5 ms·

I guess you'd have to come up with a way to explain that adding numbers (which is what you're doing with 1/3 + 1/3) is not the same as combining/averaging fract

I guess you'd have to come up with a way to explain that adding numbers (which is what you're doing with 1/3 + 1/3) is not the same as combining/averaging fractions, i.e. when you're totaling subgroups into a larger group. It's almost like we need a different "combining" operator for the latter that means to add both the numerator and denominator, because + isn't right for this. Now that I think about it, I'm surprised there is no such operator for averaging. It's not as bad of a nightmare as trying to explain the answer to the missing dollar riddle: https://en.m.wikipedia.org/wiki/Missing_dollar_riddle https://en.m.wikipedia.org/wiki/Missing_dollar_riddle. That's an absolute nightmare.

ColinWright 6y ago
I never understood why the Missing Dollar Riddle ever confuses people. As soon as they say "Add the $2 to the $27" I say, "But why are you adding something someone has to a total that people paid?" That, in turn, is like the "Age of the Shepherd" problem[0] ... people just add/subtract/multiply/divide things randomly without thinking about what they mean. [0] https://mystudentvoices.com/how-old-is-the-shepherd-the-problem-that-shook-school-mathematics-ad89b565fff https://mystudentvoices.com/how-old-is-the-shepherd-the-prob...
- wcarey 6y ago
  The more I've taught math, the more convinced I am that getting people to "think about what they mean", and to think about what mathematical words mean is 90% of the project. I remember reading long ago (I'd love to find it again) about a CS department that gave a quiz to incoming students that was very predictive of their success. The answers they wrote didn't matter; what mattered was whether their answers evinced consistent meaning applied to terms and operations.
  - ColinWright 6y ago
    As I recall, that study was debunked/retracted, so you may have trouble finding it. The paper: http://eis.mdx.ac.uk/research/PhDArea/saeed/paper1.pdf http://eis.mdx.ac.uk/research/PhDArea/saeed/paper1.pdf Retraction: http://www.eis.mdx.ac.uk/staffpages/r_bornat/papers/camel_hump_retraction.pdf http://www.eis.mdx.ac.uk/staffpages/r_bornat/papers/camel_hu...
    - wcarey 6y ago
      Thanks! I hadn't seen the retraction. That's really interesting.
    - twic 6y ago
      It's a bit more complicated than "debunked/retracted", as that second document explains: > Dehnadi, to his credit, stuck to his guns and did the meta-analysis that showed that he’d discovered a phenomenon and that his test was a worthwhile predictor. The original paper contained several linked claims: that there is an ability to make consistent mental models, that it's intrinsic and fixed, that it predicts ability to program, and that few people have it, and hence few people can learn to program. AIUI, the debunked/retracted claims are that it's intrinsic and fixed, and that few people have it. It looks like the ability exists, but it can be learned, and it is linked to programming ability. Which i think does line up with wcarey's point: > The more I've taught math, the more convinced I am that getting people to "think about what they mean", and to think about what mathematical words mean is 90% of the project. [...] The answers they wrote didn't matter; what mattered was whether their answers evinced consistent meaning applied to terms and operations.
      - ColinWright 6y ago
        Agreed. I won't go back and edit my comment, but what you say is true. There really is an interesting thing going on here, even if the original paper was an over-reach.
    - nyanpasu64 6y ago
      Excerpt from the retraction: > In autumn 2005 I became clinically depressed. My physician put me on the then-standard treatment for depression, an SSRI. But she wasn’t aware that for some people an SSRI doesn’t gently treat depression, it puts them on the ceiling. I took the SSRI for three months, by which time I was grandiose, extremely self-righteous and very combative – myself turned up to one hundred and eleven. I did a number of very silly things whilst on the SSRI and some more in the immediate aftermath, amongst them writing “The camel has two humps”. I’m fairly sure that I believed, at the time, that there were people who couldn’t learn to program and that Dehnadi had proved it. The paper doesn’t exactly make that claim, but it comes pretty close. Perhaps I wanted to believe it because it would explain why I’d so often failed to teach them. It was an absurd claim because I didn’t have the extraordinary evidence needed to support it. I no longer believe it’s true. A sad story :(
- rcxdude 6y ago
  This reminds me of this problem which made the rounds recently[0], where a word problem essentially makes a lot of people commit a fencepost error [0] https://math.stackexchange.com/questions/379927/how-long-will-it-take-marie-to-saw-another-board-into-3-pieces https://math.stackexchange.com/questions/379927/how-long-wil...
barbegal 6y ago
And the "missing dollar riddle" can be exploited as the "change raising scam" https://www.youtube.com/watch?v=1Uaw88H0AmE https://www.youtube.com/watch?v=1Uaw88H0AmE
didgeoridoo 6y ago
Someone in the comments makes a good point that the best thing to do here may be to introduce ratio notation for proportions (e.g. 2:4) which CAN be added/combined according to the kids’ intuitions — 1:2 combined with 1:2 does indeed equal 2:4, which reduces back to 1:2. You could then teach how to go from ratios to fractions by adding the ratio sides together and putting that in the denominator for each side... poof, you’ve invented averages! No way someone would have come up with that approach on the fly, though.
- CydeWeys 6y ago
  Yeah that's a good approach. The problem remains though that if you use the + operator on ratios you're still overloading it to mean something different in a way that doesn't retain its meaning when you start expressing things as fractions instead. So 1:2 + 1:2 works, but 1/3 + 1/3 doesn't. I think you still want a different operator for this. Maybe ⊕ or ⋃ or ⋓ ? I'm just spitballing here. There's definitely enough options in Unicode that an existing operator should be suitable for this purpose: https://en.wikipedia.org/wiki/Mathematical_operators_and_symbols_in_Unicode https://en.wikipedia.org/wiki/Mathematical_operators_and_sym...
  - wcarey 6y ago
    And + is overloaded in a bunch of ways students encounter in high school, and much time is spent talking about when you're allowed to add and when you're not and which rules apply when. Examples: 1 + 2 - fine. 1/2 + 1/2 - one set of rules. 1/2 + 1/3 - a subtly different set of rules. 1:20 + 0:45 - yet another set of rules. Modular. 30° + 350° - fine? But maybe modular. 15% + 20% - who knows? 15% of what? 20% of what? (1,2) + (2,4) - can't be done. a^2 + a^2 - fine. a^2 + a^3 - nope. a^2 + b^2 - nope. It would be lovely if mathematics were taught as a strongly typed language without overloaded operators, alas all our corpus is in the language it's in.
    - uryga 6y ago
      > (1,2) + (2,4) - can't be done. are you talking about points or open intervals? the notation is ambiguous ;) but addition of points is fine if you look at them as vectors > a^2 + a^2 - fine. a^2 + a^3 - nope. a^2 + b^2 - nope. what's wrong with the 2nd and 3rd ones? the 3rd one looks like half of Pythagoras theorem which is widely considered "fine" afaik ;)
yamrzou 6y ago
The confusion comes from the fact that we attach the same label "3" to two different sets in the case of bottles, so we end up saying "one out of three plus one out of three". It should be "one out of group 1 + one out of group 2", and then as you said, we'll need a special operator for combining groups.
bwestergard 6y ago
Adding numerators and denominators to find the "mediant" is sometimes called "doing a freshman sum". In some countries I gather this is introduced as a separate operation (see page 23 in the translation of a Soviet primary school textbook by Gelfand linked below). https://en.wikipedia.org/wiki/Mediant_(mathematics) https://en.wikipedia.org/wiki/Mediant_(mathematics) https://www.cimat.mx/ciencia_para_jovenes/bachillerato/libros/algebra_gelfand.pdf https://www.cimat.mx/ciencia_para_jovenes/bachillerato/libro...
- CydeWeys 6y ago
  Thanks for putting a name on this specific operation. It sounds like it definitely needs its own specific operator. And hi Bjørn!
6gvONxR4sf7o 6y ago
> I'm surprised there is no such operator for averaging. You can't have an operator for combining portions of groups with fractions alone, because 1/3 = 2/6. Combining groups of B boys and N people total, you get B1+B2 boys and N1+N2 people. Let's use @ for that operator, just to not distract from the usual addition. a/b @ c/d = (a+c)/(b+d). Let's combine a group of 1/3 boys with a group of 2/3 boys. 1/3 @ 2/3 = 3/6. But 1/3 = 2/6, so that should be the same as 2/6 @ 2/3 = 4/9. But 3/6 isn't 4/9. You end up with this issue of a/b @ c/d = p a / p b @ q c / q d = (p a + q c) / (p b + q d), which can be anything. So we have the conclusion that if this operator makes sense, then all numbers are equal. You end up with a notion of numbers that is useless for the original problem of combining groups of people of different genders. What we should do here is define it on pairs of numbers rather than fractions. A fraction and a total (p, N), or the number of boys and number of girls (b, g). The latter is super straightforward: (paul, jenny) + (bob, alice) = (paul and bob, jenny and alice), so numerically, it's (b1, g1) @ (b2, g2) = (b1+b2, g1 + g2), but (1, 3) is not (2, 6) here, unlike with fractions. Real simple. If we want to connect this back to the world of fractions, (p1, N1) @ (p2, N2) = (number of boys / total, total) = ( (p1 N1 + p2 N2)/(N1+N2), N1+N2 ). It's just a weighted average, so you need to keep track of the weights.
- deleted 6y ago
  [deleted]
- sdenton4 6y ago
  Well, you're right so long as 1/3 = 2/6. This is a mathematical equivalence... Fractions are a 2d projective space: they are pairs (n, d) where (mn, md) is equivalent to (n, d). The suggested operation is totally fine if you are working in ambient (that is, non-projective) space... ie, if you continually keep track of the 'size' of the thing being measured. The equivalence operation is hard to master, and makes the arithmetic complicated.
OscarCunningham 6y ago
> It's almost like we need a different "combining" operator for the latter that means to add both the numerator and denominator, because + isn't right for this. Now that I think about it, I'm surprised there is no such operator for averaging. This would also cause confusion, because combining 1/3 with 1/2 and 2/4 in this way would yield different results, even though 1/2 = 2/4.
GavinMcG 6y ago
You do this by including units. 1/3 + 1/3 = 2/3. And here, 1/3x + 1/3y = 2/6(x+y).

ColinWright 6y ago

I never understood why the Missing Dollar Riddle ever confuses people. As soon as they say "Add the $2 to the $27" I say, "But why are you adding something someone has to a total that people paid?" That, in turn, is like the "Age of the Shepherd" problem[0] ... people just add/subtract/multiply/divide things randomly without thinking about what they mean. [0] https://mystudentvoices.com/how-old-is-the-shepherd-the-problem-that-shook-school-mathematics-ad89b565fff https://mystudentvoices.com/how-old-is-the-shepherd-the-prob...

wcarey 6y ago
The more I've taught math, the more convinced I am that getting people to "think about what they mean", and to think about what mathematical words mean is 90% of the project. I remember reading long ago (I'd love to find it again) about a CS department that gave a quiz to incoming students that was very predictive of their success. The answers they wrote didn't matter; what mattered was whether their answers evinced consistent meaning applied to terms and operations.
- ColinWright 6y ago
  As I recall, that study was debunked/retracted, so you may have trouble finding it. The paper: http://eis.mdx.ac.uk/research/PhDArea/saeed/paper1.pdf http://eis.mdx.ac.uk/research/PhDArea/saeed/paper1.pdf Retraction: http://www.eis.mdx.ac.uk/staffpages/r_bornat/papers/camel_hump_retraction.pdf http://www.eis.mdx.ac.uk/staffpages/r_bornat/papers/camel_hu...
  - wcarey 6y ago
    Thanks! I hadn't seen the retraction. That's really interesting.
  - twic 6y ago
    It's a bit more complicated than "debunked/retracted", as that second document explains: > Dehnadi, to his credit, stuck to his guns and did the meta-analysis that showed that he’d discovered a phenomenon and that his test was a worthwhile predictor. The original paper contained several linked claims: that there is an ability to make consistent mental models, that it's intrinsic and fixed, that it predicts ability to program, and that few people have it, and hence few people can learn to program. AIUI, the debunked/retracted claims are that it's intrinsic and fixed, and that few people have it. It looks like the ability exists, but it can be learned, and it is linked to programming ability. Which i think does line up with wcarey's point: > The more I've taught math, the more convinced I am that getting people to "think about what they mean", and to think about what mathematical words mean is 90% of the project. [...] The answers they wrote didn't matter; what mattered was whether their answers evinced consistent meaning applied to terms and operations.
    - ColinWright 6y ago
      Agreed. I won't go back and edit my comment, but what you say is true. There really is an interesting thing going on here, even if the original paper was an over-reach.
  - nyanpasu64 6y ago
    Excerpt from the retraction: > In autumn 2005 I became clinically depressed. My physician put me on the then-standard treatment for depression, an SSRI. But she wasn’t aware that for some people an SSRI doesn’t gently treat depression, it puts them on the ceiling. I took the SSRI for three months, by which time I was grandiose, extremely self-righteous and very combative – myself turned up to one hundred and eleven. I did a number of very silly things whilst on the SSRI and some more in the immediate aftermath, amongst them writing “The camel has two humps”. I’m fairly sure that I believed, at the time, that there were people who couldn’t learn to program and that Dehnadi had proved it. The paper doesn’t exactly make that claim, but it comes pretty close. Perhaps I wanted to believe it because it would explain why I’d so often failed to teach them. It was an absurd claim because I didn’t have the extraordinary evidence needed to support it. I no longer believe it’s true. A sad story :(
rcxdude 6y ago
This reminds me of this problem which made the rounds recently[0], where a word problem essentially makes a lot of people commit a fencepost error [0] https://math.stackexchange.com/questions/379927/how-long-will-it-take-marie-to-saw-another-board-into-3-pieces https://math.stackexchange.com/questions/379927/how-long-wil...

barbegal 6y ago

And the "missing dollar riddle" can be exploited as the "change raising scam" https://www.youtube.com/watch?v=1Uaw88H0AmE https://www.youtube.com/watch?v=1Uaw88H0AmE

didgeoridoo 6y ago

Someone in the comments makes a good point that the best thing to do here may be to introduce ratio notation for proportions (e.g. 2:4) which CAN be added/combined according to the kids’ intuitions — 1:2 combined with 1:2 does indeed equal 2:4, which reduces back to 1:2. You could then teach how to go from ratios to fractions by adding the ratio sides together and putting that in the denominator for each side... poof, you’ve invented averages! No way someone would have come up with that approach on the fly, though.

CydeWeys 6y ago
Yeah that's a good approach. The problem remains though that if you use the + operator on ratios you're still overloading it to mean something different in a way that doesn't retain its meaning when you start expressing things as fractions instead. So 1:2 + 1:2 works, but 1/3 + 1/3 doesn't. I think you still want a different operator for this. Maybe ⊕ or ⋃ or ⋓ ? I'm just spitballing here. There's definitely enough options in Unicode that an existing operator should be suitable for this purpose: https://en.wikipedia.org/wiki/Mathematical_operators_and_symbols_in_Unicode https://en.wikipedia.org/wiki/Mathematical_operators_and_sym...
- wcarey 6y ago
  And + is overloaded in a bunch of ways students encounter in high school, and much time is spent talking about when you're allowed to add and when you're not and which rules apply when. Examples: 1 + 2 - fine. 1/2 + 1/2 - one set of rules. 1/2 + 1/3 - a subtly different set of rules. 1:20 + 0:45 - yet another set of rules. Modular. 30° + 350° - fine? But maybe modular. 15% + 20% - who knows? 15% of what? 20% of what? (1,2) + (2,4) - can't be done. a^2 + a^2 - fine. a^2 + a^3 - nope. a^2 + b^2 - nope. It would be lovely if mathematics were taught as a strongly typed language without overloaded operators, alas all our corpus is in the language it's in.
  - uryga 6y ago
    > (1,2) + (2,4) - can't be done. are you talking about points or open intervals? the notation is ambiguous ;) but addition of points is fine if you look at them as vectors > a^2 + a^2 - fine. a^2 + a^3 - nope. a^2 + b^2 - nope. what's wrong with the 2nd and 3rd ones? the 3rd one looks like half of Pythagoras theorem which is widely considered "fine" afaik ;)

yamrzou 6y ago

The confusion comes from the fact that we attach the same label "3" to two different sets in the case of bottles, so we end up saying "one out of three plus one out of three". It should be "one out of group 1 + one out of group 2", and then as you said, we'll need a special operator for combining groups.

bwestergard 6y ago

Adding numerators and denominators to find the "mediant" is sometimes called "doing a freshman sum". In some countries I gather this is introduced as a separate operation (see page 23 in the translation of a Soviet primary school textbook by Gelfand linked below). https://en.wikipedia.org/wiki/Mediant_(mathematics) https://en.wikipedia.org/wiki/Mediant_(mathematics) https://www.cimat.mx/ciencia_para_jovenes/bachillerato/libros/algebra_gelfand.pdf https://www.cimat.mx/ciencia_para_jovenes/bachillerato/libro...

CydeWeys 6y ago
Thanks for putting a name on this specific operation. It sounds like it definitely needs its own specific operator. And hi Bjørn!

6gvONxR4sf7o 6y ago

> I'm surprised there is no such operator for averaging. You can't have an operator for combining portions of groups with fractions alone, because 1/3 = 2/6. Combining groups of B boys and N people total, you get B1+B2 boys and N1+N2 people. Let's use @ for that operator, just to not distract from the usual addition. a/b @ c/d = (a+c)/(b+d). Let's combine a group of 1/3 boys with a group of 2/3 boys. 1/3 @ 2/3 = 3/6. But 1/3 = 2/6, so that should be the same as 2/6 @ 2/3 = 4/9. But 3/6 isn't 4/9. You end up with this issue of a/b @ c/d = p a / p b @ q c / q d = (p a + q c) / (p b + q d), which can be anything. So we have the conclusion that if this operator makes sense, then all numbers are equal. You end up with a notion of numbers that is useless for the original problem of combining groups of people of different genders. What we should do here is define it on pairs of numbers rather than fractions. A fraction and a total (p, N), or the number of boys and number of girls (b, g). The latter is super straightforward: (paul, jenny) + (bob, alice) = (paul and bob, jenny and alice), so numerically, it's (b1, g1) @ (b2, g2) = (b1+b2, g1 + g2), but (1, 3) is not (2, 6) here, unlike with fractions. Real simple. If we want to connect this back to the world of fractions, (p1, N1) @ (p2, N2) = (number of boys / total, total) = ( (p1 N1 + p2 N2)/(N1+N2), N1+N2 ). It's just a weighted average, so you need to keep track of the weights.

deleted 6y ago
[deleted]
sdenton4 6y ago
Well, you're right so long as 1/3 = 2/6. This is a mathematical equivalence... Fractions are a 2d projective space: they are pairs (n, d) where (mn, md) is equivalent to (n, d). The suggested operation is totally fine if you are working in ambient (that is, non-projective) space... ie, if you continually keep track of the 'size' of the thing being measured. The equivalence operation is hard to master, and makes the arithmetic complicated.

OscarCunningham 6y ago

> It's almost like we need a different "combining" operator for the latter that means to add both the numerator and denominator, because + isn't right for this. Now that I think about it, I'm surprised there is no such operator for averaging. This would also cause confusion, because combining 1/3 with 1/2 and 2/4 in this way would yield different results, even though 1/2 = 2/4.

GavinMcG 6y ago

You do this by including units. 1/3 + 1/3 = 2/3. And here, 1/3x + 1/3y = 2/6(x+y).