Can 1/3 and 1/3 = 2/6? It seemed so

Post by asdkhadsj » Tue, May 26, 2020, 3:21 PM UTC

Wowfunhappy wrote: post #111

forbiddenvoid wrote: post #40

The 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…

Post by Spivak » Tue, May 26, 2020, 3:22 PM UTC

macspoofing wrote: post #43

Spivak wrote: post #24

I can't imagine doing this to a class of fourth graders but I don't think her thinking is wrong. I think the correct way to views maths notation is that it's a language and we should treat people using it "incorrectly" as a grammar mistake and try to understand the idea they're trying to express. The fourth grader is saying "1/3 + 1/3 = 2/6" but the idea she's trying to get across is that "avg(1/3,1/3) = 1/3 but the…

>The fourth grader is saying "1/3 + 1/3 = 2/6" but the idea she's trying to get across is that "avg(1/3,1/3) = 1/3 but the size of the whole has doubled." I disagree. The fourth-grader is perfectly correct within the context of the analogy that the teacher used. The problem is the analogy is wrong which is a general problem of relying on metaphors and analogies to explain rigorous technical concepts. Fractions are no…

Post by GavinMcG » Tue, May 26, 2020, 3:24 PM UTC

CydeWeys wrote: post #2

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 surpris…

Post by p4bl0 » Tue, May 26, 2020, 3:26 PM UTC

tsimionescu wrote: post #99

p4bl0 wrote: post #63

Earlier quoted context omitted.

Yup, it's a typing problem. Do it with different things in the two sets and it becomes clearer: 1/3 apples + 1/3 oranges = 2/6 fruits . This is another instance of situations where teaching compsci or at least programming could help teaching maths rather than the other way around as it is traditionally thought.

It's not really a typing problem, it's a problem of references. You can't add 1/3 apples and 1/3 oranges. However, you can add 1/3 * x people with 1/3 * y people and get 2/6 * z people, if x people +y people =z people.

    abstract class Fruit:
      prop name: string
      prop rotten: bool

    class Apple (Fruit):
      prop name: string = "apple"
      prop rotten: bool = False

    class Orange (Fruit):
      prop name: string = "orange"
      prop rotten: bool = False

    func add (a: List, b: List) -> List:
      return foldl(lambda l, f: l.append(f), a, b)

    func filter (p: Func, l: List) -> List:
      return [f for f in l if p(f)]

    func is_rotten (f: Fruit) -> bool:
      return f.rotten

    a1 = Apple()
    a2 = Apple()
    a3 = Apple()
    a3.rotten = True
    apples = [a1, a2, a3]

    o1 = Orange()
    o2 = Orange()
    o3 = Orange()
    o3.rotten = True
    oranges = [o1, o2, o3]

    fruits = add(apples, oranges)

    print(len(filter(is_rotten, apples)), "/", len(apples))
    > "1/3"

    print(len(filter(is_rotten, oranges)), "/", len(oranges))
    > "1/3"

    print(len(filter(is_rotten, fruits)), "/", len(fruits))
    > "2/6"

Post by tsimionescu » Tue, May 26, 2020, 3:32 PM UTC

macspoofing wrote: post #43

Spivak wrote: post #24

I can't imagine doing this to a class of fourth graders but I don't think her thinking is wrong. I think the correct way to views maths notation is that it's a language and we should treat people using it "incorrectly" as a grammar mistake and try to understand the idea they're trying to express. The fourth grader is saying "1/3 + 1/3 = 2/6" but the idea she's trying to get across is that "avg(1/3,1/3) = 1/3 but the…

>The fourth grader is saying "1/3 + 1/3 = 2/6" but the idea she's trying to get across is that "avg(1/3,1/3) = 1/3 but the size of the whole has doubled." I disagree. The fourth-grader is perfectly correct within the context of the analogy that the teacher used. The problem is the analogy is wrong which is a general problem of relying on metaphors and analogies to explain rigorous technical concepts. Fractions are no…