1. building thinking classrooms
Peter Liljedahl

2. 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 ). 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 ). New York, NY: Springer.
Liljedahl, P. (2016). Flow: A Framework for Discussing Teaching. Proceedings of the 40th Conference of the International Group for the Psychology of Mathematics Education, Szeged, Hungary.
Liljedahl, P. (in press). 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.
Liljedahl, P. (in press). On the edges of flow: Student engagement in problem solving. Proceedings of the 10th Congress of the European Society for Research in Mathematics Education. Dublin, Ireland.
Liljedahl, P. (in press). Building Thinking Classrooms: A Story of Teacher Professional Development. The 1st International Forum on Professional Development for Teachers. Seoul, Korea.
Liljedahl, P. (under review). Building thinking classrooms. In A. Kajander, J. Holm, & E. Chernoff (eds.) Teaching and learning secondary school mathematics: Canadian perspectives in an international context. New York, NY: Springer. 

3.
@pgliljedahl

4. JANE’S CLASS – 13 YEARS AGO

5. If 6 cats can kill 6 rats in 6 minutes, how many cats are required to kill 100 rats in 50 minutes? - Lewis Carroll

6. If 6 cats can kill 6 rats in 6 minutes, how many cats are required to kill 100 rats in 50 minutes? - Lewis Carroll
DISASTER!

7. REALIZATIONS
At no point in the previous week had I seen students be asked to think.
Jane was planning her teaching on the assumption that students either could not or would not think.
If they are not thinking they are not learning!

9. THINGS THAT ARE COMMON There are desks/tables in the room
Teachers give notes
Teachers give students problems to do in class
Students work at their desks/tables
Students work in groups
Teachers answer questions
Teachers give hints and extensions
Teachers go over the problems
Teachers assess their students

10. THINGS THAT ARE NON-NEGOTIATED
There are desks/tables in the room
Teachers give notes
Teachers give students problems to do in class
Students work at their desks/tables
Students work in groups
Teachers answer questions
Teachers give hints and extensions
Teachers go over the problems
Teachers assess their students

11. CAN WE RE-NEGOTIATE THESE?
Is there a best way to organize the desks in the room?
Is there a best way to give notes?
Is there a best type of problems to use?
Is there a best way to give problems?
Is there a best work surface for students?
Is there a best way to group students?
Is there a best way to answer questions?
Is there a best way to give hints and extensions?
Is there a best way to go over problems?
Is there a best way to assess students?

12. CAN WE RE-NEGOTIATE THESE?
Is there a best way to organize the desks in the room?
Is there a best way to give notes?
Is there a best type of problems to use?
Is there a best way to give problems?
Is there a best work surface for students?
Is there a best way to group students?
Is there a best way to answer questions?
Is there a best way to give hints and extensions?
Is there a best way to go over problems?
Is there a best way to assess students?

13. RE-NEGOTIATING THE NON-NEGOTIATED NORMS
ACTION RESEARCH (n = 400+, t = 13 years+)

14. WE USED A CONTRARIAN APPROACH. OVER TIME WE BEGAN TO LEARN SOME THINGS.

15. IS THERE A BEST WORK SURFACE FOR STUDENTS?
WE TRIED HAVING STUDENTS IN GROUPS AND WORKING:
at their desks in their notebooks
at their desks on big sheets of paper
at their desks on big whiteboards
on whiteboards on the walls
on big paper on the walls

16. IS THERE A BEST WORK SURFACE FOR STUDENTS?
CONTROLLED EXPERIMENT:
some groups at their desks in their notebooks
some groups at their desks on big sheets of paper
some groups at their desks on big whiteboards
some groups on whiteboards on the walls
some groups on big paper on the walls

17. IS THERE A BEST WORK SURFACE FOR STUDENTS?
PROXIES FOR ENGAGEMENT
time to task
time to first mathematical notation
amount of discussion
eagerness to start
participation
persistence
knowledge mobility
non-linearity of work
0 - 3

18. N (groups) 10 9 8 time to task 12.8 sec 13.2 sec 12.1 sec 14.1 sec
vertical
whiteboard
horizontal whiteboard
vertical paper
horizontal paper
notebook
N (groups) 10 9 8
time to task
12.8 sec
13.2 sec
12.1 sec
14.1 sec
13.0 sec
first notation
20.3 sec
23.5 sec
2.4 min
2.1 min
18.2 sec
discussion
2.8
2.2
1.5
1.1
0.6
eagerness
3.0
2.3
1.2
1.0
0.9
participation
1.8
1.6
persistence
2.6
1.9
mobility
2.5
2.0
1.3
non-linearity
2.7
2.9
0.8

19. N (groups) 10 9 8 time to task 12.8 sec 13.2 sec 12.1 sec 14.1 sec
vertical
whiteboard
horizontal whiteboard
vertical paper
horizontal paper
notebook
N (groups) 10 9 8
time to task
12.8 sec
13.2 sec
12.1 sec
14.1 sec
13.0 sec
first notation
20.3 sec
23.5 sec
2.4 min
2.1 min
18.2 sec
discussion
2.8
2.2
1.5
1.1
0.6
eagerness
3.0
2.3
1.2
1.0
0.9
participation
1.8
1.6
persistence
2.6
1.9
mobility
2.5
2.0
1.3
non-linearity
2.7
2.9
0.8

20. N (groups) 10 9 8 time to task 12.8 sec 13.2 sec 12.1 sec 14.1 sec
vertical
whiteboard
horizontal whiteboard
vertical paper
horizontal paper
notebook
N (groups) 10 9 8
time to task
12.8 sec
13.2 sec
12.1 sec
14.1 sec
13.0 sec
first notation
20.3 sec
23.5 sec
2.4 min
2.1 min
18.2 sec
discussion
2.8
2.2
1.5
1.1
0.6
eagerness
3.0
2.3
1.2
1.0
0.9
participation
1.8
1.6
persistence
2.6
1.9
mobility
2.5
2.0
1.3
non-linearity
2.7
2.9
0.8

21. VERTICAL NON-PERMANENT SURFACES

22. #VNPS

23. ONE MARKER PER GROUP

24. IS THERE A BEST WAY TO GROUP STUDENTS?
WE TRIED GROUPING STUDENTS:
strategically for productivity
strategically for socialization
strategically for classroom management
by letting students self-group
randomly

25. IS THERE A BEST WAY TO GROUP STUDENTS?
WE TRIED GROUPING STUDENTS:
strategically for productivity
strategically for socialization
strategically for classroom management
by letting students self-group
visibly random groups of three

26. IS THERE A BEST WAY TO GROUP STUDENTS?
focused change study
grade 10 (15-16 years old)
racially bifurcated
50% Asian
50% Caucasian
socially stratified
jocks
nerds
facionistas
gang bangers
drama queens

27. IS THERE A BEST WAY TO GROUP STUDENTS? AFTER 3 WEEKS
students become agreeable to work in any group they are placed in
there is an elimination of social barriers within the classroom
mobility of knowledge between students increases
reliance on co-constructed intra- and inter-group answers increases
reliance on the teacher for answers decreases
engagement in classroom tasks increase
students become more enthusiastic about mathematics class

28. VISIBLY RANDOM GROUPS

29. IS THERE A BEST WAY TO ANSWER QUESTIONS?
Teachers in Canada answer ~600 questions/day. CONTRARIAN APPROACH DON’T ANSWER ANY QUATIONS This did not work with young students.

30. IS THERE A BEST WAY TO ANSWER QUESTIONS?
STUDENTS ONLY ASK 3 TYPES OF QUESTIONS:
proximity questions
stop thinking questions
keep thinking questions
ONLY ANSWER

31. IS THERE A BEST WAY TO ORGANIZE THE DESKS IN THE ROOM?
WE TRIED ORGANIZING THE ROOM WITH:
desks in single rows
desks in a circle
desks in pairs in rows
square tables
desks in threes in rows
rectangular tables
desks in twos scattered
circular tables
desks in threes scattered
lily pad tables
desks in a U-shape
no furniture
DEFRONTED GROUPS OF THREE

32. IS THERE A BEST WAY TO GIVE NOTES?
don’t keep up
n=16
yes
n=3
don’t use notes
n=27
USE NOTES TO STUDY

33. IS THERE A BEST WAY TO GIVE NOTES?
WE TRIED:
notes for less time
uploaded notes
photocopied notes
fill in the blank notes
no notes

34. IS THERE A BEST WAY TO GIVE NOTES?
WE TRIED:
notes for x minutes
uploaded notes
photocopied notes
fill in the blank notes
no notes
thoughtful notes (notes to my future dumber self)

35. IS THERE A BEST WAY TO GIVE PROBLEMS?
WE TRIED:
textbook
worksheet
printed on paper
projected on wall
written on board
orally

36. IS THERE A BEST WAY TO GIVE PROBLEMS?
orally
projected on wall
written on board
printed on paper
worksheet
textbook

37. IS THERE A BEST TYPE OF PROBLEMS TO USE?
WE TRIED:
rich problems
word problems
numeracy problems
advanced problems
review problems 

38. IS THERE A BEST TYPE OF PROBLEMS TO USE?
WE TRIED:
rich problems
word problems
numeracy problems
advanced problems
review problems 
PROBLEMS MUST HAVE SOMETHING LEFT TO THINK ABOUT AND THEY MUST BE EXTENDABLE

39. IS THERE A BEST TYPE OF PROBLEMS TO USE?
FIRST SIX PROBLEMS
(non-curricular tasks)
REST OF PROBLEMS
(curricular tasks)

40. PUTTING IT ALL TOGETHER

42. begin with good problems
use vertical non-permanent surfaces
form visibly random groups

43. use oral instructions
defront the classroom
answer only keep thinking questions
build autonomy

44. level to the bottom
use hints and extensions to manage flow
use mindful notes
use 4 purposes of assessment

45. building thinking classrooms

48. LESSON STRUCTURE Levelling to the bottom (summarizing)
VRG, VNPS, (short lesson), Task #1
Levelling to the bottom (summarizing)
Thoughtful notes
3 – 6 tasks to check understanding
WORKING ON PROBLEMS

49. THANK YOU! liljedahl@sfu.ca www.peterliljedahl.com/presentations
@pgliljedahl | #vnps | #thinkingclassroom
Global Math Department