Math rigor facilitating student understanding through process goals GRADE BAND: 6-12 Summer 2012 Slides 3-8: 5 minutes Adapted from VDOE SOL Institutes

Promoting Students’ Mathematical Understanding Five goals – for students to become mathematical problem solvers who communicate mathematically; reason mathematically; make mathematical connections; and use mathematical representations to model and interpret practical situations 5 Virginia Process Standards: “Mathematical” Problem Solving, Reasoning, Communication, Connections, and Representations These processes are intertwined and, by focusing on a subset of the five processes, all processes can be collectively addressed.

Mathematical Problem Solving Students will apply mathematical concepts and skills and the relationships among them to solve problem situations of varying complexities. Doing math

Mathematical Communication Students will use the language of mathematics, including specialized vocabulary and symbols, to express mathematical ideas precisely. Students will organize and consolidate their own mathematical thinking. Analyze and evaluate the mathematical thinking and strategies of others.

Mathematical Reasoning 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. Make and investigate mathematical conjectures. Select and use various types of reasoning and methods of proof.

Mathematical Connections Students will relate concepts and procedures from different topics in mathematics to one another and see mathematics as an integrated field of study. REAL WORLD!!! Multi-step problems that require knowledge from one idea to use in another idea.

Mathematical Representations 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. Explain mathematics through a combination of tables, graphs, algebra, diagrams, pictures, etc.

Triplet Tasks 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? Distribute 3 Tasks handout Whole-group discussion to follow table discussions. Prompts: How would you characterize level of thinking is required for each of the three tasks? Time: 10 min.

Examining Differences between Tasks What is cognitive demand? thinking required Cognitive Demand (as defined in the purple book): The kind and level of thinking required of students in order to successfully engage with and solve the task.

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 Prior to revealing the bullets on this slide, each table will share one characteristic of a low-level task. One of the presenters will use chart paper to record these contributions. Possible discussion prompt if time allows: Brainstorm examples of tasks that have a particular characteristic. Adapted 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 DOING Mathematics Prior to revealing the bullets on this slide, each table will share one characteristic of a low-level task. One of the presenters will use chart paper to record these contributions. Possible discussion prompt if time allows: Brainstorm examples of tasks that have a particular characteristic. Adapted 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

Mindstreaming How do you choose the problems, tasks or projects that you plan for your students? 2 minutes – individually/elbow partner

What should be considered when selecting tasks? Content Alignment Process Standards Cognitive Depth What are some key planning resources to use when selecting tasks?

Content Alignment Alignment is based upon: SOL & Curriculum Framework Essential Understandings Essential Knowledge & Skills Essential Questions (middle schools) Vertical Articulation Pacing Guide - HCPS Purpose - student needs Show sample of new pacing/curriculum guides Where do you find the content in the VA SOL curriculum frameworks? Essential Questions, Understandings, etc.

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 and long-term retention of content

Process Standards Communication: Talking and writing about math Problem-Solving Reasoning Multiple Representations Connections Can participants identify the 5 process standards? Which process standards were involved for you as a learner while doing the Triplet Task?

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 Which definition do you like better?

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) Now that participants have a good understanding of what defines high cognitive demand tasks, share with them other characteristics of Rich Mathematical Tasks. (Other researchers have studied additional qualities of rich tasks. See Implementing Standards-Based Mathematics Instruction, pp. 6–7.)

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. A mathematical task can be described according to the kinds of thinking it requires of students, it’s level of cognitive demand.

Let’s Recap!

Process Standards 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 structure 8) Look for and express regularity in repeated reasoning Mathematical Representations Representations 4) Model with mathematics   5) Use appropriate tools strategically 6) Attend to precision Where do the process standards fit into this statement? Problem Solving, Communication, Connections, Representations, Reasoning

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