"Well, is it the 20 or the 3?" I ask. Totally bewildered now, her face creases and she looks down. Hurriedly, I rack my brains. "Do you know about tens and units?" Her face clears. She nods yes. "Would you put 20 in the tens column or the units column?"
"Tens," she whispers.
"Good girl!" I boom heartily, causing her to flinch back in her chair. "So," I go on, with cautious optimism, "Where would you put 3?"
"Units?" - very doubtfully.
"Good. So - how do you write 23?" "Don't know."
I start again, with some coloured plastic bits and bobs. As we go doggedly on, it starts to feel as if Millie really does understand the principle of writing down numbers in columns. What she doesn't grasp, or, perhaps, doesn't like, is the way you can take apart and reconnect all the numbers. This kind of promiscuity numbers go in for, the wanton way in which 20 will meet up with 3 and straightaway become 23 and then, without a moment's hesitation, break hands and go back to being just 20 and 3 again, is not at all to her taste.
Is this, I wonder in the kind of half-mad dissociation which sometimes overcomes me in the classroom, because her father died when she was very young and she hates breakings up? Or is it because she is a very kind-hearted girl who finds it difficult to complain even when the class terror pretends he is Dracula and bites her?
Or does she just not grasp number: not place value, not the two operations so far attempted, but number itself?
Perhaps we need an intermediate concept between the counting activities of the reception class and this year's manipulation of numbers.
Of course, it is clear that some children can abstract for themselves this commutative property of number and they are the ones we say have "an aptitude for maths". Then there are those who can scarcely rely on their counting skills. But for a large middling group, like Millie, there seems to be a barrier which timidity or cast of mind forbids them to cross. This is a pity, because for all the talk about "using and applying maths in the real world", maths is about abstraction and relation and not about how many pieces of pie there are - although how many pieces of pie there are is something to do with maths.
Millie and I agree on how to write 23 but I hold back on seeing if she can also write 33. Meantime Eddie, who has been listening quietly, holds up his paper. "See, Miss," he says. "One hundred and twenty three. Same as 1,2, 3. Why?"
I gulp. I think I need some of those intermediate concepts. "Well," I begin, "you know about making a list?" He shook his head. Shall I go off on "making a list"? No, we're doing maths. So, "How do we know it's a hundred?"
He looks at me as if I'm two digits short of an answer. "Because it's in the hundreds column."
"Good," I say enthusiastically, "Good." Where do I go from here? "So, we know it's a hundred because it has numbers in the other columns after it."
He gives me a funny look just as I realise that the list "1,2,3" also has two numbers after the 1. Fortunately, the bell rings as I am struggling to formulate some answer relating to setting out, punctuation and context. It takes me all the way home to realise that I, too, find the ability of numbers to metamorphose into other numbers deeply unsettling. I want there to be answers. But, actually, there are only manipulations and relations: a truth about maths which Eddie, aged six, enjoys and which Millie and I find scary. Is it coincidence that we are both female?
Bob Jelley is head of St Giles Middle School, Warwickshire. Always keen on children's writing and books for children, his best lessons invariably come under the general heading of English.
"My real favourite is a straight crib from Sandy Brownjohn. She did a course in Warwickshire some years ago and it changed my English teaching completely, " he says. "The lesson is intended to inspire the children to write about the making of a shadow. What it does really is introduce pupils to the idea of moving from concrete to abstract ideas.
"You start by talking about shadows they might know of - famous ones such as the one in Peter Pan, where the Nurse pulls the window down and cut's Peter's shadow off. There's also an interesting shadow idea in Ursula Le Guin's A Wizard of Earthsea.
"Then you talk about the quality of shadows, and with luck someone will start to let their imagination go. They might say that a shadow has to be flexible, and it has to be able to grow and shrink quickly - perhaps as you pass a street light.
"It also has to be a good mimic, because it copies everything that you do. The children start with concrete, physical properties - that because it's black for example, it would need to be made from tar or black paper.
"After a while you start to move into more abstract ideas - and someone will say that a shadow is fast moving for example, like the dark breast of a cheetah, or that a shadow is able to change quickly, like a dream. Always, you see, there's this move from the real to the abstract.
"Then when you have these ingredients, you talk about how ingredients are mixed together. You'll ask for words and phrases from recipes - 'weigh in a little ofI' 'fold gently inI' 'a shake of...' "They do their writing then - 'The making of a shadow' , and perhaps you let them draw silhouettes on black paper and stick them to their work. I'd do this lesson with Year 5 or 6 - perhaps it works best with Year 5.
"Sandy Brownjohn's books and ideas are pure magic. They introduced me, for example, to the idea of teaching children to write in different poetic forms such as haiku, and I find that they really do assimilate these and produce them later when they are able to choose their own forms."
u Sandy Brownjohn's To Rhyme or Not to Rhyme is published by Hodder Stoughton Educational, Pounds 11.99. Her new book of poems for children, Both Sides of the Cat Flap, will be published by Hodder's Children's Books in July.