Elaborating RealWorld Contexts in Urban High School Mathematics

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.

Culturally relevant mathematics pedagogy • Teaching mathematics for understanding Connecting concepts, procedures, and representations Grappling with important mathematics • Centering instruction on students’ experiences Mathematizing everyday, familiar, or local contexts Solving abstract problems with familiar representations Creating classroom norms that facilitate broad and sustained participation • Developing critical consciousness about and with mathematics 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)

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

Culturally Relevant Mathematics Pedagogy Real-World Contexts Teaching for Understanding making connections grappling with mathematics Problem-Contexts 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 Analogies

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?

Elaboration • Of problem contexts 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) • As general discursive practice o Amplifying (e. g. , Walqui & van Lier, 2010) o Structure of elaboration (e. g. , Gibbons, 2002).

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

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

RWCs by Topic and Role 100% 90% 3 80% 70% 59 29 60% 50% 27 5 No RWC 2 Central Analogical Incidental 40% 10 18 30% 9 20% 3 10% 4 4 10 8 19 Algebra Total 2 0% Prob/Stat Geometry

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

Elaboration Across Roles • Central – Entry – Exit • Analogies – Entry – Broadening Participation

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

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

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