Boosting achievement in A2 Core Mathematics:

2. Boosting achievement in A2 Core Mathematics: Supporting lower ability students through the C3 and C4 modules Phil Chaffé 2012

3. 10.00 – 11.15am: Hitting the ground running: successful transition to A2 level 11.15 – 11.30am: DISCUSSION: COFFEE BREAK 11.30 – 12.30pm: Picking up the problems: identifying when and where students struggle 12.30 – 1.30pm: LUNCH AND INFORMAL DISCUSSION 1.30 – 2.45pm Materials and methods: teaching the difficult topics 2.45 – 3.00pm: DISCUSSION: AFTERNOON TEA 3.00 – 3.45pm: Preparing students for examinations

4. Hitting the ground running: successful transition to A2 level • What to do after AS levels are complete • The skills that are needed to make a successful start to A2 level mathematics • Developing the essential skills needed to start the A2 course • Preparing students for the challenge of the A2 course • Materials and ideas that ensure a good start (some moved to the afternoon session)

5. Putting things in context The three post 16 transitions • GCSE to AS level • GCSE algebraic manipulation techniques are expected to be used with more fluency. • Mathematical terms are expected to be a part of a student’s vocabulary. • GCSE knowledge is expected to be applied efficiently (and quickly). • Less guidance is given for solving problems. • A limited number of new techniques are introduced.

6. AS level to A2 level • Students are expected to recall, select and use their knowledge of mathematical facts, concepts and techniques with fluency in a variety of contexts. • Mathematical arguments now have to be rigorous, logical and precise. • There is more emphasis on proof. • Manipulation of mathematical expressions is expected to be fluent and precise. • Students need to be able to handle substantial problems presented in an unstructured form.

7. A2 level to university • Mathematical arguments have to be concise and relevant. • Mathematics has to be used creatively to solve complex problems. • Students are expected to question the techniques that they use. • Manipulation of mathematical expressions is expected to be fluent and precise. • There is a high emphasis on proof and an expectation that students have a number of techniques at their disposal.

8. AS level to A2 level • Students are expected to recall, select and use their knowledge of mathematical facts, concepts and techniques with fluency in a variety of contexts. • Mathematical arguments now have to be rigorous, logical and precise. • There is more emphasis on proof. • Manipulation of mathematical expressions is expected to be fluent and precise. • Students need to be able to handle substantial problems presented in an unstructured form.

9. The aims of an A level mathematics course (paraphrased from 4 specifications) • To develop a deeper understanding of the way that mathematics and mathematical processes work. • To promote confidence and foster enjoyment. • To develop a student’s ability to reason logically. • To give students the skills to recognise incorrect reasoning. • To teach students how to generalise and to construct mathematical proofs.

10. To extend the range of mathematical skills and techniques available to a student. • To give students the opportunity to use their mathematical skills in more difficult, unstructured problems. • To help students develop an understanding of coherence and progression in mathematics and of how different areas of mathematics can be connected. • To develop a student’s ability to communicate effectively with mathematics. • The help students acquire the skills needed to use technology effectively and recognise when this may be inappropriate and where there are limitations. • To encourage students to take more responsibility for their own learning and the evaluation of their mathematical development.

11. Preparing students for the challenge of the A2 course Students are expected to recall, select and use their knowledge of mathematical facts, concepts and techniques with fluency in a variety of contexts. “What we do in most traditional classrooms is require students to commit bits of knowledge to memory in isolation from any practical application—to simply take our word that they "might need it later." For many students, "later" never arrives. This might well be called the freezer approach to teaching and learning. In effect, we are handing out information to our students and saying, "Just put this in your mental freezer; you can thaw it out later should you need it." With the exception of a minority of students who do well in mastering abstractions with little contextual experience, students aren't buying that offer. The neglected majority of students see little personal meaning in what they are asked to learn, and they just don't learn it.” DALE PARNELL, Oregon State University From: High School Mathematics at Work: Essays and Examples for the Education of All Students (1998)

12. Preparation starts at AS level • Each “strand” of the specification is made clear to students • Connections between skills/techniques across strands and levels are made clear to students • Students are aware that the skills they are using will link to many other areas • Developing a toolkit mentality • Skills/techniques are taught with an indication of why they are useful and the many ways in which they may be applied

13. What to do after finishing the AS level course • Prepare students for the expectations of the A2 course. • Use activities that show the strong links between AS and A2 mathematics skills. • There should be some time to look at AS skills, ideas and techniques and develop them along the lines needed for A2 mathematics. • Skills practice exercises can be set to develop the fluency needed.

14. What skills are needed?

15. Activity: What skills are needed? • Look through the worked questions from C3/C4 • What ‘bits’ of mathematics can you identify? • Look out for • notation that you recognise • ‘normal’ mathematical skills being used

16. Transition Work • This should be used to reinforce AS skills, develop some problem solving tenacity and introduce some of the basics of the A2 core. • An example of a transition unit • This example is designed for discussion. • Some questions to ask when looking through the unit • Is the content appropriate? • Are the correct skills being reinforced? • Is the quantity appropriate? • Are there enough problem solving activities? • Is the introduction to A2 appropriate?

17. The skills needed for a successful transition to A level • Personal skills • Retention of previously acquired information and skills • Initiative in solving problems • Perseverance in solving problems • Willingness to overcome the desire for a quick trick or formula • Overcoming the aversion to ‘wordy’ problems • An understanding of why the skills are useful

18. Mathematical Skills

19. ‘Eureka’ moments, fine tuning and deep understanding • Focus on learning rather than teaching • Discovery activities • in lessons (groups or individually) • at home • Use lesson objectives that tantalise

20. ‘Eureka’ moments, fine tuning and deep understanding • Developing understanding • Working towards examination questions • Text book or activity led • Build in discussion time • Link to previous knowledge

21. ‘Eureka’ moments, fine tuning and deep understanding • Encourage thinking more about the maths • Differentiation by outcome • Challenge both able and weaker students

22. Picking up the problems: identifying when and where students struggle • Anticipating problems and preparing for difficulties • Spotting problems by everyday classroom monitoring • Diagnostic activities and instant troubleshooting • Dealing with deep seated problems

23. Anticipating problems and preparing for difficulties Look at the C3 and C4 examination questions. Pick a problem - analyse the skills required to solve the problem. Discussion questions What are the mathematical skills required? Where are these skills taught (at what stage)? Where do you think your students will have difficulty with the question? If they would simply not be able to start, what is stopping them? What would they need to be able to get started with the question?

30. Spotting problems by everyday classroom monitoring • Direct questioning • Working with groups/pairs • “Culture of explanation” Activities/exercises that can be used to monitor understanding • Focused on a sensible number of things in a topic. • Key questions asked – promote thinking.

31. Example Say as much about …. as you can You have introduced function and taught most of the initial skills including the main definitions.

32. Asking the right questions – domain and range • Be upfront about what you are doing. Let the students know that you will be assessing their responses to your questions. • Make sure that the weaker students have a some chance of answering – the idea is to find out what they do know rather that prove that they know nothing. • Have a series of options available that the student can choose from. Use these to get past the “I don’t know” response. • When supplying options, give possibilities that are at least partially correct as well as the real answer. This allows the student to show how they understand something even if they did pick the wrong option. • Have a balance of questions. Don’t keep things to easy all of the time; ask questions that will stretch the understanding of the most able students. • Think about how you will deal with zero or negative responses. • Even though you are assessing them, remember to be liberal with praise.

33. Asking the right questions activity Domain and range

34. What are the skills that are being tested? • How do you know that a student has understood those skills? • What does a student need to say to indicate that they have those skills? • What are you going to ask to check that the student really has understood? • How many questions is enough?

35. Instant troubleshooting • Small group activities • Groups of >2 allow you to help one of the students while the others get on with the task • Occasionally social engineering helps when arranging groups • Deal with the immediate problem whilst trying to assess if it is deep seated or a “quick fix” • Keep trying to tie the explanation in to what the student does know.

37. Dealing with deep seated problems • Find out what the student is thinking first. • Ask questions to break down the steps that they think in. • Avoid saying directly that the student has it all wrong. • Build the explanation from the ground up. Don’t be afraid of going back to the very basics. • Involve the student in the explanation by getting them to take you through it. • It does take time and patience. Don’t try to do it all at once.

38. Intervention • Mentoring, where possible, is an effective method. • Student study pairing has also been used effectively in schools. • BUT • It can be “after the horse has bolted”. • It almost always involves staff giving up their own time.

39. This question requires a number of skills that the students find difficult. How would you build a ground up explanation to help a student overcome a deep seated problem with it.

40. Materials and methods: teaching the difficult topics Algebra and Functions, Trigonometry, Vectors • Teaching to promote confidence and fluency in algebra • Providing focused support for those struggling with algebra • Teaching to promote confidence and fluency in trigonometry • Providing focused support for those struggling with trigonometry • Introducing vectors to lower ability students • Materials and ideas to develop the key skills

41. Algebra and Functions OCR Specification – Core Mathematics 3

42. The modulus function • Introduction using Geogebra – link to transforming graphs • Get students to identify what is happening and why. • Follow up activities • Modulus graph matching activity • How many solutions – modulus equations activity

43. Providing focused support for students struggling with functions Terminology The terminology is very important so make sure the student is as confident as possible with the language of functions. Transformations Students need to be very familiar with the effect of transformations and the links to what they have done before. The order in which transformations is applied needs to be very clear.

44. Composite functions The order for applying each function should be clearly understood. Substituting numbers into one then the other should be done first before moving on to algebraic substitution. Inverse functions These need to be though of first as “undoing” something. “I think of a number multiply by 5 and add 6 the result is..” type of questions work well initially. The links between the graph of the function and that of its inverse should be made very clear.

45. Trigonometry OCR Specification – Core Mathematics 3

46. A starting point…. 1 ϴ

47. An alternative…. 1 1

48. 1

49. A Geogebra activity that uses links to graphical transformation.

50. Providing focused support for students struggling with trigonometry Periodicity Students really need to know how to use the periodicity of the trigonometric functions so they can calculate all of the required solutions to a trigonometric equation. Sketch graphs using both degrees and radians are essential. Using the formula booklet Weaker students should have (at least a copy) of the formula booklet page with the given trig identities from the start. It makes it clear what they do need to learn and gets them used to looking in the correct place.