1. Elaborating Real-World Contexts in Urban High School Mathematics Lessons Haiwen Chu Laurie Rubel City University of New York This material is based on work supported by the National Science Foundation under grant no. 0742614. Any opinions, findings, and conclusions or recommendations expressed in this material are those of the authors and do not necessarily reflect the views of the National Science Foundation.

2. Culturally relevant mathematics pedagogy Teaching mathematics for understanding Centering instruction on students’ experiences Developing critical consciousness about and with mathematics Connecting concepts, procedures, and representations Grappling with important mathematics Mathematizing everyday, familiar, or local contexts Solving abstract problems with familiar representations Creating classroom norms that facilitate broad and sustained participation Addressing issues of power as objects of mathematical analysis Inviting students to be critical about mathematics (Ladson-Billings, 1995; Gutstein et al., 1997)  (Rubel & Chu, 2012)

3. Mathematics Real-World Contexts As central problem-context As analogy or metaphor

4. Culturally Relevant Mathematics Pedagogy Teaching for Understanding making connections grappling with mathematics Centering on Students everyday or local contexts familiar representations broadening participation Developing Criticality with and about math addressing issues with math being critical about math Real-World Contexts Problem-Contexts Analogies

5. Research Questions 1.To what extent and in what ways do mathematics lessons provide opportunities for students to connect mathematics to RWCs? a.With which mathematical topics are RWCs associated? b.What roles do RWCs play in mathematics lessons? 2.How can elaboration of RWCs support culturally relevant mathematics pedagogy?

6. Elaboration Of problem contexts As general discursive practice o Narrative and paradigmatic modes (Chapman, 2006) o Entering and exiting (Depaepe et al., 2010) o “Contextual features” of story problems (Jackson et al., 2011) o Amplifying (e.g., Walqui & van Lier, 2010) o Structure of elaboration (e.g., Gibbons, 2002).

7. Identifying Elaboration Explicitly discussing, including: – Exploring – Interpreting – Explaining – Connecting – Supplying examples or counterexamples

8. RWCs by Topic and Role TOPICS Geometry Algebra Probability & Statistics ROLES Analogical Central Incidental

9. RWCs by Topic and Role

10. Focusing in on Elaboration All Lessons (106) High Cognitive Demand (30) RWC present (14) Elaboration Present (10) Analogies (5)Central (5)

11. Elaboration Across Roles Central – Entry – Exit Analogies – Entry – Broadening Participation

12. Central Problem Contexts Addressing the underlying mathematical structure Taking realistic considerations into account Entry Identifying components of slope Maximizing fence enclosure Exit Symbolic formulas for slope Units for volume

13. Analogies Entry – “middles”, balance, and center – Negative space images Broadening Participation – Extended support for reasoning through a definition.

14. Implications For Practice For Research Incorporating analogies and elaboration. Focus on balancing entry and exit. Content and discourse analysis of elaboration. Rubrics and ratings for analogies and elaboration.