1. Putting the ‘x’ back into goal-free problems: a brainstorming approach Carina Schubert 1, Paul Ayres 2, Katharina Scheiter 1 and John Sweller 2 1 University of Tuebingen 2 University of New South Wales

2. Hypotheses (Theory follows from John Sweller talk) o No group differences for higher ability students (impact of prior knowledge) o For students with lower ability: o Goal Free with X will be more effective than the Goal Free and the Conventional groups o Goal Free will be more effective than the Conventional group Planned contrasts will be conducted accordingly

3. Method

4. Method Participants o 82 year 9 students from two Sydney Girls high schools o 44 students were of higher mathematical ability, 38 students of lower mathematical ability (according to school grading in general mathematical ability - top 20% excluded) o Randomly assigned to one of three treatment groups: o Conventional o Goal- free o Goal-free with X

5. Method Materials  Learning materials 16 geometry problems 16 geometry problems 8 different geometry theorems employed 8 different geometry theorems employed 14 two-step problems, 2 three-step problems 14 two-step problems, 2 three-step problems For Conventional and Goal-free with X group, the problems had a goal angle X For Conventional and Goal-free with X group, the problems had a goal angle X

6. Method

7. Method Materials  Testing materials ( with X for all groups) 8 Test questions: o similar to acquisition problems o same combination of theorems in same order but with different numbers o 7 two-step, 1 three-step 8 Transfer questions: o 4 with same combinations of theorems as in acquisition problems in inverted order o 4 with different combinations of theorems – different, unusual configurations ( see Ayres & Sweller, 1990) o 3 two-step, 5 three-step

8. Method Materials  Self-rating measures (after acquisition & after testing): 7-point Likert scale How difficult did you find it to answer the questions? (Sweller et al.) How difficult did you find it to answer the questions? (Sweller et al.) How much mental effort did you use to answer the questions? (Paas et al.) How much mental effort did you use to answer the questions? (Paas et al.) How much did you concentrate when you answered the questions? (Cierniak, Scheiter & Gerjets, 2008) How much did you concentrate when you answered the questions? (Cierniak, Scheiter & Gerjets, 2008) How motivated were you to answer the questions? (Ayres & Youssef, 2008) How motivated were you to answer the questions? (Ayres & Youssef, 2008)

9. Method Rationale for CL measures o Explore potential differences between difficulty and mental effort (Van Gog & Paas, in press) o Explore relation between CL measures and test performance

10. Method Instruction (acquisition): o Conventional: “For each question find the value of angle x” o Goal-free: “For each question find as many angles as you can” o Goal-free with X: “For each question, find as many angles as you can in any order you like” Instruction (test + transfer): “For each question, find the value of angle x”

11. Method Procedure SELF-RATING ACQUISITION SELF-RATING TEST TRANSFER 15 mins 1 min 10 mins 1 min

12. Results

13. Results: Scores in acquisition o max. score 32 o all planned contrasts were n.s. o Higher ability students > lower ability students (F (1, 80) = 95.81; p =.00)

14. Results: Testscore o max. score 16 o all planned contrasts were n.s. o Higher ability students > lower ability students (F (1, 80) = 23.14; p=.00)

15. Results: Transfer Score o max. score 16 o Higher ability > lower ability (F (1, 80) = 62.17; p =.00) o for higher ability students all planned contrasts were n.s. Lower ability students: o Goal-free X > Conventional (t (22) = 2.06; p =.05) o Goal-free X > Goal-free (t (24) = 2.20; p <.05) o Goal-free not better than Conventional

16. Hypotheses summary o For the higher ability group there were no differences between treatment groups (supported) o For the lower ability group there were significant group differences on the transfer problems only o Goal-free with X was superior to the other two groups (supported) o Goal-free = Conventional (not supported)

17. Further analysis o Two potential explanations of the results: o Schema acquisition (learnt the geometry problems) o Strategy acquisition (learnt the problem solving strategy) o We examine the number of angles calculated o We examine the use of extra constructions (many students used a strategy of drawing extra lines to find angles – a strategy evidently learnt at school)

18. Results: Number of Angles in acquisition o 3 x 2 ANOVA o significant group differences (F(2,76) = 7.38; p<.01): (F(2,76) = 7.38; p<.01): Goal Free, Goal Free X > Conventional o Higher ability > lower ability (F(1,76) = 57.38; p=.00) (F(1,76) = 57.38; p=.00)

19. Results: Number of angles in test o No group differences o Higher ability > lower ability (F(1,76) = 7.63; p<.01)

20. Results: Number of angles in Transfer o Higher ability > lower ability (F(1,76)=27.01; p<.01) o group x ability interaction (F(2,76)=3.78; p<.05) o Simple effects + post- hoc on lower ability students: Goal-free with X > Goal-Free and Conventional

21. Strategy acquisition o Evidence suggests that for the lower ability students the Goal-free with X group continued with the strategy of finding additional angles for the transfer problems. In contrast, the Goal-free group seemed to abandon the strategy.

22. Results: Constructions Acquisition o Higher ability > lower ability (F(1,76)=6.78;p=.01) o Group x ability interaction (F(2,76)=5.60;p<.01) o Simple effects and post-hoc: o no group effect for lower ability students o For higher ability students, Conventional > Goal Free

23. Results: Constructions Test o No group effect o No ability effect o No interaction

24. Results: Constructions Transfer o Group effect (F(2,76)=5.27; p Goal Free

25. Results: CL measures 3 x 2 x (2) ANOVA Repeated measures Between phases GroupAbility Group x Ability Motivation p=.06 ↓ n.s. p=.00 ↑ n.s. Concentrationn.s.n.s. n.s. Difficulty n.s. P=.00 ↓ n.s. Mental Effort p=.00 ↑ n.s.n.s.p=.03^ ^ Simple effects and post-hoc: high ability students invested less mental effort in Goal-free with X group than in Goal-free and Conventional group

26. Results: CL measures Correlations Difficulty 1 Difficulty 2 Effort 1 Effort 2 Difficulty 1.66**.52**.27* Difficulty 2.32**.39** Effort 1.61**

27. Results: CL measures Correlations of CL measures and Transfer Score Mot 1 Con 1 Diff 1 Eff 1 Mot 2 Con 2 Diff 2 Eff 2 Transfer.55**.57**-.58**-.05.48**.62**-.58**.12

28. Conclusions

29. Conclusions o On transfer problems, Goal-free with X was superior to Goal-free and Conventional o The data on the number of angles calculated suggest that the Goal-free with X group have learnt the problem solving strategy of generating angles o Surprisingly the Goal-free was not superior to the Conventional group. The Goal-free group rejected the problem solving strategy of generating angles.

30. Conclusions o How has adding an X to the Goal-free group influenced the results so much? o It is notable that in some previous research into goal- free problems the learning domain has been quite restricted. In studies by Owen and Sweller (1985) in trigonometry and Ayres (1990) in geometry, few sides and angles could be calculated. However, in this study many more angles could be calculated – it may be that the X gives the group more focus.