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November  2021, 1(4): 256-278. doi: 10.3934/steme.2021017

Non-routine mathematical problem-solving: Creativity, engagement, and intuition of STEM tertiary students

1. 

Department of Mathematics, University of Auckland, Auckland, New Zealand; j.novak@auckland.ac.nz (J.N.); moj.thomas@auckland.ac.nz (M.T.)

2. 

Department of Mathematical Sciences, Auckland University of Technology, Auckland, New Zealand; sergiy.klymchuk@aut.ac.nz (S.K.); priscilla.murphy@aut.ac.nz (P.E.L.M.)

3. 

School of Learning Development and Professional Practice, University of Auckland, Auckland, New Zealand; jm.stephens@auckland.ac.nz (J.S.)

* Correspondence: t.evans@auckland.ac.nz; Tel: +64-9-923-8783

Academic Editor: Zlatko Jovanoski

Received  March 2021 Revised  November 2021 Published  November 2021

This study set out to evaluate an intervention that introduced a period of non-routine problem-solving into tertiary STEM lectures at four tertiary institutions in New Zealand for 683 students. The aim was twofold: to attempt to increase student engagement and to introduce them to the kind of domain-free abstract reasoning that involves critical, creative, and innovative thinking. This study was conducted using a mixed-methods approach, utilizing different types of instruments to gather data: comprehensive student pre- and post-test questionnaires, a content validation survey for the questionnaires, focus group interviews (student participants), open-ended questionnaire (lecturer participants), and naturalistic class observations. The main findings are as follows. Students' behavioural engagement was significantly greater during the intervention. Perceptions of the utility value of the activity improved at the end of the semester for all students. There were no significant changes in students' convergent thinking (problem-solving), intuition, or creativity (originality, fluency, and elaboration traits of the divergent thinking) during the course, probably due to the relatively short timescale of the intervention. However, overall, the results of the investigation suggest that with a relatively small effort, teachers can improve STEM student engagement by devoting a few minutes per lecture on non-routine problem-solving. This is something that can be easily implemented, even by those who primarily teach in a traditional lecturing style.

Citation: Tanya Evans, Sergiy Klymchuk, Priscilla E. L. Murphy, Julia Novak, Jason Stephens, Mike Thomas. Non-routine mathematical problem-solving: Creativity, engagement, and intuition of STEM tertiary students. STEM Education, 2021, 1 (4) : 256-278. doi: 10.3934/steme.2021017
References:
[1]

Blondal, K.S. and S. Adalbjarnardottir, Student Disengagement in Relation to Expected and Unexpected Educational Pathways. Scandinavian Journal of Educational Research, 2012. 56(1): 85-100. https://doi.org/10.1080/00313831.2011.568607. doi: 10.1080/00313831.2011.568607.

[2]

Kahu, E.R. and K. Nelson, Student engagement in the educational interface: understanding the mechanisms of student success. Higher Education Research & Development, 2018. 37(1): 58-71. https://doi.org/10.1080/07294360.2017.1344197. doi: 10.1080/07294360.2017.1344197.

[3] W. Poundstone, How would you move Mount Fuji? Microsoft's cult of the puzzle - How the world's smartest companies select the most creative thinkers, Little Brown and Company, 2000. 
[4]

Falkner, N., R. Sooriamurthi, and Z. Michalewicz, Teaching puzzle-based learning: Development of basic concepts. Teaching Mathematics and Computer Science, 2012. 10(1): 183-204.

[5]

Thomas, C., et al., Puzzle-based Learning of Mathematics in Engineering. Engineering Education, 2013. 8(1): 122-134. https://doi.org/10.11120/ened.2013.00005. doi: 10.11120/ened.2013.00005.

[6]

Evans, T., M.O.J. Thomas, and S. Klymchuk, Non-routine problem solving through the lens of self-efficacy. Higher Education Research & Development, 2020: 1-18. https://doi.org/10.1080/07294360.2020.1818061. doi: 10.1080/07294360.2020.1818061.

[7]

Klymchuk, S., Puzzle-based learning in engineering mathematics: students' attitudes. International Journal of Mathematical Education in Science and Technology, 2017. 48(7): 1106-1119. https://doi.org/10.1080/0020739X.2017.1327088. doi: 10.1080/0020739X.2017.1327088.

[8]

Parhami, B., A puzzle-based seminar for computer engineering freshmen. Computer Science Education, 2008. 18(4): 261-277. https://doi.org/10.1080/08993400802594089. doi: 10.1080/08993400802594089.

[9]

Pugh, K.J. and D.A. Bergin, Motivational Influences on Transfer. Educational Psychologist, 2006. 41(3): 147-160. https://doi.org/10.1207/s15326985ep4103_2. doi: 10.1207/s15326985ep4103_2.

[10]

Ford, J.K., et al., Relationships of goal orientation, metacognitive activity, and practice strategies with learning outcomes and transfer. Journal of Applied Psychology, 1998. 83(2): 218-233. https://doi.org/10.1037/0021-9010.83.2.218. doi: 10.1037/0021-9010.83.2.218.

[11]

Usher, E.L. and F. Pajares, Sources of Self-Efficacy in School: Critical Review of the Literature and Future Directions. Review of Educational Research, 2008. 78(4): 751-796. https://doi.org/10.3102/0034654308321456. doi: 10.3102/0034654308321456.

[12]

Selden, A., et al., Why can't calculus students access their knowledge to solve non-routine problems?, in Research in collegiate mathematics education: IV. Issues in mathematical education, E. Dubinsky, A. Schoenfeld, and J. Kaput, Editors. 2000, pp. 128-153: American Mathematical Society.

[13] Z. Michalewicz and M. Michalewicz, Puzzle-based learning: An introduction to critical thinking, mathematics, and problem solving, Hybrid Publishers, 2008. 
[14]

Fredricks, J.A., Engagement in School and Out-of-School Contexts: A Multidimensional View of Engagement. Theory Into Practice, 2011. 50(4): 327-335. https://doi.org/10.1080/00405841.2011.607401. doi: 10.1080/00405841.2011.607401.

[15]

Fredricks, J.A., M. Filsecker, and M.A. Lawson, Student engagement, context, and adjustment: Addressing definitional, measurement, and methodological issues. Learning and Instruction, 2016. 43: 1-4. https://doi.org/https://doi.org/10.1016/j.learninstruc.2016.02.002.

[16]

Watt, H.M.G. and M. Goos, Theoretical foundations of engagement in mathematics. Mathematics Education Research Journal, 2017. 29(2): 133-142. https://doi.org/10.1007/s13394-017-0206-6. doi: 10.1007/s13394-017-0206-6.

[17]

Helme, S. and D. Clarke, Identifying cognitive engagement in the mathematics classroom. Mathematics Education Research Journal, 2001. 13(2): 133-153. https://doi.org/10.1007/BF03217103. doi: 10.1007/BF03217103.

[18]

Eccles, J., Expectancies, values, and academic behaviors, in Achievement and Achievement Motives: Psychological and Sociological Approaches., J.T. Spence, Ed. 1983, pp. 75-146: W.H. Freeman.

[19] A.H. Schoenfeld, Mathematical Problem Solving, Academic Press, 1985. 
[20] G. Pólya, How to Solve it?, Princeton University Press, Princeton, NJ, 1945. 
[21]

Shoenfeld, A.H., Research Commentary: Reflections of an Accidental Theorist. Journal for Research in Mathematics Education JRME, 2010. 41(2): 104-116. https://doi.org/10.5951/jresematheduc.41.2.0104. doi: 10.5951/jresematheduc.41.2.0104.

[22]

Connell, J.P. and J.G. Wellborn, Competence, autonomy, and relatedness: A motivational analysis of self-system processes, in Self processes and development. 1991, pp. 43-77. Hillsdale, NJ, US: Lawrence Erlbaum Associates, Inc.

[23]

Ahlfeldt, S., S. Mehta, and T. Sellnow, Measurement and analysis of student engagement in university classes where varying levels of PBL methods of instruction are in use. Higher Education Research & Development, 2005. 24(1): 5-20. https://doi.org/10.1080/0729436052000318541. doi: 10.1080/0729436052000318541.

[24]

Ryan, A.M., Peer Groups as a Context for the Socialization of Adolescents' Motivation, Engagement, and Achievement in School. Educational Psychologist, 2000. 35(2): 101-111. https://doi.org/10.1207/S15326985EP3502_4. doi: 10.1207/S15326985EP3502_4.

[25]

Jaworski, B., et al., An Activity Theory Analysis of Teaching Goals versus Student Epistemological Positions. The International Journal for Technology in Mathematics Education, 2012. 19: 147-152.

[26]

Burton, L., Why Is Intuition so Important to Mathematicians but Missing from Mathematics Education? For the Learning of Mathematics, 1999. 19(3): 27-32.

[27]

Fischbein, E., Intuition and Proof. For the Learning of Mathematics, 1982. 3(2): 9-24.

[28]

Attridge, N. and M. Inglis, Increasing cognitive inhibition with a difficult prior task: implications for mathematical thinking. ZDM, 2015. 47(5): 723-734. https://doi.org/10.1007/s11858-014-0656-1. doi: 10.1007/s11858-014-0656-1.

[29]

Thomas, M.O.J., Inhibiting intuitive thinking in mathematics education. ZDM, 2015. 47(5): 865-876. https://doi.org/10.1007/s11858-015-0721-4. doi: 10.1007/s11858-015-0721-4.

[30]

Trémolière, B. and W. De Neys, When intuitions are helpful: Prior beliefs can support reasoning in the bat-and-ball problem. Journal of Cognitive Psychology, 2014. 26(4): 486-490. https://doi.org/10.1080/20445911.2014.899238. doi: 10.1080/20445911.2014.899238.

[31]

MacLeod, C.M., The concept of inhibition in cognition, in Inhibition in cognition. 2007, pp. 3-23. Washington, DC, US: American Psychological Association.

[32]

Gilmore, C., et al., The role of cognitive inhibition in different components of arithmetic. ZDM, 2015. 47(5): 771-782. https://doi.org/10.1007/s11858-014-0659-y. doi: 10.1007/s11858-014-0659-y.

[33]

Kahneman, D., Thinking, fast and slow. Thinking, fast and slow. 2011, New York, NY, US: Farrar, Straus and Giroux.

[34]

Babai, R., E. Shalev, and R. Stavy, A warning intervention improves students' ability to overcome intuitive interference. ZDM, 2015. 47(5): 735-745. https://doi.org/10.1007/s11858-015-0670-y. doi: 10.1007/s11858-015-0670-y.

[35]

Frederick, S., Cognitive Reflection and Decision Making. Journal of Economic Perspectives, 2005. 19(4): 25-42. https://doi.org/10.1257/089533005775196732. doi: 10.1257/089533005775196732.

[36] S. Klymchuk, Money puzzles: On critical thinking and financial literacy, Maths Press, 2001. 
[37]

Guilford, J.P., Creativity: Yesterday, Today and Tomorrow. The Journal of Creative Behavior, 1967. 1(1): 3-14.

[38]

Thomas, M.O.J., Developing versatility in mathematical thinking. Mediterranean Journal for Research in Mathematics Education, 2008. 7(2): 67-87.

[39]

Hatano, G. and Y. Oura, Commentary: Reconceptualizing School Learning Using Insight From Expertise Research. Educational Researcher, 2003. 32(8): 26-29. https://doi.org/10.3102/0013189X032008026. doi: 10.3102/0013189X032008026.

[40]

Thomas, M.O.J. Conceptual representations and versatile mathematical thinking. in ICME-10: The 10th International Congress on Mathematical Education. 2004. Copenhagen, Denmark.

[41] G. Wallas, The art of thought, Harcourt, Brace & Company, 1926. 
[42]

Leikin, R., in Exploring Mathematical Creativity Using Multiple Solution Tasks. 2009, pp. 129-145: Brill | Sense.

[43]

Leikin, R. and D. Pitta-Pantazi, Creativity and mathematics education: the state of the art. ZDM, 2013. 45(2): 159-166. https://doi.org/10.1007/s11858-012-0459-1. doi: 10.1007/s11858-012-0459-1.

[44]

Sriraman, B. and P. Haavold, Creativity and giftedness in mathematics education: A pragmatic view, in First compendium for research in mathematics education, J. Cai, Ed. 2016: National Council of Teachers of Mathematics.

[45]

Guilford, J.P., Traits of creativity., in Creativity and its cultivation, H.H. Anderson, Ed. 1959, pp. 142-161: Harper & Brothers.

[46]

National Survey of Student Engagement, Indiana University Center for Postsecondary Research and Planning, 2000.

[47] E. Torrance, Education and the creative potential, University of Minnesota Press, 1963. 
[48]

Evans, T., et al., Engagement of undergraduate STEM students: the influence of non-routine problems. Higher Education Research & Development, 2020: 1-17.https://doi.org/10.1080/07294360.2020.1835838. doi: 10.1080/07294360.2020.1835838.

show all references

References:
[1]

Blondal, K.S. and S. Adalbjarnardottir, Student Disengagement in Relation to Expected and Unexpected Educational Pathways. Scandinavian Journal of Educational Research, 2012. 56(1): 85-100. https://doi.org/10.1080/00313831.2011.568607. doi: 10.1080/00313831.2011.568607.

[2]

Kahu, E.R. and K. Nelson, Student engagement in the educational interface: understanding the mechanisms of student success. Higher Education Research & Development, 2018. 37(1): 58-71. https://doi.org/10.1080/07294360.2017.1344197. doi: 10.1080/07294360.2017.1344197.

[3] W. Poundstone, How would you move Mount Fuji? Microsoft's cult of the puzzle - How the world's smartest companies select the most creative thinkers, Little Brown and Company, 2000. 
[4]

Falkner, N., R. Sooriamurthi, and Z. Michalewicz, Teaching puzzle-based learning: Development of basic concepts. Teaching Mathematics and Computer Science, 2012. 10(1): 183-204.

[5]

Thomas, C., et al., Puzzle-based Learning of Mathematics in Engineering. Engineering Education, 2013. 8(1): 122-134. https://doi.org/10.11120/ened.2013.00005. doi: 10.11120/ened.2013.00005.

[6]

Evans, T., M.O.J. Thomas, and S. Klymchuk, Non-routine problem solving through the lens of self-efficacy. Higher Education Research & Development, 2020: 1-18. https://doi.org/10.1080/07294360.2020.1818061. doi: 10.1080/07294360.2020.1818061.

[7]

Klymchuk, S., Puzzle-based learning in engineering mathematics: students' attitudes. International Journal of Mathematical Education in Science and Technology, 2017. 48(7): 1106-1119. https://doi.org/10.1080/0020739X.2017.1327088. doi: 10.1080/0020739X.2017.1327088.

[8]

Parhami, B., A puzzle-based seminar for computer engineering freshmen. Computer Science Education, 2008. 18(4): 261-277. https://doi.org/10.1080/08993400802594089. doi: 10.1080/08993400802594089.

[9]

Pugh, K.J. and D.A. Bergin, Motivational Influences on Transfer. Educational Psychologist, 2006. 41(3): 147-160. https://doi.org/10.1207/s15326985ep4103_2. doi: 10.1207/s15326985ep4103_2.

[10]

Ford, J.K., et al., Relationships of goal orientation, metacognitive activity, and practice strategies with learning outcomes and transfer. Journal of Applied Psychology, 1998. 83(2): 218-233. https://doi.org/10.1037/0021-9010.83.2.218. doi: 10.1037/0021-9010.83.2.218.

[11]

Usher, E.L. and F. Pajares, Sources of Self-Efficacy in School: Critical Review of the Literature and Future Directions. Review of Educational Research, 2008. 78(4): 751-796. https://doi.org/10.3102/0034654308321456. doi: 10.3102/0034654308321456.

[12]

Selden, A., et al., Why can't calculus students access their knowledge to solve non-routine problems?, in Research in collegiate mathematics education: IV. Issues in mathematical education, E. Dubinsky, A. Schoenfeld, and J. Kaput, Editors. 2000, pp. 128-153: American Mathematical Society.

[13] Z. Michalewicz and M. Michalewicz, Puzzle-based learning: An introduction to critical thinking, mathematics, and problem solving, Hybrid Publishers, 2008. 
[14]

Fredricks, J.A., Engagement in School and Out-of-School Contexts: A Multidimensional View of Engagement. Theory Into Practice, 2011. 50(4): 327-335. https://doi.org/10.1080/00405841.2011.607401. doi: 10.1080/00405841.2011.607401.

[15]

Fredricks, J.A., M. Filsecker, and M.A. Lawson, Student engagement, context, and adjustment: Addressing definitional, measurement, and methodological issues. Learning and Instruction, 2016. 43: 1-4. https://doi.org/https://doi.org/10.1016/j.learninstruc.2016.02.002.

[16]

Watt, H.M.G. and M. Goos, Theoretical foundations of engagement in mathematics. Mathematics Education Research Journal, 2017. 29(2): 133-142. https://doi.org/10.1007/s13394-017-0206-6. doi: 10.1007/s13394-017-0206-6.

[17]

Helme, S. and D. Clarke, Identifying cognitive engagement in the mathematics classroom. Mathematics Education Research Journal, 2001. 13(2): 133-153. https://doi.org/10.1007/BF03217103. doi: 10.1007/BF03217103.

[18]

Eccles, J., Expectancies, values, and academic behaviors, in Achievement and Achievement Motives: Psychological and Sociological Approaches., J.T. Spence, Ed. 1983, pp. 75-146: W.H. Freeman.

[19] A.H. Schoenfeld, Mathematical Problem Solving, Academic Press, 1985. 
[20] G. Pólya, How to Solve it?, Princeton University Press, Princeton, NJ, 1945. 
[21]

Shoenfeld, A.H., Research Commentary: Reflections of an Accidental Theorist. Journal for Research in Mathematics Education JRME, 2010. 41(2): 104-116. https://doi.org/10.5951/jresematheduc.41.2.0104. doi: 10.5951/jresematheduc.41.2.0104.

[22]

Connell, J.P. and J.G. Wellborn, Competence, autonomy, and relatedness: A motivational analysis of self-system processes, in Self processes and development. 1991, pp. 43-77. Hillsdale, NJ, US: Lawrence Erlbaum Associates, Inc.

[23]

Ahlfeldt, S., S. Mehta, and T. Sellnow, Measurement and analysis of student engagement in university classes where varying levels of PBL methods of instruction are in use. Higher Education Research & Development, 2005. 24(1): 5-20. https://doi.org/10.1080/0729436052000318541. doi: 10.1080/0729436052000318541.

[24]

Ryan, A.M., Peer Groups as a Context for the Socialization of Adolescents' Motivation, Engagement, and Achievement in School. Educational Psychologist, 2000. 35(2): 101-111. https://doi.org/10.1207/S15326985EP3502_4. doi: 10.1207/S15326985EP3502_4.

[25]

Jaworski, B., et al., An Activity Theory Analysis of Teaching Goals versus Student Epistemological Positions. The International Journal for Technology in Mathematics Education, 2012. 19: 147-152.

[26]

Burton, L., Why Is Intuition so Important to Mathematicians but Missing from Mathematics Education? For the Learning of Mathematics, 1999. 19(3): 27-32.

[27]

Fischbein, E., Intuition and Proof. For the Learning of Mathematics, 1982. 3(2): 9-24.

[28]

Attridge, N. and M. Inglis, Increasing cognitive inhibition with a difficult prior task: implications for mathematical thinking. ZDM, 2015. 47(5): 723-734. https://doi.org/10.1007/s11858-014-0656-1. doi: 10.1007/s11858-014-0656-1.

[29]

Thomas, M.O.J., Inhibiting intuitive thinking in mathematics education. ZDM, 2015. 47(5): 865-876. https://doi.org/10.1007/s11858-015-0721-4. doi: 10.1007/s11858-015-0721-4.

[30]

Trémolière, B. and W. De Neys, When intuitions are helpful: Prior beliefs can support reasoning in the bat-and-ball problem. Journal of Cognitive Psychology, 2014. 26(4): 486-490. https://doi.org/10.1080/20445911.2014.899238. doi: 10.1080/20445911.2014.899238.

[31]

MacLeod, C.M., The concept of inhibition in cognition, in Inhibition in cognition. 2007, pp. 3-23. Washington, DC, US: American Psychological Association.

[32]

Gilmore, C., et al., The role of cognitive inhibition in different components of arithmetic. ZDM, 2015. 47(5): 771-782. https://doi.org/10.1007/s11858-014-0659-y. doi: 10.1007/s11858-014-0659-y.

[33]

Kahneman, D., Thinking, fast and slow. Thinking, fast and slow. 2011, New York, NY, US: Farrar, Straus and Giroux.

[34]

Babai, R., E. Shalev, and R. Stavy, A warning intervention improves students' ability to overcome intuitive interference. ZDM, 2015. 47(5): 735-745. https://doi.org/10.1007/s11858-015-0670-y. doi: 10.1007/s11858-015-0670-y.

[35]

Frederick, S., Cognitive Reflection and Decision Making. Journal of Economic Perspectives, 2005. 19(4): 25-42. https://doi.org/10.1257/089533005775196732. doi: 10.1257/089533005775196732.

[36] S. Klymchuk, Money puzzles: On critical thinking and financial literacy, Maths Press, 2001. 
[37]

Guilford, J.P., Creativity: Yesterday, Today and Tomorrow. The Journal of Creative Behavior, 1967. 1(1): 3-14.

[38]

Thomas, M.O.J., Developing versatility in mathematical thinking. Mediterranean Journal for Research in Mathematics Education, 2008. 7(2): 67-87.

[39]

Hatano, G. and Y. Oura, Commentary: Reconceptualizing School Learning Using Insight From Expertise Research. Educational Researcher, 2003. 32(8): 26-29. https://doi.org/10.3102/0013189X032008026. doi: 10.3102/0013189X032008026.

[40]

Thomas, M.O.J. Conceptual representations and versatile mathematical thinking. in ICME-10: The 10th International Congress on Mathematical Education. 2004. Copenhagen, Denmark.

[41] G. Wallas, The art of thought, Harcourt, Brace & Company, 1926. 
[42]

Leikin, R., in Exploring Mathematical Creativity Using Multiple Solution Tasks. 2009, pp. 129-145: Brill | Sense.

[43]

Leikin, R. and D. Pitta-Pantazi, Creativity and mathematics education: the state of the art. ZDM, 2013. 45(2): 159-166. https://doi.org/10.1007/s11858-012-0459-1. doi: 10.1007/s11858-012-0459-1.

[44]

Sriraman, B. and P. Haavold, Creativity and giftedness in mathematics education: A pragmatic view, in First compendium for research in mathematics education, J. Cai, Ed. 2016: National Council of Teachers of Mathematics.

[45]

Guilford, J.P., Traits of creativity., in Creativity and its cultivation, H.H. Anderson, Ed. 1959, pp. 142-161: Harper & Brothers.

[46]

National Survey of Student Engagement, Indiana University Center for Postsecondary Research and Planning, 2000.

[47] E. Torrance, Education and the creative potential, University of Minnesota Press, 1963. 
[48]

Evans, T., et al., Engagement of undergraduate STEM students: the influence of non-routine problems. Higher Education Research & Development, 2020: 1-17.https://doi.org/10.1080/07294360.2020.1835838. doi: 10.1080/07294360.2020.1835838.

Figure 1.  Example of a non-routine problem: finding a shaded fraction of the large square
Figure 2.  Off-task behaviour patterns, reported as the average number of students per observation in a group of 30 students (approximately)

Notes: the data points correspond to the rate of off-task behaviour among a group of 30 students for each of the five lectures observed (recorded prior, during, and straight after the intervention).

Figure 3.  Level of engagement in non-routine problem-solving activities by prior achievement (N=55)

Notes. Item: With respect to this course, about how often have you worked on solving puzzles and/or creative problems during class? (4=Very Often, 3=Often, 2=Occasionally, 1=Never)
Error bars: +/-2 SE

Table 1.  Data collection and analyses (5 sub-studies)
Sub-study Design Data collection Sample Data analysis
1 Naturalistic observation Observers (detached, non-participant) used two observation protocols in class at the start and the end of the trial semester. Protocol 1 was designed to validate the fidelity of implementation; Protocol 2 to record occurrences of student off-task behaviour Over 600 students studying mathematics, astronomy and computer science in New Zealand tertiary institutions Quantitative, Friedman test
2 Quasi-experimental, comparative Two questionnaires at the start (T1) and the end (T2) of the trial semester comprising various instruments 683 students from four tertiary institutions were invited to participate, but only 64 students completed both questionnaires Quantitative, Descriptive, ANOVA tests, Regression analysis
3 Descriptive, comparative As part of the questionnaire at T2, an instrument assessing student overall engagement in the course during the trial semester based on [23] Identical sample as in phase 2 Quantitative, Descriptive, ANOVA test
4 Descriptive Survey of lecturer-participants (open-ended questions) Nine lecturers participating in the trial Qualitative
5 Descriptive Focus group interviews of student-participants (tasks and open-ended questions) Twelve volunteering students (in five small groups) Qualitative
Sub-study Design Data collection Sample Data analysis
1 Naturalistic observation Observers (detached, non-participant) used two observation protocols in class at the start and the end of the trial semester. Protocol 1 was designed to validate the fidelity of implementation; Protocol 2 to record occurrences of student off-task behaviour Over 600 students studying mathematics, astronomy and computer science in New Zealand tertiary institutions Quantitative, Friedman test
2 Quasi-experimental, comparative Two questionnaires at the start (T1) and the end (T2) of the trial semester comprising various instruments 683 students from four tertiary institutions were invited to participate, but only 64 students completed both questionnaires Quantitative, Descriptive, ANOVA tests, Regression analysis
3 Descriptive, comparative As part of the questionnaire at T2, an instrument assessing student overall engagement in the course during the trial semester based on [23] Identical sample as in phase 2 Quantitative, Descriptive, ANOVA test
4 Descriptive Survey of lecturer-participants (open-ended questions) Nine lecturers participating in the trial Qualitative
5 Descriptive Focus group interviews of student-participants (tasks and open-ended questions) Twelve volunteering students (in five small groups) Qualitative
Table 2.  Statistics of CRT by time and form
Time 1 Time 2 Difference (Time 2 - Time 1)
Variable n M SD M SD M SD t-value Cohen's d
All Participants 41 0.76 0.28 0.80 0.25 0.04 0.25 0.89 0.15
  Form A - Form A 14 0.87 0.19 0.84 0.22 (0.03) 0.19 (0.62) (0.14)
  Form B - Form B 7 0.71 0.32 0.71 0.28 0.00 0.38 0.00 0.00
  Form A - Form B 5 0.76 0.26 0.72 0.33 (0.04) 0.09 (1.00) (0.13)
  Form B - Form A 15 0.68 0.32 0.81 0.23 0.13 0.24 2.09 0.48
Notes. Parenthetical values (in red) are negative.Cohen's d = (MT2-MT1) / SQRT((SDT12 + SDT22) / 2)
Time 1 Time 2 Difference (Time 2 - Time 1)
Variable n M SD M SD M SD t-value Cohen's d
All Participants 41 0.76 0.28 0.80 0.25 0.04 0.25 0.89 0.15
  Form A - Form A 14 0.87 0.19 0.84 0.22 (0.03) 0.19 (0.62) (0.14)
  Form B - Form B 7 0.71 0.32 0.71 0.28 0.00 0.38 0.00 0.00
  Form A - Form B 5 0.76 0.26 0.72 0.33 (0.04) 0.09 (1.00) (0.13)
  Form B - Form A 15 0.68 0.32 0.81 0.23 0.13 0.24 2.09 0.48
Notes. Parenthetical values (in red) are negative.Cohen's d = (MT2-MT1) / SQRT((SDT12 + SDT22) / 2)
Table 3.  Statistics of convergent thinking by time and form
Time 1 Time 2 Difference (Time 2 - Time 1)
Variable n M SD M SD M SD t-value Cohen's d
All Participants 42 0.55 0.25 0.53 0.28 (0.02) 0.29 (0.39) (0.07)
  Form A - Form A 15 0.53 0.23 0.52 0.29 (0.02) 0.33 (0.19) (0.06)
  Form B - Form B 7 0.64 0.20 0.61 0.13 (0.04) 0.09 (1.00) (0.21)
  Form A - Form B 5 0.55 0.41 0.60 0.29 0.05 0.33 0.34 0.14
  Form B - Form A 15 0.52 0.24 0.48 0.32 (0.03) 0.33 (0.40) (0.12)
Notes. Parenthetical values (in red) are negative.Cohen's d = (MT2-MT1) / SQRT((SDT12 + SDT22) / 2)
Time 1 Time 2 Difference (Time 2 - Time 1)
Variable n M SD M SD M SD t-value Cohen's d
All Participants 42 0.55 0.25 0.53 0.28 (0.02) 0.29 (0.39) (0.07)
  Form A - Form A 15 0.53 0.23 0.52 0.29 (0.02) 0.33 (0.19) (0.06)
  Form B - Form B 7 0.64 0.20 0.61 0.13 (0.04) 0.09 (1.00) (0.21)
  Form A - Form B 5 0.55 0.41 0.60 0.29 0.05 0.33 0.34 0.14
  Form B - Form A 15 0.52 0.24 0.48 0.32 (0.03) 0.33 (0.40) (0.12)
Notes. Parenthetical values (in red) are negative.Cohen's d = (MT2-MT1) / SQRT((SDT12 + SDT22) / 2)
Table 4.  CRT Effect for Gender (transformed variable: average)
Variable n Time 1 Time 2 Test F Partial η2 Observed Power
M SD M SD
  Male 23 0.74 0.30 0.81 0.23 Time 0.53 0.014 0.110
  Female 17 0.80 0.24 0.79 0.28 Gender 0.07 0.002 0.058
Total 40 0.77 0.28 0.80 0.25 Time*Gender 1.06 0.027 0.171
Variable n Time 1 Time 2 Test F Partial η2 Observed Power
M SD M SD
  Male 23 0.74 0.30 0.81 0.23 Time 0.53 0.014 0.110
  Female 17 0.80 0.24 0.79 0.28 Gender 0.07 0.002 0.058
Total 40 0.77 0.28 0.80 0.25 Time*Gender 1.06 0.027 0.171
Table 5.  Frequency, elaboration, and originality scores by time
Time 1 Time 2 Difference (Time 2 - Time 1)
Variable n M SD M SD M SD t-value Cohen's d
Fluency 43 7.35 4.31 8.09 4.14 0.74 3.27 1.49 0.18
Elaboration 43 2.56 3.13 2.49 2.33 (0.07) 3.20 (0.14) (0.03)
Originality 43 10.52 7.38 12.00 8.36 1.48 6.89 1.57 0.19
Notes. Parenthetical values (in red) are negative.Cohen's d = (MT2-MT1) / SQRT((SDT12 + SDT22) / 2)
Time 1 Time 2 Difference (Time 2 - Time 1)
Variable n M SD M SD M SD t-value Cohen's d
Fluency 43 7.35 4.31 8.09 4.14 0.74 3.27 1.49 0.18
Elaboration 43 2.56 3.13 2.49 2.33 (0.07) 3.20 (0.14) (0.03)
Originality 43 10.52 7.38 12.00 8.36 1.48 6.89 1.57 0.19
Notes. Parenthetical values (in red) are negative.Cohen's d = (MT2-MT1) / SQRT((SDT12 + SDT22) / 2)
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