WEBVTT

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[Music]

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I am Ken Koedinger.

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I am a professor of
human-computer interaction

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and psychology at
Carnegie Mellon University.

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Recognizing notations

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and articulating them
is very important

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as a component in
problem solving.

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Students need to be able to
use those mathematical tools

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to be effective problem
solvers, and it turns

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out that those notations
are quite challenging

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for students.

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One of the reasons
that we know

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that mathematical notation
is surprisingly difficult

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for students are studies

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where we have compared
student problem solving

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and word problems to solving
those exact same problems

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when they are given
an equation.

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And what we have discovered is
that there are many elements

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of learning the
notational system

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that are surprisingly
challenging.

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This recommendation is really
important in helping teachers

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and developers to get past
their expert blind spots

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and to understand what's
really hard for students

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and develop instruction
that can then focus

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in on those student
difficulties.

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Algebraic notations
are an important part

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of problem solving.

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But sometimes we can do
activities that are going

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to facilitate problem
solving by focusing

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on the notation side.

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When you are translating
an English story problem

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into algebra, a big difficulty
for students is producing

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that new algebraic language.

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The importance of
algebraic notation

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for solving problems
comes out particularly

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as problems get more complex.

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For simpler problems,
students are often able

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to solve simpler
"algebraic problems"

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without using algebra at all.

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But that only goes so
far and we have research

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that shows particularly

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where problems involve
more complex forms

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where the unknown
is referenced more

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than once in the problem.

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Those are really hard to
solve intuitively, like,

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for instance, John bought
a coat at a 20% discount.

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He paid $38.

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What was the original price?

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You can't just divide
the 20% into 38

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and get the answer--you will
get the wrong answer--or even

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multiply 20% by 38.

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The equation there, x minus .2x
= 38 helps you understand how

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to solve that problem

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and facilitates a more
effective solution

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to those kinds of
more complex problems.

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If you use simpler
story problems early,

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which makes some sense,

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and students don't
use the notation,

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they may get those
answers right

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but you may not
be preparing them

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for the more challenging
problems.

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You want them to get

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to use those more
sophisticated strategies

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so that when they are in
more challenging situations,

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the powerful algebra
strategy--that overhand

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serve--can be applied
and be effective.

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There are some techniques
teachers can use

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to help students
understand mathematical,

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and algebraic notation,
in particular,

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and make connections
to problem solving.

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One of those is to give
a student a story problem

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and the equation that
models that story problem,

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and then ask them to explain
components of that equation

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in terms of what they refer
to in the story problem.

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For instance, you might
have a story problem

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about Joseph earning
money selling seven CDs

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and his old headphones.

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He said the headphones were
$10 and he made $40.31.

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How much did he
sell each CD for?

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So the equation for that
problem is 10 + 7x = $40.31.

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Now, you would give all
of that to the student.

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The activity for the
student now starts

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with explaining elements
of that equation.

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What does the x represent?

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What does the 7x
represent in that equation?

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What does the 10 + 7x
represent in that equation?

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And each of those components
can be referenced back

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to the equation, and
in fact the $40.31,

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that represents the same
thing--how much he made--as does

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the 10 + 7x.

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One of the tricks to these is

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that students sometimes
think they can only give an

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explanation once, but
they need to understand

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that in problems with
equations there are two ways

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to represent the same thing,

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and that's why we put an
equal sign between them.

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It's not only that level of
explanation that's important

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in these kinds of
problems; it's also important

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that each little subcomponent

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of the expression
has a meaning.

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And helping students pull
that apart, you want them

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to be able to decompose
the notation,

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make sense of the
parts, and make sense

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of how the parts fit together
to tell a story in an equation

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that matches the
story in English.

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Sometimes in class when
you give a problem you get

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different solutions from
students--oftentimes, actually.

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Let's say you give a
geometric pattern problem

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where you ask students to come

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up with some mathematical
form.

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So some students might say,
"Well, it's 3x plus six."

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Another student
might say, "No,

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I think it's x plus
2 multiplied by 3.

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They are actually both right;

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those are both good
expressions.

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Those are great opportunities
to help students think

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about equivalence of different
algebraic expressions

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and even how you can transform
one expression into the other.

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To have those first emerge
from a pattering problem

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or a story problem and
be able to say, yeah,

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both students are
right, but let's now talk

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about why those two
expressions are equivalent,

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not only because they
model the same problem

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but because we can apply
a mathematical principle

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that transforms
one into another.

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The role of algebraic
notation in problem solving is

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that it's a powerful tool
and one of the things

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that we have discovered is
that tool takes time to learn.

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We should be having students
use the language of algebra

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as early as possible in
school and repeatedly.

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[Music]