‘The most important skill we take from the study of mathematics [is] a need to explore, describe and define the universe around us.’
—Richard Schaar, Professor of Mathematics, University of Southern California

‘The mathematician and the preschooler have much in common. They approach the study of mathematics in much the same way. Like mathematicians, young children are intrigued by the mathematics in their world. Consider the intensity and determination children bring to their investigations of the mathematics that surrounds them. This comes not from a need to complete an assignment or because it will be useful to them. It comes from a need to know.’
‘Guided only by their feeling for symmetry, simplicity and generality, and an indefinable sense of the fitness of things, creative mathematicians now, as in the past, are inspired by the art of mathematics rather than by any prospect of ultimate usefulness.’
—Eric Temple Bell, mathematician

Many young children are naturally interested in numbers and in response to a situation, will ask: How many are there? How many do we need? Do we have enough? Who has the most? Are there any extras?

They are interested in geometry and explore to find out: How are these shapes alike? How are they different? Which fit together? Which ones leave spaces? What can we build with these? What other shapes can we make using these shapes?

They want to measure and compare, and they wonder: Which is … bigger? More? Heavier? Longer? Shorter? How can we find these out?

They experience the connections between maths and music when exploring rhythm and patterns, and between maths and art when working with symmetry and design.

The mathematics that engages the mathematician and the preschooler is the study of patterns and relationships, order and predictability. This is not mathematics the way most of us experienced it in school. Yet it is the mathematics that will serve our children most as they move through school, and, in time, that will await them as they enter the workforce. The mathematician and the preschooler need and want to experience the joy and excitement of the world of mathematics.

The learning of mathematics is an active endeavour. Children need to be involved in investigating, comparing, wondering and checking to see what happens. They need to think about what they are experiencing, to notice what happens and to begin to talk about what they notice. They need to listen and think about what other people have to say. They need to begin to make connections, to see relationships among mathematical ideas. They need to see how experiences can be recorded: with blocks, with pictures and sometimes with mathematical symbols. Important mathematical ideas will naturally arise through children’s play.

‘Children’s play and interests are the sources of their first mathematical experiences. These experiences becomemathematical as the children represent and reflect on them … The most powerful mathematics for a preschooler is usually not acquired while sitting down in a group lesson but is brought forth by the teacher from the child’s own selfdirected, intrinsically motivated activity.’
—Douglas H Clements, Mathematics for young children

Number

Children’s mathematics includes number experiences that ask them to discover: How many … leaves did you pick up? How many pockets do you have? Can you put three horses in the stable? Can you put four fish in the ocean? Can you give me four of those?

There are a number of ways to explore number; for example, acting out counting songs. This helps children bring meaning to the counting sequence they are learning. They learn to count beyond what they already know by counting along with the teacher. A teacher can ask, ‘How many children are here today?’ and then model writing numerals when the children tell how many. Some children will label what they counted with a numeral and show how many horses they put in the stable, and a few will even try to write the numeral on their own.

Spatial experiences

Try to provide opportunities that allow children to see how shapes fit together through experiences with puzzles, blocks and boxes. Children who play with blocks will gradually build structures that begin to look more and more organised and stay balanced a bit longer. It is also good for the children to have them hear their teacher using language that describes where objects are located: Here it is—under the table.; Let’s look behind the cubbies.; We put the calendar over the bookshelf.

And they begin to use spatial language themselves, like: The baby is in the baby bed. She is under the covers.; The biscuits are inside our play oven.

Pattern experiences

Or you can model patterns based on colour. As an example, model patterning when making a long Unifix™ train with students, with blocks positioned as red, red, yellow, red, red, yellow. Ask children what colour they think comes next?

There are usually many opportunities to show children patterns that exist in everyday life. It can be as simple as pointing at a child’s shirt and studying the pattern on the clothing: Look, I see a pattern on your shirt—red, green, white, red, green, white.

Allow children opportunities to copy repeating patterns using a variety of materials; for example: block, button, block, button. If they have pattern cards available to copy, some will continue the pattern beyond the card.

Children can experience patterns through music and become aware of rhythm and repetition when they march around the room, tap their rhythm sticks or beat on their drums. They experience patterns through stories when a teacher reads predictable books and they delight in guessing what comes next.

Download free activity pages from Developing maths concepts in the early years

Click here for more information on Developing maths concepts in the early years

In my 11 years of teaching Years 3 and 4, I have noticed, and taken an interest in, how children learn basic maths facts. This includes having conducted an action research project in which I studied approaches to teaching maths facts and concluded that students learn and retain automaticity of facts when they are taught them by using strategies and looking for patterns, and are given opportunities for meaningful practice.

Afterwards, I shopped at teacher stores, bookstores and online for a book that teaches multiplication facts in this manner. I was unable to find it! What I found was ‘drill and kill’ (worksheets that present the facts repeatedly), or speed tests. I bought and tried many of these books … and they didn’t work!

As such, I began creating my own practice pages to make up for the lack of effective materials on the market. The strategies I developed for teaching the multiplication facts are the basis for Multiplication success with algebra. My intention is to share my successful strategies with other teachers who are also looking for more efficient, practical ways to teach their students these facts.

What’s different about this approach?

The approach used in my book provides students with meaningful practice as they learn multiplication. They are exposed to and practise algebraic concepts as they master the facts. In a time when the curriculum is overwhelming and difficult to fit into a school day, an approach that synthesises concepts provides an important advantage.

The following are examples of the strategies used in my book:

1. Students use the identity property to figure out that any number (n) multiplied by 1 is equal to that number (n • 1 = n); and the zero product property, which states that any number (n) multiplied by zero is always equal to zero (n • 0 = 0).
2. Students use the doubling–halving strategy to help them learn many of the multiplication facts. For example, if you multiply a number by 2 and then double it, you will get the answer to multiplying that number by 4. If you multiply a number by 4 and halve it, you will get the answer to multiplying that number by 2: (n • 2) x 2 = n • 4 or (n • 4) ÷ 2 = n • 2
3. Students are also prompted to look for patterns to help them figure out the products. For example, all numbers multiplied by 9 have a digit sum of 9; all numbers multiplied by 5 have a 0 or 5 in the ones place of the product.

While students are learning the facts using strategies and looking for patterns, they are also practising algebraic ideas like balance, variable and function.

Balance involves the idea that the equal sign shows the relationship between numbers. As such, while students are practising the 10 and 5 multiplication tables, they are also practising balance when they solve the equation: 4 x 5 = Δ • 10

Variables are the unknowns in an equation, and this book provides students with practice in determining the value of the variable in an equation. For example, in the equation Δ x 3 = 15, the student solves for Δ (the unknown).

Function is another idea that students practise. Students use the given rule to determine the output in a function table.

Download these free activity pages from Multiplication success with algebra by Jen Rubera

Click here for more information on Multiplication success with algebra by Jen Rubera

If you are reading this, then you are one of the first to access the R.I.C. eNews and Views section of our website.

Our hope is that R.I.C. eNews and Views will provide you with interesting and thought-provoking articles regarding education in Australian primary schools, along with regular updates on what’s happening at R.I.C. Publications. Think of it as R.I.C.’s blog.

We aim to make it an easy read and one that you’ll return to with each update. Indeed, we want you to keep returning month after month, so if you have any suggestions regarding ways to improve the page we want to hear from you. We want the content to be fresh, interesting and worthwhile to you … the busy classroom teacher. So here’s a quick snapshot of some of the things we intend to offer:

• Teaching tips: We will include a monthly article that targets a subject area of current interest to teachers. Hopefully, we will give you some ideas and tips you can apply to your classroom teaching.
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• Special deals: Return to the webpage to keep a tab on any specials which we may have happening. This will allow you to keep up to date with the best resources at the best prices.

Reliable as a provider of easy-to-use reproducibles, recently R.I.C. Publications has widened our product offerings to include some great new products such as The comprehension box, Essential phonics, BrainSnack™, a large range of interactive whiteboard resources and much more. Of course, we will also continue to update our current teaching resources to keep them current with the Australian national curriculum.

And one final comment … 2012 is the year of the London Olympic Games and we are again the licensed educational publisher to the Australian Olympic Committee and will provide you with the best in educational resources to help you ride the wave of interest in this event with your classroom. Watch this space!

So, we hope this sounds as exciting to you as it is to us! And we want to hear from you. Our job is to make your life easier, so we need to know what types of problems teachers are encountering so we can help find a solution. Conversely, tell us what is working for you, so it can be shared with others.

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