Those two things are memorisation tasks. Maths is not about memorisation.
You are not supposed to remember that the area of a triangle is a * h / 2, you’re supposed to understand why it’s the case. You’re supposed to be able to show that any triangle that can possibly exist is half the area of the rectangle it’s stuck in: Start with the trivial case (right-angled triangle), then move on to more complicated cases. If you’ve understood that once, there is no reason to remember anything because you can derive the formula at a moment’s notice.
All maths can be understood and derived like that. The names of the colours, their ordering, the names of the planets and how they’re ordered, they’re arbitrary, they have no rhyme or reason, they need to be memorised if you want to recall them. Maths doesn’t, instead it dies when you apply memorisation.
Ein Anfänger (der) Gitarre Hat Elan. There, that’s the Guitar strings in German. Why do I know that? Because my music theory knowledge sucks. I can’t apply it, music is all vibes to me but I still need a way to match the strings to what the tuner is displaying. You should never learn music theory from me, just as you shouldn’t learn maths from a teacher who can’t prove a * h / 2, or thinks it’s unimportant whether you can prove it.
You are not supposed to remember that the area of a triangle is a * h / 2
Yes you are. A lot of students get the wrong answer when they forget the half.
you’re supposed to understand why it’s the case
Constructivist learners can do so, ROTE learners it doesn’t matter. As long as they all know how to do Maths it doesn’t matter if they understand it or not.
You’re supposed to be able to show that any triangle that can possibly exist is half the area of the rectangle it’s stuck in
No they’re not.
If you’ve understood that once, there is no reason to remember anything because you can derive the formula at a moment’s notice.
And if you haven’t understood it then there is a reason to remember it.
you can derive the formula at a moment’s notice
Students aren’t expected to be able to do that.
All maths can be understood and derived like that
It can be by Constructivist learners, not ROTE learners.
The names of the colours, their ordering, the names of the planets and how they’re ordered, they’re arbitrary
No they’re not. Colours are in spectrum order, the planets are in order from the sun.
Maths doesn’t, instead it dies when you apply memorisation
A very substantial chunk of the population does just fine with having memorised Maths.
As far as I know, the only reason multiplication and division come first is that we’ve all agreed to it. But it can’t be derived in a vacuum as that other dude contends it should be.
No, only multiply and divide are. 2+3 is really +2+3, but we don’t write the first plus usually (on the other hand we do always write the minus if it starts with one).
As far as I know, the only reason multiplication and division come first is that we’ve all agreed to it.
No, they come first because you get wrong answers if you don’t do them first. e.g. 2+3x4=14, not 20. All the rules of Maths exist to make sure you get correct answers. Multiplication is defined as repeated addition - 3x4=3+3+3+3 - hence wrong answers if you do the addition first (just changed the multiplicand, and hence the answer). Ditto for exponents, which are defined as repeated multiplication, a^2=(axa). Order of operations is the process of reducing everything down to adds and subtracts on a number line. 3^2=3x3=3+3+3
Very confidently getting basic facts wrong doesn’t inspire confidence in the rest of your comments.
Your example still doesn’t give a reason why 2 + 3 * 4 is 2 + 3 + 3 + 3 +3 instead of 2 + 3 + 2 + 3 + 2 + 3 + 2 + 3 other than that we all agree to it.
And you can shove the condescension up your ass until you understand the difference between unary and binary operators.
But to original point. I’m not disagreeing with anything and you’re proving my point for me. There is no fundamental law of the universe that says multiplication comes first. It’s defined by man and agreed to. If we encounter aliens someday, the area of their triangles are still going to be half the width times the height, the ratios of their circles circumference to diameter are still going to be pi, regardless of how they represent those values. But they could very well prioritize addition and subtraction over multiplication and division.
Nothing. And that’s why people don’t write equations like that: You either see
46 + ---2
or
6 + 4
-------
2
If you wrote 6 + 4 / 2 in a paper you’d get reviewers complaining that it’s ambiguous, if you want it to be on one line write (6+4) / 2 or 6 + (4/2) or 6 + ⁴⁄₂ or even ½(6 + 4) Working mathematicians never came up with PEMDAS, which disambiguates it without parenthesis, US teachers did. Noone else does it that way because it does not, in the slightest, aid readability.
Is it also lazy to learn Roy G. Biv to know the color spectrum instead of learning all the physics and optical properties behind that?
Or what about My Very Elderly Mother Just Served Us Nine Pickles to know the planets instead of learning orbital dynamics and astrophysics?
Christ man, it’s a mnemonic device for elementary schoolers.
Those two things are memorisation tasks. Maths is not about memorisation.
You are not supposed to remember that the area of a triangle is
a * h / 2, you’re supposed to understand why it’s the case. You’re supposed to be able to show that any triangle that can possibly exist is half the area of the rectangle it’s stuck in: Start with the trivial case (right-angled triangle), then move on to more complicated cases. If you’ve understood that once, there is no reason to remember anything because you can derive the formula at a moment’s notice.All maths can be understood and derived like that. The names of the colours, their ordering, the names of the planets and how they’re ordered, they’re arbitrary, they have no rhyme or reason, they need to be memorised if you want to recall them. Maths doesn’t, instead it dies when you apply memorisation.
Ein Anfänger (der) Gitarre Hat Elan. There, that’s the Guitar strings in German. Why do I know that? Because my music theory knowledge sucks. I can’t apply it, music is all vibes to me but I still need a way to match the strings to what the tuner is displaying. You should never learn music theory from me, just as you shouldn’t learn maths from a teacher who can’t prove
a * h / 2, or thinks it’s unimportant whether you can prove it.It is for ROTE learners.
Yes you are. A lot of students get the wrong answer when they forget the half.
Constructivist learners can do so, ROTE learners it doesn’t matter. As long as they all know how to do Maths it doesn’t matter if they understand it or not.
No they’re not.
And if you haven’t understood it then there is a reason to remember it.
Students aren’t expected to be able to do that.
It can be by Constructivist learners, not ROTE learners.
No they’re not. Colours are in spectrum order, the planets are in order from the sun.
A very substantial chunk of the population does just fine with having memorised Maths.
What fundamental property of the universe says that
6 + 4 / 2 is 8 instead of 5?
The fundamental property of Maths that you have to solve binary operators before unary operators or you end up with wrong answers.
But +, -, *, and / are all binary operators?
As far as I know, the only reason multiplication and division come first is that we’ve all agreed to it. But it can’t be derived in a vacuum as that other dude contends it should be.
No, only multiply and divide are. 2+3 is really +2+3, but we don’t write the first plus usually (on the other hand we do always write the minus if it starts with one).
No, they come first because you get wrong answers if you don’t do them first. e.g. 2+3x4=14, not 20. All the rules of Maths exist to make sure you get correct answers. Multiplication is defined as repeated addition - 3x4=3+3+3+3 - hence wrong answers if you do the addition first (just changed the multiplicand, and hence the answer). Ditto for exponents, which are defined as repeated multiplication, a^2=(axa). Order of operations is the process of reducing everything down to adds and subtracts on a number line. 3^2=3x3=3+3+3
Typical examples of binary operations are the addition ( + {\displaystyle +}) and multiplication ( × {\displaystyle \times }) of numbers and matrices
Very confidently getting basic facts wrong doesn’t inspire confidence in the rest of your comments.
Your example still doesn’t give a reason why 2 + 3 * 4 is 2 + 3 + 3 + 3 +3 instead of 2 + 3 + 2 + 3 + 2 + 3 + 2 + 3 other than that we all agree to it.
…says person quoting Wikipedia and NOT a Maths textbook! 😂
Yes it does., need to work on your comprehension…
You can disagree as much as you want and 3x4 will still be defined as 3+3+3+3. It’s been that way ever since Multiplication was invented.
The arithmetic operations, addition + , subtraction − , multiplication × , and division ÷
That better? Or you can find one you like all by yourself: https://duckduckgo.com/?q=binary+operator&ko=-1&ia=web
And you can shove the condescension up your ass until you understand the difference between unary and binary operators.
But to original point. I’m not disagreeing with anything and you’re proving my point for me. There is no fundamental law of the universe that says multiplication comes first. It’s defined by man and agreed to. If we encounter aliens someday, the area of their triangles are still going to be half the width times the height, the ratios of their circles circumference to diameter are still going to be pi, regardless of how they represent those values. But they could very well prioritize addition and subtraction over multiplication and division.
Nothing. And that’s why people don’t write equations like that: You either see
4 6 + --- 2or
6 + 4 ------- 2If you wrote
6 + 4 / 2in a paper you’d get reviewers complaining that it’s ambiguous, if you want it to be on one line write(6+4) / 2or6 + (4/2)or6 + ⁴⁄₂or even½(6 + 4)Working mathematicians never came up with PEMDAS, which disambiguates it without parenthesis, US teachers did. Noone else does it that way because it does not, in the slightest, aid readability.Says someone who clearly hasn’t looked in any Maths textbooks
Only if their Maths was very poor. #MathsIsNeverAmbiguous
Yes they did.
It was never ambiguous to begin with.
Says someone who has never looked in a non-U.S. Maths textbooks - BIDMAS, BODMAS, BEDMAS, all textbooks have one variation or another.