WEBVTT

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[Music]

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I am John Woodward.

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I am the dean of the School
of Education at the University

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of Puget Sound, and I was chair

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of the Math Problem Solving
Guide for grades 4 through 8.

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Grades 4 through 8
are critical periods

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for kids learning mathematics.

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We even see that by looking
at the international data.

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And one of the things
that's indicative

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of mathematics instruction in
other countries in the world,

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particularly, countries
like Japan,

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that have always done
well on math assessments,

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is that a significant portion
of their classroom instruction

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across a week's period
of time, for example,

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involves problem solving.

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Math problem solving has
traditionally been thought

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of as just word problem
solving, and that's true,

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we can look at an example
where students are asked

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to configure a pen for horses
using the same perimeter

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in different area.

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But we can also think
about math problems solving

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as symbol manipulation.

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For example, students can
look at different solutions

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to algebraic equations and talk

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about the different strategies
used to solve those equations.

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Another example is

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where students are simply given
a simple visual setup-how many

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squares on a checkerboard-and
asked to answer

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that problem using some
kind of visual analysis.

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The checkerboard problem
is a really good problem,

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in part because people
think the answer is obvious.

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Anybody who has played checkers
or chess, look at the board

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and say, "Well, okay,
let me remember.

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Yeah, that's right, they
are 8x8, or 64 squares."

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But it's obviously more
complicated that when you start

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to analyze it visually.

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You've got the larger square
that frames the checkerboard

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and then as you think about
it further, you see patterns,

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you see 2x2s, 3x3s, 4x4s.

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There are all kinds of
square patterns inside

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that checkerboard.

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The panel derived over time
five major recommendations

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that are supported by
empirical evidence.

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But for the first one, where
there isn't a lot of research,

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all of them have moderate
or strong levels of evidence

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that support their practices.

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One of the first recommendations
that's quite important is

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that teachers plan for
problem solving as part

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of whole-class instruction.

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The idea is problem solving
is not to be left to homework

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or independent seat work.

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That creates its own problem

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because we can't specify
how many minutes per day

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or how many hours per
week; we simply feel

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that mathematical problem
solving should be a more central

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portion of the instruction
throughout, say,

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a unit of instruction.

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So the recommendations begin
with planning very clearly

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for problem solving, making sure

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that the problems are
appropriate to the content

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that kids are learning,
that they vary

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in their difficulty level,
that they are sensitive

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to kids' background
knowledge, their language use,

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those aspects-before they even
begin the problem solving.

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A second recommendation
has to do

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with helping kids
monitor their strategies

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as they solve the problem.

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This may begin with a
task list of some kind,

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where kids can think about
strategies that they might use

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in solving the problems.

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We also recommend that
teachers in the initial phases

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of problem solving talk out
loud or model how they go

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about working through
a complex problem.

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A third recommendation has
to do with the visualizations

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of problems using
appropriate representations.

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These can be number strips,
these can be diagrams;

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a variety of things come
into play to help kids break

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down a problem and visualize it

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as they work their
way toward a solution.

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A fourth recommendation has

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to do-and this is really quite
critical-that kids spend time

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seeing and listening to
variety of strategies

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for solving a problem.

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That's where that "Aha,
I have never thought

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about this before" comes from.

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It makes sense that students
are immersed in their own way

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of thinking about a problem and
solving it and come to think

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that that's the only
way to solve it.

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What becomes critical
is for kids afterwards

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to hear different
solution strategies.

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The importance of this is
this builds greater cognitive

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flexibility and different
ways of looking

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at how they can solve problems.

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And a final recommendation
has to do

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with connecting the
problem solving

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with the mathematical notation.

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A lot of times people
think, well,

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they have solved the
problem, they have gone

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through the process, kids
have talked about strategies,

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but then they forget to connect
it to the formal mathematics

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that kids may be studying
at that point in time.

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I can think of a number of
key issues that teachers need

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to consider when they think
about problem solving.

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The first and foremost
one is the amount

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of time they are going to
allocate to the problem solving.

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That needs to shift.

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That's not to say again that
there is going to be 15 minutes

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or 20 minutes per day, but it
needs to have a greater presence

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across the week, across
the unit of instruction.

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The second one, which
also involves planning,

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has to do with ensuring that
you're meeting and attending

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to the linguistic background,
the cultural background

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of the students you
are working with.

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The third thing that I would
think it would be very important

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for teachers to consider is
just the types of problems;

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ask themselves what's the
purpose of the problems

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that they want to use.

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For example, should they
provide several problems

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on a new topical area, or is
the purpose of problem solving

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to develop that persistence
to get them to be able to work

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with a problem that's novel
and has an unusual set

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of characteristics,
irrelevant information

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that they have to struggle with.

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Students need to persist
in their problem solving.

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You see this very clearly by the
time they get to high school.

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So the ability to stick with
a problem, deal with aspects

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of a problem that are
unknown or unclear

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at first is a central part of
mathematical problem solving

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and success in mathematics.

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[Music]