The Intersection of Quality Math Tasks and Instruction Grades 6-12

1. The Intersection of Quality Math Tasks and Instruction Grades 6-12 Winter 2018

2. INTERSECTION OF TASKS AND INSTRUCTION The Week at a Glance

3. INTERSECTION OF TASKS AND INSTRUCTIONNorms That Support Our Learning • Take responsibility for yourself as a learner. • Honor timeframes (start, end, activity). • Be an active and hands-on learner. • Use technology to enhance learning. • Strive for equity of voice. • Contribute to a learning environment where it is “safe to not know.” • Identify and reframe deficit thinking and speaking.

4. INTERSECTION OF TASKS AND INSTRUCTIONFeedback on Feedback

5. INTERSECTION OF TASKS AND INSTRUCTIONEquity Equity isn’t giving every student the same thing; it’s giving every student what they need. It is about fairness. Ensuring all children – regardless of circumstance – are receiving high-quality and standards-aligned instruction is an equitable practice. We want to ensure standards-aligned instruction is causing the equitable practices needed to close the gaps caused by racism, bias, and poverty. All week, we will explore our learning through this lens, and we will capture those moments visibly here in our room.

6. Equity, Language, and Learners Students need well-structured opportunities to practice language to learn it.  Amplify, do not simplify, language. Content and language develop inseparably and in integrated ways; language development occurs over time and in a nonlinear manner. Scaffold students toward independence with complex tasks; do not scaffold by simplifying text language and task complexity. We are the gatekeepers of language in the classroom as teachers and leaders. Acquiring the language for the masterful use of standard English in writing and speaking benefits all students. All students bring valuable knowledge and culture to the classroom. INTERSECTION OF TASKS AND INSTRUCTION 6

7. INTERSECTION OF TASKS AND INSTRUCTIONShare Your Learning Don’t forget to jot down ideas for • Light bulb moments • Why I teach/lead

8. INTERSECTION OF TASKS AND INSTRUCTIONObjectives and Agenda Objective Participants will be able to: • Determine and modify the level of cognitive demand of a task, by examining the task itselfand student work elicited by the task. • Evaluate tasks to determine the level of cognitive demand as a condition for facilitating effective classroom discussion. • Plan for instruction using the Five Practices to build mathematical discussions and discourse Agenda • Cognitive Demand: Analysis and Student Work • Increasing Cognitive Demand • Discourse in the Math Classroom • Planning for Mathematical Discourse with the Five Practices • Observing Mathematical Discourse through the Five Practices

9. INTERSECTION OF TASKS AND INSTRUCTIONHow did we get here?

10. Analyzing the Cognitive Demand of Tasks Leveling tasks Connections to rigor and the SMPs Connections to equitable classrooms

11. INTERSECTION OF TASKS AND INSTRUCTIONNaming Levels Activity • With your table group, examine the Benchmark Tasks Grid for your grade band (middle, high). • What words or phrases could be the “header” for each column? • What words or phrases could be the “header” for each row? • When your group has agreed on the header names, please make a Post-it for each one.

12. INTERSECTION OF TASKS AND INSTRUCTIONNaming Levels Activity, continued

13. INTERSECTION OF TASKS AND INSTRUCTIONLeveling Tasks Activity • Using our new “rubric,” rate each sample task by placing it into the appropriate row. • Once you’ve given your ratings, place a Post-it with the letter of the task on our whole-group chart. • Be ready to share the rationale for two of your ratings (one from the top half of the rubric and one from the bottom half). SAMPLE PACKET

14. INTERSECTION OF TASKS AND INSTRUCTIONCognitive Demand & Rigor • Examine the tasks in Column 3 of your benchmark tasks grid. • Which of the tasks in this column are aligned to the Standards? Which standards are they? • If the alignment isn’t perfect, what additional directions, prior work, or subsequent tasks could be used to meet the full intent of the standard? • Which aspects of rigor are found in each of these standards? • Based on these sets of tasks and standards, start forming a conjecture: where do the aspects of rigor “live” on the levels of cognitive demand?

15. INTERSECTION OF TASKS AND INSTRUCTIONCognitive Demand & The SMPs • Looking at the same group of tasks (column 3): which Standards for Mathematical Practice could be addressed by each of the tasks?

16. INTERSECTION OF TASKS AND INSTRUCTIONResearch says… The task… “sets the ceiling” for implementation and for discussion. “Tasks with low cognitive demands simply do not provide fodder for teachers to engage students in thinking, reasoning, or mathematical discourse throughout the enactment of the lesson. If opportunities for high-level thinking and reasoning are not embedded in instructional tasks, these opportunities rarely materialize during mathematics lessons. This finding, robust in its consistency across several studies, suggests that Standards-based curricula and/or high-level instructional tasks are a necessary condition for ambitious mathematics instruction.” Boston & Wilhelm (2015)

17. Analyzing Cognitive Demand of Tasks in Light of Student Work

18. INTERSECTION OF TASKS AND INSTRUCTIONA Quality Tool: the IQA Rubric • Instructional Quality Assessment in Mathematics (IQA): sets of rubrics for assessing lesson observations and student work in terms of cognitive demand • Using the rubric, score each task/response, and write a brief explanation. • As you work, note any questions/disagreements that arise for you.

19. INTERSECTION OF TASKS AND INSTRUCTIONRating Tasks & Student Work • In your groups, rate each set of tasks and student work. • Your group will share: • One high-cognitive demand example (score 3 or 4), AND • One low-cognitive demand example (score 1 or 2), OR • One example that declined from high task (3-4) to low student work (1-2). • For the low cognitive demand and decline examples, consider how to revise the task or its directions to better elicit and maintain cognitive demand (or ways to engage students in the SMPs). • For the decline examples, consider what the decline might imply for (or about) instruction. SAMPLE PACKET

20. INTERSECTION OF TASKS AND INSTRUCTIONReflection What does it imply about the intersection of tasks and instruction if… • Students solved the task in more than one way even though the task directions did not specifically ask for multiple strategies? • Students consistently used “because” in their written explanations? • All or most of the students did not complete the cognitively challenging parts of the task? • Even though students were writing “explanations,” the explanations only involved procedural steps? • All students provide explanations similar in wording? • All student work samples look “template”?

21. INTERSECTION OF TASKS AND INSTRUCTIONSelecting Tasks for Equitable Classrooms In what ways can our selection of tasks for classroom use help or hinder our efforts to create equity for all students in our schools and classrooms? What are some of the challenges you see in giving every single student access to rigorous mathematical tasks?

22. Lunch Break!

23. Adapting Tasks to Increase the Cognitive Demand Adaptations to raise cognitive demand Adaptations to meet standards Introducing tasks using MLRs

24. INTERSECTION OF TASKS AND INSTRUCTIONAdapting a Task • Select a Level 1-2 task from the benchmarks task grid we used earlier. • Revise the directions, requirements, and/or information given in the task to: • Increase the cognitive demand to Level 3 or 4 • Meet the full intent of the relevant standard • Identify one mathematical language routine (MLR) that would be helpful for introducing this task. • Bonus: Describe any other tasks that might be needed to fully address the standards. (You don’t have to create actual tasks; just describe the mathematics that the tasks would need to address).

25. INTERSECTION OF TASKS AND INSTRUCTIONTechniques to Increase Cognitive Demand • Is there a way to set the task in a problem-solving context (especially “naked numbers” tasks)? • Does the task press students to represent a situation in mathematical language (e.g. an expression or equation)? • Does the task have different “intuitive” solutions or strategies (correct or incorrect) for which students could compare, contrast, debate? Can the task directions provide opportunities for students to (a) create a mathematical argument or (b) critique the reasoning of others? • Can the task include a situation that could be modeled mathematically (diagram, multiple representations)? • Could the task include tools that would help students reason (manipulatives, calculators, tracing paper, etc.)? • Could the task draw students’ attention to the importance of precision? • Is there a physical, numerical, computational, etc., structure that students could use to make generalizations? • Is there something systematic about the reasoning in the task (or based on prior tasks) that could promote reasoning and problem solving?

26. INTERSECTION OF TASKS AND INSTRUCTIONPresentations & Feedback • As you share, please think about: • Which of the techniques did your partner use to adapt the task? • How did your partner choose a particular MLR? Is there another that would also help? • Give one piece of positive feedback, and one suggestion for improving the task or its implementation even further. Push each other’s thinking so that you each walk away better at doing this.

27. INTERSECTION OF TASKS AND INSTRUCTIONAdapting Tasks for Equitable Classrooms • Which activity do you find more challenging—adapting a task to modify the cognitive demand/aspect of rigor, or planning ways to introduce it so that all students have access? • What are some ways we need to exercise caution when adapting tasks?

28. Promoting Mathematical Discourse in 6-12 Winter 2018

29. PROMOTING MATHEMATICAL DISCOURSEObjectives and Agenda Objective Participants will be able to: • Determine an modify the level of cognitive demand of a task, by examining the task itselfand student work elicited by the task. • Evaluate tasks to determine the level of cognitive demand as a condition for facilitating effective classroom discussion. • Plan for instruction using five practices to build mathematical discussions and discourse Agenda • Cognitive Demand: Analysis and Student Work • Increasing Cognitive Demand • Discourse in the Math Classroom • Planning for Mathematical Discourse with the Five Practices • Observing Mathematical Discourse through the Five Practices

30. DISCOURSEIN THE MATH CLASSROOM Classroom Discourse Picture It: Today, you visited the math classrooms you work in/with. The visits were informal and short (15 min each). When you think about the math instruction and discourse that you saw, what’s going well? What could be better? Why is math discourse important?

31. PROMOTING MATHEMATICAL DISCOURSEThe Importance of Discussion Mathematical discussions are a key part of current visions of effective mathematics teaching as they: • Encourage student construction of mathematical knowledge • Make student’s thinking public so it can be guided in mathematically sound directions • Learn mathematical discoursepractices (Smith & Stein, 2011) • Develop students’ identity as doers of mathematics (Aguirre, Mayfield-Ingram, Martin, 2013) • Learn to see things from other people’s perspectives (Smith, 2013) • Shift the mathematical authority from teacher (or textbook) to community (Webel, 2010)

32. Discourse? https://youtu.be/6yJmfN5otRU?t=17s

33. PROMOTING MATHEMATICAL DISCOURSETheLeavesand Caterpillar Problem A fourth-grade class needs five leaves each day to feed its 2 caterpillars. How many leaves would they need each day for 12 caterpillars? Use drawings, words, or numbers to show how you got your answer. Consider what a 4th grader might do. Directions: • Work out the problem as a 4th grader would individually. • With your shoulder partner, select who is Partner A and Partner B. • Partner A - read sentence stem on the handout. • Partner B - respond using sentence stem on the handout. • As a table, brainstorm as many ways students might respond to this problem. Include both correct and incorrect solutions.

34. PROMOTING MATHEMATICAL DISCOURSEADramaticReading As you read and listen, consider: • What is promising about Mr. Crane’s instruction? • In what ways might Mr. Crane improve?

35. PROMOTING MATHEMATICAL DISCOURSEThe Leaves and Caterpillar Vignette What is promising? What teachers commonly mention: • Students are working on a mathematical task that appears to be both appropriate and worthwhile • Students are encouraged to provide explanations and use strategies that make sense to them • Students are working with partners and publicly sharing their solutions and strategies with peers • Students’ ideas appear to be respected

36. PROMOTING MATHEMATICAL DISCOURSEThe Leaves and Caterpillar Vignette What can be improved? What teachers commonly mention: • Beyond having students use different strategies, Mr. Crane’s goal for the lesson is not clear for students as they work. • There is a “show and tell” feel to the presentations • not clear what each strategy adds to the discussion • different strategies are not related • key mathematical ideas are not discussed • no evaluation of strategies for accuracy, efficiency, etc.

37. PROMOTING MATHEMATICAL DISCOURSEWhat now? How can we improve our skills with orchestrating productive mathematics discussions of high quality tasks?

38. PROMOTING MATHEMATICAL DISCOURSEThe Role of Communication Communication is an essential part of mathematics and mathematics education. It is a way of sharing ideas and clarifying understanding. Through communication, ideas become objects of reflection, refinement, discussion, and amendment. The communication process also helps build meaning and permanence for ideas and makes them public. NCTM, 2000

39. PROMOTING MATHEMATICAL DISCOURSEWhat is “expert facilitation”? Skillful improvisation... • Interprets and synthesize students’ thinking on the fly • Fashions responses that guide students to evaluate each other’s thinking, and promote building of mathematical content over time ...requires deep knowledge of: • Relevant mathematical content • Student thinking about content and how to frame it • Subtle pedagogical moves • How to rapidly apply all of this in specific circumstances

40. PROMOTING MATHEMATICAL DISCOURSESomeChallenges • Lack of familiarity • Reduces teachers’ perceived level of control • Requires complex, split-second decisions • Requires flexible, deep, and interconnected knowledge of content, pedagogy, and students

41. Take a Break and Stretch

42. PROMOTING MATHEMATICAL DISCOURSETheFivePractices Anticipating Monitoring Selecting Sequencing Connecting

43. PROMOTING MATHEMATICAL DISCOURSETheFive+Practices

44. PROMOTING MATHEMATICAL DISCOURSEPurpose of the Five Practices To make student-centered instruction more manageable by moderating the degree of improvisation required by the teachers and during a discussion.

45. PROMOTING MATHEMATICAL DISCOURSEAnticipating

46. PROMOTING MATHEMATICAL DISCOURSEMonitoring • Listen, observe, identify key strategies. • Keep track of approaches. • Ask questions of students to get them back on track or to think more deeply. • Consider incorrect answers and correct answers as equally valuable

47. PROMOTING MATHEMATICAL DISCOURSEChart for the 5 Practices Column 3: Teachers Sequencethe order in which selected student work will be shared. Column 1: Teachers Anticipate the strategies and and approaches students may take. Column 2: Teachers Monitorthe work of specific students. (during and/or after the lesson) Column 4: Teachers consider how to Connectthe thinking of multiple students to draw out the big math ideas. Column 2: Teachers Select which student work to highlight in a class discussion.

48. PROMOTING MATHEMATICAL DISCOURSESelecting, Sequencing, and Connecting Directions: As a table, complete the Chart for Monitoring using the student work from Leaves and Caterpillars. Create a poster to share your work with the group. • Monitor which students used which strategy and write names and samples from their work in column 2 • Select the strategies/solution paths that you would want to have shared during a whole group discussion. • Specify the sequence in which they would be shared and explain why you selected the particular responses and how you determined the ordering of the presentations. • Determine the strategies you would want to specifically connect and explain the connection you would want students to see.

49. PROMOTING MATHEMATICAL DISCOURSEEquityMove What do we do if all students use the same strategy for solving the problem? What do we do if students come up with ineffective solution methods for solving the problem?

50. PROMOTING MATHEMATICAL DISCOURSEThinkingThrough a Lesson In what ways is this process the same or different from other planning routines teachers use?