5 ms·

The answer is either 'yes', because the 5 is actually in the hundreds place, or 'no' because the value of the 5 is actually 500. You can answer either because o

The answer is either 'yes', because the 5 is actually in the hundreds place, or 'no' because the value of the 5 is actually 500. You can answer either because of incomplete information/context to the question (which might be provided by surrounding questions or class work). The intention of the question is to find out if the child has understood the concept of place values, and I think the ambiguity of the question is there so that you can accept either answer that demonstrates that the child understands that the value of a digit changes based on its location, and that the location of the 5 in 582 notates a 'hundred' multiplier. While I think the structure of the question might be off putting, asking about place values makes plenty of sense to me. I recall being taught place values in the mid to late 90s up here in Ontario, so it's not exactly a new fangled thing. edit: used less broken sentences.

TheOtherHobbes 11y ago
Is this really a good way to teach place values? To me it's confusing as anything, because it implies that symbol values aren't really constant. Isn't it easier to learn that symbol values are constant, but they're modified by position? So the 5 in 582 is 5 X 100. It's not "500", because conceptually that implies a different unique symbol. One way emphasises a limited set of consistent symbols, and a small pool of symbolic operations. The other way suggests there's an infinite pool of symbols of varying values, and you're supposed to think of them as separate numbers when they're in different place groups. The real point of math is generalisation and symbolic and functional economy. So I think the latter obscures what's happening instead of simplifying it.
- logfromblammo 11y ago
  If you really want to be technically correct (the best kind of correct), the "5" in "582" has the symbol value five multiplied by position value ten exponent two. Symbols defined using only references to themselves are notoriously difficult to understand. When the student answers, "it's in the hundreds place", that tells me a lot. It really says, "I think I know the concept you are trying to test, and I would like to prove that I have mastered it, but the question you asked does not allow me to do so, as it is ambiguous as to whether you are testing my knowledge of the invariant symbol value of '5', the positional value of the third digit from the right in a decimal number, or the embedded multiplication inherent in decimal notation. As the value of '5' is the least complex of those three concepts, I have purposefully left it out of my answer, and I hedged the other two senses by explicitly saying the second and implying the third." Or in shorter terms, "I'm not sure exactly why you are asking me that, so I'm not sure whether to tell you five or five hundred." As a result, I would say that answer is correct, but given in a form that exposes the inherent weakness of the question. The symbol '5' has value five no matter where it appears. The radix-exponent-2 position has the same value no matter which numeral symbol appears in it. The student can best demonstrate knowledge of this generalization by performing radix conversions. The '5' in hexadecimal number 0x582 is still worth five, but it is multiplied by position value sixteen exponent two rather than ten exponent two. The '5' is in the twohundredfiftysixes place rather than the hundreds place. So the symbol has net value five touhunnerfittysixes rather than five hundreds. If you don't emphasize the separation between symbol value and positional value, you're actually encouraging a regression to the mental-math roadblocks of Roman numerals, where I, V, X, L, C, D, and M have distinct invariant values.

TheOtherHobbes 11y ago

Is this really a good way to teach place values? To me it's confusing as anything, because it implies that symbol values aren't really constant. Isn't it easier to learn that symbol values are constant, but they're modified by position? So the 5 in 582 is 5 X 100. It's not "500", because conceptually that implies a different unique symbol. One way emphasises a limited set of consistent symbols, and a small pool of symbolic operations. The other way suggests there's an infinite pool of symbols of varying values, and you're supposed to think of them as separate numbers when they're in different place groups. The real point of math is generalisation and symbolic and functional economy. So I think the latter obscures what's happening instead of simplifying it.

logfromblammo 11y ago
If you really want to be technically correct (the best kind of correct), the "5" in "582" has the symbol value five multiplied by position value ten exponent two. Symbols defined using only references to themselves are notoriously difficult to understand. When the student answers, "it's in the hundreds place", that tells me a lot. It really says, "I think I know the concept you are trying to test, and I would like to prove that I have mastered it, but the question you asked does not allow me to do so, as it is ambiguous as to whether you are testing my knowledge of the invariant symbol value of '5', the positional value of the third digit from the right in a decimal number, or the embedded multiplication inherent in decimal notation. As the value of '5' is the least complex of those three concepts, I have purposefully left it out of my answer, and I hedged the other two senses by explicitly saying the second and implying the third." Or in shorter terms, "I'm not sure exactly why you are asking me that, so I'm not sure whether to tell you five or five hundred." As a result, I would say that answer is correct, but given in a form that exposes the inherent weakness of the question. The symbol '5' has value five no matter where it appears. The radix-exponent-2 position has the same value no matter which numeral symbol appears in it. The student can best demonstrate knowledge of this generalization by performing radix conversions. The '5' in hexadecimal number 0x582 is still worth five, but it is multiplied by position value sixteen exponent two rather than ten exponent two. The '5' is in the twohundredfiftysixes place rather than the hundreds place. So the symbol has net value five touhunnerfittysixes rather than five hundreds. If you don't emphasize the separation between symbol value and positional value, you're actually encouraging a regression to the mental-math roadblocks of Roman numerals, where I, V, X, L, C, D, and M have distinct invariant values.