WEBVTT

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[Music]

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I am Francis (Skip) Fennell.

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I am professor of
education at McDaniel College

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in Westminster, Maryland.

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I direct the Elementary
Mathematics Specialists

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and Teacher Leaders Project.

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I am also a past president
of the National Council

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of Teachers of Mathematics.

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If there is one thing

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about the Common
Core State Standards,

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now adopted by over 40
states, that's really critical

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for the teachers at
particularly grades 3

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through middle school
is the tremendous impact

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and influence of
rational numbers.

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At a time when the Common Core
State Standards are really

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showing a concerted effort and
emphasis to work with fractions,

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all kinds of fractions,
we must have teachers

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who understand fractions
at a very deep level.

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The obvious is, you
sure need to know more

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than one way to do something.

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And as trivial as that sounds,
I am in far too many classrooms

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where I see teachers get stuck

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because they only know
one way to show it.

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A critical word is
representation.

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I want you to be able
to represent fractions,

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circular region, rectangular
region, fractions as part

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of a group, fractions

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on the number line-be
very conversant about,

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across all of those
representations,

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and more importantly, tell me
how this works in a context.

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It's one thing to say 1/3 x 1/2.

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Another way to think about
that is that's really saying

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one-third of a half.

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So that's really like
dividing one-half and looking

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at a third of that amount.

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And is that easier to
see with a circular region,

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or is it easier to
see a number line?

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Can you think of a situation

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where you might actually
have a third of a half.

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I would want teachers to be
comfortable with, and children

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to have access to, a number
of manipulative materials

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that would help them see
fraction as a part of region,

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fraction as a part of a set,
fraction as a part of collection

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of objects, and so forth.

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Here, let me demonstrate.

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These pattern blocks,
for instance,

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here is a hexagon shape,
and this red one can sit

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on the yellow one, and you
can establish one-half.

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You can compare that to a
situation where I am going

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to use a blue shape that
is essentially one-third

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of this amount.

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Compare the two together
and think about,

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well, what's 1/2 plus 1/3?

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We can see that it doesn't
fill the entire amount.

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We are not really sure what that
is, except this shape allows us

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to think about that
because this green shape,

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which kind of fills
it in to make a whole,

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is essentially a sixth.

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And I can confirm that by
putting that across the blue

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to ensure that there are two
greens, if you will, and a blue,

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and in the red shape
three of those.

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So if I put 1/2 and
1/3 together, it's 5/6.

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Whether it's concrete
materials or paper folding

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or drawings-and drawings are
great-whether that's number

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lines or drawings of area
regions or circular regions

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or what have you, all of
those are ways to sort

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of conceptualize, if you
will, this mathematics.

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But let's also at
least consider technology.

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Virtual manipulatives
are out there.

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There are a number of them.

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I want students to be very
comfortable in comparing

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and ordering fractions.

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That means that their
teachers need to be comfortable

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in setting that up, setting
those representations up,

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asking the right
questions about what happens

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when I compare fractions and all
the denominators are the same?

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What happens when
I compare fractions

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and all the numerators
are the same?

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When we compare a
group of fractions,

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when all the numerators
are the same

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but denominators are different?

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For instance, 5/3 and
5/8 and 5/6 and 5/7

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or whatever-when
we compare those,

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the smaller the denominator,
the larger the fraction.

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And that's always a kind of
"wow" moment for lots of kids

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and frankly lots of
teachers, because it goes

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against what they are
learning with wholes.

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I want to make a
particular point

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about multiplication
division, because when we think

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about those two operations,
first of all,

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when we teach those
two operations,

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the actual algorithms
are very simply taught,

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and unfortunately this
is part of our problem.

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If I want to think about 1/3 x
1/2, 1/3 of 1/2, I am dividing.

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This is where we have that sort
of convergence of curriculum.

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When I multiply fractions, most
of the time things get smaller.

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1/3 of 1/2 is 1/6.

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So all of a sudden, these kids
who are, again, spending a lot

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of time with whole numbers

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and generally things get
bigger-when they multiply

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fractions, generally
things get smaller.

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We owe it to children to have
them really understand well why

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they do things.

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That, frankly, is one
of the major obstacles,

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if you will, of sense of number.

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If kids were understanding about
what happens when we multiply

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and what happens when we divide,
then they would be able to look

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at their quotient and
their product and know

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that it makes sense.It's
really important

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that teachers understand
mathematics way beyond their

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particular assignment.

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The goal for me is for you

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to understand the mathematics
that's coming to you

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and goes way beyond your
particular assignment or level

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for this academic year.

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What we are actually
in a perfect position

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to do right now is to really
focus on the content background,

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the mathematical
background of the teacher,

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and that's at the
pre-service level and it's also

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in professional development.

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But it's really investing in
the mathematics that's unique

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to teaching the subject,
and that's different

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than if you have a higher-level
mathematics that one might study

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at the collegiate level.

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It's saying, let's unpack,
in this case, fractions

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and see how that plays out.

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Perfect opportunities to get
at lots of what we are talking

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about here -- a variety of
representations, different ways

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to think about fractions and
how they are contextualized

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in our society, thinking
about ways to use materials

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to help convey those
representations.

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All of those, in my
opinion, are prime territory

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for content explorations
for teachers in service,

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and I think we are going
to see more of that.

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[Music]