Because so many of us had an unhappy time in maths lessons at school, we have come to expect that everything to do with mathematics will be both boring and difficult.
Without realising it, we can be affected by this when we come to present mathematical topics to children. Or we may simply avoid situations where children have the opportunity to play with mathematical ideas and where we have to answer questions that challenge our mathematical understanding.
This then creates a vicious circle. We feel anxious and we create a boring, predictable experience for children, who in turn become bored and anxious about finding right answers.
The situation with music is not much better. Many people say
* I can't sing
* I can't play an instrument
* I can't read music.
This is despite the fact that all around the world there are singers who can't play an instrument, instrumentalists who can't read music, and other people who can read music who are neither particularly skilled in singing or playing.
Where we are reluctant to sing and play music, we can unwittingly convey to children the message that only a few special people are musical.
The truth is that we are all born with considerable musical and mathematical abilities and that we constantly use those skills in our everyday lives. In order to cook a meal or cross the road we use a lot of mathematical understanding. Each one of us knows thousands of songs and pieces of music and probably recognises thousands more.
Why link music and maths?
Music and mathematics are often linked - and unlikely as it may seem when we're bopping around to our favourite pop song, the two subjects have a great deal in common.
Mathematics has been called the science of patterns. Music (and dance, which is closely linked) is undoubtedly the art of pattern.
Enter any music shop and you will find a collection of CDs called 'Baby Mozart', 'Baby Beethoven' or 'Baby Bach'. These are sold because parents believe that babies listening to such music will have enhanced development.
The evidence for this is not entirely conclusive and the effects often appear to be short-lived. However, where effects have been found, they are often in the area of mathematics.
We claim that children learn through play, yet mathematics is rarely playful. Music, on the other hand, is always linked with play - we play pianos or flutes and we even play CDs. Making maths more playful, expressive or communicative will make learning more effective for children.
Music can help in this process of making maths more enjoyable for you and the children you teach.
Music also supports memory, so it can help children to remember aspects of mathematics. We should not forget, however, that music is also important in its own right, so it shouldn't only be used to support other subjects. Make sure you take some time to enjoy it in its own right.
Pattern
An understanding of pattern is essential to an understanding of mathematics. As you will know from discussing children's paintings and drawings with them, children often have difficulty in defining pattern.
Very often, they will describe anything that is not a representation of something in particular as a pattern.
Practitioners commonly use materials such as beads, or commercial resources such as teddy bears, to help children develop an understanding of pattern.
Including a range of musical patterns encourages children to explore pattern through sound and appreciate that it can be great fun.
Musical patterns are an integral part of the songs we sing with children.
There is pattern in the following:
* alternating verse and chorus of songs such as 'Lord of the Dance'
* repeating tunes of songs such as 'Twinkle, Twinkle Little Star', 'London Bridge', and 'The Wheels on the Bus'
* growing structure in 'One Man Went to Mow' or 'The Twelve Days of Christmas'
* scales and the regularity of musical structures themselves.
Encouraging children to make and identify patterns in their musical exploration can promote both mathematical and musical development and understanding. This can be done in an adult-led situation such as carpet time, and in child-initiated play with instruments.
It may even occur in children's informal vocal play. You may, for example, overhear a child singing something as simple as 'la-la-la-la-la/ boom boom'. By repeating that sequence with the child, we can draw it to their attention and play about with it.
Similarly, a child might be playing with a xylophone and begin by playing the lowest note, then hitting the two lowest notes, then three notes - each time beginning with the lowest and creating a growing pattern.
We can reinforce the learning by commenting on the pattern and discussing any possible extensions, in just the way we would do with a bead pattern or an ordered line of cars.
Some children will find it interesting to produce visual representations of their musical patterns, using a range of materials.
Understanding of pattern can also be extended in other ways:
* The pattern created by the child singing 'la-la-la-la-la/ boom boom'
might, for example, be represented by small sticks and buttons as:
l l l l l l ** l l l l l * l l l l l ** l l l l l **
l l l l l l ** l l l l l * l l l l l ** l l l l l **
* This interest could be developed by asking another child to play or sing their version of the visual pattern (such as that above made from sticks and buttons), which might, for example, be playing five notes on a recorder followed by two beats on a drum; singing tum, tum, tum, tum, tum/oy, oy; or clapping five times and stamping twice
* Getting children to describe the patterns in their vocal play, in their exploration with instruments and other soundmakers (such as homemade shakers, plastic bottles, pieces of paper, etc) and in rhythmic movements
* Creating dances with repeating patterns
* Identifying and describing patterns while listening to live and recorded music.
Counting and calculating
Music supports memory. We all have a clear understanding of this, since we all use so many counting songs.
Every practitioner knows dozens of counting songs, and we use them because it is vital for children to be able to recite number words in the correct order in order to be able to count.
The easiest way for young children to commit to memory a long list of strange and unconnected words, is through song - a fact that is exploited in cultures around the world.
Another factor that enables counting songs and rhymes to make numbers memorable is the use of finger play. The part of the brain that is responsible for counting is next to the part of the brain that governs the use of our fingers. So singing '1, 2, 3, 4, 5/Once I caught a fish alive'
makes number names in the right order more memorable, and the accompanying physical actions make the learning even more secure.
Familiar counting songs can also be used to support children in calculating. This happens automatically in 'Five fat sausages sizzling in a pan', since two sausages are lost each time - 'one went pop and the other went bang'. Instead of one speckled frog jumping off the log, two or three can do so. This can be made even more playful by writing your own counting songs.
Shape, space and measures
This is a very rich area of mathematical understanding to develop through music, since music and dance operate in both time and space. This means that all sorts of activities exploring long or short, high or low, loud or quiet, fast or slow contribute to both areas of learning.
In most cases, the meaning of these terms is similar whether used mathematically or musically. 'High' and 'low' provide an interesting contrast. A high number is a big number, but a high note, on for example a xylophone, is shorter or smaller than a low note.
As with counting, songs can be used to support memory for learning about shape. Helen MacGregor (in Tom Thumb's Musical Maths) has written songs about circles, squares and triangles, and just as with counting, you can make up your own as well.
One group of three- and four-year-olds got very excited about snails and wrote a long song, using a vast range of positional language (sung to the tune of 'London Bridge is Falling Down') and all based on Ruth Brown's beautiful picture book Snail Trail (Andersen Press) - under the bridge, through the tunnel, over the wall and so on.
Instruments
The shape and size of instruments can also be explored, and links made with the relationship between size and sound.
Instead of the usual wide assortment of instruments in the music area, you could try putting out only cylindrical instruments. You'll be surprised how many you can gather together (rainmakers, guiros, etc), or even some made from cylindrical cartons. You could provide a collection of big and small instruments, or drums or Boomwhackers in a range of sizes.
Recorders are useful in exploring high and low - the big ones make very low sounds, while the tiny ones make high sounds. Perhaps you can persuade a group of older children to come and play for your children. You could also just put out tiny instruments and make some fairy music (or music to soothe a baby to sleep). Outdoors, you could provide some big instruments to make some loud music for giants.
Sets of nesting boxes with lids can be used to make shakers, and children will enjoy exploring the sounds made. Does putting the same amount of, for example, pebbles in different sized boxes produce the same sound, or is it different?
Tuned percussion (such as xylophones and glockenspiels) can be bought with high notes placed above low notes. Several suppliers can provide step chime bars or step glockenspiels which make the musical meaning of high and low more visible.
There are also instruments from other countries (such as the Vietnamese trung, a sort of vertical bamboo xylophone) which demonstrate high and low in this way.
Dance and movement
Dance and movement are also excellent ways to explore shape. Encourage the children to move to music and to talk about the shape of their movements.
Stimulate discussions by providing pictures of things that bring to mind certain kinds of movement - for example, windmills or washing machines for circles or flowing shapes, icicles or robots for jagged movement.
Providing lengths of ribbon or scarves or hoops to use in the movement may also encourage this line of thought and discussion.
Outdoors
Music indoors can be intrusive if other activities are going on. However, if we want children to be able to develop an understanding of 'loud' and 'quiet', then they need to explore the former as well as the latter. The outdoor area provides an ideal context for doing this.
Create musical patterns outdoors using:
* sheets of corrugated plastic flashing or roofing for scraping
* buckets, flower pots, rain barrels or dustbins to hit
* lengths of plastic drainpipe, cut to different lengths - these are played very successfully with a thickly padded table tennis bat (or much more fun, a flipflop)
* a hanging frame such as some low branches, a clothes horse or a fence, to which are tied a variety of sound makers, such as pots and pans and chimes
* cardboard or plastic tubes of different lengths to blow
* metal containers in the water tray which change the sound they make as water is added or poured out
* something to shake, such as large plastic bottles with stones or lengths of chain inside.
These materials can also be used to undertake some mathematical (and musical) problem-solving, such as:
* how long do two pieces of drainpipe need to be in order to sound like a police siren?
* what is the optimum number of stones to put in the bottle to make the loudest possible sound?
* how far apart can two children playing the same instrument be and still hear each other?
WHAT DO WE MEAN BY 'PATTERN'?
* 'Mathematics is the classification and study of all possible patterns.
Pattern is here used to cover almost any kind of regularity that can be recognised by the mind. Life, and certainly intellectual life, is only possible because there are certain regularities in the world.' (Keith Devlin (2000), The Maths Gene, page 72).
Understanding pattern Authors and researchers are divided on how we develop our understanding of pattern. However, most would agree that we come to understand pattern through the following stages - though not necessarily in this order:
* recognising a pattern
* describing a pattern
* copying a pattern
* copying a pattern and extending it
* creating a pattern
* discussing a sequence or repeating pattern.
Key vocabulary There are many kinds of pattern, and by describing musical (and visual) patterns, we can help children to recognise and explore them. Types of pattern include:
* line patterns or sequences
* repeating patterns
* growth patterns, ascending and descending, growing or decreasing
* symmetrical patterns
* radial patterns
* cyclic patterns.
WRITING A COUNTING SONG
* Take a familiar and well-loved counting book (such as Ten Terrible Dinosaurs by Paul Stickland) and set it to a known tune. This one works well to the tune of 'Ten green bottles', while the wonderful counting book One Is a Snail, Ten Is a Crab by April Pulley Sayre and Jeff Sayre can be adapted successfully to the tune of 'One man went to mow'.
* Change the words of a familiar counting song to fit in with a particular topic or interest. Opportunities for this are endless - 'Ten yellow daffodils/sitting in a bowl'; or '1, 2, 3, 4, 5/once I had some beans for tea/6, 7, 8, 9, 10/some for you and some for me'. It won't be long before you and the children have hundreds of brainwaves - and the important thing is that you will all be playing with maths!
* Get some more ideas for starting points from Tom Thumb's Musical Maths.
Helen MacGregor has taken known tunes and rhymes and adapted them to support the development of a wide range of mathematical concepts. One example is 'Baa baa maths sheep' in which the first verse is conventional in talking about three bags full, but then goes on to talk about six bags and nine bags, with two (or three) each for the master, the dame and the little boy.
We know that thinking begins with physical action.Babies and toddlers learn about circularity by moving around and around. Rhythmic movement is, for this reason, a very useful way of promoting counting.
Music allows you to count in your head. You will have experienced this kind of internalised counting when you hear from the music when it is time to change direction or vary your step.
Children will certainly hear and respond to changes in speed, and this can be linked to counting. If you count your steps out loud you will hear (and feel) how much faster or slower you are going.
In Let's Go Zudie-o, authors Helen MacGregor and Bobbie Gargrave have included music that may be used for a parachute dance. It involves counting steps forwards and back but, in fact, children will quickly internalise the counting.
Understanding of counting and calculating can also be extended through music and dance by:
* making up dances with a certain number of steps or beats
* maintaining a beat, while fitting different rhythms into the spaces (which has the added advantage of promoting vital mathematical skills of estimation and problem-solving)
* matching actions to songs that don't have any, or changing existing ones.
References and further reading
* Ruth Brown (2000), Snail Trail, Andersen Press
* Keith Devlin (2000), The Maths Gene, Weidenfeld and Nicolson
* Helen MacGregor, Tom Thumb's Musical Maths, A&C Black
* Helen McGregor and Bobbie Gargrave, Let's go Zudie-o, A&C Black
* Linda Pound (2006) (2nd ed), Supporting Mathematical Development in the Early Years, Open University Press
* Linda Pound and Chris Harrison (2003), Supporting Musical Development in the Early Years, Open University Press
* April Pulley Sayre and Jeff Sayre, One Is a Snail, Ten Is a Crab, Walker Books
* Paul Stickland, Ten Terrible Dinosaurs, Ragged Bears