1. Teaching for Understanding: Discourse and Purposeful Questioning
March 21, 2016
Session 2A
Vickie Inge
Joleen Lambert

2. Welcome & Table Group Introductions
Who are You Where do You Work

3. Teaching for Understanding: Discourse and Purposeful Questioning Framing Questions
How can purposeful questions and patterns of questions move students to math talk that promotes reasoning and sense making for deep understanding?
What are the five mathematics talk moves that support classroom discourse?
How can mathematics specialist and teacher leaders support teachers in being purposeful in their questioning?

4. Teaching for Deep Understanding Relational Thinking
With a shoulder partner, pick one of the aqua hexagons; each person has 1 minute to share what they think it means.
Teaching for Deep Understanding
Relational Thinking
Critical Thinking
Making Sense of Mathematics
Higher Level Questioning
Proficient in Math
Higher Order Thinking Skills
With a shoulder partner, pick one of the hexagons and in 1 minute each person shares what they think it means.
--Facilitator asks for show of hand for how many people thought about their word or phrase in different or slightly differing ways? Could this be an issue when talking with administrators and parents? Could it cause it confusions for students as they work with different teachers. Could it be an issue when teachers collaborate and discuss instruction or students?

5. Mathematical Proficiency
Conceptual
Understanding
Strategic
Competence/
Problem Solving
Procedural
Fluency
Adaptive Reasoning
Productive Disposition
Mathematical Proficiency
Mathematical
Proficiency
Students with deep understanding are Mathematical Proficient.

6. A student who is Mathematical Proficient demonstrates--
conceptual understanding - comprehension 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.
Handouts, page 2

7. Effective NCTM, Principles to Actions, p. 10
Teaching and Learning is one of the essential elements for a successful mathematics program as ( p. 5)
Transition: NCTM identified a framework that reflects the learning principals and the last 20 years of accumulated knowledge about mathematics teaching.
NCTM, Principles to Actions, p. 10

8. Effective Mathematics Teaching Practices
NOVICE
APPRENTICE
EFFECTIVE
PROFICIENT
Establish mathematics goals to focus learning
Implement tasks that promote reasoning and problem solving.
Use and connect mathematical representations
Facilitate meaningful mathematical discourse
Pose purposeful questions
Build procedural fluency from conceptual understanding.
Elicit and use evidence
of student thinking
Specialist and teacher leader decisions: How to help teachers move along a teacher practice continuum.

9. Questions are teachers’ tools to promote classroom discourse and set up that lightbulb moment!

10. Question Sort
Open the envelop on your table and work as table groups to sort the questions into exactly 4 non-overlapping groups or sets.
Analyze the type of “brain engagement,” thinking each set brings out in a student and develop a word or phrase that could be used to categorize the type of questions in each set.
Ask groups to give the 4 labels they come up with

11. Group Sharing
What descriptions for the classifications did your table group identify?
What discussions did you and your partner have about how to group the questions?
Does anyone have an example of a question that caused some differences of opinion?

12. Question Type– For What Purpose
Reaching a common understanding and language for discussing the type of “brain engagement,” thinking that different questions types elicit.
Principles to Actions: Ensuring Mathematical Understanding for All (page 36-37)
Ask groups to give the 4 labels they come up with

13. Question Type– For What Purpose
Identify what a student has to think about and demonstrate for each question type.
What seems to be an important distinction between type 1 and 2 questions and type 3 and 4 questions.
Read -- Pair -- Share
Principles to Actions: Ensuring Mathematical Understanding for All (page 36-37)
Ask groups to give the 4 labels they come up with

14. Gathering Information
Probing Thinking
Making the Mathematics Visible
Encouraging Reflection and Justification
Deep Understanding

15. Making Sense of Mathematics
Teachers’ questions are crucial in helping students make connections and learn important mathematics concepts.
Teachers need to know how students typically think about particular concepts, how to determine what a particular student or group of students thinks about those ideas, and how to help students deepen their understanding
Weiss & Pasley, 2004
Facilitation Suggestion
Summarize the Teaching and Learning Principle, noting the strong emphasis on promoting students’ ability to make sense of mathematical ideas and to reason mathematically. Ask them to keep this Principle in mind throughout the session and in particular, as they watch the video clips from the kindergarten classroom.

16. Principles to Actions Professional Learning Toolkit Website with Resources
8th grade prealgebra class in Pittsburg

17. Virginia Standard of Learning 2009 Mathematical Progression for Functional Thinking
recognize and describe a variety of patterns formed using numbers, tables, and pictures, and extend the patterns, using the same or different forms recognize, create, and extend numerical and geometric patterns describe the relationship found in a number pattern and express the relationship …b) write an open sentence to represent a given mathematical relationship, using a variable; identify and extend geometric and arithmetic sequences represent relationships with tables, graphs, rules, and words make connections between any two representations (tables, graphs, words, and rules) of a given relationship graph a linear equation in two variables.
Use this slide to share the critical reasons for teachers at all grade levels to know what came before and what comes after a particular topic within a big idea. The big idea hear is functional relationships and how to represent them. Even though we will be watching an 8th grade math class, with some tweaking the task could be used in early grade levels.

18. The Calling Plans Task
Long-distance company A charges a base rate of $5.00 per month plus 4 cents for each minute that you are on the phone. Long- distance company B charges a base rate of only $2.00 per month but charges you 10 cents for every minute used.
Part 1: How much time per month would you have to talk on the phone before subscribing to company A would save you money?
Part 2: Create a phone plane, Company C, that costs the same as Companies A and B at 50 minutes, but has a lower monthly fee than either Company A or B.
The video clips featured in this session focuses on students who are trying to create phone plans for Company C (part 2).
It is suggested that you engage teachers in solving the task prior to analyzing the video. This will take minutes. The lesson guide (9. Calling Plans-LESSON GUIDE-MS-Brovey) contains suggestions for conducting the lesson and sample solutions that may be helpful to you. In addition, using the learning goals on slide 5 to guide your discussion of the task will help participants understand what the teacher featured in the video was trying to accomplish.
If time is limited you may want to tell teachers that the cost of plans A and B can be determined by use for the equations shown below and ask teachers to explain what each equation means in the context of the problem. Then tell them that the phone plans both cost $7.00 for 50 minutes and ask them how they could verify this.
Cost A= .04m + $5.00
Cost B = .10m + $2.00
You could then provide four solutions that students came up with for Company C (shown below) and ask them: a) if the equations fulfill the specified conditions; b) how they could arrive that the equations; c) if there are any other equations that fit the criteria (without rounding).
Cost C1 = .11m + $1.50
Cost C2 = .12m + $1.00
Cost C3 = .13m + $ .50
Cost C4 = .14m

19. Pose Purposeful Questions
Effective Questions should:
Gather information about and reveal students’ current understandings;
Probe thinking and encourage students to explain, elaborate, or clarify their thinking;
Make the mathematics more visible and accessible for student examination and discussion and connect mathematical structures; and
Encourage reflection and justification to reveal deeper understanding including making generalizations and developing arguments.
Deeper Understanding
Facilitation Suggestions
Summarize the key features of this Mathematics Teaching Practice.

20. The Calling Plans Task – Part 2 The Context of Video Clip 1
Prior to the lesson:
Students solved the Calling Plans Task – Part 1.
The tables, graphs and equations they produced in response to that task were posted in the classroom.
Video Clip 1 begins immediately after Mrs. Brovey explained that students would be working on the Calling Plans Task – Part 2 and read the problem to students. Students first worked individually and subsequently worked in small groups.

21. Lens for Watching Video Clip 1
As you watch the first video clip, pay attention to the teacher and student indicators associated with Pose Purposeful Questioning .
Think About’s:
What types of questions is the teacher using?
What can you say about the pattern of questions?
What do you notice about the student actions?
Lens for Watching the Video Clip 1 - Time 1
As you watch the video, make note of what the teacher does to support student learning and engagement as they work on the task.
In particular, identify any of the Effective Mathematics Teaching Practices that you notice Mrs. Brovey using.
Be prepared to give examples and to cite line numbers from the transcript to support your claims.
Here are some of the things that teachers might notice:
Many of the questions pressed students to explain what they did (lines 2, 4, 6, 25, 28, 31, 77) and probed for meaning (lines 8, 13, 16, 21, 50, 54, 57, 59). The teacher first tried to understand what the student was doing from the students’ point of view. These types of questions could be referred to as ASSESSING QUESTIONS - since the purpose is to determine what the student knows and understands about what they have done. The questions are closely tied to the what the students has produced.
Other questions serve to press students to move beyond or challenge their current thinking. For example, in lines the teacher asks “I am going to ask you to see if there is another plan that you could have” challenging the student’ statement that “1 is practically the only thing you could use.” Later in lines she challenges another group to explain where the 50¢ comes in, how the three questions are related, and to find a fourth equation. In this way she is pressing them to look for a ways to explain the pattern that they noticed. These types of questions could be referred to as ADVANCING QUESTIONS - since the purpose is to more students beyond where they currently are and to explore why things work mathematically.
In general the teacher seems to begin her interactions with a group by posing assessing questions. When these questions give her a good sense of where the students are she asks an advancing question. It is important to note that she stays with a group in order to hear their answers to the assessing questions but leaves a group to explore the advancing question on their own. In so doing she is sending an important message to students that sees them as capable of making progress without her closely monitoring their work.

22. Patterns of Questioning
Initiate-Response-Evaluate (IRE) Questioning
Teacher asks a question to quickly gather factual information with a specific response in mind. A student responds and then the teacher evaluates the response.
Student has limited opportunity to think.
Teacher has no access to whether or how students are making sense of the mathematics.
Explain the two types of questions: funneling and focusing. Funneling questions are good “quick assessments” for the teacher. Focusing questions are good for revealing misconceptions, developing reasoning skills, practicing the language of argument/conjecture/discourse. Focusing questions should be in the context of problems solving situations.
Principles to Actions, page 37

23. Patterns of Questioning:
Funneling Questioning
A teacher asks a series of questions to guide students through a procedure or to a desired result.
Teacher engages in cognitive activity about the idea and determining the next question to ask to guide or lead the student to a particular idea.
Student merely answering questions – often without seeing connections.
Principles to Actions, page 37

24. Patterns of Questioning
Focusing Questioning
A teacher listens to student responses and uses student response to probe their thinking rather than leading them to how the teacher would solve the problem.
Allows teacher to learn about student thinking.
Requires students to articulate and explain their thinking.
Promotes making connections.
Principles to Actions, page 37

25. Patterns of Questioning
Funneling Questions
How many sides does that shape have?
Which side is longer?
Is this angle larger?
How do you know?
Focusing Questions
What have you figured out?
Why do you think that?
Does that always work? If yes, why? If not, why not? When not?
Is there another way?
How are these two methods different? How are they similar?
Explain the two types of questions: funneling and focusing. Funneling questions are good “quick assessments” for the teacher. Focusing questions are good for revealing misconceptions, developing reasoning skills, practicing the language of argument/conjecture/discourse. Focusing questions should be in the context of problems solving situations.
Principles to Actions, page 37 \

26. Another reason for Purposeful Questioning!
……formative assessment requires considerable changes in what teachers do daily… More basketball and less ping-pong. Dylan Wiliam

27. Video Clip 2 focuses on the discussion between teacher and students regarding the patterns they notice.
Following individual and small group work, Mrs. Brovey pulls the class together for a whole group discussion. Several different equations that satisfy the conditions of the problem are offered by students. Jake, a student in the class then proposed a theory that every time the rate increases by 1 cent the base rate decreases by 50 cents. Mrs. Brovey records the four possible phone plans for Company C (shown below) on the board and ask the class what patterns they see. C = .14m
C = .13m + $ .50
C = .12m + $1.00
C = .11m + $1.50
Have the equations and Jake’s statement written on chart paper.

28. Lens for Watching Video Clip 2
As you watch the second video this time, pay attention to the questions the teacher asks. Specifically:
To what extent do the questions encourage students to explain, elaborate, or clarify their thinking?
To what extent do the questions make mathematics more visible and accessible for student examination and discussion?
How are the questions similar to or different from the questions asked in video clip 1?
Many of the questions pressed students to explain what they did (lines 2, 4, 6, 25, 28, 31, 77) and probed for meaning (lines 8, 13, 16, 21, 50, 54, 57, 59). The teacher first tried to understand what the student was doing from the students’ point of view. These types of questions could be referred to as ASSESSING QUESTIONS - since the purpose is to determine what the student knows and understands about what they have done. The questions are closely tied to the what the students has produced.
Other questions serve to press students to move beyond or challenge their current thinking. For example, in lines the teacher asks “I am going to ask you to see if there is another plan that you could have” challenging the student’ statement that “1 is practically the only thing you could use.” Later in lines she challenges another group to explain where the 50¢ comes in, how the three questions are related, and to find a fourth equation. In this way she is pressing them to look for a ways to explain the pattern that they noticed. These types of questions could be referred to as ADVANCING QUESTIONS - since the purpose is to more students beyond where they currently are and to explore why things work mathematically.
In general the teacher seems to begin her interactions with a group by posing assessing questions. When these questions give her a good sense of where the students are she asks an advancing question. It is important to note that she stays with a group in order to hear their answers to the assessing questions but leaves a group to explore the advancing question on their own. In so doing she is sending an important message to students that sees them as capable of making progress without her closely monitoring their work.
What are teachers doing?
Advancing student understanding by asking questions that build on, but do not take over or funnel, student thinking.
Making certain to ask questions that go beyond gathering information to probing thinking and requiring explanation and justification.
Asking intentional questions that make the mathematics more visible and accessible for student examination and discussion.
Allowing sufficient wait time so that more students can formulate and offer responses.
What are students doing?
Expecting to be asked to explain, clarify, and elaborate on their thinking.
Thinking carefully about how to present their responses to questions clearly, without rushing to respond quickly.
Reflecting on and justifying their reasoning, not simply providing answers.
Listening to, commenting on, and questioning the contributions of their classmates.

29. Managing Effective Student Discourse
Why is high level classroom discourse so difficult to facilitate?
What knowledge and skills are needed to facilitate productive discourse?
Students are reluctant to engage/ classroom environment not comfortable for students to take risks
Coverage of material - Not enough time for students to respond via discussion
Beliefs about what it means to teach
Question whether discussion promotes learning
Lack of skill in posing questions for discussion – or lack of awareness of the questions you actually ask
Teachers afraid of losing control of the class (of the thinking)
Vocabulary/ meaning often challenging to students. Teachers afraid they won’t understand students’ ideas
Teachers have preconceived notions of “the answer” and are not open to other possible responses
Establish a classroom environment supportive of risk taking
Deep content knowledge
Listening and patience
Good questioning skills
How to use a wrong answer in pedagogically productive ways
Keep the mathematical goal of the lesson in mind
Walk and Talk:
Meet up with someone from a different table to discuss the question the facilitator indicates.

30. What are teachers doing? What are students doing?
Pose Purposeful Questions Teacher and Student Actions ( Principles to Actions page 41)
What are teachers doing?
What are students doing?
Advancing student understanding by asking questions that build on, but do not take over or funnel, student thinking.
Making certain to ask questions that go beyond gathering information to probing thinking and requiring explanation and justification.
Asking intentional questions that make the mathematics more visible and accessible for student examination and discussion.
Allowing sufficient wait time so that more students can formulate and offer responses.
Expecting to be asked to explain, clarify, and elaborate on their thinking.
Thinking carefully about how to present their responses to questions clearly, without rushing to respond quickly.
Reflecting on and justifying their reasoning, not simply providing answers.
Listening to, commenting on, and questioning the contributions of their classmates.
Handout: 5-TeacherStudentActions-Questions-ES-Smith.pdf
Facilitation Suggestions
Ask each participant to identify a shoulder partner. One person will study the “teacher actions” and the other person will study the “student actions.” They should use the handout to highlight or mark key ideas and to make note of important actions. The partners should then turn and summarize some of the key ideas from their respective list for each other.
As a whole group, discuss: Based on the indicators or actions in the list, what does posing purposeful questions really entail? What will you see teachings doing? What will you see students doing?

31. “Our goal is not to increase the amount of talk in our classrooms, but to increase the amount of high quality talk in our classrooms—the mathematical productive talk.” –Classroom Discussions: Using Math Talk to Help Students Learn, 2009

32. Planning for Mathematical Discussion
What do We Talk About
Productive Talk Formats
1. Mathematical Concepts
2. Computational Procedures
3. Solution Methods and Problem-Solving Strategies
4. Mathematical Reasoning
5. Mathematical Terminology, Symbols, and Definitions
6. Forms of Representation
Whole-Class Discussion
Small-Group Discussion
Partner Talk What Do We Talk About?
Chapin S., O’Connor, C., & Canavan Anderson, N. (2003). Classroom discussions: Using math talk to help students learn. Sausalito, CA: Math Solutions.

33. A survey of multiple studies on questioning support the following:
Plan relevant questions directly related to the concept or skill being taught.
Phrase questions clearly to communicate what the teacher expects of the intent and quality of students’ responses.
Do not direct the question to anyone until after it is asked so that all students pay attention.
Allow adequate wait time to provide students time to think before responding.
Encourage and design for wide student participation.

34. How can we support teachers in purposeful questioning. (HO 7)
Bridging to Practice
How can we support teachers in purposeful questioning. (HO 7)

35. Bridging to Practice Analyzing the Challenging Situation
Some Ideas for Trouble Shooting Challenges
My students will not talk
The same few kids do all the talking
3. Should I call on students who _______
4. My students will talk, but they will not listen
5. What to do if students provide a response I do not understand
6. I have students at different levels
What to do when students are wrong
The discussion is not going anywhere--or at least not where I planned
9. Answers or responses are superficial
What if the first speaker gives the right answer
What to do for English Language Learners

36. Change is Not Easy or Comfortable
This is “Grand Avenue”
The point of the cartoon is for teachers to recognize it’ll be easy to go back to old habits because we are used to them. We all need support to adopt and maintain new habits and ways of doing things.

37. Support—Support--Support
Come on team we can do this together for the good of the students! OR
This is “Grand Avenue”

38. Transition from toword
Less Of
More Of
Rapid fire teacher questions
Questions directed to the whole class, with few students responding.
Questions that ask students to state small pieces of knowledge unrelated to the larger context.
Questions that ask what students know
Questions with quick answers.
Questions limited to current understanding
Plan activities in a lesson
Thoughtful questions that are linked to push student thinking
Questions directed to student partners or small groups.
Questions that require connections between and among concepts
Questions that ask how students know.
Questions with wait time for student thinking
Questions that extend understanding to a new context
Plan questions in the activities based on expected and unexpected student responses.

39. Question Types to Avoid
yes-no (These draw one-word -- Yes or No -- responses: "Does the square root of 9 equal 3?")
tugging (These place emphasis on rote: "Come on, think of a third reason.")
guessing (These encourage speculation rather than thought: "How many ways can ½ be written?")
leading (These tend to give away answers: "How do right angles and parallel sides help to build rectangles?")
vague (These don't give students a clue as to what is called for: "Tell us about graphs.")

40. 8 ways teachers can talk less and get kids talking more
kids-talking.html

41. "What’s The Big Idea?" November 2006 K-12 Alliance/WestEd Teacher to
Student
Student to Student
Product
"What’s The Big Idea?"
November 2006
K-12 Alliance/WestEd