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Peter Hall is an Advanced Skills
Teacher of Mathematics working at Imberhorne School, a state
secondary school in East Grinstead, West Sussex. He was one of four
NRICH Teacher Fellows who worked on embedding NRICH materials into
their teaching during the year 2008-2009. In this article, he
writes about his experiences of working with students at Key Stage
[Readers may also be interested in other
about teachers using NRICH]
Do I have to do problem solving all the time?
No! There are many ways of using the NRICH problems. I've
tried to think carefully about the classes I've worked with to see
whether an NRICH problem would make a good starter - as a way into
a new topic or as the main activity for the lesson, or as a plenary
at the end of a topic. Some NRICH problems have resulted in the
class being keen to know how to do something - for example,
subtracting with negative numbers. This has created an interest in
the topic and some motivation for the class to want to learn
But I've got a mixed ability group
I'm tempted to say that every class is a mixed ability class... but
I think one of the strengths of an open ended task is that
different students can work on it in varying degrees of depth. Many
problems work well at both a superficial and an in-depth level
(something about being a pool in which a child can paddle but also
an elephant can drown?). Perhaps you can take advantage of the
mixed ability in the way that you structure the groups. Maybe this
is an ideal time for AfL to come to life - it has worked quite
nicely to have groups present their solutions as a poster and then
have the class move around the room and rate the poster.. This also
means the students are able to see the varying responses to the
question. On several occasions I've found the better mathematicians
have not been the ones with the best ideas - and have not had the
most creative approach to an unfamiliar problem.
How can I make group-work work to my advantage?
Go and talk to your English teacher or your Humanities
teacher, go and watch the way that they get groups to co-operate
and work well. There is much more to think about than we maths
teachers tend to realise. I usually just ask the students to get
into groups of their own choosing, but I'm starting to feel this is
a fairly lazy approach and I need to dictate things a little more.
Perhaps I ought to choose groups based on their last maths test-
and even then do I want a "top group" or do I want to structure
each group with a mix of ability. Do I want to put all the naughty
ones in the same group, or all the quiet ones in the same group?
What about giving each person in the group a role - so they have a
job to do? What about a captain (to keep the group on task) a
facilitator (to ensure that everyone gets heard) a resource manager
(the only one allowed to ask for help, or get paper or scissors)
and a recorder / reporter (to do the recording). More radical ideas
- take one student's work from the group and use this to judge the
whole group, make the whole group take a test- but mark one
question from each student...
At my school we teach fractions every year. We still find year 12
seem to have "forgotten" how to add up fractions correctly. So
perhaps it is worth taking more time to "do fractions" in year 7 so
that they understand them better. Perhaps a problem solving
approach will yield other ways of approaching a topic that you
haven't thought of before. Maybe there are critical subjects that
you can identify where it would be worth taking a little more time.
If you're redesigning the KS3 curriculum perhaps you can create
more time for some topics. Of course some students seem to benefit
from constant revision and re-covering topics -maybe it is worth
saving the problem solving lesson on fractions for a later point in
the year- so that the problem solving lesson becomes a revision
lesson as well as a lesson on problem solving.
But this problem doesn't have a right answer?
Be brave! Encourage them to work on these sorts of problems. Be
excited when they find an answer you haven't thought of. Give
prizes... If you get more than one solution can you involve the
students in judging what is a good solution - or which is the
better one? From time to time it could be good for the students to
work on a problem where you don't know what the answer is -and tell
them that. Is that a nicer feel for their thinking than just trying
to discover the thing that you know already.
How is this going to engage my students?
Some students may find this approach too daunting - and others
may take some time to be comfortable working in an environment
where there isn't any immediate feedback in terms of the right /
wrong answer. Many NRICH problems have a sense of optimisation
about them - this seemed to create a nicely challenging atmosphere
in the classroom. Other times we have found a question about "make
the largest number with..." went into a new direction by a student
quietly saying "I bet no-one can find a smaller answer than me"
- and a whole new direction evolved.
What about other teachers?
I think this year I've found three groups of people - some are very
keen to work in this fashion and have been keen to discover the
NRICH website - especially the curriculum map and the topic search,
and for this group of staff I've only needed to point the way.
Others have wanted more help. This help has included me choosing
"good" problems for their classes or collecting together resources
for their use. A further group of teachers have found the whole
process too scary. Right now they're not used to teaching in this
fashion - they are still fairly didactic and uncomfortable trying
anything else. But as the new KS3 curriculum unfolds, and as we
look for more rich tasks and try to incorporate more functional
skills. ... I think more people will be converted to think in a
problem solving fashion.
How has your attitude changed?
I think working in a problem solving way has encouraged me to be
more flexible in lessons, with some problems I've had a go
beforehand and then I've expected the students to tackle them in
the same way as me. It has been interesting when they haven't. I've
seen more evidence of how they think, and the areas of maths they
don't like / aren't comfortable with when I've encouraged them to
try their own ideas. With some classes this has needed more support
to ensure they're all still doing something when they easily might
How are you going to use these problems next?
For next year some of the problems have been included in our scheme
of work, so I want to help other staff get the best from the
problems as we get to them. For my classes I want to try harder to
foster a changed learning environment and try harder to make
problem solving a more natural approach to a new area of the
curriculum. I'd also like to try harder problems with my sixth form
classes- I'd like to believe they'll make a better job of it...
I'm still wondering whether I should leave things a little
less sewn up - perhaps we should have more questions in our
plenary, more things to dwell on - for them overnight, or over a
weekend - again I'm wondering whether this will improve motivation,
or interest, or re-engage some curiosity.
I need to try harder to be more inventive and creative in the
way I use group work. I need to let some groups be quite
unsuccessful and see their reaction to that, rather than stepping
in and helping them too much. I'd like to think about moving the
chairs and tables around on a more permanent basis... but perhaps
that is a step too far!