Gather

When Gather was a child, he learned this lesson the hard way, and it left a mark.

He had grown up helping at his family's button-and-cloth stall in the market, where his particular job was to add up how much fabric a customer had brought. One busy afternoon a tailor dumped a whole pile of scraps onto the counter — some of them cut into fifths, some of them cut into quarters, all of it jumbled together in a heap.

"How much cloth altogether?" the tailor asked, already counting out coins.

Little Gather started counting the pieces, one by one. One, two, three, four, five, six. "Six pieces!" he announced proudly, certain he had been quick and accurate.

The tailor frowned down at him. "Six pieces of what, though? Look properly. These three here are fifths. These three are quarters. They aren't the same size as each other. You can't simply lump them together and call it six."

Gather looked again, and his stomach sank. The fifths were narrow, skinny slips of cloth. The quarters were noticeably wider. He had counted them all as though they were identical — but they were nothing of the sort. Six pieces, certainly. But six pieces of nothing in particular, a total that didn't actually mean anything.

His ears went hot with the discovery. He had been adding things up his entire life, and it turned out he'd been doing it wrong — or at the very least only half-right, the whole time. You could count pieces, yes. But only if every piece was the same size as the others.

"How do I fix it?" he asked, quietly.

The tailor's frown softened into something kinder. "You make them match first," he said. "And then you count."

Gather came to the academy years afterward, basket on his back, determined to teach the one thing that had once tripped him so badly.

Slice, the old tortoise who ran the place, set a test on the table the moment Gather arrived: two-sixths of a chocolate bar, and three-sixths of a chocolate bar. "Add them," he said.

Gather dropped both into his basket and gave it his customary shake. "Five-sixths," he said at once. "Both of them already cut into sixths — the pieces match — so two plus three is five. Five-sixths."

Then Slice slid a trickier pair across the table toward him: one-half of a bar, and one-third of a bar. "And these?"

Gather did not shake his basket. He stopped completely. He set the two pieces down side by side on the table and studied them with real care. The half was large. The third was distinctly smaller. They did not match, and he could see it.

"I can't add these yet," he said. "Not honestly, anyway. The pieces are different sizes, and if I just declared 'two pieces,' I'd be telling a lie about how much there really is. First I have to make them the same. I'd cut both of them into sixths — the half becomes three-sixths, the third becomes two-sixths. Now they match. Now I'm allowed to gather them. Five-sixths."

Slice nodded, slowly and with evident approval. "Most people shake the basket first and do their thinking afterward, if they bother at all," he said. "You thought first. You may stay."

In his corner of the academy, surrounded on all sides by baskets, Gather met a thoroughly frustrated otter named Wren, who dropped into a chair as though the chair had wronged her.

"I hate adding fractions," Wren announced. "I did one-fourth plus one-half, and I got two-sixths, and my teacher said it was wrong. But I added the tops together AND the bottoms together, which is exactly what you do with ordinary numbers, so how can it possibly be wrong?"

"Ah," Gather said, gently. "That's the trap, and it's a clever one — it feels completely reasonable right up until it isn't. Let's gather them for real and watch what happens." He took out a fourth-piece and a half-piece and set them both in his basket. "Look at them. They don't match — the half is plainly bigger. If we just smush the numbers together on paper, the basket has no idea what size the pieces actually are."

Then he took up the half-piece and showed Wren, patiently, how it could be cut into two equal fourths. "Watch closely. One half is the very same amount as two-fourths. Now both of our pieces are fourths. They match." He dropped them in together. "One-fourth plus two-fourths. Three pieces, every one of them a fourth. Three-fourths."

Wren stared at the basket. "So the bottom number just... stays the same, once they match?"

"The bottom number tells you the size of the pieces," Gather said. "Once the sizes match, you stop touching the bottom entirely. You only gather up the tops — how many pieces you've got. Three-fourths."

Wren let out a startled laugh. "I was adding the sizes together. That's like saying two baskets plus two baskets equals four... basket-baskets. Which is nonsense."

"Exactly so," Gather grinned. "Keep the size. Gather the count. That's the whole job."