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4/28/2015

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Improving Students’

Mathematical Problem

Solving Skills

Debi Faucette & Susan Pittman – April 28, 2015


Introductions

2

Welcome!


In this session, we will:

3

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


“Our greatest weakness lies

in giving up. The most

certain way to succeed is

always to try just one more

time.”

- Thomas Edison

4

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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


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


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


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?


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


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.


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



Use Games to Assess Students’ Math

Vocabulary


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)


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)


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


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


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!

31

“Anyone who has

never made a

mistake has never

tried anything

new.”

- Albert Einstein


Strategies for Problem Solving

K – N – W – S


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


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?


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


Same Problem – Different Problem-

Solving Heuristic

37 37


Strategies for Problem Solving

• Survey

• Question

• Read

• Question

• Compute or construct

• Question


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


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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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?


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

42


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


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


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.


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://jwilson.coe.uga.edu/ctl/ctl/resources/Algebra/Algebra.html







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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]

52

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