MATH RIGOR

FACILITATING STUDENT UNDERSTANDING THROUGH PROCESS GOALS

Adapted from VDOE SOL Institutes

GRADE BAND: 6-12Summer 2012

Promoting Students’ Mathematical Understanding

Five goals – for students to1. become mathematical problem solvers who 2. communicate mathematically; 3. reason mathematically;4. make mathematical connections; and5. use mathematical representations to model

and interpret practical situations

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Mathematical Problem Solving

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• Students will apply mathematical concepts and skills and the relationships among them to solve problem situations of varying complexities.

Mathematical Communication

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• Students will use the language of mathematics, including specialized vocabulary and symbols, to express mathematical ideas precisely.

Mathematical Reasoning

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• Students will learn and apply inductive and deductive reasoning skills to make, test, and evaluate mathematical statements and to justify steps in mathematical procedures. Students will use logical reasoning to analyze an argument and to determine whether conclusions are valid.

Mathematical Connections

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• Students will relate concepts and procedures from different topics in mathematics to one another and see mathematics as an integrated field of study.

Mathematical Representations

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• Students will represent and describe mathematical ideas, generalizations, and relationships with graphical, numerical, algebraic, verbal, and physical representations. Students will recognize that representation is both a process and a product.

Triplet Tasks

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Complete the three tasks on your handout. In your group, discuss the following questions:

• What do students need to know to solve each task?

• How are the tasks similar?• How are the tasks different?• What process standards (problem solving,

reasoning, representations, connections, communication) are used?

Examining Differences between Tasks

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What is cognitive demand?

thinking required

What is Rigor?• Google it. • Try

– “instructional rigor”– “rigor and relevance”– “academic rigor”– “rigor mortis”

Task Analysis Guide – Lower-level Demands• Involve recall or memory of facts, rules, formulae, or

definitions• Involve exact reproduction of previously seen material• No connection of facts, rules, formulae, or definitions to

concepts or underlying understandings.• Focused on producing correct answers rather than

developing mathematical understandings• Require no explanations or explanations that focus only

on describing the procedure used to solve

11Adapted from Stein, M.K., Smith, M.S., Henningsen, M.A., & Silver, E.A. (2000). Implementing standars-based mathematics instruction: A casebook for professional development. New York, NY: Teachers College Press

Task Analysis Guide – Higher-level Demands• Focus on developing deeper understanding of concepts• Use multiple representations to develop understanding

and connections• Require complex, non-algorithmic thinking and

considerable cognitive effort• Require exploration of concepts, processes, or

relationships• Require accessing and applying prior knowledge and

relevant experiences to facilitate connections• Require critical analysis of the task and solutions

12Adapted from Stein, M.K., Smith, M.S., Henningsen, M.A., & Silver, E.A. (2000). Implementing standars-based mathematics instruction: A casebook for professional development. New York, NY: Teachers College Press

DOING

Mathematics

Mindstreaming

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How do you choose the problems, tasks or projects that you plan for your students?

What should be considered when selecting tasks?

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• Content Alignment

• Process Standards

• Cognitive Depth

Content AlignmentAlignment is based upon:

• SOL & Curriculum Framework• Essential Understandings• Essential Knowledge & Skills• Essential Questions (middle schools)• Vertical Articulation

• Pacing Guide - HCPS• Purpose - student needs

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Vertical Articulation of Content

Which related prerequisites did students have previously?How will they use this concept next year?

Why is it important knowledge to have?• Consistency• Connections• Relevance• Vocabulary• Building – Not Repeating!

All these lead to deeper understanding andlong-term retention of content 16

Process Standards

Communication:Talking and writing

about math

Problem-Solving

Reasoning

Multiple Representations

Connections

Cognitive Demand…students who performed best on a project assessment designed to measure thinking and reasoning processes were more often in classrooms in which tasks were enacted at high levels of cognitive demand (Stein and Lane 1996), that is, classrooms characterized by sustained engagement of students in active inquiry and sense making (Stein, Grover, and Henningsen 1996). For students in these classrooms, having the opportunity to work on challenging mathematical tasks in a supportive classroom environment translated into substantial learning gains.

---Stein & Smith, 2010

Understanding the “increased rigor” of the new SOL comes through analysis of the SOL and the Curriculum Framework

Characteristics of Rich Mathematical Tasks• High cognitive demand (Stein et. al, 1996; Boaler & Staples, 2008)

• Significant content (Heibert et. al, 1997)

• Require Justification or explanation (Boaler & Staples, in press)

• Make connections between two or more representations (Lesh, Post & Behr, 1988)

• Open-ended (Lotan, 2003; Borasi &Fonzi, 2002)

• Allow entry to students with a range of skills and abilities

• Multiple ways to show competence (Lotan, 2003)

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Thinking About Implementation• In order for students to reason about and

communicate mathematical ideas, they must be engaged with high cognitive demand tasks that enable practice of these skills.

• BUT! … simply selecting and using high-level tasks is not enough.

• Teachers need to be vigilant during the lesson to ensure that students’ engagement with the task continues to be at a high level.

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Let’s Recap!

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Process Standards

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VIRGINIA (Process Goals)

NCTM (Process Standards)

CCSS (Mathematical Practices)

Mathematical Problem Solving

Problem Solving 1) Make sense of problems and persevere in solving them.

Mathematical Communication

Communication 3) Construct viable and critique the reasoning of others

Mathematical Reasoning Reasoning and Proof 2) Reason abstractly and quantitatively

Mathematical Connections Connections 7) Look for and make use of structure8) Look for and express regularity in repeated reasoning

Mathematical Representations

Representations 4) Model with mathematics

    5) Use appropriate tools strategically

    6) Attend to precision

Key Messages• We must not solely focus on multiple-choice assessments• We must provide students with rich, relevant, and

rigorous tasks that focus on more than one specific skill and require application and synthesis of mathematical knowledge

• We must connect mathematics content within and among grade levels and subject areas to facilitate long term retention and application

• We must reflect on our own teaching and resist the urge to blame students

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