5 ms·

>For example, 3 bundles of 5 bananas is different from 5 bundles of 3 bananas although they total to the same number of bananas. Their structures are different.

by jasode 11y ago

>For example, 3 bundles of 5 bananas is different from 5 bundles of 3 bananas although they total to the same number of bananas. Their structures are different. I don't fully buy into this justification. The "5x3" problem on the test had "pure" numbers with no annotation of "objects". It's the blog writer that inserted an additional interpretation of "bananas" or "bundles". Instead, the "5x3" can be interpreted as counting iterations of "rows" -- or -- "columns" of a rectangle. Whichever orientation the child picked in his head can yield 5+5+5 or 3+3+3+3+3. In fact, take a closer look at the photo and you'll see in Question #2 that the child had a "different rectangle orientation" than the teacher! The Q1 & Q2 should not have been marked as incorrect. As for the other justification about possibly using a commutative law that's out of sequence with the learning curriculum, it still seems possible to interpret "5x3" using plain English as "take 5 and copy it out 3 times". No jumping ahead to Commutative Law required.

tremendo 11y ago
When I learned English (second language) I remember thinking "wow, wonderful, the language of multiplication tells you exactly what to do!" which I read as, in this case 5 × 3 => "[five times] three" 3+3+3+3+3, as the teacher illustrated, but here the student apparently answered "five [three times]". In my first language (Spanish) the multiplication is read as "five by three" which conjures up rectangles or lists, which can be vertical or horizontal oriented, and in either case, less clear and unambiguous than the English version. Still I believe it's certainly teaching the wrong lesson to mark the answer as incorrect, especially when the red mark comes without explanation. Even if the problem states "Use the repeated addition strategy". The author mentions it's crucial to understand this but I don't believe important enough to discourage a young student this way. The explanation of what was requested and the method of arriving at it should be made explicit, and it may have happened in class, we just don't know.
- newjersey 11y ago
  If you look at the next question, they go over the five by three in a rectangle approach. Maybe we should do away with grading students based on exam performance altogether.
  - vixen99 11y ago
    As against what happens out there in the tough world of work.
  - vixen99 11y ago
    As against what happens out there in the tough world of work.
  - chris_wot 11y ago
    My wife is a teacher in the NSW education system (Australia) and I've seen her use the rectangle system. However, the rectangle system is used to also show that if you take the same rectangle with the items placed in it in a uniform distribution, the rotate the rectangle and its contents by 90 degrees the the number of items are the same, but the row and column numbers swap around. If anything the rectangular system shows that multiplication is commutative, which I feel is its real value. Interestingly enough, that isn't ever explained to most teachers so I'm not surprised if it's being misapplied as a technique for learning!
- mattmcknight 11y ago
  I feel the opposite way about the English reading "5 times 3". In English, the subject comes first, so I would expect the sentence to mean take 5, use "times" as a verb, and 3 as the adverb. Likewise, if you read it as "5 multiplied by 3", you would expect to take five, three times.
  - signal11 11y ago
    Exactly. And this is why we use math operators rather than English terms for math operations. If you want to specifically mean 5 times a group of 3, define a new operator for it, like 5 ○ 3. Don't 'overload' the x operator with your own arbitrary meaning.
    - chris_wot 11y ago
      Ummm... I think you just overloaded the x operator yourself!
  - deleted 11y ago
    [deleted]
dznkorxt76qb 11y ago
> "you'll see in Question #2 that the child had a "different rectangle orientation" than the teacher" Vectors can be considered identical for the same reason... [1, 0] and [9, 0] are the same arrow if you move your head in the latter case. Here, the teacher is assuming the kid's head is at [0,0], when rotation (or translation) doesn't change the arrow any more than the rectangle. Neither is wrong. The lesson: memorize the teacher's poor use of language; ignore objective reality.
jmilloy 11y ago
> The "5x3" problem on the test had "pure" numbers with no annotation of "objects" It's not the "5x3" problem but the "repeated addition strategy" problem. I think that's part of the problem. Similarly, the bananas example isn't about the 5 and the 3 but about a difference between counting "x sets of y" and "y sets of x".
- jasode 11y ago
  >a difference between counting "x sets of y" and "y sets of x". You're making the same mistake as the blog writer by overlaying a difference between "x" and "y" that was not on the test. The child did do the repeated addition strategy. It's just that the child's "shape" of the addition didn't exactly match the teacher's. If the point of the problem was the "repeated addition" instead of the final answer "15", the child still did it correctly. He/she showed his work of repeated addition! The actual test problem was stated as "5 times 3" and not "5subscriptX times 3subscriptY" or "5subscriptBundles times 3subscriptBananas". You're arguing about a test the child didn't actually take.
  - jmilloy 11y ago
    The objective and obvious difference between the 5 and the 3 is that the 5 is first and the 3 is second. The point is that because the 5 is first, as everyone can see, it has a specific job in the repeated addition technique. (The bananas and bundles just illustrates an example for why, in another context, being first or second would be important. But on the test, 5 is still first.) On the other hand, you are invoking a "repeated addition" that the student was never taught. Your repeated addition strategy is "add <one of numbers> together <the other number> times". The taught repeated addition strategy was "add <the second number> together <the first number> times".
    - signal11 11y ago
      > The point is that because the 5 is first, as everyone can see, it has a specific job in the repeated addition technique. Um, this is sophistry. The question asked for "5 x 3" using repeated addition. The x is a very well defined mathematical operator and "repeated addition" has a very well-defined meaning, and the child has demonstrated it by repeatedly adding 5 three times. Yes, the child's cardinal sin is he Did Not Do As He Was Taught(tm), but seriously, that's more the teacher's and the school board's problem in my book, not the child's.
3pt14159 11y ago
I was leaning that way, but when I tried to the plain English way, I defaulted to "Five threes" not the backwards "take five and copy it out three times". Once he made the tie back to programming he was able to change my mind that this was important enough. Although it does seem unfair for the individual child.

by jasode 11y ago

tremendo 11y ago

When I learned English (second language) I remember thinking "wow, wonderful, the language of multiplication tells you exactly what to do!" which I read as, in this case 5 × 3 => "[five times] three" 3+3+3+3+3, as the teacher illustrated, but here the student apparently answered "five [three times]". In my first language (Spanish) the multiplication is read as "five by three" which conjures up rectangles or lists, which can be vertical or horizontal oriented, and in either case, less clear and unambiguous than the English version. Still I believe it's certainly teaching the wrong lesson to mark the answer as incorrect, especially when the red mark comes without explanation. Even if the problem states "Use the repeated addition strategy". The author mentions it's crucial to understand this but I don't believe important enough to discourage a young student this way. The explanation of what was requested and the method of arriving at it should be made explicit, and it may have happened in class, we just don't know.

newjersey 11y ago
If you look at the next question, they go over the five by three in a rectangle approach. Maybe we should do away with grading students based on exam performance altogether.
- vixen99 11y ago
  As against what happens out there in the tough world of work.
- vixen99 11y ago
  As against what happens out there in the tough world of work.
- chris_wot 11y ago
  My wife is a teacher in the NSW education system (Australia) and I've seen her use the rectangle system. However, the rectangle system is used to also show that if you take the same rectangle with the items placed in it in a uniform distribution, the rotate the rectangle and its contents by 90 degrees the the number of items are the same, but the row and column numbers swap around. If anything the rectangular system shows that multiplication is commutative, which I feel is its real value. Interestingly enough, that isn't ever explained to most teachers so I'm not surprised if it's being misapplied as a technique for learning!
mattmcknight 11y ago
I feel the opposite way about the English reading "5 times 3". In English, the subject comes first, so I would expect the sentence to mean take 5, use "times" as a verb, and 3 as the adverb. Likewise, if you read it as "5 multiplied by 3", you would expect to take five, three times.
- signal11 11y ago
  Exactly. And this is why we use math operators rather than English terms for math operations. If you want to specifically mean 5 times a group of 3, define a new operator for it, like 5 ○ 3. Don't 'overload' the x operator with your own arbitrary meaning.
  - chris_wot 11y ago
    Ummm... I think you just overloaded the x operator yourself!
- deleted 11y ago
  [deleted]

dznkorxt76qb 11y ago

> "you'll see in Question #2 that the child had a "different rectangle orientation" than the teacher" Vectors can be considered identical for the same reason... [1, 0] and [9, 0] are the same arrow if you move your head in the latter case. Here, the teacher is assuming the kid's head is at [0,0], when rotation (or translation) doesn't change the arrow any more than the rectangle. Neither is wrong. The lesson: memorize the teacher's poor use of language; ignore objective reality.

jmilloy 11y ago

> The "5x3" problem on the test had "pure" numbers with no annotation of "objects" It's not the "5x3" problem but the "repeated addition strategy" problem. I think that's part of the problem. Similarly, the bananas example isn't about the 5 and the 3 but about a difference between counting "x sets of y" and "y sets of x".

jasode 11y ago
>a difference between counting "x sets of y" and "y sets of x". You're making the same mistake as the blog writer by overlaying a difference between "x" and "y" that was not on the test. The child did do the repeated addition strategy. It's just that the child's "shape" of the addition didn't exactly match the teacher's. If the point of the problem was the "repeated addition" instead of the final answer "15", the child still did it correctly. He/she showed his work of repeated addition! The actual test problem was stated as "5 times 3" and not "5subscriptX times 3subscriptY" or "5subscriptBundles times 3subscriptBananas". You're arguing about a test the child didn't actually take.
- jmilloy 11y ago
  The objective and obvious difference between the 5 and the 3 is that the 5 is first and the 3 is second. The point is that because the 5 is first, as everyone can see, it has a specific job in the repeated addition technique. (The bananas and bundles just illustrates an example for why, in another context, being first or second would be important. But on the test, 5 is still first.) On the other hand, you are invoking a "repeated addition" that the student was never taught. Your repeated addition strategy is "add <one of numbers> together <the other number> times". The taught repeated addition strategy was "add <the second number> together <the first number> times".
  - signal11 11y ago
    > The point is that because the 5 is first, as everyone can see, it has a specific job in the repeated addition technique. Um, this is sophistry. The question asked for "5 x 3" using repeated addition. The x is a very well defined mathematical operator and "repeated addition" has a very well-defined meaning, and the child has demonstrated it by repeatedly adding 5 three times. Yes, the child's cardinal sin is he Did Not Do As He Was Taught(tm), but seriously, that's more the teacher's and the school board's problem in my book, not the child's.

3pt14159 11y ago

I was leaning that way, but when I tried to the plain English way, I defaulted to "Five threes" not the backwards "take five and copy it out three times". Once he made the tie back to programming he was able to change my mind that this was important enough. Although it does seem unfair for the individual child.