Characterizing Tasks for Teaching Mathematics in Dynamic Geometry System and Modelling Environments
Abstract
:1. Introduction
2. Task and Modeling Task
3. Conceptual Framework
3.1. Modelling and Digital Tools
The word “design” carries a meaning of accomplishing goals in a particular environment satisfying a set of requirements or subject to a set of constraints; it is a strategic approach (roadmap) towards achieving a certain expectation. Design necessarily creates boundary; a structure or framework on which meaning and knowledge can grow.(p. 4)
Computational media both empower the mathematical processes involved in modeling activities by providing new “worlds” to explore and potentially shape the world we try to model.[16] (p. 79)
3.2. Teacher Knowledge for Designing Tasks
4. Methodology
4.1. Introductory Phase for Working with Dynamic Geometry Systems
4.2. Tasks Design Phase
4.3. Data Collection and Analysis
5. Discussion and Analysis of the Results
5.1. Characteristics Elements of the Tasks
Category  Code  Explanation 

Modeling objective  MO1—Describe  Intended to establish the description of the system’s characteristics 
MO2—Predict  Seeks to predict the behavior of the system  
MO3—Understand  Seeks to understand the impact of certain modifications on the characteristics of the system  
MO4—Represent  Seeks to build a reproduction of the system in a medium other than the original  
MO5—Intervene  Seeks to understand the system in order to intervene in it  
Activity intended for the student  I1—Solve  These actions are specific to each task and depend on the modeling objective and the learning objective 
I2—Identify  
I3—Explore  
I4—Apply  
I5—Build  
I6—Model  
I7—Work mathematically  
I8—Interpret  
I9—Observe 
Category  Code  Description 

Model  M1—based on theoretical knowledge  In addition to the geometric tools of the DGS, the model considers theoretical elements such as formulas or known behaviors associated with the phenomenon. 
M2—based on the characteristics of the situation  Using the geometric tools of the DGS, the model mainly considers the characteristics and behaviors of the situation.  
M3—theoretical origin  The context gives rise to building the model with theoretical elements.  
M4—situational origin  The context gives rise to building the model by selecting elements and behaviors of the situation to represent.  
Type of simulation  S1—motion controlled by the user  The user controls the motion of certain objects present in the simulation. 
S2—motions defined by the designer  The user’s actions are limited by the geometric configuration of the simulation.  
Type of solution  TS1—unique solution  Right answer determined by the design of the task. 
TS2—dynamic open solution  The dynamism of DGS favors multiple responses.  
TS3—subjective answers  Answers that are difficult to classify as correct or incorrect.  
TS4—solution path determined by the geometric configuration  The dynamic configuration favors only one type of solution strategy.  
TS5—multiple solution paths  The dynamic configuration favors different solution strategies. 
Task/Group  Task Code  Activity Intended for the Student/Code  Objective Intended to Model  Characteristic of the Model  Type of Simulation  Solutions  

Bus in tunnel G1  T1  Solve an intersection problem between geometric objects, interpret the result.  I1 I3 I4 I8  MG2 MG3 MG5  M2 M4  S1  TS2 TS4 
Planetary motion G2  T2  Identify the behavior of planetary motion. Identify properties of the ellipse.  I2 I7 I9  MG4  M1 M3  S2  TS2 TS4 
Basketball G4  T4  Build the geometric model. Explore possible answers to the task. Add distances, use axial symmetry.  I3 I5 I6  MG3 MG5  M2 M1 M4  S1  TS2 TS5 
Motion of an inner circle G5  T5  Explore, given a geometric configuration, the geometric locus described by certain defined points based on the motion of inscribed circles.  I3 I7 I9  MG1 MG3 MG4  M1 M3  S2  TS2 TS4 
Harvest G6  T6  Build and solve a situation involving areas and perimeters and proportionality ratios.  I1 I3 I5 I6  MG2 MG3 MG5  M2 M4  S1  TS2 TS3 
Height of a building G7  T7  Solve using Thales’s theorem and trigonometric ratios.  I1 I3 I4 I7  MG3  M1 M3  S2  TS1 TS4 
Launch G8  T8  Explore and relate the behavior of a parabola with the value of its parameters.  I3 I7 I9  MG3 MG4  M1 M3  S2  TS1 TS4 
Speed bump G9  T9  Simplify and build the simulation. Study possible solutions algebraically in the context of software.  I5 I6 I7 I8  MG2 MG3 MG5  M2 M4  S1  TS2 TS3 TS5 
Seat arrangement G10  T10  Explore the geometric configuration, paying attention to the combination of distances and angles to find the optimal solution.  I3 I8  MG3 MG5  M2 M1 M4  S1  TS2 TS3 TS4 
Lift gate G11  T11  Explore different cases. Find the optimal solution or determine whether there is no solution.  I3  MG3 MG5  M2 M4  S1  TS2 TS3 TS4 
Wind power G12  T12  Perform algebraic operations and calculations using a given mathematical expression.  I1 I7 I9  MG4  M1  S2  TS1 
5.2. Two Examples
5.2.1. Speed Bump
5.2.2. Height of a Building
We want to calculate the height of a building by using a laser and a convex mirror with a certain curvature, expressed in the attached GeoGebra file. If the distance between the building and the mirror is 10 (m), is it possible to calculate the height of the building? Prove your answer. If the answer is yes, how would you do it? Explain your procedure.
6. Conclusions
Author Contributions
Funding
Institutional Review Board Statement
Informed Consent Statement
Data Availability Statement
Acknowledgments
Conflicts of Interest
References
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Written Report  GeoGebra Efile 




Group  Extract from the Report  Category 

G1  The intention is for students to use these skills to find solutions to the parabolic tunnel problem in a way that is flexible and creative. Also interested in adhering to the reality that is detailed in the context regarding traffic accidents.  Skill development 
G7  The activity was not done to teach the concept of trigonometric ratios, but rather to find an application for them. Therefore, accompanying this activity should be preliminary theory classes that teach this concept […]  Applications of mathematics 
G11  The fundamental concept of modeling is to encourage students to be curious about the world and the ideas that surround them.  Motivation 
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GuerreroOrtiz, C.; CamachoMachín, M. Characterizing Tasks for Teaching Mathematics in Dynamic Geometry System and Modelling Environments. Mathematics 2022, 10, 1239. https://doi.org/10.3390/math10081239
GuerreroOrtiz C, CamachoMachín M. Characterizing Tasks for Teaching Mathematics in Dynamic Geometry System and Modelling Environments. Mathematics. 2022; 10(8):1239. https://doi.org/10.3390/math10081239
Chicago/Turabian StyleGuerreroOrtiz, Carolina, and Matías CamachoMachín. 2022. "Characterizing Tasks for Teaching Mathematics in Dynamic Geometry System and Modelling Environments" Mathematics 10, no. 8: 1239. https://doi.org/10.3390/math10081239