Many students at beginner university level math courses do the same kind of mistakes, albeit with larger numbers: 10/13 + 7/21 = 17/34. Decimals are much easier to deal with imo, but mathematicians does not like them because they are inexact.
Can 1/3 and 1/3 = 2/6? It seemed so
141–150 of 308 posts
Re: Can 1/3 and 1/3 = 2/6? It seemed so
#142The 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…
1/3 of one table = 1 student.
1/3 of two tables = 2 students.
1/3 of one table + 1/3 of one table = 2/3 of one table.
2/3 of one table = 1/3 of two tables.
1/3 of one table + 1/3 of one table = 1/3 of two tables.
--
It would take a lot longer to explain than to write, but that's how I'd be tempted to proceed.
As far as learning the basics, it sounds like this class was just getting them introduced to the concept of fractions, while the arithmetic rules would come later...
Re: Can 1/3 and 1/3 = 2/6? It seemed so
#143Earlier quoted context omitted.
> one with respect to multiplication, the other with respect to function composition And, of course, multiplication is a function so one is just a specific instance of the other. Infix notation is another blight on the mathematical notation landscape. The amount of human effort that has been put into figuring out how to parse a+b*c is staggering. All of this confusion could have been avoided if we'd just started with…
Multiplication is a function of numbers, and function composition is a function of functions, so neither is an instance of the other (unless you take the unusual perspective of thinking of numbers themselves as functions, but I don’t think that’s what you meant).
> I don’t think that’s what you meant
Why not?
Multiplication is a composition of addition, and addition is a composition of the successor function. The successor function is not a composition of anything, but you can define it in terms of set theory if you don't want to simply accept it as a primitive. But sets are functions too.
Re: Can 1/3 and 1/3 = 2/6? It seemed so
#144This 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…
Why can't we have mathematics devoid of these ambiguities? One reasons is that humans have small working memories, and novice mathematics students have even smaller ones. The ambiguity of notation, while confusing, once mastered, allows us to write shorter expressions, whose meaning we resolve from context, and which become both easier to write and to understand.
A second reason is that while mathematical logic is rigorous and precise, unequal mathematical objects are similar to each - even objects that on first glace seem nothing alike. For instance, the two types of inverses you mention, or sets and linear spaces, or groups and tetrahedrons. And because of the diffuse nature of mathematical objects, it is inevitable that the same notation will be used for unequal objects. Because, it advances our human understanding of mathematics to use the same notation for two unequal but similar objects.
This second reason, once understood, is one threshold between the mechanical mastery of the intermediate student and almost artistic use of mathematics by the advanced student.
Re: Can 1/3 and 1/3 = 2/6? It seemed so
#145Earlier 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 right response could then be expanded... "No, you can't do that, because those kids are at a different table. If you want to add Jack and Brad you have to go back and see their fraction _of both tables_. They are each 1/6 so together they are 1/6 + 1/6 = 2/6" Then you could talk about if the ratios are same at both tables 1/3. Adding both tables together keeps the ratio. 1/3 + 1/3 = 1/3.
I like the units explanation a lot more. Saying you can't put two tables together when you clearly can in the real world is seems deeply unsatisfying. Kids hate being told they can't do something.
Re: Can 1/3 and 1/3 = 2/6? It seemed so
#146Earlier quoted context omitted.
Absolutely terrifying. How did he communicate with other peer mathematicians?
Mathematicians understand how to read definitions of notation. If all of your trig is fractions of pi, writing pi redundantly everywhere is not useful.
Is that commonly done?
Re: Can 1/3 and 1/3 = 2/6? It seemed so
#147This 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…
Why can't we have mathematics devoid of these ambiguities? One reasons is that humans have small working memories, and novice mathematics students have even smaller ones. The ambiguity of notation, while confusing, once mastered, allows us to write shorter expressions, whose meaning we resolve from context, and which become both easier to write and to understand. A second reason is that while mathematical logic is ri…
I think it's a lot less beautiful. I think it has to do more with historical accidents. We've stumbled our way forward in mathematics; there is no grand plan that unifies our notation. Just look at calculus for a prime example - df/dx and f'(x) come from two different lineages and get used interchangeably; the df/dx notation can be intensely misleading when students think that (for example) they should be able to use normal fraction rules.
Re: Can 1/3 and 1/3 = 2/6? It seemed so
#148Earlier quoted context omitted.
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?…
> 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.
Re: Can 1/3 and 1/3 = 2/6? It seemed so
#149Earlier quoted context omitted.
The right response could then be expanded... "No, you can't do that, because those kids are at a different table. If you want to add Jack and Brad you have to go back and see their fraction _of both tables_. They are each 1/6 so together they are 1/6 + 1/6 = 2/6" Then you could talk about if the ratios are same at both tables 1/3. Adding both tables together keeps the ratio. 1/3 + 1/3 = 1/3.
I like the units explanation a lot more. Saying you can't put two tables together when you clearly can in the real world is seems deeply unsatisfying. Kids hate being told they can't do something.
mnsc's explanation shows that you can put the tables together, you just need to take both tables into account from the outset.
Re: Can 1/3 and 1/3 = 2/6? It seemed so
#150The 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…
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.