# 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

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**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