WEBVTT

00:00:00.136 --> 00:00:06.726
[Music] Welcome to the overview on
Effective Problem-Solving Instruction.

00:00:08.166 --> 00:00:11.076
When teaching students how to
solve mathematics problems,

00:00:11.266 --> 00:00:14.806
teachers need to be continually alert
for instructional opportunities.

00:00:14.806 --> 00:00:19.416
As students share their reasoning while
working through a challenging problem,

00:00:19.786 --> 00:00:22.956
observant teachers will have
instructional "openings"

00:00:22.956 --> 00:00:24.546
that can be used to introduce

00:00:24.546 --> 00:00:28.726
or reinforce familiar
mathematics concepts and methods.

00:00:28.726 --> 00:00:32.415
Taking advantage of these opportunities
to further students' skills

00:00:32.415 --> 00:00:35.556
and understanding is at the
core of effective teaching.

00:00:36.646 --> 00:00:40.386
There are three major strategies that
teachers can use to guide students

00:00:40.386 --> 00:00:42.896
as they learn how to
tackle mathematics problems.

00:00:42.896 --> 00:00:47.956
- Teach students how to create visual
representations of relevant information

00:00:47.956 --> 00:00:51.606
in a problem, - Encourage
students to explore multiple ways

00:00:51.606 --> 00:00:56.236
to solve a problem, and - Demonstrate
how to break a problem into steps

00:00:56.236 --> 00:00:59.656
that can then be expressed through
formal mathematical notation.

00:01:00.326 --> 00:01:02.106
Let's look at these one at a time.

00:01:02.106 --> 00:01:07.876
Research suggests that students who
develop visual representations prior

00:01:07.876 --> 00:01:11.796
to working with equations are
more effective problem solvers.

00:01:11.796 --> 00:01:15.896
This may be because visual
representations help students develop a

00:01:15.896 --> 00:01:18.526
deeper understanding of the
problems they are working with.

00:01:19.426 --> 00:01:23.096
The right type of representation can
help a student get a coherent view

00:01:23.096 --> 00:01:26.216
of the problem by identifying
and organizing pieces

00:01:26.216 --> 00:01:28.156
of relevant mathematical information.

00:01:28.216 --> 00:01:33.806
Specifically, the visualization helps
students summarize what key information

00:01:33.806 --> 00:01:37.216
is known and see what the problem
is asking them to solve for.

00:01:38.146 --> 00:01:41.676
Use of an appropriate visual can
also reveal the relationships

00:01:41.676 --> 00:01:44.246
between the quantities
identified in the problem.

00:01:44.876 --> 00:01:48.636
Once students grasp these relationships,
they can focus their attention

00:01:48.636 --> 00:01:51.846
on mathematics reasoning and
the problem-solving process.

00:01:51.846 --> 00:01:57.886
Students are also in a better position
to express a problem using equations.

00:01:57.886 --> 00:02:03.476
Teachers are advised to consistently use
a few powerful visual representations.

00:02:03.476 --> 00:02:08.136
A powerful model or representation is
one that has a variety of applications,

00:02:08.175 --> 00:02:10.866
such as a number line or strip diagram.

00:02:10.866 --> 00:02:14.726
If students work with a
particular visual representation

00:02:14.726 --> 00:02:17.956
when they encounter a certain type
of problem, they are more likely

00:02:17.956 --> 00:02:20.666
to grow comfortable with that
tool and use it on their own.

00:02:20.666 --> 00:02:25.216
Students also are then less
likely to use narrative pictures,

00:02:25.216 --> 00:02:26.266
which can distract them

00:02:26.266 --> 00:02:29.006
from the essential mathematical
information in the problem.

00:02:30.076 --> 00:02:34.286
Tables, number lines, strip
diagrams, percent bars,

00:02:34.286 --> 00:02:38.886
and schematic diagrams are among
the most frequently used visuals.

00:02:39.026 --> 00:02:44.246
Schematic diagrams use abstract graphic
symbols rather than realistic pictures

00:02:44.246 --> 00:02:47.626
and include only relevant
problem elements.

00:02:47.626 --> 00:02:51.696
Of course, some visuals are better
suited for particular types of problems.

00:02:53.126 --> 00:02:58.876
For example, strip diagrams work well
for problems that involve comparisons.

00:02:58.876 --> 00:03:02.616
In this example, Cheri is
using a strip diagram to find

00:03:02.616 --> 00:03:07.296
out how much chili was served for
dinner when the family has consumed 3/4

00:03:07.386 --> 00:03:11.496
of the chili and there
is 1 1/2 cups left.

00:03:11.496 --> 00:03:15.026
Tables work well for ratio
and proportion problems.

00:03:15.026 --> 00:03:20.046
Here, Pedro and Sally have drawn a
ratio table to figure out how much paint

00:03:20.046 --> 00:03:24.726
of each color is needed to paint five
classrooms if one gallon of yellow

00:03:24.726 --> 00:03:29.616
and five gallons of beige are
required to paint two classrooms.

00:03:29.616 --> 00:03:33.296
To help students learn how to
employ visual representations,

00:03:33.296 --> 00:03:35.766
teachers can talk aloud
about what they are thinking

00:03:35.766 --> 00:03:39.076
and the decisions they are making
as they reason through a problem.

00:03:39.826 --> 00:03:44.046
During the thinkaloud, teachers should
demonstrate how they identify what

00:03:44.046 --> 00:03:47.546
information will be placed in
the diagram and what aspects

00:03:47.546 --> 00:03:49.096
of the problem are irrelevant.

00:03:50.006 --> 00:03:53.966
It is essential that teachers explain
why the representation they are using is

00:03:53.966 --> 00:03:55.816
appropriate for the problem at hand.

00:03:55.816 --> 00:03:59.166
It is just as important
for students to explain

00:03:59.166 --> 00:04:01.666
to the teacher how they
are setting up a diagram

00:04:01.666 --> 00:04:04.046
and representing the
quantities in the problem.

00:04:04.776 --> 00:04:08.046
By listening to their students'
reasoning, teachers are better able

00:04:08.046 --> 00:04:11.786
to identify and address
possible misconceptions.

00:04:11.786 --> 00:04:15.666
As teachers grow to understand how their
students are thinking about problems,

00:04:15.666 --> 00:04:18.685
they will be able to introduce
them to the idea that there is more

00:04:18.685 --> 00:04:20.555
than one way to think about a problem.

00:04:20.555 --> 00:04:24.836
Researchers recommend that
students be taught explicitly

00:04:24.886 --> 00:04:27.586
that problems can be
solved in more than one way.

00:04:27.586 --> 00:04:30.906
Students who practice
multiple strategies

00:04:30.906 --> 00:04:34.206
and share their solutions become
more flexible and efficient

00:04:34.206 --> 00:04:37.606
in problem solving, and are
more likely to see options

00:04:37.606 --> 00:04:38.896
when approaching a problem.

00:04:39.946 --> 00:04:44.096
Teachers should routinely demonstrate
two or more ways to solve a problem.

00:04:44.816 --> 00:04:47.836
Looking at worked examples
with multiple solutions side

00:04:47.836 --> 00:04:51.876
by side gives students practice
comparing similarities and differences

00:04:51.876 --> 00:04:55.486
in the strategies, which can help
strengthen analytical thinking.

00:04:55.516 --> 00:05:00.256
Students also benefit by being
expected to use multiple methods

00:05:00.256 --> 00:05:01.986
to solve problems themselves.

00:05:01.986 --> 00:05:06.926
As students get used to using multiple
approaches, teachers should talk

00:05:06.926 --> 00:05:10.486
through the reasons why one solution
might be favored over another.

00:05:10.486 --> 00:05:13.106
This will help students understand

00:05:13.106 --> 00:05:17.666
that strategies should be chosen
based on ease and efficiency.

00:05:17.696 --> 00:05:21.236
It can also be helpful for a teacher
to demonstrate approaches to problems

00:05:21.236 --> 00:05:24.096
that are not successful
and discuss why they seem

00:05:24.096 --> 00:05:26.316
like they would work,
but why they don't.

00:05:27.386 --> 00:05:30.706
When teachers routinely focus on
students' thinking and reasoning,

00:05:30.846 --> 00:05:33.826
and not merely on the mechanics
of a particular solution,

00:05:33.826 --> 00:05:38.326
students begin to expect that there will
be multiple ways to approach any problem

00:05:38.566 --> 00:05:41.566
and that for some problems there
will be more than one solution.

00:05:42.556 --> 00:05:46.156
Comparing different strategies
does take time, as students need

00:05:46.156 --> 00:05:49.566
to comprehend each approach
before contrasting it with others.

00:05:50.556 --> 00:05:54.396
When deciding which students will share
their solutions with the whole class,

00:05:54.396 --> 00:05:56.936
it is best to choose
three or four students

00:05:56.936 --> 00:05:59.146
who have used different
approaches to the problem.

00:06:00.136 --> 00:06:03.676
Dividing the class into small
peer groups that compare solutions

00:06:03.676 --> 00:06:06.826
and explain their approaches
to each other can help students

00:06:06.826 --> 00:06:11.276
who are reluctant to discuss their
reasoning in front of the whole class.

00:06:11.306 --> 00:06:14.426
As students observe each
other's reasoning and solutions,

00:06:14.426 --> 00:06:18.566
they more clearly understand that there
are multiple ways to approach problems

00:06:18.566 --> 00:06:22.826
and they can begin to analyze
those approaches for efficiency.

00:06:22.826 --> 00:06:26.876
Mathematical notation helps students
organize the information in a problem,

00:06:27.196 --> 00:06:29.456
articulate mathematics concepts,

00:06:29.456 --> 00:06:31.926
and think about their
options for solving a problem.

00:06:32.016 --> 00:06:37.116
Students may prefer to approach
problems intuitively or informally,

00:06:37.286 --> 00:06:39.806
rather than spend the time
to develop an equation.

00:06:40.716 --> 00:06:43.856
Intuitive approaches may
work for simple problems,

00:06:43.856 --> 00:06:48.856
but notational tools are needed for
more complex and challenging problems.

00:06:48.856 --> 00:06:52.736
By explicitly linking the ideas
in a word problem to an equation,

00:06:53.076 --> 00:06:57.226
teachers demonstrate how to express
problems through mathematical notation,

00:06:57.366 --> 00:06:59.466
including algebraic notation.

00:07:00.646 --> 00:07:02.756
One way to help students
become comfortable

00:07:02.756 --> 00:07:07.866
with mathematical notation is to provide
worked examples of word problems along

00:07:07.866 --> 00:07:11.596
with related mathematical
expressions or equations.

00:07:11.686 --> 00:07:15.016
A teacher can challenge
students to match key information

00:07:15.016 --> 00:07:18.396
in the problem statement with the
related component in the equation.

00:07:19.566 --> 00:07:23.076
Problem solving provides a teacher
with many opportunities to review

00:07:23.076 --> 00:07:26.146
or explain relevant mathematical
concepts

00:07:26.146 --> 00:07:27.966
and introduce new ways of reasoning.

00:07:28.706 --> 00:07:31.146
When preparing problems,
teachers should work

00:07:31.146 --> 00:07:33.756
through several approaches in advance.

00:07:33.756 --> 00:07:37.326
This will help them anticipate the
mathematical concepts students may

00:07:37.326 --> 00:07:39.696
attempt to use in order
to solve a problem.

00:07:40.786 --> 00:07:43.626
A goal of problem solving
is to help students learn

00:07:43.626 --> 00:07:47.576
to articulate mathematically
valid explanations.

00:07:47.656 --> 00:07:51.176
Teachers may need to support
students in organizing their ideas

00:07:51.176 --> 00:07:54.596
and rewording explanations so
they are mathematically correct.

00:07:55.456 --> 00:07:59.166
The teacher's probing questions can
help students refine their thinking

00:07:59.166 --> 00:08:02.886
and develop explanations that are
logical and can be generalized

00:08:02.886 --> 00:08:05.296
and applied in other problem situations.

00:08:05.326 --> 00:08:08.616
Students will need time
to become familiar

00:08:08.616 --> 00:08:11.456
with the abstract symbolic
notation of algebra.

00:08:12.306 --> 00:08:15.826
Teachers may use arithmetic
problems as a first step,

00:08:15.826 --> 00:08:19.286
drawing on students' prior math
experience to frame a solution

00:08:19.286 --> 00:08:23.266
to a problem before translating
the same problem into an equation

00:08:23.266 --> 00:08:26.256
with variables representing
the problem's components.

00:08:27.136 --> 00:08:32.076
These three core strategies-- visual
representations, multiple approaches

00:08:32.076 --> 00:08:34.645
to problems, and mathematical notation--

00:08:34.905 --> 00:08:37.666
are the teacher's primary
tools for taking advantage

00:08:37.666 --> 00:08:41.076
of the instructional opportunities
available when problem solving.

00:08:41.775 --> 00:08:46.956
All three can be applied at all grade
levels and with all mathematical topics.

00:08:48.046 --> 00:08:51.366
To learn more about Effective
Problem-Solving Instruction,

00:08:51.366 --> 00:08:53.976
please explore the additional resources
on the Doing What Works website.

00:08:53.976 --> 00:08:54.000
[Music]