; y = x2 - 6x + 10; y = x2 - 6x + 9; y = x2 - 5x + 6

26. Modifying and changing How can you change: this recurring decimal into a fraction? the equation y = 3x + 4, so that it passes through (0,-1)? Pythagoras’ theorem so that it works for triangles that are not right-angled? the formula for the area of a trapezium into the formula for the area of a triangle? Improving learning in mathematics

27. Generalising and conjecturing What are these special cases of? 1, 4, 9, 16, 25.... Pythagoras’ theorem. A circle. When are these statements true? A parallelogram has a line of symmetry. The diagonals of a quadrilateral bisect each other. Adding two numbers gives the same answer as multiplying them. Improving learning in mathematics

28. Explaining and justifying Use a diagram to explain why: a2 − b2 = (a + b)(a − b) Give a reason why: a rectangle is a trapezium. How can we be sure that: this pattern will continue: 1 + 3 = 22; 1 + 3 + 5 = 32…? Convince me that: if you unfold a rectangular envelope, you will get a rhombus. Improving learning in mathematics

29. Workshop 2Designing Appropriate Questions Use the worksheet provided to write 1 question in each category which relates to a topic you are teaching at the moment. Share your questions with the others at your table Improving learning in mathematics

30. What is a good question? Robert Fisher Prof of Education at Brunel University Improving learning in mathematics is an invitation to think, or to do. It stimulates because it is open-ended. A good question: is productive – it looks for a response will generate more questions.

31. Improving learning in mathematics Resources Bowland TrustBetter MathsNI Curriculum AfLYoutube

32. Improving learning in mathematics Personal reflection Revisit your thoughts on questioning ref: Bowland Trust

33. Improving learning in mathematics Lunch

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35. Improving learning in mathematics Session 3 Learning from Mistakes and Misconceptions

36. Improving learning in mathematics Workshop 3a Analysing Learner’s Work Consider the samples of work and record the errors made and possible thinking which may have led to them. Share your thinking around the table.

37. Improving learning in mathematics Workshop 3b • Interpreting Multiple Representations • Working in groups of 3 take turns to match pairs of cards and place them on the table side by side. • Explain why they make a pair. • Partners should challenge thinking if necessary. • When finished place the cards in order of size – smallest to largest.

38. Improving learning in mathematics Personal reflection

39. Improving learning in mathematics Session 4 Sharing good practice electronically

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48. Improving learning in mathematics 3-Day Programme • Day 2 - in-school (sub-cover provided) for planning/preparation • In your classroom - use one (or more) of the ideas/activities from today • Online - visit the LNI site and post a comment (relating to your experience) on the discussion board • Day 3 - out-centre – share with colleagues: • Monday, 9 March 2009 – NWTC • Tuesday, 10 March 2009 – TEC

49. Improving learning in mathematics Session 5 Opening the boxes

50. Opening the box Active Learner Centred Approaches To the Teaching & Learning of Mathematics 5 Packs Improving learning in mathematics