Creative approaches to Mathematics across the curriculum.
This article first appeared in Mathematics Co-ordinators File No. 20 published by pfp publishing in 2005. It is reproduced here with their permission. (Although the File is no longer produced, some articles may be available through Optimus Publishing http://www.teachingexpertise.com/-821.)
All too often we ignore the creative aspects of mathematics in our classroom teaching and focus on routine exercises and the repetition of procedures. In doing so we are unlikely to spark enthusiastic responses to the subject from the children. Not many children say that maths is their favourite lesson and yet it could be so easily. On the NRICH website we offer many ideas for making connections
between mathematics and other subjects as well as links to other valuable resources on the internet. Mathematics is often described as the study of pattern which might help to explain its strong connections with art, leading to some of the most creative aspects of mathematics. Let us consider tessellations as just one aspect of pattern. We offer a structured investigation of shapes that
tessellate on the website called Tessellating Triangles
. It starts by looking at tessellating equilateral triangles and then goes on to look at various other kinds of triangles and then quadrilaterals.
Whenever I think of tessellations I always think of Escher, an artist who has certainly explored this area thoroughly. It is indeed questionable whether he was more of an artist or more of a mathematician. Exploring how his tessellations link to underlying repetitions of quadrilaterals with 'bits' cut out and added on gives children a creative way into developing their own more elaborate
tessellating patterns. It might be worth avoiding the more sinister patterns though! Websites presenting Escher's work abound and can easily be found through search engines. The official website
is a good place to start.
One living mathematician who has done a lot of work in this area is Roger Penrose. He has explored tiling patterns that cover the whole plain like a tessellation but are not regular in the way in which the pattern repeats itself. His designs are stunning in their beauty and underlying simplicity, and we have developed one of them into an interactive activity called Building Stars
. This design uses two quadrilateral shapes: a kite and a dart and they are repeated over and over again but do not fit together in a regular way. As well as exploring this pattern interactively on the internet, you could also use the templates to create your own kites and darts either cut out in sticky paper
or in sponge so that you could print a pattern using the shapes.
One of the key features of the kites and darts that are used in this pattern is the ratio between the lengths of the sides which is the Golden Ratio. Investigating the importance of this number in art and architecture as well as more mundane things like the shape of A4 sheets of paper is yet another fruitful avenue of mathematical study and once again it links to many different curriculum
The mention of architecture takes us into Islamic art , of which decorative patterns are a central feature, owing to the prohibition of representation of natural forms in Islam. As is often the case with constraints of one kind or another, this restriction has led Islamic artists to become masters of abstract geometrical patterns and tilings. One frequently used tiling is a stars and crosses
design which can be constructed using a ruler and a pair of compasses. The template can be found on the NRICH website as well as in the form of an interactive game to play with the pattern. Constructing the pattern oneself is quite a challenge even for the highest attainers.
Another kind of pattern that is found in art and that has mathematical significance is the spiral. In fact there are many different spirals and they arise out of different contexts. Archimedes, the Greek mathematician has one named after him which can be constructed
using a cotton reel, strip of paper and a pencil.
What other connections can we make with mathematics across the curriculum? One approach to teaching maths that I and many others find useful is to contextualise it within a story and there are many stories that we can tell that have mathematics in them. One of my favourites is a story about Teddy Town
bears of different colours live in different streets. It offers a context for an interesting problem about shape, space and combinations.
There are plenty of other stories that you may be familiar with that have mathematics in them for example Eric Carle's picture books such as 'The Hungry Caterpillar' and 'The Bad Tempered Ladybird' which can help to introduce young children to counting and the concept of time on the clock. For older children, Hans Ensenbergerg's 'The Number Devil' is a delightful story that describes a young boy
dreaming about numbers which he is introduced to by the 'Number Devil'. It includes all sorts of different numbers from Fibonacci numbers to triangle numbers and prime numbers with a chapter on each. Wouldn't it be refreshing to have a shared reading of a mathematics book for a change?
A couple of years ago I was involved in a project in Lewisham in which I worked with an expert Theatre in Education team called 'Make Believe Arts' (phone Trisha Lee on 020 8692 8886) to develop some mathematical stories that had maths built into them as part of the story itself: a particular mathematical problem had to be resolved in order for the story to progress. It was tremendous fun and
proved a memorable experience for the children. A traditional story that is similar is an Indian tale called Sissa's Reward
I have already mentioned quite a few mathematicians of historical significance and the connections between maths and history are very strong and well worth investigating. There is the history of mathematics itself which is fascinating in its own right. It is hard to imagine how the Romans did arithmetic when one looks at their number system and the development of our own conventions are
fascinating. How did they manage without a symbol for zero for so long? Delving back into ancient history, the Egyptians were brilliant at geometry and their methods helped them to construct the Pyramids, but what about fractions? They had an interesting method which only allowed them to express fractions as unit fractions with one as the numerator. Expressing fractions as the sum of unit fractions
is tricky but is an interesting mathematical problem.
What about the great mathematicians? I have already mentioned Archimedes but there are so many others: Pythagoras, Escher, Euler and of course Fibonacci. St Andrew's University in Scotland has a website dedicated to the history of mathematics and it is possible to find a different mathematician for every day of the year there. There are a number of sources of information on the web about
Fibonacci but one of the best
must be that developed by an enthusiast, Ron Knott. He offers enough ideas to fill your maths lessons for weeks as well as your history, science and art lessons! There are lesson ideas, challenging questions and links to other websites galore. Look at his example about Fibonacci's
The Fibonacci connection brings me more or less to the end of this article although I have only scratched the surface of the possibilities for linking maths with other curriculum areas in exciting and challenging ways. I haven't even mentioned geography with maps, bearings, contours just as starters or physics where we could explore speed, time, rates of change, measurement and a host of other
things. I will leave those for another time. The creative possibilities are endless and a fruitful field for exploring.