Q&A #522

Teachers' Lounge Discussion: Teaching subtraction

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From: gail

To: Teacher2Teacher Public Discussion
Date: 1999020721:16:10
Subject: Re: Re: teaching substraction(sic)  & in refrence(sic) to barrowing(sic)

From: Ray M <raypublk@san.rr.com>
>You don't have to know a thing about 10's to learn about borrowing. 

You are right here...   I will bet what she MEANT to say is that you
needed to know about how place values are arrived at for a determined
base...   but you and I know that most arithmetic is taught using base
ten, so that is probably what she assumed we would think about as we
read her post...

>It is rather easy to teach most four year olds how to count in
>binary:   put 1 dot on your r. thumb, 2 dots on r. index, 4 dots on
>right middle, etc.  Now you can uniquely represent 0 to 31 on the
>right hand.  Using both hands, you can uniquely represent 0 to 1023. 
>Transfering the finger notation to paper is easy: finger up =1,
>finger down = 0.  Once counting in binary is second nature, you can
>add in binary.  Dots can be used to visualize numbers and carrying. 
>There are close parallels to all the rules in base ten.  Once you
>know binary, octal and hex come almost for free and their rules
>parallel those of base ten too.  So there are three useful bases
>other than ten in which to learn about borrowing.  Most spreadsheets,
>symbolic math programs, and scientific calculators can help you check
>your work.

Would you agree that counting and understanding quantity are not
always the same thing?  I know young children who can count in a rote
manner, but haven't quite reached an understanding that the one-to-one
correspondence of those words they are saying and a set of items will
tell them the amount in the set.  We take it for granted, but it is a
critical thinking skill that has to be developed.  

You idea about dots on fingers is an interesting one...   I think I
would put it in the semi-concrete stage if I were thinking about a
child's development.  Concrete level learners need to experience the
ideas and concepts through their senses, feeling, touching and
manipulating materials as they make connections between what they
already understand and what they are learning.  At the semi-concrete
stage learners are able to move past three dimensional objects they
can move to drawings and pictures they can use to build their
understanding.  Your dots represent real world sets for the various
place values.  They enable a person to determine value of digits
without having real objects to touch and group.  What a great idea!  

>It is simply not true that "all the numbers on the top represent 10"
>and to say so will conflict with trying to teach place value.

I agree with you...   making a generalization like that doesn't help a
child understand what is happening when the regrouping takes place.  I
like to sue unit blocks ( ones, tens and hundreds) to demonstrate what
you are explaining with your examples in base ten.  that way my
students can look at the examples, find the similarities tht are
occurring, and tell ME what the rule is...   They discover, and it is
theirs...  I tell them, and maybe they will remember it, maybe not... 

>     In the example given:
>     458900
>     -235774
>     the statement is made that you have to start with borrowing from
>the 9.  In some sense that is true, but it seems like bad advice to
>give to  a second grader.  I would always give the advice to start in
>the ones place, note that I need to borrow, look at the next place
>value (10s), note the need to borrow, look in the next place(100s),
>etc. (I kind of like Judy's story!)   But that is certainly not the
>only algorithm.  

>While gifted 7 year olds can certainly do that, I don't recommend
>trying it in an average 2nd grade classroom.  But if you, as the
>teacher, will take the time to learn how to do it, you will
>understand what borrowing is really about.

I once saw a demonstration that suggested that the place values be
grouped in a meaningful way for a problem like this...   fro example,
looking at 900 instead of looking at isolated digits...   and deciding
that to begin subtracting in the ones place wouldn't work without some
regrouping, so we might take a ten from the 90 tens (see?) and leave
89 tens     900 -->  89 (10)

Another workshop by a group that puts out Math Their Way suggested we
present math in base 6, helping students computer by using beans, cups
and boxes...    1 - 5 beans could roll around loose, but as soon as
there was a 6th bean, they needed to be transferred to a cup...  and
as soon as there were 6 cups they needed to be transferred to a box,
and then to a super cup, and then a super box, etc.  

While I am not sure I am ready to teach my fifth graders that way as a
whole group, I still vividly remember the week I spent doing this, and
how lost I felt ( and I can computer quite well in base ten) every
time I tried to leave the cups and beans alone and work with paper and
pencil.  I just wasn't ready to leave the manipulative help at that
point.  It drove home the point that this means of computing we teahc
our children may be second -nature to us, but it is all new to our
children, and oftern doesn't seems logical to them at all, even though
we might think it should...

So , to make a long story just a bit longer, I will tell you I agree
with the spirit of you message.  I think there ARE a variety of
algorithms being used a ll over the world to teach children to
compute, and that we are way off base as educators if we think there
is only one "correct" method.  I also think that as adults we
sometimes lose sight of how foreign some of these methods appear at
first sight.  

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