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Topic: Fractions, concept and calculations

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Subject:   RE: Fractions, concept and calculations
Author: ten frame lady
Date: Oct 28 2004
OOPS! There is a typo in the division explanation.

When we divide 8 by 4 to get 2, the quotient means that there are 2 groups of 4
in 8 (NOT 2 groups of 8)!!

On Oct 27 2004, ten frame lady wrote:
> Oh, my goodness! Don't give up on your students' ability to make
> sense of fractions! Study after study has shown that memorizing
> without some clue as to the concept does not last very long.

> One problem we teachers have to overcome is the impulse to  do too
> much "giving the best explanation  possible" at the expense of too
> little encouragement to students that they themselves can make sense
> of fractions.

If we as teachers do not take the time and trouble
> to dig into fractions concepts with our students, we send them along
> into algebra with half-baked notions and a firm belief that they are
> "no good at math." AND If we want our students to make sense of
> fractions, then we also have to dig deeper into our own
> understanding.

2. There are some excellent resources to help
> teachers guide their students through activities that make sense of
> fractions. One of the best ones I know is "Understanding Fractions"
> by Christine Losq (available from Great Source Education Group).
> This resource not only provides concept-building activities that dig
> into the meaning of fractions, but also shows you how students
> responded, what insights they gained, what insights the teachers
> gained from the students' work (misconceptions as well as
> discoveries).

The original query on this web site involved
> comparing the fractions 6/7 and 7/6. Before getting into any kind of
> arithmetic strategies, I ask kids to simply picture what these
> fractions represent. I need to know that my students can translate
> 6/7 and 7/6 into meaningful and accurate pictures and have the
> language to describe those pictures. For example, I first have
> students "help me see" the meaning of 6 out of 7 equal parts (they
> usually draw a pie, simulate 7 equal parts, and shade 6 of them).
> Then I say, "show me 1 sixth, 2 sixths, 3 sixths" etc. until kids
> realize that 7/6 is 6/6 plus one more sixth. So they realize the 7/6
> is greater than 1. After that, the comparison between 6/7 and 7/6 is
> easy. I usually follow with some similar examples (comparing a
> fraction less than 1 with a fraction that represents a quantity that
> is greater than 1). I want them to recognize that when the numerator
> is greater than the denominator, the expression is greater than 1.
> We then point out that  there are  several things to look out for
> when dealing with a fraction: all the possible the relationships
> between the numerator and the denominator; the relative size of the
> fractional part represented by each denominator.
NOTE: I find
> number line models less useful because you already have to know a
> lot about fractions in order to successfully situate a fraction on a
> number line. And many fractions we run into in our textbooks (like
> sevenths) are really hard to show on a number line for comparison
> purposes.

3. Be sure to emphasize the importance of language in
> developing an understanding of fractions.
For example, when you
> teach division of a fraction by a fraction, start with a whole
> number example to set up the language.
8÷4 = ? means "how many
> groups of 4 can I take out of 8?" (There are 2 groups of 4 in 8, so
> the quotient is 2 and means 2 groups of 8.) Be sure to remind
> students that the = means two different ways to show the same
> quantity and that the quotient is not just a random number but a
> unit of measure, in this case "groups of 4."

Apply the same
> language to division of a whole number by a fraction and the result
> makes sense. For example,

8 ÷ 1/2 means "how many halves can I
> take out of 8 wholes?" The answer, 16, now makes sense because there
> are 16 halves in 8 wholes. When you have student model this kind of
> division problem with simple manipulatives like paper fraction
> wheels or fraction bars that they can cut up and mark, they will
> focus on the underlying meaning of the numerical expression and of
> the operation.

The application of language to division of a
> fraction by a fraction also works.
3/4 ÷ 1/2 means "how many one-
> half size pieces can I take out of a 3/4 size piece?" (The quotient,
> 1 1/2  (half-size pieces), then makes sense. Again, give kids the
> time to model the logic of the division to find out that you can
> take out 1 one-half size piece and half of a one-half size piece.)
> It does take close attention to what the quotient means because the
> idea of a "whole unit" has a double meaning when you get into
> division of fractions by fractions: You are dealing with less than
> one whole unit, since you only have 3/4 to start. The divisor, 1/2,
> functions both as a quantity related to the original whole and as
> the unit that defines the quotient (1 and 1/2 half-size pieces are
> in a 3/4 size piece of a whole).

4. Pie models and pictures
> really do help to reinforce the language connection and the
> underlying concept, although I have a hard time sometimes convincing
> my stressed middle school colleagues that this is so. So don't skimp
> on the time it takes to let the concept sink in.

5. If we take
> the time to really explore fractions concepts, students will be able
> to answer for themselves why, when you divide a whole number by a
> fraction you seem to end up with "more than what you started with"
> and why, when you divide a fraction by a fraction, the quotient
> sometimes appears to be "more" than what you started with.

7. The
> mental math involved is the ultimate goal, of course. And the
> "invert and multiply" rule is the shortcut we want all of our
> students to eventually be masterful with. However, if we just "give
> the best explanation possible" and focus too quickly on the
> underlying factoring and computational skills, we will NEVER help
> our student truly understand fractions.

Most important: Kids
> really will rise to the challenge if you constantly communicate your
> belief in their abilities to dig into and understand the language
> and logic of math.

On Oct 27 2004, ihor wrote:
> Hi,
> > All I've been able to do is to give the best
> explanation
> possible, and let their individual intuition work and develop. I
> > don't think I can teach to improve that.  I can show different,
> perhaps
> better
> ways to do one thing or another, but the
> arithmetic of fractions [not
> necessarily the physical
> representation] does eventually boil down to
> memorising
> method, with or without deeper understanding.

I'm with you all
> > the way until you got to memorizing with or without understanding.
> > Unfortunately, most people come away with the latter. I think that
> > is collectively our fault that we have not been able to come up
> with
> better ways to help students learn it. And you may be right,
> it may
> not be possible to have the majority of students
> understand what
> they are doing. Most of them will go through the
> motions to pass the
> test just like all the generations before
> them. But I'm an optimist
> I think more kids can understand and
> appreciate the math they are
> learning and its not just the kids
> that are good at it or like math.
> I watched a lot of teachers
> over the years who find ways to do it
> and I think their teaching
> secrets (their pedagogical content
> knowledge) ought to be shared
> as much as possible. Otherwise we give
> up too soon and let
> average kids go through those same semi-
> productive hoops. As a
> good friend of mine once told me when he was
> arguing the point of
> just getting by: "Look at me I was terrible at
> math, but I
> mangaged to be successful as a lawyer." I would say he
> became a
> lawyer despite the handicap of his math skills and it
> didn't have
> to be that way. (What he really meant to say was that he
> was
> "terrible" because he didn't understand it, not because he
> couldn't "do it."

>Ability to recognise the same
> pattern and
> > thus the same approach to solution in rational algebra pays off
> > as
> the skill through practice [at what point DO they
> understand?]
> puts the
> method
> into automode as the larger
> problem develops
> into a solution.  I'm comparing
> that to
> practicing scales on the
> piano. .... however intuitive it
> becomes, and not having to think
> about where to place fingers
> allows for other, better things to
> develop more naturally in the
> long run.  All can learn to play, but
> not all can learn to play
> well,
> no
> matter the teacher.

> of the most formative
> experiences of my life was taking piano
> lessons for 3 years: It
> was the ultimate drill and kill experience
> because the teaching
> was so uninspiring that I actually avoided
> contact with any form
> of music for years. Fortunately, it was not
> life threatening... I
> do enjoy all kinds of music today and even
> learned to play a
> little guitar in my later years. I don't have kids
> of my own but
> I did watch closely my niece and nephews grow up all
> taking years
> of music lessons from a good teacher. Only one of them
> became
> reasonably good at it, but the others enjoyed the experience
> of
> learning to play. (I'm jealous.)


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