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Why do pupils need to know a written method for long division by the end of year 6?

Discussion in 'Mathematics' started by Anonymous, Feb 3, 2015.

  1. Anonymous

    Anonymous New commenter

    This is a very specific question.

    Understanding division is important. Knowing when to divide is important. Knowing if your answer is correct by using estimation skills is important.

    But ---- why is understanding a long written method for long division useful?

    I know it's handy for A-level maths and polynomials.

    But - what about the rest of the children. Could the time spent on learning this algorithm be better spent elsewhere in maths learning the basics and reinforcing knowledge.

    Is spending time on getting children to have a written method for long division going to make numerate children? Will it help secondary children?
     
  2. Vince_Ulam

    Vince_Ulam Star commenter



    Prepare yourself.





    Understanding division does not result in a quotient. Understanding and proficiency are not coterminous.





    There is no point knowing when to divide if division cannot be achieved.



    An estimate is not a quotient. The point of estimation is to check if a solution is sensible, not to anticipate division. Too many teachers put the cart before the horse because they are not proficient in division. They should be proficient in division.




    Why is any written method useful? It builds the skill. Understanding and proficiency are not coterminous.





    Division is basic.





    Yes, and yes.



    Why do you ask these things? I do not wish to presume.


     
  3. Anonymous

    Anonymous New commenter

    You haven't answered why long division is needed.

    Short division - useful. Why long division?

    Explain that to an 11 year old child when they say "Why can't I use a calculator to work out 1278 / 193? It's what an adult would do?"
     
  4. PaulDG

    PaulDG Occasional commenter

    Short division does not 'scale'. When the numbers involved are large, there simply isn't the space to write all the necessary working.

    They only do that because you let them. If you've had them doing long division since year 3 they'll be asking you "why do adults use calculators to do stuff that's so easy to do with a pencil?"
     
  5. Vince_Ulam

    Vince_Ulam Star commenter



    I did. It builds the skill.



    Now, back to my question: Why do you ask? Are you uncomfortable teaching long division?


     
  6. OK. I can do: mental division, short division, long division, polynomial division etc. I can teach them all too, quite happily.

    I encourage pupils to work without a calculator as far as possible. I'm very much in favour of developing strong calculating skills..

    BUT

    I don't think that it's necessary or even desirable to teach long division.

    But division can be achieved using a calculator if necessary.

    I'm probably misunderstanding you here: What skill? The skill of being able to do long division?
     
  7. Bensusan

    Bensusan New commenter

    Not only should they learn long division but they should only do it with quill and ink. ;-)
     
    kate_drew likes this.
  8. Vince_Ulam

    Vince_Ulam Star commenter



    When did long division cease to be calculation?





    If you have assimilated arithmetical algorithms as schemata through practice then a calculator is unnecessary.




    The schema which the long division algorithm represents and precipitates.


     
  9. Bensusan

    Bensusan New commenter

    roughly translated , a self-fulfilling algorithm. lol.
     
  10. Vince_Ulam

    Vince_Ulam Star commenter



    [​IMG]


     
  11. [QUOTE user="Vince_Ulam"]When did long division cease to be calculation?[/QUOTE]

    You know I didn't say that.

    Do you think that strong calculating skills should include every type of calculation? Probably not, so we can leave some out and still be considered to have strong calculating skills. I'd choose to leave out long division.

    Would you like to see pupils taught to find square roots without a calculator? I could teach that too. [​IMG]

    [QUOTE user="Vince_Ulam"]If you have assimilated arithmetical algorithms as schemata through practice[/QUOTE]

    Do you mean "if you've learnt to do them well"?

    [QUOTE user="Vince_Ulam"]then a calculator is unnecessary.[/QUOTE]

    I'm not so sure. Faced with something like 54325837831345 / 47539824324 a calculator seems very desirable if not absolutely necessary to me.

    [QUOTE user="Vince_Ulam"]The schema which the long division algorithm precipitates and represents.[/QUOTE]

    I'm not great with English as I'm sure you can tell. Could you explain that more simply for me? As it is I'm stuck thinking that a skill is not a schema. Still, I think you're saying that learning long division helps you understand long division.

    [​IMG]
     
  12. Bensusan

    Bensusan New commenter

    @Vince Yes, well, I’m polymerized tree sap and you’re an inorganic adhesive, so whatever verbal projectile you launch in my direction is reflected off of me, returns to its original trajectory and adheres to you. ;-)
     
  13. Bensusan

    Bensusan New commenter

    yup. Nailed it. Vince's MO seems to be to obfuscate things by referencing a thesaurus rather than provide a reason. As you said, there are other algorithms we could teach for the sake of teaching algorithms and there are many we don't teach every child - why not teach Euclid's algorithm, for example? We also don't teach children how to use a quill despite it being a skill That is arguably just as useless in the modern day as the long division algorithm. Teach it to those who need it (advanced algebra) and keep the focus on problem solving (the REALLY important skill) for the rest. IMO.
     
    kate_drew likes this.
  14. frustum

    frustum Star commenter

    I've never taught long division with all the remainder stuff underneath. If they are competent at adding/subtracting two/three digit numbers mentally (which they should be), then I think it works much better to jot the relevant multiplication table down at the side, and work out the remainders mentally. I tell them that the only difference between dividing by 7 and dividing by 37 is that they don't know the 37 times table, so we fix that. In the example above, jotting down the 193 times table is going to work better for most than trying to work out what number to put underneath 1278 without, anyway. You need to write the calculation out larger, so there is room to write 2/3 digit carries in, but that's doable.

    I'm with Robyn.

    I am extremely competent at long division. I rarely do it. Do you?

    If I want to know 1278/193, if I need a rough answer I estimate. I might then improve on "about 6" by saying that it will be a bit more, or even by saying that there will be 7 left over from each 200 plus the extra 78, making 120, so the bit is 120/nearly 200, giving us just over 6.6. If I want a more exact answer, I will generally use a calculator. As a matter of interest, if you would do that one by hand, how many decimal places would you go to?

    I would love all students to be able to use those mental skills, to estimate, to know whether the estimate is going to be too big or too small, and to know to use a calculator when it is appropriate. Chunking, too - that's how I divide a bill, and most people I know.

    I really do think that long division should have had its day. I don't mind doing it with abler pupils, especially to explore recurring decimals, but I'd really rather work on those mental skills with the weaker ones. Apart from anything else, when I'm getting a costing from someone, if they can't do the basic arithmetic, I'd rather see them use a calculator for the hard stuff.
     
    kate_drew likes this.
  15. Vince_Ulam

    Vince_Ulam Star commenter

    You know I didn't say that.

    [/quote]



    Then would you like to address the contradiction in the section of your post I quoted?





    The four operations are not optional to numeracy.



    Do you mean "if you've learnt to do them well"?

    [/quote]



    No. Learning to do long division well implies only that a person has learnt to do long division well.



    I'm not so sure. Faced with something like 54325837831345 / 47539824324 a calculator seems very desirable if not absolutely necessary to me.

    [/quote]



    It may be so for you at this time, but there are algorithms which will enable you to resolve this yourself with little effort. In the same way, nobody is born knowing long division but so many people are capable of learning it that we would be wrong to accept your excuse from schoolchildren.



    I'm not great with English as I'm sure you can tell. Could you explain that more simply for me? As it is I'm stuck thinking that a skill is not a schema. Still, I think you're saying that learning long division helps you understand long division.

    [/quote]

    A schema is a cognitive machine. You feed it a problem, it gives you a result. Practice an algorithm long enough and you develop the corresponding schema as a reflex - you don't need to think about it and, in this case, you don't need to reproduce long division in your head. Long division is only the way that we represent its corresponding schema in order to precipitate that schema through practice.

     
  16. There's no contradiction. I do teach the algorithm and I don't think division is optional. It's really just that particular way of writing down the steps that I do not bother with. I recommend the use of a calculator when dividing by numbers with more than 2 or 3 significant figures.

    [QUOTE user="Vince_Ulam"]It may be so for you at this time, but there are algorithms which will enable you to resolve this yourself with little effort.[/QUOTE]

    Seriously, LOL! I could already if I had to but I would get no benefit from doing so and I have better things to do.
     
  17. Vince_Ulam

    Vince_Ulam Star commenter



    There is.





    So you do think that teaching long division is necessary and desirable.



    Seriously, LOL! I could already if I had to but I would get no benefit from doing so and I have better things to do.

    [/quote]



    Then it was a stupid example if you wish your students to reach your proficiency.


     
  18. [QUOTE use="Vince_Ulam"]So you do think that teaching long division is necessary and desirable.[/QUOTE]

    Short division and long division are the same algorithm. I don't teach long division. I do teach the algorithm.
     
  19. Vince_Ulam

    Vince_Ulam Star commenter

    Short division and long division are the same algorithm. I don't teach long division. I do teach the algorithm.

    [/quote]



    If after learning what you are calling 'short division' your students are then, immediately and without further tuition, capable of performing long division then you are teaching them long division via the long division algorithm.


     
  20. Anonymous

    Anonymous New commenter

    What skill have they learnt?

    If -as has been argued elsewhere - it's good for number manipulation, then why not use chunking. It teaches division nicely - and you can tell that division is inverse multiplication.

    No one is really going to use long division / chunking later. I never used it in a career where I used maths a lot.

    I can see no justification for expecting every year 6 child to learn how to do long division. It's like expecting every year 6 child to use log tables.
     
    kate_drew likes this.

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