#
Dynamic and Interactive Tools to Support Teaching and Learning^{ †}

^{1}

^{2}

^{†}

## Abstract

**:**

## 1. Introduction

- 1.
- Posing the right questions;
- 2.
- Real world ⟶ math formulation;
- 3.
- Computation;
- 4.
- Math formulation ⟶ real world, verification.

## 2. Materials and Methods

## 3. Results

- Khan Academy ↪ non-profit educational organization, conceived by Sal Khany, that creates a set of online tools that help educate students. The organization produces short lessons in the form of videos and its website also includes supplementary practice exercises and materials for educators. All resources are available for free to users of the website and application;
- Photomath ↪ app for solving mathematical problems. The app instantly scans, accurately solves and intuitively explains math problems to users through step-by-step explanations;
- Wolfram|Alpha ↪ computational knowledge engine for computing answers and providing knowledge, developed by Wolfram Research and it was released in 2009. It works by using its vast store of expert-level knowledge and algorithms to automatically answer questions, do analysis and generate reports. Refs. [32,34] contain some examples of the possible use of Wolfram|Alpha as a technological learning tool;
- Wolfram Demonstrations Project ↪ open-code resource, conceived by Stephen Wolfram, as a way to bring computational exploration to the widest possible audience. Includes lots of computable document format files for various areas of knowledge;

#### 3.1. Dynamic and Interactive Tools

#### 3.1.1. Designing Dynamic and Interactive Tools with Mathematica

`Manipulate`command (https://reference.wolfram.com/language/tutorial/IntroductionToManipulate.html, accessed on 21 December 2021) (Figure 1), whose output is not just a static result but a running program that we can interact with.

`Manipulate`command, it is possible to observe some of the Mathematica functions that allow solving the problem involving the analysis, and possible resolution, of an exact differential Equation (calculating the auxiliary primitives needed to obtain the desired output).

`Manipulate`. This command is responsible for creating an interactive object that contains one or more functional controls, such as the sliders for the parameters’ values (Figure 3) and checkboxes for the plots’ options. Through dynamic changes of the parameters’ values, it is possible to obtain static and non-static visual information [26]. It is through this kind of dynamic interaction that “computer algebra systems present new opportunities for teaching and learning” [37].

#### 3.1.2. Precalculus, Differential Calculus and Integral Calculus

- Left Panel ↪ In this panel the user can vary the values of one or more parameters, choose which “transformations” of the main function are to be presented, and wether or not to show the tangent line in a chosen tangent point. In addition, the user has the option to see the results in the exact or approximate forms;
- Middle Panel ↪ In this panel all the functions are plotted, according to the options made by the user in the left panel;
- Right Panel ↪ In this panel it is presented all the analytical information concerning the main function and its “transformations”, again in accordance with the options chosen by the user in the left panel.

`Manipulate`command that can be used as active learning tools, either in a remote or classroom environment (Figure 5 and Figure 6, Figure 9).

#### 3.1.3. Riemann Sums

#### 3.1.4. Ordinary Differential Equations

#### 3.2. Dynamic and Interactive Active Learning Tools

#### 3.2.1. Promoting the Dialog and Reflection

“What kind of functions will appear if we …?”.

#### 3.2.2. Promoting the Acquisition of New Knowledge and the Transmission of Acquired Knowledge

#### 3.2.3. Promoting Self Evaluation through Faculty Evaluation

#### 3.2.4. Technological Learning Tools and Students’ Autonomous Work

#### 3.3. Evaluation Questions

## 4. Discussion

## Supplementary Materials

## Funding

## Conflicts of Interest

## References

- Li, Y.; Schoenfeld, A.H. Problematizing teaching and learning mathematics as “given” in STEM education. Int. J. Stem Educ.
**2019**, 6, 1–13. [Google Scholar] [CrossRef] - Conceição, A.C.; Coelho, A.C.; Gonçalves, C.D. Estratégia pedagógica no Ensino Superior baseada no conceito de aprendizagem ativa. In Ensino-Aprendizagem de Ciências e Suas Tecnologias; Schimiguel, J., Frenedozo, R.C., Coelho, A.C., Eds.; Edições Brasil: Jundiaí, Brasil, 2019; pp. 9–25. [Google Scholar]
- Conceição, A.C.; Pereira, J.C.; Silva, C.M.; Simão, C.R. Software educacional em pré-cálculo e cálculo diferencial: O conceito F-Tool. In Proceedings of the Encontro Nacional da SPM 2012, Faro, Portugal, 9–11 July 2012; pp. 57–60. [Google Scholar]
- Costanzo, F.; Gray, G. On the implementation of interactive dynamics. Int. J. Eng.
**2000**, 16, 385–393. [Google Scholar] - Foertsch, J.; Moses, G.; Strikwerda, J.; Litskow, M. Reversing the lecture/homework paradigm using eTEACH web-based streaming video software. Int. J. Eng.
**2002**, 91, 267–274. [Google Scholar] [CrossRef] - Fogler, H.S.; Montomery, S.M.; Zipp, R.P. Interactive computer modules for undergraduate chemical engineering instruction. Comput. Appl. Eng. Educ.
**1996**, 1, 11–24. [Google Scholar] [CrossRef] - García, O.; Laredo, M. Comunidades académicas virtuales como medio en la enseñanza y aprendizaje usando software matemático. Revista Iberoamericana de Producción Académica y Gestión Educativa
**2014**, 1. Available online: https://www.pag.org.mx/index.php/PAG/article/view/93 (accessed on 21 December 2021). - Gray, G.; Costanzo, F. The interactive classroom and its integration into the mechanics curriculum. Int. J. Eng.
**1999**, 15, 41–50. [Google Scholar] - Guamán, L.R.B.; Córdova, C.C. Using Wolfram software to improve reading comprehension in mathematics. In Proceedings of the 2016 EBMEI International Conference on Education, Information and Management (EBMEI-EIM 2016), São Paulo, Brazil, 31 August–1 September 2016; pp. 53–58. [Google Scholar]
- Morales, F.; Valencia, A.; Valencia, R.; Mario, J. Análisis de software matemático usados en nivel superior. Rev. Vínculos
**2013**, 10, 299–307. [Google Scholar] - Prado, J.L.; Freira, A.M.; Albuquerque, I.; Júior, P.P. Experienciando o software Mathematica na sala de aula. In Proceedings of the IV Colóquio Internacional Educação e Contemporaneidade, Laranjeiras, Brasil, 22–24 September 2010. [Google Scholar]
- Randow, C.L.; Miller, A.J.; Costanzo, F.; Gray, G.L. Mathematica Notebooks for Classroom Use in Undergraduate Dynamics: Demonstration of Theory and Examples. In Proceedings of the 2003 American Society for Engineering Education Annual Conference & Exposition, Nashville, TN, USA, 22–25 June 2003. [Google Scholar]
- Silva, J.; Astudillo, A. CbL-Cálculo: Curso b-learning para el apoyo de la enseñanza y aprendizaje de cálculo en ingeniería. Rev. Educ. Distançia
**2012**, 30. Available online: https://revistas.um.es/red/article/view/232581 (accessed on 21 December 2021). - Ellis, J.; Wieselmann, J.; Sivaraj, R.; Roehrig, G.; Dare, E.; Ring-Whalen, E. Toward a productive definition of technology in science and STEM education. Contemp. Issues Technol. Teach. Educ.
**2020**, 20, 472–496. [Google Scholar] - Andraz, J.M.; Conceição, A.C. Dynamic and interactive mathematical tools in socio-economics sciences classrooms. In Proceedings of the 4th International Conference on Numerical and Symbolic Computation Developments and Applications (SYMCOMP2019), Porto, Portugal, 11–12 April 2019; Loja, M.A., Barbosa, J.I., Rodrigues, J.A., Vasconcelos, P.B., Eds.; APMTAC—Associação Portuguesa de Mecânica Teórica, Aplicada e Computacional: Lisboa, Portugal, 2019; pp. 321–336. [Google Scholar]
- Macintyre, T.; Forbes, I. Algebraic skills and CAS—Could assessment sabotage the potential? Int. J. Comput. Algebra Math. Educ.
**2002**, 9, 29–56. [Google Scholar] - Buchberger, B. Should students learn integration rules? ACM Sigsam Bull.
**1990**, 24, 10–17. [Google Scholar] [CrossRef] - Kilicman, A.; Hassan, M.A.; Said Hussain, S.K. Teaching and learning using mathematics software “The New Challenge”. Procedia Soc. Behav. Sci.
**2010**, 8, 613–619. [Google Scholar] [CrossRef] - Mason, J. A comprehensive mathematics curriculum with Mathematica. In Proceedings of the Joint Mathematics Meeting, Wolfram Technology Conference, Champaign, IL, USA, 21–23 October 2004. [Google Scholar]
- Meyers, C.; Jones, T.B. Promoting active learning: Strategies for the college classroom. Biochem. Educ.
**1994**, 2, 192. [Google Scholar] - Ramos, A.; Delgado, F.; Afonso, P.; Cruchinho, A.; Pereira, P.; Sapeta, P.; Ramos, G. Implementação de novas práticas pedagógicas no Ensino Superior. Rev. Port. Educ.
**2013**, 26, 115–141. [Google Scholar] [CrossRef] - Bonwel, C.C.; Eison, J.A. Active Learning: Creating Excitement in the Classroom; ASHEERIC Higher Education Report No. 1; George Washington University: Washington, DC, USA, 1991. [Google Scholar]
- Chan, M.M.; Amado-Salvatierra, H.R.; Plata, R.B.; Hernández Rizzardini, R. La efectividad del uso de simuladores para la construcción de conocimiento en un contexto MOOC. In Proceedings of the II International Conference MOOC-Maker (MOOC-Maker 2018), Medellín, Colombia, 11–12 October 2018; pp. 42–53. [Google Scholar]
- Cross, K.P.; Angelo, T.A. Classroom Assessment Techniques: A Handbook for Faculty; National Center for Research to Improve Postsecondary Teaching and Learning: Ann Arbor, MI, USA, 1988. [Google Scholar]
- Conceição, A.C. Software educativo em pré-cálculo e cálculo diferencial. Rev. Ciência Elem.
**2018**, 6, 36–38. [Google Scholar] [CrossRef] - Conceição, A.C.; Pereira, J.C.; Silva, C.M.; Simão, C.R. Mathematica in the classroom: New tools for exploring precalculus and differential calculus. In Proceedings of the 1st National Conference on Symbolic Computation in Education and Research (CSEI 2012), Lisboa, Portugal, 2–3 April 2012. [Google Scholar]
- Drake, E.; Battaglia, D. Teaching and Learning in Active Learning Classrooms; The Faculty Center for Innovative Teaching, Central Michigan University: Mount Pleasant, MI, USA, 2014. [Google Scholar]
- Fernandes, S.; Pereira, J.C. Providing an Active Learning Environment for Introducing Linear Programming. In Advances in Operations Research Education; Springer: Cham, Switzerland, 2018. [Google Scholar]
- Andraz, J.M.; Candeias, R.; Conceição, A.C. Bridging Symbolic Computation and Economics: A Dynamic and Interactive Tool to Analyze the Price Elasticity of Supply. Math. Comput. Appl.
**2019**, 24, 87. [Google Scholar] [CrossRef] - Andraz, J.M.; Candeias, R.; Conceição, A.C.; Serafim, I. An interactive way of analyzing economic concepts using symbolic computation. In Proceedings of the 4th International Conference on Numerical and Symbolic Computation Developments and Applications (SYMCOMP2019), Porto, Portugal, 11–12 April 2019; Loja, M.A., Barbosa, J.I., Rodrigues, J.A., Vasconcelos, P.B., Eds.; APMTAC—Associação Portuguesa de Mecânica Teórica, Aplicada e Computacional: Lisboa, Portugal, 2019; pp. 343–356. [Google Scholar]
- Conceição, A.C. Personal Communication: Utilização de Recursos Pedagógicos Dinâmicos e Interativos em Unidades Curriculares da Licenciatura em Matemática Aplicada à Economia e à Gestão, II Encontro deReflexão e Partilha Pedagógica em Ciências Sociais (ERPP). 2020. Available online: https://www.youtube.com/watch?v=5O7q6DCJCAU (accessed on 15 December 2021).
- Conceição, A.C.; Fernandes, S. Wolfram|Alpha: Uma ferramenta de aprendizagem ativa em Cálculo I. In Proceedings of the II International Congress on Interdisciplinarity in Social and Human Sciences, Faro, Portugal, 11–12 May 2017; pp. 301–306. [Google Scholar]
- Conceição, A.C.; Fernandes, S.; Pereira, J.C. Prática pedagógica com o software educacional F-Tool em Cálculo I. In Proceedings of the Congresso Nacional de Práticas Pedagógicas no Ensino Superior (CNaPPES 2015), Leiria, Portugal, 3 July 2015; pp. 99–104. [Google Scholar]
- Conceição, A.C.; Martins, P.V. Prática pedagógica em Engenharia Informática: Análise da utilização do Wolfram|Alpha. In Proceedings of the Congresso Nacional de Práticas Pedagógicas no Ensino Superior (CNaPPES 2016), Lisboa, Portugal, 14–15 July 2016; pp. 389–394. [Google Scholar]
- Pereira, J.C.; Conceição, A.C. F-Tool 2.0: Exploring the Logistic function in the classroom. In Proceedings of the 1st International Conference on Algebraic and Symbolic Computation (SYMCOMP2013), Lisboa, Portugal, 9–10 September 2013; pp. 149–158. [Google Scholar]
- Andraz, J.M.; Conceição, A.C.; Ponte, R. Dinamismo e interatividade na análise do conceito de elasticidade da oferta. UAlgoritmo
**2021**, 3, 35–40. [Google Scholar] - Hayden, M.B.; Lamagna, E.A. NEWTON: An interactive environment for exploring mathematics. J. Symb. Comput.
**1998**, 25, 195–212. [Google Scholar] [CrossRef] - Menges, R.J.S. Research on Teaching and Learning: The Relevant and the Redundant. Rev. High. Educ.
**1988**, 11, 259–268. [Google Scholar] [CrossRef]

**Figure 1.**Part of the code of the tool “Riemann Sums”, used in the examples presented in Figures 7 and 14.

**Figure 2.**Part of the code of the tool “Christmas Scene Method for identifying and solving exact differential equations”, used in the examples presented in Figures 8, 16 and 17.

**Figure 3.**Part of the code of the “F-Logistic” tool [35] responsible for some checkboxes for the plots’ options and for the parameter values (including the choice of styles and sizes).

**Figure 8.**Some of the possible functions for the function M when using the tool Christmas Scene Method for identifying and solving exact differential equations.

**Figure 9.**Example of an exercise proposed to encourage discussion, dialogue and reflection, available in the Computable Document Format: What kind of functions will appear if we change dynamically the parameter A?

**Figure 10.**Example of a problem that can be projected and asked to be resolved by a student, in a remote or in a classroom learning environment, concerning the invertibility concept.

**Figure 11.**A student’s response to a question about invertibility, raised in a classroom learning environment (during the global pandemic caused by the coronavirus SARS-CoV-2).

**Figure 12.**Solution of the exercise displayed on Figure 10 through the technological F-Logistic tool.

**Figure 13.**Video on the invertibility concept produced by the author (using the F-Exponential tool [26]) and available in the learning management system moodle of the class.

**Figure 14.**Example of three possible short-problems proposed to be solved individually, concerning the Riemann Sum concept.

**Figure 15.**Image that illustrate a challenging problem that can be solved as autonomous students’ work.

**Figure 16.**Image that illustrate an activity to be done in autonomous students’ work. The student must justify why it is an exact ODE and confirm all the computed integrals.

**Figure 17.**Image that illustrate an activity to be done in autonomous students’ work. The student must justify why it is not an exact ODE.

**Figure 18.**Image that illustrate part of an evaluation task: graphical representation of the inverse function whose function is drawn and analytical determination of that function.

**Figure 19.**Image that illustrate part of another evaluation task: graphical representation of the inverse function whose function is drawn and analytical determination of that function.

**Figure 20.**Image that illustrate the solution of the evaluation task that can be associated to Figure 18.

**Figure 21.**Image that illustrate the solution of the evaluation task that can be associated to Figure 19.

Publisher’s Note: MDPI stays neutral with regard to jurisdictional claims in published maps and institutional affiliations. |

© 2021 by the author. 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 (https://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Conceição, A.C.
Dynamic and Interactive Tools to Support Teaching and Learning. *Math. Comput. Appl.* **2022**, *27*, 1.
https://doi.org/10.3390/mca27010001

**AMA Style**

Conceição AC.
Dynamic and Interactive Tools to Support Teaching and Learning. *Mathematical and Computational Applications*. 2022; 27(1):1.
https://doi.org/10.3390/mca27010001

**Chicago/Turabian Style**

Conceição, Ana C.
2022. "Dynamic and Interactive Tools to Support Teaching and Learning" *Mathematical and Computational Applications* 27, no. 1: 1.
https://doi.org/10.3390/mca27010001