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The purpose of this article is to share some thoughts about
different aspects of helping children with their development of
I was involved in looking at the large collection of activities
that were under the heading of "area" on NRICH when I noticed a big
difference that would divide the activities into two. There was
Inside Seven Squares
Sending a Parcel
. (You can find these resources by clicking on
the links or by typing their title into the keyword search box in
the top right-hand corner of the NRICH page, then clicking on
"Title search".) All these activities were about 'calculating'
areas in different, and sometimes difficult, situations. They
involved having knowledge of various formulae and an understanding
of when and how to apply them. In a way they are activities which
involve arithmetic in a similar way to using arithmetic to work out
the cost of shopping.
4 articles @ £1.99 each
4.5 metres of material @ £5.39 a metre
3 bottles @ £2.49 each [special offer 3 for 2]
What change would there be from a £50 note?
The remaining activities were
Transformations on a Pegboard
A Square in a Circle
Tiles on a Patio
My New Patio
The Big Cheese
. These problems require no particular knowledge of
formulae to do with the calculation of area in different
situations. They require pupils to be able to count, and probably,
cut paper, draw, use a ruler, look out for patterns and start to
observe systems that they use, to question and to feel the freedom
to think and be creative.
I call the first group the "Calculation Style" and the second group
the "Exploration Style". I believe that at the end of the
Calculation ones the pupils may feel more secure about their
knowledge of formulae and, be more confident to tackle similar
problems in other areas of the subject. Some of the best pupils
would feel proud that they managed it all and got them "right".
You, the teacher, may have used them for reinforcement and/or
assessment reasons. For the Exploration activities I believe that
most pupils would have their concepts of area enhanced, would have
discovered something new, would have been creative and would have
sustained their work for a prolonged period of time. You the
teacher, would have used the activities for these kinds of
I would suggest that you try the two kinds with your pupils and see
what results they bring. When I was taught in the 50's it was a
matter of "Here's how you do it; Copy it; Repeat it many times; Use
it" and then you'll know it for life! How untrue that was unless
you had a natural flair for it, in which case you got bored. The
Calculation activities are still quite close to the work in those
days but have an important role to play in keeping pupils focussed,
readily supplying assessment and building up confidence. Whereas
the Exploration ones have got much more of the 'Using and Applying'
part of the Curriculum at their heart and at the same time help the
pupils to develop concepts in that part of Mathematics. The other
important aspects of the Exploration activities are that they are
accessible to a very wide range of ability. For example The Big
Cheese is accessible from year two pupils and has the potential to
excite postgraduate students too.
Thinking and creativity are so very important aspects of learning
that it seems to be a huge advantage to working with the
explorations. Therefore when starting on some new topics of the
Mathematics curriculum I would encourage the exploration approach
to enable thinking, creativity, enhanced concept development,
enjoyment and motivation. Whereas starting off with the Calculation
style is more likely to please only a few and dis-engage many. So
we need a mixture of both kinds of activity and a balance that
helps the pupils in every area of their learning.