WEBVTT

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[Music]

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I am Dave Geary,
psychology professor

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at the University of Missouri.

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I contributed to the development
of the fractions Practice Guide

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and was a member

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of the President's National
Mathematics Advisory Panel.

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The basic operations for
fractions-that is, addition,

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subtraction, multiplication,
and division-are the same

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as with whole numbers,
but the procedures

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to get those results differ,

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and that is what can
be confusing for kids.

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It's important to understand the
concepts underlying procedures

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in addition to correctly
using the procedures.

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Understanding the concepts
allows kids to recognize errors

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when they make errors.

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It allows them to estimate
what is a reasonable answer

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and what's an unreasonable
answer.

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And it's also important, too,

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for them to transfer their
knowledge to new problems,

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that is, use fractions in
areas where it's appropriate,

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but that they haven't
actually had before.

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If you understand that
fractions are numbers

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and that seven-eighths
is very close to one

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and that seven-sixths
is a little over one,

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then if you are adding
seven-eighths and seven-sixths,

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your answer should
be very close to two.

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And so, if you get an answer
very different from that,

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then you should recognize

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that you have done the
adding incorrectly.

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The same is true for
multiplication --

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that if you are multiplying
two fractions, say,

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one-half times one-third,
you should know

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that the outcome is going to
be less than both of those.

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It will be one-half
of one-third.

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Conceptually understanding
what multiplication does

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and that fractions
really are numbers,

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just like whole numbers,
allows kids to recognize

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when they make these mistakes.

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Misconceptions in fractions
is very, very common.

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Many of those misconceptions
result from kids trying

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to apply what they already know

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to what they are
trying to learn.

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For fractions it
often leads to errors.

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Children don't really
understand the need

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for a common denominator, but
they do understand related types

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of real-world examples.

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They understand that you can't
add one quarter and one dime

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and get two quarters, or
add one quarter and one dime

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and get two dimes, because they
know that one quarter is 25,

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or one-fourth of a dollar,
and one dime is ten,

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or one-tenth of a dollar.

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And they know that adding
them up will give you 35.

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I prefer using the number line over other types

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of pictorial or concrete examples.

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The number line is easily
used to illustrate the need

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for a common denominator.

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So, for example,
three-fourths plus one-half.

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Kids will often add the numerators

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and add the denominators-incorrectly coming

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up with four-sixths or two-thirds.

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If you take the number line
and convert the one-half

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to two-fourths, then you
can easily take that segment

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of two-fourths, add it
to the other number line

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with the segment of three-fourths,

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and come up with the correct
answer of five-fourths.

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At times, it's easier
to use different types

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of representations other
than the number line.

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A good example of this is
multiplication of fractions,

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particularly with
values less than one.

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The fraction [Practice] Guide
presents an example of a cake

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where the frosting available
only covers two-thirds

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of the cake, and then you
want to cut that frosted part

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into quarters, or one-fourths.

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The result then being you're
getting not one-fourth

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of the cake, but one-fourth
of two-thirds of the cake.

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Multiplying that through,
you get two-twelfths,

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or one-sixth of the total
cake, or one-fourth

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of the two-thirds of the cake.

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It's a good way of illustrating
how multiplying two numbers less

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than one gives you an
even smaller number

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than the two you started with.

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When learning to
divide fractions,

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children are taught the
invert-and-multiply rule,

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which works, but they don't understand why it works.

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They don't understand the concept underlying it.

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We can approach teaching this two ways.

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One is a few steps to prove
mathematically why it works,

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and the other is to use a number
line to illustrate why it works.

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The first step would be to
multiply the denominator,

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one-fourth, by it's reciprocal.

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So one-half times four over one
divided by one-fourth times four

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over one, and that gives us
four-halves divided by one.

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Any number divided by one is itself,

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and that gives us
four-halves, which equals two.

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Now, if we took a number line
and had the same problem,

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one-half plus one-fourth, and we
broke the one-half into fourths,

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we could then ask the question:

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How many one-fourths
fit into one-half?

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Then you could demonstrate
on the number line

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that that would be two.

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For kids to solve
fractions problems correctly,

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they have to know the
procedures and they have

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to know the concepts
underlying those procedures.

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If they don't understand the
concepts, they are not going

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to be able to use the procedures

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in different contexts.Kids
will make mistakes,

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and their mistakes will
reflect their misconceptions.

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Use these errors
and misconceptions

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to teach the concepts.

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It is a perfect way

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to understand why they
are making mistakes

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and a perfect avenue for
correcting those mistakes.

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[Music]