1. Purposeful Questioning 6-8 September 24, 2015
The Virginia Council of Mathematics Specialist Conference
NASA Langley Research Center,
Hampton, VA
Purposeful Questioning 6-8 September 24, 2015
Vickie Inge

2. Overview of the Session
Examining the types of questions and the patterns of questioning teachers use in the classroom.
Watch video clips of an 8th grade pre-algebra class working on the Calling Plans Task.
Relate teacher and student actions in the video to the effective teacher and student actions as well as the types of questions for purposeful questioning.
Consider how to support teachers in using purposeful questioning.
Session Outline
Overview of the Session – 2 minutes
Question Sort minutes
Overview of Types of Questions and Patterns of Questions Matrix– 10 minutes
Solve and discuss the Donuts Task making connections to the VASOL – 10 minutes
Watch and discuss Donut Task Video Clip #1 – 15 minutes (NCTM Website
Watch the video
Make initial observations of teacher and student actions
Discuss as a whole group
Watch and Video Clip #2 – 15 minutes
Discuss observations of teacher and student actions
Discuss next steps in supporting teachers in applying the ideas to teachers’ own classroom – 10 minutes
TOTAL TIME – 75 minutes

3. 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

4. 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 engagement and thinking each set brings out in the class room 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

5. Making Sense of Mathematics
An excellent mathematics program requires effective teaching that engages students in meaningful learning through individual and collaborative experiences that promote their ability to make sense of mathematical ideas and reason mathematically.
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.
National Council of Teachers of Mathematics. (2014). Principles to actions: Ensuring mathematical success for all. Reston, VA: Author. (p. 7)

6. Work with a partner to explore the information contained in he table.
In the 2nd Column you will see the identifying name for each type of question that the authors of Principles to Actions: Ensuring Mathematical Success for All use.
Column 3 provides a description or purpose of each type of question.
Column 4 provides examples of specific questions representing each type.
Column 1 identifies the two patterns of questioning observed in most classrooms and the set up of the table suggests the type of question that typically is used in each pattern.
Work with a partner to explore the information contained in he table.
Overview of the Relationship Between Question Types and Patterns of Questioning Table, HO page 3
Note that different researchers use various names for the categories, how every the purpose of the categories is very similar across researchers.

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

8. Mathematics Learning Goals
Ms. Bovery’s
Mathematics Learning Goals
Students will understand that:
the point of intersection is a solution to each equation (Companies A, B, and C);
the rate of change (cost per minute) determines the steepness of the line;
if the y-intercept (monthly base rate) is lowered then the rate of change (cost per minute) must increase in order for the new equation to intersect the other two at the same point.

9. 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

10. Pose Purposeful Questions
Effective Questions should:
Reveal students’ current understandings;
Encourage students to explain, elaborate, or clarify their thinking; and
Make the mathematics more visible and accessible for student examination and discussion.
Facilitation Suggestions
Summarize the key features of this Mathematics Teaching Practice.

11. Pose Purposeful Questions Teacher and Student Actions (HO page 2)
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?

12. The Context of the Video Segment
The Calling Plans Task
The Context of the Video Segment
School: Pittsburgh Classical Academy, Pittsburgh, PA
Principal: Valerie Merlo
Teacher: Mrs. Elizabeth Brovey, Math Coach
Class: Pre-Algebra 8th Grade Class
Curriculum: Connected Mathematics Project 2
Size: 27 students
At the time the video was filmed, Elizabeth Brovey was a coach at Classical Academy in the Pittsburgh Public School District. The students are mainstream eighth grade Pre-Algebra students. The lesson occurred in April.
Facilitation Suggestions
Summarize the context of the video.
To establish a norm for the viewing and discussion of the video, remind teachers that Mrs. Brovey was willing to share her classroom instruction so that we could use the video to analyze and discuss the teaching of mathematics. It is important to be respectful of her work. We need to remember that we are only seeing a “snapshot” of her work.

13. 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.

14. 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.

15. 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

16. 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.

17. Managing Effective Student Discourse
Why is high level classroom discourse so difficult to facilitate?
What knowledge and skills are needed to facilitate productive discourse?
Why is it important? (i.e. Why do we care?)
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

18. Questioning Techniques
Funneling Questions
How many sides does that shape have?
Is this angle larger?
What is the product?
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.

19. Questioning: Funneling or Focusing
Funneling occurs when a teacher asks a series of questions to guide students through a procedure or to a desired result.
Teacher engages in cognitive activity
Student merely answering questions – often without seeing connections

20. Questioning: Funneling or Focusing
Focusing requires the teacher to listen to student responses and guide them based on what students are thinking rather than how the teacher would solve the problem.
Allows teacher to learn about student thinking
Requires students to articulate their thinking
Promotes making connections

21. “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

22. Planning for Mathematical Discussion
Productive Talk Formats
What do We Talk About
Whole-Class Discussion
Small-Group Discussion
Partner Talk What Do We Talk About?
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
Chapin S., O’Connor, C., & Canavan Anderson, N. (2003). Classroom discussions: Using math talk to help students learn. Sausalito, CA: Math Solutions.

23. 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.

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

25. 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

26. 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.

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

28. Resources
Chapin S., O’Connor, C., & Canavan Anderson, N. (2003). Classroom discussions: Using math talk to help students learn. Sausalito, CA: Math Solutions. Huinker, D., & Freckmann, J. L. (2004). Focusing conversations to promote teacher thinking. Teaching Children Mathematics, 10(7) National Council of Teachers of Mathematics. (2014). Principles to Actions: Ensuring Mathematical Success for All. Reston, VA: National Council of Teachers of Mathematics. National Council of Teachers of Mathematics. (n.d.) Principles to Actions Professional Learning Toolkit. Retrieved September From Reinhart, S. D. (2000). Never say anything a kid can say. Mathematics Teaching in the Middle School, 5(8) 478–483.
Author. (Date published if available; n.d.--no date-- if not). Title of article. Title of web site . Retrieved date. From URL.

29. Resources
Sullivan, P., & Lilburn, P. (2002). Good questions for math teaching: Why ask them and what to ask. Grades K-6. Sausalito, CA: Math Solutions. Schuster, L., & Anderson, N. C. (2005). Good Questions for math teaching: Why ask them and what to ask. Grades 5-8. Sausalito, CA: Math Solutions. Small, M. (2012). Good Questions – Great Ways to Differentiate Mathematics Instruction. New York, NY: Teachers College Press. Smith, M. S., & Stein, M. K. (2011). 5 Practices for Orchestrating Productive Mathematics Discussions. Reston, VA: National Council of Teachers of Mathematics and Thousand Oaks, CA: Corwin Press. Smith, M.S., Hughes, E.K., & Engle, R.A., & Stein, M.K. (2009). Orchestrating discussions. Mathematics Teaching in the Middle School, 14 (9),