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Can 1/3 and 1/3 = 2/6? It seemed so

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Re: Can 1/3 and 1/3 = 2/6? It seemed so

#161

Post by tsimionescu »

CydeWeys wrote: post #113
tsimionescu wrote: post #99

Earlier quoted context omitted.

It's not really a typing problem, it's a problem of references. You can't add 1/3 apples and 1/3 oranges. However, you can add 1/3 * x people with 1/3 * y people and get 2/6 * z people, if x people +y people =z people.

You definitely can add 1:2 apples and 1:2 oranges, you just need to do so using a common base type (such as fruits or objects). Note that I'm using ratio notation because the answer for the above is not the same as adding 5:10 apples and 1:2 oranges; in other words, the exact numerator and denominator both matter, so it's not really a simple fraction; a simple fraction can be reduced to its lowest terms (e.g. 5/15 be…

You can't perform operations with apples, oranges and fruits unless you use them completely interchangeably ,i.e. 1 apple = 1 orange. Otherwise, you will break algebraic equalities.

You can convert apples to fruit and oranges to fruit and then do operations on fruit, but you can't go back from the result to apples and/or oranges. For example, 1/7 of (3 apples + 4 oranges) is 1 fruit, but we can't tell if it's one orange, or 1 apple, or 1/3 apple and 2/3 orange.

Re: Can 1/3 and 1/3 = 2/6? It seemed so

#163

Post by munificent »

Wowfunhappy wrote: post #111
forbiddenvoid wrote: post #40

The units are missing, and I think that's a key factor here. Both of the equations up on the board at the end are correct because they are counting different things. This is a huge miss if you use a numeric only approach to fractions. The student came up and wrote 1/3 + 1/3 = 2/6. What they meant by that is 1/3 (of the students at a table) + 1/3 (of the students at a different table) = 2/6 (of the students at those t…

You're right, but the tricky part is, how do you explain that to children who are just being introduced to the concept of adding fractions, without leading them off track? That's hard! > The confusion comes because no one calls out that they're talking about fractions of different things. But she did: "When thinking about fractions, it’s important to keep your attention on what the whole is. [...] you’re thinking abo…

> You're right, but the tricky part is, how do you explain that to a group of children who is just being introduced to the concept of adding fractions, without leading them off track? That's hard!

She's already introduced bottles of water and pencils. So one interesting question is, "If you take three bottles of water, and add three pencils, do you now have all of the bottles of water?"

Re: Can 1/3 and 1/3 = 2/6? It seemed so

#164

Post by macspoofing »

tsimionescu wrote: post #130
macspoofing wrote: post #43

Earlier quoted context omitted.

>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 no…

> 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'…

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.

Re: Can 1/3 and 1/3 = 2/6? It seemed so

#165

Post by abdullahkhalids »

spenczar5 wrote: post #147
abdullahkhalids wrote: post #144

Earlier quoted context omitted.

Why can't we have mathematics devoid of these ambiguities? One reasons is that humans have small working memories, and novice mathematics students have even smaller ones. The ambiguity of notation, while confusing, once mastered, allows us to write shorter expressions, whose meaning we resolve from context, and which become both easier to write and to understand. A second reason is that while mathematical logic is ri…

I think it's a lot less beautiful. I think it has to do more with historical accidents. We've stumbled our way forward in mathematics; there is no grand plan that unifies our notation. Just look at calculus for a prime example - df/dx and f'(x) come from two different lineages and get used interchangeably; the df/dx notation can be intensely misleading when students think that (for example) they should be able to use…

Of course, there is no grand plan, and yes, a lot of notation is historical accidents. But my point is that attempting to "fix" it will probably yield a better notation, but not the "perfect" notation.

As an aside, df/dx treated a fraction is used as soon as second or third semester of uni when they learn how to solve differential equations using separation of variables, and physicists/chemists start using total derivatives for thermodynamics. I am not aware of notation different from df/dx where these subjects would be just as clear.

Re: Can 1/3 and 1/3 = 2/6? It seemed so

#166

Post by [deleted] »

dejj wrote: post #105
forbiddenvoid wrote: post #40

The units are missing, and I think that's a key factor here. Both of the equations up on the board at the end are correct because they are counting different things. This is a huge miss if you use a numeric only approach to fractions. The student came up and wrote 1/3 + 1/3 = 2/6. What they meant by that is 1/3 (of the students at a table) + 1/3 (of the students at a different table) = 2/6 (of the students at those t…

They sneakily replaced scalar addition by point-wise vector addition. [1,3] [1,3] = [2,6]

[deleted]

Re: Can 1/3 and 1/3 = 2/6? It seemed so

#167

Post by JadeNB »

Wowfunhappy wrote: post #111
forbiddenvoid wrote: post #40

The units are missing, and I think that's a key factor here. Both of the equations up on the board at the end are correct because they are counting different things. This is a huge miss if you use a numeric only approach to fractions. The student came up and wrote 1/3 + 1/3 = 2/6. What they meant by that is 1/3 (of the students at a table) + 1/3 (of the students at a different table) = 2/6 (of the students at those t…

You're right, but the tricky part is, how do you explain that to children who are just being introduced to the concept of adding fractions, without leading them off track? That's hard! > The confusion comes because no one calls out that they're talking about fractions of different things. But she did: "When thinking about fractions, it’s important to keep your attention on what the whole is. [...] you’re thinking abo…

> You're right, but the tricky part is, how do you explain that to children who are just being introduced to the concept of adding fractions, without leading them off track? That's hard!

I think it's the opposite: if you are taught the procedure for adding fractions without understanding what's going wrong here, then you just know a procedure, not a concept. Taking the detour into the detailed discussion of what 'the whole' is takes longer, but cements that crucial idea, and so lays the foundation for understanding addition of fractions more generally.

Re: Can 1/3 and 1/3 = 2/6? It seemed so

#168

Post by wcarey »

abdullahkhalids wrote: post #144
wcarey wrote: post #4

This is a lovely (edit: having been in similar shoes, also terrifying-in-the-moment) example of a broader problem in teaching mathematics: the language we use to describe mathematical reasoning is a natural language, like English or Latin, and therefore full of the sorts of bizarre irregularities you'd find in a natural language. Mathematics is also a language about rigorously and precisely defined objects. The conce…

Why can't we have mathematics devoid of these ambiguities? One reasons is that humans have small working memories, and novice mathematics students have even smaller ones. The ambiguity of notation, while confusing, once mastered, allows us to write shorter expressions, whose meaning we resolve from context, and which become both easier to write and to understand. A second reason is that while mathematical logic is ri…

As you say, the artistic use of cross-object synthesis and analogy definitely distinguishes the advanced students from the mechanical. Advanced students develop a sort of dialect that harmonizes the mathematical objects they encounter in a way that illuminates them all.

More than any of that, though, I think what you can see (even here in this thread!) is that, while we pretend that mathematics is a single cultural practice of rational humans communicating with other rational humans, it's really many smaller communities of mathematicians, all of varying skills trying to communicate with each other. My mathematical language as a teacher of 12-18 year olds is very different from my mathematical language when I did computational geometry for a living.

Because you have many communities of mathematicians producing new notation, you end up with dialects that all sort of meld together in the same way that reading Shakespeare is very different from reading Hemingway or Eco (in translation).

The closest programming analogue would be C++, where you have several mutually unintelligible dialects spoken by different communities of programmers with different concerns.

Re: Can 1/3 and 1/3 = 2/6? It seemed so

#169

Post by rmtech »

"+" is the wrong mathematical operation for combining fractions of distinct sets.

The right operation is a weighted sum:

1/3×1/2 + 1/3×1/2 = 1/3

The "1/2" is there because each of the two sets that we are combining is the same size.

Re: Can 1/3 and 1/3 = 2/6? It seemed so

#170

Post by tsimionescu »

asdkhadsj wrote: post #121
tsimionescu wrote: post #99

Earlier quoted context omitted.

It's not really a typing problem, it's a problem of references. You can't add 1/3 apples and 1/3 oranges. However, you can add 1/3 * x people with 1/3 * y people and get 2/6 * z people, if x people +y people =z people.

I think you both are saying the same thing, in this context at least.

There is a very important difference between types and quantities.

For example, 1/3 + 1/3 = 1 is true, if I mean 1/3 of 3 + 1/3 of 0. 1/3 + 1/3 = 2/6 can only be true if we add some quantities, we can't make it true by choosing the right types (we could fix it by choosing a different definition of +, or =, though).

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