Julia Aguirre, Ph.D. University of Washington Tacoma jaguirre@u.washington

1. Rethinking Transition Mathematics for Advancement: A Teaching Analysis Tool for Lesson Planning and Practice Julia Aguirre, Ph.D. University of Washington Tacoma jaguirre@u.washington.edu Transition Mathematics Project Summer Faculty Institute Leavenworth, WA August 24, 2010

2. Session Goals • Introduce a math teaching framework and student learning outcomes that promote advancement in math competence, confidence, and equity. • Introduce a teaching tool to analyze and enhance math instruction to support mathematics advancement of all students.

3. Overview: • 9:00-10: 30 • ACTIVITY 1: Framing the issues for mathematical advancement • ACTIVITY 2: Analyzing teaching from multiple dimensions • 10:30-11:00 • BREAK • 11:00-12:30 • ACTIVITY3: Analyzing our own teaching practice – Lesson plan analysis • Reflections & Next steps • LUNCH

4. Group Share – Poster 1 • What are some reasons you have heard of (from media, research, colleagues) that explain why some students do well in mathematics and others struggle?   • Any particular areas of mathematics that come to mind? • Any particular demographic groups?

5. Learning Outcomes • Students learn that mathematics is an essential analytical tool to understand complex issues/problems and potentially change the world. • Students deepen their mathematical understanding and skills through analyzing complex social issues and problems that are important to them and their community. • Students become more motivated to learn and engage with important rich mathematics. • Students develop a intellectual and cultural competence that enable them to maintain their cultural integrity while succeeding academically, particularly in mathematics. Greer et al (2009); Gutierrez (2007, 2009); Gutstein (2006); Gutstein & Peterson (2005).

6. Teaching MathematicsFramework (Aguirre, 2009) Pedagogy of Access Problem-solving Classical Mathematics Knowledge Community Knowledge Conceptual understanding, adaptive reasoning, strategic competence, productive disposition, procedural fluency, problem solving, academic language, math discourse Funds of knowledge, linguistic knowledge, Informal/everyday mathematics, country of origin school mathematics Critical Knowledge & Critical Mathematics Knowledge Critical Knowledge Pedagogy of Transformation Problem-posing

7. Classical Mathematics Knowledge Mathematical power(core mathematical ideas, conceptual understanding, procedural fluency, problem-solving; standards; academic language; mathematical discourse) Passing the gates(standardized tests, high school graduation, college, etc)

8. Community Knowledge Informal Math Knowledge/Funds of Knowledge: people have and produce math knowledge outside of school tied to specific cultural/community practices (e.g. household activities, commerce activities, tiendas, games) Formal Math Knowledge: people have and produce formal math knowledge within schools that is culturally constructed (e.g. symbolic notation, algorithms; mathematical discourse). ABC ABC <ABC

9. Critical Knowledge Critical Mathematical Knowledge: To use mathematics as an analytical tool to understand power relations, decisions, social issues and sociopolitical context of reality. To use mathematics to foster positive change and/or take action to challenge injustice. Critical Knowledge in General: Knowledge beyond mathematics needed to understand the sociopolitical context.(e.g.multiple histories; structures, policies, and practices that create equity and inequity in society)

10. Pedagogy of Access –– Transformation Beyond access to investigate, challenge and change institutional structures, policies, and practices that may perpetuate inequity (e.g. low cognitive demand curriculum, student tracking and placement practices, resource allocation) Access to classical math knowledge, high cognitive demand tasks, academic language, and discourse practices is key to advancement in mathematics Access to community knowledge as a resource to learn rich and rigorous mathematics Access to high expectations, high quality mathematics content, and strong student-teacher relationships

11. Pedagogy of Problem-Solving –– Problem-Posing Problems are derived from learners and their contexts (i.e. authentic problems; issues that affect them and increasingly compel them to respond and change.) Shared intellectual authority; co-investigators Teacher plays an active role in helping to mathematize those contexts Connects explicitly to critical knowledge and guides transformative inquiry and action Adaptive reasoning, strategic competence (NRC, 2001) “is learning to grapple with new and unfamiliar tasks when relevant solution methods (even if only partially mastered) are not known.” (Schoenfeld, 1992) Challenges traditional role of teacher as sole intellectual authority - (i.e. knows all the answers). Requires flexibility with uncertainty and “experience, confidence, and self-awareness” on part of the teacher

12. Analyzing Mathematical Tasks Papi’s 70th Birthday A true story It was Señor Aguirre’s 70th Birthday. His three children wanted to throw him a big party to celebrate. The hall rental, mariachi, food, and decorations will cost a total of $4,500. The brother, a special medical doctor (anesthesiologist) who makes about $20,000 per month, suggested that the three children split the cost equally. One of the sisters, a university professor who makes about $6,000 per month, said that would not be fair. She suggested the following: the brother pays $3150. She would pay $900, and the other sister, a partner in the family business and single mom with 2 boys who makes about $3000 per month, should pay $450. TASK*: Write a position statement using mathematical evidence (e.g. proportions, ratios, percent) to support your conclusion to the following questions: •Which person do you agree with and why? •What is fair in this situation? •Can you think of an alternative financial arrangement that might be better (more fair)?

13. Analyze Math Task • Work on the Papi’s birthday problem • Analyze math task for the following components: • Cognitive Demand (high/low) • Classical Math Knowledge • Community Knowledge • Critical Knowledge • Prepare to summarize main discussion points about: strengths limitations of the task, evidence, and questions/concerns

14. Papi’s birthday • Mathematical leverage (additive/absolute to multiplicative/ relative thinking) • For middle school students • Pre-service teachers • Familiar context that is culturally grounded • Explores facets of mathematical and social conceptions of fairness • High cognitive demand activity (Stein et al, 2000) • Task is offered in two languages (e.g. English, Spanish)

15. Culturally Responsive Math Teaching: Lesson Analysis Tool • Intellectual Support • Depth of Knowledge and Student Understanding • Mathematical Analysis • Mathematics Discourse & Communication • Student Engagement • Academic Language Support for ELL • Use of L1 (home language) • Use ESL scaffolding strategies • Funds of Knowledge/Culture/Community Support • Use of Critical knowledge/Power/Social Justice

16. Video Lesson Analysis: Division of Fractions • Use the rubric to rate the lesson 1-5 on a specific dimension • Provide evidence from the lesson to support your rating • Discuss your rating with your table mates • Be prepared to share your rating and evidence with the whole group

17. Rethinking Math Teaching: Analyze own lesson • Rate your math lesson/unit based on the rubric criteria • Provide specific evidence from your lesson to support your rating. • Reflect on Activity: • What are the strengths and limitations of your lesson according to the rubric? • What strategies or areas would you like to strengthen as a result of this analysis? Give an example of how you might strengthen one area (this can be in this lesson or in subsequent lessons). • How does this analysis help, if at all, your math lesson planning process to meet the math learning needs of your students? • Is there anything you would change about the rubric in relation to helping you facilitate mathematics learning of your students? Why.

18. Reflection & Next Steps • What are some key takeaways from morning activities about advancing math for all students? • In your own courses • As a department/institution