Maths Setting – moving towards whole class teaching and away from sets:
An overview was provided to staff by L Alexander which began this proposal.
External advice was sought and a national report by Mike Askew was shared with staff
and is provided. A report by Tom Collins on removing setting and the fears
surrounding this has also been attached.
Findings from a discussion with teaching staff: the majority were in favour of teaching
maths by class.
External advice was sought at a local cluster meeting where around 24 schools were
in attendance. Only one other school set and that is for Year 6 (however, changing
from September). Elsewhere, in the Gravesend cluster, of those schools which did
set, all are moving away from sets for September. In Maidstone, the change is about
50-50. One teacher had moved away from sets last September, much to her
reluctance but on her Head Teacher’s wishes. She is teaching in Year 6 and reported
that she would now never want to go back to sets. She found that she had so much
more progress teaching in this way. The advisor also went on a 2-day conference in
Oxford, where speakers ranged from Singapore experts to Jane Jones from HMI,
Cassy Turner from America and some OUP specialists. All talked about the needs for
mastery and the curriculum; and all talked about the need for taking away the labels
and the ceiling we put on children in setting. One speaker described how if he was
someone struggling with a concept, why would he want another struggling student to
help him?
(The points made in the following paragraph come from the cluster maths meeting.)
The Importance of Mastery
Procedural fluency is not mastery. Teaching for mastery is the belief that every
student can achieve. A detailed, carefully planned curriculum should secure the
fundamental skills. Carefully crafted lessons should develop small conceptual steps.
Questions should be varied. Mastery is a change of mind-set. Higher achievers
thrive when exposed to maths of a deeper understanding. The philosophy of the new
curriculum is to broaden our pupils’ understanding and experiences not to just push
them onto the next rung of the ladder. The philosophy of the new curriculum is to
get all children working at the expected standard. All children should be ‘kept’
together in what they are taught. All children are encouraged to master their
understanding of mathematics. Mastery is a different approach to previous teaching
for some people. It is a change of mind-set.
During a meeting (22/6/2016) with Linda Pritchard, the School Improvement Partner,
Mrs Alexander discussed the issue of setting. She didn’t believe sets had a place with
the new curriculum and explained that in her experience schools have moved away
from these. This is therefore another external source which supports the movement
towards whole class teaching.
Research based on Tom Collins:
Removing setting from maths teaching in a three-form entry primary school
Tom Collins (Maths Lead at St Joseph’s Catholic Primary School in Guildford, and one of the
Primary Mastery Specialists being trained under the Maths Hubs programme), exudes
enthusiasm about the effects that teaching for mastery is having on pupils across the
attainment range. Impressed by his observations of Shanghai teachers in a local school, he
has thought carefully about the practicalities of implementing teaching for mastery in his
school. “We aren’t about ‘becoming Shanghai’” he says; “it’s about taking the elements of
teaching for mastery that can be applied in our own schools, in our own classrooms, and
implementing them, one considered, sure-footed step at a time.”
Teaching All-Attainment Classes
Tom felt that removing the setted structure in his three-form entry school, was a strong initial
step in encouraging teachers to try new methods to improve mathematical learning. “We had
traditionally taught ability sets in Years 5 and 6 for many years, believing that the gap was so
vast by this point we couldn't conceivably support and challenge children in mixed ability
classes. However, consistently, attainment in English (where we continued to teach in mixed
ability classes) was always better. As a result, we phased out maths ability sets in Year 5 last
year (2014/15) and have brought the same year group into Year 6 without ability setting this
year. It was an unpopular move with a handful of parents at first; however, we have been
careful to ensure the most confident children are always challenged and engaged in class and
so any resistance was short-lived.”
Teachers accustomed to working with ability sets often express concerns about whether it is
possible to stretch the quickest learners in mixed-attainment classes. Encouragingly, Tom has
found that high attaining students are demonstrating a greater depth of understanding that
he had not always seen previously and finding maths lessons more enjoyable. “Previously my
high-attaining Y6 pupils would be siphoned off to tackle Level 6 material and whilst they are
able to answer the L6 questions at a procedural level, they often do not recognise the
concepts and connections in what they are doing. Now I see Y6 high-attainers enjoying maths
more as they really unpick the concepts.”
So how
is Tom achieving this depth with high attaining children, whilst scaffolding learning for the
lower attainers?
Pair-talk
Notably, Tom uses pair-talk effectively and frequently, interspersed with individual pupils
explaining their understanding to the class. During pair-talk episodes, Tom minimises his
intervention, viewing this as a time for him to listen and learn about their depth of
understanding, only intervening if he hears a misconception that needs to be addressed to
avoid it being reinforced.
Tom arranges his desks in rows, with the pupils paired up, and those children that most often
need help in seats that the teacher can physically get to most easily. “The pairings
themselves are only selected for one reason (not perceived ability) - I ask myself, ‘Who will
this child work alongside in a focused, supportive and collaborative manner?’” Pairings change
every half term.
Importance of Questioning Style
Continual encouragement to verbalise understanding to the rest of the class appears to be
deepening meaning for all children. Children that have grasped the concept give deeper,
more precise explanations. In a Y3 lesson, forming additive equations with coins, some
children were able to confidently explain proportional errors in bar models such as the one
below, using precise language and proportional reasoning.
Other children are encouraged by Tom’s persistent, supportive and delving questioning style.
Local teachers observing Tom’s lesson commented on how long he was prepared to wait
patiently for a child to get to an answer, helping along the way, without giving up or moving
on. Tom explains that he is attempting to overcome the familiar phenomenon of children
‘freezing up’ when put on the spot. He has observed that it is too easy for such children to
say nothing and learn that the teacher will eventually provide the answer, or stop asking
them. He makes it clear that he expects all to contribute.
Tom recognises two types of question in his teaching – questions to help probe and deepen
understanding of new concepts and questions to assess children’s understanding of prior
concepts. “I aim the first type of question at children I think can offer some helpful, insightful
observations of a new concept (often based on what I've overheard during paired discussion
beforehand) and I aim the second type of question at children who, based on my formative
assessment, are likely to need to revisit a concept in order to secure new learning.” He tries
particularly hard to ask questions that expose misconceptions and address them head on.
Carefully Crafted Lessons
Another tactic Tom uses, to ensure stretch for high attainers and scaffolding for others is to
very carefully craft his lessons, thinking precisely about every example he uses and every
question set for pupils. This ensures slow but steady progression, by varying just one
component of the questions at a time. Pupil work time is regularly punctuated by teacher-led
discussion and examples, so Tom frequently ‘reels the kite back in’ after letting children run
with it in their own work. This technique was evident in the final exercise (below) that Tom
set for the Y3 lesson, allowing children to reason that the final equation wasn’t possible
(letting the kite out). He then demonstrated that an ‘=’ sign in the second box, as well as the
necessary + and – signs, could make the equation true (reeling the kite back in).
This task was set in a Y3
lessons on forming additive equations. Tom asked the children to insert mathematical
symbols to make the equations true.
Research by Mike Askew
The best way to deal with the wide range of maths ability in your class is rarely to place
children in groups of high, middle and low attainers, says Mike Askew. A more subtle
approach is required...
When running professional development sessions I’m often asked, “Do you believe in ability
grouping for mathematics?” The sentiment doesn’t surprise me, but couching it in terms of
‘belief’ does. Saying one believes (or not) in ability grouping leaves the door open for a
response of, ‘Well, I believe otherwise,’ and neither the answer nor the response does much
to change practices.
However, I do know there is sufficient evidence clearly demonstrating the benefits and
drawbacks of different types of grouping to show that grouping practices should not be left to
beliefs.
Between class grouping
Some primary schools choose to organise children into ‘sets’ for mathematics teaching, often
organised around three levels of mathematical attainment. You can see the logic to this. The
National Numeracy Strategy advocated dividing classes into three or four attainment groups;
so if a school is large enough, why not put together all the high attainers, all the middle
attainers and all the low attainers?
Well, the research evidence is clear: setting pupils does not lead to higher attainment for all.
At best, the children who gain from such an arrangement are the higher attainers, but even
then the gains for that group are not that impressive. And study after study has shown the
gap between higher and lower attaining groups gets wider when children are put into sets
(and there are detrimental effects on the attitudes of lower attaining children).
One study, for example, tracked 1,000 KS2 pupils who were taught using the same teaching
materials. In test results, the pupils in mixed ability classes significantly outperformed those
taught in sets. Another study of 12 primary schools found the KS2 results of pupils grouped
by ability were rarely higher than the average for the local authority, or England overall.
Within-class groupings
If children are in heterogeneous classrooms, then what is the best way of grouping them to
deal with the diversity of mathematical attainment? In contrast to research into betweenclass grouping, the research into within-class grouping is more positive: it can raise
attainment.
But the picture of what makes for effective withinclass grouping is complex. There is no onesize-fits-all model of grouping and it is certainly not as simple as organising children into
high, medium or low attainment groups. Research findings indicate the importance of
grouping students in particular ways for particular purposes; the type of grouping depends on
the type of learning outcomes being worked on and the learning tasks set. For mathematics,
it is helpful to think about groupings for fluency, problem solving and reasoning.
Fluency groups
When it comes to practice and consolidation activities, the research shows that pupils are
generally best off working individually. This makes sense – learners are going to be diverse in
what they are fluent in, practice activities need to be individually tailored, and time on task is
more focused when children practise individually. But, as I noted in a previous article, (Teach
Primary 6.3 - tinyurl.com/tpfluency) the important thing here is that children are actually
practising – consolidating what they are reasonably fluent in – rather than learning new
material. Perhaps such tasks can be set as homework, since practice should not require a
teacher to hand.
Problem solving groups
If we accept that problem solving can be a powerful way of engaging with new mathematical
ideas (as opposed to applying mathematics previously learnt in a decontextualized fashion)
then research shows paired work as the best grouping for developing understanding.
The findings here rest on extending the idea of ‘cognitive conflict’ into ‘socio-cognitive
conflict’. Piaget introduced the idea of cognitive conflict: that learning comes about through
an individual becoming aware of a contradiction in their understanding. (For example, the
conflict between water poured from a wide container into a thin one looking as though it has
increased in volume, but logic telling you that the amount cannot really have changed.)
Researchers now have shown that such conflict need not only be an individual act of
cognition, but can be provoked by pairs bringing different perspectives to a problem.
Differences in perspectives come to be resolved by the development of a joint perspective
that is more complex than either child originally thought.
It seems common sense to assume that mixed attainment pairs working together may lead to
the lower attaining partner advancing towards the level of the higher attaining partner, but
this partner not gaining as much from the experience. Research does show, however, that
even when pairs have differing levels of attainment, the more advanced child can progress as
much as his or her less advanced peer – the old saw of ‘two heads being better than one’
appears to hold true.
A key issue in effective pair work is the power dynamic between the children. If one child is
dominant and the other acquiescent, then working through to a shared perspective is unlikely
to come about. Paired work only works well when the partners trust each other and can work
well together. In fact, trust and cooperation seem to be more important considerations when
selecting pairs to work together than factors such as matching attainment levels, or
friendships.
Reasoning groups
Larger groups appear to be best suited to the development of reasoning and, when everyone
in the class has been working on similar versions of a task, whole class dialogue can provide
a suitable arena for practising this skill – even with a wide range of attainment in the
participants. Just as pairs working on a problem can reach a level of understanding higher
than either could attain on their own, so a larger group (including the whole class) can reach
a higher level of reasoning and understanding.
One of the reasons why this is so comes from studies revealing that the person who is likely
to learn most from group work is the participant asking the most questions; with the person
answering the most questions making the next highest learning gains. When children present
their work in a whole class setting, they can set up a dialogue that provokes questions about
their solutions. These questions can then be answered both by the presenters and other
children in the group.
Collectively, the class establishes a more sophisticated level of reasoning, and in the process,
individual understanding improves.
Thus the plenary becomes important and, in order to provoke dialogue, needs to be more
than a show and tell.
Ideas need to bounce off each other for the mathematics to emerge, which means children
carefully listening to, building on – or arguing against – each other’s explanations.
A shift in perspective
Putting this into practice requires a shift in how we think about and plan for classes. One way
of thinking about a class is as a collection of 30 individuals, each of whom has a specific level
of mathematical understanding that has to be catered for. As I’ve noted above, this way of
viewing the class probably does need to be adopted when planning activities to develop
fluency.
An alternative view of a class is as a collective: the class as a whole has ‘understanding’ and
‘needs’. From this perspective, tasks for pairs or groups to work on need to be carefully
chosen so they are beyond the grasp of any individual member of the pair or group. If pair or
group tasks are chosen on the basis of being appropriate for the level of individual
attainment, they may not be sufficiently challenging to provoke socio-cognitive conflict and
the consequent deep learning. Tasks need to be chosen so that the resources required to
solve them – knowledge, skills, problemsolving strategies and so forth – are not within the
grasp of any single individual and accomplishing the task requires input from others.
At the same time, tasks need to be chosen so that everyone in the group can become
engaged. I find the idea from NRich (nrich.org) of ‘low threshold, high ceiling’ tasks
particularly helpful here – tasks that have an easy entry point (a low threshold) but the
potential to engage learners in deep mathematical reasoning (high ceilings). As Lynne
McClure points out in an article on this idea (nrich.maths.org/7701), one of the great
advantages of such tasks is that they allow learners to show what they can do rather than
what they cannot do.
The key shift in perspective regards how we view diversity. Too often diversity is cast as a
‘problem’ in classrooms, as something that needs to be reduced or managed in some way.
The research into effective group work shows that diversity in classrooms is actually a good
thing. Diversity – of ideas, of strategies, of reasoning – is essential to levering up
understanding. We need to embrace diversity.
Much of the research I draw on here is summarised in ‘An extended review of pupil
grouping in schools’ (DfES, Kutnick P, Sebba J, et al, 2005). The full report can be
downloaded from tinyurl.com/tpgroups
BBC report on Maths Mastery