# Teachers’ Use of Technology Affordances to Contextualize and Dynamically Enrich and Extend Mathematical Problem-Solving Strategies

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## Abstract

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## 1. Introduction

## 2. Digital Technologies and Mathematical Reasoning

## 3. Essentials of a Conceptual Framework

## 4. Methodological Elements, Tasks, and Procedures

## 5. Presentation of Results

“…Often they (warm-up activities) are based on a math-rich newspaper article, or on a math-rich web site, which is followed by a half dozen questions aimed at helping students read with a mathematical eye while also familiarizing them with the context of the model-eliciting activity”[34] (pp. 45–46).

**Comment**: Viéte’s strategy of considering limiting cases of involved objects not only helped the participants to reduce the complexity of approaching the problem via traditional methods, but also provided a path to focus on special cases of the task. In this process, the dynamic model associated with each case was constructed in terms of geometric meaning of the involved concepts and was useful to solve each case. Thinking of both approaches, as a whole, provided the route and method to solve the task. In terms of the participants’ tool appropriation, there is evidence that approaching the simpler cases dynamically demanded that they thought of both cases through the tool’s affordances and they recognized and applied the methods and strategies they used in those cases to solve the general Apollonius problem (Figure 13).

## 6. Discussion of Results

## 7. Instructional Implications

- (a).
- The construction of a dynamic model of the problem. The goal is that teachers and students rely on technology affordances to construct a model of the situation or problem. Within this model, they can move or drag orderly objects with the idea of observing behaviors of some objects’ attributes in order to detect invariants or patterns associated with those attributes. The use of GeoGebra allows problem solvers to move objects continuously and observe an immediate effect or change in their attributes’ behaviors [37]. In this study, the participants observed that the initial objects (point, circle or line) represent, indeed, an object family since they can be moved to different positions or change the length of the initial radius to show and analyze different arrangements. This stage is important for learners since they can pass or transit from analyzing properties of a particular case to the consideration and analysis of a family that represents those cases.
- (b).
- The importance of searching mathematical relations. The purpose in moving objects within the model is to identify patterns and formulate conjectures regarding mathematical properties or relationships among objects’ attributes. The process involved in the formulation of relations might be achieved by problem solvers through getting and analyzing information that comes from measuring length of segments, angles, registering areas, perimeters, slopes, etc. and using affordances to trace loci of attributes, define relations or functions through points, or using a table to organize data.
- (c).
- The presentation of conjectures demands that subjects develop a language and notation to communicate and express them. For example, in tracing the locus of point A when point L is moved along line EF, they conjectured initially that such a locus was a parabola. Then, they explored empirically, by measuring distances, AL & AQ, that these distances were the same for any position of point L on EF. Thus, they identified point Q as the focus and line EF as the directrix of the parabola.
- (d).
- The transition from an empirical and a visual approach in formulating and supporting conjectures to the use of geometric or algebraic arguments. In general, dynamic models of problems maintain structural properties when main objects are moved or enlarged/shorted within the model. Then, it is common that changing the dimensions or positions of some objects contributes to the process of validating a conjecture. Furthermore, the geometric analysis often provides formal arguments to validate conjectures. For example, in Figure 9, point F (the point that generates the locus) lies on the perpendicular bisector of PE and therefore, d(F, P) = d(F, E). Likewise, d(F, C) − d(F, P) = d(E, C) which is always constant. Then, the locus generated by point F when point E is moved along a circle with center C is a hyperbola whose foci are points P & C. This geometric argument, although inspired and initially conjectured empirically, does not depend on measuring and moving objects within the configuration; it relies on the use of geometric properties of the involved objects.

## 8. Concluding Remarks

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 1.**Ten cases of Apollonius’ problem (P-point, L-line, C-circle) (https://en.wikipedia.org/wiki/Problem_of_Apollonius, accessed on 19 January 2021).

**Figure 3.**The center C of a tangent circle to a given line lies on the perpendicular to the line at the tangency point L.

**Figure 4.**The intersection of the perpendicular bisector of QL and the perpendicular to EF at L is the center of the tangency circle to L that passes through point Q.

**Figure 5.**Any point on the locus can be the center of the tangent circle to EF that passes through point Q.

**Figure 6.**The loci of points A & B when point L is moved along line EF are parabolas that intersect at the centers of tangent circles to line EF and pass through points P & Q (https://www.geogebra.org/m/vycaymhz, accessed on 18 January 2021).

**Figure 7.**Finding the center of the tangent circle to line EF that passes through points P & Q (https://www.geogebra.org/m/cunnsdav, accessed on 2 January 2021).

**Figure 8.**How to draw a tangent circle to line AB and to circle with center C and that passes through point P?

**Figure 10.**Drawing four tangent circles that solve the problem (https://www.geogebra.org/m/xkf9g6xs, accessed on 2 January 2021).

**Figure 13.**Drawing eight tangent circles to three given circles (https://www.geogebra.org/m/vmd8s4xs, accessed on 3 January 2021).

Participants | Resources | Topic | Main Ideas |
---|---|---|---|

Pair A | Wikipedia, YouTube | Apollonius’ problem and Treatise on Conic sections | Definition of conic sections, the importance of focusing on simpler problems. |

Pair B | Wikipedia, Khanacademy | Apollonius’ problem, Hellenistic period | The context of the problem and the Greek period. |

Group C | Wikipedia, Woframalpha | Apollonius life and problems | Methods for solving the Apollonius problem and the construction of dynamic models of simpler cases |

Participants | Involved Objects | Dynamic Model | Concepts and Strategies |
---|---|---|---|

Pair A | Lines AB & CD and point P (LLP) | Two solutions: Moving point Q | Drawing a tangent circle to two given lines. Perpendicular line to line CD through point Q. Perpendicular bisector of QP. Locus of H when point Q is moved along CD |

Pair B | Line EF and points P &Q (LPP) | Two solutions using parabolas | The consideration of a simpler case: drawing a tangent circle to a line that passes through a given point. Perpendicular bisector, parabola. |

Group C | Circle, Line, and a Point (CLP) | Four solutions using conic sections | Perpendicular bisector, hyperbola, parabola, locus of points, perpendicular bisector, joining two simpler cases. |

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**MDPI and ACS Style**

Santos-Trigo, M.; Barrera-Mora, F.; Camacho-Machín, M. Teachers’ Use of Technology Affordances to Contextualize and Dynamically Enrich and Extend Mathematical Problem-Solving Strategies. *Mathematics* **2021**, *9*, 793.
https://doi.org/10.3390/math9080793

**AMA Style**

Santos-Trigo M, Barrera-Mora F, Camacho-Machín M. Teachers’ Use of Technology Affordances to Contextualize and Dynamically Enrich and Extend Mathematical Problem-Solving Strategies. *Mathematics*. 2021; 9(8):793.
https://doi.org/10.3390/math9080793

**Chicago/Turabian Style**

Santos-Trigo, Manuel, Fernando Barrera-Mora, and Matías Camacho-Machín. 2021. "Teachers’ Use of Technology Affordances to Contextualize and Dynamically Enrich and Extend Mathematical Problem-Solving Strategies" *Mathematics* 9, no. 8: 793.
https://doi.org/10.3390/math9080793