James Olsen, Ph.D. and Joe Illichman Western Illinois University JR-Olsen@wiu

1. 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/

2. 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

3. 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.

4. 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.

5. 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.

6. 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.

7. 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.

8. Why are Fractions, Decimals, and Percents Important? • Common Core State Standards (handout) • NCTM Focal Points (handout) • National Math Advisory Panel (handout) Fractions are pervasive

9. 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.

10. 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 !

11. Why “PFD” ? A PFD is…… (a Personal Flotation Device!) Logo for the website (under construction):

12. Background for theLandscape of Problems PFD Concept Map (flowchart) – see next slide and handout.

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

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

15. 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.

16. Dimensions of the Landscape • The Number Variation Dimension. What number variations can give students difficulties? • See handout.

18. Dimensions of the Landscape • The Application Dimension. How can each objective be applied? • See handout.

20. 3D: Objective, Number Variation, Application

21. 3D: Objective, Number Variation, Application From the back.

22. 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 <

23. [Some Example ProblemsSee back of handout]

24. Percent Change • Where the rubber meets the road! • It all comes together. • Not surprisingly, it’s how the media describes our world.

25. Taking Requests ! What problems do you want to see?

26. 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.

27. Thank You • Websites • faculty.wiu.edu/JR-Olsen/wiu/ • faculty.wiu.edu/JR-Olsen/wiu/BootCamp/landscape/front.html

28. 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?