multiplication with open arrays

Discussion in 'Elementary Education' started by bella84, Dec 18, 2016.

  1. a2z

    a2z Maven

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    Well, if your district and state assessments use the convention, you will be setting yourself up for failure just as any one who ignores conventions for the SAT, GRE, or AP tests would be setting themselves up for failure.
     
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  2. Pashtun

    Pashtun Fanatic

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    I need an example to help me understand this better.
    When would not knowing the convention, hinder your ability to solve a problem?
     
  3. bella84

    bella84 Aficionado

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    Fair enough. I don't have a problem with telling my students "You have to do it [this way] on a test, but, mathematically, either way is a correct representation." And, if I'm the one grading the test, I'm not going to count it wrong.

    My first priority is that they understand the math. Passing a test is secondary.
     
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  4. Pashtun

    Pashtun Fanatic

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    Wouldn't it be, "if your district or state is assessing the convention itself"?
     
  5. czacza

    czacza Multitudinous

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    It hinders a students ability to score well on standardized tests. Outside of this particular math question, not adhering to conventions can lead to miscommunication and misunderstandings.
     
  6. Leaborb192

    Leaborb192 Enthusiast

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    ,
     
    Last edited: Feb 13, 2019
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  7. Pashtun

    Pashtun Fanatic

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    Right, a pure assessment of the convention versus using it in context to solve a problem?
     
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  8. Pashtun

    Pashtun Fanatic

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    This example of convention seems out of place to me and I am with Bella, I don't see its value outside of a pure convention.

    Can you give me some examples outside of this example of the conventions you are referring to?
     
  9. bella84

    bella84 Aficionado

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    I'm also still wondering, if this convention is really as important to standardized tests as has been suggested here, why would reputable sources, like the NCTM and Khan Academy, not make a point to follow the convention? Their work clearly shows that they have written expressions that show [columns] x [rows].

    Here is the NCTM link again for anyone who doesn't want to go back to the beginning of the thread: http://forums.atozteacherstuff.com/index.php?threads/multiplication-with-open-arrays.199095/

    Here is the Khan Academy link: https://www.khanacademy.org/math/ar...t-intro/v/multiplication-as-groups-of-objects
     
  10. a2z

    a2z Maven

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    on an aside....
    In the original questions were all of the units all smooshed together or were there clear rows or columns?
     
  11. bella84

    bella84 Aficionado

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    This:
    Screenshot 2016-12-18 19.38.54.png
    And this:
    Screenshot 2016-12-18 19.40.45.png

    The directions are: "Write an expression that matches the box above."
     
  12. a2z

    a2z Maven

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    You notice Khan uses the phrase groups of and always puts it on the left hand size. He uses the convention of groups. The layout may be a grid pattern with rows and columns but he consistently uses the term groups of.
     
  13. a2z

    a2z Maven

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    Thanks for the pictorial.
     
  14. a2z

    a2z Maven

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    8 x 3 = 24 Heck, why not? It is right depending on how you group it? 2 x 12? 12 x 2? 24 x 1?
     
  15. bella84

    bella84 Aficionado

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    Yes, I did notice that. My point was that a person could see the groups either way - in columns or rows - and that either expression would be correct, depending on how that person saw the groups. If we are not told how we are supposed to see the groups, are we not able to see it how we choose and write the expression that aligns to our thinking?
     
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  16. bella84

    bella84 Aficionado

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    Perhaps... They could definitely use partial products or the the double and halve strategy to solve it that way. Although, we've been learning this as a precursor to area, which is why [rows] x [columns] or [columns] x [rows] would be more appropriate in this particular case.

    The goal here has been to represent the dimensions of the box with an expression.
     
  17. Pashtun

    Pashtun Fanatic

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    Can you explain this? I am not understanding what you are trying to say here?
     
    Last edited: Dec 18, 2016
  18. gr3teacher

    gr3teacher Phenom

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    I think this is one of those things that is likely to end up making kids mad more than anything, given that an array with 4 rows of 6 is also, by necessity, an array of 6 columns of 4. If you want kids to use a visual model which requires 4 x 6 to be 4 x 6, rather than 6 x 4, either use a number line model or a groupings model. Don't use the one sort of visual which by definition will work either way.

    Heck, look at the area model (which students learn, in Virginia at least, the same year they work with arrays), which uses length x width. Going by an array, that would be columns x rows, not rows x columns.
     
  19. Leaborb192

    Leaborb192 Enthusiast

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    ,
     
    Last edited: Feb 13, 2019
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  20. mathmagic

    mathmagic Enthusiast

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    Admittedly, I didn't read 100% of the words, but did skim the three pages. Here are my thoughts (having grown up with math as my big love, gotten a degree in math, K-12 endorsement):

    I think there's importance to students being able to explain their thinking of what they write. That is, if they write 4 x 6, that they can say it is 4 rows of 6. Or, if they write 6 x 4, that they can say it is 6 columns/groups of 4. Do I care about the order? Not yet. If they are successful with that verbalization, then I may work with them and say that we often think about rows first. But forcing that upon a student who cannot verbalize an understanding is just going to muddy the waters, so I don't even mention it for them. Even then, if a student still makes a mistake and is mentioning columns first or whatnot, I'm not going to worry about it. Heck, I realize how many things I learned about math concepts I had previously learned once I went through college/math ed courses. Many of them wouldn't have registered way back when.

    In my classroom, we focus on meaning, and then I'll mention "norms" as they come up, are developmentally appropriate, and I know they can be successful with without making current concepts more difficult.
     
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