Maths Without Limits
Opening Young Minds to Endless Possibilities

## Using Dialogue

Many teachers will introduce a new idea in a Maths lesson by presenting a problem and then getting the class to suggest possible answers. These will be discussed and the teacher will lead the class to the correct answer.

The problem with this approach is that only a few of the pupils are involved. Others listen to the debate without really engaging with it, simply waiting to find out at the end of the discussion ‘how you do it’.

What is needed is an approach where, instead of telling pupils whether their answers are right or wrong, the teacher challenges the pupils to engage with each others’ suggestions and develop their own understanding of the rightness or wrongness of each idea put forward.

Benefits

The key benefits of this process, compared with the typical question and answer approach are as follows:

 Traditional Question and Answer Teacher-led Dialogue Only a proportion of pupils are fully involved. Less confident pupils opt out of the thinking because they have not understood. All pupils are challenged to participate in the thinking process. The teacher is aware of the fact that some pupils are finding the discussion stimulating and some are finding it difficult or appear disinterested. The teacher is fully aware of the level of understanding and thinking blocks of individual pupils. The thinking goes at the pace of those who are participating in the discussion. The thinking goes at a pace which allows all pupils to remain on board in the learning process. Learning is superficial, dealing only with the problem presented. Learning is deep and engages with a whole raft of related concepts and ideas.

How it Works

There are two main ingredients in this process

1   The “Why” Question

• The teacher puts a question to the class.

• When a pupil gives an answer to the question the teacher does not declare whether the answer is right or wrong.

• Instead, the teacher asks the question “Wny? Why do you think that?” and gets the pupil to make their thinking explicit.

• The teacher then follows up the explanation by asking another pupil by name, “N____, What do you think?” and again asks “Why?”

• The teacher gets the buy in of a less confident pupil, (who was perhaps already beginning to tune out and lose interest) by again asking “N____, what do you think?” and again asking “Why?”, supporting this pupil in verbalising whatever understanding they might have.

• The teacher gets the input of other pupils as necessary.

• The teacher gets the general agreement of the whole class or group on whether the new idea should be kept or discarded.

2   Playing the Daft Laddie or Daft Lassie

• In the course of the discussion the teacher makes a suggestion /puts forward an idea which is clearly wrong and asserts “that’s right isn’t it?”

• If the class is unresponsive or acquiescent (being unused to the teacher ever saying anything that is wrong!) then the teacher spells it out more carefully, making the nonsense of the idea more obvious.

• Pupils challenge the teacher with a chorus of “No”s.

• The teacher responds in pantomime style with a questioning “Nooo??”

• Pupils then eagerly explain why the teacher is wrong and what they think the correct idea is.

• The teacher can then act further as daft laddie or lassie to elicit further ideas, or come out of role to clarify the current agreed thinking with further use of “Why?” as explained above.

Note on Effective Use

As with all teaching tools, the teacher-led dialogue process outlined above should be used in combination with other teaching methods. If used to excess in every lesson, it would make learning very tedious and laborious. Each teacher will want to develop their own style, assessing for themselves and for the group of pupils they are working with, when the process should be used, and the best combination of the two ingredients. However, if used well and in appropriate measure, teacher-led dialogue can have a radical effect, creating a real engagement in the learning process among less enthusiastic and less confident pupils and taking all pupils, able and less able alike, on a journey of discovery together.

Four Activities

The four activities which follow are designed to support many of the core mathematical topics being taught in class with pupils in the middle years in Primary School.  All make use of the pupil-led dialogue approach.

They are intended for use in a spiral fashion, with each activity being returned to regularly at different levels with the same group of pupils. The same activity could be revisited several times over successive days, or returned to at different points throughout the year as different teaching topics are introduced. Most activities can also be adapted for work with older or younger age groups, so could be revisited with the same pupils over a number of years. With each new use of the activity, different ideas can be explored and the activity can be developed to accommodate different levels of understanding.

The activities are thus presented with an ‘initial problem’ and then with suggested lines of development. These should not be seen as prescriptive, but simply as pointing the way to creative avenues of exploration. Which avenues are explored, and in which order, will be up to the individual teacher and will depend on the needs of the particular group of children with whom he or she is working.

Some of the activities are specific to a particular maths topic (Eg ‘Numbers to 100’) and therefore should be used when that topic is being tackled in class. Others, such as ‘How many each and altogether?’ can be recast to support a number of different topics – in this case, multiplication and division – and so will be revisited in different contexts.  They should not be consigned to a special problem-solving lesson!

The principle behind all the activities is the same:

• Begin with the whole class in a giant group by working ‘big’ using large scale equipment – hoops, bean bags, metre sticks, class whiteboard etc.

• Then move to using smaller scale equipment with the pupils working in pairs at their own desks. (Hoops and bean bags can be replaced with jam jar lids and counters, metre sticks can be replaced with individual number lines, class whiteboard can be replaced with small dry wipe boards)

• Finally move to the pupils working individually, drawing their own diagrams, tables etc in their jotters or workbooks to support their own learning.

It is desirable that children, through working with problems, develop an awareness of different problem solving strategies:

• Use a diagram

• Make an organised list

• Make a table or chart

• Look for a pattern

• Etc

Most interesting problems or investigations will require the use of several of these. The teacher should point these out by name each time they become useful and encourage the pupils to use the well-known terms when discussing the strategies that they decide to employ.

Key Equipment

The activities rely on the teacher having certain basic equipment. Most of this will be found in any primary school. A couple of items may need to be specially ordered.

Equipment from the gym cupboard:

• Large plastic hoops

• Bean bags

• Coloured stacking cones in three colours

Equipment from the maths cupboard (some home-made):

• A set of large numbers which can be stood on (eg on cork tiles) – 0 to 20.

• A copious number of red and white Metre Sticks – enough for one for every two pupils in the class. These can be ordered from NES Arnold. They have the particular feature of having a marking for each number from 0 to 100, but only having the numbers 0, 5, 10, 15, 20 etc actually written on the stick.

• Counters

• A series of jam jar lids or similar (enough for 4 between each pair of children) – small pieces of paper can be used if necessary.

Items from the stationery cupboard:

• Small envelopes

• Plenty of paper

## Thinking Teams

Once pupils have had some experience of teacher-led dialogue, you can introduce collaborative thinking teams.  As with whole-class dialogue, the objective is to engage all pupils in the thinking process.

Pupils find it easier to express their ideas in a group than they do in whole class discussion.  They have more confidence voicing ideas and there is more opportunity for each individual to participate.

Collaborative thinking teams can be used in any context where you want pupils to engage with written material.  They can be used for tackling a page of a Maths textbook, or with materials specifically designed for group work.  They lend themselves especially well to the online investigations.

The benefits of group working are multifold:

• There is no room for passengers.  Pupils cannot hide from each other in a group!  Everyone has to participate in the thinking.

• No-one gets left behind.  The group has to make sure that everyone understands so everyone can keep up with the group’s thinking.

• Thinking can be recorded.  Pupils can explore ideas initially using individual whiteboards and then record their ideas more permanently in their exercise books or jotters.

As with many kinds of team working their are several different roles which are allocated at the start:

• The team leader makes sure the group is on task and that everyone is included in the thinking.

• The monitor makes sure the chairperson is doing the job properly.

• The recorder reminds the group to record their thinking or answers.

When starting out with the process, you may wish to use the ‘group thinking diagram’, which shows the 6 steps in the thinking process: Read, Suggest, Discuss, Agree, Record, Check.  An A3 copy goes in the middle of the group’s table with a ‘mascot’ on it that walks around the diagram as the discussion progresses so that everyone (including the teacher) knows which step they are on.  This can be dispensed with once children are comfortable with the process.

When introducing the diagram it can be useful to have a conversation with the class about the kind of language that might be used by the chairperson or recorder in each step, in particular, the two ‘thinking questions’ which the team leader can use:

• “What do you think, N, about M’s idea?” ( this ensures that everyone is listening actively to others’ ideas and that the members build on each others’ thinking)

• “Why do you think that, N?” ( this forces speakers to think more deeply and to explain their thinking to others)

It is also worth discussing with the class the huge opportunity given by ‘wrong answers’   It is when someone in the group has a different answer or idea that the interesting discussion takes place.  The group has to investigate the different idea or answer to find out whether:

• one person got the wrong answer or idea – if so, HOW and WHY?

• both answers or ideas are right

• the ‘wrong answer’ is actually right and everyone else is wrong

To make it clear to pupils what you are looking for in their discussions, you should model the process for the pupils.  Gather the pupils together on the carpet and give them individual white boards and pens.  You act as team leader and draw pupils’ attention to your use of the two ‘thinking questions’.

During the class discussion, observe carefully to identify the most confident pupils for the different roles.  Ask for volunteers for the team leader role and select from these for the number of teams you want. (Teams of six work well.)  Select children to play the other roles and group pupils strategically!

When trying out the teams, ensure that the pupils are sitting close enough to all be fully involved in debate.  This will probably involve grouping pupils around one end of a standard table of six.  Use some simple subject matter which the pupils are familiar with, such as a page of a Maths textbook.   Teams discuss each section of the page, record their answers individually, and then mark them together before moving on to the next section.  Circulate and spend quality time with each group in turn, listening in to their discussion and modelling the language that the team leader, monitor and recorder need to use to ensure everyone is staying on task, participating maturely in the discussion and recording what and when they are meant to be!

Stay relaxed if it doesn’t seem to be working!  Calmly call the class to order and involve them in discussion about what difficulties teams are having with making the process work effectively.  Praise the teams that have been having success, agree the way forward and try again.  Repeat this as necessary, stick at it, and trust that your pupils will make it work!

Once you have it working well, ask for feedback from the class as to what they like about this way of working and ask them for suggestions as to how the process could be improved.

Good luck!  Have fun!