# The City as a Tool for STEAM Education: Problem-Posing in the Context of Math Trails

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## Abstract

**:**

## 1. Introduction

#### 1.1. Math Trails: Situated Learning in Mathematics Education

#### 1.2. Problem-Solving and Problem-Posing: Fostering Mathematical Creativity in Initial Teacher Training

#### 1.3. Didactic Proposal: Sevilla Math City

- Adequacy of the total itinerary (duration and distance) to its real use as a Math Trail.
- Identification of the resolution of the task with the chosen space or physical element (monuments, buildings, pavement, views, nature, etc.) in such a way that the presence of the solver is compulsory.
- Correct definition of the objectives pursued, and the contents worked on in the area and adaptation of the statement and materials provided to the proposed objectives.
- Adequacy of the complexity of the task to the target primary education cycle.
- Quality of the oral presentation, design, materials and texts.

- How are the tasks and the Math Trails created by the students characterized according to different classification variables and research variables?
- What shortcomings or biases can be detected in the students’ productions after having worked freely on the design of the tasks?
- Are there any relationships between the different variables in the tasks produced by the students?

## 2. Materials and Methods

- (i)
- Proc-PS: a distinction between procedural and problem-solving tasks.
- (ii)
- (iii)
- Context: a distinction between academic, semi-real and real-life tasks.
- (iv)
- Openness: a distinction between open, open-ended and closed tasks in relation to the type of answer of the task.
- (v)
- Creativity: a distinction between the problem-posing categories accepting data and what-if-not proposed in Brown and Walter [29], considering task design as a creative activity.

#### 2.1. Classification Variables: Grade, Mathematical Content and Objetc

#### 2.2. Investigation Variables

#### 2.2.1. Procedural vs. Problem-Solving (Proc-PS) and Cognitive Demand of Tasks

- Memorization: tasks which need to reproduce previous learnings and to memorize facts, formulas or definitions.
- Procedures without connections (to concepts or meanings): tasks which are algorithmic, reproducing procedures that are explicitly specified or previously known from prior instruction or experience;
- And the two high level demand categories of:
- Procedures with connections (to concepts or meanings): tasks where the use of procedures are closely connected to the underlying mathematical concepts and ideas. Tasks are usually represented in multiple ways (visual, manipulatives, symbols and problem situations), making connections among multiple representations.
- Doing mathematics: tasks which require complex thinking which is not algorithmic, and the solving approach is not known nor explicit in the statement of the task. These tasks create the need for students to impose their own structure and procedure to solve the task.

#### 2.2.2. Context: Academic, Semi-Real and Real-Life Tasks

#### 2.2.3. Openness: Closed, Open-Ended and Open Tasks

#### 2.2.4. Creativity: Accepting Data vs. What-If-Not

## 3. Results

## 4. Discussion

## 5. Conclusions

- -
- An initial stage where students experience a pre-designed math trail. In one hand it would serve as a model for students of different types of tasks and a variety of urban elements on which one can create and do mathematical problems. On the other hand, it is important that pre-service teachers first take the role of users of a Math Trail since most of them did not experience this activity before.
- -
- The use of mobile technological tools, such as MathCityMap, which give students the opportunity to practice with the advantages of their inclusion into Math Trails: GPS localization of the trail and tasks, immediate feedback or gamification capabilities among others [14].
- -
- A selection of the Math Trails created by the pre-service teachers are proposed to be tested by primary school students. This last step seeks to provide students with a self-assessment, which will help them to redesign their mathematical tasks and as a final reflection for their future professional practice.

## Author Contributions

## Funding

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 1.**Interactive map of Seville with all the Math Trails developed by the students. The lines with different colors represent the itineraries and the markers are the stops of the Math Trails. Available at: https://bit.ly/3IT25ZT (accessed on 19 July 2022).

**Figure 2.**“The well of the garden has a bucket hanging on it. Imagine that there is an ant alone inside the bucket and wants to get out by climbing up the wall. If it walks 10 cm in 1 min, how much time will it take to walk the whole bucket?”.

**Figure 3.**“Look at the door of the facade of the building and recognize the geometrical figures that appear on it. Can you think of a different way of dividing the door with different geometrical figures?”.

**Figure 4.**“Consider the patio as a Cartesian plane, as if the X-axis were the short side and Y-axis the long side (each column is a point of the axis). Represent the coordinates (3,3) (0,0) (3,6) (0,4) and (2,7) locating yourselves in the correct point”.

Object | Examples |
---|---|

Urban elements | Bench, streetlight, flowerpot, fountain. |

Architectural elements | Facade, courtyard, window, door. |

Buildings | University main building, tower. |

Interior elements | Ceiling, furniture, well. |

Natural elements | Trunk, tree, flower, bush. |

2D or 3D space | Square, street. |

Other | Food, people, cars. |

Grade | Trails (% of Total) | Grade | Trails |
---|---|---|---|

1st | 0 (0%) | 4th | 1 (9%) |

2nd | 1 (9%) | 5th | 1 (9%) |

3rd | 2 (18%) | 6th | 6 (55%) |

Area | Content | Tasks | Area % | Total % |
---|---|---|---|---|

Numbers | Elementary operations | 24 | 53.3% | 24.6% |

Measure | Ordinals | 0 | 0% | 19.1% |

Geometry | Fractions and decimals | 1 | 2.2% | 54.1% |

Probability and Statistics | Percentages, proportionality | 4 | 8.9% | 2.2% |

Divisibility | 0 | 0% | ||

Counting | 14 | 31.1% | ||

Estimation | 1 | 2.2% | ||

Sequences and series | 1 | 2.2% | ||

Algebra | Equations, patterns, relations, functions | 0 | 0% | 0% |

Measurement | Geometrical magnitudes: length, area, volume | 24 | 68.6% | 19.1% |

Weight | 0 | 0% | ||

Capacity | 1 | 2.9% | ||

Currency | 1 | 2.9% | ||

Time | 1 | 2.9% | ||

Angular units | 2 | 5.7% | ||

Estimation | 6 | 17.1% | ||

Geometry | 2D figures | 49 | 49.5% | 54.1% |

3D figures | 17 | 17.2% | ||

Orientation | 5 | 5.1% | ||

Transformations, symmetry | 9 | 9.1% | ||

Perimeter, area, volume | 16 | 16.2% | ||

Coordinates | 3 | 3% | ||

Probability and Statistics | Tables and graphs | 3 | 75% | 2.2% |

Statistical measures: mean, mode, range | 1 | 25% | ||

Random experiments | 0 | 0% | ||

Probability of events | 0 | 0% |

Object | Percentage |
---|---|

Urban elements | 28.2% |

Architectural elements | 27.2% |

Buildings | 12.8% |

Interior elements | 7.7% |

Natural elements | 4.3% |

2D or 3D space | 6% |

Other | 13.7% |

Proc-PS | % | Trail | Procedural | Problem-Solving |
---|---|---|---|---|

Procedural | 56.4% | 1 | 30.8% | 69.2% |

Problem-solving | 43.6% | 2 | 62.5% | 37.5% |

3 | 55.6% | 44.4% | ||

4 | 38.5% | 61.5% | ||

5 | 72.7% | 27.3% | ||

6 | 70% | 30% | ||

7 | 80% | 20% | ||

8 | 92.3% | 7.7% | ||

9 | 50% | 50% | ||

10 | 22.2% | 77.8% | ||

11 | 44.4% | 55.6% |

Demand | % | Trail | Level 1 | Level 2 | Level 3 | Level 4 |
---|---|---|---|---|---|---|

Low Level 1. Memorization | 17.9% | 1 | 7.7% | 23.1% | 53.8% | 15.4% |

Low Level 2. Procedures without connections | 38.5% | 2 | 37.5% | 25% | 37.5% | 0% |

High Level 3. Procedures with connections | 40.2% | 3 | 33.3% | 22.2% | 44.4% | 0% |

High Level 4. Doing mathematics | 3.4% | 4 | 15.4% | 23.1% | 46.2% | 15.4% |

5 | 0% | 72.7% | 27.3% | 0% | ||

6 | 40% | 30% | 30% | 0% | ||

7 | 30% | 50% | 20% | 0% | ||

8 | 38.5% | 53.8% | 0% | 0% | ||

9 | 0% | 50% | 50% | 0% | ||

10 | 0% | 22.2% | 77.8% | 0% | ||

11 | 0% | 44.4% | 55.6% | 0% |

Context | % | Trail | Academic | Semi-Real | Real |
---|---|---|---|---|---|

Academic | 6.8% | 1 | 7.7% | 30.8% | 61.5% |

Semi-real | 17.1% | 2 | 25% | 12.5% | 62.5% |

Real | 76.1% | 3 | 11.1% | 33.3% | 55.6% |

4 | 7.7% | 23.1% | 69.2% | ||

5 | 0% | 9.1% | 90.9% | ||

6 | 10% | 20% | 70% | ||

7 | 10% | 0% | 90% | ||

8 | 0% | 0% | 100% | ||

9 | 8.3% | 25% | 66.7% | ||

10 | 0% | 11.1% | 88.9% | ||

11 | 0% | 22.2% | 77.8% |

Openness | % | Trail | Closed | Open-Ended | Open |
---|---|---|---|---|---|

Closed | 75.2% | 1 | 61.5% | 30.8% | 7.7% |

Open-ended | 21.4% | 2 | 87.5% | 0% | 12.5% |

Open | 3.4% | 3 | 88.9% | 0% | 11.1% |

4 | 84.6% | 15.4% | 0% | ||

5 | 100% | 0% | 0% | ||

6 | 80% | 20% | 0% | ||

7 | 50% | 50% | 0% | ||

8 | 84.6% | 15.4% | 0% | ||

9 | 66.7% | 25% | 8.3% | ||

10 | 55.6% | 44.4% | 0% | ||

11 | 66.7% | 33.3% | 0% |

Creativity | % | Trail | Accepting Data | What-if-Not |
---|---|---|---|---|

Accepting data | 95.7% | 1 | 92.3% | 7.7% |

What-if-not | 4.3% | 2 | 87.5% | 12.5% |

3 | 88.9% | 11.1% | ||

4 | 100% | 0% | ||

5 | 100% | 0% | ||

6 | 90% | 10% | ||

7 | 100% | 0% | ||

8 | 100% | 0% | ||

9 | 91.7% | 8.3% | ||

10 | 100% | 0% | ||

11 | 100% | 0% |

Proc-PS | Demand | Context | Openness | Creativity | |
---|---|---|---|---|---|

Proc-PS | 1.00 | 0.87 * | −0.21 * | 0.15 | 0.24 * |

Demand | 0.87 * | 1.00 | −0.16 | 0.08 | 0.19 * |

Context | −0.21 * | −0.16 | 1.00 | 0.03 | −0.18 |

Openness | 0.15 | 0.13 | 0.03 | 1.00 | 0.21 * |

Creativity | 0.24 * | 0.19 * | −0.18 | 0.21 * | 1.00 |

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**MDPI and ACS Style**

Martínez-Jiménez, E.; Nolla de Celis, Á.; Fernández-Ahumada, E.
The City as a Tool for STEAM Education: Problem-Posing in the Context of Math Trails. *Mathematics* **2022**, *10*, 2995.
https://doi.org/10.3390/math10162995

**AMA Style**

Martínez-Jiménez E, Nolla de Celis Á, Fernández-Ahumada E.
The City as a Tool for STEAM Education: Problem-Posing in the Context of Math Trails. *Mathematics*. 2022; 10(16):2995.
https://doi.org/10.3390/math10162995

**Chicago/Turabian Style**

Martínez-Jiménez, Enrique, Álvaro Nolla de Celis, and Elvira Fernández-Ahumada.
2022. "The City as a Tool for STEAM Education: Problem-Posing in the Context of Math Trails" *Mathematics* 10, no. 16: 2995.
https://doi.org/10.3390/math10162995