WEBVTT

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>> Hi, my name is Dr.
Jonathan Brendefur.

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I am a math education professor
at Boise State University.

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Fractions is a really difficult
topic for a lot of elementary

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and middle school teachers.

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Students-I think we see from
across United States-have a lot

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of difficulty achieving
well with fractions.

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And so we use professional
development,

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which is a really critical
aspect of getting teachers

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to understand what are fractions
and how you teach fractions,

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to frame fractions as a number

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of different types
of interpretations.

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When teachers are solely
focused on fractions

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as a part-whole relationship,
that does becomes a hindrance

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for teaching fractions.

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And so we would like to push
them to look at a couple

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of things within that
part-whole relationship

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for both the teachers
and for the students.

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One is, the part-whole, the
three-fourths, really consists

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of unit fractions, and so
kids begin to start talking

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about three one-fourth
pieces in there.

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So then when you
switch and say, "Well,

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what would five-fourths be?",
that's when you want students

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to represent or talk
about it as, "Hey,

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that's five one-fourth pieces."

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So that's a very different take
on looking at the relationship

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in the fractions, and then
you can ask them to draw it.

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So one way would be a
bar model where they say,

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"I filled up four one-fourth
pieces; there's one.

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And I have an additional
one-fourth piece."

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But then pushing them to the
number line, the double line,

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where they are actually
representing this continuous

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idea of multiple fourths.

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So you get to one-fourth,
two one-fourth pieces,

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three one-fourth,
four one-fourth,

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five one-fourth pieces,
and then you can continue.

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And you can actually represent

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on a number line
those multiple namings

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because that becomes a
difficult idea as well,

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that we have multiple
names for fractions.

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So when we get to four-fourths,
four one-fourth pieces,

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we can also represent
that as one whole,

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and then we can also
continue that.

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Teachers can press
them to say, well,

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"What if we had twice
as many parts?

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So now, there's eight parts."

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So that's also a representation
of eight over eight.

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So it really starts changing
our idea of what our focus is

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on the part-whole to
something much greater,

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so focusing on the idea
of equivalent pieces

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and then also this idea
of a unit fraction.

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So after the fair share and
the part-whole interpretation,

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then we really start looking
at fraction as number.

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What does that mean?

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A fraction's a point
on the number line,

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and that's how mathematicians
tend to look at it.

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But also, at that same
time, we look at, well,

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where would a fraction
be on a number line?

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How do you find it?

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And that starts moving
toward continuing this idea

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of a partitive look
from the fair share

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to also a measurement
look at fractions.

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So another example would
be, with measurement,

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you have four cups of flour

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and each recipe takes
three-fourths of a cup.

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And so you have this measurement
idea of removing three-fourths

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of a cup each time
for each recipe.

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But then that changes again
when you start looking

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at a partitive division
type of an idea

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within the context of fractions.

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And so that would be where
you have four cups of flour,

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but that only makes
three-fourths of a recipe.

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So now even though the answer
is going to be the same,

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in context those are two very
different looks at a fraction.

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Now, and then the last way
of looking at fractions,

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the last interpretation
that we pose

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in our professional development,

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is as fractions move
into ratios.

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We really work with the teachers
to help build their knowledge

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to see, given a certain
situation, how would you enact,

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how would you use
different manipulatives,

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whether it's paper
folding and paper cutting,

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to using the fraction rods,
to then moving into iconic?

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In the iconic, we really
focus not just on drawings

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of fractions, but really
moving to different types

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of mathematical diagrams-usually
the bar model

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and the number line

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or the double number line-to
get a visual representation

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of the fractions.

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And then finally, attaching
symbols and notation to all

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of that work in the
enactive and iconic stage,

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then we have tied together those
three modes of representing.

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And that tends to
be really powerful.

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So I think to really deepen
teachers' understanding

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of fractions is very critical,

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and we do it really
in two parts.

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We focus on the situation,
the context.

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So is there a contextualized
problem,

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or is it just bare numbers?

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So what is the type of numbers;

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the situation becomes
very important.

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And then we also
focus on a second part

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that is sometimes missed: How
do children develop these ideas?

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So it's two-part: What
are the situations,

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and then what do students
know or need to know

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to understand fractions?

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By opening up teachers' ideas
of all these interpretations,

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all these different situations
where you could be looking

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at fractions, then that
becomes the first starting point

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where teachers say, "Ah, I
understand what are fractions,

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how to teach fractions."

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And then they can now change
their instructional strategies

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in the classroom to accommodate
kids that are struggling.