Flow in the Classroom and Optimal Experience
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FLOW IN THE CLASSROOM: LOOKING AT TEACHER MOVES
AND STUDENTS' AUTONOMOUS ACTIONS
- Peter Liljedahl
liljedahl@sfu.ca
www.peterliljedahl.com/presentations
@pgliljedahl
Liljedahl, P. (2016). Flow: A Framework for Discussing Teaching.
Proceedings of the 40
th
Conference of the International Group for the
Psychology of Mathematics Education,
Szeged, Hungary.
Liljedahl, P. (under review). On the edges of flow: Student problem
solving behavior. In S. Carreira, N. Amado, & K. Jones (eds.),
Broadening
the scope of research on mathematical problem solving: A focus on
technology, creativity and affect
. New York, NY: Springer.
MIHALY CSÍKSZENTMIHÁLYI
T
H
E
O
P
T
I
M
A
L
E
X
P
E
R
I
E
N
C
E
OPTIMAL EXPERIENCE
1.
There are clear goals every step of the way.
2.
There is immediate feedback on one’s actions.
3.
There is a balance between challenges and skills.
4.
Attention is focused on one’s actions.
5.
Distractions are excluded from consciousness.
6.
There is no worry of failure.
7.
Self-consciousness disappears.
8.
The sense of time becomes distorted.
9.
The activity becomes satisfying in its own right.
-
Csíkszentmihályi (1990)
OPTIMAL EXPERIENCE - internal
1.
There are clear goals every step of the way.
2.
There is immediate feedback on one’s actions.
3.
There is a balance between challenges and skills.
4.
Attention is focused on one’s actions.
5.
Distractions are excluded from consciousness.
6.
There is no worry of failure.
7.
Self-consciousness disappears.
8.
The sense of time becomes distorted.
9.
The activity becomes satisfying in its own right.
-
Csíkszentmihályi (1990)
OPTIMAL EXPERIENCE - external
1.
There are clear goals every step of the way.
2.
There is immediate feedback on one’s actions.
3.
There is a balance between challenges and skills.
4.
Attention is focused on one’s actions.
5.
Distractions are excluded from consciousness.
6.
There is no worry of failure.
7.
Self-consciousness disappears.
8.
The sense of time becomes distorted.
9.
The activity becomes satisfying in its own right.
-
Csíkszentmihályi (1990)
OPTIMAL EXPERIENCE
1.
There are clear goals every step of the way.
2.
There is immediate feedback on one’s actions.
3.
There is a balance between challenges and skills.
ANXIETY
BOREDOM
FLOW
FLOW
EXPERIENCE
1.
There are clear goals every step of the way.
2.
There is immediate feedback on one’s actions.
3.
There is a balance between challenges and skills.
ANXIETY
BOREDOM
FLOW
FLOW EXPERIENCE
1.
There are clear goals every step of the way.
2.
There is immediate feedback on one’s actions.
3.
There is a balance between challenges and skills.
ANXIETY
BOREDOM
FLOW
=
E
N
G
A
G
E
M
E
N
T
FLOW EXPERIENCE
ANXIETY
FLOW
BOREDOM
FLOW EXPERIENCE
ANXIETY
FLOW
BOREDOM
FLOW EXPERIENCE
ANXIETY
FLOW
BOREDOM
MOMENTS OF IMBALANCE
ANXIETY
FLOW
BOREDOM
TWO STUDIES
STUDY 1
What do teachers do in a
moment of imbalance
?
STUDY 2
What do student do in a
moment of imbalance
?
STUDY 1
What do teachers do in a
moment of imbalance
?
METHODOLOGY
•
Purposive sampling –
effective
mathematics
teachers
•
15 teachers → 6 participants
•
Video of a
typical
lesson
Claire
•
teaches primarily the senior (grades 11 and 12) academic
courses
Connor
•
teaches all levels of high school mathematics
•
teaches some junior (grades 8 -10) science
CLAIRE
CLAIRE
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00:00 Claire begins with a brief
review of the previous lesson
.
06:30 Claire
delivers a 'lesson' on calculating the angle θ (0
o
< θ ≤ 360
o
) given a trigonometric ratio r
.
This lesson involves her giving several examples of how to solve such tasks.
22:00 Claire
asks the class to solve for θ: sin θ = 0.8, cos θ = 0.32, and tan θ = 1.2
. During this activity
Claire circulates and checks on how students are doing. When a student puts up their hand she
quickly moves to them and answers their question. The first two questions asked by students
pertained to the fact that the ratio for the third question (tan θ = 1.2) is greater than 1.
26:15 Claire
stops the activity to re-explain the limitations on the ratios
for each trigonometric
relationship.
31:00 Claire calls the class to attention and
goes over the solutions
to each of the three questions.
36:30 Claire
gives the next question for the students to solve (solve for θ: 3sin θ + 1 = 2.8; 0
o
< θ ≤
360
o
).
Almost immediately many students put up their hands. Claire helps two students to
understand the task and begin to solve it.
40:00 Claire calls the class to attention and
reviews how to solve the equation 3
x
+ 1 = 2.8
.
42:30 Claire
refocuses the students on the original task: 3sin θ + 1 = 2.8
.
50:00 Claire calls the class to attention and
goes over the solutions to the question
.
55:30 Claire
assigns homework.
CONNOR
Liljedahl, P. (2016).
Building thinking classrooms: Conditions for problem solving
. In P.
Felmer, J. Kilpatrick, & E. Pekhonen (eds.),
Posing and Solving Mathematical
Problems: Advances and New Perspectives
. (pp. 361-386). New York, NY: Springer.
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CONNOR
a classroom that is not only conducive to thinking but also occasions
thinking, a space that is inhabited by thinking individuals as well as
individuals thinking collectively, learning together and constructing
knowledge and understanding through activity and discussion. It is a
space wherein the teacher not only fosters thinking but also expects
it, both implicitly and explicitly
(Liljedahl, 2016b, p.364)
CONNOR
a classroom that is not only conducive to thinking but also occasions
thinking, a space that is inhabited by thinking individuals as well as
individuals thinking collectively, learning together and constructing
knowledge and understanding through activity and discussion. It is a
space wherein the teacher not only fosters thinking but also expects
it, both implicitly and explicitly
(Liljedahl, 2016b, p.364)
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CONNOR
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00:00 Connor
reviews how to multiply two first degree binomials
on the board: (
x
+ 2)(
x
+ 3) =
x
2
+ 5
x
+ 6.
He then asks the question, "
what do the binomials have to be if the answer is x
2
+ 7x + 6?
"
01:30 Connor
places the students into random groups and asks them to work on vertical whiteboards
to
find the answer. He then
begins to circulate
amongst the groups as they begin to work.
05:00 Connor
stops to speak with a group who is having trouble
understanding the task. He re-writes the
example as follows:
(
x
+ 2)(
x
+ 3) =
x
2
+ 5
x
+ 6 (?)(?) =
x
2
+ 7
x
+ 6
He then points to the question marks and asks, "what has to go in here so that the product of the two
binomials is this (pointing at the quadratic expression)?
I'll give you a hint – look at the last number
."
07:00 Connor
asks a group who has an answer to check their solution
by multiplying the binomials. Once
the solution is confirmed he
gives the group a new task:
x
2
+ 6
x
+ 8
.
08:30 Connor gives a new task to two groups asking them first to check their answer.
09:00
For the next 32 minutes Connor moves around the room giving new tasks and helping groups that
are stuck
. Eventually
Connor projects a list of 20 progressively challenging tasks onto a wall
. These
range from the initial task of
x
2
+ 7
x
+ 6 to tasks as complex as 6
x
2
+ 10
x
- 4. The groups start to
move through these tasks one by one solving each and checking their answers.
41:00
Connor
gathers the students around one whiteboard and asks them to walk him through how to
solve the question
x
2
+ 5
x
– 24
. Connor forces the students to articulate their thinking at each step.
47:00
Connor
suggests that the students sit down and write down some notes for themselves
.
52:00
Connor
projects five more tasks on the wall and asks the students to solve them on their own
.
REGULATING CHALLENGE
CLAIR - SYNCHRONOUS
ANXIETY
BOREDOM
FLOW
CONNOR - ASYNCHRONOUS
ANXIETY
FLOW
BOREDOM
STUDY 2
What do student do in a
moment of imbalance
?
METHODOLOGY – TEACHING STYLE
Liljedahl, P. (2016).
Building thinking classrooms: Conditions for problem solving
. In P.
Felmer, J. Kilpatrick, & E. Pekhonen (eds.),
Posing and Solving Mathematical
Problems: Advances and New Perspectives
. (pp. 361-386). New York, NY: Springer.
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METHODOLOGY – TEACHING STYLE
Liljedahl, P. (2016).
Building thinking classrooms: Conditions for problem solving
. In P.
Felmer, J. Kilpatrick, & E. Pekhonen (eds.),
Posing and Solving Mathematical
Problems: Advances and New Perspectives
. (pp. 361-386). New York, NY: Springer.
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METHODOLOGY – TEACHING STYLE
BUILDING THINKING CLASSROOMS
1.
visibly random groups
2.
vertical non-permanent surfaces
3.
good problems
4.
flow to guide hints and extensions (Connor)
5.
levelling to the bottom
METHODOLOGY - THE DATA
•
two classrooms (grade 11 &12 pre-calculus)
•
watching for moments of imbalance
•
video didn’t work (too narrow a field of vision)
•
field notes
•
photographs
•
interviews
NOTEABILITY
TM
METHODOLOGY – THE CLASSROOMS
CAMERON
•
uses only curricular tasks (progressively harder)
•
levels to the bottom after 2-3 questions
•
levels to the bottom at end of lesson
•
no homework
CHARMAINE
•
uses non-curricular tasks 2-3 times a week as warm-up
•
uses curricular tasks (progressively harder)
•
levels to the bottom with 25 minutes to go
•
assigns 3-5 questions to do for the remainder of class with whom
you want and where you want
CREATE AND MAINTAIN FLOW
ANXIETY
FLOW
BOREDOM
MOMENTS OF IMBALANCE
ANXIETY
FLOW
BOREDOM
CHALLENGE TOO LOW
ANXIETY
FLOW
BOREDOM
CHALLENGE TOO LOW: CASE 1 (n=4)
Researcher
I notice you are not working on the assigned questions. What's up?
Mikaela
We did some of them.
Researcher
I saw that. I noticed that you did two very quickly. Took a little break
from the math and then went back and did another one. I was sort of
waiting to see if you would get back to it.
Allison
This stuff is easy. I'll finish it at home on my own.
Mikaela
Its actually too easy. I don't even think I will bother finishing it at home.
Allison
… Yeah. I probably won't either.
Researcher
It's easy? Is that why you stopped working on it?
Mikaela
Yeah.
Researcher
I saw you two work together before at the end of class. I don't recall
seeing you two giving up before.
Allison
We aren't giving up. Sometimes we don't finish all the questions
because they are hard and we run out of time. But these are easy.
Researcher
What makes them easy.
Mikaela
They just are. The first three are exactly the same and we could do
them no problem.
QUITTING
CHALLENGE TOO LOW: CASE 2 (n>20)
Researcher
So, I notice that you guys are now on question 5 and your teacher
has not visited you once. How are you getting your questions?
Ameer
We just look around and see what the next question is and do that
one.
Researcher
What would your teacher say about that?
Carl
Um … he'd probably want to check to see that we got the previous
one before giving us the next one …
Ameer
… but we are doing that.
Researcher
Why don't you just wait for your teacher to get here and give you
the next question?
Carl
We're on a roll. And sometimes we have to wait a long time.
Researcher
Do you realize that you are doing the problems out of sequence
from the order your teacher is giving them?
Colton
Oh really? That’s probably why some were so hard.
AUTONOMOUSLY
SEEKING INCREASED
CHALLENGE
CHALLENGE TOO LOW: CASE 3 (n=3)
Researcher
I have been watching you while I have been here. I notice that you
always do a lot of questions. Can you tell me about that?
Jennifer
Yeah. I like to do a lot of questions. It's good practice. It's how I
learn.
Researcher
So, are you looking for harder and harder questions to challenge
yourself.
Jennifer
Not really. I just do all of them. So, if the teacher asks us to do 4a, I
will also do 4bc and d and so on.
Researcher
Do you find them easy.
Jennifer
Yeah..
Researcher
How many do you do?
Jennifer
I just work the whole time at the end of class and then for maybe an
hour at home.
TOLERANCE FOR
THE MUNDANE
CHALLENGE TOO LOW
FLOW
BOREDOM
TOLERANCE
CHALLENGE TOO LOW: CASE 1
FLOW
BOREDOM
TOLERANCE
CHALLENGE TOO LOW: CASE 2
FLOW
BOREDOM
TOLERANCE
CHALLENGE TOO LOW: CASE 3
FLOW
BOREDOM
TOLERANCE
CHALLENGE TOO HIGH
ANXIETY
FLOW
BOREDOM
CHALLENGE TOO HIGH: CASE 1 (n=6)
Researcher
I have been watching your group for a bit and I notice that you aren’t
working?
Robert
We gave up. This question is stupid.
Katrina
We tried, but we weren't getting anywhere. So we gave up.
Researcher
What do you think the problem is?
Shannon
This question is too hard.
Robert
… too hard. We don't get it.
Katrina
And the teacher hasn’t come over to help us.
Researcher
What kind of help are you looking for?
Shannon
You know, a hint or something.
Researcher
What would a hint do for you.
Shannon
Help us understand the question.
Katrina
… or remind us a little bit about how to do it.
QUITTING
CHALLENGE TOO HIGH: CASE 2 (n>20)
Researcher
I notice that you have been moving about the room a bit. Why?
Michael
Oh. We were just stuck so we went over there to get some ideas.
Researcher
Did it help?
Michael
Oh yeah. We got it now.
Mikaela
Michael got it. It took me a little longer, but I'm good now.
Researcher
You were also moving around a little bit earlier in the class?
Mikaela
Oh, you mean when we were checking answers? Yeah, we thought we
were doing something wrong, but we were good.
Researcher
Lena, you like to look around a lot.
Lena
I do? What do you mean?
Researcher
You know, when Michael was working on the board you look around a
lot at the other groups.
Lena
Right. I am just making sure we are on the right track.
AUTONOMOUSLY
SEEKING HELP
CHALLENGE TOO HIGH: CASE 3 (n=2)
Researcher
Question #5 was a tough one, huh?
Oliver
Yeah, that one took us a while.
Connor
In the end it wasn't that hard though. We were just missing something.
Researcher
Oh really. How did you figure it out?
Connor
We just kept at it and then we saw it.
Researcher
I noticed that your teacher came over to help. Did she help you?
Oliver
No, we wouldn't let her. We knew we knew how to do it and we wanted
to figure it out ourselves.
PERSEVERANCE
IN THE FACE OF
CHALLENGE
CHALLENGE TOO HIGH
FRUSTRATION
FLOW
BOREDOM
PERSEVERANCE
TOLERANCE
CHALLENGE TOO HIGH: CASE 1
FRUSTRATION
FLOW
BOREDOM
PERSEVERANCE
TOLERANCE
CHALLENGE TOO HIGH: CASE 2
FRUSTRATION
FLOW
BOREDOM
PERSEVERANCE
TOLERANCE
CHALLENGE TOO HIGH: CASE 3
FRUSTRATION
FLOW
BOREDOM
PERSEVERANCE
TOLERANCE
DISCUSSION & CONCLUSIONS
FRUSTRATION
FLOW
BOREDOM
DISCUSSION & CONCLUSIONS
FRUSTRATION
FLOW
BOREDOM
PERSEVERANCE
TOLERANCE
DISCUSSION & CONCLUSIONS
FRUSTRATION
FLOW
BOREDOM
PERSEVERANCE
TOLERANCE
DISCUSSION & CONCLUSIONS
FRUSTRATION
FLOW
BOREDOM
PERSEVERANCE
TOLERANCE
students
autonomously
sought to
increase their
ability or
decrease the
challenge
DISCUSSION & CONCLUSIONS
FRUSTRATION
FLOW
BOREDOM
PERSEVERANCE
TOLERANCE
students
autonomously
sought to
maintain flow
DISCUSSION & CONCLUSIONS
FRUSTRATION
FLOW
BOREDOM
PERSEVERANCE
TOLERANCE
perseverance
and tolerance
serve as a buffer
to allow this
autonomy to
function
DISCUSSION & CONCLUSIONS
FRUSTRATION
FLOW
BOREDOM
PERSEVERANCE
TOLERANCE
vertical surfaces
facilitates this
autonomy to
function
DISCUSSION & CONCLUSIONS
Student autonomous actions are VERY important in
maintaining flow (engagement).
•
Connor, Charmaine, Cameron had classroom
structures that allowed for this
•
Connor encouraged this explicitly
•
Claire prevented this from happening
•
synchronous feedback, hints, and extensions
•
work was invisible to other students
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H
A
N
K
Y
O
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liljedahl@sfu.ca
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This content delves into the concepts of flow in the classroom and optimal experience, focusing on teacher moves, students' autonomous actions, and the key elements that contribute to an optimal experience according to Csikszentmihalyi. The research by Peter Liljedahl sheds light on how clear goals, immediate feedback, the balance between challenges and skills, and other factors can enhance the learning environment. Understanding flow and optimal experience can provide valuable insights for improving teaching practices and promoting student engagement.
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FLOW IN THE CLASSROOM: LOOKING AT TEACHER MOVES AND STUDENTS' AUTONOMOUS ACTIONS - Peter Liljedahl
liljedahl@sfu.ca www.peterliljedahl.com/presentations @pgliljedahl
Liljedahl, P. (2016). Flow: A Framework for Discussing Teaching. Proceedings of the 40thConference of the International Group for the Psychology of Mathematics Education, Szeged, Hungary. Liljedahl, P. (under review). On the edges of flow: Student problem solving behavior. In S. Carreira, N. Amado, & K. Jones (eds.), Broadening the scope of research on mathematical problem solving: A focus on technology, creativity and affect. New York, NY: Springer.
THE OPTIMAL EXPERIENCE MIHALY CS KSZENTMIH LYI
1. There are clear goals every step of the way. 2. There is immediate feedback on one s actions. 3. There is a balance between challenges and skills. 4. Attention is focused on one s actions. 5. Distractions are excluded from consciousness. 6. There is no worry of failure. 7. Self-consciousness disappears. 8. The sense of time becomes distorted. 9. The activity becomes satisfying in its own right. - Cs kszentmih lyi (1990) OPTIMAL EXPERIENCE
1. There are clear goals every step of the way. 2. There is immediate feedback on one s actions. 3. There is a balance between challenges and skills. 4. Attention is focused on one s actions. 5. Distractions are excluded from consciousness. 6. There is no worry of failure. 7. Self-consciousness disappears. 8. The sense of time becomes distorted. 9. The activity becomes satisfying in its own right. - Cs kszentmih lyi (1990) OPTIMAL EXPERIENCE - internal
1. There are clear goals every step of the way. 2. There is immediate feedback on one s actions. 3. There is a balance between challenges and skills. 4. Attention is focused on one s actions. 5. Distractions are excluded from consciousness. 6. There is no worry of failure. 7. Self-consciousness disappears. 8. The sense of time becomes distorted. 9. The activity becomes satisfying in its own right. - Cs kszentmih lyi (1990) OPTIMAL EXPERIENCE - external
1. There are clear goals every step of the way. 2. There is immediate feedback on one s actions. 3. There is a balance between challenges and skills. ANXIETY BOREDOM OPTIMAL EXPERIENCE
1. There are clear goals every step of the way. 2. There is immediate feedback on one s actions. 3. There is a balance between challenges and skills. ANXIETY BOREDOM FLOW EXPERIENCE
1. There are clear goals every step of the way. 2. There is immediate feedback on one s actions. 3. There is a balance between challenges and skills. ANXIETY = ENGAGEMENT BOREDOM FLOW EXPERIENCE
ANXIETY BOREDOM FLOW EXPERIENCE
ANXIETY BOREDOM FLOW EXPERIENCE
ANXIETY BOREDOM FLOW EXPERIENCE
ANXIETY BOREDOM MOMENTS OF IMBALANCE
STUDY 1 What do teachers do in a moment of imbalance? STUDY 2 What do student do in a moment of imbalance? TWO STUDIES
Purposive sampling effective mathematics teachers 15 teachers 6 participants Video of a typical lesson Claire teaches primarily the senior (grades 11 and 12) academic courses Connor teaches all levels of high school mathematics teaches some junior (grades 8 -10) science METHODOLOGY
Claire's Lesson (grade 11) solving trigonometric ratio problems 00:00 Claire begins with a brief review of the previous lesson. 06:30 Claire delivers a 'lesson' on calculating the angle (0o < 360o) given a trigonometric ratio r. This lesson involves her giving several examples of how to solve such tasks. 22:00 Claire asks the class to solve for : sin = 0.8, cos = 0.32, and tan = 1.2. During this activity Claire circulates and checks on how students are doing. When a student puts up their hand she quickly moves to them and answers their question. The first two questions asked by students pertained to the fact that the ratio for the third question (tan = 1.2) is greater than 1. 26:15 Claire stops the activity to re-explain the limitations on the ratios for each trigonometric relationship. 31:00 Claire calls the class to attention and goes over the solutions to each of the three questions. 36:30 Claire gives the next question for the students to solve (solve for : 3sin + 1 = 2.8; 0o < 360o). Almost immediately many students put up their hands. Claire helps two students to understand the task and begin to solve it. 40:00 Claire calls the class to attention and reviews how to solve the equation 3x + 1 = 2.8. 42:30 Claire refocuses the students on the original task: 3sin + 1 = 2.8. 50:00 Claire calls the class to attention and goes over the solutions to the question. 55:30 Claire assigns homework. CLAIRE
Liljedahl, P. (2016). Building thinking classrooms: Conditions for problem solving. In P. Felmer, J. Kilpatrick, & E. Pekhonen (eds.), Posing and Solving Mathematical Problems: Advances and New Perspectives. (pp. 361-386). New York, NY: Springer. Liljedahl, P. (2014). The affordances of using visibly random groups in a mathematics classroom. In Y. Li, E. Silver, & S. Li (eds.), Transforming Mathematics Instruction: Multiple Approaches and Practices. (pp. 127-144). New York, NY: Springer. CONNOR
a classroom that is not only conducive to thinking but also occasions thinking, a space that is inhabited by thinking individuals as well as individuals thinking collectively, learning together and constructing knowledge and understanding through activity and discussion. It is a space wherein the teacher not only fosters thinking but also expects it, both implicitly and explicitly(Liljedahl, 2016b, p.364) CONNOR
levelling assessment flow answering questions oral instructions defronting good problems vertical non- permanent surfaces visibly random groups a classroom that is not only conducive to thinking but also occasions thinking, a space that is inhabited by thinking individuals as well as individuals thinking collectively, learning together and constructing knowledge and understanding through activity and discussion. It is a space wherein the teacher not only fosters thinking but also expects it, both implicitly and explicitly(Liljedahl, 2016b, p.364) CONNOR
Connor's Lesson (grade 11) factoring quadratics 00:00 Connor reviews how to multiply two first degree binomials on the board: (x + 2)(x + 3) = x2 + 5x + 6. He then asks the question, "what do the binomials have to be if the answer is x2 + 7x + 6?" 01:30 Connor places the students into random groups and asks them to work on vertical whiteboards to find the answer. He then begins to circulate amongst the groups as they begin to work. 05:00 Connor stops to speak with a group who is having trouble understanding the task. He re-writes the example as follows: (x + 2)(x + 3) = x2 + 5x + 6 (?)(?) = x2 + 7x + 6 He then points to the question marks and asks, "what has to go in here so that the product of the two binomials is this (pointing at the quadratic expression)? I'll give you a hint look at the last number." 07:00 Connor asks a group who has an answer to check their solution by multiplying the binomials. Once the solution is confirmed he gives the group a new task: x2 + 6x + 8. 08:30 Connor gives a new task to two groups asking them first to check their answer. 09:00 For the next 32 minutes Connor moves around the room giving new tasks and helping groups that are stuck. Eventually Connor projects a list of 20 progressively challenging tasks onto a wall. These range from the initial task of x2 + 7x + 6 to tasks as complex as 6x2 + 10x - 4. The groups start to move through these tasks one by one solving each and checking their answers. 41:00 Connor gathers the students around one whiteboard and asks them to walk him through how to solve the question x2 + 5x 24. Connor forces the students to articulate their thinking at each step. 47:00 Connor suggests that the students sit down and write down some notes for themselves. 52:00 Connor projects five more tasks on the wall and asks the students to solve them on their own. CONNOR
CLAIR in the first activity the challenge of the tasks is reduced for the whole class through the explicit lesson on, and examples of, how to solve these types of tasks Clair then lowers the challenge even further in the middle of the activity by reviewing the range of ratios possible everyone was given the second more challenging activity at the same time the challenge of this activity is then reduced for the whole class when Claire shows how to solve an analogous algebraic task CONNOR Connor regulates the challenge of each task according to the ability of each individual group groups who need help receive individualized help groups whose ability allows them to solve a task are given a more challenging task to work on eventually the groups are shown how to check their own solutions and from then on they are able to increase the challenge of the activity as their ability increases REGULATING CHALLENGE
ANXIETY BOREDOM CLAIR - SYNCHRONOUS
ANXIETY BOREDOM CONNOR - ASYNCHRONOUS
Liljedahl, P. (2016). Building thinking classrooms: Conditions for problem solving. In P. Felmer, J. Kilpatrick, & E. Pekhonen (eds.), Posing and Solving Mathematical Problems: Advances and New Perspectives. (pp. 361-386). New York, NY: Springer. Liljedahl, P. (2014). The affordances of using visibly random groups in a mathematics classroom. In Y. Li, E. Silver, & S. Li (eds.), Transforming Mathematics Instruction: Multiple Approaches and Practices. (pp. 127-144). New York, NY: Springer. METHODOLOGY TEACHING STYLE
levelling assessment flow answering questions oral instructions defronting good problems vertical non- permanent surfaces visibly random groups Liljedahl, P. (2016). Building thinking classrooms: Conditions for problem solving. In P. Felmer, J. Kilpatrick, & E. Pekhonen (eds.), Posing and Solving Mathematical Problems: Advances and New Perspectives. (pp. 361-386). New York, NY: Springer. Liljedahl, P. (2014). The affordances of using visibly random groups in a mathematics classroom. In Y. Li, E. Silver, & S. Li (eds.), Transforming Mathematics Instruction: Multiple Approaches and Practices. (pp. 127-144). New York, NY: Springer. METHODOLOGY TEACHING STYLE
BUILDING THINKING CLASSROOMS 1. visibly random groups 2. vertical non-permanent surfaces 3. good problems 4. flow to guide hints and extensions (Connor) 5. levelling to the bottom METHODOLOGY TEACHING STYLE
two classrooms (grade 11 &12 pre-calculus) watching for moments of imbalance video didn t work (too narrow a field of vision) field notes photographs interviews NOTEABILITYTM METHODOLOGY - THE DATA
CAMERON uses only curricular tasks (progressively harder) levels to the bottom after 2-3 questions levels to the bottom at end of lesson no homework CHARMAINE uses non-curricular tasks 2-3 times a week as warm-up uses curricular tasks (progressively harder) levels to the bottom with 25 minutes to go assigns 3-5 questions to do for the remainder of class with whom you want and where you want METHODOLOGY THE CLASSROOMS
ANXIETY BOREDOM CREATE AND MAINTAIN FLOW
ANXIETY BOREDOM MOMENTS OF IMBALANCE
ANXIETY BOREDOM CHALLENGE TOO LOW
Researcher I notice you are not working on the assigned questions. What's up? Mikaela We did some of them. Researcher I saw that. I noticed that you did two very quickly. Took a little break from the math and then went back and did another one. I was sort of waiting to see if you would get back to it. Allison This stuff is easy. I'll finish it at home on my own. Mikaela Its actually too easy. I don't even think I will bother finishing it at home. Allison Yeah. I probably won't either. Researcher It's easy? Is that why you stopped working on it? Mikaela Yeah. Researcher I saw you two work together before at the end of class. I don't recall seeing you two giving up before. Allison We aren't giving up. Sometimes we don't finish all the questions because they are hard and we run out of time. But these are easy. Researcher What makes them easy. Mikaela They just are. The first three are exactly the same and we could do them no problem. CHALLENGE TOO LOW: CASE 1 (n=4)
Researcher So, I notice that you guys are now on question 5 and your teacher has not visited you once. How are you getting your questions? Ameer We just look around and see what the next question is and do that one. Researcher What would your teacher say about that? Carl Um he'd probably want to check to see that we got the previous one before giving us the next one Ameer but we are doing that. Researcher Why don't you just wait for your teacher to get here and give you the next question? Carl We're on a roll. And sometimes we have to wait a long time. Researcher Do you realize that you are doing the problems out of sequence from the order your teacher is giving them? Colton Oh really? That s probably why some were so hard. CHALLENGE TOO LOW: CASE 2 (n>20)
Researcher I have been watching you while I have been here. I notice that you always do a lot of questions. Can you tell me about that? Jennifer Yeah. I like to do a lot of questions. It's good practice. It's how I learn. Researcher So, are you looking for harder and harder questions to challenge yourself. Jennifer Not really. I just do all of them. So, if the teacher asks us to do 4a, I will also do 4bc and d and so on. Researcher Do you find them easy. Jennifer Yeah.. Researcher How many do you do? Jennifer I just work the whole time at the end of class and then for maybe an hour at home. CHALLENGE TOO LOW: CASE 3 (n=3)
ANXIETY BOREDOM CHALLENGE TOO HIGH
Researcher I have been watching your group for a bit and I notice that you arent working? Robert We gave up. This question is stupid. Katrina We tried, but we weren't getting anywhere. So we gave up. Researcher What do you think the problem is? Shannon This question is too hard. Robert too hard. We don't get it. Katrina And the teacher hasn t come over to help us. Researcher What kind of help are you looking for? Shannon You know, a hint or something. Researcher What would a hint do for you. Shannon Help us understand the question. Katrina or remind us a little bit about how to do it. CHALLENGE TOO HIGH: CASE 1 (n=6)
Researcher I notice that you have been moving about the room a bit. Why? Michael Oh. We were just stuck so we went over there to get some ideas. Researcher Did it help? Michael Oh yeah. We got it now. Mikaela Michael got it. It took me a little longer, but I'm good now. Researcher You were also moving around a little bit earlier in the class? Mikaela Oh, you mean when we were checking answers? Yeah, we thought we were doing something wrong, but we were good. Researcher Lena, you like to look around a lot. Lena I do? What do you mean? Researcher You know, when Michael was working on the board you look around a lot at the other groups. Lena Right. I am just making sure we are on the right track. CHALLENGE TOO HIGH: CASE 2 (n>20)
Researcher Question #5 was a tough one, huh? Oliver Yeah, that one took us a while. Connor In the end it wasn't that hard though. We were just missing something. Researcher Oh really. How did you figure it out? Connor We just kept at it and then we saw it. Researcher I noticed that your teacher came over to help. Did she help you? Oliver No, we wouldn't let her. We knew we knew how to do it and we wanted to figure it out ourselves. CHALLENGE TOO HIGH: CASE 3 (n=2)
