Regional Project Report Form
Applicant School/College.
Please include name, name of organisation, contact address, telephone and preferred email address.
Enter the name of your organisation, contact address, telephone and preferred email address.
Angela Nicolas, Hornsey School for Girls, Inderwick Road, Hornsey, London N8 9JF; 020 8348 6191;
- A brief outline of your project.
.Please give a brief outline of your project.
You may insert photos if you wish
This project used rich tasks and group activities to help students to deepen their understanding of functions and also to take greater responsibility for their learning and become more open in their thinking and development of problem-solving skills.
- Why did you choose this project? Please outline briefly why you chose this theme for study, your rationale for developing your learning through this project and what you hoped to achieve.
Explain why you chose this project.
The aim of this activity was to use rich tasks, group activities and independent learning strategies to improve the teaching and learning of an A level topic, and thus improve the students’ attitude and approach to mathematical learning.
The topic chosen was Functions as we felt this was a topic that we had taught in a traditional way for a long time. Many students had real difficulty in understanding the intricacies and subtleties of this subject, and this was reflected in their answers to examination questions.
We wanted to help students to take greater responsibility for their learning and to be more open in their thinking, to develop their problem-solving skills and most importantly, to develop our teaching in order to encourage discussion and peer learning.
- Describe what happened as you carried out the project.
Explain what happened as you carried out the project.
We planned to use five different activities over a period of two and a half lessons (2.5 hours) to teach, practise and revise key concepts on Functions from the Edexcel C3 specification.
We consulted the DCSF ‘Standards Unit’ resources as examples of ‘good learning’ and tried to emulate this style of teaching in our activities.
The Head of Department and I worked closely throughout the process to research, choose and plan the activities for the sequence of lessons.
THE ACTIVITIES AND THEIR LEARNING OUTCOMES
Activity 1 – Starter activity on composite functions: We will combine you
We found this free activity on the Kangaroo Maths web-site in the AS/A2 Schemes of Work section. The resource came with excellent teacher notes.
I asked students to define what a composite function was and for the students who could give me an example of a composite function but not a definition, I asked them to think back to GCSE and ‘composite shapes’.
I printed off slides 11 to 20 on A4 coloured paper and scattered them around the room. As students came in, I gave them each a random single copy of slides 1 to 10 on different coloured paper and asked them to consider the three functions (f, g and h) at the bottom of their A4 slide sheet (f(x) = 4x – 3; g(x) = x2; h(x) = 1/(x-2).
I asked students to look for their ‘paper buddy’ – their buddy being the value of their composite function when x = -2. For any student that found their buddy quickly, they were given the task of checking that other buddy pairs were correct and supporting students who had yet to find their buddy.
This was a nice kinaesthetic and colourful start to the lesson in which students were keen to participate! All the slides were conveniently labelled in the bottom right hand corner so that correct buddies occurred if the sum of the two slides was 21 making it easy for me to check the answers!
Activity 2 – Problem-solving functions activities
I found this activity from a published resource. The flexible, problem-solving activities are designed to encourage pupils to interpret and analyse mathematics, work collaboratively and communicate decisions. Each activity provides a problem with twelve possible solutions. When a correct solution is clicked on, a letter appears (incorrect solutions are indicated by a cross). When all possible solutions have been found, the letters can be rearranged to find a key mathematical word or a Mystery Mathematician. Variations on this theme include 'find a friend', 'find the logic' and 'find the group'. All activities are also available in PDF form.
Students were given the composite functions activity and inverse functions activity and were asked to work in groups, discuss and find the solutions. Students found the inverse functions activity challenging as they had to recall their graph sketching skills from AS Maths. Also students had very different ways of finding the inverse function algebraically. This activity led to the most intensive mathematical discussion.
Activity 3 – Group work on multiple choice activities
From another published source we selected a multiple choice activity; the students worked as a group to answer the questions.
The question on range took the longest to do, as students had to think for themselves; they realised that making a quick sketch of the function would be a useful strategy. A further question that provoked discussion was a quadratic function that needed careful solving. This activity involved students having to reason to find the correct answers, to confront difficulties rather than avoid them and to develop mathematical language through their communication with one another.
Activity 4 – Assessment activity on Functions
From the same source we selected an activity consisting of 9 questions which really delved into misconceptions that arise in considering functions. Students found the first question most difficult as it forced them to separate and understand the difference between mappings and functions.
Other questions linked composite and inverse functions e.g. given f(x) and g(x), find (gf)-1 and f-1g-1(x). Again this activity forced students to understand, separate and then combine their knowledge of inverse and composite functions.
Activity 5 – Activity on composite functions that cannot always be evaluated
For this activity, students were put into pairs with a printed copy of slides 1 to 10 (from the starter activity) and also slides 21 to 30. Students had to pair up composite functions (slides 1 to 10) with their solutions (slides 21 to 30) for x = 2. The functions f, g and h were written at the bottom of each slide.
Most students were soon able to realise, understand and explain to one another and to me why some slides on the worksheet were blank, with little help from me. For example, one of the functions, namely 1/(x-2), could not be evaluated for x = 2 due to the restricted nature of the domain.
It was interesting to see that some students preferred to find the composite function algebraically before substituting in the x = 2 value and how some automatically substituted in x = 2 into the starter function before taking their answer and placing it into the second function. There was excellent discussion between the students about which way was right/wrong and which was easiest/hardest. It raised questions that forced students to explain and justify their thought-flows.
- What did you actually achieve as a result of doing this project? Please include both qualitative and quantitative results where possible to substantiate your claims.
Explain what you achieved.
Student evaluations showed me that students had enjoyed the activities and learnt a lot through the team activities. The students said things like:
‘I can now link functions to their graphs quickly’
‘I enjoyed the teamwork with people I have not worked with before’
‘I now know why my calculator says error for 1/0’
‘I thought multiple choice activities were easy at GCSE but now it seems that all the answers could be right and I have to think hard to cross off the wrong ones’
‘I think I can manipulate any function now – rather than functions manipulating me’
Our combined teacher evaluations showed me that we had reached our aims: teachers had witnessed students taking greater responsibility for their learning and being more open in their thinking and developing their problem-solving skills. It was certainly the case that there is a direct correlation between the number of probing questions that the teacher asks and the number the students then ask of themselves and one another!
‘All students were able to access the learning materials but were still challenged. Students were thoroughly engaged, working in twos and threes, checking one another's answers, referring to examples in their notes and text books. Students were encouraged to justify their answers to one another. Angela prompted students with the lightest of touches. Probing questions facilitated learning. The range of activities reinforced understanding. Teaching and learning resources and techniques were very similar to those based on the DCSF standards unit’.
- What impact has the project had on:
- you and your teaching;
- your pupils/students and their learning;
- colleagues in your school/college;
- parents, governors and the wider community?
Explain what impact the project has had.
These activities and the students’ responses to the activities both in their evaluation and in the actual lesson have reminded us that we need to be a facilitator of learning in the A-level classroom if we are to nurture strategic problem solvers as well as examination passers.
In our discussions, we also considered how it is almost too easy to show and tell students what to do in the A-level mathematics classroom and present mathematics as a series of algorithms or, worse still, chapters of sums that need to be completed – something that I am sure we are all guilty of with the pressures of daily teaching.
The pressures of exams tend to lead us down the path of ‘chalk and talk’. This activity took considerable planning and thinking time but the thinking, teamwork, negotiation and evaluation skills the students developed will serve them well for future mathematics lessons and the mathematical world around them.
Remember to include the evidence on which you are basing your claims!
- Did everything go as you expected?
If it didn’t, why was that? What would you change if you were going to do a similar project in the future?
The project progressed as we expected. The only thing we would change is to give it more time (about an hour) to give students more time to reflect and discuss and maybe ask each group of students to prepare a group presentation on their knowledge of functions.
- Who was involved? Please provide information about colleagues involved in the project. Did you find any research, enquiry or resources useful in supporting your project - if so, what? Explain who was involved.
Three teachers were involved in the project in school (the Head of Department, the Deputy Head of Department and me - Head of Key Stage 5). Outside of school, we worked with the Head of Mathematics at Quintin Kynaston School, who advised us and supported us throughout the project.
- What did you find useful to support your project?e.g. research, previous National Centre funded projects, resources, people, etc.
The resources referenced in this project were very useful in helping us design the mathematical activities on functions.The discussions we had with one another were paramount in designing how the students were to learn about functions. We decided very early on that the activities would be investigational in nature so that the students could lead themselves and be facilitated by the teacher.
- Who have you told about your project? Please provide dates, names and school/college/organisation of anyone to whom you have talked about this project.
We shared our work with the Head of Mathematics at Quintin Kynaston School and are using the outcomes of this project to revise and review teaching and learning styles at Key Stage 5 in our learning area.
- What advice would you give to any other group of teachers doing a similar project?
The advice I would give to other groups of teachers is, firstly, to do a grouped teacher review on learning in the chosen Key Stage of the project first to bring to the surface the outcomes that need to be achieved. For example, we wanted to develop our Key Stage 5 learners’ investigational and discussion techniques and this gave us the focus for the activities that we chose.
Secondly, it is important that the outcomes are shared and discussed, not just in the departmentbut also as a learning tool to other teachers of that Key Stage in other subjects across the school. Once the students have developed a learning skill in one area (eg discussion, investigational learning etc) this can be applied to other subjects to develop their intra personal skills.
- What will change as a result of this project? e.g. curriculum, organisation, pedagogy.
As a result of this project we will be reviewing our Schemes of Learning to incorporate as many investigational activities at Key Stage 5. These activities should encourage students to become more open in their thinking and therefore develop their problem-solving skills and take learning away from the more traditional teacher-led knowledge.
We will also be incorporating more structured time in our department to share, discuss and record our learning experiences from activities we undertake so that the outcomes of the project have a ripple effect throughout the year.We will share this knowledge with other teachers of Key Stage 5 in other subject areas as appropriate
- How would you like to develop the work of this project in the future? e.g. curriculum developments, further funded projects, training for teachers. How could other National Centre funded projects support you?
We would like to develop this work by considering the other Key Stages in addition to Key Stage 5. With the continual changes to examination structure (modular, linear, coursework), it will be necessary to constantly develop students learning styles to maximise their chances of success. In terms of Key Stage 5, I would like to work with other teachers to develop our Further Maths AS curriculum as it is a course we have recently introduced.