'The only way to avoid the formation of entrenched misconceptions is through discussion and interaction. A trouble shared, in mathematical discourse, may become a problem solved.' D.Wood (1988) 'How children think and learn'

It is therefore essential that we offer learners opportunities to share and discuss what they are working on with one another and to encourage them to work collaboratively. The 'Improving learning in mathematics' box takes collaborative learning as its base line and the approaches that it advocates are intended to get learners and teachers working together collaboratively and communicating about learning in mathematics. The book that comes in the box is short and an easy read and worth every minute that you will spend on it. A copy of the book can be downloaded by clicking on the link in the section above.

There are many different activities in the box which you can use to get started and their place in an A level scheme of work is clear. There are also many different accounts on the NCETM portal about how teachers have used the activities to encourage discussion and collaborative working ( see NCETM Annual Conference - Workshop: Raising and improving the standards in mathematics through active learning and

Using Improving Learning in Mathematics with diverse post-16 groups to change their relationship with mathematics

The 'Improving learning in mathematics' pack also has an excellent CPD section. There is a section which suggests activities that teachers can use before trying out the activities in the classroom. Click here to download this section.

There are many excellent activities in the pack which typify the approaches for some which fit well with the AS and A2 syllabus see:

Matching functions and derivatives

Using binomial probablities

Interpreting distance time graphs

Part of the pack contains the software to make many of the activities. It is easy to use and is called Tarsia - it can be downloaded free from the internet. Alternatively look at such web sites as Kangaroo Mathematics where there are many examples of activities which will encourage collaborative learning.

It is one thing to agree that discussion is essential to learning but another to encourage it in your classroom. If your learners are new to discussing mathematics you may find that quality discussion does not come easily at first. Some ground rules may need to be established and some activities used which help discussion to take place.

Case Study - Developing discussion with Further Mathematics students

A teacher was working with a group of AS students - all female and all Bangladeshi. The girls did not find it easy or comfortable to discuss their learning or opinions. The teacher started by using several strategies for encouraging confidence:

• Using mini whiteboards to show answers to questions. This is the least threatening way to begin to discuss and the girls become used to contributing and find out that their answers are treated seriously and that often there is more than one right answer
• Using a poster activity in groups. The girls have to communicate in order to create the poster, they only have one pen so they have to communicate. It is not easy the first time they try it and the teacher encourages them with prompts and spends some time talking to them about how they can tackle the task.
• Using some of the approaches from the Standards box. For example the girls are given a series of statements about matrices and are asked to decide if they are - always, sometimes, never' true - again this cannot be done in silence and gradually the girsl get used to discussing meaningfully what they are doing.
• The teacher works with the girls when they are discussing and uses such prompts as, 'That's a good point - say that again', 'If you don't think that's OK you have to say - how could you do that?', 'What about you - I've noticed that you are not saying much - what do you think?'
• Over a period of weeks the girls get better at discussion.

' After two months I can't keep them quiet but my proudest moment was when they were at a revision day and contributed more than other students and were prepared to argue the toss with the visiting lecturer as well as asking for clarification.'

It takes time to develop good high quality discussion. There is an excellent PD session in the Standards Box called, 'Managing Discussion' it encourages teachers to:

• experience discussion
• reflect on how discussion can be used to promote learning
• explore the characteristics of purposeful discussion
• explore the management skills that are needed to implement purposeful discussion

To download the unit (PD4) click here

See also Mathematical Discussion

Mathematics uses words,symbols and images in precise ways and it is important that we expect this precision from our learners. Using mathematical language does not come easily and it needs to be encouraged from an early age.

The approaches used by the Standards Box help to develop this language if the teacher is aware of the need to model the language required. Take for example the task 'Linking properties and forms of quadratic functions' (to download the task click here) this requires students to match up equatioins, properties and graphs. If the teacher works hard on the vocabulary in her questions and in the prompts then the learners will follow suit.

Doing poster activities are invaluable to develop precise language as learners will have to express themselves in terms that allows others to grasp exactly what is being meant. Similarly getting students to write notes from a chapter or asking them to write instructions for explaining a set of operations will develop not only their understanding but also their ability to commincate precisely.

See also

Volleyball NOT Ping Pong

Reading is something that students need to do if they are to develop the ability to use mathematical language well. The Maths Association has a list of excellent mathematics books suitable for sixth formers on its website www.m-a.org.uk . Another excellent source of reading for sixth formers is the online magazine Plus Magazine. http://plus.maths.org/ . Textbooks themselves can be used creatively as well with reading being set for homework and used as a focus for a class discussion. See 'Making the most of your textbook' - Laurinda Brown in 'Aspects of Teaching Secondary Mathematics - Perspectives on Practice' - Ed Linda Haggarty.

See also Connections between topics both within and beyond mathematics and the real world - AS and A2 and Reading about Mathematics