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.
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.