Number patterns - what use are they anyway?

Maths in the early years is a lot of playing with numbers to discover what they do - turn them this way and that, add 10, take 5, make it as big as you can, what happens if... 

Essentially it's about patterns. Recognising number patterns is an important problem-solving skill. We want our children to see a pattern when they look at examples because they can then use that pattern to generalise and predict in another circumstance. Transferring the play to mathematical understanding is the goal.

Parents often ask how they can help their children with mathematical understanding. This is often prefaced with something along the lines of 'I couldn't do maths...' But everyone can play with patterns. They can be fun and creative and lead to every child finding mathematics accessible and enjoyable.

Here are some examples our Year 2s and Year 3s are playing with at the moment.

 Fluency, understanding, reasoning and problem solving with patterns like this is expected by mid Term 2 for all Year 3 students.

I don't get it...

This week's post is a guest post from Rose Marszalek,  our Head of School: Curriculum, who writes about her experiences learning mathematics as a child.

I don’t know about you, but when I went to school Maths involved blindly following rules such as A = (L + B) x 2, C = π r² , and putting a zero at the start of the second line when you were multiplying by 2 digits.  I had no idea why, but I did it because my Teacher told me to and it worked.  I loved Maths right up until Year 9.  In fact, until then I was determined that I was going to be a high school Maths Teacher.  But what happened in Year 9?  Well, that’s when I was asked to explain, justify and prove my mathematical thinking and reasoning.  And could I? No, I couldn’t because I’d just been following the processes and formulae drilled into me for the previous 8 years of my schooling.  In primary school in Queensland in the 1980s I was not taught or exposed to rainbow facts or near doubles or place value or the split strategy.  I dutifully learned all my number facts (and spelling for that matter) by rote and regurgitated them at the weekly tests and did pretty well.  I trusted the processes my Teachers drilled into me for addition, subtraction, multiplication and even the dreaded long division.  I loved it all.  But all this showed was that I had a good memory.  I made the connections and saw the patterns myself because my brain was naturally looking for them (and perhaps because my mother was a Maths teacher).  However, I couldn’t truly explain many maths concepts I thought I understood, until I had to teach them many years later to the students in front of me.

Today though, as we implement the Australian Curriculum Mathematics, we teach kids the patterns and expect them to reason, justify and prove their thinking, starting in Prep. We guide them to make the connections between related number facts, to use place value to multiply and to measure the area or a shape without a formula – controversial I know!  There is a time and a place for a mathematical formulae and vertical algorithms (or ‘sums’ as I called the pages and pages of them I did and loved in primary school).  This place however is not till later in the primary school years and in some cases not till secondary school.  Racing through number facts and rote learning them day after day is not the way of Maths anymore.  We need our children to make the connections and see the patterns, so that they know that 8 + 9 is 17 (near double or doubles plus 1) and can extend this to know that 80 + 90 is 170 and 800 + 900 is 1700.   Don’t be in a hurry for your child to get through their number facts.  Take the time to re-learn them yourselves with the strategies that your child’s Teacher is teaching them.  You just never know, something might make sense for you for the first time!

Number sense - what is it and how do we get it?

I've blogged about number sense before ( see The times they are changing - your times tables, that is.).  It's an area we have put a lot of emphasis on over the last 18 months. And we've seen that students with number sense approach maths with a 'can do' attitude (no political pun intended here), they enjoy maths more, they start problem solving tasks with purpose, and most simply, they just know what to do. (Hands up everyone who felt like this in a maths class at school!)

Last week a parent asked Kathleen Gordon, one of our Year 4 teachers, for some assistance in helping his child with maths. As luck would have it, Kathleen had spent the Friday before with some of our other teachers at the uni learning some new techniques and understandings for parent education in maths, and more specifically number sense.  Kathleen's reply is a 'must read' for every parent (edited below).

'There is a tension between wanting to reach year level benchmarks and making sure students have established a strong foundation in the basics first. For example, pushing students to memorise multiplication and division number facts before they have a sound understanding of addition and subtraction number facts will result in poor quality understanding of both concepts.

Research shows that rushing on leads to poor performance and anxiety about maths. In the words of a highly regarded maths educator - it's like building the roof of a house before the foundations are complete. You can do it but chances are it will fall over before too long.

This may not be what parents want to hear. There is no quick fix for any child. 

Research shows that both using timed practise drills and using strategies to learn number facts are effective. However, when students know and use the strategies, they can recall facts long after tests and use this information to work out extended facts (3 x 2 = 6 so 3 x 20 = 60) and approximation tasks, and hence problem solve effectively.

Our school program supports the learning of number facts in multiple ways including teaching strategies for learning particular number facts and doing regular timed practise drills. The strategies are taught at school and form part of the take home package to practise with as homework.

If students regularly use these strategies and repetitive drills at home and school (rather than just trying to rote learn and memorise the facts) they should in time develop the required fluency.

Some students prefer to learn with visual aids and others with auditory ones. We teach students to find their preferred approach and use it repeatedly. Repetitive activities will help students learn. The kind of repetitive activities you can try include:
- saying the strategy (when appropriate) when writing down the number fact e.g. 'I know that 5 + 5 is 10 so 5 + 6 is one more which is 11' and doing this repeatedly
- recording the strategies for particular number facts (see number 1) on a mobile device such as an iPod and listening to it repeatedly
- make a poster of the tricky facts and put them up where they will be seen (bedroom, on the back of the toilet door etc)
- playing with flash cards (you can make your own, download them from the Internet or buy them from the supermarket for about $10) or fact family cards (available from class teachers)
- playing ‘beat the timer’ games with a set of number facts (allow four seconds for each – ten number facts in forty seconds) play these on paper, on the Internet, CD Rom or iPod (or similar) device
- learning rhyming chants or listening to and signing along to multiplication songs (these can be purchased on the Internet)
 

Fact family cards which we use in the classroom are available from class teachers. You may like to use them too. They help students see the relationships between addition and subtraction and between multiplication and division. They are also useful because they appear to reduce the amount of number facts to learn - (learn one and get three more for free – i.e.. If I know 2 x 6 = 12, I also know 6 x 2 = 12 and 12 ÷ 6 = 2 and 12 ÷ 6 = 2 ).'
 
This is fantastic advice for any parent wanting to help their child 'know what to do' when faced with a mathematics problem - and these problems present themselves everywhere in life, too - at the supermarket (or anywhere money is used in a transaction), when calculating quantities of an item required for a project, when estimating time for bus timetables or the number of days until the holidays.  Maths is everywhere - and fluent number sense kids can navigate these situations with ease.

The times, they are a changing...your times tables, that is!

When I was in Year 4 my teacher would line us all up on a Friday morning for the times tables competition - you were the winner if you were the last one standing. I knew my tables VERY well - I practised them over and over and on a Friday morning was fast and accurate and most weeks was one of the last left standing.

I also had multiple routes for getting right answers to 'problems' such as 73-49=...well, 3-9 you can't do so you add 10 here, and you add 10 there, and then it's 13-9 and 7-5 and the answer is 24 - simple!

My strategies for calculating and memorising stood me in good stead until senior secondary school.

Year 11 and 12 maths was a nightmare as I struggled to apply what I had memorised to problems presented by the teacher or the text book. I could no longer memorise everything I needed to know to be successful, and like many kids, I grew to hate maths passionately.

Modern mathematics teaching focuses on children developing conceptual understanding and flexible thinking, that is, fluency - they need to understand why, for example, double 5 is 10, and how knowing that is important to understanding larger more complex numbers.

I sat with a couple of Year 3s the other day and talked with them about numbers. I wrote down on paper a 6 and a 9, and asked them - 'what is the sum of 6 and 9?' One went straight to his fingers by counting out 6 fingers, folding them over and counting on another 9 fingers. This is certainly not a strategy a student who is fluent with addition facts would use. The other child studied the numbers for a bit and then said, 'It's 15...yes, 15.' I asked her how she came up with that without counting. She looked at me and said 'I just took 1 off the 6 and added it to the 9 to make 10; that made a sum 10+5 and that's easy - it's 15'.

The second child demonstrated the thinking our teachers are striving to develop - she was fluent, she understands place value, addition and the associative property between numbers in order to arrive at a correct response - she was efficient, accurate and flexible in her thinking, all in a matter of a few seconds. The first child may well have arrived at the right answer eventually but without the key ingredient of actually fully understanding the properties of the numbers he was working with.

Developing fluency in numeracy has been a major strategy for development at Peregian Springs State School this year. We have been lucky enough to work with an esteemed educator in the field, Rob Profitt-White. If you are interested in these ideas you may like to listen to one of our teachers, Chris Wise, interviewing him on the topic - Fluency in Numeracy.

So when you are next helping your child with their number facts homework, ask them to explain how they arrived at their answer, rather than simply focussing on whether they give you the 'right' answer in the shortest amount of time. Thinking deeply about numbers takes time to develop and if you can do this, being the last one left standing on a Friday morning is not all that important.