WEBVTT

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[Music]

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I'm Tom Carpenter.

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I'm an emeritus professor at the
University of Wisconsin-Madison

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and a researcher at
the Wisconsin Center

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for Education Research.

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I was a member of the What
Works Clearinghouse panel

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that produced the report
Developing Effective

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Fraction Instruction.

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I'm going to talk about
the first recommendation

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in the report, which is to
build upon children's intuitive

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knowledge of sharing to
develop fraction concepts.

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The overarching theme
of the report is

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that effective instruction
is grounded

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in developing understanding.

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The defining feature of
learning with understanding is

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that knowledge is connected.

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In particular, it's important
that knowledge is connected

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to the things that
children already understand.

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Young children have a lot
of intuitive knowledge

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about sharing situations
and can solve problems

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involving sharing.

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And these sharing
problems can be used then

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to develop basic
fraction concepts.

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We usually start,
essentially, with division.

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Children as young

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as kindergarten can solve
division problems involving

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sharing that will
ultimately lead

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up to developing fraction ideas.

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For example, you might
pose the following problem

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to young children: "I
just baked 12 cupcakes,

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and I want to put
them on three plates.

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How many cupcakes would
be on each plate?"

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So children might solve
that problem by starting out

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and counting out
counters of some kind

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to represent the 12 cupcakes.

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Children might draw some
representation of the plates

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or they might use some other
way of representing the plates,

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but very commonly they'll
draw plates like that.

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And then they will
essentially say, "Well,

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we've got to have the
same number on each plate,

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so I will deal the counters
to the plate, one at a time."

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And then they would count
the number of cupcakes

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on each plate-one, two, three,
four; one, two, three, four;

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one, two, three,
four-probably count them all.

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They could just count one plate,
since they're all the same,

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but they'll count them all.

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So they start with
problems like that,

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and that's a basic division
problem that children

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in kindergarten can solve.

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They don't solve that because
they have been shown how

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to solve them; these are
sort of intuitive solutions.

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And it's really much more
effective to pose the problems

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to the children and
let them come

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up with their specific
ways of solving them.

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After children have had some
experience partitioning sets

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that can be partitioned
exactly, then we pose a problem

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to them like, "We have
two children sharing

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three cupcakes."

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Now you can't give
all the cupcake;

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you can't take the
three cupcakes

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and put them into even piles.

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Children have to figure out what
to do with the leftover cupcake.

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A solution that the child
herself drew-this is a

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first-grade child-drew the two
people sharing the cupcakes,

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drew the three cupcakes,
and then gave one cupcake

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to each child, and then divided
the remaining cupcake in half.

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The notion of halving
can be used

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to solve quite a few
problems, like we've just seen,

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but it's somewhat
limited in terms

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of developing fraction concepts.

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So here's another example,
where three children were asked

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to share five cakes, and each
child was given one cake.

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The other two cakes
were cut in half,

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so each child could
get another half.

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But this particular child-a
first grader-didn't know what

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to do with the remaining part.

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The remaining part, he would
give that to the teacher

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or give that to someone else.

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One of the things children
need to get is some sort

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of an anticipatory notion-to
anticipate how many pieces

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they're going to cut it into
and sort of start out thinking

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about the number of pieces
that they want to get.

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One of the central ideas

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in understanding fractions
is the notion of equivalence.

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Equivalence underlies
most of the operations

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that you do with fractions.

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If you really understand
equivalence, then adding

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and subtracting fractions
becomes fairly straightforward.

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In this example, we have eight
children sharing five pancakes.

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And one child did it by
cutting each of the pancakes

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up into eight pieces, and so
each child got five-eighths.

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In the other example, another
child cut each of the pancakes

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into halves, and each
child got a half.

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But then there was one
pancake that needed to be cut

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up some more, and so the
child cut it up into halves

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of the halves, and halves
of the halves again,

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so there were eight pieces,
and wound up with the solution

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of one-half plus one-eighth.

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And so there the issue
then became who gets more,

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the child who got
five-eighths or the child

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who got one-half
plus one-eighth?

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What's the relationship
between one-half

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and the four-eighths
that are compared?

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So again, you're comparing
equivalent fractions.

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[Sharing and Equivalencies]

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Sharing can also be
used with older children

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to further develop the
idea of equivalence.

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Older children won't
necessarily draw pictures

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to represent the problems;

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they can use numerical
representation of the problems.

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Here is a representation
that was developed

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by a Dutch researcher by
the name of Leen Streefland

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for a problem he posed of 24
children going to a pizza parlor

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and ordering 16 pizzas.

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So the question is, "How
can these children sit

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at different table arrangements

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so that they all get the
same amount of pizza?"

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So one possibility is that
24 children sit at one table

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and share the 16 pizzas.

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But another possibility is that
we divide the table down so

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that we have two
tables of 12 children

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with eight pizzas on each table.

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Or we can divide those
tables down further

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so that six children are sharing
four pizzas, and ultimately

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that three children
are sharing two pizzas.

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Each of these situations are
essentially representations

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of equivalent fractions
and linked

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into the ideas of sharing.

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What we see in these examples
are children solving problems

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in ways that make sense to them.

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They have some intuitive
knowledge of sharing.

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And the teacher has posed
fraction problems in terms

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of sharing situations that the
children are allowed to solve

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in ways that they
devise for themselves.

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You pose the problems
and get out of the way

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and let the children solve
them and build on the knowledge

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that they bring to instruction.

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[Music]