Paired and group work
for
secondary school
students
in
mathematics
Roger Ray
Teaching and Learning Consultant
School Improvement Service
East Riding of Yorkshire Council
One of the most successful ways for students to learn is through peer talk
and interaction. How often have you seen the latest craze sweep through a
school like wildfire? I never mastered the Rubik’s Cube but remember
many students who could solve the puzzle with amazing speed and
dexterity. Who taught them the skill? It certainly wasn’t me. Peer
interaction is a great motivator where pupils communicate within their own
levels of understanding, using their own familiar common language.
Vygotsky described parents of children, who frame or simplify a child’s
immediate world in such a way, to facilitate learning, as mediators. As
teachers of mathematics we do this by setting up scenarios and
conjectures carefully framed for students to explore within their present
level of knowledge, perceptions, prejudices and experience. As mediators
we intervene to help students make sense of a situation especially when the
experience is just beyond their present level of understanding. Vigotski
also went on to describe changes in mediation for students, as they get
older. By adolescence the main mediators are their peers. Although they
still do some work developing their thinking for themselves, they often see
or hear a peer complete or display a skill which is just beyond their own
competence level. They will then, often with discussion and interaction (and
sometimes without) make the skill their own. The small steps in the
development of skills, mental, practical, oral and social are taken from the
outside the social arena that adolescents share with their peers, to inside as their own possessions. In discussions, the more able students (in
particular skills) benefit from explaining and externalising their ideas and
understanding to other students, who, in turn, benefit from peer
interaction with students working above their present level of skill or
understanding.
If you are still not convinced of the power of this type of activity, consider
the following ‘hidden rules’ when pupils work with a teacher.
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Never interrupt a teacher when they are explaining something (even
for clarification) they might think you are rude.
After the first explanation by a teacher, it is okay to ask if you don’t
understand, but if you still don’t understand after a second
explanation, just nod and try and sort it out yourself or ask a friend.
These are common ‘hidden rules’ discovered by researchers talking to
students about classroom interactions.
Of course students being allowed to talk to each other in lessons instead of
listening to a teacher may take some teachers outside their normal style of
working and indeed their comfort zone. As teachers we often believe that
absolute control is paramount and indeed it often is particularly in the
earlier part of our career. So how do we maintain control and yet give
students more time to discuss ideas and engage in purposeful talk?
Paired talk
Imagine looking at a colleague working in a classroom through a window,
unobserved. You see the teacher in control at the front of the class asking
questions. The questions are followed by students raising their hands to
answer – sometimes a few hands are raised – sometimes lots. Sometimes
after a poor showing of hands the teacher appears to ask another question
and more hands are raised before a student is selected to respond. Some
students never raise their hands and some students never get the
opportunity to give their response. The teacher seems unaware of the
variety of answers that may be available effectively only ‘scratching the
surface’.
It is a common image and since the teacher is in control and some pupils are
responding, learning appears to be taking place. But is it - and is it learning
for all? Have you ever seen (or dare I say experienced) the situation when
after such an activity lots of students are unable to engage in the activity
that follows or cannot attempt the homework based on the lesson, whilst it
appeared that the question and answer session was successful?
The
question and answer session is often described as ping-pong (table tennis to
the more refined) where a question is asked by the teacher and a pupil
responds with a return followed by another teacher question. The model I
want you to think of is not ping-pong but volleyball. Here the server sends
across the ball and participants on the other side pass the ball between
them to set up a quality return. This is how paired talk can work in
mathematics. Is it reasonable to expect instant responses to questions in
mathematics? Quality responses come after thinking time and often
discussion.
Imagine asking this question to a class.
“Why when I double both the numerator and denominator of a fraction
does it not get twice as big?”
The best response I got from a pair of Y7 students was that if you make
the denominator twice as big you make the fraction half as big so you need
twice as many pieces - so you double the numerator.
It was followed by questions from other pairs of students asking for
clarification. The pair of students did this using diagrams and examples.
This was one of the answers taken from pairs of students after two
minutes’ thinking and discussion time.
Two important ideas lie within the last statement.
 Give students an allotment of talking /thinking time (and stick to the
timing).
 Take responses from several pairs of students (often without
comment).
A warning here in terms of control – once students become heavily involved
in discussion it may be hard to regain their attention without shouting.
Shouting tends to make even more noise and may give the impression of an
unruly class to passers by (remember our image through the window from
earlier). An agreed signal to end discussion or even better a countdown to
allow students to finish their sentence or make a final point is useful. In
longer discussion periods, some trainers/teachers use a piece of music to
signal the end of discussion time.
Asking students a good probing question followed by saying something like
‘thirty seconds with a partner – come up with an answer or idea’ will give:
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students a change of activity
students a chance to talk
the prospect of some quality answers/ideas
the teacher a break and time to plan the next episode in the lesson.
Often when observing teachers I get the feeling they are working very
hard but the students are not. Paired work can resolve this imbalance.
Setting up paired work
If you are intending to use paired work make sure students are in pairs at
the start. ‘Isolates’ are easier to place early on as they arrive by inviting
them to sit with……because you’ll need a partner - rather than during the
lesson.
Friendship pairs work usually, but you may want to encourage students to
work with different students or in boy girl combinations. Warn them of
this don’t spring it upon them and if possible justify your reasons. If you
number mini-whiteboards, repeating each number, you can ask students to
find their matching number. If you number from one to thirty, you can ask
complements to 31 to pair off e.g. 1 and 30, 2 and 29 etc.
By careful arrangement you can ask pupils to pair off with someone behind
them or in front of them instead of next to them. In this way students
interact with different people.
If students are not used to working in this way it may take a while for them
to acclimatise – persevere, talk to teachers from other subjects who work
in this way with your students.
Taking responses
Since students have worked in pairs, you have halved the number of
possible responses but they should be of better quality. Trying to take
responses from all pairs takes time. You can speed this process up
however, by asking for written responses on mini- white boards and then
pairing similar responses or inviting comments from particularly good
responses (since you can scan them all). Remember white boards give
maximum feedback and help your assessment for learning but writing on
pieces of scrap paper can be just as effective. Thumbs up to agree,
thumbs down to disagree or thumbs horizontal if you can’t decide, are good
ways for all pairs to give feedback.
Remember our statement earlier “Why when I double both the numerator and denominator of a fraction
does it not get twice as big?” You could follow this with the question –
“Is this always true?”- followed (after paired discussion) with a thumbs
response.
Planning
With practice most teachers can work flexibly, slipping with ease from the
usual questioning set up, to paired responses. Planning can help in the early
stages particularly with phrasing questions that will, promote thinking and
discussion and probe understanding.
Pupils working in larger groups
The main benefit of group and paired work is students discussing their
thoughts and ideas.
In this way students externalise their ideas,
misconceptions and clarify their thinking. They also have the opportunity
to experience how other people think and work – this will facilitate
opportunities to begin to explore metacognition (thinking about thinking) in
plenary sessions – a vital step in development of learning.
The ideal, in my experience in mathematics, is students working in groups
of four. This often works best when an initial question or problem is
presented to pairs of students who, after considering and discussing their
ideas, then join other pairs to discuss in groups of four.
Beginning an activity in groups of four, with students unfamiliar with group
work can initially be unproductive e.g.
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Students work as individuals not as a group
One student dominates, does all the work and makes all the decisions
One student seems isolated and does not contribute - becoming a
passenger
Pupils argue, fail to listen to others and cause resentment
A whole school guidance policy on students working groups way may be
useful and if group work features across the curriculum frequently,
students should become more familiar with how to work in this way.
However, you may initially have to agree and prominently display a few
simple guidelines or ‘golden rules’ for group work.
 Always listen to other people’s views and ideas.
 Respect other people’s views – to disagree politely
is okay - but give reasons why.
 Discuss don’t argue (debate).
 Decide how you will show that you want to speak.
 It is okay to change your mind.
 It is okay to agree with people and praise their
ideas.
 Jot down some key ideas.
 Note how long your group has got for the
discussion or task.
 Make some decisions that you all agree on.
 Keep focussed on the task.
 Group work is team work – work as a team.
 Keep reviewing what you have done … so we’ve
decided that…….
 Prepare for how you will let your teacher or class
know what you have done or decided.
Classroom layout
Classrooms come in all shapes and sizes - so do desks, chairs and even
students. Some teachers rarely move furniture and often with good reason
if another classroom is below the one you are working within. Classrooms
often have an ‘owner’ and if you are ‘visiting’ the classroom, protocol may
have to be observed before rearrangement takes place. However, surely if
you are timetabled in a room you should decide the arrangements to best
suit the teaching style and activities matched to your lesson.
Below (Fig.1) is a typical arrangement for a classroom. Paired work can
easily take place here. The teacher is the larger circle standing in front of
the board and to the side of the teacher’s desk
Fig.1
Notice in Fig.2, how by simply inviting the first and third rows to turn
around, pairs can move to fours for discussion. Notice where the teacher
is now standing to have an overview of the groups. This is fine for
discussion where a limited amount of paper and equipment is needed and
where students are required to quickly return to the more formal seating
arrangement.
Fig.2
A more radical arrangement might be as below (Fig. 3). Here students have
their own workspace and two easily arranged ways of working in different
pairs with minimum movement are possible.
Fig.3
To work in ‘intimate’ groups of fours students could move to positions as in
Fig.4 with two tables on which to place materials.
Fig.4
Whatever arrangements you make try and consider
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Do the students have enough space for the activity?
Can they see the board and you comfortably?
Can you see every face?
Could you still evacuate the room quickly in an emergency?
Taking feedback
feedback
This can be done in a variety of ways.
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Select one group to describe their ideas and work. Invite other
groups to add new information or question the work of a group.
Give each group an acetate sheet and an OHT pen and ask them to
summarize, in one sentence, their work so far. Display the sentences
of several groups on the OHP (these can be overlaid) and discuss.
Use A3 or larger flip chart paper for groups to jot down ideas and
progress –groups swap or share sheets or display them. Groups can
be given time to question ideas or read other groups’ jottings.
Groups write ideas ‘post-its’ and post them on a board for other
groups to view.
Other ways of working
Rainbowing
Students work in groups of 4 and discuss a problem or task. Each group
member has a colour or number (this could be the colour of a home made
mini-white board or a number on the board). Only four colours or numbers
are used. After discussion new groups are formed getting together
students with the same colour or number. This means that in a class of 28,
new groups of 7 members are formed and the findings or ideas from each
group of 4 can be shared. Pupils can then return to their original groups of
4 armed with new ideas. You could of course begin with 4 groups of 7,
each with a colour of the rainbow and then form smaller groups of 4 in the
same way.
Jigsawing
Groups of students work on different parts of a problem e.g. planning the
cost of a holiday or building a house. Groups then share information with
the rest of the class to complete the jigsaw. This can be done at a simple
level e.g. students exploring 2D shapes – Which have line symmetry? Which
have rotation symmetry? Which can have obtuse internal angles? Which for
a given perimeter give the biggest area? etc.
Snowballing
Paired work combines, after initial discussion, to become work in groups of
4. The groups of 4 then share ideas with other groups of 4 or become a
group of 8.
Envoying/Experting
One person (an envoy) from a group of say 4 students moves to another
group to share their expertise or the group’s ideas, with another group.
Notes or jottings should be taken with the envoy.
Listening Quads
In groups of four, one student takes the role of speaker and explains a key
idea from the lesson, without interruption. Afterwards the listeners can
ask for clarification or correct misconceptions. Examples from a session on
fractions might be: How cancelling a fraction works; which fractions are
easy to add and which are not; the proper thing to do with top heavy
fractions! This is a good pre-plenary activity to allow students to clarify
their understanding before you ask them probing questions.
Planning a lesson
I was asked to deliver a handling data, demonstration lesson to a Y7 middle
to low attainment group in a school, fairly early on in the introduction of
the KS3 strategy. Here is my lesson plan (always in greater detail than in
reality since teachers have to observe and make sense of the planning
process).
Objectives
 Given a problem that can be addressed by statistical methods suggest
possible answers.
 Decide which data would be relevant to an enquiry and possible sources.
 Plan how to collect and organise data.
 Construct diagrams to represent data.
 Interpret diagrams and charts.
 Compare two sets of data
The conjecture used in the lesson is from section 4 of the framework for
teaching mathematics Y7, 8 & 9 page 248.
Equipment/resources
A tabloid and broadsheet newspaper.
Photocopies of articles on the same news item from each of the above enough for one of each news item per four students. (Identify each news
item as only A and B not tabloid and broadsheet).
Paper – lined, plain and graph.
Mini white boards and pens or scrap pieces of A4/A5 paper.
Settler activity
As students arrive direct them in pairs to the challenge on the board.
‘In pairs I bet you can’t write down the names of ten newspapers’
Starter activity.
On white boards pupils write down how many newspaper names they could
think of. Ask pairs with same number on their boards to group together.
How does this help us view the class results – why?
Can we improve on this? In pairs, 1 minute to come up with an idea?
Collect ideas (pupils display ideas on back of mini-white boards).
Make a human bar chart with the white boards each column of boards
having the same number.
Try a human pictogram.
One person from each pair to stand with others in a circle making a circle
of the mini-white boards (numbers the same on the whiteboards together).
What type of chart are we making? (a sort of pie/doughnut chart)
How is it different from the other charts? (give thinking time)
Main activity
Explain and display the difference in appearance of tabloid and broadsheet
newspapers. Record and ask students to say the words ‘tabloid’ and
‘broadsheet’.
Write the statement on the board
‘Tabloid newspapers are easier to read than broadsheets’
One minute in pairs – why is this true/untrue? Count down last 15 seconds
to end of task.
Take brief feedback.
Bring out the following:
What do we mean by easier to read – (easier to hold) – (big clear font)?
Move towards how groups could judge the difficulty in reading the articles.
Distribute to groups of four students a recent article on the same piece of
news from both a tabloid and broadsheet newspaper.
Give pairs and fours time to read and decide how they could analyse the
texts to address:
‘Tabloid newspapers are easier to read than broadsheets’
Ask one group to feedback ideas.
Invite other groups to add new ideas.
In discussion bring out ideas like:
Is word length a measure of difficulty in reading the word?
What makes a word hard to read?
How do teachers judge/test your reading age?
Depending on outcomes pupils’ ideas may have to be directed.
Set the task
In fours decide how you will analyse the texts to decide how difficult
they are to read.
I would like your group to produce some data (a table), some charts
(graphs perhaps) and a few sentences to explain what you found out.
You may not finish all the tasks by the end of the lesson but I expect you
to have some data and be able to tell the class about your progress and
difficulties.
Tip 1 – think carefully about how your data will be used to make a chart.
Tip 2 – remember the golden rules for group work.
Circulate, collect ideas, misconceptions, and listen in on discussions.
At a convenient point ask one member of each group (an envoy) to visit
another group and explain what they have done so far and also bring back
information on what the group they visited was doing.
Plenary
It is often difficult to break students away from a group task
especially when they have not finished a task?
Try and rearrange seating positions or bring students to the front of
the class to focus the plenary.
Stress we have 10 minutes to share our progress in this the most important
part of the lesson.
Who thinks A is easiest to read?
Who thinks B?
Who is still undecided?
Why was this task hard to do?
How did we tackle the task?
If my next class was going to do the same activity – in your groups come up
with one sentence of advice (give time to decide on a sentence).
Go through the handling data cycle and decide where we are?
Who would be interested in our findings and why?
What type of chart did you/will you use and why?
Revisit key words like data, conjecture, analyse, tabloid and broadsheet.
Encourage students to ‘sentence complete’ using the words or repeat the
words after the teacher.
Footnotes
Notice the italicised, underlined text to identify paired/group activity.
Notice how the lesson is planned to engage the students as they arrive with
a challenge ‘I bet you can’t………………..’ Boys in particular enjoy tasks phrased
in this way.
Notice how little instruction is given in the lesson and how pupils have to
make decisions and think things out.
Be particularly aware of the notes in the plenary. Getting that quality
attention focus and interaction in a plenary in my experience is hard after
group work. It would have been easy to let the students work almost to the
end of the lesson before packing away and continuing next lesson. In
primary schools, teachers are acutely aware of this and often gather
children around them on a carpet to conduct an ‘intimate’ plenary.
In the plenary, time is still given for volley ball and not just ping-pong.
The above lesson served as a good way of integrating much of the good
practice I’ve tried to outline in this document.
However, a consultant colleague from another authority used my lesson plan
to demonstrate a lesson, to a whole department of six teachers, utilizing
the power of paired and group work. The feedback was good and most of
the teachers were surprised by the insight they gained into the pupils
understanding and how the students progressed. The observing teacher
who normally taught the class was open enough to comment about how
motivated, interested and well behaved the group were. Another teacher
immediately replied.
“What do you expect with seven teachers in the classroom?”
Perhaps this teacher missed some important observations!
Some further reading is listed below. This is followed by a prompt sheet
which, if displayed in your classroom or reviewed frequently may help in
your daily work. Helping you make the most of your interactions with
students is the intention of this booklet. I hope it proves useful.
C.A.M.E. Thinking skills
Muhlewr Adhami, David Johnson
and Michael Shayer.
King’s College London School of
Education
Assessment for learning
beyond the black Box
Dylan Wiliam and Paul Black
University of Cambridge School of
Education
Closing the learning gap
Mike Hughes
Network Educational Press
The framework for teaching
Mathematics Years 7, 8 & 9
DfES publications 0020/2001
Literacy across the curriculum
(Section 7 Managing group talk).
DfES publications 0235/2001
Training Materials for
Foundation subjects
DfES publications 0350/2002
Interactive teaching using
mini-whiteboards in secondary
mathematics + ideas for group
work.
Roger Ray,
Teaching & Learning Consultant
School Improvement Service
East Riding of Yorkshire Council
 Play more volleyball and less ping-pong.
 If you plan nothing else plan your questions.
 Are you working harder than the students and if so
why?
 Good answers take time to produce – give time.
 Practise this phrase ‘Take 30 seconds to talk to
your partner and come up with an idea’.
 It’s hard to opt out when you’re with a partner.
 Pupils often learn better from talking to each
other.
 Pupils talking together does not always indicate a
teacher is not in control.
 Furniture can be moved.
 Pupils who have to move, increase their heart rate
and their brains then work better.
 Group work is not students sitting together working
individually.
 Sharing equipment, paper, diagrams etc. focuses
group work and social skills.
 Human beings enjoy the social aspect of learning.