Fractions Addressing a Stumbling Block for Developmental Students

1. FractionsAddressing a Stumbling Block for Developmental Students Wade Ellis, Jr. West Valley College (retired)

2. A Problem There are 135 students in a class. There are 25% more boys than girls. How many boys and how many girls are in the class?

3. Possible Solutions

4. A Problem There are 135 students in a class. There are 25% more boys than girls. How many boys and how many girls are in the class? 100% 135 5/9 5:4 4/9 1/9 55.5% 44.4% 1% 1.35

5. Outline • Setting the stage • AMATYC Crossroad and Beyond Crossroads • MathAMATYC Educator • What is a Fraction? and Equivalent Fractions • Using Technology ― Action/Consequence Principle • Questions that Advance Student Learning • A Progression for Learning Fractions • Ratios and Proportions & Percents • Comments and Suggestions

6. 1988 NCTM Yearbook on Algebra Common Mistakes in Algebra(Marquis, 1988) 10 of 22 were related to fractions

7. Learning Fractions • If you are training someone to be a retail clerk, and you believe that that person will never need to know much more math than a retail clerk knows, then you can teach fractions using standard algorithms for doing common fraction problems.  But, if you think that the person you are teaching might need to know more advanced mathematics later, then you should teach fractions in a different way.  Jim Pellegrino Distinguished Professor of Cognitive Psychology at the University of Illinois at Chicago

8. Learning Fractions (Cont’d) • In math, you can teach arithmetic by simply teaching the most efficient arithmetical algorithms or you can teach it in a way that greatly facilitates the learning of algebra – so you understand the idea of equivalence . . . , not just what you need to do to execute procedures.  . . . Research shows what kids understand and what they don’t understand depends very much on how we teach the material. Jim Pellegrino

9. Crossroads in Mathematics • First, technology can be used to aid in the understanding of mathematical principles. • Second, students will use technology naturally and routinely as a tool to aid in the solution of realistic problems.

10. Beyond Crossroads • Inquiry. Effective mathematics instruction should require students to be active participants. Students learn through investigation. Advances in neuroscience confirm that students’ active involvement in learning mathematics is important in the process of building understanding and modifying the structure of the mind.

11. James Stigler in the MathAMATYC Educator • Students who have failed . . .[might succeed]if we can first convince them that mathematics makes sense . . . • . . . key concepts in the mathematics curriculum . . . included comparisons of fractions, placement of fractions on the number line, operations with fractions/decimals/percents, ratio, . . . • . . . the ability to correctly remember and execute procedures . . . is a kind of knowledge that is fragile without deeper conceptual understanding of fundamental mathematical ideas. • Finally, when students are able to provide conceptual understanding, they also produce correct answers.

12. Technology: What’s a Fraction?

13. Technology: Equivalent fractions

14. Fractions in the Common Core • Grade 3 • Develop understanding of fractions as numbers. • Grade 4 • Extend understanding of fraction equivalence and ordering. • Build fractions from unit fractions. • Understand decimal notation for fractions, and compare decimal fractions. • Grade 5 • Use equivalent fractions as a strategy to add and subtract fractions. • Apply and extend previous understandings of multiplication and division. • Grade 6 • Apply and extend previous understandings of multiplication and division to divide fractions by fractions.

15. CCSS Mathematical Practices • Make sense of problems and persevere in solving them. • Reason abstractly and quantitatively. • Construct viable arguments and critique the reasoning of others. • Model with mathematics. • Use appropriate tools strategically. • Attend to precision. • Look for and make use of structure. • Look for and express regularity in repeated reasoning.

16. A Progression for Learning Fractions (Prof. Wu) • What is a Fraction?* • Equivalent Fractions* • Fractions and Unit Squares • Creating Equivalent Fractions • Adding & Subtracting Fractions with Common Denominators* • Adding Fractions with Unlike Denominators • Fractions as Division • Mixed Numbers* • Multiplying Whole Numbers and Fractions • Fraction Multiplication* • Dividing a Fraction by a Whole Number • Division of Whole Numbers by a Fraction • Dividing a Fraction by a Fraction • Units Other Than Unit Squares • Comparing Units

17. What is a Fraction? • Teacher Guidance Document • I. The Mathematical Focus References Common Core Standards Covered • II. About the File • III. Possible Objectives • IV. Sample Questions

18. Engaging in a concrete experience Observing reflectively Developing an abstract conceptualization based upon the reflection Actively experimenting/testing based upon the abstraction People learn by Zull, 2002

19. Conceptual Knowledge: • Makes connections visible, • enables reasoning about the mathematics, • less susceptible to common errors, • less prone to forgetting. • Procedural Knowledge: • strengthens and develops understanding • allows students to concentrate on relationships rather than just on working out results NRC, 1999; 2001

20. Take an action on a mathematical object Observe the mathematical consequences and Reflect on the mathematical implication of those consequences Conceptual Understanding

21. Action Consequence Principle Interactive Dynamic Technology

22. Dynamic Interactive Technology: Action Consequence Principle Students • take an action on a mathematical object, • observe the consequences of that action, and • reflect on the mathematical implications of those consequences Burrill & Dick, 2008

23. A/C documents & Learning Take an action on a mathematical object Observe the consequences Reflect on the mathematical implications Engage in concrete experience Observe reflectively Develop abstract conceptualization Experiment and test concepts

24. Technology as aTool for Developing Understanding Key is asking good questions Predict consequence in advance of action (what would happen if…?) Consider action that would produce a given consequence (what would make … happen?) Conjecturing/Testing/Generalization (When…?) Justification (Why…?)

25. What teachers do: • The only reasons to ask questions is to:(Black et al., 2004) • Probeto uncover students’ thinking • discover misconceptions that exist • Pushto advance students’ thinking • make connections • justify or prove their thinking

26. Possible Questions • Handout

27. What is a Ratio?

28. Ratios and Fractions

29. Ratio Table

30. Ratio and Slope

31. Percent

32. Questions for What Is a Fraction? Describe where three fifths will be. How will three fifths differ from seven fifths? Explain your thinking, then check your answer using the tns file. Where will 4/8 be? b) 0/8? c) Is eleven eighths closer to one or to two? How do you know? If the number of 1/5’s is larger than the 5, what can you say about the size of the fraction? Explain. Suppose the unit fraction was 1/5 and the numerator was between 11 and 14. Where is the fraction? If the unit fraction were 1/6, where would fractions with a numerator between 25 and 29 be?

33. Questions for What Is a Fraction? (cont’d) How many copies of ½ are in 2? Use the file to make a conjecture about whether the following sentences are correct. a) 0 is a fraction. b) A whole number cannot be a fraction. c) A fraction can have many names.

34. Problem • At a dance, 2/3 of the girls dance with 3/5 of the boys. What proportion of the students are dancing?

35. What does fraction as a point on a number line buy us? • A constant way to think: k/p is k copies of 1/p - the length of the concatenation of k segments each of which has length 1/p. • Behavior similar to whole numbers: • k/3 is a multiple of 1/3 • Larger fraction is to the right on the number line • Connection of whole number to fractions. • One number has many names and none more important than another. • No difference between proper and improper fractions

36. Closing Discussion • Questions • Comments

37. References • Burrill, G. & Dick, T. (2008). What state assessments tell us about student achievement in algebra. Paper presented at NCTM 2008 Research Presession • Dick, T. & Burrill, G. (2009). Technology and teaching and learning mathematics at the secondary level: Implications for teacher preparation and development. Presentation at the Association of Mathematics Teacher Educators, Orlando FL. • National Research Council. (1999). How People Learn: Brain, mind, experience, and school. Bransford, J. D., Brown, A. L., & Cocking, R. R. (Eds.). Washington, DC: National Academy Press • Zull, J. ( 2002). The Art of Changing the Brain: Enriching the Practice of Teaching by Exploring the Biology of Learning. Association for Supervision and Curriculum Development, Alexandria, Virginia.

38. References • What Does it Really Mean to be College and Work Ready?: The Mathematics Required of First Year Community College Students, National Center on Education and the Economy, 2013. ,

39. wellis@ti.com