Addressing the Challenges of Murphy’s Law of Fractions, Decimals, and Percents

James Olsen, Ph.D. and Hayley Moore

Western Illinois University

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

International Conference onCollege Teaching and Learning, April 8, 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– Including student errors

• (A few Strategies for Percent Problems– Handout)• 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 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)

*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 Calculus is hard:– Highly interconnected ideas. • Derivative|rate of change|slope of tangent line.• Fraction|decimal|percent.

– Technical details can trip you up.• Quotient rule, chain rule.• Common denominators, regrouping with mixed

numbers.

– Concepts are hard to think about.• Acceleration.• A fractional part.Both are based on the ratio !

Why “PFD” ?

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

Landscape of Problems

• Background: – PFD Concept Map (flowchart) – see handout.– 16 PFD Objectives (handout).

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

3D: Objective, Number Variation, Application

3D: Objective, Number Variation, Application

From the back.

Landscape of ProblemsSee Handout – Grids

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

Landscape of ProblemsSee Handout – Grids

• The Application Dimension. Q: How can each objective be applied?A: In many ways!

Some Example ProblemsLandscape of Problems

See Handout

Future and ongoing efforts

• Develop the PFD Landscape problem.– Many are in DRAFT form (on the website).faculty.wiu.edu/JR-Olsen/wiu/BootCamp/landscape/front.html

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

• Find a way to automate the PFD Mastery Quizzes to 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?

Addressing the Challenges of Murphy’s Law of Fractions, Decimals, and Percents

Olsen/Moore