Navigating the Landscape of Fraction Decimal and Percent

Navigating the Landscape of Fraction, Decimal, and Percent Problems: Understanding and Addressing Student Difficulties ICTM – Springfield Oct. 22, 2011 James Olsen, Ph. D. and Joe Illichman Western Illinois University JR-Olsen@wiu. edu http: //faculty. wiu. edu/JR-Olsen/wiu/

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 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 the Landscape 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.

3 D: Objective, Number Variation, Application

3 D: Objective, Number Variation, Application From the back.

A Look at the Website • 216 PFD Landscape problems to download. faculty. wiu. edu/JR-Olsen/wiu/Boot. Camp/landscape/front. html You may copy and paste into worksheets, POW’s, quizzes, etc. *Go to website* > and Example Problems <

[Some Example Problems See 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/Boot. Camp/landscape/front. html

Navigating the Landscape of Fraction, Decimal, and Percent Problems: Understanding and Addressing Student Difficulties Olsen/Illichman 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?