I still don't agree. Sure, there are limits to particular intuitions, but all of the rules make perfect sense with real world quantities.
For example, 1/3 of an orange + 1/3 of an orange is actually 2/3 of an orange, not 2/6 of one. And 1/3 of 1 kg of flour is exactly 2/6 or 300/900 of that kg of flour. Sure, it's hard to talk about 1 Graham's number / 3 graham's number of 1kg of flour, so it does break down at some point, but unless you go overboard with quantities, all of the rules for fractions are in fact intuitive, and important for day to day things like cooking and money management. In fact, fractions and their operations are probably older than the idea of abstract rules, because they are fundamentally useful things.
The child in this example wasn't even making the mistake of thinking the rule for + is the rule for your @ operation. They were confused because they were trying to apply the intuition they had built up for how to translate real-world problems into fractions in the wrong way. Their result was in fact physically true: it was true that 1/3 of the children at one table + 1/3 of the children at the other table was equal to 2/6 of the children at both tables. This was confusing them because it suggested a different way of manipulating the numbers than they had just been shown.
The right solution, again, was to teach them how to translate 'a fraction of something' to rational numbers - that is, to multiply the fraction by the something, with only a special notational case when that something is 1. If they had known to do this, their intuition would have translated directly into the correct algebraic formula. No need to learn the abstract rules yet.
For example, 1/3 of an orange + 1/3 of an orange is actually 2/3 of an orange, not 2/6 of one. And 1/3 of 1 kg of flour is exactly 2/6 or 300/900 of that kg of flour. Sure, it's hard to talk about 1 Graham's number / 3 graham's number of 1kg of flour, so it does break down at some point, but unless you go overboard with quantities, all of the rules for fractions are in fact intuitive, and important for day to day things like cooking and money management. In fact, fractions and their operations are probably older than the idea of abstract rules, because they are fundamentally useful things.
The child in this example wasn't even making the mistake of thinking the rule for + is the rule for your @ operation. They were confused because they were trying to apply the intuition they had built up for how to translate real-world problems into fractions in the wrong way. Their result was in fact physically true: it was true that 1/3 of the children at one table + 1/3 of the children at the other table was equal to 2/6 of the children at both tables. This was confusing them because it suggested a different way of manipulating the numbers than they had just been shown.
The right solution, again, was to teach them how to translate 'a fraction of something' to rational numbers - that is, to multiply the fraction by the something, with only a special notational case when that something is 1. If they had known to do this, their intuition would have translated directly into the correct algebraic formula. No need to learn the abstract rules yet.