James Calleja
©2015
Plan for the Session
Thinking about Collaboration
Task
Topic
Time
Introduction
Introduction to the topic of discussion What do you expect to get from today’s session? Aims of the session
¼ h
Working on a Task
Collaborative work on the ‘Human Tower’ assessment task
½ h
Reflecting about your experience working in a small-‐group
½ h
Follow-‐up Reflection Lesson Video: Students’ Talk
Teacher Concerns Watching and Analyzing a Lesson
Watch and discuss a video showing Year 7 students discussing the ‘Human Tower’ task
½ h
Dylan Williams Video: Group goals Obstacles and concerns with collaborative work When should I engage my students in a collaborative activity?
½ h
Setting the ‘ground rules for discussion’ Watching and analyzing a discussion lesson The teacher’s role during small-‐group work
¾ h
What do we Reflections and follow-‐up discussion about the learn from this classroom setting, the phases of the lesson, the lesson? teacher’s role and student learning
½ h
Aims of the session For today’s session we will have the following aims: o To explore opportunities in which collaborative work may benefit student learning o To reflect upon concerns in introducing and managing small-‐group collaboration and discussion o To understand the roles of students and the teacher within a collaborative environment o To reflect critically on a small-‐group collaboration based lesson o To experience features and aspects that promote effective small-‐group discussion
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Teaching and Learning Mathematics through Inquiry
WORKING ON A TASK IN SMALL-‐GROUPS
20 min
Work on the ‘Human Tower’ task (see pages 4 and 5), looking at ways in which students might try to solve the problem posed. You will: •
First reflect on the problem individually
•
Then work as a group (of 4) to solve the problem
(12 minutes)
•
Finally present your solutions to the whole group
(5 minutes)
(3 minutes)
REFLECTING ON YOUR EXPERIENCE WORKING IN A SMALL-‐GROUP
20 min
Some of the questions below are adapted from the PRIMAS PD materials. PD Module 5: Students working collaboratively available online: www.primas-‐project.eu As a whole group you are asked to reflect on the following questions: 1. Was it helpful for you to first have some time to think about the problem before you discussed it in your group? 2. How would you describe your role in the group? Did someone take over? Was someone more of a passenger? Were you given the opportunity to contribute your ideas to the group? Did you consider the alternative views of everyone in the group? 3. Did each member assume a different role? Why do you think this happened? What role did you assume while working on the task? Did anyone decide about this? 4. Did you feel uncomfortable or threatened? If so, why? 5. Did the discussion stay ‘on task’ or were you ‘wandering’ at times? 6. How do you see this collaborative task in your classroom?
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Imagine your group of friends is asked to stand on each other’s shoulders to build a human tower like the one similar to the one in the picture below.
How high would your tower be? The challenge for your group is to design the best structure for building your tower. Your tower needs to be made up of at least three to four people. Think about every detail. Show your design and describe how high this tower would be.
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Teaching and Learning Mathematics through Inquiry
The photo shows a ‘CASTELL’. A Castell is a human tower built traditionally in festivals by people in Catalonia (Spain). Explain how you would estimate the following: a) The number of people involved in building the castell shown.
b) The height of the human tower in the picture.
__________________________________________________________________________
Fill in Group Members’ Names
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LOOKING AT STUDENT TALK IN SMALL-‐GROUP DISCUSSIONS Neil Mercer (1995) identified three typical ways of talking among students as they work in small-‐groups – disputational talk, cumulative talk and exploratory talk. ü
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Disputational talk Involves disagreements and individual rather than collective decision-‐ making. Exchanges are usually brief and consist of assertions or counter-‐ assertions. Cumulative talk Represents a building of ideas based on each other’s suggestions aimed at providing a common consensus. Exchanges in this type of talk are usually repetitions, confirmations and elaborations Exploratory talk Characterised by critical but constructive engagement with each other’s ideas. Challenges are justified and alternatives suggested. Joint agreement in decision-‐making is the end result.
Disputational talk, in which students simply disagree and go on to make individual decisions, is not beneficial. Cumulative talk, in which students build uncritically on what each other has said, is also undesirable. For true collaborative work, students need to develop exploratory talk consisting of critical and constructive exchanges, where challenges are justified and alternative ideas are offered. The most helpful talk appears to be that where the participants work on and elaborate each other’s reasoning in a collaborative, rather than competitive atmosphere (PRIMAS, 2011). Mercer (1995) argues that in planning collaborative activities in the mathematics classroom, we should be aiming to promote exploratory talk: By incorporating both conflict and the open sharing of ideas, represents the more 'visible' pursuit of rational consensus through conversation. More than the other two types, it is like the kind of talk which has been found to be most effective for solving problems through collaborative activity. (p. 105)
Reference: Mercer, N. (1995). The guided construction of knowledge: Talk amongst teachers and learners. Clevedon: Multilingual Matters. PRIMAS (2011). Students Working Collaboratively: How can we foster scientific discussion? PD Module 5: Students working collaboratively. Available online on: www.primas-‐project.eu/artikel/en/1221/Professional+development+modules/view.do
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Teaching and Learning Mathematics through Inquiry
WATCHING A LESSON VIDEO
10 min
You will now watch a video of a teacher, Joanne, using the ‘Human Tower’ task with her Year 7 class. Pay particular attention to how the students work in their groups. You are asked to determine whether students’ talk is more disputational, cumulative or exploratory. You may want to write down some notes/points that you might consider important. ___________________________________________________________________________________________________ ___________________________________________________________________________________________________ ___________________________________________________________________________________________________ ___________________________________________________________________________________________________ ___________________________________________________________________________________________________ ___________________________________________________________________________________________________
REFLECTING ON STUDENTS’ TALK
15 min
What are your comments on the lesson? Who generates the mathematical ideas that get discussed? Who evaluates and/or responds to these ideas? In your opinion, do the discussions help or hinder learning? Can you identify groups of students in the video in which this occurred? Would you say that there is evidence of disputational, cumulative and/or exploratory talk within the groups?
VIDEO: DEVELOPING GROUP GOALS
10 min
Dylan Williams explains that collaborative learning is effective when teachers create ‘group goals’ – getting students working as a group rather than working in a group. To achieve this, teachers need to create two conditions for students in their classrooms: collective responsibility and individual accountability. How can students become collectively responsible and individually accountable?
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TEACHER CONCERNS WITH IMPLEMENTING COLLABORATIVE WORK
15 min
While working in pairs:
•
Think about potential obstacles and concerns that may hinder teachers from engaging students with collaborative work.
•
Pick out one particular concern that a teacher may have. Think about how you would respond to that.
•
Share your choice together with your response.
USING COLLABORATIVE WORK As a whole group, think and share your ideas about this question:
15 min
For which lessons may collaborative work be a suitable pedagogy to use in the mathematics class?
TEACHING STUDENTS HOW TO PARTICIPATE IN DISCUSSION
30 min
Have a look at this scenario! Stephan has been teaching mathematics for twelve years. He is very enthusiastic about teaching and likes to try out new ideas in his class. Stephan has learned that his students always seem to lack conceptual understanding. They seem to find it hard to tackle the challenging problems that he usually presents them with. This year Stephan is willing to address this by using collaborative work with his class. He thinks that it would be a good idea to establish some ‘ground rules for discussion’ with his class. What ‘ground rules for discussion’ would you suggest? How would it be best for Stephan to establish these rules? How could Stephan use these rules to engage students in valuable discussion?
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Teaching and Learning Mathematics through Inquiry
Now let’s have a look at Stephan’s list of ground rules… The list below is taken from the PRIMAS PD materials available online: www.primas-‐project.eu Here are some suggested 'ground rules' for students to use as they work in groups. These could be displayed and reinforced over time. Maybe you could involve your class in drawing up a similar list. 1.
Give everyone in your group a chance to speak
"Lets take it in turns to say what we think". "Claire, you haven't said anything yet."
2.
Listen to what people say
"Don't interrupt – let Sam finish". "I think Sam means that..."
3.
Check that everyone else listens
"What did Sue just say?" "I just made a deliberate mistake – did you spot it?
4.
Try to understand what is said
"I don't understand. Can you repeat that?" "Can you show me what you mean?"
5.
Build on what others have said
"I agree with that because..." "Yes and I also think that..."
6.
Demand good explanations
"Why do you say that?" "Go on... convince me."
7.
Challenge what is said
"That cannot be right, because..." "This explanation isn't good enough yet."
8.
Treat opinions with respect
"That is an interesting point." "We all make mistakes!"
9.
Share responsibility
"Let's make sure that we are all able to report this back to the whole class."
10. Reach agreement
"We've got the general idea, but we need to agree on how we will present it."
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WATCHING A DISCUSSION LESSON BY STEPHAN
15 min
This 15-‐minute video captures a 40-‐minute lesson in which my Year 7 students (11 year-‐olds) work on an inquiry-‐based task, dealing with properties of triangles. Before you watch the video, I would like to give you some useful information about my class. The 26 girls taking part are the highest set class. They are, by and large, considered to be average to high ability students. This is not the first experience that my students had working in small groups. Throughout the year, I provided students with numerous opportunities of learning mathematics within a collaborative setting. The topic was ‘new’ to the students – in the sense that I planned this task to serve as a topic starter. Students’ prior knowledge relied within what they had done and recalled from the previous years, that is, while they were in primary. Indeed, I knew my students well enough to believe that they would, at least, be able to remember the names of particular triangles and identify some basic properties. This discussion lesson highlights three main phases: 1.
Task Presentation – I first assess students’ prior knowledge of triangles and then explain the purpose of the task – classifying triangles in a two-‐way table.
2.
Small-‐Group Discussion – Students work in groups of 4 to 5 (I prefer a heterogeneous group composition). I like to define my role as a ‘guide by the side’ as students work on the task.
3.
Students Presentation – Students communicate their work to the whole-‐class justifying their conclusions.
Enjoy watching the video!
This video is also available on YouTube.
Stephan Teacher of Mathematics St Clare College, Malta
Follow the link below: http://www.youtube.com/watch?v=dT5NLZ2GQQo&feature=youtu.be
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Teaching and Learning Mathematics through Inquiry
ANALYZING STEPHAN’S DISCUSSION LESSON
30 min
The list provide below is taken from the PRIMAS PD materials available online: www.primas-‐project.eu
THE TEACHER’S ROLE DURING SMALL-‐GROUP DISCUSSION
Make the purpose of the task clear
Explain what the task is and how they should work on it. Also, explain why they should work in this way. ‘Don’t rush, take your time. The answers are not the focus here. It’s the reasons for those answers that are important. You don’t have to finish, but you do have to be able to explain something to the rest of the class.’
Keep reinforcing the ‘ground rules’
Try to ensure that students remember the ground rules that were discussed at the beginning. Encourage students to develop a responsibility for each other’s understanding. ‘I will pick one of you to explain this to the whole class later – so make sure all of you understand it’.
Listen before intervening
When approaching a group, stand back and listen to the discussion before intervening. It is all too easy to interrupt a group with a predetermined agenda, diverting their attention from the ideas they are discussing. This is not only annoying and disruptive (for the group), it also prevents students from concentrating.
Join in, don’t judge
Try to join in as an equal member of the group rather than as an authority figure. When teachers adopt judgmental roles, students tend to try to ‘guess what’s in the teacher’s head’ rather than try to think for themselves: ‘Do you want us to say what we think, or what we think you want us to say?’
Ask students to describe, explain and interpret
The purpose of an intervention is to increase the depth of reflective thought. Challenge students to describe what they are doing (quite easy), to interpret something (‘can you say what that means?’) or to explain something (‘can you show us why you say that?).
Make students do the thinking
Many students are experts at making their teachers do the work! They know that if they ‘play dumb’ long enough, then the teacher will eventually take over. Try not to fall for this. If a student says that he or she cannot explain something, ask another student in the group to explain, or ask the student to choose some part of the problem that she can explain. Don’t let them off the hook! When a student asks the teacher a question, don’t answer it (at least straight away). Ask someone else in the group to do so.
Don’t be afraid of leaving discussions unresolved.
Some teachers like to resolve discussions before they leave the group. When the teacher leads the group to the answer, then leaves, the discussion has ended. Students are left with nothing to think about, or they go on to a different problem. It is often better to reawaken interest with a further interesting question that builds on the discussion and then leave the group to discuss it alone. Return some minutes later to find out what has been decided.
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QUESTIONS FOR ANALYZING STEPHAN’S DISCUSSION LESSON
Think and talk about the following: •
How does the teacher introduce the lesson?
•
Does he emphasise specific ground rules for discussion during the lesson? Why, do you think, he did/didn’t do that?
•
What is the teacher’s role during each the three phases of the lesson? Does his role change? How?
•
How does the teacher assist students during the small-‐group activity?
•
To what extent, do you think, was setting students to work in heterogeneous groups of 4 to 5 students beneficial?
•
In your opinion, what works and what would you improve?
ADDITIONAL MATERIAL
There is a very good video that demonstrates how teachers may set up a collaborative learning community in their classroom. Click the link below. It is worth watching! http://www.youtube.com/watch?v=kZxNldBEU6o
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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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