That’s because (strictly speaking) they aren’t teaching math. They’re teaching “tricks” to solve equations easier, which can lead to more confusion.
Like the PEMDAS thing that’s being discussed here. There’s no such thing as “order of operations” in math, but it’s easier to teach by assuming that there is.
Edit:
To the people downvoting: I want to hear your opinions. Do you think I’m wrong? If so, why?
Yes and no. You teach how to solve equations, but not the fundamentals (and if you do then kudos to you, as it’s not a trivial accomplishment). Fundamentals, most of the time, are taught in universities. It’s so much easier that way, but doesn’t mean it’s right. People call it math, which is fair enough, but it’s not really math in a sense that you don’t understand the underlying principles.
Yes there is!
Nope.
There’s only commutation, association, distribution, and identity. It doesn’t matter in which order you apply any of those properties, the result will stay correct.
2×2×(2-1)/2 = 2×(4-2)/2 = 1×(4-2) = 4-2 = 2
As you can see, I didn’t follow any particular order and still got the correct result. Because no basic principle was broken.
Or I could also go
2×2×(2-1)/2 = 4×(2-1)/2 = 4×(1-0.5) = 4×0.5 = 2
Same result. Completely different order, yet still correct.
My response to the rest goes back to the aforementioned.
You teach how to solve equations, but not the fundamentals
Nope. We teach the fundamentals. Adults not remembering them doesn’t mean they weren’t taught. Just pick up a Maths textbook. It’s all in there. Always has been.
Fundamentals, most of the time, are taught in universities
No they’re not. They only teach order of operations from a remedial point of view. Most of them forget about The Distributive Law. I’ve seen multiple Professors be told by their students that they were wrong.
it’s not really math in a sense that you don’t understand the underlying principles
The Constructivist learners have no trouble at all understanding it.
Nope.
Yep!
There’s only commutation, association, distribution, and identity.
And many proofs of other rules, which you’ve decided to omit mentioning.
It doesn’t matter in which order you apply any of those properties, the result will stay correct
But the order you apply the operations does matter, hence the proven rules to be followed.
2×2×(2-2)/2
Notably you picked an example that has no addition, subtraction, or distribution in it. That’s called cherry-picking.
Completely different order, yet still correct
Yep, because you cherry-picked a simple example where it doesn’t matter. It’s never going to matter when you only pick operations which have the same precedence.
My response to the rest goes back to the aforementioned
Sure. They are, however, not the focus. At least that’s not how I’ve been taught in school. You’re not teaching kids how to prove the quadratic formula, do you? No, you teach them how to use it instead. The goal here is different.
They only teach order of operations.
Again, with the order of operations. It’s not a thing. I’ve given you two examples that don’t follow any.
The constructivist learners…
That’s kinda random, but sure?
And many proofs of other rules…
They all derive from each other. Even those fundamental properties are. For example, commutation is used to prove identity.
But the order you apply operators does matter
2+2-2 = 4-2 = 2+0 = 0
2 operators, no order followed.
If we take your example
2+3×4 then it’s not an order of operation that plays the role here. You have no property that would allow for (2+3)×4 to be equal 2+3×4
Look, 2+3×4 = 1+3×(2+2)+1 = 1+(6+6)+1 = 7+7 = 14
Is that not correct?
Notably you picked…
It literally has subtraction and distribution. I thought you taught math, no?
2-2 is 2 being, hear me out, subtracted from 2
Same with 2×(2-2), I can distribute the value so it becomes 4-4
No addition? Who cares, subtraction literally works the same, but in opposite direction. Same properties apply. Would you feel better if I wrote (2-2) as (1+1-2)? I think not.
Also, can you explain how is that cherry-picking? You only need one equation that is solvable out of order to prove order of operation not existing. One is conclusive enough. If I give you two or more, it doesn’t add anything meaningful.
At least that’s not how I’ve been taught in school
If you had a bad teacher that doesn’t mean everyone else had a bad teacher.
You’re not teaching kids how to prove the quadratic formula, do you?
We teach them how to do proofs, including several specific ones.
No, you teach them how to use it instead.
We teach them how to use everything, and how to do proofs as well. Your whole argument is just one big strawman.
Again, with the order of operations
Happens to be the topic of the post.
It’s not a thing
Yes it is! 😂
I’ve given you two examples that don’t follow any
So you could not do the brackets first and still get the right answer? Nope!
2×2×(2-2)/2=0
2×2×2-2/2=7
That’s kinda random, but sure?
Not random at all, given you were talking about students understanding how Maths works.
2+3×4 then it’s not an order of operation that plays the role here
Yes it is! 😂 If I have 1 2-litre bottle of milk, and 4 3-litre bottles of milk, there’s only 1 correct answer for how many litres of milk of have, and it ain’t 20! 😂 Even elementary school kids know how to work it out just by counting up.
They all derive from each other
No they don’t. The proof of order of operations has got nothing to do with any of the properties you mentioned.
For example, commutation is used to prove identity
And neither is used to prove the order of operations.
2 operators, no order followed
Again with a cherry-picked example that only includes operators of the same precedence.
You have no property that would allow for (2+3)×4 to be equal 2+3×4
And yet we have a proof of why 14 is the only correct answer to 2+3x4, why you have to do the multiplication first.
Is that not correct?
Of course it is. So what?
It literally has subtraction and distribution
No it didn’t. It had Brackets (with subtraction inside) and Multiplication and Division.
I thought you taught math, no?
Yep, and I just pointed out that what you just said is wrong. 2-2(1+2) has Subtraction and Distribution.
2-2 is 2 being, hear me out, subtracted from 2
Which was done first because you had it inside Brackets, therefore not done in the Subtraction step in order of operations, but the Brackets step.
Also, can you explain how is that cherry-picking?
You already know - you know which operations to pick to make it look like there’s no such thing as order of operations. If I tell you to look up at the sky at midnight and say “look - there’s no such thing as the sun”, that doesn’t mean there’s no such thing as the sun.
That’s because (strictly speaking) they aren’t teaching math. They’re teaching “tricks” to solve equations easier, which can lead to more confusion.
Like the PEMDAS thing that’s being discussed here. There’s no such thing as “order of operations” in math, but it’s easier to teach by assuming that there is.
Edit: To the people downvoting: I want to hear your opinions. Do you think I’m wrong? If so, why?
Yes we are. Adults forgetting it is another matter altogether.
Yes there is! 😂
No, I know you’re wrong.
If you don’t solve binary operators before unary operators you get wrong answers. 2+3x4=14, not 20. 3x4=3+3+3+3 by definition
Yes and no. You teach how to solve equations, but not the fundamentals (and if you do then kudos to you, as it’s not a trivial accomplishment). Fundamentals, most of the time, are taught in universities. It’s so much easier that way, but doesn’t mean it’s right. People call it math, which is fair enough, but it’s not really math in a sense that you don’t understand the underlying principles.
Nope.
There’s only commutation, association, distribution, and identity. It doesn’t matter in which order you apply any of those properties, the result will stay correct.
2×2×(2-1)/2 = 2×(4-2)/2 = 1×(4-2) = 4-2 = 2
As you can see, I didn’t follow any particular order and still got the correct result. Because no basic principle was broken.
Or I could also go
2×2×(2-1)/2 = 4×(2-1)/2 = 4×(1-0.5) = 4×0.5 = 2
Same result. Completely different order, yet still correct.
My response to the rest goes back to the aforementioned.
Nope. We teach the fundamentals. Adults not remembering them doesn’t mean they weren’t taught. Just pick up a Maths textbook. It’s all in there. Always has been.
No they’re not. They only teach order of operations from a remedial point of view. Most of them forget about The Distributive Law. I’ve seen multiple Professors be told by their students that they were wrong.
The Constructivist learners have no trouble at all understanding it.
Yep!
And many proofs of other rules, which you’ve decided to omit mentioning.
But the order you apply the operations does matter, hence the proven rules to be followed.
Notably you picked an example that has no addition, subtraction, or distribution in it. That’s called cherry-picking.
Yep, because you cherry-picked a simple example where it doesn’t matter. It’s never going to matter when you only pick operations which have the same precedence.
…cherry-picking.
Sure. They are, however, not the focus. At least that’s not how I’ve been taught in school. You’re not teaching kids how to prove the quadratic formula, do you? No, you teach them how to use it instead. The goal here is different.
Again, with the order of operations. It’s not a thing. I’ve given you two examples that don’t follow any.
That’s kinda random, but sure?
They all derive from each other. Even those fundamental properties are. For example, commutation is used to prove identity.
2+2-2 = 4-2 = 2+0 = 0
2 operators, no order followed.
If we take your example
2+3×4 then it’s not an order of operation that plays the role here. You have no property that would allow for (2+3)×4 to be equal 2+3×4
Look, 2+3×4 = 1+3×(2+2)+1 = 1+(6+6)+1 = 7+7 = 14
Is that not correct?
It literally has subtraction and distribution. I thought you taught math, no?
2-2 is 2 being, hear me out, subtracted from 2
Same with 2×(2-2), I can distribute the value so it becomes 4-4
No addition? Who cares, subtraction literally works the same, but in opposite direction. Same properties apply. Would you feel better if I wrote (2-2) as (1+1-2)? I think not.
Also, can you explain how is that cherry-picking? You only need one equation that is solvable out of order to prove order of operation not existing. One is conclusive enough. If I give you two or more, it doesn’t add anything meaningful.
If you had a bad teacher that doesn’t mean everyone else had a bad teacher.
We teach them how to do proofs, including several specific ones.
We teach them how to use everything, and how to do proofs as well. Your whole argument is just one big strawman.
Happens to be the topic of the post.
Yes it is! 😂
So you could not do the brackets first and still get the right answer? Nope!
2×2×(2-2)/2=0
2×2×2-2/2=7
Not random at all, given you were talking about students understanding how Maths works.
Yes it is! 😂 If I have 1 2-litre bottle of milk, and 4 3-litre bottles of milk, there’s only 1 correct answer for how many litres of milk of have, and it ain’t 20! 😂 Even elementary school kids know how to work it out just by counting up.
No they don’t. The proof of order of operations has got nothing to do with any of the properties you mentioned.
And neither is used to prove the order of operations.
Again with a cherry-picked example that only includes operators of the same precedence.
And yet we have a proof of why 14 is the only correct answer to 2+3x4, why you have to do the multiplication first.
Of course it is. So what?
No it didn’t. It had Brackets (with subtraction inside) and Multiplication and Division.
Yep, and I just pointed out that what you just said is wrong. 2-2(1+2) has Subtraction and Distribution.
Which was done first because you had it inside Brackets, therefore not done in the Subtraction step in order of operations, but the Brackets step.
You already know - you know which operations to pick to make it look like there’s no such thing as order of operations. If I tell you to look up at the sky at midnight and say “look - there’s no such thing as the sun”, that doesn’t mean there’s no such thing as the sun.