1. Fractions Don’t Need to be Fatal

2. Session Goals Participants will understand: Why the Focus on Fractions? What are ‘fractions’? Intro to KNAER (grounded in research and in classrooms) Classroom dynamics

3. Ordering Coffee ½ Cream ¼ Sugar ¼ Coffee

4. Fractions Can Be Fatal… pediatricians, nurses, and pharmacists… were tested for errors resulting from the calculation of drug doses for neonatal intensive care infants… Of the calculation errors identified, 38.5% of pediatricians' errors, 56% of nurses' errors, and 1% of pharmacists' errors would have resulted in administration of 10 times the prescribed dose. (Grillo, Latif, & Stolte, 2001, p.168) (Bruce & Ross, 2009)

5. What the Research Says… The mathematics education literature is resounding in its findings that understanding fractions is a challenging area of mathematics for North American students to grasp (National Assessment of Educational Progress, 2005). Students also seem to have difficulty retaining fractions concepts (Groff, 1996). (Bruce & Ross, 2009)

6. Implications beyond Childhood Adults continue to struggle with fractions concepts (Lipkus, Samsa, & Rimer, 2001; Reyna & Brainerd, 2007) even when fractions are important to daily work related tasks. (Bruce & Ross, 2009)

7. Why a Focus on Fractions? Fractions are a difficult to learn concept because they require deep conceptual knowledge of –part-whole –measurement –ratios the College Math Project identified fractions as an area requiring further attention 7

8. What fraction would you use to represent this?

9. Math For Teaching: Fractions

10. Why use multiple models? Pictures Written Symbols Oral Language Real World Connections Manipulative Models Children who have difficulty translating a concept from one representation to another are the same children who have difficulty solving problems and understanding computations. Strengthening the ability to move between and among these representations improves the growth of children’s concepts.

11. Why Focus on Content Knowledge? “Teachers must understand their subjects deeply and flexibly, and skillfully represent them in intellectually honest ways to a wide range of students.” Deborah Lowenberg Ball and Francesca M. Forzan 11

12. How can you name this? 6 green:4 yellow Part-Part (set) 6 green:10 shapes Part-Whole (set) One fifth of the total area is green Part-Whole (area)

13. Math Knowledge for Teaching Mark Hoover Thames and Deborah Lowenberg Ball 13 CCK – knowing if an answer is correct, knowledge of definitions and procedures PCK – knowing the most useful ways of representing and formulating the subject in order to make it comprehensible to others SCK – knowing more math than CCK but distinct from PCK Horizon Knowledge – ‘mathematical peripheral vision’

14. Some Background Funded by Knowledge Networks for Applied Educational Research (KNAER) Collaborative Action Research Project involving: –Teachers and administrators from KPRDSB, OCDSB, SCDSB –Laurie Moher, Suhana Kadoura, Trish Steele –Dr. Cathy Bruce, Tara Flynn and Rich McPherson of Trent University 14

15. What did it look like in practice? The teams explored student and teacher fraction understanding co-planned, co-implemented, and observed lessons analysed student responses sustained a focus on fractions 15

16. One Task Which way would you cut the cake so that you could share it with a friend fairly. Explain. 16

17. Let’s talk about it Share your thinking. Let’s hear what the students did. 17

18. Fractions Can be Fun! We learned that: –fractions ARE complex –learning is struggle, and sense-making is both challenging and rewarding –learning together alongside colleagues and students is energizing –thinking about thinking results in understandings that extend beyond fractions

19. For you in the near future… a Digital Paper on EduGAINS.ca entitled Professional Learning about Fractions 19