ATM People • Kath Cross

Kath Cross is a long-standing contributor to Mathematics Teaching and to mathematics education generally. Here we collect some articles and anecdotes from her and about her.

The Weight of a Snowflake
A Comic Dilemma...
A Cross to Bear
Engagement and Excitement in Mathematics

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Kath Cross Research Day 2003

Mathematics Teaching MT189 (December 2004) is almost entirely devoted to the research day that was held in November 2003 to celebrate Kath Cross’s contribution to mathematical education over the last 40 years. Over 50 friends and colleagues from all over the country converged on a hotel near to where Kath has always lived in Lancashire.

<< Read Mathematics Teaching 189

The Weight of a Snowflake

Tell me the weight of a snowflake, a coal mouse asked a wild dove.

Nothing more than nothing, was the answer.

In that case I must tell you a marvellous story, the coalmouse said. I sat on the branch of a fir, close to its trunk, when it began to snow - not heavily, not in a raging blizzard: no, just like a dream, without a sound and without any violence. Since I did not have anything better to do, I counted the snowflakes settling on the twigs and needles of my branch. Their number was exactly 3,741,952.

Having said that the coalmouse flew away. The dove, since Adam’s time an authority on the matter, thought about the story for a while, and finally said to herself: Perhaps there is only one more person’s voice lacking for peace to come to the world.

A good story but what is it about? Mathematics? The importance of the individual? Not leaving decisions to others?

When David Wheeler spoke at Lancaster in 1975 with that call to humanise mathematical education, he said the following, which is a quote also used by David Lingard in a closing lecture, but I have amended it to remove ‘he’ and I do not think David will mind:

“I don’t expect, and I don’t want, all children to find mathematics an engrossing study, or one that they want to devote themselves to either in school or in their lives. Only a few will find mathematics seductive enough to sustain a long term engagement. But I would hope that all children could experience at a few moments in their careers...the power and excitement of mathematics...so that at the end of their formal education they at least know what it is like and whether it is an activity that has a place in their future.”

An exciting challenge lies ahead as we take on board the National Curriculum and other aspects of the Act. We owe it to the children to continue to work together professionally, to share ideas, to discuss and to argue, to sharpen our thoughts about how we relate mathematics to the students and their needs and aspirations in order to be able to respond, not only to them, but to others who have a right to know.

I guess we have to be better at communicating, which could imply that we have to be clearer ourselves about what mathematical education is.

Amidst all the pressures and the rapid speed of change, let us not lose sight of what we believe mathematical education is really about.

A few weeks ago at church we sang a hymn which seems particularly apt for those in education at this time: ‘Grant us wisdom, grant us courage, for the living of these days’.

A Comic Dilemma...

Visit to a boy’s secondary modern school

Joined a class half way through and went to sit on a chair, at the back of the first aisle, that had been placed between two desks. The boys were sitting in single rows. The teacher was working at the board and the pupils were copying things down. There were no questions to the boys and the teacher rarely turned round.

The boy on my right (in the back corner) was coming to the end of his exercise book and was looking anxious. He would need a new book, but it would be difficult to attract the teacher’s attention - and embarrassing to interrupt the exposition.

As I could see that the boy was getting increasingly anxious I offered him a sheet of my paper. This was early in my HMI career and HMI were part of the DES. The paper in my clipboard was the official one with the Civil Service crest. He accepted it, but kept looking at the crest and at me. This paper was a bit posh! I gestured that it was OK to use it, and he did. After a few minutes he lifted up his desk lid, took out a comic and gave it to me whispering: “Read this; it’s always boring in here!”

Well I did not disagree with that - but what do you do? When you were at school and someone passed you a note, you had a desk in front of you to hide the note. But I was sitting on a chair in the aisle between desks. Do I read it and risk the teacher turning round and looking at the class for the first time in ages -imagine the headlines - or do I ignore it and lose my street cred with this young man who certainly deserved better?

I leave you to think about what I did!

A Cross to Bear

The following are the four pages of a leaflet by Lyndon Baker and Ian Harris.

Engagement and Excitement in Mathematics

This was the title chosen by those involved in organising the research day. I have to say that I thought this to be an excellent focus as well as the one on paragraph 243 of the Cockcroft report ‘Mathematics Counts’ [1]. This created a lot of discussion between teachers and others involved in mathematics education at the time. It is interesting that it still managed to do this 21 years later.

Inevitably, as I retire after more than 37 years as teacher and an HMI, this is going to be a very personal note. As someone who has not written much under my own name - because HMI could not do that - it looks pretty poor if, when I do write, I only say the things I have said before. But then does it matter? I decided not to look back over previous articles, booklets or talks, but to rely on my memory for key statements that have been significant me.

Not surprisingly, it turns out that all the things that were important when I started to teach and really began to think about how pupils learn are still important now. The things we are grappling with are still the same. These include the questions:

How do we involve pupils in their own learning?
How do we engage them with the mathematics and encourage them to want to know more?
How do we provide the necessary knowledge, skills and understanding for the future?

Several of my talks over the years have been based around the theme: children learning mathematics. These three words are each significant. That is:

children (or young people and adults);
learning;
mathematics.

Firstly, the way we teach is dependent on what we think about the children, (young people or adults) we are trying to teach. Are they ‘empty vessels’, or do they bring something with them? If so, how is this built on? How do we use what we know about pupils to help them to progress?

Secondly, the way we teach also depends on what we know about the way people learn. The decisions made by teachers, which are designed to help them to learn, are determined by this.

Thirdly, what the teacher thinks about mathematics and the purpose of mathematics teaching are crucial elements. When I worked in a tertiary college (under FE regulations) there were those who believed that mathematics was a service subject only. It was taught to give the young people the necessary basic knowledge and skills to be able to do whatever they needed to do in their engineering courses. There are still tensions between the provision of courses to help students move onto the immediate next steps and those which might provide knowledge, skills and understanding for the longer term. Does it really matter if pupils are excited by the work or not? I think it does.

An important document with which I was involved was the Non-Statutory Guidance written to accompany the first version of the national curriculum in 1989 [2]. It states that: ‘Mathematics is not only taught because it is useful. It should be a source of delight and wonder, offering pupils intellectual excitement and an appreciation of its essential creativity’.

But let’s go back to the Cockcroft report. Paragraph 243 focused on teaching (rather than learning), because that is what the teacher ‘controls’. It was, however, directly linked to learning because it grew out of consideration of what the teacher wanted the pupil to learn. (See paragraphs 240 and 241.) Different strategies are needed if the teacher is developing understanding compared with imparting factual information. The comments in the paragraphs were based on a review of research carried out for the committee.

A few years later, when I became an HMI, I thought there was a simpler reason for suggesting a range of teaching styles. This was to give variety to the lessons and more stimulus to the learning. In those days, the pattern of secondary mathematics lessons in any one school was pretty predictable; there was either exposition by the teacher followed by individual practice, or pupils worked individually through a scheme with little or no whole class interaction. When pupils turned up at a mathematics lesson the format was very predictable. There rarely seemed to be an element of surprise. This was similar in primary schools where work was predominantly based on an individualised approach.

A question for us now is whether pupils feel that their lessons are predictable and whether this leads to a rather mundane approach.

Of course, there is something to be said for structure: pupils like to know where they are and it is good for teachers to think about the time available and what they would like their pupils to achieve. I well remember my HMI colleague David Hale, in his closing address at the 1992 Easter conference in Lancaster, saying that many lessons did not have a proper beginning or end; they tended to just filter out. My own evidence supported this. But can we expect to tie things down at the end of every lesson? Some ideas take time to digest and need to be mulled over. This is where the teacher has to make a decision.

Paragraph 243 was the most famous paragraph and grew out of the consideration of a lot of evidence to the committee as well as the research. There were many other paragraphs that were important for me. One was paragraph 34, which is in the section on ‘The mathematical needs of adult life’. It stated that:

“Most important of all is the need to have sufficient confidence to make effective use of whatever mathematical skill and understanding is possessed, whether this is little or much.”

I hope we remember this when we discuss any proposals for the 14 - 19 or 16+ ranges.

The phrase ‘at-homeness’ with numbers and with measurement was key to getting across the messages. “That’s not mathematics, it’s common sense” (paragraph 65) was also key in trying to sort out the mathematics people did use in their ‘everyday’ lives. The person who said that was so ‘at home’ with the work that it was ‘just common sense’.

Earlier I mentioned ‘surprise’. I remember visiting a primary school with my HMI colleague Doreen Penn in the early days of the national curriculum. There was, understandably, concern about all the assessment involved. Many LEAs had produced tick sheets to help teachers to record attainment. Inevitably, many pupils' profiles, in the same year group, looked the same and it seemed almost like a waste of time filling in the forms. “But how do you record differences?” asked Doreen, concerned about the relatively small proportion who had attained higher than the expected levels, or those who had not reached them. She then asked: “How do you record surprises? What about the unexpected?” That made us all think - me as well as the headteacher! I wondered whether the tasks set enabled pupils to surprise the teacher or whether they restricted them.

Paragraph 420 in Cockcroft began with the sentence: ‘Testing, whether written, oral or practical, should never be an end in itself but should be a means of providing information which can form the basis of future action’. Lots of progress has been made but I wonder how much all the end-of-module tests we have nowadays help to determine future action.

In the early days of Ofsted, a limited number of reviews of research were commissioned. I was very pleased that the first such review was in mathematics; it was carried out by Mike Askew and Dylan Wiliam of King’s College, London [3]. The most significant quotation for me is:

“‘Knowing by heart’ and ‘figuring out’ support each other in pupils' progression in number.”

Others were:

“Moving from practical to formal work is often far from straightforward.”
“Learning is more effective when common misconceptions are addressed, exposed and discussed in teaching.”

Kath Hart was one of my HMI colleagues for a few years: we joined HMI on the same day in September 1984. The research she and her colleagues carried out on misconceptions and errors was very influential and thought provoking, and I also think of work carried out at the Shell Centre in Nottingham in this connection.

My first involvement in an ATM working group was as part of a research group, led by Alan Bishop, preparing for ICME in Exeter in 1972. The theme was: teachers as researchers in their own classrooms. It was good to be told that teachers had something to contribute. Pupils' attitudes to learning were also an important part of this work. Anne Haworth, with whom I taught, and I worked together on this and I know we learned a lot about ourselves as teachers and about the pupils we were trying to teach.

It was good for me to work with other teachers after the publication of ‘Mathematics Counts’ on translating paragraph 243 into practice. It is OK to write something in committee - but how does it work in practice? Barbara Ball and George Knights write about that in this issue. The work on ‘blobs and links’ was particularly illuminating and one of those things I hold on to. How do we help pupils to make links with what they already know? How do we help them to keep these links strong [4].

A quotation that is embedded in my mind was made by David Wheeler, who was my tutor on my PGCE course. It was made in his address to the 1975 ATM Easter conference and cited in MT 71. This was later quoted by David Lingard in his Easter address (reported in MT 107) and then by me in my own closing lecture, Easter 1989. I made it plural so that we can avoid saying ‘he’ [5].

“I don’t expect, and I don’t want, all children to find mathematics an engrossing study, one that they want to devote themselves to either in school or in their lives. Only a few will find mathematics seductive enough to sustain a long term engagement. But I would hope that all children could experience at a few moments in their career...the power and excitement of mathematics … so that at the end of their formal education they at least know what it is like and whether it is an activity that has a place in their future.” David Wheeler, 1975

I look back on a full-time career in education knowing that I owe a great deal to a lot of people. I have particularly valued my ATM links and this is why I have endeavoured to attend the annual conference every year, although sometimes it was difficult because of pressures of work and lack of school holidays. It is good to have the batteries recharged at least once a year and to go away fired up to tackle new things. Thanks to all those who have challenged me to think afresh about what was going on in mathematics classrooms.

Thanks to old friends for thinking of, and then organising, the research day. Thanks to those who were able to make the journey to North East Lancashire, and thanks to those who could not but sent their best wishes and greetings. Thanks also for the gifts. I am learning a lot about digital photography from the magazines I receive as part of the gift and the two courses attended.

Retirement has inevitably caused lots of reminiscing. The most important thing is that we do not just look back but move forward from where we are now. Much has been achieved but could we help today’s children and young people to learn their mathematics even more effectively so that they are engaged and excited by it? How do we help teachers, as well as pupils, to be more engaged in mathematics?

That remains a challenge.

Kath Cross is retired but doing occasional work for the DfES, Ofsted and LEAs.

References:

(1) W. H. Cockcroft: Mathematics Counts (The Cockcroft report); HMSO, 1982
(2) Mathematics Non-Statutory Guidance; HMSO, 1989
(3) Mike Askew and Dylan Wiliam: Recent research in mathematics education 5-16; HMSO, 1995
(4) Barbara Ball and George Knights: ‘Cockcroft 243 today’; MT189, December 2004, p 28.
(5) David Wheeler: ‘Humanising mathematical education’; MT 71, June 1975
David Lingard: ‘Myths’; MT107, June 1984
Kath Cross: ‘Sharing Perspectives... People learning mathematics’; MT130, March 1990