Improving Students` Mathematical Problem

Transcription

4/28/2015
Introductions
Welcome!
Improving Students’
Mathematical Problem
Solving Skills
Debi Faucette & Susan Pittman – April 28, 2015
GEDtestingservice.com • GED.com
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Session Objectives
In this session, we will:
Discuss the impact of effective reading skills
on students’ problem-solving ability
“Our greatest weakness lies
in giving up. The most
certain way to succeed is
always to try just one more
time.”
Identify and apply problem-solving
strategies for a given problem
Engage in problem solving
- Thomas Edison
Share resources and ideas
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4/28/2015
What does reading have to do with
math problem solving?
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
5GEDtestingservice.com
• GED.com
Assumption
Students’ Common Experiences
in Math Classrooms
“ 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.”
• 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”
Draper, Smith, Hall, & Siebert, 2005; Kane, Byrne, & Hater,
1974; O’Mara, 1982
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– “being told what to do”
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4/28/2015
Reading in Math
(Barton & Heidema, 2002)
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
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.
words, numbers, and symbols.
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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
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Know the Language of Math
– 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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4/28/2015
Example of a Vocabulary Strategy
Know the Language of Math
Verbal and Visual Word Association – (Barton & Heidema, 2002)
Vocabulary Term(s)
Visual Representation
Definition(s)
Personal Association or a
Characteristic
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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Verbal and Visual Word Association – (Barton & Heidema, 2002)
Frayer Model – (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
Definition (in own words)
x= -2
f(x)
-Point (a,0) is the x-intercept of the graph
Just find the answer to the function
of y=f(x)
and that will be the zero. If I graph it,
-number a is a zero of the function f
the zeros are where the function crosses
- number a is a solution of f(x)=0
-(x- the x-axis.
a) is a factor of polynomial f(x) -Root is
Special Note: this is just for real
the function on the TI for this
solutions.
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Facts/Characteristics
x= 3
Examples
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WORD or
SYMBOL
Non-Examples
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Frayer Model – (Barton & Heidema, 2002)
Definition (in own words)
Facts/Characteristics
An expression in this form is called
a radical, b is called the radicand
and the n is called the index of the
radical.
n
a is the positive square root of a
 a is the negative square root of a
b
RADICAL
Non-Examples
Examples
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81  3 because 3  81
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1  3 1  4 1  5 1 1
n
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0 0
9 3
 9  3
9  can ' t do
Not a radical – this is
3
a division sign
2205
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Use Games to Assess Students’ Math
Vocabulary
Problem Solving
Developing Quantitative and Algebraic Reasoning Skills
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Why Problem Solving?
What is a
heuristic,
and why is it
important?
“The single best way to grow a better brain is
to engage in challenging problem solving.”
~Jensen (1998)
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What are heuristics?
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How do we use heuristics in problem
solving?
A heuristic is a thinking strategy,
To give a
representation
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
• Draw a
diagram/bar
model
• Make a list
• Create
equations
To make a
calculated guess
• Guess and
check
• Look for
patterns
• Make
suppositions
simply heuristics, are ways for making progress on
difficult problems. Heuristics are components for
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
problem solving. (Polya, 1973)
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4/28/2015
Explicit instruction matters
• Solve problems out loud
• Explain your thinking process
Using Graphic Organizers for
Mathematical Problem Solving
• Graphic organizers allow students to:
• Allow students to explain their thinking process
– sort information as essential or non-essential
• Use the language of math and require students to do so as well
– structure information and concepts
• Model strategy selection
– identify relationships between concepts
• Make time for discussion of strategies
– organize communication about an issue or problem
• Build time for communication
– utilize experiences as a starting point of the problemsolving process
• Ask open-ended questions
Zollman, 2011; 2009a; 2009b
• Create lessons that actively engage learners
Jennifer Cromley, Learning to Think, Learning to Learn
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Value of Teaching with Problems
Polya’s Approach to Problem Solving
• Places students’ attention on
mathematical ideas
Understand
the problem
• Develops “mathematical power”
• Develops students’ beliefs that
they are capable of doing
mathematics
Look back
(reflect)
Devise a
plan
• Provides ongoing assessment data
Carry out
the plan
• Allows an entry point for students
Polya, George. How To Solve It, 2nd ed. (1957).
Princeton University Press.
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4/28/2015
Let’s Get Started!
Remember Your Heuristics
(problem-solving strategies)
“Anyone who has
never made a
mistake has never
tried anything
new.”
•
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
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- Albert Einstein
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Understand
the problem
Strategies for Problem Solving
Look back
(reflect)
K–N–W–S
K
What facts
do I KNOW
from the
information
in the
problem?
N
What
information
do I NOT
need?
Devise a
plan
Carry out
the plan
W
S
What does
the problem
WANT me
to find?
What
STRATEGY
or
operations
will I use to
solve the
problem?
Look back
(reflect)
Devise a
plan
Carry out
the plan
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
What facts do I
KNOW from the
information in the
problem?
What information
do I NOT need?
Reading and Writing to Learn in Mathematics: Strategies to Improve Problem
Solving by Clare Heidema at www.ohiorc.org/adilit
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Understand
the problem
How Does It Work?
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W
What does the
problem WANT
me to find?
S
What
STRATEGY or
operations will I
use to solve the
problem?
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4/28/2015
Sometimes, one needs to think “within the box,”
but not necessarily in a step-by-step approach
Connect
Same Problem – Different ProblemSolving Heuristic
Brainstorm
What do I know?
Brainstorm ways to solve this
problem.
What additional information is
needed?
What possible strategies
could be used?
What formulas are needed?
Main Idea
What do you
need to find?
Solve
Try it here.
Underline key
words/phrases in the
problem and say what they
mean.
Is the answer reasonable?
What do you
need to know to
answer the
question?
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?
Zollman, A. (2006a, April). Annual Conference of the National Council of Teachers of Mathematics, St. Louis, MO
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Strategies for Problem Solving
• Survey
Understand
the problem
Look back
(reflect)
Survey
Scan the problem to get a general ideas of
what it’s about. Clarify terms
Devise a
plan
Question
What is the problem about, and what is the
information in the problem?
Carry out
the plan
• Question
Read
Identify relationship and facts needed to
solve the problem.
• Read
• Question
Question
What to do? How to solve the problem?
• Compute or construct
Compute (or construct)
Do the calculations or construct a solution.
• Question
Question
Is the algebra correct? Are the calculations
correct? Does the solution make sense?
Reading and Writing to Learn in Mathematics: Strategies to Improve
Problem Solving by Clare Heidema at www.ohiorc.org/adilit
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4/28/2015
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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Remember, one way doesn’t fit
every student!
Frayer Model
Examples
Facts/characteristics
Word
Study the problem
Organize the facts
Line up a plan
Verify your plan with action
Examine the results
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
Red
Blue
Yellow
Total
20
10
2
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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)
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?
Nonexamples
SOLVE
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M&Ms are in 3 colors. There are conditions on the 96
M&Ms.
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Definition in your own words
Survey
Scan the problem to get a general
ideas of what it’s about. Clarify terms.
• How will your instructional practices need
to change as you integrate mathematical
modeling into the classroom?
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4/28/2015
The Challenge
Real-World Algebra
• Increase instruction on problem-solving strategies
My Ford Bronco was fitted at the factory with 30 inch diameter
• Increase emphasis on algebraic thinking
tires. That means its speedometer is calibrated for 30 inch
• Provide instruction in higher-order mathematics
diameter tires. I "enhanced" the vehicle with All Terrain tires that
have a 31 inch diameter. How will this change the speedometer
• Shift focus from “rules or processes” of
mathematics to deeper understanding of “why”
readings? Specifically, assuming the speedometer was accurate
• Incorporate close-reading strategies into the math
classroom
drive with my 31 inch tires so that the actual speed is 55 mph?
in the first place, what should I make the speedometer read as I
CTL Resources for Algebra. The Department of
Mathematics. Education University of Georgia
http://jwilson.coe.uga.edu/ctl/ctl/resources/Algebra/Al
gebra.html
• Have high expectations of all students
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Instruction has to move from…
• Cursory approach to teaching
math, like the following:
To conceptual teaching . . .
Conceptual teaching is:
• Using schema to organize new knowledge
– introduce a skill, such as the
Pythagorean Theorem;
• Developing units around concepts
– provide students with the formula;
• Teaching knowledge/skill/concept in context
– review a few sample problems from
the textbook;
What it’s not!
– have students complete a few
problems on their own; and
– move to the next skill or concept.
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• Providing schema based on students’ prior knowledge
• Worksheets
• Drill
• Memorization of discrete facts
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4/28/2015
http://www.gedtestingservice.com/
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
Debi.Faucette@gedtestingservice.com
Susan Pittman
skptvs@aol.com
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Improving Students` Mathematical Problem

Transcription

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