5 ms·

> 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 t

by 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.
    - jmilloy 11y ago
      It doesn't seem like a cardinal sin so much as a small quantitative note that the process was taught a different way that the teacher thinks is important. Let's suppose one student can follow the procedure when asked but can't actually multiply in application, one student can't follow the procedure correctly but can multiply when needed, and a third can do both. Probably the first student will get questions on this quiz correct but will struggle on much of the rest of the unit, maybe get a low grade or hopefully get the help they need. The second (with the paper shown in the OP), will probably get an high grade because they got partial credit on a silly quiz. The third will get a higher high grade. What's so bad about that? BTW, appealing to definitions won't work here, because the x does have a very well-defined mathematical meaning: a x b := b + ... + b.
      - signal11 11y ago
        > ... does have a very well-defined mathematical meaning: a x b := b + ... + b. Please complete the definition. which is that a x b = b x a, so a x b can also be written as a x .. x a. There is nothing special about the order.
        jmilloy 11y ago
        That's not part of the definition. That's a separate property. I'm not sure we should care about that in elementary school, so the point is not to defend the teacher but only that you can't use the definition as an argument against the teacher.
        chris_wot 11y ago
        No, I'm afraid you've not given a complete definition of multiplication. You need to also show that multiplication is commutative, which is indeed a property of multiplication but MUST be included in the definition. At the child's level (primary age child, NOT high-school) then it is unnecessary to introduce the distributive property. But you honestly have to make the associative property very, very clear of the child will potentially have problems down the track! (Edit: brain fart - I said associative when I meant commutative. Oops!)
  - jasode 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. If you(royal-you) insist that the 5 being the first factor has a specific job and you teach such nonsense to a child, it means you're not teaching actual mathematics. In _real_ math, the factors/mutiplicands have no notion of ordinal rank such as "first" or "second" or "specific jobs". Even if the child was not formerly taught The Commutative Law, it's not impossible for him to see multiplication tables[1]. (In fact, many are hung as big posters in elementary classrooms.) Any child with pattern recognition abilities beyond a chimpanzee would notice that the cells of XY have the same answer as YX. He/she would ask mom/dad/teacher "is xy always same as yx?". In the world of _pseudo_ math that stresses bizarre hoop jumping, we overlay non-mathematical concepts such as "specific job" to factors. Maybe this skill is important and transferable to the enlisted man to make sure he makes his bed before cleaning his machine gun instead of the other way around so everyone in the squad doesn't get punished with 50 pushups. But don't pass it off as "teaching math." [1]https://www.google.com/search?q=multiplication+table&es_sm=93&tbm=isch https://www.google.com/search?q=multiplication+table&es_sm=9...
    - jmilloy 11y ago
      I only have a bachelors degree in math, focusing on theory, but I think we have a different understanding of what actual mathematics is. For example, in some "real math", definitions, properties, and axioms are well-distinguished and mixing them up can get you in trouble. More importantly, are we even trying to teach "real" math to elementary kids (I wish we did, but I don't think we do) or "computation"? Both are useful and interesting.
      - chris_wot 11y ago
        Yeah, but the problem is: repeated addition is attempting to take something very concrete like I give four children three marbles each, how many marbles does each child have? You then use that addition technique to have them add up the number of marbles (in essence it's as if you are asking them to count on their hands, which is a valid technique at this level). But that helps the child understand the concept of addition in a very literal and concrete fashion, because at the age of 4-5 years old (sometimes older), children don't think at a higher level of abstraction. And using symbols to represent multiplication IS a higher level of abstraction. It seems to me, a non-educator, that the counting technique has value in word problems. But as soon as the child shows they understand the concept, then you introduce the notation (e.g. 5 x 3), explain the numbers can be added up either as five values added up three times, or three values added up five times. That the test talked about a "strategy" is not really maths, and frankly it seems to be misapplying a solid teaching technique, leading to confusion, anger and a lack of confidence in the child. If that's happening, then I'd suggest the technique is not all that solid and teachers and other educators should seriously consider whether it is causing more harm than good. P.S. If you have a Bachelors in Mathematics, then surely you can see that there is a fundamental problem if a child is taught that 5x3 is not the same as 3x5?

by jmilloy 11y ago

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.
  - jmilloy 11y ago
    It doesn't seem like a cardinal sin so much as a small quantitative note that the process was taught a different way that the teacher thinks is important. Let's suppose one student can follow the procedure when asked but can't actually multiply in application, one student can't follow the procedure correctly but can multiply when needed, and a third can do both. Probably the first student will get questions on this quiz correct but will struggle on much of the rest of the unit, maybe get a low grade or hopefully get the help they need. The second (with the paper shown in the OP), will probably get an high grade because they got partial credit on a silly quiz. The third will get a higher high grade. What's so bad about that? BTW, appealing to definitions won't work here, because the x does have a very well-defined mathematical meaning: a x b := b + ... + b.
    - signal11 11y ago
      > ... does have a very well-defined mathematical meaning: a x b := b + ... + b. Please complete the definition. which is that a x b = b x a, so a x b can also be written as a x .. x a. There is nothing special about the order.
      - jmilloy 11y ago
        That's not part of the definition. That's a separate property. I'm not sure we should care about that in elementary school, so the point is not to defend the teacher but only that you can't use the definition as an argument against the teacher.
        chris_wot 11y ago
        No, I'm afraid you've not given a complete definition of multiplication. You need to also show that multiplication is commutative, which is indeed a property of multiplication but MUST be included in the definition. At the child's level (primary age child, NOT high-school) then it is unnecessary to introduce the distributive property. But you honestly have to make the associative property very, very clear of the child will potentially have problems down the track! (Edit: brain fart - I said associative when I meant commutative. Oops!)
- jasode 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. If you(royal-you) insist that the 5 being the first factor has a specific job and you teach such nonsense to a child, it means you're not teaching actual mathematics. In _real_ math, the factors/mutiplicands have no notion of ordinal rank such as "first" or "second" or "specific jobs". Even if the child was not formerly taught The Commutative Law, it's not impossible for him to see multiplication tables[1]. (In fact, many are hung as big posters in elementary classrooms.) Any child with pattern recognition abilities beyond a chimpanzee would notice that the cells of XY have the same answer as YX. He/she would ask mom/dad/teacher "is xy always same as yx?". In the world of _pseudo_ math that stresses bizarre hoop jumping, we overlay non-mathematical concepts such as "specific job" to factors. Maybe this skill is important and transferable to the enlisted man to make sure he makes his bed before cleaning his machine gun instead of the other way around so everyone in the squad doesn't get punished with 50 pushups. But don't pass it off as "teaching math." [1]https://www.google.com/search?q=multiplication+table&es_sm=93&tbm=isch https://www.google.com/search?q=multiplication+table&es_sm=9...
  - jmilloy 11y ago
    I only have a bachelors degree in math, focusing on theory, but I think we have a different understanding of what actual mathematics is. For example, in some "real math", definitions, properties, and axioms are well-distinguished and mixing them up can get you in trouble. More importantly, are we even trying to teach "real" math to elementary kids (I wish we did, but I don't think we do) or "computation"? Both are useful and interesting.
    - chris_wot 11y ago
      Yeah, but the problem is: repeated addition is attempting to take something very concrete like I give four children three marbles each, how many marbles does each child have? You then use that addition technique to have them add up the number of marbles (in essence it's as if you are asking them to count on their hands, which is a valid technique at this level). But that helps the child understand the concept of addition in a very literal and concrete fashion, because at the age of 4-5 years old (sometimes older), children don't think at a higher level of abstraction. And using symbols to represent multiplication IS a higher level of abstraction. It seems to me, a non-educator, that the counting technique has value in word problems. But as soon as the child shows they understand the concept, then you introduce the notation (e.g. 5 x 3), explain the numbers can be added up either as five values added up three times, or three values added up five times. That the test talked about a "strategy" is not really maths, and frankly it seems to be misapplying a solid teaching technique, leading to confusion, anger and a lack of confidence in the child. If that's happening, then I'd suggest the technique is not all that solid and teachers and other educators should seriously consider whether it is causing more harm than good. P.S. If you have a Bachelors in Mathematics, then surely you can see that there is a fundamental problem if a child is taught that 5x3 is not the same as 3x5?