WEBVTT

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[Music]

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My name is Robert Siegler.

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I'm the Teresa Heinz Professor
of Cognitive Psychology

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at Carnegie Mellon University.

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I was the chairman of the panel
that wrote the Practice Guide

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on teaching fractions, and
on this panel we had people

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with a variety of
kinds of expertise.

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We had a mathematician.

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We had several teachers of
mathematics who had won awards

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for their excellence
in teaching.

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We had several math educators
who taught at universities.

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And we had several psychologists

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who studied how children
learn fractions.

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When children come to school
they already have some basic

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understanding of fractions.

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For example, if you ask them
to give you half and me half,

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they can do that by counting
one for you, one for me,

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one for you, one
for me, and so on.

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The next advances come
when schools start

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to teach children
more about fractions

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in a somewhat more
formal way, and depending

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on the school district, this
might occur in second grade

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or third grade or fourth grade.

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And the children start
learning some general ideas

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about fractions, such as
what the numerator means

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and what the denominator means

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and that increasing
numerators means

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that the fraction gets bigger

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and increasing denominators
means

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that the fraction gets smaller

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if you keep the numerator
the same size.

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So they learn these when
they're taught the concepts,

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and that ranges from
second through fourth grade.

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By about fourth grade
and continuing into fifth

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and sixth grade,
children are learning

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about fraction arithmetic.

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They tend to learn addition and
subtraction of fractions earlier

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and multiplication

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and especially division
of fractions later.

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So by about sixth grade,
children have been exposed

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to not only what
fractions are but how

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to combine them in
arithmetic ways.

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Children also, when they
get into sixth grade

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and increasingly in middle
school, are going to be exposed

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to concepts like proportions and
ratios and rates, and they learn

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to solve those kind of problems
primarily in middle school.

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There are a variety of reasons
why fractions are receiving a

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great deal of emphasis
right now.

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Probably the most important is
that U.S. children do poorly

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in this aspect of
mathematics learning.

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U.S. children have a great deal

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of difficulty understanding
fractions.

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This is true in fraction
arithmetic,

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where all four operations
present difficulty,

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where the children often
confuse the operations

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and make mistakes.

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But it also includes
simpler aspects of fractions,

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for example, locating a fraction
on a number line or deciding

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which of two fractions
is bigger than the other.

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The major misconception that
children have with fractions is

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that they treat them
like whole numbers.

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For example, a problem
like 3/5 plus 5/6,

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they might give an
answer like 8/11

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because they add the numerators
and add the denominators

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as if they were whole numbers.

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We don't stay with any
one concept long enough

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for the children
to really master it

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and understand it deeply.

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Compared with other
countries, the U.S. teaches

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about four times as many
mathematical concepts

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in any one year, but
obviously when you look

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at the bottom-line result,
they're not learning four times

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as much math; they're
actually learning less math.

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In the panel's research we
found that there were a variety

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of reasons why teachers need
professional development

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in fractions in particular.

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One reason is that, for many
teachers, their understanding

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of fractions is relatively
shallow.

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They understand that
a given procedure

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such as invert-and-multiply
works,

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but they have little
understanding of why it works,

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why that's a legitimate
thing to do

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in solving fraction
division problems.

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Without this kind of
understanding, they are unable

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to convince many students that
this is a legitimate thing

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to do, and the students
are reduced

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to a kind of rote memorization.

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The problem with
rote memorization is

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that people tend to forget it.

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Children cannot learn
the procedures very well

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because they lack the basic
conceptual understanding

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of what fractions are.

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So if you have no idea how big
12/13 is and how big 7/8 are,

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you're as likely to say that
12/13 plus 7/8 = 19 or 21

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as you are to say
that it's about 2.

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And, in fact, that's
exactly what happened

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in a National Assessment of
Practice [Progress] question.

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Children very often made
mistakes on the procedures

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because they didn't understand
the underlying concepts.

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In addition to getting a much
better conceptual understanding

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of fractions than children
in the U.S. typically get,

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we also need to help
children translate

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between the conceptual
understanding and how that plays

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out in the arithmetic procedure.

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And we probably need to give
the children more practice

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with the procedures
that are now being built

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on this solid conceptual base
than we typically do at present

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[Music]