Navigating the Landscape of Fraction, Decimal, and Percent Problems: Understanding and

Addressing Student Difficulties

James Olsen, Ph.D. and Joe Illichman

Western Illinois University

JR-Olsen@wiu.eduhttp://faculty.wiu.edu/JR-Olsen/wiu/

ICTM – Springfield Oct. 22, 2011

Outline• History – where this began• Goals– Overall goals.– Goals for this session.

• Importance of PFD– What it means to be fluent. Why are fractions so

hard?• Landscape of Problems– 3 Dimensions

• Example Problems• Percent Change Problems• Future and ongoing efforts

History – Where this Began• Developed four mastery quizzes on Percents,

Fractions, and Decimals (for a 300-level teaching methods course). I thought it would be an easy exercise. Found many holes in student understandings.

• Began to study student misunderstandings and difficulties (and what I could do about it).

• Started developing a PFD Boot Camp website.• The project got bigger.

Overall Goals for our “PFD” Work• The goals are to help students:

– Understand the meanings of, and relationships between, fractions, decimals, and percents,

– Understand the uses of fractions, decimals, and percents in the real world,

– Develop computational skills and confidence working with fractions, decimals, and percents.

– Apply computational skills to fluently solve real-world problems.

• Intended Audience(s):– Teacher education students,– College students,– Junior or senior high students,– Anyone who wants to improve their understanding and skills with

fractions, decimals, and percents.

Goals for this Session• Show the importance of Percent, Fraction, and

Decimal (PFD) concepts and procedures for helping students be College and Career Ready (CCR).

• Describe a (3-dimensional) Landscape of PFD word problems, which should help teachers help their students develop competency in PFD applications.

Assumptions• We recognize the importance of beginning with

hands-on activities—the Bruner process Physical Pictorial Symbolic.

• We are at the symbolic level—actually solving real problems.

• We are emphasizing procedural knowledge. Both conceptual knowledge and procedural knowledge are important. They enhance one another.

Assumptions• We believe to understand something well it

should be connected in the brain to many other ideas.

• We believe that mathematics is a mental activity and that most of the PFD problems we are writing should be done mentally or with paper-and-pencil.

Why are Fractions, Decimals, and Percents Important?

• • • • • Common Core State Standards (handout)• NCTM Focal Points (handout)• National Math Advisory Panel (handout)

Fractions are pervasive

Define Fluency

Fluency is being able to use a variety of skills flexibly in varied settings or

contexts.

Basketball metaphor.Use of a second language.

Why are fractions so hard?• For the same 3 reasons that Algebra and Calculus are hard:

– Highly interconnected ideas. • Fraction - decimal - percent.• Factors – zeros - x-intercepts.• Derivative - rate of change - slope of tangent line.

– Technical details can trip you up.• Common denominators, regrouping with mixed numbers.• Like terms, laws of exponents.• Quotient rule, chain rule.

– Concepts are hard to think about.• A fractional part, percents greater than 100%.• Functions, domain, range.• Rate of change, acceleration.

The ubiquitous ratio !

Why “PFD” ?

A PFD is……(a Personal Flotation Device!)Logo for the website (under construction):

Background for theLandscape of Problems

PFD Concept Map (flowchart) – see next slide and handout.

This Concept Map led to the 16 PFD Objectives.Objective being the first dimension of the Landscape of Problems.

Landscape of Problems• Three Dimensions:–Objective (16), –Number Variation,–Application.

Dimensions of the Landscape

• The Objective Dimension. What should students be able to do with fractions, decimals, and percents?• See handout of the 16 Objectives.

Dimensions of the Landscape

• The Number Variation Dimension. What number variations can give students difficulties?• See handout.

Dimensions of the Landscape

• The Application Dimension. How can each objective be applied?

• See handout.

3D: Objective, Number Variation, Application

3D: Objective, Number Variation, Application

From the back.

A Look at the Website

• 216 PFD Landscape problems to download.faculty.wiu.edu/JR-Olsen/wiu/BootCamp/landscape/front.html

You may copy and paste into worksheets, POW’s, quizzes, etc.

*Go to website*> and Example Problems <

[Some Example ProblemsSee back of handout]

Percent Change

• Where the rubber meets the road!• It all comes together.• Not surprisingly, it’s how the media

describes our world.

Taking Requests !

What problems do you want to see?

Future and ongoing efforts

• Develop the PFD Boot Camp.– Including videos and worked examples.

• Find a way to automate the PFD Mastery Quizzes so that all faculty can use them.

Thank You

• Websites

• faculty.wiu.edu/JR-Olsen/wiu/• faculty.wiu.edu/JR-Olsen/wiu/BootCamp/landscape/front.html

Examples

• We have 78 million dollars budgeted this year for TV advertising. If each TV advertising contract is 1.3 million dollars, how many advertising contracts can we buy?

• If the field is three-quarter miles by one and a half miles, find the area of the field.

• The Facebook Group had an increase of 560 people join the group from September to October. There is now a total of 910 people. What is the percent change?

Navigating the Landscape of Fraction, Decimal, and Percent Problems: Understanding and

Addressing Student DifficultiesOlsen/Illichman