Julia Aguirre, Ph.D.University of Washington Tacoma

jaguirre@u.washington.edu

Transition Mathematics ProjectSummer Faculty Institute

Leavenworth, WAAugust 24, 2010

Session GoalsIntroduce 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.

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 analysisReflections & Next steps

LUNCH

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?

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).

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Problem-solving

Problem-posing

Conceptual understanding, adaptive reasoning, strategic competence, productive disposition, procedural fluency, problem solving, academic language, math discourse

Critical Knowledge &

Critical Mathematics Knowledge

Funds of knowledge, linguistic knowledge, Informal/everyday mathematics, country of origin school mathematics

Pedagogy ofAccess

Pedagogy of Transformation

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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)

Classical Mathematics Knowledge

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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

Community Knowledge

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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)

Critical Knowledge

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Pedagogy of Access –– Transformation

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

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)

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Pedagogy of Problem-Solving –– Problem-Posing

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

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

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Analyzing Mathematical Tasks  

Papi’s 70th BirthdayA 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)?

Analyze Math Task

Work on the Papi’s birthday problemAnalyze math task for the following

components:Cognitive Demand (high/low)Classical Math Knowledge Community KnowledgeCritical Knowledge

Prepare to summarize main discussion points about: strengths limitations of the task, evidence, and questions/concerns

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Papi’s birthdayMathematical 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)

Culturally Responsive Math Teaching: Lesson Analysis Tool

Intellectual SupportDepth of Knowledge and Student

UnderstandingMathematical AnalysisMathematics Discourse & CommunicationStudent EngagementAcademic 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

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 matesBe prepared to share your rating and

evidence with the whole group

Rethinking Math Teaching: Analyze own lessonRate your math lesson/unit based on the rubric

criteriaProvide 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.

Reflection & Next StepsWhat are some key takeaways from morning

activities about advancing math for all students?In your own coursesAs a department/institution