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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Danny and the Decimals

Spinach pizza

It was too good a chance to miss.  The Co-op over the road was offering two small pizzas for £1 and the note I’d scrawled to myself during my last tutoring session with Danny said, “Do decimals – pizzas!”

(If you haven’t encountered Danny in my blog before, you can find out more about him by checking this post.)

Oh we’d tried decimals together many times, Danny and me.  I’m sure various school teachers and teaching assistants had made similar attempts.  The result was always the same.  He’d start to twitch, writhe and wave his hands around.  His eyes became both wide and wild.

“I don’t DET decimals!” he would complain frantically.  “All weird points and stuff.  I don’t det dat!”

As we all know, trying to master a new skill while in a state of blind panic is next to impossible.  That’s where the pizzas came in.

As I ushered Danny into the kitchen rather than the study and began unwrapping them, he deduced that Christmas had been delayed but was now well and truly here.

“We doing dooting today?” he asked, hopefully.

“Well we need to cook these before we start on the maths, yes,” I told him. “Because we’re going to be doing maths with pizzas.”

We stood watching the pizzas as they browned and bubbled.  It took a mere six minutes – just the right amount of time to prepare Danny for the lesson.

“How many pizzas did we put in?” I asked.

“Two,” he grinned, salivating like one of Mr Pavlov’s canine subjects.

“Well when they’re cooked we’ll be leaving one of them whole and cutting the other one up,” I explained.

“Otay,” Danny nodded.  “Day smellin’ dood.”

“Yes, and do you think you’ll have room to eat some of the pieces?” I enquired casually.

Danny expressed a certainty that he would.

“But you will have to concentrate hard on the maths, Danny.”

“Oh, I will.  I promise I will,” he assured me.

“Because when we cut one up, it will be in pieces – decimal parts of a pizza.”

He didn’t even flinch.  “Otay.  Do you dint dey’re done now?”

Česky: Pizza

Danny was as good as his word.  He sat in the study, watching silently as I transferred a whole pizza to the first plate and carefully cut the second into ten slices.

“This,” I explained, wafting one piece of pizza in front of him, “is smaller than a whole pizza, isn’t it?”

He agreed that it was.

“There are ten pieces, so each one is a tenth.  We also call that zero point one of a pizza.  Two tenths would be zero point two.  Could you show me zero point six?”

No problem.  Danny went on to identify all fractions of a pizza to one decimal place.

Next I places three of the tenths onto the plate with the the whole pizza, showing him the plate now held 1.3 pizzas, while the other plate held 0.7.  He caught on instantly.  Once he’d shown he could move slices around to create 1.5 and 0.5 pizzas respectively, I asked him to lay out 1.6 and 0.3.  He held the left-over slice in his hand.  I suggested eating it, since there was nowhere else to put it.  Danny happily complied.

The next step was to divide one of the slices into ten minuscule slivers of pizza.  A couple disintegrated, so that they too had to be eaten.  However once he had been shown that these were hundredths of a pizza, he was able to deal with decimals to two places with no difficulty.  Within the space of half an hour, Danny was happily recording and constructing numbers and quantities such as 1.72 and 0.53.

“I think you get decimals now, Danny,” I remarked.

His eyes widened, but he was still calm.  “I dint I do,” he agreed.

A few more mouthfuls of pizza and several other maths activities later, I set him a final task.  We have been playing the ‘ten game’ for months, with little discernible progress.  It involves laying ten counters on the table.  I swiftly remove some and he is supposed to tell me – instantly – how many I’ve taken by counting those that remain. Number bonds to ten – another long-term sticking point.  Tonight, though, his answers were fast and accurate, for the first time ever.

“How did you suddenly get so good at this, Danny?” I asked.

He smiled slowly.  “I’m tellin’ myself dat dey’re bits of pizza!” he announced triumphantly.  “When I dee dem lite pizza I dan do dem easily.”

Staring happily into the middle distance he remarked dreamily, “Pizzas dan solve any problem.  Dey’re brilliant at maths!”

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