Really great anecdote! That first plus sign that Addison wrote is actually more of an "And" instead of an arithmetic "plus"
Can 1/3 and 1/3 = 2/6? It seemed so
251–260 of 308 posts
Re: Can 1/3 and 1/3 = 2/6? It seemed so
#252The 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…
I think she could have continued by adding more tables until they got to 4/3. That is obviously wrong - so it is easier to convince the kids that something is wrong. Then back track and explain what went wrong and explain the rules for adding fractions.
Re: Can 1/3 and 1/3 = 2/6? It seemed so
#253The 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…
The units aren’t the issue IMO. You said: > 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 tables). That’s not what ‘+’ means. Addition doesn’t mean “I have this thing and the other thing; please describe the result”; addition means a specific operation on numbers (or on elements of an additive group, or on numbers with units, e…
I think you guys actually agree more than you think --
You say "you cannot fully describe 1 student at table of three people as 1/3", which is true: What's missing is the unit (or dimension, I'm ignoring the difference here).
You can only add two things if they have the same units, as per "dimensional analysis" [1].
So this is an entirely meaningless statement:
[students]/[seats at table 1] + [students]/[seats at table 2]
But you can fix the units with some multiplication (because dimensions do form an Abelian group under multiplication):
([students]/[seats at table 1]) * ([seats at table 1]/[total seats]) + ([students]/[seats at table 2]) * ([seats at table 2]/[total seats])
Which simplifies to:
[students]/[total seats] + [students]/[total seats]
Now that's a statement with meaning!
Since I know that
[seats at table 1]/[total seats] = 1/2
[seats at table 2]/[total seats] = 1/2
I've just derived the calculation that I really wanted to do:
(1/3)(1/2) + (1/3)(1/2) = (2/6)
[1] https://en.wikipedia.org/wiki/Dimensional_analysis#Dimension...
Re: Can 1/3 and 1/3 = 2/6? It seemed so
#254I think that one key issue is that the students are confusing addition with the more general concept of considering two things together.
Re: Can 1/3 and 1/3 = 2/6? It seemed so
#255Earlier quoted context omitted.
there's not much evidence that most humans would ever discover the concept of fractional arithmetic by themselves. this is a generic mathematics class with young children who by themselves are not likely to walk or even stumble into fractional arithmetic by themselves. there's also no evidence that teaching people complex sophisticated stuff before they have grasped basic concepts enhances their curiosity or learning…
There's incredibly strong evidence that the way to teach mathematics and the only way to learn it is to foster the environment in which you can rediscover the key insight for yourself. Obviously (to the younger students, to me, Lockhart, Dewey, and others, but not to most teachers, administrators, voters) there is no track and no curriculum for this. They are not stumbling into it by themselves! They have a guide! Th…
You're stretching what I said. I should note that I'm a lifelong fan of radical education (particularly John Holt), and I don't think that the way we teach children most things has a lot to recommend it.
However, your conception of how this could work starts from the supposition that the kids are actually interested enough to wonder. I don't doubt that there's something that will get every child wondering (and likely, more than one thing). But if we're going to actually require the teaching of fractional arithmetic (implying that we're requiring the learning of it, really) then we need to accept that we'll be teaching it a great many children who are not interested in the conundrum, and who even in the presence of a great teacher will remain not interested in it.
Self-led discovery learning is without doubt the best kind, but its not compatible with the current goals of education in a (post)industrial highly structured society, because it will naturally lead to people who for their own reasons chose never to learn things that we consider vital. I might be entirely willing to agree that they are not vital, and even that a (post)industrial highly structured society may also be a bit of an issue, but pretending that every child will just be naturally curious about 1/3+1/3=2/6 vs 1/3+1/3=2/3 is, IMO, not ground in reality.
In the original article, the guide isn't "standing front of the art". She's just taken them over to a corner that has a piece called "no mammals lay eggs", and then noticed that right next door to it, there's a platypus. She's wondering what to say next, or whether to say anything.
Re: Can 1/3 and 1/3 = 2/6? It seemed so
#256Earlier quoted context omitted.
there's not much evidence that most humans would ever discover the concept of fractional arithmetic by themselves. this is a generic mathematics class with young children who by themselves are not likely to walk or even stumble into fractional arithmetic by themselves. there's also no evidence that teaching people complex sophisticated stuff before they have grasped basic concepts enhances their curiosity or learning…
I have a hard time accepting this. My daughter is in 3rd grade and seems to have a reasonable grasp of fractions. They teach her about them at (public) school and we've discussed them at home. Sure, "1/3 + 2/7" is outside her reach, but simpler stuff, things she can visualize, are well within her grasp. > why is 2/3 different from 2/6 anyway Because 2 pieces of a pie that you cut into 3 pieces is more than 2 pieces o…
No disrespect, but you read HN and think that your daughter's take on math and the home context you provide for it is a sensibly representative starting point?
My daughter's now 25 and has been quite the nerd herself through the years (eventually landing in linguistics and speech pathology), but I'd never assume that the fascinations she had instrinsically and that I helped foster as a parent were really typical. I wish they were - and hey, here I am reading HN too :)
Re: Can 1/3 and 1/3 = 2/6? It seemed so
#257The 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…
I think we really need to start teaching people paying attention to units. Even in math classes.
1/3 + 1/3 is not a correct mathematical description of the problem.
1/3 [students at table A] + 1/3 [students at table B] is.
Let's shorten this to: 1/3 sTA + 1/3 sTB. Then, you factor out the 1/3, to get: 1/3 * (sTA + sTB). And now it's it's impossible to give the wrong answer "2/3".
Or, in other words, it's best to "keep your attention on what the whole is" by keeping it explicitly written out in the equation.
Re: Can 1/3 and 1/3 = 2/6? It seemed so
#258Earlier quoted context omitted.
> 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?"
This is interesting. I wonder if kids can then take that and transfer it over. Bottles and Pencils are obviously different but kids at table A and kids at table B are both kids. Even more problematically, the equation appears to work. I suspect there'll be some amount of dissatisfaction here and the kids won't understand.
Not sure if this is at the level of comprehension of these kids, but I'd explain it this way: "kids at table A" and "kids at table B" are both "kids", but different amount of kids. You can treat them as the same only if you have a conversion factor. So, if you know that there are 12 kids at table A, and 20 kids at table B, you can multiply your variables by these amounts, and now both expressions have the unit "kids", and you can add them together. But if you don't know the conversion factor, they're like Bottles and Pencils.
Re: Can 1/3 and 1/3 = 2/6? It seemed so
#259Earlier quoted context omitted.
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…
The issue is that it's ratios, not fractions. 1:3. You take 1:3 and 1:3 and it's still 1:3 or 2:6. You haven't changed the ratio for this at all by showing it as fractions, it's simply being represented and presented without correct context.
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?
Re: Can 1/3 and 1/3 = 2/6? It seemed so
#260Earlier quoted context omitted.
There's incredibly strong evidence that the way to teach mathematics and the only way to learn it is to foster the environment in which you can rediscover the key insight for yourself. Obviously (to the younger students, to me, Lockhart, Dewey, and others, but not to most teachers, administrators, voters) there is no track and no curriculum for this. They are not stumbling into it by themselves! They have a guide! Th…
You're stretching what I said. I should note that I'm a lifelong fan of radical education (particularly John Holt), and I don't think that the way we teach children most things has a lot to recommend it. However, your conception of how this could work starts from the supposition that the kids are actually interested enough to wonder. I don't doubt that there's something that will get every child wondering (and likely…
Yes. Every child I've ever met is curious about something, and that's what they should be learning. The idea that in year X every student should learn Y during hour Z every day is one of the reasons I said the system was irreparably broken. That is how schooling gets in the way of education.
Does every student learn to understand fractions under the current regime? Not in my experience.
Edit: And yes, you should say something about the platypus, because understanding that models are simplifications and incomplete can be enough of an escape hatch for the smart kids (the ones that usually hate math class) to notice that even the teacher knows that there's always more, and that can be enough to keep them from throwing it all away in disgust as a useless mishmash of arbitrary, conflicting, and incorrect rules.