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### Boosting achievement in AS level Core Mathematics

Supporting lower ability students through the C1 and C2 modules

Phil Chaffé 2012

Programme

10.00 – 10.45am Boosting initial subject knowledge and understanding

10.45 – 11.15am Boosting algebra skills

11.15 – 11.30am Discussion: coffee break

11.30 – 12.15pm Boosting coordinate geometry skills

12.15 – 1.00pm Boosting sequences and series skills

1.00 – 2.00pm Lunch and informal discussion

2.00 – 2.45pm Boosting trigonometry skills

2.45 – 3.00pm Discussion: afternoon tea

3.00 – 3.45pm Boosting calculus skills

Ensuring a smooth transition from GCSE to AS level

Preparation for initial discussion:

Find the exact value of

A proper factor of an integer N is a positive integer, not including 1 or N, that divides N.

(i) show that has exactly 10 proper factors

(ii) find how many other integers of the form (where and are integers) have exactly 10 factors.

When is sundered by , the residue is . Use the detritus principium to find the appraisal of .

Discussion – Initial experiences of AS level Mathematics

You have just started an AS level mathematics course and have been presented with these questions…..

- What mathematical things do you recognise in the questions?
- What is initially stopping you from answering the questions?
- What would help you to answer the questions?
- What skills will you need to be able to answer the questions?
- Which phrases cause you difficulty?

Comments from thestudentroom.co.uk

“Hey, just wanted to know:

How difficult did you find module C1 (in AS level maths) compared to GCSE Maths?

What grade did you get in C1 module and did you do the exam in January? If so, how much revision do you think you did for the C1 module up until you had to take the exam in January?

What books did you find helpful to revise from if you're one of the people who got a grade A in the C1 module maths AS level exam?

Was the transition from GCSE Maths to C1 module AS level maths hard/difficult? or easy or the same difficulty as with GCSE (when you did GCSE)?

ESPECIALLY MEAN EDEXCEL C1 MODULE AS LEVEL”

I did Edexcel... Umm... At 1st I struggled because I didn't work much for GCSE and so I was used to being lazy. lol. With some effort the transition should bridge up nicely. The transition really depends on the person. It was different for everyone in my class and we were a Further Maths Class so we had to finish the spec in 8 weeks. A lot of people who had nearly perfect scores at GCSE thought it was a little challenging at the time but when we did C3 near the end of the year 12 we realised that C1 is not difficult at all. I found I had to put some effort in at AS and A2, it was not plain sailing to say the least. I think differentiation and integration are the hardest things that come up in C1 and you MUST get rid of ALL your silly mistakes by past papering. I used to make so many really dumb mistakes and think to myself wth am I doing but it's all part of the learning process. Maths is a thoroughly enjoyable A-Level.

I’m finding it hard! One day left to revise and I’m bricking it, I got an A at GCSE, I was in the bottom set for Maths then. All my class now have done additional maths and I’m finding it hard, I’m actually scared for Monday!

C1 to C2 was the more difficult transition I found!

I thought C1 was ok at the time. I got 81/100 first time round, but I didn’t revise much. Second time round I got 94 I think. If you’re going to do maths, make sure you know your stuff. Revise as early as possible.

One crucial thing is to perfect your exam technique. This applies to any maths exam, as they can be rather challenging!

Do the Solomon papers; they’re kinda difficult, but will make the normal papers look easy. The more examples you do the better is for you, and the higher your chances of getting an A.

I know I’m against the majority here, but I thought it was a pretty big step up from gcse

Personally, I did find the shift difficult at first, but I don't think it was anything like GCSE Maths. Much more like Add Maths, but I did the CCEA exams for GCSE, so who knows. It seems daunting at first (especially since my teacher liked to mix in C2 stuff with it, ah~) but after about a month or so I just learned to go with it. Personally, I hated C1, but that's because I hate non-calc papers; can get the hard stuff but then can't add 7 and 5 properly!

I did my exams in June and so had to remember it and C2 after doing my mechanics module, but I did survive and I think I did reasonably well. I found myself using the revision books much more than the textbooks for revision, if any of that helps.

C1 is basically gcse maths + differentiation. There's little to no jump in difficulty, it's just slightly different I found. As long as you do plenty of past papers and ask about anything you're unsure about it should be fine. I did it in January- (ocr, so I know nothing about Edexcel I’m afraid), and got 100%

I didn't do any maths for almost 18 months and got 96/100 in C1 with around 4 months of preparation - but it took around 2 months to "click" and for me to start understanding things (I imagine that 2 months figure would be lower had I started it straight from GCSE) the trick is just to do lots and lots of practice questions and past papers

I did OCR. A lot of C1 duplicated GCSE maths (surds, quadratics etc) so it was fairly easy to move from GCSE to A-level. I did the exam in January and got 96%. I found the best way to revise was to spend a few weeks before the exam doing lots of past papers to get used to the style of questions you would be asked.

C1 is most probably going to be the easiest maths module you will take in AS/A2 Maths.

Only a little bit tougher than GCSE A/A* maths questions - and you only have to learn basic differentiation, integration and some stuff on arithmetic series, but the rest is pretty simple stuff.

However, don't be put off if you're not getting 95%+ in the first month of sixth form - I wasn't reaching that until Nov/Dec. Although C1 is an easy module, you'll find out that at times you'll make silly mistakes that could cost you a few marks especially as it’s a non-calculator paper.

Discussion – What do you want from the transition period from GCSE to AS level

- What basic mathematical skills do they need to start the course with?
- What initial obstacles to learning are you hoping to avoid by doing transition work?
- What student characteristics are you trying to promote?
- What does a steady AS level student have that a GCSE student doesn’t?

Transition

- Recognition that AS level is a step up from GCSE
- in understanding
- in competence with skills
- in pace
- in expectations of students
- Reminding students of what they have forgotten during the time after their GCSEs
- Introducing the language used at AS level
- Taking the first steps in mathematical thinking (introducing rigour)
- Removing the bad habits from GCSE

Activities and approaches for the first few weeks – settling into the course

- Use quick games/starters
- Indices game in PowerPoint
- Numbers ladder
- Definitions game
- Involve every student
- Keep the pace up
- Make sure all can achieve some success
- Use activities that allow for a baseline assessment of understanding
- Use lesson checklists to keep students focused on the key information

Bringing the subject to life

- Each topic should have a clear introduction that puts it in its mathematical context
- Every skill, no matter how mundane, should be introduced as important
- Mathematical discussion should be encouraged where possible
- An idea of where a topic lies in the wider world of mathematics is important
- Practical examples can be used where they have immediate relevance but this should not be ‘forced’
- Lessons should have a beginning, middle and end
- Extra examples from nrich, plus magazine etc. should be used where appropriate

Five golden rules for engaging weaker students

- 1. Be openly enthusiastic about what is being taught
- Avoid cynicism, it will be picked up on readily by students
- Don’t expect a great deal back in terms of enthusiasm but don’t be disillusioned by this

- 2. Focus on learning rather than teaching.
- Avoid giving too much away to start with
- Aim for ‘eureka’ moments

- 3. Differentiate but don’t ‘water down’
- Carefully select who you ask each question to
- Assign tasks ensuring that all will gain some success
- Take the content to the level that is required for the exam

Five golden rules for engaging weaker students

- 4. Be firm about what you want but keep the pressure off
- Expect high standards
- Avoid backing students into a corner
- Make the expectations ‘just the way it is’ rather than ‘because I say so’

- 5. Have more than one way of explaining everything
- Try to aim for three different sounding explanations for each thing taught (even if they boil down to being the same thing really)
- Stick to the main explanation when dealing with the whole class but be prepared to use one of the others with students who are struggling
- After you’ve exhausted all of your explanations, go back to the first in the hope that it has sunk in a bit more!

‘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

‘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

‘Eureka’ moments, fine tuning and deep understanding

- Encourage thinking more about the maths
- Differentiation by outcome
- Challenge both able and weaker students

Asking the right questions – 7 hints and tips

- Keep the pressure off by only directing questions at weaker students that they have a good chance of answering – the idea is to engage them in the lesson rather than lock them out.
- 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 achieve some degree of success even if they 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 whole class.
- Remember that you are trying to engage the learner with the simpler questions. Questions that are designed to really stretch the whole classes understanding can be aimed at the most able.
- Think about how you will deal with zero or negative responses.
- Remember to be liberal with praise.

Addressing a variety of learning styles

“The approaches and methods used for teaching mathematics in schools can have a huge impact on how much students learn in the classroom as well as on the quality of the learning that takes place.”

Mathematics Education in Europe (Eurydice 2012)

“it is not possible to identify a single best method, but found that there are many different types of learning and many different methods that should be applied, appropriate to the learner and the particular learning outcome required”

Mathematics Matters (Swann, NCETM 2008)

Four valuable types of learning (NCETM)

- fluency in recalling facts and performing skills;
- conceptual understanding and interpretations for representations;
- strategies for investigation and problem solving;
- appreciation of the power of mathematics in society.

“different methods are appropriate in developing these different types of

learning, including, as an example, the use of higher order questions, encouraging reasoning rather than 'answer getting', and developing mathematical language through communicative activities”

Hiebert and Grouws (2009)

“particular methods are not, in general, effective or ineffective. All teaching methods are ‘effective for something’ ”

“different methods are appropriate in developing these different types of learning, including, as an example, the use of higher order questions, encouraging reasoning rather than 'answer getting', and developing mathematical language through communicative activities”

Two important features of teaching when developing conceptual understanding:

- discussions around mathematics including examining relationships between different areas of maths, exploring why different procedures work as they do and examining differences between different approaches
- requiring students to work on complex, open problems in mathematics

Hiebert and Grouws (2009)

“when developing skill efficiency…… clear and fast paced presentation and modelling by the teacher, followed by practice by the students, worked well.”

“this is not a simple dichotomy…… it is not true that one approach works

in one area only”

“a weighted balance between the two teaching approaches might

be appropriate, with a heavier emphasis on the features related to conceptual understanding”

Fleming (VARK)

- There is no real evidence that teaching for any particular teaching style leads to improved results. Fleming’s VAK (more accurately VARK) model has been widely used in schools but often leads to undesirable outcomes, for example
- Teachers tending to teach to the learning style they find most exiting (so, lots of kinaesthetic running around then – possibly not) in a bid to appear to Ofsted that they are taking different learning styles into account.
- Teachers ignoring the existence of different learning styles and sticking to the approach they are most confident with.
- Teachers trying to do a bit of everything but being out of their comfort zone with some styles and so being less than effective when employing them.
- This is not a problem with the model itself but rather with the way it has been used.

Individualized Learning

It is important that students develop an ability to cope with learning styles that are not necessarily their favoured ones.

In other words – let the situation dictate the learning style that is encouraged…

BUT make sure that there are situations that address different learning styles…

REMEMBER to give each student the opportunity to achieve some success – limiting learning styles may not let this happen.

Individualized learning is about tailoring the approach to meet a student’s needs rather than wants.

Individualized Learning

- The key to this is:
- Identifying areas where understanding is not secure – this does not mean constant testing. Teaching has to include time to listen to each student to assess their grasp of the material.
- Giving each student a ‘balanced diet’ of education - including the experience of learning using a variety of styles.
- Reacting in good time when a student does not have a secure understanding of the material.
- Any approach used must take these things into account whilst being focused on the mathematics being taught.

Getting the algebra up to speed

- The key algebra skills that need to be embedded before C1 and C2 are
- Collecting like terms
- Laws of Indices
- Changing the subject of a formula
- Expanding expressions
- Factorising quadratic expressions including the difference of two squares
- Solving quadratic equations by factorisation
- Using the quadratic formula
- Manipulating surds
- Solving simultaneous equations by elimination and substitution

Transition materials

Notes page 17 – example copied from a transition booklet given to year 11 students

Know your indices

The aim of the activity is for students (and the teacher) to asses their ability to find equivalent indices.

This game works particularly well on a smart board but any projection method is fine.

Two teams line up, one each side of the whiteboard.

The teacher clicks on the cloud in the centre of the screen and a ‘question’ comes up.

The first player in each team has to tap the equivalent expression from their ‘answers’ list.

The teacher taps the board again and the correct answer is highlighted.

The front player moves to the back of the queue and the process is repeated.

This runs until all of the questions have been asked.

Quadratics and other polynomials

* Initial activity – classifying functions – polynomials and other functions (Excel)

* Algebraic division

* The factor theorem – Excel file

* Different forms of quadratics – sorting activity

The discriminant

* Investigating quadratic curves

Discussion – The key ideas in coordinate geometry

Coordinate geometry forms a large part of the core mathematics course at AS level.

What are the key ideas in coordinate geometry?

What skills should the students already possess?

What problems do you anticipate that students will have with coordinate geometry?

What do you eventually expect students to be able to achieve?

Straight lines and points

* Using gradient to find points.

What is the equation of a straight line?

* Straight lines and their gradients (perpendicular lines) activity.

* Excel student self test

* Group work activity – what does the design look like?

* Fine tuning – find the equations of the “objects” in each design

Circles and other curves

* Transformations of the unit circle.

* Group work activity – what does the design look like?

* Fine tuning – find the equations of the “objects” in each design.

* Tangents and normals (introducing ideas for differentiation).

Defining sequences and series

* Sorting out the sequences activity

Generating a sequence by description

* Definitions matching game

How many clues do you need?

Arithmetic and geometric sequences and series

* Excel file - the sum of an arithmetric series

* Excel file - the sum of a geometric series

Convergence, divergence and infinite series

* Excel file - convergence and divergence

Escaping right angled triangles

* Circular functions using geogebra

Using trig graphs effectively

* Symmetries of trig functions - activity

* Card sort – true, false, sometimes

* True or false trig solutions cards

Differentiation, a lesson checklist

Before the lesson, students are given the student sheet and spend a couple of minutes reading it.

They should be told that the idea of the checklist is to make sure they are confident that everything that should be covered in the lesson has been covered by the end.

They can tick off the boxes as and when they think that section of the lesson is finished.

They should also read the points to consider and listen out for section thoroughly.

The lesson

A teacher led reminder of how to calculate the gradient of straight line

A demonstration of what is meant by the tangent to the curve using Geogebra(or an equivalent piece of software) and showing that the gradient of the tangent changes as the point of contact is moved along the curve

A discussion of the way that a chord can be used to find a gradient close to that of a tangent – measuring the gradient of the tangent directly using Geogebra is not done as that particular shortcut will not help with understanding that we are looking for a limit

An activity in which the students use Geogebra to draw a polynomial curve (different curves given to different groups) and calculate the gradient of a chord between a fixed point and a variable point further up the curve. The variable point is moved closer to the fixed point in equal steps until the students in the group can predict the limit. Other points are then considered.

A collection of each group’s results with a discussion of what they seem to be showing

A plenary showing (or hopefully summarising) the algebraic method for differentiating a polynomial and a question session to mop up any areas the students do not feel have been covered

Checklist

- Gradients of straight lines
- Gradients of tangents to curves
- Gradients of chords
- The limit of the gradient of a chord
- Numerical answers to gradients of chords
- An algebraic method – differentiation
- Some things to listen out for or consider during the lesson
- Do you know how to calculate the gradient of a straight line?
- Have you been reminded of this?
- What does a positive gradient mean?
- What does a negative gradient mean?
- Which curve is being used to demonstrate tangents?
- Why do you think this curve has been chosen?

What is the tangent to a curve?

When is the gradient of the tangent to a curve positive?

When is the gradient of the tangent to a curve negative?

What is a chord?

Why are chords being used to find the gradient of a tangent?

What is meant by a limit?

Some things to consider during the activity

Why are you finding the gradient of a set of chords from a set starting point?

Why are you making the distance between the start and end point of the chord closer together each time?

Are you able to predict the limit of the gradient of the chord?

Why are you changing the starting point?

Why have other groups been given different curves?

What do these curves have in common with your curve?

Some things to listen out for or consider during the follow up to the activity

Why are each group’s results being placed in a table?

Can you see how the limit relates to the equation of the original curve in each case?

Can you think of a way to predict what the gradient of the tangent will be?

Some things to listen out for or consider during the plenary

Do you know what is meant by ‘differentiation’?

Do you know how to find the gradient of the tangent to a simple curve at a given point?

Could you write a rule that gives the algebraic method for differentiating a polynomial function?

If you do not know the answers to any of these questions at the end of the lesson, please ask about them.

Introducing and teaching differentiation

* Tangents and chords Geogebra activity

* Using tangents and chords to get the idea of differentiation Geogebra and Activ Inspire

* Tangents of curves to check gradient formulae

Untangling the ‘meanings’ of integration

Teaching indefinite integration

* Using tangent fields – Activ Inspire (or Autograph)

Teaching definite integration

Why does the “antiderivative” give the area under a curve?

Plenary

philchaffe@furthermaths.org.uk

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