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I'll bite. Here is a sample from my daughter's math homework this week: Bobby's teacher asked him what the value of the 5 in 582 is. He answered that it was

I'll bite. Here is a sample from my daughter's math homework this week: Bobby's teacher asked him what the value of the 5 in 582 is. He answered that it was in the hundreds place. Did he answer the teachers question? If yes explain why. If no what is the right answer? Bonus points if you can explain the intention of asking the value of a numeral in a larger integer.

dsfyu404ed 11y ago
That bonus question.... wow
twixy 11y ago
No, I would say the right answer is 500. However, if he said it's in the hundreds place then I would say he understands the concept of place values.
icegreentea 11y ago
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.
DanBC 11y ago
They're making sure the child knows the 5 in 582 represents ten times more than the 5 in 852, which represents ten times more than the 5 in 825. This is standard 1 (4.NBT.A.1) https://books.google.co.uk/books?id=RvpzCQAAQBAJ&pg=PT160&lpg=PT160&dq=common+core+what+is+the+value+of+5+in+528&source=bl&ots=ZNO_3MGlEQ&sig=u-BDrlP7GNmhdBa-TeQxlSnrJeI&hl=en&sa=X&ved=0ahUKEwjfzcWRzaTJAhXI1iwKHWCVDRMQ6AEIITAA#v=onepage&q=common%20core%20what%20is%20the%20value%20of%205%20in%20528&f=false https://books.google.co.uk/books?id=RvpzCQAAQBAJ&pg=PT160&lp...
r00fus 11y ago
What grade is your daughter in?
- droopybuns 11y ago
  2nd.
  - twixy 11y ago
    This is really funny because my son is in the 2nd grade, his name is Bobby and I actually went through the same exact misunderstanding a few weeks ago.
MarkMc 11y ago
This is an excellent question because it allows the student to demonstrate an understanding that (a) digits have different 'meaning' depending on the context - the 5 'means 500' and the 8 'means 80'; and (b) there is a difference between replying with a true statement and answering correctly. Bobby has made a true statement that is related to the question, but hasn't completely answered the question. The correct is that the value of 5 in 582 is 500, and an even better answer would be "the value of 5 in 582 is 500 because the 5 is in the hundreds place". This distinction is something that many teachers overlook. There is an excellent handbook for teachers called "Teach Like A Champion" [1] which emphasises that "Right is Right" - here is an excerpt: --------------------------------- Right Is Right is about the difference between partially right and all-the-way right—between pretty good and 100 percent. The job of the teacher is to set a high standard for correctness: 100 percent. The likelihood is strong that students will stop striving when they hear the word right (or yes or some other proxy), so there's a real risk to naming as right that which is not truly and completely right. When you sign off and tell a student she is right, she must not be betrayed into thinking she can do something that she Cannot. Many teachers respond to almost-correct answers their students give in class by rounding up. That is they'll affirm the student's answer and repeat it, adding some detail of their own to make it fully correct even though the student didn't provide (and may not recognize) the differentiating factor. Imagine a student who's asked at the beginning of Romeo and Juliet how the Capulets and Montagues get along. “They don't like each other,” the student might say, in an answer that most teachers would, I hope, want some elaboration on before they called it fully correct. “Right,” the teacher might reply. “They don't like each other, and they have been feuding for generations.” But of course the student hadn't included the additional detail. That's the “rounding up.” Sometimes the teacher will even give the student credit for the rounding up as if the student said what he did not and what she merely wished he'd said, as in, “Right, what Kiley said was that they don't like each other and have been feuding. Good work, Kiley.” Either way, the teacher has set a low standard for correctness and explicitly told the class that they can be right even when they are not. Just as important, she has crowded out students' own thinking, doing cognitive work that students could do themselves (e.g., “So, is this a recent thing? A temporary thing? Who can build on Kiley's answer?”). When answers are almost correct, it's important to tell students that they're almost there, that you like what they've done so far, that they're closing in on the right answer, that they've done some good work or made a great start. You can repeat a student's answer back to him so he can listen for what's missing and further correct—for example, “You said the Capulets and the Montagues didn't get along.” Or you can wait or prod or encourage or cajole in other ways to tell students what still needs doing, ask who can help get the class all the way there until you get students all the way to a version of right that's rigorous enough to be college prep: “Kiley, you said the Capulets and the Montagues didn't get along. Does that really capture their relationship? Does that sound like what they'd say about each other?” In holding out for right, you set the expectation that the questions you ask and their answers truly matter. You show that you believe your students are capable of getting answers as right as students anywhere else. You show the difference between the facile and the scholarly. This faith in the quality of a right answer sends a powerful message to your students that will guide them long after they have left your classroom. ---------------------------------------- [1] http://www.amazon.com/gp/product/1118901851 http://www.amazon.com/gp/product/1118901851
rayiner 11y ago
That's a great question. Figuring out how to identify why answers are partially but not fully correct is a great way to understand the ins-and-outs of a concept.

dsfyu404ed 11y ago

That bonus question.... wow

twixy 11y ago

No, I would say the right answer is 500. However, if he said it's in the hundreds place then I would say he understands the concept of place values.

icegreentea 11y ago

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.

DanBC 11y ago

They're making sure the child knows the 5 in 582 represents ten times more than the 5 in 852, which represents ten times more than the 5 in 825. This is standard 1 (4.NBT.A.1) https://books.google.co.uk/books?id=RvpzCQAAQBAJ&pg=PT160&lpg=PT160&dq=common+core+what+is+the+value+of+5+in+528&source=bl&ots=ZNO_3MGlEQ&sig=u-BDrlP7GNmhdBa-TeQxlSnrJeI&hl=en&sa=X&ved=0ahUKEwjfzcWRzaTJAhXI1iwKHWCVDRMQ6AEIITAA#v=onepage&q=common%20core%20what%20is%20the%20value%20of%205%20in%20528&f=false https://books.google.co.uk/books?id=RvpzCQAAQBAJ&pg=PT160&lp...

r00fus 11y ago

What grade is your daughter in?

droopybuns 11y ago
2nd.
- twixy 11y ago
  This is really funny because my son is in the 2nd grade, his name is Bobby and I actually went through the same exact misunderstanding a few weeks ago.

MarkMc 11y ago

This is an excellent question because it allows the student to demonstrate an understanding that (a) digits have different 'meaning' depending on the context - the 5 'means 500' and the 8 'means 80'; and (b) there is a difference between replying with a true statement and answering correctly. Bobby has made a true statement that is related to the question, but hasn't completely answered the question. The correct is that the value of 5 in 582 is 500, and an even better answer would be "the value of 5 in 582 is 500 because the 5 is in the hundreds place". This distinction is something that many teachers overlook. There is an excellent handbook for teachers called "Teach Like A Champion" [1] which emphasises that "Right is Right" - here is an excerpt: --------------------------------- Right Is Right is about the difference between partially right and all-the-way right—between pretty good and 100 percent. The job of the teacher is to set a high standard for correctness: 100 percent. The likelihood is strong that students will stop striving when they hear the word right (or yes or some other proxy), so there's a real risk to naming as right that which is not truly and completely right. When you sign off and tell a student she is right, she must not be betrayed into thinking she can do something that she Cannot. Many teachers respond to almost-correct answers their students give in class by rounding up. That is they'll affirm the student's answer and repeat it, adding some detail of their own to make it fully correct even though the student didn't provide (and may not recognize) the differentiating factor. Imagine a student who's asked at the beginning of Romeo and Juliet how the Capulets and Montagues get along. “They don't like each other,” the student might say, in an answer that most teachers would, I hope, want some elaboration on before they called it fully correct. “Right,” the teacher might reply. “They don't like each other, and they have been feuding for generations.” But of course the student hadn't included the additional detail. That's the “rounding up.” Sometimes the teacher will even give the student credit for the rounding up as if the student said what he did not and what she merely wished he'd said, as in, “Right, what Kiley said was that they don't like each other and have been feuding. Good work, Kiley.” Either way, the teacher has set a low standard for correctness and explicitly told the class that they can be right even when they are not. Just as important, she has crowded out students' own thinking, doing cognitive work that students could do themselves (e.g., “So, is this a recent thing? A temporary thing? Who can build on Kiley's answer?”). When answers are almost correct, it's important to tell students that they're almost there, that you like what they've done so far, that they're closing in on the right answer, that they've done some good work or made a great start. You can repeat a student's answer back to him so he can listen for what's missing and further correct—for example, “You said the Capulets and the Montagues didn't get along.” Or you can wait or prod or encourage or cajole in other ways to tell students what still needs doing, ask who can help get the class all the way there until you get students all the way to a version of right that's rigorous enough to be college prep: “Kiley, you said the Capulets and the Montagues didn't get along. Does that really capture their relationship? Does that sound like what they'd say about each other?” In holding out for right, you set the expectation that the questions you ask and their answers truly matter. You show that you believe your students are capable of getting answers as right as students anywhere else. You show the difference between the facile and the scholarly. This faith in the quality of a right answer sends a powerful message to your students that will guide them long after they have left your classroom. ---------------------------------------- [1] http://www.amazon.com/gp/product/1118901851 http://www.amazon.com/gp/product/1118901851

rayiner 11y ago

That's a great question. Figuring out how to identify why answers are partially but not fully correct is a great way to understand the ins-and-outs of a concept.