Deepening mathematical understanding for 'A' level students
Students will be able most effectively to use – and build on – their mathematical knowledge, both at university and in the work place, if they have a deep understanding of the topics and techniques and how they relate to each other, rather than just knowing ‘tricks’ to get the right answers in their AS- and A2-level examinations.
Many materials designed to promote such deep understanding can be used at a range of levels. (For example, the Bowland materials were originally developed for Key Stage 3, but are often used effectively at Key Stages 4 and 5, too.)
The key to effective use of these materials is to give the students time and stimulation for exploration, active learning, discussing and explaining to each other: not just to focus on ‘getting the right answer’.
Some teachers find this challenging – fearing what might happen if they cannot answer the resultant questions – so we start the Resources section below with professional development resources to help develop appropriate teaching techniques and confidence.
Students’ mistakes can be used to build understanding, as shown in the NCETM Departmental Workshop on Use of mistakes and misconceptions to increase understanding.
The NCETM Departmental Workshop on Effective day-to-day provision for able, gifted and talented students has ideas for working with high fliers.
This report from Keele University describes how using interactive whiteboards can deepen students’ understanding.
Improving Learning in Mathematics (ILIM) contains CPD activities, including CDs on Thinking about Learning. Here a teacher describes why he likes these materials and how he uses them.
The Mathemapedia article Using rich collaborative tasks – AS and A2 discusses how rich, collaborative tasks can address many AS and A2 students’ problems and suggests sources of such tasks.
The FE Magazines suggest many sources of subject knowledge support.
In the Further Education section of the What Makes a Good Resource microsite you will find favourite resources shared by teachers, along with explanations of how they used the activities in their classrooms.
The microsite, Raising Achievement in GCSE Mathematics, also offers teachers the opportunity to share favourite resources and explain how they have used them for deepening mathematical understanding. They have selected resources from NRICH, Bowland, ILIM and AMP Nuffield. These resources can be used effectively with ‘A’ level students, too.
The Mathematical Moment in FE Magazine Issue 26 considers Functions and their derivatives. This activity helps students make connections between algebraic and graphical representations of functions and their derivatives.
Have you considered encouraging able students to follow the Further Mathematics AS or A-level?
The Further Mathematics microsite includes Ways To Offer Further Mathematics.
Explore a piece of mathematics
Braids was inspired by a PhD student at Exeter University.
The topic is not directly on most syllabuses, but provides a thought-provoking activity that can be used for maths clubs or as an introduction to group theory, encouraging the development of problem-solving techniques.
It has been said by some recently that ‘teachers learn best when they learn from each other’. If you know of another link to an activity or resource on the NCETM website that you think would be useful or appropriate for readers of this guidance, please use the comments box below to let us and them know of your idea.
Several NCETM communities
are used by teachers of ‘A’ level to discuss new ideas and current development. For example, the ICT in Mathematics community
has a thread on effective classroom use of iPods and iPads and another one on Interactive Whiteboards (IWBs).
“Activity by itself is not enough; it is the sense that is made of it that matters” (DRIVER, R. (1983) The Pupil as Scientist?, Milton Keynes, Open University Press.
For the things you have tried out for yourself in your own classroom to become useful pieces of professional learning, there is a need to capture them, reflect upon them and remark on them. The NCETM Personal Learning Space (PLS) allows you to do this.
Use My Learning Journal to collect your thoughts and reflections as well as to log actions; documents can be attached to your entries. You could do this now by visiting your own PLS.
Use My Favourites and Notes to take note of and organise interesting things you have found (like this page) and want to return to easily in future.
Use the Sharing and Contacts facility to share elements of your PLS with colleagues, selecting them from your own list of contacts
Use the “Request a reminder” function
Find out more details of these and other functions of the PLS.