The Final Report of the National Mathematics Advisory Panel

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CREATING A CLASSROOM OF
MATHEMATICS PROBLEM SOLVERS

Grades 3
-
8

BAIN FAMILY

This is our family at the summit of Mt.
Ellinor

in Olympic National Park. It was a great family
adventure. We saw many adults turn around before reaching the summit, but we pressed on and
enjoyed the summit with a few mountain goats.

Tricia

Mahreah

Grace

Ben

Jonah

Ruby

MY BACKGROUND


1998 graduate of Martin Luther College with a
double major in education and music.


2004 earned post
-
baccalaureate Wisconsin
state teaching license.


Ten
-
time classroom supervisor of student
teachers.


2010 graduate of Martin Luther College with a
Master of Science in Education degree

instruction emphasis.


THE IMPORTANCE OF MATHEMATICS

& DEFINING PROBLEM SOLVING

TIMSS


International studies;
comparing eighth
graders


TIMSS = Trends in
International
Mathematics and
Science Study


1995: U.S. ranked 28
th

out of 41 countries


1999: U.S. ranked 19
th

out of 34 countries


2003: U.S. ranked 15
th

out of 45 countries


2007: U.S. ranked 9
th

out
of 47



Why is math important for our country?


Why is math important for our church?


Why is math important for our students?

NATIONAL MATHEMATICS ADVISORY PANEL

Foundations for Success: The Final Report of the
National Mathematics Advisory Panel

(2008)

“The eminence, safety, and well
-
being of nations
have been entwined for centuries with the ability of
their people to deal with sophisticated quantitative
ideas. Leading societies have commanded
mathematical skills that have brought them
advantages in medicine and health, in technology
and commerce, in navigation and exploration, in
defense and finance, and in the ability to understand
past failures and to forecast future developments.”
(p. xi)

IS THIS PROBLEM SOLVING?


In the numeral 78,965,
what does the 8 mean?


These were the scores for
the spelling tests: 25, 19,
16, 25, 18, 19, 25, 24, 25,
23. What is the median?


70 * 18 = ______


Tatiana gets her teeth
cleaned every 6 months. If
her last appointment was in
February, when is her next
appointment?


Beth’s allowance is $2.50
more than
Kesia’s
. Beth’s
allowance is $7.50. What is
Kesia’s

allowance?


3/8 of 40 is ______.


Josie has 327 photographs.
She can put 12 photos on
each page of her scrapbook.
Estimate the number of
scrapbook pages she will
need.


How can you find the value of
18
3

using your calculator?

DEFINITIONS OF PROBLEM SOLVING

NCSM

(NATIONAL COUNCIL OF SUPERVISORS OF
MATHEMATICS)

NCTM

(NATIONAL COUNCIL OF TEACHERS OF
MATHEMATICS)

“the process of applying
previously acquired knowledge
to new and unfamiliar
situations”

“problem solving means
engaging in a task for which the
solution method is not known in
advance”

FEATURES OF PROBLEM SOLVING

MEIR BEN
-
HUR

JOAN M. KENNEY

“Problem solving requires
analysis, heuristics, and
reasoning toward self
-
defined
goals”

a process that involves such
actions as modeling,
formulating, transforming,
manipulating, inferring, and
communicating

TEACHING PROBLEM SOLVING

OVERVIEW


Key Words


Strategies


Process by George
Pólya


Teacher’s Role


What words tell a person to multiply?


What words tell a person to subtract?


What words do you notice are particularly
troublesome to your students?

AN EXAMPLE OF TEACHING KEY WORDS

“Two flags are similar. One flag is three times as
long as the other flag. The length of the smaller
flag is 8 in. What is the length of the larger flag?”

WHAT DOES RESEARCH SAY?


Undermines real problem solving


Make problem solving a mechanical process
which makes students prone to errors


Understanding the language of mathematics is
important.


“Two flags are similar. One flag is three times as long
as the other flag. The length of the larger flag is 8 in.
What is the length of the smaller flag?”



Ben
-
Hur
, M. (2006).
Concept
-
rich mathematics instruction: Building a strong foundation for reasoning and problem solving.

Alexandria, VA:
Association for Supervision and Curriculum Development.

Xin
, Y. P. (2008). The effect of schema
-
based instruction in solving mathematics word problems: An emphasis on
prealgebra

conceptualization
of multiplicative relations.
Journal for Research in Mathematics Education
,
39
(5), 526
-
551.


LITERATURE STRATEGIES FOR MATH WORDS


Word Wall


Math Word Dictionary


Vocabulary Cards


Semantic Feature Analysis

STRATEGIES INSTRUCTION

WAYS TO INTRODUCE

ONE WAY

ANOTHER WAY


Teacher models the strategy.


Students work on problems
using that strategy.


Teacher models the strategy.


Students work on problems
which may or may not use
the modeled strategy.

TYPES OF STRATEGIES


Make a Model or
Diagram


Make a Table or List


Look for Patterns


Use an Equation or
Formula


Consider a Simpler Case


Guess and Check/Test


Work Backward


Others?

USING A FORMULA

MAKE A MODEL OR DIAGRAM

Perimeter of a square : P = 4S.

Using this formula, students could
determine the side lengths for
each of the squares as 10 inches
and 9 inches.

Area of a square: A = S
2
.

Larger square = 100 in.
2

Smaller square = 81 in.
2

The difference between the areas
of the two squares is found by
subtracting the smaller area from
the larger area.

Use graph paper to draw one
square inside the other.

Count the squares to find the
difference.

“One square has a perimeter of 40 inches. A second square has a
perimeter of 36 inches. What is the positive difference in the
areas of the two squares?”

WHAT DOES RESEARCH SAY?


Teaching strategies improves mathematics
problem solving abilities.


Teaching strategies does not improve overall math
achievement.


Teachers need to avoid teaching strategies as an
algorithm.

Rickard, A. (2005). Evolution of a teacher’s problem solving instruction: A case study of aligning teaching practice with ref
orm

in
middle school mathematics.
Research in Middle Level Education Online
,
29
(1), 1
-
15.

Higgins, K. M. (1997). The effect of year
-
long instruction in mathematical problem solving on middle
-
school students’
attitudes, beliefs, and abilities.
Journal of Experimental Education
,
66
(1).

Jitendra
, A.,
DiPipi
, C. M., &
Perron
-
Jones, N. (2002). An exploratory study of schema
-
based word
-
problem
-
solving
instruction for middle school students with learning disabilities: An emphasis on conceptual and procedural
understanding.
Journal of Special Education
,
3
6(1), 23
-
38.

Mastromatteo
, M. (1994). Problem solving in mathematics: A classroom research.
Teaching and Change
,
1
(2), 182
-
189.

Schoenfeld
, A. H. (1988). When good teaching leads to bad results: The disasters of “well
-
taught” mathematics courses.
Educational Psychologist
,
23
(2), 145
-
166.

PROCESS BY GEORGE
PÓLYA

THE ELEMENTS OF THE PROCESS


Understand the Problem


Make a Plan


Follow/adjust the Plan


Look Back

UNDERSTANDING THE PROBLEM


Define important words.


Identify necessary and unnecessary information.


Stating what is known and unknown.


Determine if other information is needed.


Decide if calculations need to be made prior to
another calculation.


Rephrase the problem.


Consider this: pose a problem situation without a
question.

MAKE A PLAN


Select a strategy


Use what is known to determine how to find a
solution


The goal would be that students be able to
explain, with reasons, why they think their
strategy could work.

FOLLOW/ADJUST THE PLAN


Students carry out the plan they made.


Students show the work that they do, and they
may be asked to write explanations.


Students adjust their plan if they notice
something isn’t working or they have
determined a better way to solve the problem.

LOOK BACK


Very valuable!


Check that solution fits problem.


Consider strategy choices and their consequences.


Create related problem(s) that could be solved the
same way.


Offer changes to the problem and infer their affect
on the solution.


Connect to other problems already studied.


Make generalizations.

THE TEACHER’S ROLE

ADDRESS MISCONCEPTIONS


Undergeneralizations


Ex.: 3:4 and ¾ are different things


Ex.: “=“ only means perform operations to find
answer


Overgeneralizations


Ex.: Multiplying two numbers makes a bigger
number


Ex.: Misapplication of regrouping


Ex.: Finding common denominators when
multiplying fractions.

THE ENVIRONMENT


Foster a classroom environment friendly to asking
questions.


Adjust content to meet student needs.


Use a wide variety of activities.


Allow time for exploration.


Organize and represent concepts in different ways.


Pose probing questions to foster meta
-
cognition.


Model meta
-
cognition.


Promote dialogue.

ASSESSMENT/EVALUATION

PROBLEM SOLVING FORM


Creates a framework to help students see the
importance of explaining why they are doing
what they are doing.


Use in groups or individually.


Can be used in portfolios to share with parents
at conferences or for student self
-
reflection of
progress.

ANECDOTAL RECORDS


Teacher notes while observing problem solving


Listen for evidence the student seeks information
to fully understand the problem.


Consider a student’s ability to persevere.


Note use of appropriate strategies.


Listen to student oral explanations for
misconceptions or proper conceptual
understanding.


Look for algorithmic errors.

CHECKLIST OR RUBRIC


Much the same as anecdotal records, but this
may be scored from viewing written work.


A rubric may be formed using
Pólya’s

process
or specific to the learning goals of the lesson.


Share whatever rubric you use with students
and make sure they understand it.


Provide opportunities for self
-
evaluation.


QUESTIONS OR COMMENTS?