«Welcome to Issue 60 of the Secondary Magazine. Contents From the editor What are the natural thinking powers of human beings? How can we help ...»
Welcome to Issue 60 of the Secondary Magazine.
From the editor
What are the natural thinking powers of human beings? How can we help students use and develop them
It’s in the News!
A long journey home
The fortnightly It's in the News! resources explore a range of mathematical themes in a topical context.
During March 2010 the Icelandic volcano Eyjafjallajokull started to erupt. By 14 April, the volcano entered
an explosive phase and released clouds of mineral ash which formed a cloud which rose up to 10 km into the atmosphere. Air Traffic Controllers decided that as ash can lead to engine failure they would close the airspace, leaving thousands of travellers stranded across the globe. This resource invites pupils to reconstruct a journey home from Prague to England as a result of the ash cloud.
The Interview – Adrian Pinel Adrian created the ATM Mathematical Activity Tiles and Loop Cards. He chose mathematics because it was presented to him at school in a mind-numbingly boring way, and he was amazed when his eight-year-old son seemed to know intuitively how to build a 60-pentagon space-shape.
Focus on…numbers in colour How can Cuisenaire materials help students use their natural thinking skills to learn to do arithmetic competently?
An idea for the classroom – digital operations Working with digits can stimulate mathematical discovery and creativity in all learners.
Historical snapshots: some figurate numbers at Key Stages 3 and 4 Seeing figurative numbers, such as triangulo-triangular numbers - which are triangular numbers in four dimensions, in their historical contexts brings new life to students’ explorations.
5 things to do What is an Uncanny Cube and a Canny Uncube? And have you read Alex’s Adventures in Numberland? Also find out what the mathematics subject associations are up to.
Diary of a subject leader Issues in the life of an anonymous subject leader The subject leader chooses classes to use for demonstration lessons, drops off work for his lessons near a recycling bin, and helps to appoint a new teacher.
Contributors to this issue include: Snezana Lawrence, Sue Madgwick, Mary Pardoe, Adrian Pinel, Peter Ransom and Heather Scott.
.www.ncetm.org.uk A Department for Education initiative to enhance professional development across mathematics teaching From the editor What do your students believe that mathematics is? What do they think that doing mathematics is all about?
Mathematicians create, invent, conjecture, and experiment. And it is only by doing in school those kinds of things, by thinking mathematically, that students can use mathematics effectively outside and beyond school.
Students who believe that mathematics is a given body of knowledge and standard procedures, a set of truths and rules that they need to be shown, struggle with mathematics in school and in the outside world. They oftenfear and dislike mathematics, avoiding it if they can. Inspectors reported – in Mathematics: understanding the score (Ofsted, 2008) – that many pupils say ‘I don’t like maths because I’m no good at it. It’s boring. I prefer active or creative subjects.’ For generations, teaching mathematics was concerned with communicating established results and methods. Teachers tried to prepare students for dealing with mathematics in life by giving them ‘a bag of facts’. Consequently avoiding mathematics has become respectable. Many otherwise successful people are happy to announce that they ‘cannot do maths’, whereas few people would say publicly that they cannot read or write.
In their paper, The Mathematics Education Landscape in 2009, the Advisory Committee on Mathematics Education (ACME) expressed their belief that many students are not doing mathematics as well as they could, or not really doing it at all. We were reminded that mathematics teaching too often depends on telling students methods, rules and facts, and too rarely on helping them to make sense of what they can find out so that they can use mathematics independently.
In the 21st century we want to empower students; we want to help them develop genuinely mathematical ways of thinking. And these can be developed from the natural abilities that all people are born with.
Caleb Gattegno explained in his book What we owe children that we all used natural thinking powers when, as babies, we taught ourselves how to speak a language.
So, what are these powers? Well, as human beings we naturally:
imagine and picture things in our minds, and talk about what we imagine notice differences and similarities between things find what is the same about things we come across that in other ways are different generalise specialise conjecture – make an ‘educated guess’ about what is true use various forms of reasoning, or argument, to try to convince others that what we believe is true.
One of the best ways to help students to think mathematically is to look out for opportunities for them to make use of their natural abilities naturally. Then you can point out to them what they did by themselves.
While being interviewed in 1985 Caleb Gattegno said: “Everybody wants to work on weakness, but I work on strengths.” I hope that this issue helps us think about how we can implement these ideas.
During March 2010 the Icelandic volcano Eyjafjallajokull started to erupt. By 14 April, the volcano entered an explosive phase and released clouds of mineral ash which formed a cloud which rose up to 10 km into the atmosphere. Air Traffic Controllers decided that as ash can lead to engine failure they would close the airspace, leaving thousands of travellers stranded across the globe. This resource invites pupils to reconstruct a journey home from Prague to England as a result of the ash cloud.
The activity gives students the opportunity to deal with some unfamiliar and, in some ways, complex data – it uses real data gathered (at great expense) during the journey. Students could be asked to suggest some questions they would like to ask about the situation which could be answered using the data given.
This resource is not year group specific and so will need to be read through and possibly adapted before use. The way in which you choose to use the resource will enable your learners to access some of the Key Processes from the Key Stage 3 Programme of Study.
Download this It’s in the News! resource - in PowerPoint format
About you: I’m originally from Jersey, in the Channel Islands. Born at the end of the Occupation, I’m old enough to have experienced the 60s as a young teacher – yes, I was there, and no, I do remember it! I’ve since taught across the board in schools and at university. I was Head of Department at two schools when I found that Mode 3 CSE was a great way to break out of the mundane. Moving into college work, I found myself running a Maths Centre for the LEA. Highlights of this period were devising Loop Cards in 1978, and ATM MATs in 1980. Both of these were in response to urgent teaching needs that I had observed and identified. Such needs have motivated me throughout my career.
Why did I choose mathematics?
Because it isn’t there – yet it is to be found in everything of interest in the world and in the universe.
Finding it and playing with it is such a great way to earn one’s living. Also, because it was presented to me in such a mind-numbingly boring way at school that even at primary level I was forced to invent my own games and puzzles based upon mathematics. I guess that the invention of Loop Cards was in part the result of my revolt against that uninspiring teaching.
Who has inspired me?
At first, Bill Brookes, who was my PGCE tutor and later my Masters tutor. He is sorely missed. In the late 1980s, Adri Treffers and Ed de Moor, with their Empty Number Line, and Erich Ch. Wittmann and his great group of colleagues at Dortmund for their imagery-based approaches to number. Jeni, for being arguably the best teacher I have ever seen – most naturally talented, and vigilantly true to her principles. ‘Reader, I married her’.
The best book on teaching mathematics that I have ever read is...
Starting Points by Dick Tahta and Ray Hemmings. Indeed, I return often to almost everything I have that was written by Dick Tahta, and fondly recall the many insights he gave me in the long conversations we shared over the years.
Some mathematics that has amazed me?
There is so much – but what my sons, aged eight and six, did with ATM MATs when I brought home the first sets that were produced takes some beating. Matthieu (eight) arranged 10 regular pentagons to fit into a flat circular pattern, then he joined these with latex glue, so that they flexed into 3D. He then asked me for another 50 pentagons, and with Oliver’s help produced the 60-pentagon space-shape that was later put on the cover of the first ATM MATs Handbook. It was many years later that Paul G worked out the geometrical basis for this figure. I still don’t know how Matthieu ‘just knew’ he needed exactly 60 pentagons!
A significant mathematics-related incident in my life...
was visiting the Alhambra and the Real Alcázar with Kev Delaney during ICME in 1996. We spent many hours in the Alhambra rooms absorbing the geometry, as I subtly, permanently reorganised my whole picture of the world of tessellations, and of their projections into domed ceilings.
The most recent use of mathematics in my job was...
devising the activities and sessions for a four-day conference on Problem Solving and Problem Posing for
A mathematics joke that makes me laugh is...
since the late 90s, the National Numeracy Strategy. Well, a hollow laugh, that is! This is tinged with sadness that so much human effort and expensive resources were used to so little good effect, meanwhile blighting the breadth of the primary curriculum.
OK, there are numerous mathematical jokes in Douglas Adams’ The Hitchhiker’s Guide to the Galaxy, but the observational humour of Dave Allen, explaining how he tried to teach his son to tell the time is ten minutes of humorous, awareness-raising narrative that always cracks me up. For example: “On each clock there are three hands. The first hand is the hour hand. The second hand is the minute hand, and the third hand is the second hand…” What has kept me motivated?
The dire straits we got into over not having enough mathematics teachers. For the past seven years my main focus has been on developing the best possible Subject Knowledge Enhancement courses, utilising the talents of some old friends and new colleagues in making these happen.
If I was not doing this job...
I would probably spend far too long finishing fiendish crossword puzzles – Azed used to be my mainstay. I doubt I could kick the habit of inventing mathematical puzzles and problems – you see, I couldn’t really stay away! However, without groups of students or teachers to work with, I’m not sure if the source of my inspiration would survive for long. It always seems to be sparked off by what they are doing, saying, asking. I still want to write down more of the mathematical activities and problems I have devised or adapted over the years. I drew great satisfaction from the Mad About Maths set of three books, as well as from my many articles and more recent publications. As I’ve just retired to become a freelance consultant and author, watch this space!
“Georges Cuisenaire showed in the early fifties that students who had been taught traditionally, and were rated ‘weak’, took huge strides when they shifted to using the (Cuisenaire) material. They became ‘very good’ at traditional arithmetic when they were allowed to manipulate the rods.” The Science of Education Part 2B: the Awareness of Mathematization by Caleb Gattegno.
Because students who cannot do arithmetic cannot function effectively in most mathematical situations that arise in life, it is worth exploring any resource that may help them. And teachers have found that Cuisenaire materials can help all learners, not only young children. They have found that when students think about and manipulate unmarked Cuisenaire rods the students use their natural thinking skills to learn to do arithmetic competently and naturally. For example, learners see in a single arrangement of
rods how numbers are related at the same time by both addition and subtraction:
This arrangement reveals naturally three interchangeable ways of ‘seeing’ the relationship – that red + green = yellow, yellow – red = green and yellow – green = red. Students observe that no one of those three ‘ways of seeing’ dominates, and if you can ‘see’ it in one way you know that you will also be able to ‘see’ it in either of the other two ways.
And multiplication, division, and fraction facts, and relationships between those ideas, appear naturally to
students when they think normally about an arrangement such as this:
(Three red blocks are the same length as a green block, the number of red lengths that there are in a green length is three, and a red block is one third of a green block.) The teacher’s role is to prompt students to focus on what they are naturally aware of, to help them realise how much they know naturally so that they develop confidence in their own thinking.
During 1953, Dr Caleb Gattegno saw young students in Georges Cuisenaire’s classroom learning to do arithmetic competently and rapidly using only coloured rods. Many years later, Gattegno told English mathematics teachers that “Children could improve their mathematics if they worked with Cuisenaire rods instead of notation and verbiage… …there was something in the manipulation of the rods which made their mind clear about conventions and notations and other things….” (you can listen to Gattegno on the ATM website).
To see for yourself how Georges Cuisenaire worked with pupils using his rods (and nothing else) in an Alpine primary school, it is well worth persevering 07:18 minutes into this French film.