# Active Learning via Problem-Based Collaborative Games in a Large Mathematics University Course in Hong Kong

^{*}

## Abstract

**:**

## 1. Introduction

#### 1.1. Active Learning

#### 1.2. Different Approaches to Active Learning in STEM Education

#### 1.3. Conceptions of Teaching and Learning Within the Hong Kong Context

## 2. Literature Review and Research Hypotheses

#### 2.1. Active Learning in Mathematics in Tertiary Education in Asia

#### 2.2. Student Performance and Active learning in Large University Math Classes

#### 2.3. Active Learning Through a Problem-Based Learning Methodology in Mathematics Education

#### 2.4. Applying Technology-Enabled Active Learning in Mathematics Education

#### 2.5. Game-Based Learning in Mathematics in Tertiary Education in Asia

#### 2.6. Embedding Formative Assessment into Game-Based Learning

#### 2.7. Research Hypotheses

**H1:**

**H2:**

**H3:**

## 3. Research Methodology

#### 3.1. Research Setting and Activity

#### 3.2. Pre and Post Concept and Midterm Tests

#### 3.3. Active Learning Procedures and Interventions in Large Classes

#### 3.3.1. Question and Answer

^{x}can we simply apply the power rule?”

#### 3.3.2. Student Response Systems

^{,}(a)(x − a) to approximate y = f (x) is

#### 3.3.3. Cooperative Problem-Based Learning with Games

“Claim: If the function f (x) is an odd degree polynomial, then f (x) has a root. The possible four answers are: (A) Claim is true; (B) Claim is false; (C) If the claim is true depends on the odd number; (D) None of the above.”

## 4. Results and Analyses

#### 4.1. Pre-CCI Test and Pre-Calculus Knowledge

#### 4.2. Post-CCI Test and Students’ Perceptions of Active Learning

#### 4.3. The Normalized Gain

^{2}indicated that there could be other latent factors to normalized gain and hence, future studies could examine other variables that could affect normalized gain with a higher influence.

#### 4.4. Midterm Test Scores

#### 4.5. Active Learning Groups and Normalized Gain

#### 4.6. Hypotheses Testing

**H1:**

**H2:**

**H3:**

#### 4.7. Comparison of North American Versus Asian CCI Scores

- The average gain over all six sections was 0.23815 (0.35).
- Two sections have a gain of 0.37 to 0.46 (0.4 to 0.44).
- The range of gain scores was 0.12 to 0.46 (0.21 to 0.44).

## 5. Discussion

## 6. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## References

- Kim, K.; Sharma, P.; Land, S.M.; Furlong, K.P. Effects of active learning on enhancing student critical thinking in an undergraduate general science course. Innov. High. Educ.
**2013**, 38, 223–235. [Google Scholar] [CrossRef] - Marton, F. Towards a pedagogical theory of learning. In Deep Active Learning; Springer: Singapore, 2018; pp. 59–77. [Google Scholar]
- Prince, M. Does active learning work? A review of the research. J. Eng. Educ.
**2004**, 93, 223–231. [Google Scholar] [CrossRef] - Roehl, A.; Reddy, S.L.; Shannon, G.J. The flipped classroom: An opportunity to engage millennial students through active learning strategies. J. Fam. Consum. Sci.
**2013**, 105, 44–49. [Google Scholar] [CrossRef] - Ní Raghallaigh, M.; Cunniffe, R. Creating a safe climate for active learning and student engagement: An example from an introductory social work module. Teach. High. Educ.
**2013**, 18, 93–105. [Google Scholar] [CrossRef] - Shroff, R.; Ting, F.; Lam, W. Development and validation of an instrument to measure students’ perceptions of technology-enabled active learning. Australas. J. Educ. Technol.
**2019**, 35, 109–127. [Google Scholar] [CrossRef] - Koohang, A.; Paliszkiewicz, J. Knowledge construction in e-learning: An empirical validation of an active learning model. J. Comput. Inf. Syst.
**2013**, 53, 109–114. [Google Scholar] [CrossRef] - Lumpkin, A.; Achen, R.M.; Dodd, R.K. Student perceptions of active learning. Coll. Stud. J.
**2015**, 49, 121–133. [Google Scholar] - Savery, J.R. Overview of problem-based learning: Definitions and distinctions. In Essential Readings in Problem-Based Learning: Exploring and Extending the Legacy of Howard S. Barrows; Purdue University Press: West Lafayette, IN, USA, 2015; Volume 9, pp. 5–15. [Google Scholar]
- Baepler, P.; Walker, J.; Driessen, M. It’s not about seat time: Blending, flipping, and efficiency in active learning classrooms. Comput. Educ.
**2014**, 78, 227–236. [Google Scholar] [CrossRef] - Sharples, M. The design of personal mobile technologies for lifelong learning. Comput. Educ.
**2000**, 34, 177–193. [Google Scholar] [CrossRef] - Blasco-Arcas, L.; Buil, I.; Hernández-Ortega, B.; Sese, F.J. Using clickers in class. The role of interactivity, active collaborative learning and engagement in learning performance. Comput. Educ.
**2013**, 62, 102–110. [Google Scholar] [CrossRef] - Lai, C.-L.; Hwang, G.-J. A self-regulated flipped classroom approach to improving students’ learning performance in a mathematics course. Comput. Educ.
**2016**, 100, 126–140. [Google Scholar] [CrossRef] - Lucas, A. Using peer instruction and i-clickers to enhance student participation in calculus. Primus
**2009**, 19, 219–231. [Google Scholar] [CrossRef] - Hmelo-Silver, C.E.; Barrows, H.S. Goals and strategies of a problem-based learning facilitator. Interdiscip. J. Probl.-Based Learn.
**2006**, 1, 21–39. [Google Scholar] [CrossRef] - McCarthy, J.P.; Anderson, L. Active learning techniques versus traditional teaching styles: Two experiments from history and political science. Innov. High. Educ.
**2000**, 24, 279–294. [Google Scholar] [CrossRef] - Michael, J. Where’s the evidence that active learning works? Adv. Physiol. Educ.
**2006**, 30, 159–167. [Google Scholar] [CrossRef] [PubMed] - Gormally, C.; Brickman, P.; Hallar, B.; Armstrong, N. Effects of inquiry-based learning on students’ science literacy skills and confidence. Int. J. Scholarsh. Teach. Learn.
**2009**, 3, 16. [Google Scholar] [CrossRef] - Smith, K.A.; Douglas, T.C.; Cox, M.F. Supportive teaching and learning strategies in STEM education. New Dir. Teach. Learn.
**2009**, 2009, 19–32. [Google Scholar] [CrossRef] - Freeman, S.; O’Connor, E.; Parks, J.W.; Cunningham, M.; Hurley, D.; Haak, D.; Dirks, C.; Wenderoth, M.P. Prescribed active learning increases performance in introductory biology. CBE-Life Sci. Educ.
**2007**, 6, 132–139. [Google Scholar] [CrossRef] - Freeman, S.; Eddy, S.L.; McDonough, M.; Smith, M.K.; Okoroafor, N.; Jordt, H.; Wenderoth, M.P. Active learning increases student performance in science, engineering, and mathematics. Proc. Natl. Acad. Sci. USA
**2014**, 111, 8410–8415. [Google Scholar] [CrossRef][Green Version] - Haak, D.C.; HilleRisLambers, J.; Pitre, E.; Freeman, S. Increased structure and active learning reduce the achievement gap in introductory biology. Science
**2011**, 332, 1213–1216. [Google Scholar] [CrossRef] - Kennedy, P. Learning cultures and learning styles: Myth-understandings about adult (Hong Kong) Chinese learners. Int. J. Lifelong Educ.
**2002**, 21, 430–445. [Google Scholar] [CrossRef] - Pham, T.T.H.; Renshaw, P. How to enable asian teachers to empower students to adopt student-centred learning. Aust. J. Teach. Educ.
**2013**, 38, 65–85. [Google Scholar] [CrossRef] - Fendos, J. Us experiences with stem education reform and implications for asia. Int. J. Comp. Educ. Dev.
**2018**, 20, 51–66. [Google Scholar] [CrossRef] - Bureau, H.K.E. Promotion of STEM Education—Unleashing Potential in Innovation; Curriculum Development Council: Hong Kong, 2015; pp. 1–24. [Google Scholar]
- Cheng, X.; Ka Ho Lee, K.; Chang, E.Y.; Yang, X. The “flipped classroom” approach: Stimulating positive learning attitudes and improving mastery of histology among medical students. Anat. Sci. Educ.
**2017**, 10, 317–327. [Google Scholar] [CrossRef] [PubMed] - Wu, W.-H.; Yan, W.-C.; Kao, H.-Y.; Wang, W.-Y.; Wu, Y.-C.J. Integration of RPG use and ELC foundation to examine students’ learning for practice. Comput. Hum. Behav.
**2016**, 55, 1179–1184. [Google Scholar] [CrossRef] - Chien, Y.-T.; Lee, Y.-H.; Li, T.-Y.; Chang, C.-Y. Examining the effects of displaying clicker voting results on high school students’ voting behaviors, discussion processes, and learning outcomes. Eurasia J. Math. Sci. Technol. Educ.
**2015**, 11, 1089–1104. [Google Scholar] - Kaur, B. Towards excellence in mathematics education—Singapore’s experience. Procedia-Soc. Behav. Sci.
**2010**, 8, 28–34. [Google Scholar] [CrossRef] - Chen, C.-H.; Chiu, C.-H. Collaboration scripts for enhancing metacognitive self-regulation and mathematics literacy. Int. J. Sci. Math. Educ.
**2016**, 14, 263–280. [Google Scholar] [CrossRef] - Li, Y.-B.; Zheng, W.-Z.; Yang, F. Cooperation learning of flip teaching style on the MBA mathematics education efficiency. Eurasia J. Math. Sci. Technol. Educ.
**2017**, 13, 6963–6972. [Google Scholar] [CrossRef] - Rosenthal, J.S. Active learning strategies in advanced mathematics classes. Stud. High. Educ.
**1995**, 20, 223–228. [Google Scholar] [CrossRef] - Radu, O.; Seifert, T. Mathematical intimacy within blended and face-to-face learning environments. Eur. J. Open Distance E-learn.
**2011**, 14, 1–6. [Google Scholar] - Isabwe, G.M.N.; Reichert, F. Developing a formative assessment system for mathematics using mobile technology: A student centred approach. In Proceedings of the 2012 International Conference on Education and e-Learning Innovations (ICEELI), Sousse, Tunisia, 1–3 July 2012; pp. 1–6. [Google Scholar]
- Kyriacou, C. Active learning in secondary school mathematics. Br. Educ. Res. J.
**1992**, 18, 309–318. [Google Scholar] [CrossRef] - Keeler, C.M.; Steinhorst, R.K. Using small groups to promote active learning in the introductory statistics course: A report from the field. J. Stat. Educ.
**1995**, 3, 1–8. [Google Scholar] [CrossRef] - Stanberry, M.L. Active learning: A case study of student engagement in college Calculus. Int. J. Math. Educ. Sci. Technol.
**2018**, 49, 959–969. [Google Scholar] [CrossRef] - Carmichael, J. Team-based learning enhances performance in introductory biology. J. Coll. Sci. Teach.
**2009**, 38, 54–61. [Google Scholar] - Wanous, M.; Procter, B.; Murshid, K. Assessment for learning and skills development: The case of large classes. Eur. J. Eng. Educ.
**2009**, 34, 77–85. [Google Scholar] [CrossRef] - Rotgans, J.I.; Schmidt, H.G. Situational interest and academic achievement in the active-learning classroom. Learn. Instr.
**2011**, 21, 58–67. [Google Scholar] [CrossRef] - Chi, M.T.; Wylie, R. The ICAP framework: Linking cognitive engagement to active learning outcomes. Educ. Psychol.
**2014**, 49, 219–243. [Google Scholar] [CrossRef] - Isbell, L.M.; Cote, N.G. Connecting with struggling students to improve performance in large classes. Teach. Psychol.
**2009**, 36, 185–188. [Google Scholar] [CrossRef] - Roh, K.H. Problem-based learning in mathematics. In ERIC Clearinghouse for Science Mathematics and Environmental Education; ERIC Pubications: Columbus, OH, USA, 2003. [Google Scholar]
- Padmavathy, R.; Mareesh, K. Effectiveness of problem based learning in mathematics. Int. Multidiscip. E-J.
**2013**, 2, 45–51. [Google Scholar] - Triantafyllou, E.; Timcenko, O. Developing digital technologies for undergraduate university mathematics: Challenges, issues and perspectives. In Proceedings of the 21st International Conference on Computers in Education, Bali, Indonesia, 18–22 November 2013; pp. 971–976. [Google Scholar]
- Duffy, T.M.; Cunningham, D.J. Constructivism: Implications for the design and delivery of instruction. In Handbook of Research for Educational Communications and Technology; Jonassen, D.H., Ed.; Simon & Schuster: New York, NY, USA, 1996; pp. 170–198. [Google Scholar]
- Mellecker, R.R.; Witherspoon, L.; Watterson, T. Active learning: Educational experiences enhanced through technology-driven active game play. J. Educ. Res.
**2013**, 106, 352–359. [Google Scholar] [CrossRef] - Domínguez, A.; Saenz-De-Navarrete, J.; de-Marcos, L.; Fernández-Sanz, L.; Pagés, C.; Martínez-Herráiz, J.-J. Gamifying learning experiences: Practical implications and outcomes. Comput. Educ.
**2013**, 63, 380–392. [Google Scholar] [CrossRef] - Keengwe, J. Promoting Active Learning through the Integration of Mobile and Ubiquitous Technologies; IGI Global: Hershey, PA, USA, 2014. [Google Scholar]
- Ciampa, K. Learning in a mobile age: An investigation of student motivation. J. Comput. Assist. Learn.
**2014**, 30, 82–96. [Google Scholar] [CrossRef] - Ally, M.; Prieto-Blázquez, J. What is the future of mobile learning in education? Int. J. Educ. Technol. High. Educ.
**2014**, 11, 142–151. [Google Scholar] - Abdul Jabbar, A.I.; Felicia, P. Gameplay engagement and learning in game-based learning: A systematic review. Rev. Educ. Res.
**2015**, 85, 740–779. [Google Scholar] [CrossRef] - Park, H. Relationship between motivation and student’s activity on educational game. Int. J. Grid Distrib. Comput.
**2012**, 5, 101–114. [Google Scholar] - Pivec, M. Play and learn: Potentials of game-based learning. Br. J. Educ. Technol.
**2007**, 38, 387–393. [Google Scholar] [CrossRef] - Eseryel, D.; Law, V.; Ifenthaler, D.; Ge, X.; Miller, R. An investigation of the interrelationships between motivation, engagement, and complex problem solving in game-based learning. J. Educ. Technol. Soc.
**2014**, 17, 42–53. [Google Scholar] - Shroff, R.H.; Keyes, C. A proposed framework to understand the intrinsic motivation factors on university students’ behavioral intention to use a mobile application for learning. J. Inf. Technol. Educ. Res.
**2017**, 16, 143–168. [Google Scholar] [CrossRef] - Snow, E.L.; Jackson, G.T.; Varner, L.K.; McNamara, D.S. Expectations of technology: A factor to consider in game-based learning environments. In Proceedings of the International Conference on Artificial Intelligence in Education, Memphis, TN, USA, 9–13 July 2013; Springer: Berlin/Heidelberg, Germany; pp. 359–368. [Google Scholar]
- Yoon, H.S. Can i play with you? The intersection of play and writing in a kindergarten classroom. Contemp. Issues Early Child.
**2014**, 15, 109–121. [Google Scholar] [CrossRef] - Ku, O.; Chen, S.Y.; Wu, D.H.; Lao, A.C.; Chan, T.-W. The effects of game-based learning on mathematical confidence and performance: High ability vs. Low ability. J. Educ. Technol. Soc.
**2014**, 17, 65–78. [Google Scholar] - Cheng, Y.-M.; Lou, S.-J.; Kuo, S.-H.; Shih, R.-C. Investigating elementary school students’ technology acceptance by applying digital game-based learning to environmental education. Australas. J. Educ. Technol.
**2013**, 29, 96–110. [Google Scholar] [CrossRef] - Yang, J.C.; Chien, K.H.; Liu, T.C. A digital game-based learning system for energy education: An energy conservation pet. TOJET Turk. Online J. Educ. Technol.
**2012**, 11, 27–37. [Google Scholar] - Sung, H.-Y.; Hwang, G.-J. Facilitating effective digital game-based learning behaviors and learning performances of students based on a collaborative knowledge construction strategy. Interact. Learn. Environ.
**2018**, 26, 118–134. [Google Scholar] [CrossRef] - Sung, H.-Y.; Hwang, G.-J. A collaborative game-based learning approach to improving students’ learning performance in science courses. Comput. Educ.
**2013**, 63, 43–51. [Google Scholar] [CrossRef] - Hwang, G.-J.; Wu, P.-H.; Chen, C.-C. An online game approach for improving students’ learning performance in web-based problem-solving activities. Comput. Educ.
**2012**, 59, 1246–1256. [Google Scholar] [CrossRef] - Chang, K.-E.; Wu, L.-J.; Weng, S.-E.; Sung, Y.-T. Embedding game-based problem-solving phase into problem-posing system for mathematics learning. Comput. Educ.
**2012**, 58, 775–786. [Google Scholar] [CrossRef] - Bai, H.; Pan, W.; Hirumi, A.; Kebritchi, M. Assessing the effectiveness of a 3-D instructional game on improving mathematics achievement and motivation of middle school students. Br. J. Educ. Technol.
**2012**, 43, 993–1003. [Google Scholar] [CrossRef] - Kebritchi, M.; Hirumi, A.; Bai, H. The effects of modern mathematics computer games on mathematics achievement and class motivation. Comput. Educ.
**2010**, 55, 427–443. [Google Scholar] [CrossRef] - Hung, C.-M.; Huang, I.; Hwang, G.-J. Effects of digital game-based learning on students’ self-efficacy, motivation, anxiety, and achievements in learning mathematics. J. Comput. Educ.
**2014**, 1, 151–166. [Google Scholar] [CrossRef] - Belland, B.R. The role of construct definition in the creation of formative assessments in game-based learning. In Assessment in Game-Based Learning; Springer: New York, NY, USA, 2012; pp. 29–42. [Google Scholar]
- Law, V.; Chen, C.-H. Promoting science learning in game-based learning with question prompts and feedback. Comput. Educ.
**2016**, 103, 134–143. [Google Scholar] [CrossRef] - Charles, D.; Charles, T.; McNeill, M.; Bustard, D.; Black, M. Game-based feedback for educational multi-user virtual environments. Br. J. Educ. Technol.
**2011**, 42, 638–654. [Google Scholar] [CrossRef] - Tsai, F.-H.; Tsai, C.-C.; Lin, K.-Y. The evaluation of different gaming modes and feedback types on game-based formative assessment in an online learning environment. Comput. Educ.
**2015**, 81, 259–269. [Google Scholar] [CrossRef] - Ismail, M.A.-A.; Mohammad, J.A.-M. Kahoot: A promising tool for formative assessment in medical education. Educ. Med. J.
**2017**, 9, 19–26. [Google Scholar] [CrossRef] - Wang, T.-H. Web-based quiz-game-like formative assessment: Development and evaluation. Comput. Educ.
**2008**, 51, 1247–1263. [Google Scholar] [CrossRef] - Huang, S.-H.; Wu, T.-T.; Huang, Y.-M. Learning diagnosis instruction system based on game-based learning for mathematical course. In Proceedings of the 2013 IIAI International Conference on Advanced Applied Informatics (IIAIAAI), Los Alamitos, CA, USA, 31 August–4 September 2013; pp. 161–165. [Google Scholar]
- Huang, Y.-M.; Huang, S.-H.; Wu, T.-T. Embedding diagnostic mechanisms in a digital game for learning mathematics. Educ. Technol. Res. Dev.
**2014**, 62, 187–207. [Google Scholar] [CrossRef] - Chen, Y.-C. Empirical study on the effect of digital game-based instruction on students’ learning motivation and achievement. Eurasia J. Math. Sci. Technol. Educ.
**2017**, 13, 3177–3187. [Google Scholar] [CrossRef] - Etikan, I.; Musa, S.A.; Alkassim, R.S. Comparison of convenience sampling and purposive sampling. Am. J. Theor. Appl. Stat.
**2016**, 5, 1–4. [Google Scholar] [CrossRef] - Cohen, J. Statistical Power Analysis for the Behavioral Sciences; Lawrence Earlbaum Associates: Hillsdale, NJ, USA, 1988. [Google Scholar]
- Cohen, J. Statistical power analysis. Curr. Dir. Psychol. Sci.
**1992**, 1, 98–101. [Google Scholar] [CrossRef] - Krejcie, R.V.; Morgan, D.W. Determining sample size for research activities. Educ. Psychol. Meas.
**1970**, 30, 607–610. [Google Scholar] [CrossRef] - Epstein, J. The calculus concept inventory-measurement of the effect of teaching methodology in mathematics. Not. Am. Math. Soc.
**2013**, 60, 1018–1027. [Google Scholar] [CrossRef] - Epstein, J. Development and validation of the calculus concept inventory. In Proceedings of the Ninth International Conference on Mathematics Education in a Global Community, Charlotte, NC, USA, 7–12 September 2007; pp. 165–170. [Google Scholar]
- Derera, E.; Naude, M. Unannounced quizzes: A teaching and learning initiative that enhances academic performance and lecture attendance in large undergraduate classes. Mediterr. J. Soc. Sci.
**2014**, 5, 1193. [Google Scholar] [CrossRef] - Chi, M.T. Active-constructive-interactive: A conceptual framework for differentiating learning activities. Top. Cogn. Sci.
**2009**, 1, 73–105. [Google Scholar] [CrossRef] [PubMed] - Hwang, A.; Ang, S.; Francesco, A.M. The silent Chinese: The influence of face and kiasuism on student feedback-seeking behaviors. J. Manag. Educ.
**2002**, 26, 70–98. [Google Scholar] [CrossRef] - Hwang, A.; Arbaugh, J.B. Seeking feedback in blended learning: Competitive versus cooperative student attitudes and their links to learning outcome. J. Comput. Assist. Learn.
**2009**, 25, 280–293. [Google Scholar] [CrossRef] - Good, T.L.; Slavings, R.L.; Harel, K.H.; Emerson, H. Student passivity: A study of question asking in K–12 classrooms. Sociol. Educ.
**1987**, 60, 181–199. [Google Scholar] [CrossRef] - Harrington, C.L. Talk about embarrassment: Exploring the taboo-repression-denial hypothesis. Symb. Interact.
**1992**, 15, 203–225. [Google Scholar] [CrossRef] - Hubbard, J.K.; Couch, B.A. The positive effect of in-class clicker questions on later exams depends on initial student performance level but not question format. Comput. Educ.
**2018**, 120, 1–12. [Google Scholar] [CrossRef] - Lewin, J.D.; Vinson, E.L.; Stetzer, M.R.; Smith, M.K. A campus-wide investigation of clicker implementation: The status of peer discussion in stem classes. CBE-Life Sci. Educ.
**2016**, 15, ar6. [Google Scholar] [CrossRef] - Hoffman, C.; Goodwin, S. A clicker for your thoughts: Technology for active learning. New Libr. World
**2006**, 107, 422–433. [Google Scholar] [CrossRef] - Gauci, S.A.; Dantas, A.M.; Williams, D.A.; Kemm, R.E. Promoting student-centered active learning in lectures with a personal response system. Adv. Physiol. Educ.
**2009**, 33, 60–71. [Google Scholar] [CrossRef] [PubMed][Green Version] - Mamun, M.R.A.; Kim, D. The effect of perceived innovativeness of student response systems (SRSS) on classroom engagement. In Proceedings of the 2018 Twenty-Fourth Americas Conference on Information Systems, New Orleans, LA, USA, 16–18 August 2018; pp. 1–5. [Google Scholar]
- Markowski, C.A.; Markowski, E.P. Conditions for the effectiveness of a preliminary test of variance. Am. Stat.
**1990**, 44, 322–326. [Google Scholar] - de Winter, J.C.; Gosling, S.D.; Potter, J. Comparing the Pearson and Spearman correlation coefficients across distributions and sample sizes: A tutorial using simulations and empirical data. Psychol. Methods
**2016**, 21, 273–290. [Google Scholar] [CrossRef] [PubMed] - Spearman, C. The proof and measurement of association between two things. Am. J. Psychol.
**1987**, 100, 441–471. [Google Scholar] [CrossRef] [PubMed] - Hestenes, D.; Wells, M.; Swackhamer, G. Force concept inventory. Phys. Teach.
**1992**, 30, 141–158. [Google Scholar] [CrossRef] - Coletta, V.P.; Phillips, J.A. Interpreting FCI scores: Normalized gain, preinstruction scores, and scientific reasoning ability. Am. J. Psychol.
**2005**, 73, 1172–1182. [Google Scholar] [CrossRef][Green Version] - Cohen, J.; Cohen, P. Applied Multiple Regression/Correlation Analysis for the Behavioral Sciences; Erlbaum: Hillsdale, NJ, USA, 1983. [Google Scholar]
- Dodhia, R.M. A review of applied multiple regression/correlation analysis for the behavioral sciences. J. Educ. Behav. Stat.
**2005**, 30, 227–229. [Google Scholar] [CrossRef] - Gelman, A. Analysis of variance—Why it is more important than ever. Ann. Stat.
**2005**, 33, 1–53. [Google Scholar] [CrossRef] - Keselman, H.J.; Rogan, J.C. A comparison of the modified-Tukey and Scheffe methods of multiple comparisons for pairwise contrasts. J. Am. Stat. Assoc.
**1978**, 73, 47–52. [Google Scholar] - Marascuilo, L.A.; Levin, J.R. Appropriate post hoc comparisons for interaction and nested hypotheses in analysis of variance designs: The elimination of type IV errors. Am. Educ. Res. J.
**1970**, 7, 397–421. [Google Scholar] [CrossRef]

**Figure 1.**An example of a Kahoot! question asked at the end of an introduction to the definition of a limit section.

Section | Male | Female | Bachelor Level | Higher Diploma | Total Number of Enrolment | N | Test Score Mean | SD |
---|---|---|---|---|---|---|---|---|

Elephant | 144 | 34 | 83 | 95 | 178 | 168 | 69.9583 | 24.08427 |

80.90% | 19.10% | 46.60% | 53.40% | 100.00% | ||||

Dog | 120 | 29 | 143 | 6 | 149 | 143 | 74.7937 | 19.99545 |

80.50% | 19.50% | 96.00% | 4.00% | 100.00% | ||||

Cat | 111 | 56 | 109 | 58 | 167 | 158 | 60.9241 | 24.62574 |

66.50% | 33.50% | 65.30% | 34.70% | 100.00% | ||||

Mouse | 130 | 21 | 31 | 120 | 151 | 136 | 58.6544 | 24.90995 |

86.10% | 13.90% | 20.50% | 79.50% | 100.00% | ||||

Lion | 128 | 47 | 174 | 1 | 175 | 170 | 72.3941 | 21.7977 |

73.10% | 26.90% | 99.40% | 0.60% | 100.00% | ||||

Tiger | 121 | 76 | 93 | 104 | 197 | 184 | 57.5761 | 21.91837 |

61.40% | 38.60% | 47.20% | 52.80% | 100.00% | ||||

Total | 754 | 263 | 633 | 384 | 1017 | 959 | 65.6439 | 23.86943 |

74.10% | 25.90% | 62.20% | 37.80% | 100.00% |

**Table 2.**Pedagogical interventions featured under the interactive, constructive, active, and passive (ICAP) framework.

Passive (Receiving) | Active (Manipulating) | Constructive (Creating) | Interactive (Social Exchange) |
---|---|---|---|

Sitting still and listening to explanations | Summarizing throughout lectures; highlighting sentences | Building concept maps | Cooperative groups—facilitation of student involvement |

Sitting and reading slides silently | Activating existing knowledge | Asking questions | Student-student discussions |

Sitting and watching a video | Assimilating, encoding or storing new information | Explaining concepts and integrating new information with existing knowledge | Small group activities + feedback |

Information is stored in encapsulated form without embedding it in a relevant schema | Applying knowledge to similar but non-identical contexts (i.e., similar problems or concepts that need to be explained | Comparing and contrasting to prior knowledge or other materials | Cooperative/team-based and collaborative/group learning; |

Minimal understanding | Selected information activates prior knowledge and schema | Justifying or providing rationales; deep understanding | Group interaction—communication between group members, including cooperative problem solving exercises |

**Table 3.**Demographic information of students who completed both pre- and post-calculus concept inventory (CCI) test (N = 365).

Section | N | Male (%) | Female (%) | Level B*(%) | HD** (%) | Pre-CCI Mean | SD | Post-CCI Mean | SD | N | Test Mean | SD |
---|---|---|---|---|---|---|---|---|---|---|---|---|

Elephant | 62 | 50 | 12 | 34 | 28 | 7.18 | 3.85 | 8.73 | 3.53 | 61 | 79.41 | 19.65 |

100% | 80.60% | 19.40% | 54.80% | 45.20% | ||||||||

Dog | 49 | 41 | 8 | 49 | 0 | 9.14 | 3.46 | 11.37 | 4.54 | 48 | 77.2 | 20.97 |

100% | 83.70% | 16.30% | 100.00% | 0.00% | ||||||||

Cat | 50 | 36 | 14 | 36 | 14 | 6.32 | 3.65 | 8.48 | 4.06 | 50 | 71.37 | 21.45 |

100% | 72.00% | 28.00% | 72.00% | 28.00% | ||||||||

Mouse | 59 | 47 | 12 | 10 | 49 | 5.98 | 2.58 | 7.47 | 3.57 | 58 | 58.88 | 23.86 |

100% | 79.70% | 20.30% | 16.90% | 83.10% | ||||||||

Lion | 81 | 59 | 22 | 81 | 0 | 7.14 | 3.54 | 11.99 | 4.35 | 79 | 76.47 | 21.44 |

100% | 72.80% | 27.20% | 100.00% | 0.00% | ||||||||

Tiger | 64 | 26 | 38 | 33 | 31 | 4.33 | 2.52 | 11.56 | 6.02 | 64 | 56.96 | 20.34 |

100% | 40.60% | 59.40% | 51.60% | 48.40% | ||||||||

Total | 365 | 259 | 106 | 243 | 122 | 6.62 | 3.57 | 10.07 | 4.76 | 360 | 70.05 | 23.01 |

100% | 71.00% | 29.00% | 66.60% | 33.40% |

Calculus Background | 1 | 2 | 3 |
---|---|---|---|

Pre CCI-Score Taken calculus before the course | –0.21** | – | |

Taken pre-calculus before the course | 0.32** | 0.45** | - |

Likert Scale | Value | Frequency | Percent (%) | Valid Percent | Cumulative Percent | |
---|---|---|---|---|---|---|

Valid | Very active | 1 | 49 | 9.2 | 14.2 | 14.2 |

Active | 2 | 117 | 21.9 | 33.9 | 48.1 | |

Somewhat active | 3 | 127 | 23.8 | 36.8 | 84.9 | |

A little active | 4 | 41 | 7.7 | 11.9 | 96.8 | |

Not active | 5 | 11 | 2.1 | 3.2 | 100 | |

Total Count | 345 | 64.6 | 100 | |||

Missing | 189 | 35.4 | ||||

Total | 534 | 100 |

Percentage of Time | Value | Frequency | Percent | Cumulative Percent |
---|---|---|---|---|

76–100% | 1 | 79 | 23.2 | 23.2 |

51–75% | 2 | 149 | 43.8 | 67.1 |

26–50% | 3 | 77 | 22.6 | 89.7 |

1–25% | 4 | 31 | 9.1 | 98.8 |

0% | 5 | 4 | 1.2 | 100 |

Total | 340 | 100 |

1 | 2 | |
---|---|---|

1. Post-CCI Score | ||

2. Level of active engagement | −0.30** | |

3. Time spent in active learning | −0.22** | 0.67** |

**Table 8.**Correlation among normalized gain, engagement in active learning and math background (N = 365).

1 | 2 | 3 | 4 | 5 | |
---|---|---|---|---|---|

1. Normalized Gain | - | ||||

2. Level of active engagement | −0.18** | - | |||

3. Time spent in active learning | −0.19** | 0.64** | - | ||

4. Taken calculus before the course | −0.06 | 0.01 | −0.05 | - | |

5. Taken pre-calculus before the course | −0.1 | −0.02 | −0.11 | 0.43** | - |

Predictor Variables | R^{2} | ∆R^{2} | b | F | p |
---|---|---|---|---|---|

Step 1 | 0.01 | 1.42 | 0.25 | ||

Taken calculus before the course | −0.02 | ||||

Taken pre-calculus before the course | −0.11 | ||||

Step 2 | 0.08 | 0.06 | 3.19** | 0.009 | |

Taken calculus before the course | −0.02 | ||||

Taken pre-calculus before the course | −0.14 | ||||

Level of active engagement | −0.03 | ||||

Time spent in active learning | −0.22* | ||||

Interest to take any online preparatory course | −0.02 |

**Table 10.**Correlation between midterm test scores, post-CCI scores, level of active engagement and time spent in active learning (N = 470).

1 | 2 | 3 | 4 | |
---|---|---|---|---|

1. Midterm test scores | – | |||

2. Post CCI score | 0.24** | – | ||

3. Level of active engagement | −0.25** | −0.27** | – | |

4. Time spent in active learning | −0.19** | −0.20** | 0.64** | – |

Predictor Variables | R2 | DR2 | b | F | p | |
---|---|---|---|---|---|---|

Step 1 | 0.26 | 25.24** | < 0.001 | |||

Pre CCI Score | 0.22** | |||||

Taken calculus before the course | 0.14* | |||||

Taken pre-calculus before the course | 0.31** | |||||

Step 2 | 0.3 | 0.04 | 18.30** | < 0.001 | ||

Pre CCI Score | 0.21** | |||||

Taken calculus before the course | 0.15* | |||||

Taken pre-calculus before the course | 0.30** | |||||

Level of active engagement | −0.16* | |||||

Time spent in active learning | −0.05 |

Variables | F | p |
---|---|---|

Pre CCI scores | 13.13** | <0.001 |

Post CCI scores | 11.73** | <0.001 |

Normalized gain | 10.37** | <0.001 |

Midterm test scores | 12.92** | <0.001 |

Level of active engagement | 1.28 | 0.28 |

Time spent in active learning | 0.83 | 0.53 |

Sections | n | Mean | SD | Tukey 1 | 2 | 3 | Scheffe 1 | 2 | 3 |
---|---|---|---|---|---|---|---|---|---|

Elephant | 62 | 0.074 | 0.36 | 0.07 | 0.07 | ||||

Mouse | 59 | 0.1002 | 0.25 | 0.1 | 0.1 | ||||

Cat | 50 | 0.1612 | 0.21 | 0.16 | 0.16 | 0.16 | 0.16 | ||

Dog | 49 | 0.2217 | 0.33 | 0.22 | 0.22 | 0.22 | 0.22 | 0.22 | |

Lion | 81 | 0.3386 | 0.35 | 0.34 | 0.34 | 0.34 | 0.34 | ||

Tiger | 64 | 0.4234 | 0.46 | 0.42 | 0.42 | ||||

Sig | 0.18 | 0.06 | 0.08 | 0.36 | 0.16 | 0.07 |

Section | N | Pre-CCI Mean | SD | Post-CCI Mean | SD | <g> |
---|---|---|---|---|---|---|

Tiger | 64 | 4.33 | 2.52 | 11.56 | 6.02 | 0.4614 |

Lion | 81 | 7.14 | 3.54 | 11.99 | 4.36 | 0.3771 |

Mouse | 59 | 5.98 | 2.58 | 7.47 | 3.57 | 0.1063 |

Cat | 50 | 6.32 | 3.65 | 8.48 | 4.06 | 0.1579 |

Dog | 49 | 9.14 | 3.46 | 11.37 | 4.54 | 0.2053 |

Elephant | 62 | 7.18 | 3.85 | 8.73 | 3.53 | 0.1209 |

Total | 365 | 6.62 | 3.57 | 10.07 | 4.76 | 0.2382 |

© 2019 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Ting, F.S.T.; Lam, W.H.; Shroff, R.H.
Active Learning via Problem-Based Collaborative Games in a Large Mathematics University Course in Hong Kong. *Educ. Sci.* **2019**, *9*, 172.
https://doi.org/10.3390/educsci9030172

**AMA Style**

Ting FST, Lam WH, Shroff RH.
Active Learning via Problem-Based Collaborative Games in a Large Mathematics University Course in Hong Kong. *Education Sciences*. 2019; 9(3):172.
https://doi.org/10.3390/educsci9030172

**Chicago/Turabian Style**

Ting, Fridolin Sze Thou, Wai Hung Lam, and Ronnie Homi Shroff.
2019. "Active Learning via Problem-Based Collaborative Games in a Large Mathematics University Course in Hong Kong" *Education Sciences* 9, no. 3: 172.
https://doi.org/10.3390/educsci9030172