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

marilynburnsmathblog.com

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

#291

Post by rocqua »

JadeNB wrote: post #204
rocqua wrote: post #80

Earlier quoted context omitted.

It's not because 'function powers' make sense, yhey are just iterated function application. That's how they work for the natural numbers and when you extend that to the integers, you immediately get f^-1 for the inverse. Notation in higher level maths is almost always very ambiguous. Because many concepts are analogues of each other and to reflect that notation is just taken from the analogue. Within a single domain,…

> It's not because 'function powers' make sense, yhey are just iterated function application. To be clear, I think you aren't saying "It's not because 'function powers' make sense" (which seems to apply that's not the reason it's done, and possibly that the reason isn't correct), but rather "It's not [an appalling notation], because 'function powers' make sense"—to me, that extra comma changes the meaning!

Indeed!

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

#292

Post by 08-15 »

wcarey wrote: post #281
08-15 wrote: post #181

That's what happens when you try to "build an intuition" for a concept instead of defining it. Okay, I get what a third of a sixpack is, it's two bottles. And half a sixpack is three bottles. I can even add them: one third (of a sixpack) plus one half (of a sixpack) is five bottles, hence five sixths! Now what's a quarter of a sixpack? That don't make no sense!! The whole point of common fractions is to close integer…

As someone who has pushed pretty hard towards precise definition and rigor as a teacher, you're in the (happy!) minority being able to approach mathematics that way. For lots of students, the process of moving from definitions and simple rules to other statements is extraordinarily difficult. I teach the properties of exponents to 14 year olds as a unit on logical necessity - the properties all flow necessarily from…

> About a third continue to maintain that, say, a^2 + b^2 = (a+b)^2

...and I bet, their justification is "It looks right!" or "Why not?" I've seen this, too.

A friend of mine had a habit of cancelling sums, such as (a+c)/(b+c) = a/b. Why? Because it looks right, and why not? And it isn't so different from (ac)/(bc) = a/b, is it? Of course, she never asked why a certain manipulation is legal. Arithmetic manipulation is legal because the teacher said so.

I suspect, this isn't a property of their personality; rather it's a failure of their math teachers to convey the idea, that everything has a reason, and if you don't know the reason why some manipulation would be allowed, it very likely isn't. By the time you tried to teach exponentiation to these students, the damage was long done.

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

#293

Post by boomlinde »

boomlinde wrote: post #190

Of course 1/3 + 1/3 = 2/6, and in this case you can make an error in reasoning that is probably common among children who are just learning fractions that will still arrive at the same answer. In this case I would have demonstrated a case where the error in reasoning that was possibly made does not lead to the correct answer, for example 1/8 + 1/4. The erroneous intuition of simply adding the groups and their subsets…

And now I feel stupid for obvious reasons

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

#294

Post by simiones »

macspoofing wrote: post #59
twic wrote: post #45

Earlier quoted context omitted.

Fractions are like groups of pencils or tables of girls and boys. Mathematicians have come up with rigorously defined approximations of them so that can work on them using formal methods.

>Fractions are like groups of pencils or tables of girls and boys. No, they aren't. I can have a fraction like -1/3... what's a negative one-third of a table of boys and girls? I can multiply -1/3 * -100/3 and get a positive fractional value - if we're talking about tables of boys and girls - what the heck happened there? How about 1/0 or 0/1? >Mathematicians have come up with rigorously defined approximations of the…

You're taking a strange ahistorical perspective. Most Natural numbers, Integers, Rationals/Fractions, and even some Irrational numbers are much older than the concept of formal mathematics.

Natural numbers were discovered as a solution to counting. Negative numbers appear naturally when you need to subtract quantities, especially when you end up with debts. Rationals appear from the need for division. Pi appears when you want to relate lengths of circles with lengths of straight lines. Square roots also appear from basic geometry used in construction, as the naming suggests.

All of these are real-world concepts that were used in some form or another at least 6000 years ago. A lot of the time, the closest formal definitions they had at the time were actually based on geometry, as algebra was probably a few thousand years away from being invented/discovered.

The modern formal definition is a post-hoc formalization of an existing intuitive concept. If the formal definition did not agree with the intuitive concept, then obviously the formal definition would not be useful and no one would use it.

There are mathematical constructs that first appeared in mathematics, and don't have intuitive concepts attached, or even don't have any real-world interpretation. A lot of irrational numbers may be in this area, as would the transfinite numbers, and probably complex numbers as well, though even some of those are usually defined based on some real-world applicability.

> I can have a fraction like -1/3... what's a negative one-third of a table of boys and girls?

It's a bit contrived, but say I ask the pupils at the table to send someone away. They can compute that they will soon have 3 people + (-1/3 * 3 people) = 2/3 * 3 people at the table. A less contrived example is "I have 2/3kg of flour, and I owe you 1/3kg, while someone else owes me 1/4kg. So, I own 2/3kg + (-1/3kg) + 1/4kg = 7/12kg".

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

#295

Post by macspoofing »

simiones wrote: post #294
macspoofing wrote: post #59

Earlier quoted context omitted.

>Fractions are like groups of pencils or tables of girls and boys. No, they aren't. I can have a fraction like -1/3... what's a negative one-third of a table of boys and girls? I can multiply -1/3 * -100/3 and get a positive fractional value - if we're talking about tables of boys and girls - what the heck happened there? How about 1/0 or 0/1? >Mathematicians have come up with rigorously defined approximations of the…

You're taking a strange ahistorical perspective. Most Natural numbers, Integers, Rationals/Fractions, and even some Irrational numbers are much older than the concept of formal mathematics. Natural numbers were discovered as a solution to counting. Negative numbers appear naturally when you need to subtract quantities, especially when you end up with debts. Rationals appear from the need for division. Pi appears when…

>The modern formal definition is a post-hoc formalization of an existing intuitive concept

Sure, but we're aiming to teach those 'post-hoc' formalizations. What fourth-graders are learning is supposed to be a foundation for future abstract mathematics. And we are using the operators and syntax of the modern formalizations as well.

Again we're not just teaching kids how to think about ratios and do practical arithmetic. If that's all we wanted, we could just give them an algorithm and an abacus and that's all they would ever need. Fractions can be used to think about ratios and things like that but they come with WAY more baggage because they are more than that. They are abstract concepts that underpin further abstract mathematics.

>It's a bit contrived, but say I ask the pupils at the table to send someone away. They

You can always add an ad-hoc extension to whatever the analogy you're using. I'm not saying that's wrong. I'm making no judgement call on how to teach mathematics to fourth-graders either (I don't know what the best way is). But I do want to point out that your extension to the analogy (negative fractions being equivalent to sending a student away) is because you've reached the limits of your initial abstraction. It's not a clean mapping either. For one thing, you just assumed the numerator is negative (student being away), but it is just as valid that the denominator may be negative (or both!). And it isn't clear that a negative numerator and a positive denominator is equivalent to a positive numerator and negative denominator. You either have to prove that from first principles using formal methods or you simply have to assert that to the fourth-grader and justify it with "it's just the way things are"

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

#296

Post by dwild »

romwell wrote: post #266
TeMPOraL wrote: post #259

Earlier quoted context omitted.

Isn't this why you end up seeing these "silly" units like kg/kg in chemistry? So that, while the value is technically dimensionless, it doesn't get added to another dimensionless value (e.g. l/l) that's a ratio of values of a different dimension?

This hits the nail on the head: 1 (person at table A) / 3 (people at table A) can't be added to 1 (person at table B) / 3 (people at table B) without conversion of units.

And then it's easy to make them see how her answer was right in its own way by adding another unit, (person at table a+b and people at table a+b) and show why it may be harder to works like that for now.

I heard so many people complains that each year in maths they would essentially learn that everything they learned the year before was wrong... can we fix that please?!

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

#297

Post by dwild »

Wowfunhappy wrote: post #138
forbiddenvoid wrote: post #127

Earlier quoted context omitted.

I don't think I agree with everything here. Glossing over critical mental models while explaining new concepts is what creates the confusion. Absolutely using drawings would have made things easier. That's a good way to establish those correct mental models. "No you can't do that" is probably not the right approach. The kid's instinct was spot on. The calculation they did was correct, just out of context. Rather than…

I'd absolutely agree if you were tutoring the child one-on-one. My concern is about leading the other 19+ kids astray with a confusing example before they're ready for it. Ignoring it does feel wrong, but I don't know that it is. There's only so much time in a class period, etc. That said, I really liked where mnsc went with it: https://news.ycombinator.com/item?id=23312266

> My concern is about leading the other 19+ kids astray with a confusing example before they're ready for it.

Except that the kids were all agreeing with her. Whether she did made that statement or not, they still believed that made sense.

I agree with you about the solution from mnsc, that was a good way to skip that without confusion.

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

#298

Post by CydeWeys »

LyndsySimon wrote: post #207
CydeWeys wrote: post #94

Earlier quoted context omitted.

I suspect we skew young enough on here that most of us don't (yet?) have kids.

I very strongly suspect that’s not true.

It would be interesting to see a demographic survey of HN commenters/contributors. Wonder if one has ever been done?

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

#299

Post by billfruit »

saagarjha wrote: post #228
billfruit wrote: post #92

Earlier quoted context omitted.

Another is sin^2(x) meaning (sin(x))^2, but by a more intuitive reading it should mean sin(sin(x)). I don't know what exactly is being gained by the usual notation: surely better clarity should be preferred over the time/effort saved in writing the extra pair of paranthesis, and I would prefer it be written as (sin(x))^2 always. Thing is mathematics, and mathematical pedagogy seems hardly concerned with such rampant…

When actually working with trigonometric identities, using parentheses typically gets unwieldily very quickly. Same reason sin(x) often becomes sin x.

Then perhaps we should drop all parens and move to prefix notation.

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

#300

Post by lonelappde »

mcguire wrote: post #177
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…

One of the article's comments: " Janelle Schorg says: " This is why students are confused and have misconceptions about ratios in middle school. When we teach fractions it is part(s) of a whole (Water bottles and pencils context) and when we teach ratios they are sets (boys and girls). It is actually okay to add ratios (as fractions) by combining the numerators and denominators, no common denominators needed. In my o…

That's totally wrong. It's not OK to add ratios. It's OK to add populations.

You can't add 1/3 and 1/3 to get 2/6 if the first 1/3 was reduced from 3 of 9 and th second was 1/3. Well, you can, but that only works in the degenerate case where the items you add (actually, average) are equal and there's no point in adding in the first place.

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