Addressing the Challenges of Murphy’s Law of Fractions, Decimals, and Percents James Olsen, Ph.D. and Hayley Moore Western Illinois University International Conference on College 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

From the back.

Landscape of Problems See Handout – Grids The Number Variation Dimension. What number variations can give students difficulties?

Landscape of Problems See Handout – Grids The Application Dimension. Q: How can each objective be applied? A: In many ways!

Some Example Problems Landscape 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