1. Fraction Applets for Developmental Mathematics Students Wade Ellis West Valley College (retired) wellis@ti.com

2. Fraction Applets for Developmental Mathematics Students Developmental Mathematics students have problems learning algebra based in part on their misconceptions about fractions and fraction operations. This presentation will demonstrate the instructional use of fraction and proportional reasoning applets along with inquiry questions that enhance and deepen student understanding of fractions sufficient to improve student performance in algebra.

3. A Learning Trajectory Algebra Fractions Ratios Rates Coordinate Axes

4. Outline Introduction Procedures and Understanding Research Basis for an Approach to Fractions Examples of Applets (Lua and Nspire) Basic Ratio Concepts Require Fractions Research Basis for Applet & Activity Development Examples of Activities for Applets Comment and Questions

5. 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. 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 Jim Pellegrino Distinguished Professor of Cognitive Psychology Cognitive Psychology University of Illinois at Chicago University of Illinois at Chicago

6. 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 Jim Pellegrino

7. James Stigler: UCLA Psychology Dept. in May 2011 Math AMATYC 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.

8. James Stigler: UCLA Psychology Dept. Author of The Learning Gap 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.

9. Research Basis for an Approach to Fractions

10. Fraction Constructs (Behr, Lesh, Post and Silver, 1983)

11. Fractions According to Prof. Wu A fraction is a point on the number line Unit fractions are emphasized (1/b is a unit fraction, b is a positive integer) (1/b is a unit fraction, b is a positive integer) Common denominators Improper fractions presented long before mixed numbers

12. Fraction Applets What is a Fraction? Creating Equivalent Fractions Adding and Subtracting Fractions with Common Denominators Fractions and Unit Squares Adding Fractions with Unlike Denominators Division of a Fraction by a Fraction*

13. Fraction Applets

14. Basic Ratio Concepts Require Fractions

15. Basic Concepts 1. Ratio as a ratio relationship between two quantities 2. Ratio and rate – ratio as a relationship – rate as a fraction with units 3. Unit rate b/a associated with a ratio a:b with a ≠ 0 4. Equivalent ratios 5. Percent of a quantity as a rate per 100 6. Constant of proportionality (unit rate) in tables, graphs, equations, diagrams, and verbal descriptions

16. Ratio of Quantities a:b Part to Whole Part to Part Fraction Number Point on the Number Line Length Area Percent Unit Rate Proportion Rate

17. What do we gain? Ratios are much more than just a different notation for fractions; ratios communicate a relationship between quantities The story emphasizes paired quantities over and over. Paired quantities leads naturally to graphs and to proportional relationships The constant of proportionality relates to both the graphical idea of slope, the physical idea of rate, geometrical notion of scaling Emphasis on variety of strategies to solve ratio/proportion problems (ratio tables, double number lines, graphs, …)

18. Algebraic Use of Fractions and Ratios Students use fractions and ratios in algebra when they study similar figures, slopes of lines, solving linear equations and proportional reasoning problems (and later when they study sine, cosine, tangent, and other trigonometric ratios in high school).

19. Research Basis for Applet & Activity Development

20.  Evidence from many sources suggests students often do not understand fundamental mathematical concepts. Our hypothesis:  Consider another approach rather than continuing what has been unsuccessful for many  Use interactive dynamic technology to support the development of understanding, especially of “tough to teach/tough to learn” fundamental concepts. Building Concepts Burrill, Dick, & Ellis, 2013

21. 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

22. As a tool for doing mathematics - a servant role to perform computations, make graphs, … As a tool for developing or deepening understanding of important mathematical concepts The Role of Technology Dick & Burrill, 2009

23. Principles for effectively integrating interactive dynamic technologies in the classroom: The Action-Consequence Principle The Questioning Principle The Reflection Principle Burrill, Dick, & Ellis, 2013

24. 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

25. Focus of an Activity On fundamental concepts One or two ideas per activity Follow a learning trajectory supported by research Recognize student misconceptions/difficulties

26. Ratio Activity Examples What is a Ratio? Ratio Tables Building a Table of Ratios Connecting Ratios to Equations Variables and Expressions

27. Ratio Applets

28. Teacher Notes The Mathematical Focus of the Activity Objectives of the Activity About the Applet Sample Questions

29. A Ratio Problem 7. Suppose the ratio was 5 to 3. If there were a total of 120 circles and squares, how many squares would there be? Explain how you found your answer.

30. An Algebraic Solution to A Ratio Problem

31. The Learning Trajectory Algebra Fractions Ratios Rates Coordinate Axes 3 rd Grade 5 th Grade 7 th Grade What you teach. How you teach it.

32. 1.What is a Fraction? 2.Equivalent Fractions 3.Fractions and Unit Squares 4.Creating Equivalent Fractions 5.Adding & Subtracting Fractions with Common Dens. 6.Adding Fractions with Unlike Denominators 7.Fractions as Division 8.Mixed Numbers Building Concepts: Fractions 9.Multiplying Whole Nos. and Fractions 10.Fraction Multiplication 11.Dividing a Fraction by a Whole Number 12.Division of Whole Numbers by a Fraction 13.Dividing a Fraction by a Fraction 14.Units Other Than Unit Squares 15.Comparing Units

33. Building Concepts: Ratios 1. Introduction to Ratios 2. Introduction to Rates 3. 3.Building a Table of Ratios 4. Ratio Tables 5. Comparing Ratios 6. Connecting Ratios and Fractions 7. Double Number Line 8. Connecting Ratio to Rate of Change 9. Adding Ratios 10. Proportions 11. Proportional Relationships 12. Solving Proportions 13. Ratio and Scaling 14. Ratio and Similarity

34. Questions and Comments wellis@ti.com Wu, H. (2011). Understanding Numbers in Elementary School Mathematics, American Mathematical Society. http://www.ams.org/bookstore-getitem/item=mbk-79 http://www.ams.org/bookstore-getitem/item=mbk-79 www.education.ti.com

35. Additional Material

36. Understand ratio concepts and use ratio reasoning to solve problems. 1. Understand the concept of a ratio and use ratio language to describe a ratio relationship between two quantities. CCSSM, 2010 What is a ratio?

37. a. Make tables of equivalent ratios relating quantities with whole number measurements, find missing values in the tables, and plot the pairs of values on the coordinate plane. Use tables to compare ratios. CCSSM, 2010 Ratio Tables/ Connection to Graphs

38. From ratios to rates to proportions Building Concepts: Ratios, 2014

39. b. Identify the constant of proportionality (unit rate) in tables, graphs, equations, diagrams, and verbal descriptions of proportional relationships. c. Represent proportional relationships by equations. d. Explain what a point (x, y) on the graph of a proportional relationship means in terms of the situation, with special attention to the points (0, 0) and (1, r) where r is the unit rate. CCSSM, 2010 Proportions

40. Connection to Geometry They can apply a scale factor that relates lengths in two different figures, or they can consider the ratio of two lengths within one figure, find a multiplicative relationship between those lengths, and apply that relationship to the ratio of the corresponding lengths in the other figure. When working with areas, students should be aware that areas do not scale by the same factor that relates lengths.

41. References Behr, M., Lesh, R., Post, T., & Silver E. (1983). Rational Number Concepts. In R. Lesh & M. Landau (Eds.), Acquisition of Mathematics Concepts and Processes, (pp. 91-125). New York: Academic Press. Building Concepts: Ratios & Proportional Reasoning. (2014). Texas Instruments Education Technology. education.ti.com Burrill, G., Dick, T., & Ellis, W. (2013). Design principles for interactive learning technology. Presentation at Learning Forward. Dallas TX Chance, B., Ben-Zvi, D., Garfield, J., & Medina, E. (2007). The role of technology in improving student learning in statistics.” Technology Innovations in Statistics Education 1. pp. 1-26. Retrieved from: http://eschlarship.org/uc/item/8sd2t4rrhttp://eschlarship.org/uc/item/8sd2t4rr Charalambos, Y., & Pitta-Pantazi, D. (2005). Revisiting a theoretical model on fractions: Implications for teaching and research. 2005. In Chick, H. L. & Vincent, J. L. (Eds.). Proceedings of the 29th Conference of the International Group for the Psychology of Mathematics Education, Vol. 2, pp. 233-240. Melbourne: PME Common Core Standards. College and Career Standards for Mathematics 2010). Council of Chief State School Officers (CCSSO) and (National Governor’s Association (NGA). delMas, R., Garfield, J., & Chance, B. (1999) A model of classroom research in action: Developing simulation activities to improve students’ statistical reasoning. Journal of Statistics Education, 7(3), www.amstat.org/publications/jse/secure/v7n3/delmas.cfm www.amstat.org/publications/jse/secure/v7n3/delmas.cfm

42. 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. Empson, S., & Knudsen, J. (2003). Building on children’s thinking to develop proportional reasoning. Texas Mathematics Teacher, 2, 16–21. Johanning, D. (2008). Learning to use fractions: Examining middle school students’ emerging fraction literacy. Journal for Research in Mathematics Education. 39(3), 281-310. Lamon, S. (1999). Teaching fractions and ratios for understanding: Essential content knowledge and instructional strategies for teachers. Hillsdale, NJ: Erlbaum. Lamon, S. (2007). Rational numbers and proportional reasoning: Toward a theoretical framework for research. In K. Lester Jr. (Ed.), Second handbook of research on mathematics teaching and learning (pp. 629–667). Charlotte, NC: Information Age Publishing. Lane, D. M., & Peres, S. C. (2006). Interactive Simulations in the Teaching of Statistics: Promise and Pitfalls. In A. Rossman and B. Chance (Eds.), Proceedings of the Seventh International Conference on Teaching Statistics. [CD-ROM]. Voorburg, The Netherlands: International Statistical Institute. National Assessment for Educational Progress (2013). Released Item. Grade 8. National Center for Educational Statistics.

43. National Research Council (2001). Adding It Up. Kilpatrick, J., Swafford, J., & Findell, B. (Eds.) Washington DC: National Academy Press. Also available on the web 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. www.nap.edu. www.nap.edu Program for International Assessment (PISA). 2012 Release items. Organization for Economic Co-operation and Development. http://www.oecd.org/pisa/pisaproducts/pisa2012-2006-rel-items-maths-ENG.pdf Progressions for the Common Core State Standards in Mathematics (2011). 6-7, Ratio and Proportional Reasoning. Singh, P. (2000). Understanding the concepts of proportion and ratio constructed by two grade six students, Educational Studies in Mathematics, 43(3), 271-292. Wu, H. (2011). Understanding Numbers in Elementary School Mathematics, American Mathematical Society. http://www.ams.org/bookstore-getitem/item=mbk-79 James Zull, ( 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.