1. The Intersection of Quality Math Tasks and Instruction
Grades K-5
As designed:
116 min before lunch;
After lunch, 77 min before afternoon break;
125 before closing (10’ survey time included)
For the majority of the day, participants will be working with either middle school or high school mathematics and student work samples. They should sit in grade band-alike tables: either grades 6-8 or grades
Materials:
Participant Handout Packets
Packets of Student Work Samples for each grade band – please be sure to collect these at the end of the activity/session
Facilitator Resource handout – for facilitators only
includes Leveling Tasks activity Answer Key and Video Transcript
Summer 2018

2. INTERSECTION OF TASKS AND INSTRUCTION Feedback on Feedback
Plus
Delta
5 min
Speaker’s Notes:
Note for facilitators – Because today needs more framing for the learning, start with the pluses and deltas and norms before the Week at a Glance or Today’s Agenda Objectives.
Highlight a couple of pluses and a couple of deltas; choose them based on impact on participant learning and/or so they feel heard.
Tell how we will respond today or what they can expect.
For the pluses or deltas that are about group behavior, encourage the group to keep doing the positive and to monitor or minimize those things that could help the learning environment improve.

3. INTERSECTION OF TASKS AND INSTRUCTION Norms 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 min.
Speaker’s Notes:
We have norms for learning together this week. At different points in the week, we may remind you of a norm if we think it has been slipping—or you can remind one another.
You can read these for yourselves, but let me expand on a couple:
Take responsibility of yourself as a learner:
Keep an open mind (esp. about what don't know or thought you knew).
Stay in learning orientation vs. performance orientation—growth mindset.
Be an active and hands-on learner.
Be active during video observation by capturing evidence in writing.
Use technology to enhance learning.
Be present (monitor multi-tasking, technology, honoring timeframes).
Equity of voice:
Share ideas and ask questions, one person at a time (airtime).
Contribute to a learning environment where it is “safe to not know.”
Appreciate everyone's perspective and journey.
Be okay with discomfort and focus on growth.
Remember to think of students and teachers from an assets-lens: one that recognizes and honors everything that they bring to the classroom and that you can build upon.

4. The Week at a Glance Day Ideas Monday
INTERSECTION OF TASKS AND INSTRUCTION
The Week at a Glance
Day
Ideas
Monday
8:30–4:30
Focus on Language, Equity and Learners
Tuesday
Standards Aligned Writing to Build Knowledge,
Language and Vocabulary
Wednesday
Adapting Curriculum for Equity
Thursday
The Intersection of Quality Math Tasks and Instruction
Promoting Mathematical Discourse
Friday
8:30–2:30
Systems Thinking for Leaders Who Want Different Results
3 min
Speaker’s Notes:
Note to facilitators: you will need to do a bit of extra framing today to connect the learning to yesterday.
Yesterday, we adapted curriculum as a key skill in order to increase access for students with unfinished learning because we’re moving to planning. It was an all-encompassing planning strategy to offset the most frequently observed adaptation habits: educators tend to make adaptation moves like the extensive blanket review at beginning of year or unit.
Unfortunately, the result is that by the time the students need to access the reviewed learning, they don’t have it anymore because the context is gone.
Yesterday, you learned a just-in-time approach in which we deliver just the right amount of scaffolding to get students to be able to engage in the grade level work of the day.
And all of this work is heavily dependent on the standards. Without the coherence of the standards, all this adaptation happens in a non-coordinated way.
Today – we are going to focus on planning new instruction. What are the moves that teachers need to plan to put in place to provide access for students?
Today, the access is about leading the learning from a place of conceptual understanding. What we know from research is that students need to interact with rich mathematical tasks. And they need to communicate to make meaning of the mathematics and make their mathematical thinking visible.

5. INTERSECTION OF TASKS AND INSTRUCTION Objectives and Agenda
Participants will be able to:
Determine and modify the level of cognitive demand of a task, by examining the task itself and 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
2 min.
Speaker’s Notes:
So today, we’re going to look at mathematical tasks from the lens of Cognitive Demand. Once you know the cognitive demand of a task, you also then know better what it takes for students to make sense of them.
For tasks with high cognitive demand, students need to engage in meaningful mathematical discourse so that they can make sense of the concepts they must understand
Yesterday, we focused on high level curriculum adaptations. Today, We want you to focus on instructional planning, delivery and technique.
Note to facilitators: This day is a heavier lift for the learners. It’s new content and quite a bit of mathematics. There is not a perfect example video or artifact. Work to help participants manage that today.

6. INTERSECTION OF TASKS AND INSTRUCTION Share Your Learning
Don’t forget to jot down ideas for
Light bulb moments
Why I teach/lead
1 min.
Speaker’s Notes:
Ask participants to continue to think about our statements about equity throughout the day and call out and/or add stickies to the Equity Wall as ideas arise. If time allows, review comments added the previous day and/or ask for any ideas about equitable practices that participants had since yesterday’s session.

7. INTERSECTION OF TASKS AND INSTRUCTION How did we get here?
Selecting, Adapting, and Preparing Tasks
Rigor
Standards for Mathematical Practice
Aligned Curriculum
Strategies for ELs
< 1 minute
Speaker’s Notes:
Briefly explain how previous learning in Pathways 1 and 2 relates to this morning’s session.
We’ve learned about the Shifts and rigor, and today we’ll be thinking about how to select tasks that involve all the aspects of rigor.
We’ll also be talking about selecting tasks that engage students in the SMPs.
We’ve talked about aligned curriculum, and today we’ll be talking about how to ensure alignment to standards in our selection of tasks, materials, and lessons.
Finally, we’ve talked about strategies for English learners, which we’ll be using to increase access to challenging tasks.
All along the way, we will be deepening your mathematical content knowledge and your understanding of the demands of the Standards.

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

9. INTERSECTION OF TASKS AND INSTRUCTION Naming Levels Activity
With your table group, examine the Benchmark Tasks Grid.
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.
10 min
Speaker’s Notes:
Materials:
Participant Handout packet pages 3-5- Benchmark tasks grid and recording sheet
Middle School is on page 3
High School is on page 4
Post-it notes
Format: activity within table groups.
Give participants 5 minutes to read through the tasks; think about what is needed to solve them (but no need to actually solve them).
Participants work individually first, then within groups. (<10 min)
Circulate while participants work. If needed, ask questions that focus their attention on the “type of thinking” the row of tasks would elicit from students, or that would be necessary to solve the task.
Observe the words/phrases groups are writing on their sticky notes so that you know what to expect during the whole-group discussion.
Activity continued on next slide.

10. Level Characteristics
INTERSECTION OF TASKS AND INSTRUCTION Naming Levels Activity, continued
Level
Characteristics 4 3 2 1
10 min
Speaker’s Notes:
Lead a whole-group sharing of responses and discussion. First, quickly consider the columns (i.e. topics). Then consider the rows (i.e. levels of cognitive complexity). Start at the bottom row and move up. Consider creating a poster with a table similar to the one in this slide where participants can hang their sticky notes as they share.
Anticipated row titles:
Memorization/recall; “either you know it or you don’t”; nothing to help the learner make sense of the task/math ideas; recall of facts, formula, definitions
Procedures without connections; rote procedure or computation (difference between this and row 1 is that there is a procedure involved)
Procedures with connections or doing mathematics; potential to elicit thinking and reasoning; opportunity for sense-making BUT does not explicitly prompt for an explanation, justification, etc.
Potential to elicit thinking and reasoning; explicitly requests explanation, reasoning, reflection, etc.
Possible sticking points:
Participants may be stuck on content and not see the importance of “type of thinking.”
Participants may want to alter the tasks by adding directions, representations, strategies. Acknowledge that these are great ideas for ‘adapting’ the task as written (that we will consider later) and suggest that they consider what is required to produce an adequate response to the task as written.
Participants may also ask about grade level or prior knowledge. Acknowledge that those are important considerations, and for this activity, assume that the task is appropriate for the grade-level (students have the necessary prior knowledge to solve the task as written).
(Note: these four levels come from Smith, M. S., & Stein, M. K. (2011). 5 practices for orchestrating productive mathematics discussions. Reston, VA: National Council of Teachers of Mathematics. At this point, it’s not necessary for participants to have a nuanced understanding of the levels as described by Smith & Stein, but you might want to mention where this framework comes from.)

11. INTERSECTION OF TASKS AND INSTRUCTION Leveling 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).
10 min
Speaker’s Notes:
5 minutes to rate tasks
5 minutes for partner/table sharing of ratings
10 minutes for whole-group discussion of ratings
Materials:
Participant Handout packet pages 3-5
Packets of Student Task and Work Samples
Middle School Tasks on pages 4-7
High School Tasks on pages 14-17
Task set answer keys (for facilitator only)
Post-it notes
Facilitation Notes:
Suggest that participants first identify a solid example of each level—tasks they are confident in scoring. Then, if time in small groups, suggest they discuss tasks that were hard to rate. (Participants will want to rate all tasks – this is fine if there is time.)
Note that there could be “mitigating circumstances” where some tasks might be routine or procedural in one situation, but could be used to elicit thinking/reasoning in another situation (e.g. classroom norms for explanation, justification, etc.).
It is not important to get perfect agreement on levels at this point, but to help participants understand the importance of the cognitive demand of the tasks they use for introducing concepts, practicing procedures and skills, assessment, etc.
SAMPLE PACKET

12. INTERSECTION OF TASKS AND INSTRUCTION Cognitive 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?
30 min
Speaker’s Notes:
Materials: standards tools (e.g. standards app or documents)
10 minutes of partner/table work on first two questions
5 minutes of sharing and whole-group discussion
10 minutes of partner/table work on third and fourth questions
MAIN POINTS:
The two goals of this activity are:
For participants to understand that students may have to engage in a range of tasks in order to meet the expectations of a given standard, cluster, or domain.
For participants to relate the framework of rigor in the CCSSM to Smith & Stein’s framework of cognitive demand. They should see, for example, that tasks which build conceptual understanding fall into Levels 3-4 of the cognitive demand framework. This is the crucial link between the content of this presentation and their learning thus far in Pathways 1 and 2.
Anticipated responses:
Middle (linear relationships/functions): Levels 3-4 tasks involve modeling linear relationships and multiple representations, as well as rate of change and initial value (i.e. 8.F.A.2, 8.F.B.4). However, tasks would benefit from opportunities to compare representations within a given task (functions represented different ways). Levels 1-2 tasks don’t clearly align to a standard; participants could reflect on why the standards don’t focus on these sorts of activities.
High (trigonometric rations and the Pythagorean Theorem: Level 2 task provides practice with HSG.SRT.C.8, but isn’t “applied”; Levels 3-4 tasks provide applied problems. Level 4 task aligns with HSF.TF.A.2-3, although students will need experiences with other special triangles and more opportunities to explain.
Possible misconceptions include:
“Higher is better.” This is salient for the elementary tasks, each of which aligns to a standard (more or less). The idea of “levels” sometimes implies that “higher-level” is better. However, the Shifts in the CCSSM remind us that students actually need to pursue all three aspects of rigor (and therefore work with tasks on all levels of the cognitive demand framework) with equal intensity.
“One task should do it all.” Participants may focus too much on “this task doesn’t…” Ask them whether the task is completely unaligned (i.e. relates to no part, or doesn’t target the intent of, any standard), or whether it’s somewhat aligned (i.e. meets part of a standard, if not the entire thing). If it’s somewhat aligned, point them to the second question on the slide.

13. INTERSECTION OF TASKS AND INSTRUCTION Cognitive 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?
5 min
Speaker’s Notes:
5 minutes of whole-group sharing
This slide is more straightforward than the last. The idea is that the high-level tasks provide opportunities for one or more of the SMPs, while the low-level tasks do not.

14. INTERSECTION OF TASKS AND INSTRUCTION Research 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)
1 min
Speaker’s Notes:
Optional slide to reinforce the previous two activities. Discuss if time allows.
Citation: Boston, M. D., & Wilhelm, A. G. (2015). Middle school mathematics instruction in instructionally focused urban districts. Urban Education, 1–33.

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

16. INTERSECTION OF TASKS AND INSTRUCTION A 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.
min
Speaker’s Notes:
Materials in packets:
IQA rubrics and benchmark sample tasks in Participant Handout packet pages 6-9
Using pages 6-7, Introduce “Instructional Quality Assessment in Mathematics” (IQA) rubrics for rating tasks (task potential) and student work (implementation), and provide a benchmark sample task with student’s work for each level (samples will span grade bands) Talk through the main points of each score level, using the sample tasks on the Participant Handout pages 6-7:
Level 1: Vocabulary; memorized knowledge.
Level 2: Rote procedures (distributive property); not memorization, because there are procedures involved.
Level 3: Students must makes sense of the graph, identify the ratio relationship between cream and chocolate, and write a rule. However, they are not asked to explain their thinking or justify the rule. Level 3 often contain representations or diagrams.
Level 4: Everything in Level 3, plus explicitly prompt for explanations. In this task, the prompt “relate the description to the diagram” requires students to provide justification of their description/rule.
Using pages 8-9 of the Participant Handout , ask participants to justify the score level.
Possible responses:
Level 1: Shape identification; vocabulary; brief responses
Level 2: Rote procedures; not memorization because there are procedures involved.
Level 3: Students must makes sense of a fraction times a whole number in order to write a number sentence that goes with the diagram. However, they are not asked to explain their thinking.
Level 4: Students are asked to “justify your answer.” The student solved the task in 2 ways and explains how each one is relevant to the task: “costs less per can and if you buy 30 cans…” Level 4 must explicitly prompt for explanations.
Transition: It’s your turn to practice on your own and deepen your internalization of the use of this tool for determining the cognitive demand of tasks and the resulting student work.

17. INTERSECTION OF TASKS AND INSTRUCTION Rating 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.
20 min
Speaker’s Notes:
Materials: Packets of Student Work Samples for each grade band – please be sure to collect these at the end of the activity/session
Facilitation Notes:
15 minutes with partners/groups to rate sample tasks and student work for their grade band:
Middle School pages 8-11
High School pages 18-23
Circulate while participants work and see who has good examples to present for high, low, and decline. Make sure that at least one example of each is shared during the discussion.
10 minutes whole-group concluding discussion
During the whole group discussion, each group will present one high-cognitive demand example and one low-cognitive demand example or one example that declined from high task to low student work.
If time allows, ask participants to discuss how to revise one of the low-demand or decline examples to better elicit and maintain cognitive demand. Encourage small changes to tasks or task directions that provide greater opportunities for students to engage in the SMPs (this will also be addressed in the next section).
SAMPLE PACKET

18. INTERSECTION OF TASKS AND INSTRUCTION Reflection
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”?
5 min
Speaker’s Notes:
Select 2-3 bullets to discuss whole-group. Participants should begin to see that the way a task is implemented in the classroom is just as important as the elements of the task itself.
These questions taken from Boston, M. D., & Wilhelm, A. G. (2015). Middle school mathematics instruction in instructionally focused urban districts. Urban Education, 1–33.

19. INTERSECTION OF TASKS AND INSTRUCTION Selecting 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?
5 min
Speaker’s Notes:
Use these questions as the basis for a written reflection that considers how the material in this section relates to participants’ thinking about educational equity.
The second question invites participants to reflect on the idea that equity isn’t always easy; giving every student access to rigorous grade-level work is often challenging. The next sections of this presentation should offer ways to address some of these challenges.
IMAGE CREDIT
Amy Rudat

20. Lunch Break! <1 min Speaker's Notes:
Remind participants that we will start the PM session promptly at 1 pm, back in this room.
IMAGE CREDIT

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

22. INTERSECTION OF TASKS AND INSTRUCTION Adapting 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).
15 min
Speaker’s Notes:
Materials:
Benchmark tasks grid (from earlier in the session)
Mathematical language routines posters (from previous session) and/or the pre-work reading for reference
Clearly frame the rationale for this activity:
We’re not doing this because Levels 1-2 are “bad;” as we’ve mentioned, they have a place in a well-rounded curriculum.
However, we know that many curricula offer only exercises at Levels 1-2, so teachers need to modify the cognitive demand/aspect of rigor in these cases.
Also, we may need to modify tasks so that students are able to access them more easily, while still engaging in the type of thinking specified by the standards.
We want you to be able to support teachers to increase cognitive demand and deepen their math content knowledge while doing so.
You may want to create an example of an adapted task to illustrate the type of product you’re looking for participants to create.
It may be helpful to give participants some time to work on adaptations, pause to focus on the MLR component of the activity, and then given them a little more time to work on that part. It’s important they engage in both parts of the activity.
As participants work, watch out for changes that make the task unnecessarily complex, or change the mathematical focus of the task.

23. INTERSECTION OF TASKS AND INSTRUCTION Techniques 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?
OPTIONAL
Speaker’s Notes:
If participants get stuck while adapting tasks, consider projecting these ideas.
Return to this slide during or after participants present, and point to a few examples of these adaptations.
(Note: the items in this list correspond with the SMPs. If time allows, this is an opportunity to reinforce the connection between rigor, the SMPs, and the new idea of cognitive demand.)

24. INTERSECTION OF TASKS AND INSTRUCTION Presentations & 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.
15 min
Speaker’s Notes:
If participants haven’t seen it yet, show them the list on the previous slide so they can consider which techniques they see, and to highlight examples of some of the techniques during the whole-group discussion.
10 minutes for partner sharing
5 minutes for whole-group discussion
Any constructive feedback that participants have for their partners will help them think about the second question on the next slide.

25. What are some ways we need to exercise caution when adapting tasks?
INTERSECTION OF TASKS AND INSTRUCTION Adapting 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?
5 min
Speaker’s Notes:
Use these questions as the basis for a written reflection that considers how the material in this section relates to participants’ thinking about educational equity.
Summary Statements to make to the second question and before transitioning to next section:
As we add elements (directions, context, etc.) to a task, we need to make sure we’re not unintentionally lowering the cognitive demand by imposing too much structure or scaffolding.
We need to consider the ways we’ll introduce the task (not just the content of the task itself).
We might need a number of tasks in order to meet the expectations of a given standard.
IMAGE CREDIT
Amy Rudat

26. Promoting Mathematical Discourse in 6-12
<1 min
Speaker’s Notes:
SAY: We will continue to build on the work we have been doing this week by connecting this morning’s work on the cognitive demands of mathematical tasks to building effective mathematical discourse.
Winter 2018

27. PROMOTING MATHEMATICAL DISCOURSE Objectives and Agenda
Participants will be able to:
Determine an modify the level of cognitive demand of a task, by examining the task itself and 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
1 min.
Speaker’s Notes:
Note where we are in our learning agenda

28. DISCOURSE IN 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?
5 min
Speaker’s Notes:
Before we dive into the “five practices,” let’s think about our own students, who are the reason we’re actually here. What is the state of discourse currently in our classrooms?
1 min: to think about the “picture it” prompt;
3 min: Turn & Talk
1 min: ask for 1 example of going well and one needs improvement
<Click> Transition: let’s first talk about why it is important
IMAGE CREDIT
Chairs in a Classroom – Flickr

29. PROMOTING MATHEMATICAL DISCOURSE The 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 discourse practices (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)
2 min
Speaker’s Notes:
”Here are some other ideas I heard you discuss.”
First 3 bullets come from Smith & Stein
References:
Smith, M. S. & Stein, M. K. (2011). Five Practices for Orchestrating Productive Mathematics Discussions. NCTM/Corwin
Aguirre, J., Mayfield –Ingram, K., & Martin, D. B. (2013). The Impact of Identity in K-8 Mathematics. Rethinking Equity-Based Pratices.
Webel, C. (2010). Shifting mathematical authority from teacher to community.  Mathematics Teacher ,  104 (4),

30. Discourse? https://youtu.be/6yJmfN5otRU?t=17s 10 min Speaker’s Notes:
Set up the problem that was proposed by the teacher by reading below.
If the link on the slide does not connect, use the URL: to access the video beginning on the 17th second (skip introduction, which gives away the gist).
The problem:
“It takes me ½ minute to do my hair and 4/6 minutes to make breakfast.
How much longer in seconds does making breakfast take than doing my hair?”
Debrief:
Based on what we learned this morning, is this a high cognitive demand task?
Are students engaged in discussion?
Are students engaged in an activity that allows them to learn from one another and promote high levels of mathematical proficiency?
This is a sample of what most teacher assessment rubrics (Danielson) would consider a successful engagement of all students in a discussion with each other. However, it is not enough to engage students in whole class discussion with opportunity to delve deeply into meaningful mathematics (SMP.1, SMP.2, SMP.7, for example)
Transition: So to this point, I present to you why discussion matters and a comfortable example of what the majority of us think is a good example of math discourse. This is OK but its not what we’re talking about when we describe “productive mathematical discourse” looks like. Let’s look at a second artifact of a teacher’s plan and compare the discourse. We’ll look for what is promising and in what ways Mr. Crane can improve.
Source:
ENGAGENY – Teacher ensures that all students contribute to the discussion – Example 1. Available online at
Image: Captured from website at

31. PROMOTING MATHEMATICAL DISCOURSE The Leaves and 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 task as a 4th grader would individually.
Now pair up to share solutions.
As a table, brainstorm as many ways students might respond to this problem. Include both correct and incorrect solutions.
17 min
Speaker’s Notes:
We will be using the student hat and the teacher hat in this activity. First, we will work as the student. You will pair up with your shoulder partner, then select who is partner A and partner B. Complete the question independently first, then using the sentence stem that was handed out, discuss your strategy with partner.
Next you will work with the teacher hat. Once all at your table have shared their strategy with a partner, brainstorm and generate some different ways students might respond to this problem. Compare and share with your neighbors to help generate a larger range of possible approaches.
ALSO: What are your thoughts on the level of rigor in this problem?
Does this count as “doing mathematics” based on what we recently discussed?
What evidence do you have for this?
Is this still applicable?

32. PROMOTING MATHEMATICAL DISCOURSE What now?
How can we improve our skills with orchestrating productive mathematics discussions of high quality tasks?
<1 min
Speaker’s Notes:
Use this question for a transition
IMAGE CREDIT
ttps://pixabay.com/p /?no_redirect

33. PROMOTING MATHEMATICAL DISCOURSE The 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
1 min
Speaker’s Notes:
How do you see this quote (from the year 2000) as connected to the Standards for Mathematical Practice?
This is a strong rationale for a share-and-discuss phase of the lesson that is more than just a show-and-tell of answers.
Not only are students sharing the mathematics and making connections, but they are also creating permanence for the mathematical ideas.
This quote connects to Mathematical Practice 3: Construct viable arguments and critique the reasoning of others.

34. PROMOTING MATHEMATICAL DISCOURSE What 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
3 min
Speaker’s Notes:
Review these points.
Now we’ll do “fist to five.” Show a fist if this set of ideas is going to be extremely challenging for you to implement. Show five fingers if this is something you already do regularly with great skill, ease, and dexterity. Show 1, 2, 3 or 4 fingers if you’re somewhere in between.

35. PROMOTING MATHEMATICAL DISCOURSE Some Challenges
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
1 min
Speaker’s Notes:
NOTE that these responsibilities are tough! We are tasked with complex, challenging work that demands the best from us!
Luckily, we can draw from the “5 practices” to support our work! Each of these concepts is drawn from a deep and robust research base.
Transition: We’ll take a 15 min break now. And when we come back, we will give you a 5-Practice framework to address these challenges in a way that promotes true mathematical discourse in the classroom.

36. Take a Break and Stretch
15 min break (approx 2:15 pm)
IMAGE CREDIT
Take a Break and Stretch

37. PROMOTING MATHEMATICAL DISCOURSE The Five Practices
Anticipating
Monitoring
Selecting
Sequencing
Connecting
1 min
Speaker’s Notes:
Margaret Smith says: Productive mathematical discourse does not happen by accident. It is a deliberate and planned-for instructional act. As we said before the break, we can draw from the “5 practices” to support our work. Each of these concepts is drawn from a deep and robust research base.
Anticipating (e.g., Fernandez & Yoshida, 2004; Schoenfeld, 1998)
Monitoring (e.g., Hodge & Cobb, 2003; Nelson, 2001; Shifter, 2001)
Selecting (Lampert, 2001; Stigler & Hiebert, 1999)
Sequencing (Schoenfeld, 1998)
Connecting (e.g., Ball, 2001; Brendehur & Frykholm, 2000)
Background for Facilitators:

38. PROMOTING MATHEMATICAL DISCOURSE The Five+ Practices
PreWork
Establish the learning goal
Select challenging mathematical tasks
Anticipating
Likely student responses to challenging mathematical tasks
Monitoring
Students' actual responses to the tasks
Uses effective questioning to support student productive struggle
Selecting
Students' responses that will be displayed in a specific order
Sequencing
Students' responses to be displayed during whole class debrief
Used to further conceptual development
Connecting
Students' responses to key mathematical ideas
2 min
Speaker’s Notes:
The animation will bring up the titles of each phase first: one-by-one. Name each. Then it will bring up the description of each, talk through each
SAY:
PreWork: However, note that it does not just begin with anticipating student responses. The first part of our day together was invested in the first two points—establishing the learning goal and selecting an appropriate task.
QUESTION 1: Where does the learning goal come from ?
SAMPLE ANSWER 1: Establishing the learning goal comes from the standards we are focusing on.
QUESTION 2: What are the characteristics of an appropriate task?
SAMPLE ANSWER 2: One with a high level of cognitive demand (3 or 4). Selecting an appropriate task is essential before starting to anticipate student responses.
SAY: We will look into each Practice in detail.

39. PROMOTING MATHEMATICAL DISCOURSE Purpose 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.
1 min
Speaker’s Notes:
The practices we have identified are meant to make student-centered instruction more manageable by moderating the degree of improvisation required by the teacher during a discussion. Rather than focusing on in-the-moment responses to students contributions, the practices instead emphasize the importance of planning.
Through planning, teachers can anticipate likely student contributions, prepare responses they might make to them, and make decisions about how to structure students’ presentations to further their mathematical agenda for the lesson. As leaders and coaches, we want you to be able to guide this planning.

40. PROMOTING MATHEMATICAL DISCOURSE Anticipating
3 min
Speaker’s Notes:
Doing the task in advance allows the teacher to fill out the strategy column on the Chart for Monitoring. This work helps the teacher quickly identify the student work that demonstrates different strategies for solving the problem. It also raises the awareness of strategies that students are not using to solve. (Participants did this when they solved the problem earlier.) Note that we might not always have the “label” for each method at the ready (like unit rate, scale factor, etc.) and this is fine.

41. Listen, observe, identify key strategies. Keep track of approaches.
PROMOTING MATHEMATICAL DISCOURSE Monitoring
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
2 min
Speaker’s Notes:
What does this LOOK LIKE in a classroom?
What strategies could YOU use to “monitor” the work of your students?
This is what teachers do who are skilled at Monitoring based on their Anticipating preparation.
<CLICK> This last bullet is really important. Often when teachers are monitoring student mathematical discussions and problem solving, they will circulate and ask questions. When a students provide incorrect answers, teachers often simply ask the question again of the students, looking for a correct answer. And then when a student answers correctly, they respond to that correct answer. Teachers who understand the mathematics and the task really well will respond equally to both incorrect and correct answers, probing and asking students to make meaning of the mathematics. They will ask students to compare and contrast solution paths, which contributes to discourse and deepens understanding of the mathematics simultaneously.
Make the case for engaging in other MLRs to support student engagement (i.e.: MLR 6: 3 Reads or MLR 2: Gather and Show Student Discourse)

42. PROMOTING MATHEMATICAL DISCOURSE Chart for the 5 Practices
Column 3: Teachers Sequence the order in which selected student work will be shared.
Column 1: Teachers Anticipate the strategies and and approaches students may take.
Column 2: Teachers Monitor the work of specific students. (during and/or after the lesson)
Column 4: Teachers consider how to Connect the thinking of multiple students to draw out the big math ideas.
5 min
Speaker’s Notes:
While students are working the task, the teacher can MONITOR student responses and record student names/work in the Work of Specific Students column <CLICK> in the appropriate strategy row. Based on what teachers see in student work, then they can SELECT <CLICK> specific student solutions the teacher would like to have shared.
Lastly, the teacher will need to choose the sequence for students to share their work <CLICK> and the connections between the types of student solutions. The connections <CLICK> give the teacher ideas for the types of questions that he/she would like to ask of the students. The goal is to move the students toward the big ideas of mathematics represented in the task.
Column 2: Teachers Select which student work to highlight in a class discussion.

43. PROMOTING MATHEMATICAL DISCOURSE Selecting, 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.
15 min
Speaker’s Notes:
Remind that we completed the Anticipation Practice when we solved the task earlier.
Some things to consider when selecting and sequencing:
Correct answers
Different strategies
Different representations
Incorrect answer
Correct pathway
Pairs that work together (e.g., student #1 and Student #2 might be thought of as a pair for comparison of similarities and differences. Why might you do this?)
Involve justification and generalization, etc.
Give 10 minutes for poster creation.
Give another 10 min: choose one group to present their work and the facilitate a discussion of the similarities and differences between other groups’ work. You can use this facilitation to model Monitoring, Selecting, Sequencing and Connecting.

44. PROMOTING MATHEMATICAL DISCOURSE Equity Move
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?
5 min
Speaker’s Notes:
Have participants do a Think-Pair-Share to discuss these two questions. Come back as a group to popcorn out ideas.
Address student discourse as an equity move.
-We are not lowering the cognitive demand of the task during implementation by changing value in the problem.
-We are offering support.
EQUITY:
Discussion increase student learning by providing students with a structure in which to communicate their understanding of math concepts
Communication Principle, NCTM, Slide 15
SMP.1 – MAKE SENSE of problems and persevere in solving them
SMP.2 – Reason abstractly and quantitatively
SMP.3 – Construct viable arguments and critique those of others.
SMP.4 – Attend to Precision
Accept reference to other SMPs if Participants justify the connections
Sample answer:
Use strategy of “in another class I saw… What do we think about this strategy?”
Start with a concrete model of the problem.
You can give students the opportunity to solve the problem again?
Move to a representational model and help students make connections between the concrete and representational solutions.

45. PROMOTING MATHEMATICAL DISCOURSE Thinking Through a Lesson
In what ways is this process the same or different from other planning routines teachers use?
8 min
Speaker’s Notes:
Possible points of discussion may include:
*Planning in lesson study
*Methodology for lesson study: Launch, explore, summarize questions from Connected Math Project
*Research on Mathematics tasks (e.g., what it takes to keep tasks at a high level)
Different from the 3-part sequence of teacher Initiation, student Response, and teacher Evaluation (IRE) pattern of interaction (NCTM, n.d.) available Discussion/
IMAGE CREDIT
Discussion groups at Ngateu – Flikr -

46. PROMOTING MATHEMATICAL DISCOURSE The Intersection of Tasks and Instruction
Worthwhile tasks alone are not sufficient for effective teaching. Teachers must also decide what aspects of a task to highlight, how to organize and orchestrate the work of the students, what questions to ask to challenge those with varied levels of expertise, and how to support students without taking over the process of thinking for them and thus eliminating the challenge.
NCTM, 2000
3 min
Speaker’s Notes:
This quote comes from the NCTM Principles & Standards for School Mathematics and emphasizes the role of teacher in orchestrating discussion.
ASK: Where do you see the 5 Practices in this quote?
NOTE that this is essential to our work of supporting independent thinkers!

47. Practicing the Five Practices
< 1 min (44 min elapsed = 3:15)
Speaker Notes:
“Let’s take some time to practice with this planning method.”
Goals & Objectives: The Activities on Slides 56–69 are meant to allow participants opportunity to build knowledge and skill through practice of the Five Practices
Participants will:
Evaluate the effectiveness of the use of the Three Reads MLR in creating linguistic access to a mathematical task.
Practice with each of the five stages of the Five Practices protocol.
Facilitate a task using the Five Practices.
Offer feedback on one another’s facilitation.
Increasing access

48. PROMOTING MATHEMATICAL DISCOURSE Moving for 10 Seconds
Lin and Diego both ran for 10 seconds, each at a constant speed. Lin ran 40 meters and Diego ran 55 meters. 
Who was moving faster? Explain how you know.
How far did each person move in 1 second? If you get stuck, consider using diagrams to represent the situations.
Han ran 100 meters in 20 seconds at a constant speed. Is this speed faster, slower, or the same as Lin’s? Diego’s?
5 min
Speaker Notes:
Facilitator moves: (1–3 min) Invite participants to consider this task. Monitor their solutions/approaches/strategies.
ASK: What MRL would you consider to increase access to this task? (2 min)
Possible Responses:
MLR 6: Three Reads – Example 1
MLR 2: Collect & Display – Example 1: Gather and Show Student Discourse
NOTE: Accept participants’ suggestions that can be justified. Make sure to ask participants to read MLR2 – Example 1 since this was not an MLR that was previously demonstrated. Point out that this MLR provide greater depth to the Monitoring Practice as it calls specifically for a focus on the language students are using to make meaning of the mathematics.
Activity Source: Open Up Resources, Grade 6, Unit 2, Lesson 9, Activity 9.3 Moving for 10 seconds. Available online at

49. PROMOTING MATHEMATICAL DISCOURSE MLR6: Three Reads
Supports the development of “reading like a mathematician”
Purpose of reads:
1st read: What is the problem about? (Context)
2nd read: What is the question? (Purpose of work)
3rd read: What information is important? (Discerning important information)
1 min
Speaker Notes:
Facilitator
We will be experiencing the Three Reads with this problem to practice how to anticipate student solutions/strategies and engagement using the Three Reads MLR.
Read: How do you use the Three Reads to support access to the problem?
*1st read: Teacher reads aloud the problem and the student creates a picture in their mind. Read problem out loud first before moving on to next slide.
2nd Read: Project problem and all follow along as one student reads it aloud.
Tip: You can hide the actual question and have students hypothesize what kinds of question could be asked with that information. Then show the question.
3rd Read: Together identify the important pieces of information in the problem.

50. What are some potential benefits to masking select
PROMOTING MATHEMATICAL DISCOURSE MLR6: Three Reads—Moving for 10 Seconds
Lin and Diego both ran for 10 seconds, each at a constant speed. Lin ran 40 meters and Diego ran 55 meters. 
Who was moving faster? Explain how you know.
How far did each person move in 1 second? If you get stuck, consider using diagrams to represent the situations.
Han ran 100 meters in 20 seconds at a constant speed. Is this speed faster, slower, or the same as Lin’s? Diego’s?
5 min
Speaker Notes:
Facilitator moves:
Say: Masking the numbers is not necessary to engage students in the Three Reads MLR.
CLICK
Turn & Talk: However, what benefits do you see in masking select portions of the problem as we did here?
Possible answers include helps students:
Focus on decoding the context of the problem,
Identify information needed to make meaning of the problem, and
Focus on the design of possible solution strategies.
Unmask numbers in the problem so students can solve it.
REFERENCE: How do you use the Three Reads access to the problem?
1st read: Teacher reads aloud the problem and the student creates a picture in their mind.
2nd Read: Project problem and all follow along as one student reads it aloud.
3rd Read: Together identify the important pieces of information in the problem.
Activity Source: Open Up Resources, Grade 6, Unit 2, Lesson 9, Activity 9.3 Moving for 10 seconds. Available online at
TURN & TALK:
What are some potential benefits to masking select
portions of the problem?

51. PROMOTING MATHEMATICAL DISCOURSE Anticipating
Brainstorm ways the problem might be solved.
What are students likely to produce?
Which problems will most likely be the most useful in addressing the mathematics?
How will you elicit deeper student thinking?
5 min
Speaker Notes:
Still referring to the Moving for 10 Seconds task.
Now let’s extend this by actively anticipating what students will think, record, and do.
Share within your table group ways students may have solved this problem and record your observations on the chart.
Work through the first and second column on misconception/strategy template for Moving for 10 Seconds.
Then turn attention to column 3, questions and responses, for the Moving for 10 Seconds.
What questions could you ask to elicit student thinking and deepen understandings?
Handout: 5 Practices Chart.
9.3 Moving for 10 seconds

52. PROMOTING MATHEMATICAL DISCOURSE Monitoring
A. Model
What do I know that the student understands?
What am I unsure whether the student understands?
What do I know the student does not understand?
What is the level of cognitive demand for this strategy?
What questions can I ask to determine where the understanding stops? 4
Lin
10 s 5
Diego
3 min
Speaker Notes:
SAY: This is one way I thought students could model this problem. I was also wondering about these things and I wrote questions to help me remember.
Are there any other ways you have come up with?

53. PROMOTING MATHEMATICAL DISCOURSE Anticipating & Monitoring
Model
3 min
Speaker Notes:
Popcorn Table Group Answers

54. Exchange with your partner, and offer other strategies.
PROMOTING MATHEMATICAL DISCOURSE Anticipating & Monitoring: Let’s Practice
Pick one lesson you brought with you and select the main task for that lesson.
Work out the task yourself. Determine if it is a high cognitive demand task that would lead to productive math discourse.
Fill in as many rows of the Misconception/Strategy Template as you can.
Exchange with your partner, and offer other strategies.
20 min
Speaker Notes:
Depending on group size, consider mixing the group to have participants talk with someone new.
SAY: Now it’s your turn. Let’s practice.
Read through the expectations on the slide and release the group. They should work with the unit they brought with them and that they worked with yesterday. If the choose a task with low cognitive demand, tell them to skip it and go back into the lessons to find one with high cognitive demand: at least level 3.
Monitor Time and and transition to the next step.

55. How will you grow their practice?
Think about what you’ve learned today about the impact of quality tasks and productive discourse in mathematics.
How does the use of mathematical language routines support access for all students to tasks with high levels of cognitive demand?
How does engaging students in math discourse contribute to an equitable math environment in math classes?
What are your development moves with the educators under your influence?
How will you grow their practice?
7 min
Speaker’s Notes:
Give them timeto reflect and/or write. Then if time, open it up to a whole group discussion.
IMAGE CREDIT

56. Feedback
Please fill out the survey located here:
Click “Explore the Agenda” on the top of the page.
Click “Details” on the center of the page.
Runtime: 5 minutes
Big Ideas:
Guide participants to the survey on the StandardsInstitutes.org website.
Say: Please fill out the survey to help us improve! 49

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