Addressing the Challenges of Murphy’s Law of Fractions, Decimals, and Percents
James Olsen, Ph.D. and Hayley Moore
Western Illinois University
[email protected]://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
