Four Features of a Productive Classroom Culture 1. Ideas are the currency of the classroom 2. Students have autonomy with respect to the methods used to solve problems. 3. The classroom culture exhibits an appreciation for mistakes as opportunities to learn. 4. The authority for reasonability and correctness lies in the logic and structure of the subject, rather than in the social status of the participants. . Ideas, expressed by any participant, have the potential to contribute to everyone’s learning and consequently warrant respect and response. Students must respect the need for everyone to understand their own methods and must recognize that there are often a variety of methods that will lead to a solution. Copyright © 2010 by Pearson Education, Inc. All rights reserved.

Learning Theory: Implications for Instruction 1. Build new knowledge from prior knowledge 2. Provide opportunities to talk about mathematics 3. Build in opportunities for reflection 4. Encourage multiple approaches 5. Treat errors as opportunities for learning 6. Scaffold new content 7. Honor diversity

Mathematics Proficiency The five “strands” of mathematics proficiency (NRC, 2001): Conceptual Understanding – comprehension of mathematical concepts, operations, and relations Procedural Fluency – skill in carrying out procedures flexibly, accurately, efficiently, and appropriately Strategic Competence – ability to formulate, represent, and solve mathematical problems Adaptive Reasoning – capacity for logical thought, reflection, explanation, and justification Productive Disposition – habitual inclination to see mathematics as sensible, useful, and worthwhile, coupled with a belief in diligence and one’s own efficacy The five process standards (NCTM, 2000): Problem Solving Reasoning and Proof Communication Connections Representations

Diversity in the Math Classroom

Ways to Ensure Equitable Teaching Highly qualified teacher Examining achievement gaps vs. instructional or expectation gaps “[H]igh expectations, respect, understanding, and strong support for all students” (NCTM, 2008) Copyright © Allyn and Bacon 2010

Creating Equitable Instruction It is not enough to provide equal opportunity for all students to learn math It is being sensitive to individual differences It is treating students fairly and impartially It is examining your beliefs about students’ abilities to learn (especially those in poverty) Copyright © Allyn and Bacon 2010

Mathematics for All Children (Diversity in Today’s Classroom) Diversity includes students who are: Identified as having a specific learning disability From different cultural backgrounds English language learners Mathematically gifted Copyright © Allyn and Bacon 2010

Mathematics for All Children (Tracking and Flexible Grouping) Is responsible for lower expectations for students in the “slow” track Frequently denies students access to challenging materials For “slower” tracks is often remedial drill Exaggerates differences instead of bridging them Makes it almost impossible to move to a higher track Does not benefit higher-achieving students Copyright © Allyn and Bacon 2010

Copyright © Allyn and Bacon 2010 Mathematics for All Children (Instructional Principles for Diverse Learners) Learning with understanding is based on connecting and organizing knowledge around big conceptual ideas Learning builds on what students already know Instruction in school takes advantage of the children’s informal knowledge of mathematics Don’t forget about accommodations and modifications Copyright © Allyn and Bacon 2010

Copyright © Allyn and Bacon 2010 Providing for Students with Special Needs Response to Intervention (RTI) Source: Scott, T., and Lane, H. (2001). Multi-Tiered Interventions in Academic and Social Contexts. Unpublished manuscript, University of Florida, Gainesville. Copyright © Allyn and Bacon 2010

RTI: Common Features Across All Tiers Research-based practices Data-driven Instructional Context-specific Copyright © Allyn and Bacon 2010

Students with Mild Disabilities Students in Tier 3 May Have Difficulty with: Memory General strategy use Attention Ability to speak or express ideas Perception of auditory, visual, or written information Integration of abstract ideas Copyright © Allyn and Bacon 2010

Research-Based Strategies (to Be Used with Tier 3 Students) Explicit strategy instruction Peer-assisted learning Student think-alouds Copyright © Allyn and Bacon 2010

Modifications and Accommodations (for Tier 3 Students) Before Structure the environment Identify potential barriers During Provide clarity After Consider alternative assessments Emphasize practice and summary Copyright © Allyn and Bacon 2010

Students with Significant Disabilities Students are expected to learn the mathematical content based on the NCTM standards Students need the content connected to real-life skills and possible features of jobs Not all facts must be mastered before progressing further in the curriculum Copyright © Allyn and Bacon 2010

Copyright © Allyn and Bacon 2010 Additional Strategies for Supporting Students with Moderate and Severe Disabilities Systematic instruction Visual supports Response prompts Task chaining Problem solving Self-determination and independent self-directed learning Copyright © Allyn and Bacon 2010

Strategies for Teaching Mathematics for ELLs Write and state the content and language objectives Build background Encourage use of native language Comprehensible input Explicitly teach vocabulary Plan cooperative/interdependent groups to support language Create partnerships with families Copyright © Allyn and Bacon 2010

Working Toward Gender Equity Although there is no discrepancy in boys’ and girls’ math scores, we need to be aware of and address gender equity in the classroom Many more males enter into graduate-level fields with a heavy emphasis on math than do females Copyright © Allyn and Bacon 2010

Copyright © Allyn and Bacon 2010 Gender Inequity Possible Causes Belief systems related to gender Teacher interactions and gender Possible Solutions Awareness Involve all students Copyright © Allyn and Bacon 2010

Reducing Resistance and Building Resilience Give children choices and capitalize on their unique strengths Nurture traits of resilience Demonstrate an ethic of caring Make mathematics irresistible Give students some leadership in their own learning Copyright © Allyn and Bacon 2010

Providing for Students Who Are Mathematically Gifted Strategies to Avoid More of the same Allowing free time when they complete their work Routinely assigning them to teach other students Strategies to Incorporate Acceleration Enrichment Sophistication Novelty Copyright © Allyn and Bacon 2010

Copyright © Allyn and Bacon 2010 Final Thoughts Identify current knowledge and build upon it Push students to high-level thinking Maintain high expectations Use a multicultural approach Recognize, value, explore, and incorporate the home culture Use alternative assessments Measure progress over time Promote the importance of effort and resilience Copyright © Allyn and Bacon 2010

Integrating Assessment into Instruction Assessment should enhance student learning Assessment is a valuable tool for making instructional decisions Copyright © Allyn and Bacon 2010

Copyright © Allyn and Bacon 2010 Why Do We Assess? To monitor student progress To make instructional decisions To evaluate student achievement To evaluate programs Source: Adapted from NCTM, Assessment Standards for School Mathematics, 1995, p. 25. Used with permission. Copyright © Allyn and Bacon 2010

Thoughts about Assessment Tasks In some instances, the real value of the task will come in the discussion that follows Explanations need to be a regular practice in every classroom Copyright © Allyn and Bacon 2010

Rubrics and Performance Indicators Scoring: Comparing students’ work to criteria or to rubrics that describe what we expect the work to be Grading: The result of accumulating scores and other information about students’ work for the purpose of summarizing and communicating to others Rubric: A framework that can be designed or adapted by the teacher for a particular group of students or particular math task, using a three- to six-point scale to rate performance Copyright © Allyn and Bacon 2010

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Copyright © Allyn and Bacon 2010 Observation Tools Anecdotal notes Observation rubric Checklists for individuals Checklists for full class Copyright © Allyn and Bacon 2010

Copyright © Allyn and Bacon 2010 Tests Will always be a part of assessment Do not have to test low-level skills Can be designed to assess understanding of concepts Should go beyond just knowing how to perform an algorithm Should allow and require a student to demonstrate a conceptual basis for the process Copyright © Allyn and Bacon 2010