1. Ours IS the Reason Why! Stop Invert and Multiply! NCSM Annual Conference 2016 Jason Gauthier @jgauthier13 Mathematics Consultant, Allegan, MI Handouts: www.whatsupwithmath.com/ncsm-2016.html

2. “No one has time to do more, but everyone has a responsibility to do things differently.” via @ideaguy42 #edchat #sxswedu@ideaguy42#edchat#sxswedu

3. Why should you stay in the room? ›Justify the need to change the algorithm for division of fractions. ›Doing some math ›Propose a different path to proficiency with division of fractions (grounded in research and the CCSSM). ›Provide an example of a way to build teachers’ understanding of division of fractions without the invert and multiply procedure. ›Talk about division and context ›Talk about resources ›A hypothetical learning trajectory

4. Should we change algorithms? ›What’s wrong with invert and multiply? –Strictly procedural, divorced from sense making –Does not link to (required) representations well –Requires advanced algebraic knowledge to prove –Students misapply and misremember the procedure –Does not link to concepts of multiplication or division ›What’s right with other methods? –Develop from sense making efforts of students –Tend to develop spontaneously from problems –Generalizable from representations –Link to conceptualizations of division and multiplication

5. From the field... ›“... students might overgeneralize this rule to other operations with fractions. Additionally, these mnemonics and sayings do not promote conceptual understanding, making it challenging for students to apply them in a problem-solving context.” (Karp, Bush, & Dougherty, 2015, p. 210-211) ›“‘Invert the divisor and multiply’ is probably one of the most mysterious rules in elementary mathematics. To avoid this mystery, students should first examine division with fractions from a more familiar perspective.” (Van de Walle, Karp, Lovin, & Bay-Williams, 2014, p. 248)

6. From the field... again ›“Do you know why division of fractions is such a nightmare for kids?... without a picture, all we provide our students is a rule about inverting and multiplying that is quickly forgotten, is misused, or just doesn’t help a lot of them.” (Leinwand, 2009, p. 20). ›“An alternative approach involves helping students understand the division of fractions by building on what they know about division of whole numbers.” (NCTM, 2000, p. 219)

7. Characteristics of an Alternative Approach ›Builds upon what students know about the division of whole numbers (NCTM, 2000; NGACBP & CCSSO, 2010; Sharp & Adams, 2002). ›Lends itself to a visual interpretation and to visual modeling of the operation (NCTM, 2000; NGACBP & CCSSO, 2010; Van de Walle, et al, 2014; Fosnot & Dolk, 2002). ›Allows students to make use of multiple representations and Mathematical Practices (NGACBP & CCSSO, 2010; Sharp & Adams, 2002; Leinwand, 2009). ›Connects to the other algorithms and methods students use to operate with fractions (Cramer, et al., 2010; NCTM, 2000). ›Allows students to use it when interpreting both measurement and partition division (Van de Walle, et al, 2014; Tirosh, 2000).

8. Why worry about Fraction Division? ›5.NF.B.7 - Apply and extend previous understandings of division to divide unit fractions by whole numbers and whole numbers by unit fractions. ›6.NS.A.1 - Interpret and compute quotients of fractions, and solve word problems involving division of fractions by fractions, e.g., by using visual fraction models and equations to represent the problem. ›7.NS.A.2 - Apply and extend previous understandings of multiplication and division and of fractions to multiply and divide rational numbers. ›7.NS.A.3 - Solve real-world and mathematical problems involving the four operations with rational numbers. ›Grade 6 is the first time students are required to divide fractions by fractions.

9. Which representations are “required?” ›“Number line” (Grade 3) ›“Visual fraction model” (Grade 3, Grade 4, Grade 5, Grade 6) ›“Equation” (Grade 4, Grade 5, Grade 6) ›“rectangular area” (Grade 5) ›“Relationship between multiplication and division” (Grade 5)

10. Number Lines and Area Models ›Estimate the answer to each problem (DO NOT SOLVE). ›What representation would you choose to use if you had to solve each problem? –Why would you choose that representation? –What are its advantages? Disadvantages? ›How would you phrase each expression as a question? –What is division really asking?

11. “Student” Solutions

12. Do the math! ›The Rules –You CANNOT use invert-and- multiply! –You must represent each problem visually. –You must relate your representation to your solution so that others would understand. ›A tip: –Be mindful of your accounting, and make it visible in your models.

13. Representations...

14. The Big Question... ›In an area model of fraction division, what is the second partitioning in the process actually doing (in a mathematical sense)? ›The second partition (re)produces the dividend as an equivalent fraction with the same denominator as the divisor (i.e. Common Denominators)!

15. Adding a little context... ›Partitive division –Number of groups is known, number in each group is unknown ›Quotative division (measurement) –Number in each group is known, number of groups is unknown

16. Problem String 3 ›Create a context for each problem. ›Make sure you have two partitive and two quotative contexts. ›Solve each problem using our new approach. ›How do your contexts support/conflict with the new approach?

17. Thoughts on classroom use... ›Partitive and quotative division must be addressed. ›Have students devise situations for each of these problems (before or after solving them) ›Concrete-Representational- Abstract continuum ›Use student work samples ›Debate representations –When are they good? –Which is the best? Why? –Are some good for some problems but not for others? Which? How do you tell? ›Pattern blocks, Cuisenaire Rods, fraction circles

18. Hypothetical Learning Trajectory ›Problem String 1 – Representations (Pattern Blocks, Number Lines, fraction strips, Cuisenaire Rods) ›Context – Partitive and Quotative division contexts for Strings 1 and 2 ›Problem String 2 – Representations to consistent approach (common denominators) ›Context – Partitive and Quotative division contexts for String 3 ›Problem String 3 – Common denominator approach connected to representations (area models and number lines)

19. References ›Cramer, K., Monson, D., Whitney, S., Leavitt, S., & Wyberg, T. (2010). Dividing Fractions and Problem Solving. Mathematics Teaching in the Middle School. 15 (6). 338-346. ›Fosnot, C. T. & Dolk, M. (2002). Young mathematicians at work: Constructing fractions, decimals, and percents. Portsmouth, NH: Heinemann. ›Harris, P. W. (2011). Building powerful numeracy for middle and high school students. Portsmouth, NH: Heinemann. ›Hiebert, J., Carpenter, T., Fennema, E., Fuson, K., Human, P., Murray, H., Olivier, A., & Wearne, D. (1996). Problem Solving as a Basis for Reform in Curriculum and Instruction: The Case of Mathematics. Educational Researcher. 25 (4). pp. 12-21. ›Karp, K. S., Bush, S. B., & Dougherty, B. J. (2015). 12 math rules that expire in the middle grades. Mathematics Teaching in the Middle School, 21(4). 208-215. ›Leinwand, S. (2009). Accessible mathematics: 10 instructional shifts that raise student achievement. Portsmouth, NH: Heinemann. ›NCTM. (2000). Principles and standards for school mathematics. Reston, VA: National Council of Teachers of Mathematics. ›National Governors Association Center for Best Practice & Council of Chief State School Officers. (2010). Common core state standards for mathematics. www.corestandards.org.www.corestandards.org ›Sharp, J. & Adams, B. (2002). Children’s constructions of knowledge for fraction division after solving realistic problems. The Journal of Educational Research, 95(6). 333-347. ›Smith, M. S., Bill, V., & Hughes, E. K. (2008). Thinking through a lesson: Successfully implementing high-level tasks. Mathematics Teaching in the Middle School, 14(3). 132-138. ›Tirosh, D. (2000). Enhancing prospective teachers’ knowledge of children’s conceptions: The case of division of fractions. Journal for Research in Mathematics Education, 31(1). 5-25. ›Van de Walle, J. A., Karp, K. S., Lovin, L. H., & Bay-Williams, J. M. (2014). Teaching student-centered mathematics: Developmentally appropriate instruction for grades 3-5. Vol. 2. (2 nd ed.). Pearson.

20. Thank you! jgauthier@alleganaesa.org @jgauthier13 www.whatsupwithmath.com