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4/28/2015
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Improving Students’
Mathematical Problem
Solving Skills
Debi Faucette & Susan Pittman – April 28, 2015
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Introductions
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Welcome!
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In this session, we will:
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Session Objectives
Discuss the impact of effective reading skills
on students’ problem-solving ability
Identify and apply problem-solving
strategies for a given problem
Engage in problem solving
Share resources and ideas
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“Our greatest weakness lies
in giving up. The most
certain way to succeed is
always to try just one more
time.”
- Thomas Edison
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What does reading have to do with
math problem solving?
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Effective Readers = Effective Problem Solvers
They can:
• Locate key information
• Distinguish between main ideas and supporting details
• Modify reading based on difficulty of text
• Ask questions before, during, and after reading
• Monitor their comprehension
– Evaluate new information
– Connect new information with existing ideas
– Organize information in ways that make sense
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Assumption
“ When a student is not successful in math, teachers
usually assume the difficulty is with the student’s
mathematical ability or possibly the student’s dislike
of mathematics, but the truth may more likely lie with
the student’s poor ability to read the mathematics
textbook.”
Draper, Smith, Hall, & Siebert, 2005; Kane, Byrne, & Hater,
1974; O’Mara, 1982
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Students’ Common Experiences
in Math Classrooms • Students find math textbooks to be intimidating and
confusing and therefore just skip past the
explanations. (Draper, 1997)
• Students expect the teacher to be the expert, do all
the talking, and be the center of the classroom.
• Students say the best means of learning math are
(Stodolsky, Salk, & Glaessner, 1991)
– “hearing an explanation”
– “asking someone”
– “being told what to do”
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Reading in Math
(Barton & Heidema, 2002) • Requires unique knowledge and skills not taught
in other content areas.
• Math textbooks contain more concepts per word,
per sentence, and per paragraph than any other
text type or content area textbook.
• Students need to be proficient at decoding
words, numbers, and symbols.
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Reading in Math (Barton & Heidema, 2002)
• Writing style in math textbooks is compact and succinct with little redundancy of text.
• Students often skip over the worded parts looking for examples, graphics, or exercises.
• Math textbooks are often written above grade level.
• Overlap between math and everyday English vocabulary can cause confusion.
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Content Area Reading Strategies
• Reading strategies are NOT for
students to learn-to-read the math
textbook but to read-to-learn from
the math textbook.
• Reading Strategies are really
Learning Strategies
– Students can use strategies to help
them comprehend what they read
– Teachers can use strategies to
check on students’ comprehension
of what they read
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Know the Language of Math
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Know the Language of Math
Which one is
the “right
triangle”?
View normally seen
in textbooks. Are students able to
recognize the properties of a right triangle?
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Example of a Vocabulary Strategy Verbal and Visual Word Association – (Barton & Heidema, 2002)
Vocabulary Term(s)
Visual Representation
Definition(s)
Personal Association or a
Characteristic
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Example of a Vocabulary Strategy Verbal and Visual Word Association – (Barton & Heidema, 2002)
Root, Zero, Factor,
Solution, x-intercept
Each word can represent the answer to the function y=f(x) where f(a)=0 and a is a root, zero, factor, solution, and x-intercept
-Point (a,0) is the x-intercept of the graph of y=f(x)
-number a is a zero of the function f -number a is a solution of f(x)=0 -(x-a) is a factor of polynomial f(x) -Root is the function on the TI for this
x= -2 x= 3
f(x)
Just find the answer to the function and that will be the zero. If I graph it, the zeros are where the function crosses the x-axis.
Special Note: this is just for real
solutions.
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Example of a Vocabulary Strategy Frayer Model – (Barton & Heidema, 2002)
Definition (in own words) Facts/Characteristics
Examples Non-Examples
WORD or
SYMBOL
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Example of a Vocabulary Strategy Frayer Model – (Barton & Heidema, 2002)
n b
Definition (in own words) Facts/Characteristics
Examples Non-Examples RADICAL
An expression in this form is called a radical, b is called the radicand
and the n is called the index of the
radical.
44 81 3 3 81because
3 541 1 1 1 1
a is the positive square root of a
a is the negative square root of a
0 0n 9 3 9 3
9 'can t do
3 2205Not a radical – this is a division sign
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Use Games to Assess Students’ Math
Vocabulary
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Problem Solving
Developing Quantitative and Algebraic Reasoning Skills
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Why Problem Solving?
“The single best way to grow a better brain is
to engage in challenging problem solving.”
~Jensen (1998)
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What is a
heuristic,
and why is it
important?
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What are heuristics?
A heuristic is a thinking strategy,
something that can be used to
identify further information about a problem and thus
help you figure out what to do when you don't know
what to do. Heuristic methods, heuristic strategies, or
simply heuristics, are ways for making progress on
difficult problems. Heuristics are components for
problem solving. (Polya, 1973)
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How do we use heuristics in problem
solving?
To give a representation
• Draw a diagram/bar model
• Make a list
• Create equations
To make a calculated guess
• Guess and check
• Look for patterns
• Make suppositions
To go through the process
• Act it out
• Work backwards
• Before-after concept
To change the problem
• Restate the problem in another way
• Simplify the problem
• Solve part of the problem
• Think of a related problem
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Explicit instruction matters
• Solve problems out loud
• Explain your thinking process
• Allow students to explain their thinking process
• Use the language of math and require students to do so as well
• Model strategy selection
• Make time for discussion of strategies
• Build time for communication
• Ask open-ended questions
• Create lessons that actively engage learners
Jennifer Cromley, Learning to Think, Learning to Learn
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Using Graphic Organizers for
Mathematical Problem Solving • Graphic organizers allow students to:
– sort information as essential or non-essential
– structure information and concepts
– identify relationships between concepts
– organize communication about an issue or problem
– utilize experiences as a starting point of the problem-
solving process
Zollman, 2011; 2009a; 2009b
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Value of Teaching with Problems
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• Places students’ attention on
mathematical ideas
• Develops “mathematical power”
• Develops students’ beliefs that
they are capable of doing
mathematics
• Provides ongoing assessment data
• Allows an entry point for students
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Polya, George. How To Solve It, 2nd ed. (1957).
Princeton University Press.
Understand the problem
Devise a plan
Carry out the plan
Look back (reflect)
Polya’s Approach to Problem Solving
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Remember Your Heuristics (problem-solving strategies)
• Look for patterns
• Consider all possibilities
• Make an organized list
• Draw a picture
• Guess and check
• Write an equation
• Construct a table or graph
• Act it out
• Use objects
• Work backward
• Solve a simpler (or similar) problem 30 GEDtestingservice.com • GED.com
Let’s Get Started!
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“Anyone who has
never made a
mistake has never
tried anything
new.”
- Albert Einstein
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Strategies for Problem Solving
K – N – W – S
Understand the problem
Devise a plan
Carry out the plan
Look back (reflect)
K N W S
What facts
do I KNOW
from the
information
in the
problem?
What
information
do I NOT
need?
What does
the problem
WANT me
to find?
What
STRATEGY
or
operations
will I use to
solve the
problem?
Reading and Writing to Learn in Mathematics: Strategies to Improve Problem
Solving by Clare Heidema at www.ohiorc.org/adilit
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How Does It Work?
Video-Online rents movies for $3 each per night.
They also offer a MAX Movie plan for $100 per year with
two free rentals per month and unlimited rentals at $1 per
movie each per night. How many movies must you rent in a
year to make the club deal worthwhile?
K N W S
What facts do I
KNOW from the information in the
problem?
What information
do I NOT need?
What does the
problem WANT me to find?
What
STRATEGY or operations will I
use to solve the problem?
Understand the problem
Devise a plan
Carry out the plan
Look back (reflect)
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Sometimes, one needs to think “within the box,”
but not necessarily in a step-by-step approach
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Brainstorm
Brainstorm ways to solve this
problem.
What possible strategies
could be used?
Connect
What do I know?
What additional information is
needed?
What formulas are needed?
Solve
Try it here.
Underline key
words/phrases in the
problem and say what they mean.
Is the answer reasonable?
Write
What steps do I need to follow to solve the problem?
How is the problem relevant to me?
How could the problem be extended?
Main Idea
What do you
need to find?
What do you
need to know to answer the
question?
Zollman, A. (2006a, April). Annual Conference of the National Council of Teachers of Mathematics, St. Louis, MO
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Same Problem – Different Problem-
Solving Heuristic
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Strategies for Problem Solving
• Survey
• Question
• Read
• Question
• Compute or construct
• Question
Understand the problem
Devise a plan
Carry out the plan
Look back (reflect)
Reading and Writing to Learn in Mathematics: Strategies to Improve
Problem Solving by Clare Heidema at www.ohiorc.org/adilit
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Survey
Scan the problem to get a general ideas of what it’s about. Clarify terms
Question
What is the problem about, and what is the information in the problem?
Read
Identify relationship and facts needed to solve the problem.
Question
What to do? How to solve the problem?
Compute (or construct)
Do the calculations or construct a solution.
Question
Is the algebra correct? Are the calculations correct? Does the solution make sense?
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How Does It Work? Let’s Start Easy
A bag of M&Ms has 96 pieces in three
colors, red, blue, and yellow. The bag has
twice as many red M&Ms as blue and five
times as many blue as yellow. How many
M&Ms of each color are in the bag?
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Survey
Scan the problem to get a general
ideas of what it’s about. Clarify terms.
M&Ms are in 3 colors. There are conditions on the 96
M&Ms.
Question
What is the problem about, and what
is the information in the problem?
How many M&Ms of each color are there?
Read
Identify relationship and facts needed
to solve the problem.
96 M&Ms – red = 2x blue, blue is 5x yellow
Red + blue + yellow = 96
Question
What to do? How to solve the
problem?
Write an equation. Use substitution.
r + b + y = 96
Make a table and try numbers
Compute (or construct)
Do the calculations or construct a
solution.
Algebra: r + b + y = 96 (r= 2b and b = 5y)
2b + b + y = 96 (substitute r = 2b)
2(5y) + 5y + y = 96 (substitute b = 5y)
10y + 5y + y = 16y = 96 so y = 6
Question
Is the algebra correct? Are the
calculations correct? Does the
solution make sense?
y = 6, b = 30, r = 60
(check using the table)
Red Blue Yellow Total
20 10 2 32
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Remember, one way doesn’t fit
every student!
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Frayer ModelDefinition in your own words Facts/characteristics
Examples NonexamplesWord
SOLVE Study the problem
Organize the facts
Line up a plan
Verify your plan with action
Examine the results
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Reflection
• What problem-solving approaches do you
find most effective with students?
• What pose the greatest concerns for you
in integrating higher-order reasoning
strategies into your classroom?
• How will your instructional practices need
to change as you integrate mathematical
modeling into the classroom?
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The Challenge
• Increase instruction on problem-solving strategies
• Increase emphasis on algebraic thinking
• Provide instruction in higher-order mathematics
• Shift focus from “rules or processes” of
mathematics to deeper understanding of “why”
• Incorporate close-reading strategies into the math
classroom
• Have high expectations of all students
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Real-World Algebra
My Ford Bronco was fitted at the factory with 30 inch diameter
tires. That means its speedometer is calibrated for 30 inch
diameter tires. I "enhanced" the vehicle with All Terrain tires that
have a 31 inch diameter. How will this change the speedometer
readings? Specifically, assuming the speedometer was accurate
in the first place, what should I make the speedometer read as I
drive with my 31 inch tires so that the actual speed is 55 mph?
CTL Resources for Algebra. The Department of
Mathematics. Education University of Georgia http://jwilson.coe.uga.edu/ctl/ctl/resources/Algebra/Al
gebra.html
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Instruction has to move from…
• Cursory approach to teaching
math, like the following:
– introduce a skill, such as the
Pythagorean Theorem;
– provide students with the formula;
– review a few sample problems from
the textbook;
– have students complete a few
problems on their own; and
– move to the next skill or concept.
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To conceptual teaching . . .
Conceptual teaching is:
• Using schema to organize new knowledge
• Developing units around concepts
• Providing schema based on students’ prior knowledge
• Teaching knowledge/skill/concept in context
What it’s not!
• Worksheets
• Drill
• Memorization of discrete facts
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http://www.gedtestingservice.com/
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Don’t Forget the PLDs (Performance Level
Descriptors)
• Provides descriptors for each
performance level
– Below Passing
– Passing
– Passing with Honors
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“High achievement always occurs in
the framework of high expectation.”
Charles F. Kettering (1876-1958)
Debi Faucette [email protected]
Susan Pittman [email protected]
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