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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. That's an absolute nightmare.




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-prob...


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.


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

Retraction: http://www.eis.mdx.ac.uk/staffpages/r_bornat/papers/camel_hu...


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.


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.


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 :(


Thanks! I hadn't seen the retraction. That's really interesting.


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-wil...


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://www.cimat.mx/ciencia_para_jovenes/bachillerato/libro...


Thanks for putting a name on this specific operation. It sounds like it definitely needs its own specific operator.

And hi Bjørn!


> 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.


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.


> 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.


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.


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_sym...


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.


> (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 ;)


Indeed. And if they're cartesian vectors, you're good. But if that second number is an angle measured in radians, you use yet a different set of rules for the addition.

The conversation with 16 year olds when you explain that their previous teacher who told them that you couldn't add points wasn't lying, but was, perhaps oversimplifying things to make their life easier, is a fun and fraught one.

I've had to reason kids through the fact that a^2 + b^2 is not equal to (ab)^2 or even (a+b)^2 more times than I can count. What's particularly difficult is that, confronted with the fact that 25 and 49 are manifestly different numbers, many still cling to the rule that a^2 + b^2 = (a+b)^2, because of the "law of distribution", which they haven't learned as the "law of distribution of multiplication of monomials over addition, and only that".


How did I go all of my life so far without hearing the super useful word "monomial"? It's such an obvious concept to have a meaningful name, and yet I don't recall anyone ever having said that word.


I also didn't encounter it until I was relatively older. Our school has been pushing to introduce more rigorous language and definition in our 7-12th math program. For some students, it really seems to help. For others, it's really hard.


> 15% + 20% - who knows? 15% of what? 20% of what?

That's just notational sugar on fractions though (15/100, 20/100) - so why is that less troubling than the general case?


I mean, the general case is certainly trouble for kids. But 15% and 20% look very similar to 15x and 20x. That the two (sometimes!) operate using different rules causes confusion for many students. The implicitness of the fractions conceals something important that's explicit if you write out the fractions. For example, multiplying percents does not do what most students intuitively think it does.

15% of 8 times 20% of 10 isn't 35% of any nice arithmetic combination of 8 and 10. That's hard to communicate to many students.


The main thing I find myself explaining with percentages (IANA educator) is that 'grossing up' != adding the same percentage back, e.g. 80% * 1.2 != 100%, which looks pretty obvious like that, but it's a common mistake among adults talking about real life percentages like taxes.

Another common, and I suppose related, one (but that I don't bother correcting often) is 'percent' != 'percentage point'. Talk of 47% of something being '3% less than' half of it really winds me up - and it's stupidly common among journalists - but it's too common to bother pointing out IMO. Live and let get wrong.


I think this just adds to the confusion. Is "adding" ratios really the same as averaging them? (I would just say that adding ratios is simply not defined.) Can you only average things by moving from fractions to ratios and then back again?

I think better to address the problem directly in fractions by saying that they're two different ways of combining them, addition and averaging, like the parent comment says. Feel free to make up a different symbol for the averaging operator, just let the kids know that it's not standard.


There's a lot more than 2 ways to combine things!

Arithmetic average or geometric average, to start.


Unlike 95% of the other answers here, this may have been received And been processed by children who are just learning fractions.

I think the most basic answer to the “what do you do next” is unfortunately to explain they can’t do 1/3+1/3=1/6 but that why will be a future lesson.

The article gets a “fair” stopping point for the day at being mindful of the whole. And that you just can’t always add them.


> ... to explain they can’t do 1/3+1/3=1/6 ...

Except that 1/3 + 1/3 = 2/3, not 1/6.

You probably just mis-typed, but in this context it's an interesting mistake.


>> they can’t do 1/3+1/3=1/6

> Except that 1/3 + 1/3 = 2/3, not 1/6.

That's why they can't do it.


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.


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


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




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