Inspiring Curiosity With Numbers (Part 1)
Isaac Howarth - Assistant Headteacher
A prime example:
It may sound like hyperbole, but I think it is fair to say that maths is a polarising subject. Ask someone if they like (or liked) maths and you will hear very different responses. I don’t plan on venturing into the debate on how much factual information do children need to learn in school, but I am sure that for maths to capture the imagination of maths we need to do more than learn facts and answer prescribed calculations.
Over the next three posts, I hope to shift some attitudes towards maths and show that teaching maths can be an inspiring and exciting subject if you ask a few off-the-wall questions. Starting with prime numbers in the grocery store, I will also introduce some different ways to look at zero and then go to infinity. Whether you are a passionate maths geek, someone who has tried to love maths but found it difficult or a teacher looking for ideas, you can find something to talk about here. Especially since maths is an essential skill for the 21st century.
I chose to start with prime numbers because they are odd-ball numbers - well, except for 2, it’s an even-ball number. Primes have many remarkable properties and do all kinds of things from protecting your banking information to explaining why different musical instruments sound different form each other.
All prime numbers share one common feature that sets them apart from other numbers. Primes can only be made by multiplying two unique numbers and no other way. A quick example, 31 is a prime number because only the factors 1 and 31 can be multiplied to make the product 31. On the other had 32 is a called a composite number because it has more than one unique pair of factors. Can you name all the factors of 32?
I’m going to head down a different path, hunting for prime numbers in the grocery store. Packaging in grocery stores is a great example of composite, but not prime, numbers. Prime numbers are conspicuous by their absence. Think of 12 drinks, they could be packed six by two or three by four. They make a neat rectangle…or “array” to be mathematically more accurate. That doesn’t seem very spectacular, but think about all the different quantities drink packs come in: 8, 12, 15, 20, 24, 30. All composite numbers! Where are the primes? Prime numbers in packaging are rare because there is only one way to pack them as an array. There would only one way to pack 17 drinks as an array that could easily be shipped in a crate; and an awkward shape at that.
This may seem like an odd quirk that only a maths geek would find fascinating. Does anyone else care? Yes, Ikea does. Ikea managed to eliminate 30% of the empty space they had when shipping tea lights simply by arranging them in neat arrays…which means composite, not prime numbers. It looks like being able to separate prime and composite numbers has stopped Ikea from shipping so much empty space.
After reading this, and thinking about where all time prime numbers have gone, I hope you’ve found something interesting you didn’t know before. Maybe you see how maths can be made exciting by asking “why is…,” “why not…” of “what if…”? If we can inspire people to ask these types of questions and look at even simple things with renewed interest, then we’re on the right track for building a more mathematically curious society.
Did you find primes fascinating? How interesting could zero be? Be sure to check back later when it will be a post about nothing.
Inspiring Curiosity With Numbers (Part 2)
Isaac Howarth - Assistant Headteacher
Much ado about nothing:
First, let me apologise about the cringe-worthy pun in the title.
In a previous post, I wrote about the importance of sparking curiosity in maths if we are going to raise a future generations that are keen to explore the unknown as well as be secure in their mathematical skills. In this adventure, it will be all about the number zero. We all know zero as that round number that means nothing. As simple zero may seem (what could be less complicated than nothing?) but there is more to this oval than there seems. The origins of our Hindu-Arabic numeral system are fascinating, and I am pleased that studying the historical development of our current number system is a statutory requirement in Year 4, but this post will be more about my own thoughts on using simple questions and simple topics to fire interest in maths.
“What is the meaning of zero?” sounds like a deep philosophical issue, however, when presented to children, they are more than happy to explore “what zero means”. To the credit of the teaching profession, it has grasped this complex issue as part of the Year 4 maths curriculum. Imagine a class of eight and nine year olds discussing “What does zero really mean? Where did it come from?” It is certainly a positive sign that the national curriculum recognises the importance of viewing maths as a human creation and not a static set of rules handed down since time immemorial.
One way I have seen this topic approached is to have children create their own number systems without a zero and without some sort of equivalent symbol to represent nothing. They usually develop a convoluted system of blank spaces to indicated a place value with nothing or invent another symbol to represent a basic grouping of ones, similar to the X for a Roman 10. I have even had class debates about whether or not we really need to digit zero. The important lesson is that children are challenged to think about mathematics differently. They are encouraged to think about it as something that people have created as a way to represent ideas and solve problems. This type of maths certainly looks and sounds different from what you may expect from “maths” but maybe that’s a change for the better.
If you teach in lower key stage 2 or have children interested in this topic, I would recommend Angela Sparagna Lopresti’s A Place for Zero.
Next time, the topic of conversation will be infinity. I’ll leave you with one last thought…zero and infinity may be two of the most interesting ideas in maths. Infinity is colloquially used to mean an inconceivable amount, but think about holding an infinite amount of nothings in your hand. I can hold zero pencils, zero pens, zero rulers, zero books, zero paper clips,…and so on, all in my hand at once. I can hold an infinite list of nothings! Next time someone says they have been doing nothing, I hope you tell them there is a lot more going on with zero than just nothing.
Inspiring Curiosity With Numbers (Part 3)
Isaac Howarth - Assistant Headteacher
Infinity and beyond: inspiring the mathematics of the massive and the miniscule:
The future health of our society needs children who are capable and flexible thinkers with mathematics. That sounds quite obvious, but how exactly can we go about doing this? In our school, I have collected together a series of problems from both curricular and non-curricular resources to create challenge booklets that will appeal to all kinds of maths. Whether your preference is for visual, geometry- based problems or for number problems, there is something interesting to be tried. This idea of a collection of interesting problems to spark curiosity was inspired by the Wild Maths website, run by the Nrich project at Cambridge University. Sometimes trying something out just to see what would happen is the best way to find something you didn’t know. It reminds me of an old saying about getting lost is the best way to get somewhere you have never been before.
On the topic of infinity, it is an idea everyone is familiar with, yet many people would have a hard time giving a precise definition. It is common in popular imagery. Just recently, when I got my new Canadian passport, I noticed that it makes an appearance as a background image on one of the pages. However, what exactly counts as infinity and what does not? The name Gregor Cantor stands out as one mathematician who tried to formalise “infinite” beyond a broad idea of something being unknowably expansive. The detail of his role in the history of mathematics is too much to fit here, but he is a character worth reading about.
To help young children grasp this concept of how something can continue forever, the basic idea of a number line with no end is a simple introduction to infinity. As children reach Year 5, they are getting a better handle on decimal place value and the idea of continually subdividing a number into further groups of ten. They would be familiar with dividing a whole into ten tenths, and then dividing a tenth into ten hundredths, and then being able to divide a hundredth into ten thousandths. As this point, a pattern becomes visible and it comes easier to explain why you can carry on dividing decimal values into further groups of ten. Representing a gap from 0 to 1 on a number line, then marking 0.1, 0.01, 0.001, 0.0001, etc… reveals the interesting conclusion that it is possible to explain infinity without every reaching the number 1. Infinity is not always enormous, it can mean infinitely small! That really throws their understanding for a loop! Try it for yourself. It is simple yet, not often explored. (This is a take on one of Zeno’s paradoxes, which makes for a great cross-curricular link if you can manage to fit in a study of ancient Greece at the same time.)
As a side note, you can do this by continuing the pattern of whole number place value columns (10 ones make a 10, 10 tens make 100 10 hundreds make 1000, etc…), but that reinforces the idea of infinity as a massive number and misses the concept-shaking point of infinity fitting in between 0 and 1.
As the summer holidays approach, teachers are presented with a golden opportunity to do some exploratory mathematics of their own. Check out this video on how to make a Möbius strip and what it can demonstrate. This is a great way to give a concrete representation of infinity for those who are not at the stage of seeing the number patterns above.
Over the past few months, I have had the chance to collect together some thoughts on what I think makes for interesting mathematics. As a professional teacher, but non-professional mathematician, I firmly believe that setting the foundations for a bright future mean helping children grapple with concepts like “Why should I care about prime numbers,” “What does zero really mean” and “Can I put a limit on infinity?” With the right models and carefully planned steps, we can help them find fascination in these big ideas. This is a starting point for moving beyond the image of maths as a competition to yell out times tables answers or a race to complete of page of calculations. If we can instil the attitude that maths is an exploration and not reaching a destination where the answer has been pre-determined, then we are on the right track.
Marking webs - from cold task to test task
Nicky Willis - Headteacher
How do you get children applying what they learn in different contexts? So often the cry of ‘but they could do that’ accompanies any unaided, timed task when it is clear that what was thought to be learned is not truly secure. As a staff we often despaired when we have analysed test outcomes or tried to maintain standards achieved in May and June only to discover that they have diminished significantly by September. So we carried out research on memory, trying to find the key that will unlock the mystery of retention of skills and knowledge trying to find tools and strategies that the children can use to help it stick.
We think we might have hit upon something surprisingly easy. We have developed marking webs to structure feedback and give children targets to work towards, which they can then use to assess their own progress. Lots of schools are using a ‘cold’ task at the beginning of a unit to see where the children are in relation to what the unit is aiming to teach and a ‘hot’ task at the end of the unit to assess progress. What we have also introduced is a ‘test’ task which takes that unit marking web and places it into their theme work a few weeks later. Test task marking web can then be used to see how much they have retained and applied in a very different context.
Take this example from Year 5. The criteria for the marking web are decided by the year team and a cold task (limited input) is given before any teaching of the unit starts. Matthew and his teacher have used the school criteria (based on SOLOs taxonomy) to determine how independently and consistently he applies the criteria and to set some targets for the unit. A hot task at the end of unit is marked by Matthew and his peers to see how much progress he has made. A few weeks later a task in theme work - making clay angels and kings from their Victorian theme – is marked using the same English criteria from the unit on instructional writing. Matthew can clearly see that his learning is now secure in all areas, apart from his new constant target, which he didn’t get a chance to demonstrate.
All year groups are using marking webs in writing and in maths. Year 1 and 2 webs look simpler with less strands for the children to focus on and overall the impact has been substantial. Writing is above average at Key Stage 1 and Key Stage 2, the quality of work is a joy and children are really engaged in making their webs grow. Maths is on a strong upward trajectory and the buzz around visibly being able to demonstrate learning is tangible. There’s work still to do but we are pleased with the way our marking webs are helping children take ownership of their targets and their progress.
Overseas Trained Teachers – a credit or a curse?
Nicky Willis - Headteacher
There are many tales to trade over the current recruitment situation in schools and a topic that consistently raises its head in Slough is that of the use of overseas trained teachers (OTT for short). For some it is a most expensive last resort with agencies taking a cut of anything from £7K to £12K for placing a teacher who could potentially leave within 1- 2 years. For others it’s an exciting opportunity to head off to the far reaches of the commonwealth and engage directly with a pool of recruits. For us, at Cippenham Primary School, it has been an interactive and dynamic process that has enriched our school and created a stream of talented and dedicated teachers.
But you have to bear with the pain, invest the time, and reap the rewards.
Over the last 3 years we’ve recruited five Canadian teachers, two Australian teachers and one Canadian LSA. Initially we found the mismatch between educational languages difficult to assimilate and felt as though we were caught in a constant induction loop. Support and training felt disproportionately geared to our overseas recruits, homesickness and practical living needs often needing as much support as understanding the national curriculum and assessment expectations. Our first three Canadian teachers seemed overwhelmed by the demands of marking, homework, behaviour and meeting parental expectation. The school team worked together to support them by opening their classrooms, spending time sharing books and plans, giving supportive feedback and generally welcoming them into the school community.
Gradually we sensed a slight relaxation into new ways of working and by the October half term, confidence creeping into their teaching and, with confidence came a sharing of ideas from Canada. We learned how Canadian pedagogy shone a light on our development of teaching and learning with the brain in mind - how the Canadian teachers instinctively provided for active learning in their classroom, making links explicit to other learning, incorporating rhythm and music and using problem solving in creative and real life ways. We learned about life in Canada and Australia and how national values are part of the classroom and school routine. The children even learned the British National Anthem from an Australian teacher so they could celebrate the Queen’s 90th birthday.
We have now successfully applied to the Home Office to become a sponsoring academy licensed for visa applications and are proudly sponsoring one of our teachers to switch to a tier 2 general working visa. This will keep her with us for the next 4 years, maintaining her excellent teaching and developing her leadership. Another Canadian trained teacher is one of our Assistant Headteachers, developing exciting and innovative approaches to maths teaching that has seen impressive gains in progress for almost all pupils. Another Canadian trained teacher with dual nationality and who remains with us after two years, is an exceptional teacher and is looking forward to a third year developing her leadership and a bright career in the UK system
It isn’t always possible to retain overseas trained teachers; one of ours has returned because she missed home so much, another because she missed her family but what we gain as a school community from these teachers is immeasurable. Canadian teachers come because 40% of graduates in Canada don’t have a long term job, because most graduates will not get consistent work after university and because they have little opportunity to build relationships with a class of pupils. With the correct support they will stay for more positive reasons and become an integral, valued and productive member of a school, contributing to the teaching profession at both local and national levels.
With the whole school support, patience and understanding that underpin our ethos in Slough, overseas teachers can, like our home-grown variety, achieve their potential to be the very best teachers they can be. And that ultimately means we can give our pupils the best education we can.
Year 5 Class Teacher Ren James talks about working overseas in a Slough School.