# Relations between Generalization, Reasoning and Combinatorial Thinking in Solving Mathematical Open-Ended Problems within Mathematical Contest

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## Abstract

**:**

## 1. Introduction

## 2. Materials and Methods

#### 2.1. Assignment of Mathematical B-Day 2018: Snake Nest

- Limit what type of snakes and what type of blankets do you are consider.
- Experimenting.
- Give a positioning strategy for the all shapes for those blankets.
- Find the corresponding minimum dimensions.
- Explain that the all forms remain under the blanket with the strategy from the third step.
- Cut bits away from the blanket to make it even smaller.

#### 2.2. Statistical Analysis

## 3. Results

#### 3.1. Example of the Authentic Team Solutions of the Subtask 2a

#### 3.1.1. Solution 2a.A

#### 3.1.2. Solution 2a.B

#### 3.2. Example of the Authentic Team Solutions of the Subtask 6d

#### 3.2.1. Solution 6d.A

#### 3.2.2. Solution 6d.B

#### 3.3. Example of the Authentic Team Solutions of the Subtask 5f

#### 3.3.1. Solution A

#### 3.3.2. Solution B

#### 3.3.3. Solution C

## 4. Discussion

#### Limitations of the Study

## 5. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

## References

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**Figure 1.**Visual aid for subtask aimed at the combinatorial thinking (adapted from [55]).

**Figure 7.**Students’ solution of the subtask 5f (

**A**) the generative strategy for expressing the formula and (

**B**) the table of expressed general formula. Translations: Obsah obdĺžnikovej prikrývky = Area of the rectangular blanket; Obsah najmenšej prikrývky = Area of the smallest blanket; Párne n = Even n; Nepárne n = Odd n.

**Table 1.**Description of the levels of algebraic pattern generalization, reasoning and proving, and combinatorial thinking.

Level | Algebraic Generalization | Reasoning and Proving | Combinatorial Thinking |
---|---|---|---|

0 | Observing particular examples | Without any argumentation or reasoning | Listing of the elements in random order, without looking for systematic strategy. |

1 | Noticing a commonality | Reasoning by one or several particular examples | Using a trial-error strategy, discovery of some generative strategies for small sets of outcomes. |

2 | Formulating a hypothesis | Correct mathematical evidence, but not formalized in a form of proof | Adopting generative strategies for bigger sets or three- and more-stage case. |

3 | Producing the expression of ${p}_{n}$ | Formal mathematical proof | Applying generative strategies and use of formulas. |

**Table 2.**Success rate and levels of reasoning skills and algebraic generalization manifested in team solutions.

Task | Description | Success Rate | Level of Reasoning Skills (M ^{1} ± SD ^{2}) | Level of Algebraic Generalization (M ± SD) | Level of Combinatorial Thinking (M ± SD) |
---|---|---|---|---|---|

2a | Proving the impossibility | 66% ^{a} | 2.45 ± 1.32 ^{a} | NA | NA |

5a | Introductory combinatorics (enumeration) | 55% ^{agh} | NA | NA | 1.42 ± 1.07 ^{ab} |

5b | 45% ^{aehi} | NA | NA | 1.00 ± 1.26 ^{a} | |

5c | 36% ^{bchijk} | NA | NA | NA | |

5d | Combinatorics (optimization) | 18% ^{kl} | NA | NA | NA |

5e | Higher lower limit | 12% ^{gl} | NA | NA | NA |

5f | Generalization | 18% ^{bcdfg} | 1.33 ± 1.53 ^{bce} | 1.15 ± 1.37 ^{a} | 1.61 ± 1.25 ^{b} |

6a | Proving the inequality | 39% ^{befj} | 1.15 ± 1.39 ^{b} | NA | NA |

6b | Proving the impossibility | 21% ^{cfgi} | 1.18 ± 1.48 ^{bc} | NA | NA |

6d | Proving the coverage | 9% ^{dfg} | 0.30 ± 0.63 ^{d} | NA | NA |

FA | Mathematical modeling | 48% ^{aej} | 0.75 ± 0.90 ^{bc} | 0.66 ± 0.92 ^{a} | NA |

^{1}M mean,

^{2}SD standard deviation. The values assigned by the same letter (a,b,c,… etc.) do not differ significantly based on the McNemar test (correctness of solution) and the median test (levels of reasoning skills and algebraic generalization) (p ≤ 0.05), e.g. Success rate in 5a does not differ significantly from 5b, as both are assigned by string containing letters a and h, but 5a differs significantly from 5c, as they do not have any letter in common.

**Table 3.**Correlation matrix of correctness of the team solution in selected subtask (phi coefficient).

Correctness | ||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|

Task | 2a | 5a | 5b | 5c | 5d | 5e | 5f | 6a | 6b | 6d | FA | |

Correctness | 2a | NA | 0.290 | 0.129 | −0.013 | 0.000 | 0.066 | 0.000 | 0.175 | 0.367 | −0.224 | 0.043 |

5a | 0.474 | NA | 0.344 | 0.311 | 0.273 | 0.339 | 0.430 | −0.011 | 0.176 | −0.077 | −0.155 | |

5b | 0.474 | 0.050 | NA | 0.449 | 0.359 | 0.034 | 0.043 | 0.260 | −0.027 | 0.346 | 0.332 | |

5c | 0.458 | 0.079 | 0.009 | NA | 0.297 | 0.298 | 0.297 | 0.164 | 0.084 | 0.199 | 0.023 | |

5d | 2.000 | 0.125 | 0.040 | 0.093 | NA | 0.547 | 0.389 | 0.263 | −0.052 | 0.124 | 0.014 | |

5e | 0.717 | 0.054 | 0.851 | 0.092 | 0.001 | NA | 0.547 | 0.081 | 0.034 | −0.117 | 0.011 | |

5f | 1.000 | 0.012 | 0.812 | 0.093 | 0.025 | 0.001 | NA | 0.414 | 0.326 | 0.199 | −0.160 | |

6a | 0.329 | 0.950 | 0.143 | 0.362 | 0.139 | 0.656 | 0.016 | NA | 0.340 | 0.392 | 0.211 | |

6b | 0.036 | 0.327 | 0.881 | 0.642 | 0.772 | 0.849 | 0.064 | 0.053 | NA | 0.164 | 0.090 | |

6d | 0.211 | 0.670 | 0.448 | 0.266 | 0.491 | 0.515 | 0.268 | 0.024 | 0.362 | NA | 0.326 | |

FA ^{1} | 0.813 | 0.389 | 0.059 | 0.899 | 0.937 | 0.950 | 0.372 | 0.239 | 0.619 | 0.064 | NA |

^{1}FA—final assignment.

**Table 4.**Correlation matrix of the level of reasoning, algebraic generalizations and combinatorial thinking manifested in the team solutions of selected subtasks (Spearman’s rho).

Level of | ||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|

Reasoning | Generalization | Combinatorial Thinking | ||||||||||

Item | 2a | 6a | 6b | 6d | FA | 5f | FA | 5a | 5c | 5f | FA | |

Reasoning | 2a | NA | 0.266 | 0.210 | 0.239 | 0.043 | 0.081 | 0.128 | 0.122 | 0.259 | 0.313 | 0.209 |

6a | 0.135 | NA | 0.814 | 0.475 | 0.204 | 0.282 | 0.089 | −0.002 | 0.229 | 0.211 | −0.020 | |

6b | 0.240 | 0.000 | NA | 0.534 | 0.243 | 0.292 | 0.182 | 0.124 | 0.181 | 0.237 | 0.219 | |

6d | 0.180 | 0.005 | 0.001 | NA | 0.077 | 0.160 | −0.089 | −0.011 | 0.270 | 0.190 | −0.085 | |

FA | 0.813 | 0.255 | 0.173 | 0.668 | NA | 0.283 | 0.724 | 0.140 | 0.299 | 0.213 | 0.247 | |

Gen ^{1} | 5f | 0.653 | 0.112 | 0.099 | 0.372 | 0.111 | NA | 0.090 | 0.185 | 0.536 | 0.393 | 0.242 |

FA | 0.479 | 0.623 | 0.311 | 0.624 | 0.000 | 0.617 | NA | −0.072 | −0.027 | 0.018 | 0.453 | |

Combinatorial thinking | 5a | 0.499 | 0.989 | 0.491 | 0.952 | 0.439 | 0.302 | 0.689 | NA | 0.450 | 0.799 | 0.095 |

5c | 0.146 | 0.201 | 0.313 | 0.129 | 0.091 | 0.001 | 0.884 | 0.009 | NA | 0.693 | −0.141 | |

5f | 0.076 | 0.239 | 0.185 | 0.288 | 0.233 | 0.024 | 0.922 | 0.000 | 0.000 | NA | 0.056 | |

FA | 0.242 | 0.914 | 0.220 | 0.636 | 0.166 | 0.175 | 0.008 | 0.600 | 0.434 | 0.759 | NA |

^{1}Gen—generalization.

**Table 5.**Correlation matrix of the solution correctness of and the levels of reasoning, algebraic generalization and combinatorial thinking manifested in the team solution of selected subtasks (Cramer’s V).

Level of | ||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|

Reasoning | Generalization | Combinatorial thinking | ||||||||||

Item | 2a | 6a | 6b | 6d | FA | 5f | FA | 5a | 5c | 5f | FA | |

2a | $V$ | 0.542 | 0.265 | 0.351 | 0.034 | 0.241 | 0.079 | 0.235 | 0.100 | 0.205 | 0.291 | 0.125 |

$p$ | 0.046 | 0.509 | 0.186 | 0.930 | 1.000 | 0.729 | 0.537 | 0.861 | 0.539 | 0.358 | 0.473 | |

5a | $V$ | 0.178 | 0.012 | 0.072 | 0.141 | 0.025 | 0.463 | 0.067 | 0.644 | 0.291 | 0.587 | 0.161 |

$p$ | 0.653 | 0.547 | 0.711 | 0.805 | 0.590 | 0.024 | 0.596 | 0.001 | 0.242 | 0.001 | 0.354 | |

5b | $V$ | 0.335 | 0.077 | 0.177 | 0.335 | 0.181 | 0.303 | 0.067 | 0.376 | 0.388 | 0.481 | 0.194 |

$p$ | 0.222 | 0.030 | 0.302 | 0.037 | 0.319 | 0.108 | 0.199 | 0.018 | 0.137 | 0.012 | 0.266 | |

5c | $V$ | 0.268 | 0.192 | 0.121 | 0.238 | 0.006 | 0.335 | 0.138 | 0.464 | 0.602 | 0.589 | 0.134 |

$p$ | 0.115 | 0.477 | 0.416 | 0.394 | 0.466 | 0.246 | 0.103 | 0.039 | 0.005 | 0.008 | 0.443 | |

5d | $V$ | 0.257 | 0.062 | 0.166 | 0.023 | 0.129 | 0.412 | 0.086 | 0.180 | 0.250 | 0.273 | 0.375 |

$p$ | 0.367 | 0.597 | 0.798 | 0.504 | 0.191 | 0.110 | 0.126 | 0.372 | 0.381 | 0.398 | 0.357 | |

5e | $V$ | 0.155 | 0.041 | 0.046 | 0.180 | 0.206 | 0.439 | 0.238 | 0.199 | 0.296 | 0.265 | 0.476 |

$p$ | 0.357 | 0.821 | 0.799 | 0.542 | 0.039 | 0.071 | 0.045 | 0.028 | 0.147 | 0.435 | 0.006 | |

5f | $V$ | 0.137 | 0.177 | 0.102 | 0.228 | 0.040 | 0.645 | 0.086 | 0.106 | 0.188 | 0.148 | 0.375 |

$p$ | 0.439 | 0.441 | 0.305 | 0.373 | 0.083 | 0.001 | 0.126 | 0.803 | 0.527 | 0.242 | 0.031 | |

6a | $V$ | 0.099 | 0.543 | 0.577 | 0.501 | 0.429 | 0.414 | 0.363 | 0.028 | 0.198 | 0.204 | 0.219 |

$p$ | 0.774 | 0.003 | 0.014 | 0.015 | 0.096 | 0.098 | 0.166 | 0.568 | 0.338 | 0.366 | 0.208 | |

6b | $V$ | 0.274 | 0.644 | 0.745 | 0.222 | 0.225 | 0.326 | 0.109 | 0.140 | 0.059 | 0.104 | 0.341 |

$p$ | 0.307 | 0.009 | 0.001 | 0.089 | 0.361 | 0.260 | 0.268 | 0.447 | 0.490 | 0.189 | 0.049 | |

6d | $V$ | 0.132 | 0.119 | 0.105 | 0.520 | 0.086 | 0.199 | 0.000 | 0.071 | 0.336 | 0.267 | 0.056 |

$p$ | 0.936 | 0.015 | 0.722 | 0.002 | 0.884 | 0.530 | 0.838 | 0.131 | 0.160 | 0.490 | 0.131 | |

FA ^{1} | $V$ | 0.013 | 0.019 | 0.128 | 0.208 | 0.606 | 0.160 | 0.355 | 0.294 | 0.290 | 0.305 | 0.182 |

$p$ | 0.083 | 0.165 | 0.189 | 0.383 | 0.003 | 0.309 | 0.235 | 0.302 | 0.382 | 0.167 | 0.295 |

^{1}FA—final assignment.

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**MDPI and ACS Style**

Medová, J.; Bulková, K.O.; Čeretková, S. Relations between Generalization, Reasoning and Combinatorial Thinking in Solving Mathematical Open-Ended Problems within Mathematical Contest. *Mathematics* **2020**, *8*, 2257.
https://doi.org/10.3390/math8122257

**AMA Style**

Medová J, Bulková KO, Čeretková S. Relations between Generalization, Reasoning and Combinatorial Thinking in Solving Mathematical Open-Ended Problems within Mathematical Contest. *Mathematics*. 2020; 8(12):2257.
https://doi.org/10.3390/math8122257

**Chicago/Turabian Style**

Medová, Janka, Kristína Ovary Bulková, and Soňa Čeretková. 2020. "Relations between Generalization, Reasoning and Combinatorial Thinking in Solving Mathematical Open-Ended Problems within Mathematical Contest" *Mathematics* 8, no. 12: 2257.
https://doi.org/10.3390/math8122257