> Those rules say that you cannot add fractions like "1/3 + 1/3 = 2/6", not because of any intuitive reason, but because it's disallowed by the 'rules' of fraction addition - that's it.
That is absolutely incorrect. There is an intuitive reason why 1/3 + 1/3 != 2/6. That reason is that 1/3 + 1/3 = 2/3, and 2/3 != 2/6.
The important thing here is to help students build the intuition that mathematical notation shouldn't be treated mechanically, you should think about what the notation represents. The temptation to say 1/3 + 1/3 = 2/6 only comes when you're blindly applying operators to notation.
Now, the example from the classroom is more subtle, because it deals with an improper translation of the English into mathematical notation. If I say 'one third of the students at table 1 are girls', that should be translated to 1/3 * 3, not 1/3. Applying this rule gives 1/3 * 3 + 1/3 * 3 = 2/6 * 6,which is perfectly correct. Similarly, 1/3 * 3 + 1/3 * 3 = 2/3 * 3 is obviously correct.
My only point is that analogies are flawed. The abstraction that they provide hides complexity that at some point will leak out.
>There is an intuitive reason why 1/3 + 1/3 != 2/6.
You and I have different ideas of what 'intuitive' means. It's 'intuitive' once you understand the rules of fractions and what they mean. It's not so easy to derive this rule if you're working in the space of real world things.
And sure, I agree you can ad hoc extend the analogy of tables to bring it in line with the underlying mathematical rules, but then your analogy is no longer as simple as it was. The complexity is leaking out of the abstraction you had it under.
>The important thing here is to help students build the intuition that mathematical notation shouldn't be treated mechanically, you should think about what the notation represents
Sure, using abstractions and analogies is a powerful way of teaching. All I did was point out that analogies have limits and at some point they can become detrimental to understanding the fundamental concepts.
This is a common complaint by Physicists when doing public lectures on Quantum Mechanics and then having people extrapolate from the metaphors to derive incorrect physical rules (e.g. faster-than-light communication from a shallow understanding of quantum entanglement).
>because it deals with an improper translation of the English into mathematical notation.
It isn't just about the improper translation to English. It is also about the improper mapping of fractions to real-world things. 2 girls out of 6 kids in a table maps nicely to the fraction 2/6. But even though 2/6 is equivalent to 80/240, the latter is a little harder to map to a table of 6 kids and 2 girls - don't you think?
>The temptation to say 1/3 + 1/3 = 2/6 only comes when you're blindly applying operators to notation.
I disagree with that in context of learning how fraction operators work. The fourth-grader logically extended the analogy that they were given because conceivably, there could have been an operator defined that matched their intuition, for example, let's call it
'@' and define it (not rigorously) as "a/b @ c/d = (a+c)/(b+d)". This operator, if existed, would work very well for combining tables of boys and girls and getting the fraction of girls to match the fourth grader's intuition. The fourth-grader is learning fractions for the first time, and that operation could have conceivably existed - so the only reason they were wrong is that they haven't been told what the rules of fraction addition are and NOT that they misunderstood the analogy. The problem is that the "+" operator does not work that way because it isn't defined this way as per axioms for fractions.
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.
That is absolutely incorrect. There is an intuitive reason why 1/3 + 1/3 != 2/6. That reason is that 1/3 + 1/3 = 2/3, and 2/3 != 2/6.
The important thing here is to help students build the intuition that mathematical notation shouldn't be treated mechanically, you should think about what the notation represents. The temptation to say 1/3 + 1/3 = 2/6 only comes when you're blindly applying operators to notation.
Now, the example from the classroom is more subtle, because it deals with an improper translation of the English into mathematical notation. If I say 'one third of the students at table 1 are girls', that should be translated to 1/3 * 3, not 1/3. Applying this rule gives 1/3 * 3 + 1/3 * 3 = 2/6 * 6,which is perfectly correct. Similarly, 1/3 * 3 + 1/3 * 3 = 2/3 * 3 is obviously correct.