“Seems fraught with philosophical gobbledygook and circular reasoning to specify what about “because the teacher said so” it is that isn’t as “mathematical” as “because you’re summing ones and tens separately”.”
“Because you’re summing ones and tens separately” isn’t really a complete gears level explanation, but a pointer or hint to one. In particular, if you are trying to explain the phenomenon formally, you would begin by defining a “One’s and ten’s representation” of a number n as a tuple (a,b) such that 10a + b = n. We know that at least on such representation exists with a=0 and b=n.
Proof (warning, this is massive, you don’t need to read the whole thing)
You then can define a “Simples one’s and ten’s representation” as such a representation such that 0<=b<=9. We want to show that each number has at least one such representation. It is easy to see that (a, b) = 10a + b = 10a +10 + b − 10 = 10(a+1) + (b-10) = (a+1, b-10). We can repeat this process x times to get (a+x, b-10x). We know that for some x, b-10x will be negative, ie. if x=b, b-10x = −9x. We can decide to look at the last value before it is negative. Let this representation be (m,n). We have defined that n>=0. We also know that n can’t be >=10, else, (m+1, n-10) still has the second element of the tuple >=0. So any number can be written in the tuple form.
Suppose that there are two simple representations of a number (x, y) and (p, q). Then 10x+y = 10p + q. 10(x-p) =y-q. Now, since y and q are between 0 and 9 inclusive, we get that y-q is between 9 and −9, the only factor of 10 in this range is 0. So 10(x-p) =0 meaning x=p and y-q=0, meaning y=q. ie. both members of the tuple are the same.
It is then trivial to prove that (a1, b1) + (a2, b2) = (a1+a2, b1+b2). It is similarly easy to show 0<=b1+b2<=18, so that b1+b2 or b1+b2-10 is between 0 and 9 inclusive. It then directly follow that (a1+a2, b1+b2) or (a1+a2-1, b1+b2-10) is a simple representation (here we haven’t put any restriction on the value of the a’s).
Observations
So a huge amount is actually going on in something so simple. We can make the following observations:
“Because you’re summing ones and tens separately” will seem obvious to many people because they’ve been doing it for so long, but I suspect that the majority of the population would not be able to produce the above proof. In fact, I suspect that the majority of the population would not even realise that it was possible to break down the proof to this level of detail—I believe many of them would see the above sentence as unitary. And even when you tell them that there is an additional level of detail, they probably won’t have any idea what it is supposed to look like.
Part of the reason why it feels more gear like is because it provides you the first step of the proof (defining one’s and ten’s tuples). When someone has a high enough level of maths, they are able to get from the “hint” quite quickly to the full proof. Additionally, even if someone does not have the full proof in their head, they can still see that a certain step will be useful towards producing a proof. The hint of “summing the one’s and tens separately” allows you to quite quickly construct a formal representation of the problem, which is progress even if you are unable to construct a full proof. Discovering that the sum will be between 0 and 18, let’s you know that if you carry, you will only ever have to carry the one. This limits the problem to a more specific case. Any person attempting to solve this will probably have examples in the past where limiting the case in such a way made the proof either easier or possible, so whatever heurestic pattern matching which occurs within their brain will suggest that this is progress (though it may of course turn out later that the ability to restrict the situation does not actually make the proof any easier)
Another reason why it may feel more gear like is that it is possible to construct sentence of a similar form and use them as hints for other proofs. So, “Because you’re summing ones and tens separately” is linguistically close to “Because you’re summing tens and hundreds separately”, although I don’t want to suggest that people only perform a linguistic comparison. If someone has developed an intuitive understand of these phenomenon, this will also play a role.
I believe that part of the reason why it is so hard to define what is or what is not “gears-like” is because this isn’t based on any particular statement or model just by itself, but in terms of how this interacts with what a person already knows and can derive in order to produce statements. Further, it isn’t just about producing one perfect gears explanation, but the extent to which a person can produce certain segments of the proof (ie. a formal statement of the problem or restricting the problem to the sub-case as above) or the extent to which it allows the production of various generalisation (ie. we can generalise to (tens & hundreds, hundreds and thousands… ) or to (ones & tens & hundreds) or to binary or to abstract algebra). Further, what counts as a useful generalisation is not objective, but relative to the other maths someone knows or the situations that someone knows in which they can apply this maths. For example, imaginary numbers may not seem like a useful generalisation until a person knows the fundamental theorem of algebra or how it can be used to model phases in physics.
I won’t claim that I’ve completely or even almost completely mapped out the space of gears-ness, but I believe that this takes you pretty far towards an understanding of what it might be.
“Seems fraught with philosophical gobbledygook and circular reasoning to specify what about “because the teacher said so” it is that isn’t as “mathematical” as “because you’re summing ones and tens separately”.”
“Because you’re summing ones and tens separately” isn’t really a complete gears level explanation, but a pointer or hint to one. In particular, if you are trying to explain the phenomenon formally, you would begin by defining a “One’s and ten’s representation” of a number n as a tuple (a,b) such that 10a + b = n. We know that at least on such representation exists with a=0 and b=n.
Proof (warning, this is massive, you don’t need to read the whole thing)
You then can define a “Simples one’s and ten’s representation” as such a representation such that 0<=b<=9. We want to show that each number has at least one such representation. It is easy to see that (a, b) = 10a + b = 10a +10 + b − 10 = 10(a+1) + (b-10) = (a+1, b-10). We can repeat this process x times to get (a+x, b-10x). We know that for some x, b-10x will be negative, ie. if x=b, b-10x = −9x. We can decide to look at the last value before it is negative. Let this representation be (m,n). We have defined that n>=0. We also know that n can’t be >=10, else, (m+1, n-10) still has the second element of the tuple >=0. So any number can be written in the tuple form.
Suppose that there are two simple representations of a number (x, y) and (p, q). Then 10x+y = 10p + q. 10(x-p) =y-q. Now, since y and q are between 0 and 9 inclusive, we get that y-q is between 9 and −9, the only factor of 10 in this range is 0. So 10(x-p) =0 meaning x=p and y-q=0, meaning y=q. ie. both members of the tuple are the same.
It is then trivial to prove that (a1, b1) + (a2, b2) = (a1+a2, b1+b2). It is similarly easy to show 0<=b1+b2<=18, so that b1+b2 or b1+b2-10 is between 0 and 9 inclusive. It then directly follow that (a1+a2, b1+b2) or (a1+a2-1, b1+b2-10) is a simple representation (here we haven’t put any restriction on the value of the a’s).
Observations
So a huge amount is actually going on in something so simple. We can make the following observations:
“Because you’re summing ones and tens separately” will seem obvious to many people because they’ve been doing it for so long, but I suspect that the majority of the population would not be able to produce the above proof. In fact, I suspect that the majority of the population would not even realise that it was possible to break down the proof to this level of detail—I believe many of them would see the above sentence as unitary. And even when you tell them that there is an additional level of detail, they probably won’t have any idea what it is supposed to look like.
Part of the reason why it feels more gear like is because it provides you the first step of the proof (defining one’s and ten’s tuples). When someone has a high enough level of maths, they are able to get from the “hint” quite quickly to the full proof. Additionally, even if someone does not have the full proof in their head, they can still see that a certain step will be useful towards producing a proof. The hint of “summing the one’s and tens separately” allows you to quite quickly construct a formal representation of the problem, which is progress even if you are unable to construct a full proof. Discovering that the sum will be between 0 and 18, let’s you know that if you carry, you will only ever have to carry the one. This limits the problem to a more specific case. Any person attempting to solve this will probably have examples in the past where limiting the case in such a way made the proof either easier or possible, so whatever heurestic pattern matching which occurs within their brain will suggest that this is progress (though it may of course turn out later that the ability to restrict the situation does not actually make the proof any easier)
Another reason why it may feel more gear like is that it is possible to construct sentence of a similar form and use them as hints for other proofs. So, “Because you’re summing ones and tens separately” is linguistically close to “Because you’re summing tens and hundreds separately”, although I don’t want to suggest that people only perform a linguistic comparison. If someone has developed an intuitive understand of these phenomenon, this will also play a role.
I believe that part of the reason why it is so hard to define what is or what is not “gears-like” is because this isn’t based on any particular statement or model just by itself, but in terms of how this interacts with what a person already knows and can derive in order to produce statements. Further, it isn’t just about producing one perfect gears explanation, but the extent to which a person can produce certain segments of the proof (ie. a formal statement of the problem or restricting the problem to the sub-case as above) or the extent to which it allows the production of various generalisation (ie. we can generalise to (tens & hundreds, hundreds and thousands… ) or to (ones & tens & hundreds) or to binary or to abstract algebra). Further, what counts as a useful generalisation is not objective, but relative to the other maths someone knows or the situations that someone knows in which they can apply this maths. For example, imaginary numbers may not seem like a useful generalisation until a person knows the fundamental theorem of algebra or how it can be used to model phases in physics.
I won’t claim that I’ve completely or even almost completely mapped out the space of gears-ness, but I believe that this takes you pretty far towards an understanding of what it might be.