James Calleja
©2015
Plan for the Session
Thinking about Questioning
Task
Topic
Time
Introduction
Introducing the session: What do you expect to get from today’s session? Aims of the session
¼ h
Working on a Task
Individual work followed by collaborative work on the ‘Chicken Run Problem’
¼ h
A look at Vignettes
Reflections and discussion on how two teachers, Mark and Amy, introduced the problem to their classes
½ h
Lesson Video: Towards Divergent Thinking
Watch and discuss a video showing Raymond working on the advice provided to him by Barbara to improve his questioning techniques
¾ h
Principles for Effective Questioning
Group discussion on research-‐based principles that may support teachers in effectively using questioning in their classrooms
½ h
Watching and Analyzing a Lesson
Watch and analyze the questioning techniques used by Joe with his Year 11 ‘high ability’ class
½ h
Planning a Lesson
Plan a lesson focusing on the effective use of questions
¾ h
Aims of the session For today’s session we will have the following aims: o To develop self-‐awareness through a critical analysis of questioning techniques o To identify key features and purposes of effective questioning o To enhance planning for and use of divergent/open questions in the mathematics class o To reflect on questioning techniques and principles in planning lessons that support inquiry-‐based learning
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Teaching and Learning Mathematics through Inquiry
WORKING ON AN INQUIRY-‐BASED TASK
20 min
THE CHICKEN RUN PROBLEM A farmer was putting a new chicken run up against a brick wall. He had 20 m of wire to put round the run. If he made a rectangle, investigate the biggest area that he could enclose.
Looking into ways in which students might try to solve the problem posed. You are asked to: •
First work individually on the problem
•
Then work in pairs to solve the problem
•
Finally present your solutions to the whole group
4 minutes 11 minutes 5 minutes
Space for working
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EXPLORING CLASSROOM VIGNETTES
30 min
Two secondary school teachers, Mark and Amy, presented ‘The Chicken Run Problem’ to their Year 11 (Form 5) mathematics classes. The following two vignettes show how each teacher introduced this problem.
Read the vignettes. Examine and discuss the types of questions that each teacher uses during the initial whole-‐class discussion of the problem. Vignette from Mark’s class
Mark projects the slide on the interactive board and then asks students to read the problem. After about thirty seconds, Mark starts with a whole-‐class discussion as follows… Mark: How long is the wire? Student A: 20 m Mark: (Pointing to the longest side) Do we know the length of the rectangle? Student B: No Mark: W hat can w e do about this? Student C: Mark the length y Mark: Good… and w hat about the w idth? Student D: Mark it x Mark: Very good… now w hat can w e find? Student E: The perimeter Student A: I can’t understand what we are doing sir! Mark: It’s easy look… let’s write d own an equation with x and y now. W hat w ill the equation look like? Student C: 2x + y = 20 Mark: Great job! Student A: I still don’t get w hat w e are doing sir! Weren’t we asked to work out the area? Mark: Yes, we will w ork that out now… Does anyone have an idea? Student C: W e can multiply the sides Mark: That’s right… and so the equation is? Student A: I don’t understand sir! Mark: W hat’s the formula for the area o f a rectangle? Student A: You multiply the length by the breadth. Mark: Good… you see… so what’s the equation for the area o f this rectangle? Student C: Area = x y
And the class went on solving the equations simultaneously and finally plotting a graph.
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Teaching and Learning Mathematics through Inquiry
Vignette from Amy’s class
Amy projects the slide on the interactive board and then asks students to read the problem. She encourages students to work on their own first, thinking about the problem for two minutes. After that Amy starts a whole-‐class discussion as follows… Amy: How can w e get started with this? Student A: I tried out some numbers. Amy: Can you say a little bit more on that? Student A: I put numbers on the sides that make up 20 when you add them up. Amy: Can anyone think of some examples? Student B: Yes miss… I did 1 and 18! Amy: W hy? Student B: Because 1 + 18 + 1 = 20 Amy: Can anyone think of some other numbers? Student C: I did 3 and 14… because 3 + 14 + 3 = 20 Amy: And is this a better guess? Student C: Yes miss… because the area w ill be 42 now! Student A: I think 5 and 15 w ould give the correct answer… as these two numbers give the largest area. It’s 75! Amy: Ok… any other ideas that you managed to come up with? Student D: I drew a rectangle but I didn’t mark one of the sides as it is touching the wall (as the one previously drawn on the board) Amy: Can anyone add anything to this? Student E: The three sides w ill then add up to 2 0 m Amy: Does anyone else agree with this? Student F: I do! Amy: Can you tell us why? Student F: Because when the farmer is trying to make a rectangle, there is no need to use any wire against the w all Amy: What do you notice about this? Can anyone elaborate on this? Student G: That the wire is used for three sides not four. Student A: But isn’t this the same idea as I had Student E: No… because w e do not know the lengths of the sides Amy: What else? Can someone else say a little bit more? Student H: It’s like adding those two equal sides to the other… without knowing the lengths Student D: How can w e do that miss w hen w e do not know the lengths? Amy: That’s an interesting observation! Can anyone provide some help here? Student F: Yes… if those two sides are equal then we can mark each side w ith an x, and we can mark the other side y Amy: Do you agree with this? Student I: W e will be working with algebra here. Miss, am I right? Amy: (Addressing Student J) What do you think about this comment? Student J: I agree because we only know the length of the wire… and nothing else! Amy: Ok… shall we fill in the sides then? Student F: I can do that! x x Amy: So, what equations can w e come up with? y Student I: I have one using the perimeter… it’s x + y + x = 20. Amy: Do you all agree w ith that? Student C: It can be simplified… 2x + y = 20 Amy: Any other idea? Student K: I think we need an equation about the area… because that’s what we are asked to find. Amy: I suggest that you now join in pairs and try to sort this out. Then, I will ask you to report on your work to the whole-‐class.
And the class went on working in pairs trying to solve the problem.
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REFLECTIONS ON THE VIGNETTES
The questions below are adapted from the PRIMAS PD materials available online: www.primas-‐project.eu For the next 15 minutes reflect on the following questions:
What different types of questions do Mark and Amy use? What different functions do their questions serve? What kind of discussion have their questions stimulated? Which of these types of questions do you usually use? Why? What mistakes do teachers tend to make when asking questions?
TOWARDS DIVERGENT QUESTIONS – A VIDEO
45 min
Raymond is a mathematics teacher who values professional development. In this video, Barbara, a mathematics education expert, observes Raymond. Barbara intends to provide Raymond with some advice about his teaching. We will watch the first 9 minutes of this video ‘Divergent Questions in 8th Grade Math’, and then discuss some advice that we would provide Raymond with. 20 minutes
Space for some notes you would like to take while watching the video. _____________________________________________________________________________________________ _____________________________________________________________________________________________ _____________________________________________________________________________________________ _____________________________________________________________________________________________ _____________________________________________________________________________________________ _____________________________________________________________________________________________ _____________________________________________________________________________________________ _____________________________________________________________________________________________
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Teaching and Learning Mathematics through Inquiry
Let’s now watch Barbara’s evaluation of the lesson (3 minutes). Be watchful and take note of her advice. We will be discussing that next. 10 minutes
_____________________________________________________________________________________________ _____________________________________________________________________________________________ _____________________________________________________________________________________________ _____________________________________________________________________________________________ _____________________________________________________________________________________________ _____________________________________________________________________________________________ _____________________________________________________________________________________________ _____________________________________________________________________________________________ Let’s look at the feedback Raymond got from Jerry, his mentor, and the suggestions that he tried to implement. Also note what Barbara is expecting to observe next time she visits his class and her final evaluation. 15 minutes
_____________________________________________________________________________________________ _____________________________________________________________________________________________ _____________________________________________________________________________________________ _____________________________________________________________________________________________ _____________________________________________________________________________________________ _____________________________________________________________________________________________ _____________________________________________________________________________________________ _____________________________________________________________________________________________
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PRINCIPLES FOR EFFECTIVE QUESTIONING
30 min
As a group, reflect on the following: What types of questions promote inquiry-‐based learning? Why? Provide examples of questions and strategies to do this effectively. _____________________________________________________________________________________________ _____________________________________________________________________________________________ _____________________________________________________________________________________________ _____________________________________________________________________________________________ _____________________________________________________________________________________________ _____________________________________________________________________________________________ The principles below are taken from the PRIMAS PD materials available online: www.primas-‐project.eu The following are five research-‐based principles for effective questioning
• • • • •
The teacher plans questions that encourage thinking and reasoning. Every student is included. Students are given time to think. The teacher avoids judging students’ responses. Students’ responses are followed up to encourage deeper thinking.
Working as a group, discuss the questions: Which of these principles do you usually implement in your teaching? Which principles do you consider most difficult to implement? Why?
_____________________________________________________________________________________________ _____________________________________________________________________________________________ _____________________________________________________________________________________________ _____________________________________________________________________________________________ _____________________________________________________________________________________________ _____________________________________________________________________________________________
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Teaching and Learning Mathematics through Inquiry
USING EFFECTIVE QUESTIONING
The questions below are taken from the PRIMAS PD materials available online: www.primas-‐project.eu Consider the following questions used at different phases of an inquiry-‐based lesson.
Beginning an Inquiry
• • • • • •
What do you already know that might be useful here? What sort of diagram might be helpful? Can you invent a simple notation for this? How can you simplify this problem? What is known and what is unknown? What assumptions might we make?
Progressing with an Inquiry
• • • • • • • • • • • • • •
Where have you seen something like this before? What is fixed here, and what can we change? What is the same and what is different here? What would happen if I changed this… to this...? Is this approach going anywhere? What will you do when you get that answer? This is just a special case of ... what? Can you form any hypotheses? Can you think of any counterexamples? What mistakes have we made? Can you suggest a different way of doing this? What conclusions can you make from this data? How can we check this calculation without doing it all again? What is a sensible way to record this?
Interpreting and evaluating the results of an Inquiry
• • • • • • • • •
How can you best display your data? Is it better to use this type of chart or that one? Why? What patterns can you see in this data? What reasons might there be for these patterns? Can you give me a convincing argument for that statement? Do you think that answer is reasonable? Why? How can you be 100% sure that is true? Convince me! What do you think of Anne's argument? Why? Which method might be best to use here? Why?
Communicating conclusions and reflecting
• • • • • • •
What method did you use? What other methods have you considered? Which of your methods was the best? Why? Which method was the quickest? Where have you seen a problem like this before? What methods did you use last time? Would they have worked here? What helpful strategies have you learned for next time?
Teachers plan effective questions beforehand. It is usually helpful to plan sequences of questions that build on and extend students' thinking. The teacher needs to remain flexible and allow time for students to follow up responses.
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FOCUSING ON AND ANALYZING JOE’S QUESTIONING
30 min
Watch this video of Joe’s lesson with his Year 11 group of students. Joe engages his students in discussion by assigning a matching cards task where students need to decide about matching equations to graphs. Watch and analyse the questions Joe uses to present the task, and later to assess students’ understanding as they work in small-‐groups. It is suggested that you analyse this clip through the five principles for effective questioning. That is, to what extent, does the teacher? 1. Plan questions that encourage thinking and reasoning _____________________________________________________________________________________________ _____________________________________________________________________________________________ _____________________________________________________________________________________________ 2. Include every student _____________________________________________________________________________________________ _____________________________________________________________________________________________ _____________________________________________________________________________________________ 3. Allow think-‐time _____________________________________________________________________________________________ _____________________________________________________________________________________________ _____________________________________________________________________________________________ 4. Avoid judging student responses _____________________________________________________________________________________________ _____________________________________________________________________________________________ _____________________________________________________________________________________________ 5. Follow-‐up responses to encourage deeper thinking _____________________________________________________________________________________________ _____________________________________________________________________________________________ _____________________________________________________________________________________________
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Teaching and Learning Mathematics through Inquiry
PLANNING A LESSON
45 min
Decide on a problem to try with your class. Use the ‘Planning for Effective Questioning’ table, provided in the next page, to plan your lesson that will engage students in thinking and reasoning. You may find the following questions taken from the PRIMAS PD materials useful. www.primas-‐project.eu
•
How will you organise the classroom?
•
How will you introduce the questioning session?
•
Which ground rules will you establish?
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What will be your first question?
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How will you give time for students to think before responding?
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Will you need to intervene at some point to refocus or discuss different strategies that they are using?
•
What questions will you use in the plenary discussion?
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PLANNING FOR EFFECTIVE QUESTIONING
The table below is taken from the PRIMAS PD materials available online: www.primas-‐project.eu
Arrange students so that they can see and hear one another as well Plan how you will as the teacher. You may need to rearrange chairs in a U shape or the arrange the room students could move and ‘perch’ closer together. Or maybe you will and the move to the back of the room so that the question is the focus of resources needed attention and not the teacher. Plan how you will Silence will be hard for you to bear in the classroom but the students introduce the may find it confusing or even threatening. questioning Explain why there will be times of quiet. For example: session
If you are using ‘No hands up’ then you will need to explain this to the students. Some teachers have had to ask their students to sit on their hands so that they remember not to put their hands up. The Plan how you will students will be allowed to put their hands up to ask a question, so if establish the a hand shoots up remember to ask them what question they would ground rules like to ask. The students may also be used to giving short answers so you could introduce a minimum length rule e.g. ‘your answer must be five words in length as a minimum’.
Plan the first question that you will use
Plan the first question and think about how you will continue. You cannot plan this exactly as it will depend on the answers that the students give but you might, for example, plan to take
• One answer and then ask others what they think about the reasoning given
• Two or three answers without comment then ask the next
person to say what is similar or different about those answers
• Will you allow 3-‐5 seconds between asking a question and expecting an answer?
Plan how you will give thinking time
• Will you ask the students to think – pair – share, giving 30
seconds for talking to a partner before offering an idea in whole class discussion?
• Will you use another strategy that allows the students time to think?
Plan how and when you will intervene
Will you need to intervene at some point to refocus students' attention or discuss different strategies they are using? Have one or two questions ready to ask part way through the lesson to check on their progress and their learning.
Plan what questions you Try not to pass judgments on their responses while they do this or could use for the this may influence subsequent contributions. plenary at the end of the lesson
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Teaching and Learning Mathematics through Inquiry
SESSION EVALUATION
10 min
Ø Briefly describe your experience during today’s session. _____________________________________________________________________________________________ _____________________________________________________________________________________________ _____________________________________________________________________________________________ _____________________________________________________________________________________________ Ø What did you feel un/comfortable doing during the session? Comfortable: _____________________________________________________________________________ _____________________________________________________________________________________________ _____________________________________________________________________________________________ Uncomfortable: __________________________________________________________________________ _____________________________________________________________________________________________ _____________________________________________________________________________________________ Ø I used to think... but now I know… I used to think ____________________________________________________________________________ _____________________________________________________________________________________________ Now I know ______________________________________________________________________________ _____________________________________________________________________________________________ Ø What will you take with you and try to implement in your class? _____________________________________________________________________________________________ _____________________________________________________________________________________________ _____________________________________________________________________________________________ Ø Any other comments/suggestions that you would like to add. _____________________________________________________________________________________________ _____________________________________________________________________________________________ Thank you for your participation and reflections.
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