1. Milano 2016 POLITECNICO di MILANO PhD COURSE day 1

2. Milano 2016 EMPIRICAL RESULTS

3. Milano 2016 VERTICAL NON-PERMANENT SURFACES

4. Milano 2016 EFFECT ON STUDENTS five high school classrooms two grade 12 (n=31, 30) two grade 11 (n=32, 31) one grade 10 (n=31) students were put into groups of two to four assigned to one of five work surfaces vertical non-permanent surface (whiteboard, blackboard) horizontal non-permanent surface (whiteboard) vertical permanent surface (flipchart paper) horizontal permanent surface (flipchart paper) notebook

5. Milano 2016 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 EFFECT ON STUDENTS 0 - 3

6. Milano 2016 vertical non-perm horizontal non-perm vertical permanent horizontal permanent notebook N (groups)10 998 time to task12.8 sec13.2 sec12.1 sec14.1 sec13.0 sec first notation20.3 sec23.5 sec2.4 min2.1 min18.2 sec discussion2.82.21.51.10.6 eagerness3.02.31.21.00.9 participation2.82.31.81.60.9 persistence2.6 1.81.9 mobility2.51.22.01.31.2 non-linearity2.72.91.01.10.8 EFFECT ON STUDENTS Liljedahl, P. (in press). Building thinking classrooms: Conditions for problem solving. In P. Felmer, J. Kilpatrick, & E. Pekhonen (eds.) Posing and Solving Mathematical Problems: Advances and New Perspectives. New York, NY: Springer.

7. Milano 2016 VISIBLY RANDOM GROUPS

8. Milano 2016 EFFECT ON STUDENTS grade 10 90% Asian or Caucasian February – April (linear system Sept - June) field notes observations interactions conversations interviews teacher students

9. Milano 2016 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 Liljedahl, P. (in press). The affordances of using visually random groups in a mathematics classroom. In Y. Li, E. Silver, & S. Li (eds.) Transforming Mathematics Instruction: Multiple Approaches and Practices. New York, NY: Springer. EFFECT ON STUDENTS

10. Milano 2016 THEORIES

11. Milano 2016 BUT … ARE THESE THEORIES? It seems reasonable that the practice of teaching mathematics can and should draw on our depth of knowledge of mathematical learning, and learning theory, but to theorise teaching is a problem with which most educators are struggling. While theory provides us with lenses for analysing learning, the big theories do not seem to offer clear insights to teaching and ways in which teaching addresses the promotion of mathematics learning. Jaworski (2006)

12. Milano 2016 BUT … ARE THESE THEORIES? Theories help us to analyse, or explain, but they do not provide recipes for action; rarely do they provide direct guidance for practice. We can analyse or explain mathematics learning from theoretical perspectives, but it is naive to assume or postulate theoretically derivative models or methods through which learning is supposed to happen. Jaworski (2006)

13. Milano 2016 BUT … ARE THESE THEORIES? Theories help us to analyse, or explain, but they do not provide recipes for action; rarely do they provide direct guidance for practice. We can analyse or explain mathematics learning from theoretical perspectives, but it is naive to assume or postulate theoretically derivative models or methods through which learning is supposed to happen. Jaworski (2006) … there are NO theories of teaching!

14. Milano 2016 THEORIES a theory needs to be:  substantiated  explanatory  predictive  testable

15. Milano 2016 THEORIES a belief, policy, or procedure proposed or followed as the basis of action (Merriam-Webster, online edition, 2016) a system of rules, procedures, and assumptions us ed to produce a result (Collins English Dictionary – Complete and Unabridged, 12th Edition, 2014) a belief or principle that guides action or assists co mprehension or judgment (American Heritage Dictionary of the English Language, Fifth Edition, 2011)

16. Milano 2016 THEORY OF vs. THEORY FOR THEORY OF LEARNING vs. THEORY FOR TEACHING

17. Milano 2016 THEORY OF vs. THEORY FOR a theory of needs to be:  substantiated  explanatory  predictive  testable a theory for needs to be:  ?  ?  ?  ?

18. Milano 2016 DIVERSE WORKSHOP OUTCOMES POSSIBLE WORKSHOP OUTCOMES: 1.enjoy the learning experience 2.learn some things about teaching 3.convert a theory of learning into a theory for teaching 4.extract a theory for teaching from empirical work 5.define requirements of a theory for teaching WORKSHOP EXPECTATION: you will achieve at least two of the above outcomes quick and dirty presentation on Wednesday afternoon