Pavlov’s Danny

English: A St. Louis-style pizza in its delive...

Regular readers of this blog may already be familiar with Danny, a ten year old boy I tutor in maths.  You may recall how, by the  judicious use of a few mini pizzas, Danny was finally able to work with decimals without hyperventilating at the very mention of them.

This week it was time to do some revision and to move him further in his studies.

By now, I reasoned (correctly, as it turned out) Danny should be able to work with printed pictures of pizza.  He had reached a stage where the pictures alone had him salivating as effectively as Mr Pavlov’s little bell did for his canine subjects.

We had images of  stacks of ten pizza delivery boxes to represent tens, whole pizza images to represent units and tenths and hundredths cut from a spare one of these.  As long as there was something to remind him of the pizza experience, Danny was able to pick up or identify 31.34 pizzas.  Even 20.25, 1.72 and 3.06 were well within his grasp.

From here we moved to an image of three boys eating pizza in front of the TV.  I had written down how much pizza each had consumed and asked Danny to rank them in order of  who had eaten the most.  He poured over the numbers with the most intense concentration.

“Tim dot the least,” he announced, “‘Tos he only dot 1.23 pizzas.  Then it’s Ed, ‘tos he had 3.6 and – oh I wish I was Sam! He’s dot 23.6 pizzas!”

We tried several similar questions.  He didn’t make a single mistake.  For Danny, motivation is everything.  Numbers don’t motivate him.  In fact they often terrify him.  Pizzas, however, are benign and desirable.  It’s important, in Danny’s mind, to know who has the most.  He comes from a large family.  To him, this is a survival skill.

Half way down the sheet, he noticed that the questions changed.  No comforting tales of pizza-snacking friends – just a request to order a set of 5 decimal numbers from smallest to largest.  The kind of question he’ll be asked to do battle with in the SATs tests in a few short weeks.

He glanced at me in panic.

“What’s these?” he asked.

“They’re still decimals, Danny,” I reassured him.  “Just think of them like pizzas.  Every time you see decimals, just think pizza, OK?”

“Right,” he said, relaxing instantly.

To my amazement and delight, he continued to order the numbers correctly.  I showered him with praise as he sorted out this group:    14.8             18.4             41.8             4.18             81.4.

“You know we’re doing these sums at sdool at the moment,”  he said thoughtfully, as he munched on the chocolate biscuit I’d given him as a reward.  “And I’m no dood at it.”

“Do you think you might do better tomorrow if you think of them as pizzas?” I wondered.

“Yes, I’m sure I tould do it then,” he smiled.

When he left, I sat down to prepare next week’s lesson.  It would be yet another attempt to encourage him to learn his multiplication bonds.  ‘If only,’ I mused, ‘I could find a way of motivating him to do that.’

Well, it’s far from perfect, but maybe this will help…Danny's maths sheet


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Why some children don’t get numbers

This post is strictly for educators and parents with children who hate maths.  No esoteric stuff this time…

When I was a teacher, for some reason I always used to be given the top set for maths – year after year.  (I’m not going to get into the ‘Is streaming children for maths a good idea?’ debate, by the way; just saying that’s the way it was.)

Well I enjoyed working with the school’s brightest and best very much, but then one year, the head teacher told me he’d like me to work with the lowest set.  That got me really excited!

I always loved a challenge.  I spent most of the summer holidays pouring through the finished maths books of my new group, trying to work out why a bunch of hard-working and well-intentioned 10 and 11 year olds had apparently failed to understand the very basics of number, while their classmates had made such excellent progress.

Finally, I had it.  There was one simple step that this group of youngsters had somehow missed – and this was the key that would help them to understand.  It goes something like this:

In English, we have 26 letters:         a b c d e f g h i j k l m n o p q r s t u v w x y z       which we use to make words.

The words can be one or more letters long:        a  my  box  daft  every  garden  quickly  unlikely  difficult…

In maths, we have 9 digits:        1  2  3  4  5  6  7  8  9  and one gap-filler  0               which we use to make numbers.

The numbers can be one or more digits long:                 3    27    154    2013    53196   100481…

So how was I to get that message across to a bunch of kids who by now were approaching some kind of number phobia? They were happy to work with numbers up to about 20.  From 40 onwards they got somewhat panicky, and if a teacher tried to introduce three- or four-digit numbers, they’d start shaking and ask to go to the toilet.

Close-up of a bucket full of midi-sized Hama b...

My solution was to give each child an abacus.  The ones in educational catalogues were hugely expensive, coloured to appeal to 3 year olds and far too big to fit on their tables.  I paced around the local shopping centre for a while and ended up with a couple of packs of self-hardening clay, a bargain tub of those little fusible plastic beads and a stiff yard broom.  The bristles made sturdy but flexible and non-dangerous abacus sticks.  I set four of them into a little strip of clay and cut them off at exactly the height of 9 beads.  That way, the children would be able to make the numbers 1 to 9 on the ‘unit’ stick, but would be forced to take them off and start again on the ‘tens’ stick when they wanted to show the number 10.

These are my updated versions - note boy and girl-friendly versions to help children feel relaxed.

These are my updated versions – note boy- and girl-friendly versions to help children feel relaxed around numbers.

The picture shows how I now make them for individual students, using a loop of wire, with beads safely trapped into the abacus.  There are, of course, only 9 beads on each wire.

We spent many days ‘building’ numbers from 1 to 99 on the abacuses and writing down the equivalent number.  Suddenly the 4 in 46 was understood as ‘four tens’ and the purpose of the zero in 20 was clear.

I insisted that they were forbidden to use the other two sticks, until they were completely begging me to allow them to make ‘hundreds numbers’.
“Nah,” I grinned.  “You don’t like big numbers, remember?”
“We do!” they insisted.  “We can manage it,  We promise.  It’ll be easy!”
“Well, if you’re sure…” I said, looking suitably doubtful and working hard to suppress a triumphant grin.

Within weeks, these children were working confidently with three- and four-digit numbers; not just building and writing them, but using them in calculations.

The next stage was to turn the abacuses round and use them to show decimals. Instead of labelling our four sticks as Thousands, Hundreds, Tens and Units, we now had Units, Tenths, Hundredths and Thousandths.  There was a clear decimal point marked between the whole numbers and the decimals and I showed them how, in this looking-glass world, zeros had to be used as gap-markers from the left, not the right.  Otherwise, they were in familiar territory.

By the end of term, the class was happily creating numbers to four decimal places, and comments like, “I get maths now!” and even, “Actually, I LIKE maths!” were heard around the room.

Now, to my great delight, I’ve been asked to work with Simeon: 14 years old, great at English, clueless at numbers.  Here we go again!