Abstract
This chapter addresses opportunities and challenges posed by the teaching and learning of mathematics through digital learning platforms basically developed using Moodle (see https://moodle.org). Specifically, we review and discuss the design and implementation of several different mathematics learning environments. Results indicate the existence of new teaching and learning opportunities—and challenges—when working with secondary or middle school students in hybrid learning environments where teaching and learning of mathematics are mediated by technology.
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Notes
- 1.
According to Dunham and Hennessy (p. 388), disadvantaged students are who traditionally do less well than the general population: “In effect, technology ‘leveled the playing field’ so that previously disadvantaged groups, who—because of different cognitive styles, learning disabilities, or special circumstances—had usually achieved less, performed as well or better than the main group on outcome measures when using computers or calculators.”
- 2.
Such figural patterns, “whether constructed ambiguously, or in a well-defined manner, consist of stages whose parts could be interpreted as being configured in a certain way” (Rivera 2010, p. 298). According to Rivera, he preferred «to use the term figural pattern to convey what I assume to be the “simultaneously conceptual and figural” (Fischbein, 1993, p. 160) nature of mathematical patterns. The term “geometric patterns” is not appropriate due to a potential confusion with geometric sequences (as instances of exponential functions in indiscrete mathematics). Also, I was not keen in using the term “pictorial patterns” due to the (Peircean) fact that figural patterns are not mere pictures of objects but exhibit characteristics associated with diagrammatic representations. The term “ambiguous” shares Neisser’s (1976) general notion of ambiguous pictures as conveying the “possibility of alternative perceptions”, (p. 50)».
- 3.
In accordance with Mason and collaborators (1985), a main idea for initiate students in the learning of algebra is that students identify a pattern in a succession of figures (or numbers) and then communicate and record by writing the common characteristics perceived between them, or the relationships that might be established initially with examples. From there teacher can drive some math questions as for example: will there be any formula that could define this pattern? Also, Mason et al., established that once agreed what defines the pattern, the regularities and relationships between its components must be translated from one natural language into a rule or general formula, which will result from a cognitive evolution of the student, such transition is not a simple cognitive exercise, but it could be supported by drawings, diagrams or words, which lead later to describe the key variables in the problem and move to the achievement of its expression in symbolic form.
- 4.
In his work on a pedagogical approach of teaching, Simon (1995) founded a constructivist learning of mathematics. He developed a model of decision-making for teachers considering the design of math tasks. Its core consists in “the creative tension between the teacher's goals about student learning and his responsibility to be sensitive and responsive to the mathematical thinking of the students” (see Simon, 1995, p. 114). Simon’s work presents a diagram (Simon, 1995, p. 136) of a—constructivist—cycle of teaching.
- 5.
See last part in the previous final note, in this section.
- 6.
According to Rivera (2010, p. 300) “meaningful pattern generalization involves the coordination of two interdependent actions, as follows: (1) abductive–inductive action on objects, which involves employing different ways of counting and structuring discrete objects or parts in a pattern in an algebraically useful manner; and (2) symbolic action, which involves translating (1) in the form of an algebraic generalization. The idea behind abductive–inductive action is illustrated by a diagram [it appeared in Rivera’s work published in 2010, p. 300, in Fig. 5], an empirically verified diagram of phases in pattern generalization that I have drawn from a cohort of sixth-grade students who participated in a constructivist-driven pattern generalization study for two consecutive years (Rivera & Becker, 2008).”
- 7.
Both modes are at the beginning of the incorporation of innovation at the school, according to the PURIA model. Following this model implies that teachers should experiment with the mentioned modes to advance toward successfully incorporating technology into classrooms (Hoyos, 2009, 2012; Zbiek & Hollebrands, 2008).
Briefly, the PURIA model consists of five stages named the Play, Use, Recommend, Incorporate, and Assess modes: “When [teachers are] first introduced to a CAS… they play around with it and try out its facilities… Then they realize they can use it meaningfully for their own work… In time, they find themselves recommending it to their students, albeit essentially as a checking tool and in a piecemeal fashion at this stage. Only when they have observed students using the software to good effect they feel confident in incorporating it more directly in their lessons… Finally, they feel they should assess their students’ use of the CAS, at which point it becomes firmly established in the teaching and learning process” (Beaudin & Bowers, 1997, p. 7).
- 8.
According to Duval (2006, p. 112): “Conversions are transformations of representations that consist of changing a register without changing the objects being denoted: for example, passing from the algebraic notation for an equation to its graphic representation, …”.
References
Balacheff, N. (1994). La transposition informatique, un nouveau problème pour la didactique des mathématiques, In M. Artigue, et al. (Eds.), Vingt ans de didactique des mathématiques en France (pp. 364–370). Grenoble: La pensée sauvage éditions.
Balacheff, N. (2004). Knowledge, the keystone of TEL design. In Proceedings of the 4th Hellenic Conference with International Participation, Information and Communication Technologies in Education. Athens, Greece. Retrieved on August 2015 at https://halshs.archives-ouvertes.fr/hal-00190082/document.
Balacheff, N. (2010a). Bridging knowing and proving in mathematics. An essay from a didactical perspective. In G. Hanna, H. N. Jahnke, & H. Pulte (Eds.), Explanation and proof in mathematics (pp. 115–135). Heidelberg: Springer. Retrieved on November 2016 at https://arxiv.org/pdf/1403.6926.pdf.
Balacheff, N. (2010b). Note pour une contribution au débat sur la FOAD. Distances et Mediations des Savoirs (DMS), 8(2), 167–169). Retrieved on April 24, 2016 at http://ds.revuesonline.com/article.jsp?articleId=15229
Balacheff, N. (2012). A propos d’un debat provocatrice…. Retrieved from http://nicolas-balacheff.blogspot.fr/2012/06/note-pour-un-debat-provocateur-sur-la.html.
Balacheff, N. (2015). Les technologies de la distance a l’ere des MOOC. Opportunités et Limites. Seminar on January 9, 2015, at the National Pedagogical University (UPN), Mexico (Unpublished ppt presentation).
Balacheff, N., & Soury-Lavergne, S. (1996). Explication et préceptorat, a propos d’une étude de cas dans Télé-Cabri. Explication et EIAO, Actes de la Journée du 26 janvier 1996 (PRC-IA) (pp. 37–50). Paris, France: Université Blaise Pascal. Rapport LAFORIA (96/33) <hal-00190416> . Retrieved at https://hal.archives-ouvertes.fr/hal-00190416/document.
Balacheff, N., & Sutherland, R. (1994). Epistemological domain of validity of microworlds: the case of Logo and Cabri-géomètre. In R. Lewis & P. Mendelsohn (Eds.), Proceedings of the IFIP TC3/WG3. 3rd Working Conference on Lessons from Learning (pp. 137–150). Amsterdam: North-Holland Publishing Co.
Beaudin, M., & Bowers, D. (1997). Logistic for facilitating CAS instruction. In J. Berry (Ed.), The state of computer algebra in mathematics education. UK: Chartwell-Bratt.
Cuban, L., Kirkpatrick, H., & Peck, C. (2001). High access and low use of technologies in high school classrooms: Explaining an apparent paradox. American Educational Research Journal, 38(4), 813–834.
Dede, C., & Richards, J. (2012). Digital teaching platforms: Customizing classroom learning for each student. New York & London: Teachers College Press.
Dunham, P., & Hennessy, S. (2008). Equity and use of educational technology in mathematics. In K. Heid & G. Blume (Eds.), Research on Technology and the Teaching and Learning of Mathematics: Vol. 1. Research syntheses (pp. 345–347). USA: Information Age Publishing.
Duval, R. (1994). Les représentations graphiques: Fonctionnement et conditions de leur apprentissage. In A. Antibi (Ed.), Proceedings of the 46th CIEAEM Meeting (Vol. 1, pp. 3–14). Toulouse, France: Université Paul Sabatier.
Duval, R. (2006). A cognitive analysis of problems of comprehension in a learning of mathematics. Educational Studies in Mathematics, 61, 103–131.
Fischbein, E. (1993). The theory of figural concepts. Educational Studies in Mathematics, 24(2):139–162.
Heffernan, N., Heffernan, C., Bennett, M., & Militello, M. (2012). Effective and meaningful use of educational technology: Three cases from the classroom. In C. Dede & J. Richards (Eds.), Digital teaching platforms: Customizing classroom learning for each student. New York & London: Teachers College Press.
Hoyos, V. (2009). Recursos Tecnológicos en la Escuela y la Enseñanza de las Matemáticas. En M. Garay (coord.), Tecnologías de Información y Comunicación. Horizontes Interdisciplinarios y Temas de Investigación. México: Universidad Pedagógica Nacional.
Hoyos, V. (2012). Online education for in-service secondary teachers and the incorporation of mathematics technology in the classroom. ZDM—The International Journal on Mathematics Education, 44(7). New York: Springer.
Hoyos, V. (2016). Distance technologies and the teaching and learning of mathematics in the era of MOOC. In M. Niess, S. Driskell, & K. Hollebrands (Eds.), Handbook of research on transforming mathematics teacher education in the digital age. USA: IGI Global.
Hoyos, V. (2017). Epistemic and semiotic perspectives in the analysis of the use of digital tools for solving optimization problems in the case of distance teacher education. In Proceedings of CERME 10. Dublin: Retrieved on 7 August 2017 at https://keynote.conference-services.net/resources/444/5118/pdf/CERME10_0572.pdf.
Hoyos, V., & Navarro, E. (2017). Ambientes Híbridos de Aprendizaje: Potencial para la Colaboración y el Aprendizaje Distribuido. Mexico: UPN [in Press].
Matus, C., & Miranda, H. (2010). Lo que la investigación sabe acerca del uso de manipulativos virtuales en el aprendizaje de la matemática. Cuadernos de Investigación y Formación en Educación Matemática, 5(6), 143–151.
Neesam, C. (2015). Are concrete manipulatives effective in improving the mathematics skills of children with mathematics difficulties? Retrieved on August 06, 2017 at: http://www.ucl.ac.uk/educational-psychology/resources/CS1Neesam15-18.pdf.
Neisser, U. (1976). Cognitive psichology. New York: Meredith.
OECD. (2015). Students, computers and learning. Making the connection. Pisa: OECD Publishing. Retrieved on August 7, 2017 at http://dx.doi.org/10.1787/9789264239555-en.
Rivera, F. (2010). Visual templates in pattern generalization activity. Educational Studies in Mathematics, 73, 297–328.
Rivera, F. (2011). Toward a visually-oriented school mathematics curriculum. Netherlands: Springer.
Rivera, F., & Becker, J. (2008). From patterns to algebra. ZDM, 40(1), 65–82.
Rodriguez, G. (2015). Estudiantes en desventaja resolviendo tareas de generalización de patrones con la mediación de plantillas visuales y manipulativos virtuales (Ph.D. thesis in mathematics education). Mexico: UPN [National Pedagogical University].
Rojano, T. (2002). Mathematics learning in the junior secondary school: Students’ access to significant mathematical ideas. In L. D. English (Ed.), Handbook of international research in mathematics education. New Jersey, USA: LEA.
Ruthven, K. (2007). Teachers, technologies and the structures of schooling. En Proceedings of CERME 5. Larnaca: University of Cyprus (Retrieved from http://ermeweb.free.fr/CERME5b on 02/15/09).
Simon, M. A. (1995). Reconstructing mathematics pedagogy from a constructivist perspective. Journal for Research in Mathematics Education, 26(2), 114–145.
Soury-Lavergne, S. (1997). Etayage et explication dans le préceptorat distant, le cas de Télé-Cabri (Thèse du Doctorat). Grenoble: Université Joseph Fourier. Retrieved at https://tel.archives-ouvertes.fr/tel-00004906/document.
Sutherland, R., & Balacheff, N. (1999). Didactical complexity of computational environments for the learning of mathematics. International Journal of Computers for Mathematical Learning, 4, 1–26.
Zbiek, R., & Hollebrands, K. (2008). A research-informed view of the process of incorporating mathematics technology into classroom practice by in-service and prospective teachers. In K. Heid & G. Blume (Eds.), Research on Technology and the Teaching and Learning of Mathematics: Vol. 1. Research syntheses (pp. 287–344). USA: Information Age Publishing.
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Hoyos, V., Navarro, M.E., Raggi, V.J., Rodriguez, G. (2018). Challenges and Opportunities in Distance and Hybrid Environments for Technology-Mediated Mathematics Teaching and Learning. In: Silverman, J., Hoyos, V. (eds) Distance Learning, E-Learning and Blended Learning in Mathematics Education. ICME-13 Monographs. Springer, Cham. https://doi.org/10.1007/978-3-319-90790-1_3
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