Can 1/3 and 1/3 = 2/6? It seemed so
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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
#2I 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.
Re: Can 1/3 and 1/3 = 2/6? It seemed so
#3That’s the same as saying: if you put 1/3rd pizza slice on top of an other 1/3rd pizza slice, you get 2/6 pizza. Layered math.
Re: Can 1/3 and 1/3 = 2/6? It seemed so
#4This 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 conceptual shear between those two things is murder for lots of students.
Just like Tacitus omits his verbs (!), when we describe fractions we often omit the implicit definition of the whole. Turns out that's a problem for many students.
It's a bit like trying to learn a context-dependent programming grammar with an inconsistent API, but worse, because it's your first "mathematical" language so you're also trying to learn what the abstract objects the language manipulates are.
Some other lovely examples:
3(5) means three times five. 3(x) means three times x. 35 means three times ten plus five. 3x means three times x. x(3) means that x is the name of a function taking, in this instance, 3 as its input.
x^{-1} means \frac{1}{x}, but f^{-1}(x) doesn't mean \frac{1}{f(x)}.
\sin{30}. Radians or degrees? Probably the writer means degrees, but there's no way to tell.
There are many more.
Re: Can 1/3 and 1/3 = 2/6? It seemed so
#5I 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 surpris…
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...
Re: Can 1/3 and 1/3 = 2/6? It seemed so
#6One third of one half, plus one third of the other half, is one third of the whole?
Re: Can 1/3 and 1/3 = 2/6? It seemed so
#7This 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…
I think x(3) would more commonly mean x times 3. f(3) would be function application. Even more context dependence.
Re: Can 1/3 and 1/3 = 2/6? It seemed so
#8"If we reduce 2/6, we get 1/3, so 1/3 + 1/3 = 1/3, that seems weird, what did we do wrong?"
Re: Can 1/3 and 1/3 = 2/6? It seemed so
#9This 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…
I also remember having constant confusion of what were assumed associations in written bigger formulas especially involving the functions and the practice of not writing the braces, and before that, if 3 1/5 meant three times 1/5 or 3 wholes and 1/5th.
I thought it was only on these low levels of educations that people settle on such confusing practices.
Later on the university I've discovered that it's the same on all levels. The notations used outside of programming are simply very context-dependent, and it's pity that that dependency is not made clear more often. Also, some practices directly come from some historical uses, and different paths to the current notations still end up to the similarly looking but different meanings.
Re: Can 1/3 and 1/3 = 2/6? It seemed so
#10I 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 surpris…
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-pr…
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.