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>The fourth grader is saying "1/3 + 1/3 = 2/6" but the idea she's trying to get across is that "avg(1/3,1/3) = 1/3 but the size of the whole has doubled." I di

>The fourth grader is saying "1/3 + 1/3 = 2/6" but the idea she's trying to get across is that "avg(1/3,1/3) = 1/3 but the size of the whole has doubled." I disagree. The fourth-grader is perfectly correct within the context of the analogy that the teacher used. The problem is the analogy is wrong which is a general problem of relying on metaphors and analogies to explain rigorous technical concepts. Fractions are not like tables of girls and boys. There are rules for how you add fractions that flow from the underlying axioms. 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.

wcarey 6y ago
How would you explain that successfully to fourth graders? (Or, for that matter, seventh graders?)
- bradjohnson 6y ago
  I would avoid multiplication entirely and say that if you are combining tables, you need to update the bottom with the new number of total students. So something like: (T1 girls) + (T2 girls) 1 + 1 _______________________ = _____ (T1 total) + (T2 total) 3 + 3 I think that's more intuitive than averages, and would still work if the tables were of different sizes.
  - wcarey 6y ago
    This is definitely a good explanatory strategy (and one that I'd use). I might label the units of the right side as well. Would you attempt to explain the abstract rules of fraction arithmetic as an system?
Spivak 6y ago
You can't just ignore the fact that people didn't conceive of fractions as the smallest field containing an integral domain with a specific equivalence relation. They are an intuitive idea that has had its grammar formalized. It is absolutely beautiful to see fractions from the ten thousand foot abstractions but you can't skip to the end. Also, looking at maths from this pure beautiful abstract lens is something that some people (hiii!) enjoy but it's also the common language for modeling problems in the real world. And sure, we do sometimes intentionally confuse the fact that modeling systems with maths is supposed to be descriptive but we have no choice because otherwise we're betraying the fact that the attempt to describe these systems is where the notation came from in the first place. All this is to say that teaching math is hard and that the abstract "rules" view doesn't do it justice. It's practical for sure but it's also the reason you have lots of students who are good at pushing symbols around but not much else.
tsimionescu 6y ago
> 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.
- macspoofing 6y ago
  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.

wcarey 6y ago

How would you explain that successfully to fourth graders? (Or, for that matter, seventh graders?)

bradjohnson 6y ago
I would avoid multiplication entirely and say that if you are combining tables, you need to update the bottom with the new number of total students. So something like: (T1 girls) + (T2 girls) 1 + 1 _______________________ = _____ (T1 total) + (T2 total) 3 + 3 I think that's more intuitive than averages, and would still work if the tables were of different sizes.
- wcarey 6y ago
  This is definitely a good explanatory strategy (and one that I'd use). I might label the units of the right side as well. Would you attempt to explain the abstract rules of fraction arithmetic as an system?

Spivak 6y ago

You can't just ignore the fact that people didn't conceive of fractions as the smallest field containing an integral domain with a specific equivalence relation. They are an intuitive idea that has had its grammar formalized. It is absolutely beautiful to see fractions from the ten thousand foot abstractions but you can't skip to the end. Also, looking at maths from this pure beautiful abstract lens is something that some people (hiii!) enjoy but it's also the common language for modeling problems in the real world. And sure, we do sometimes intentionally confuse the fact that modeling systems with maths is supposed to be descriptive but we have no choice because otherwise we're betraying the fact that the attempt to describe these systems is where the notation came from in the first place. All this is to say that teaching math is hard and that the abstract "rules" view doesn't do it justice. It's practical for sure but it's also the reason you have lots of students who are good at pushing symbols around but not much else.

tsimionescu 6y ago

> 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.

macspoofing 6y ago
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.