Playing with Patterns and Relationships #1

Mathematics can be thought of as a search for patterns and relationships. To immerse in mathematical thinking we want children to look for patterns, find relationships and use them as much as they can to solve problems. Patterns are all around us and often become an integral part of our daily lives. We find them in language, nature, buildings, music, art, sport and cultural traditions to name a few. Children love to notice these patterns and make their own. Our intentional role, then, is to promote opportunities to describe, represent and explain a variety of patterns across multiple modes. This is the first article in our Patterns and Relationships series.

Repeating, Growing and Symmetrical Patterns

Research suggests 6-9 month-old babies can notice irregularities when presented with repeated elements that suddenly change (Starkey, Spelke & Gelman, 1990). This innate ability provides a frame for exploring mathematical ideas based on patterns. Such experiences start with noticing and exploring simple patterns, later moving onto patterns in numbers and our number systems, units of measurement, computation strategies, and spatial arrangements.

Patterns provide us with a way to predict what comes next based on what has happened before.

A repeating pattern has 2 or more repeated elements; in a 2-pattern there are two elements that repeat, in a 3-pattern there are three elements that repeat and so on.

There are many ways to describe repeating patterns. In the picture below, the two elements are blue star, red circle. This is the unit that repeats. We can predict what will come after the last red circle based on the pattern that occurs beforehand. Using our knowledge of patterns, we can also tell what will come before the first blue star.

The pattern below could also be described and labelled as an A-B pattern: the star is A and the red circle is B. So A-B is the unit. If we read this pattern from left to right it says A, B, A, B, A, B. Reading it aloud like this helps us hear the pattern and inform our prediction about what comes next. Recognising this sequence and being able to reason about it are key components in the development of a deep understanding of pattern concepts.

In a growing pattern each section of the pattern grows or changes consistently, either in its shape or quantity. Growing patterns have a relationship between elements. For example, the number pattern 2, 4, 6, 8 is a growing pattern that increases by 2 each time. 100, 95, 90, 85 is a decreasing pattern that reduces by 5 each time. Growing patterns can also increase or decrease by different but predictable quantities, for example 1, 1, 2, 3, 5, 8. In this growing pattern the next element (let's choose the number 3) is the sum of the previous two elements (1 + 2). So the next element in the pattern will be 5 + 8 which is 13. This famous growing pattern is known as the Fibonacci sequence.

Symmetrical patterns have a repeated element, either in its reflection or its rotation. We can think of line symmetry as repetition by reflection. Objects with 1 line of symmetry are very common, including chairs, tables, cars, dolls, balls, and our bodies! This reflective line of symmetry is often referred to as a mirror image. The chair in this picture has one line of symmetry drawn vertically down the middle.

A shape or object can have more than 1 line of symmetry.

A shape has rotational symmetry if an outline of the turning figure matches its original shape.

I was getting some staging ready to take

Patterns in songs and sounds

Very young children have been immersed in patterns since infancy as they listen to and acquire language. Nursery rhymes, songs and rhyming texts are some of the many ways that children learn about repetition. Consider the following:

Old MacDonald had a farm, E I E I O, And on his farm he had a cow, E I E I O.
With a moo moo here and a moo moo there, Here a moo, there a moo, everywhere a moo moo.
Old MacDonald had a farm, E I E I O.

Old MacDonald had a farm, E I E I O, And on his farm he had a pig, E I E I O.
With an oink oink here and an oink oink there, Here an oink, there an oink, everywhere an oink oink.
Old MacDonald had a farm, E I E I O.

The repeating unit of this song is in bold. There is repetition inside this stanza but this it is the whole unit that appears again in the next part of the song. This repetition allows very young children to predict what will come next, whether it is via the words or the the tones of the person singing it to, or with, them.

Making simple rhythmic sound patterns is a playful way to approach a multimodal approach to exploring patterns. This could start with an A-B pattern structure (2-pattern) that is modelled first using hand clapping, finger snapping or knee tapping, for example:

CLICK CLAP CLICK CLAP CLICK CLAP
CLAP TAP CLAP TAP CLAP TAP
SNAP CLICK SNAP CLICK SNAP CLICK

Children can then create their own for others to copy and continue.

Later, an extra element can be added to the pattern structure, perhaps an A-B-C or 3-pattern.

Patterns that incorporate body movements, and finding ways to record these will enhance this multimodal approach.

Talking about patterns

shallow focus photography of yellow sunf

Repeating patterns occur in the everyday lives of young children. It may be on the stripy clothing they wear (blue, white, blue, white), in their morning routines (get up, get a bath, get dressed), as part of their outings (red light, yellow light, green light, red light), or in their dance routines (step forward, step back, step forward).

Their prevalence in nature also provides many opportunities to put on our mathematical lenses and discuss the unique and beautiful patterns we see.

This Striped Sturgeonfish found in reefs off the coast of Australia has yellow-black-blue-black as a repeating unit across its body. This would be a 4-pattern, or an ABCB pattern. Asking children what they notice about the colours on this fish is one way to start a conversation about repeating patterns. Stopping at a part of this pattern and asking what would come next will present an opportunity to consider "how do you know"?

Growing patterns also appear in nature, like the Fibonacci sequence mentioned above. The patterns of the seeds in a sunflower and the spirals in pine cones are some of the examples of the Fibonacci sequence.

This sunflower also has symmetrical patterns.

Tell me

Tell me about what you have done

Tell me about your pattern

These are helpful prompts when assisting children to distinguish between our everyday descriptions of patterns as nice pictures and patterns that have predictable and repeating elements.

A child who tells you they have made 'a pattern' like the one in this artwork is likely to be using a common translation of 'pattern' from everyday conversation. While this picture has colours that appear repeatedly, there doesn't seem to be a predictable set of elements in any direction that helps us consider this a repeating pattern. In this case we may prompt with Tell me about what you have done.

When responding to children's work that seems to portray a repetitive, predictable sequence, Pengelly (1992) suggests we use the prompt Tell me about your pattern to reinforce the mathematical focus.

Having said this, it is important to be open minded when observing children and asking them about their work. The 'pattern' they are creating may not be immediately recognisable to us, making actively listening to the child's description of their construction important.

Paintings from my twelve year old nephew

Copying, continuing and creating patterns

Copying patterns requires children to be able to identify the individual parts of the pattern, making a 1:1 discrimination of its components (McDonald, 2018). Continuing patterns goes beyond sensory recognition as the child reasons to consider what comes next based on the perceived structure.

When children create patterns with materials, sounds, movements or drawings, our intentional teaching should draw attention to the features of their patterns, and whether they are repeating or growing (Pengelly, 1992; Siemon, 2017).

Questions to guide these conversations may include:

What do you think is repeating? What is happening over and over again?
Can you separate or show me the part that repeats?
If your pattern continued, what would the next part be?
If I cover up this part of the pattern, what would come next?
If I cover up your work, can you tell me what objects are hidden and in what order they occur/what they look like?
How could we describe this pattern? Does it have a rule?
How could we record your pattern so that we can share it with others?

References:

McDonald, A. (2018). Mathematics in Early Childhood. Oxford.

Pengelly, H. (1992). Making Patterns. Ashton Scholastic.

Siemon, D., Beswick, K., Brady, K., Clarke, J., Faragher, R. & Warren, E. (2017). Teaching Mathematics: Foundations to Middle Years. Oxford