61–70 of 308 posts
I don't understand why this simple 2nd grade math is featured on hacker news. I guess the next topic will be a long discussion what would be 1/2 divided by 2/3.
A lot of people are at home right now trying to teach their kids who are also at home.
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…
https://en.wikipedia.org/wiki/Mediant_(mathematics) https://www.cimat.mx/ciencia_para_jovenes/bachillerato/libro...
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…
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.
( 1 girl / 3 students at table L + 1 girl / 3 students at table R ) can't be added, because the denominators are different.
To fix that, you can multiply by the factors ( 3 students at table L / 6 students at both tables ) and ( 3 students at table R / 6 students at both tables ) so you get ( 1 girl / 6 students at both tables + 1 girl / 6 students at both tables ) = ( 2 girls / 6 students at both tables ).
To hammer it home, you point to the table that has 2 or 4 students at it, and ask someone how to add the proportion of girls at that table to the two already under consideration.
This is a great example of the downside of over-relying on metaphors and analogies to teach mathematical concepts. The reality is that fractions are not like groups of pencils or tables of girls and boys. They are rigorously defined mathematical constructs that occasionally can be mapped to real-world things (and usually with severe constraints). 1/3 + 1/3 isn't 2/6 because it doesn't follow from the underlying axiom…
Except mathematical concepts emerged from real-world observation and problems. We didn't invent fraction out of nowhere. The moment when you have to stop relying on intuition is a pretty delicate matter, but it's still interesting to try to rely on metampho, and then understand when and why a particular metaphor stops working. Much better (imho) than teaching math as a purely formal and transcendent topic that happen…
But we did. When mathematicians provided a rigorous definition of fractions (rational numbers) they separated them from the real world. Rational numbers do not exist in the real world. Real-world does not have infinities. It does not have negative values. In the real world 1/3+1/3 does equal 2/6 in the way that the fourth-grader applied the analogy.
>The moment where you have to stop relying on intuition is a pretty delicate matter
I didn't argue that metaphors and analogies shouldn't be used. I argued that analogies and metaphors are intrinsically flawed and this article provides a great example. At some point, you have to give up on the analogy and fallback on the underlying axioms. You can't do "1/3+1/3=2/6" not because it doesn't make sense for tables of boys and girls (because it does) but because it's against the rules for adding fractions.
Earlier quoted context omitted.
I think x(3) would more commonly mean x times 3. f(3) would be function application. Even more context dependence.
I think x(3) would more commonly imply a function of 3, since you'd virtually always otherwise just write 3x.
There's other context too, based on what is known. Up to a certain point in first year at my university, most engineering students haven't ever seen functions named x and y, and so they'd mostly interpret x(3) as multiplication. Then we show them parametric curves, and suddenly x(3) looks like the x coordinate of a point on the (x(t), y(t)) curve.
Earlier quoted context omitted.
I think x(3) would more commonly imply a function of 3, since you'd virtually always otherwise just write 3x.
This is one of the reasons I annoy people by following Wolfram’s convention in Mathematica of using square braces to denote arguments passed to a function: f ( x )= fx = f × x while f [ x ] means “apply the function f to the argument x ”. An unusual convention it may well be, but at least it’s one devoid of ambiguity.
This 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 it's important to know that this really isn't true. Maths at all times is a subjective language. Maths notation is imprecise "intentionally confused", or just ad-hoc defined all the time. When you see something like. f(x) = summation(x^n, n=0, 10) We conveniently ignore that this polynomial is defined at x=0 despite 0^0 not making any sense by ad-hoc defining 0^0 = 1 in this context.
What do you think the polynomial is? I ask, because in all situations similar to this that I've encountered it made sense to define 0^0 as 1, not as 0. If you genuinely have a case where 0^0 = 0 makes consistent sense then I'd be interested in understanding it.
So, what do you think the summation actually is when expanded?
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…
This 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 it's important to know that this really isn't true. Maths at all times is a subjective language. Maths notation is imprecise "intentionally confused", or just ad-hoc defined all the time. When you see something like. f(x) = summation(x^n, n=0, 10) We conveniently ignore that this polynomial is defined at x=0 despite 0^0 not making any sense by ad-hoc defining 0^0 = 1 in this context.