Abstract
This chapter focuses on the particularities of mathematical work in the digital age. It opens with historical considerations that relate to the development of mathematical work to do arithmetic when symbolic or mechanical tools and algorithmic methods were used. He goes on to define the new mathematical work by introducing concepts of reference in the interaction between human and machine. The notion of valence of mathematical work makes it possible to account for the operating domain of interactions as well as for the possible adaptations of an evolving subject-milieu system, whether for the accomplishment of a task or in the course of learning mathematics. A difference between the work of the designer and that of a user is established, especially in terms of the effects on reasoning, proof, modelling activity and the creation of digital artefacts. The adaptation in a process of idoneity between the teaching and the learning project, as well as between the intention of the designer and the realisation by a user, is discussed.
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Notes
- 1.
The etymological meaning of the word jet in English is a borrowing straddling, on the one hand, from Anglo-French jet, ‘a cast’, from Old French jeter, ‘to cast’ and from Late Latin jectare, ‘to cast’, and on the other hand, borrowing from Anglo-French geet, ‘jet’ meaning ‘stone’, from Imperial Latin gagates, ‘jet’ meaning ‘stone’ and from Ancient Greek gagatēs, ‘lignite’. The jet à la main could then be translate into something as casting to the hand with tokens on a reckoning table, while the jet à la plume, casting to the quill using an algorithm on paper. A long struggle began in the 12th century between the abacists and the algorists. Although Leonardo of Pisa recommended the use of calculation algorithms for multiplication as early as 1202, abacus calculation remained in use until the 17th century, notably by Gottfried Wilhelm Leibniz, who still used it for long calculations (Piguet & Hügli, 2004).
- 2.
In our text, we use «situation/task/activity» as a compound word to avoid having to distinguish between the situation as a didactic project or staging by the teacher, the task as a job or obligation that the pupil has to carry out in a given time and under certain conditions, and the activity that refers to the actions and operations of the pupil or the teacher, possibly finalised or directed. In the conclusion, we propose to use the word task as a generic term.
- 3.
In French known as multiplication par jalousie in reference to the jalousie windows (jalousie coming from the Italian for jealousy, gelosia), in which louvres are used to allow the lookout without being seen, hence the observer «jealously» guards his intimacy.
- 4.
See Machine à multiplier de Léon Bollée, 1889 - La machine en fonctionnement, consulted on 15 December 2020 at https://youtu.be/Le7QIhO_7vM.
- 5.
For the sake of space, we have only illustrated part of these software programs in the following sections.
- 6.
Consulted on 15 December 2020 from https://scratch.mit.edu/projects/34251632/.
- 7.
In the Pan-Canadian study of Buteau, Jarvis & Lavicza (2014), it was shown that 43% of more than 300 Canadian mathematicians reported using computer programs in their research, but only 18% said they used it as part of their teaching. See also the research of Broley (2015) of computer programs for research and university education in mathematics, or the work of Pérez-Sanz (2006), which is based on historical data.
- 8.
The scale of this task can easily be seen by consulting the proceedings of the Espace mathématique francophone (EMF, 2021), the Conferencia Interamericana de Educación Matemática (CIAEM, 2021), the International Congress on Mathematical Education (ICME, 2021), the Reunión Latinoamericana de Educación Matemática (RELME, 2021) and the European Society for Research in Mathematics Education (CERME, 2021).
- 9.
In Brousseau, the milieu is a system that is antagonistic to the subject participating in it, like a partner, in the development of mathematical knowledge. The milieu can be material, virtual, social or symbolic, and it is most often a hybridization of these modes. Certain reasoning models of the conceptions of learners, such as that of Balacheff and Margolinas (2005), place the subject-milieu interaction at the center of conceptualization. See also Coutat et al. (2016) regarding the link between the teacher and the student-milieu system when the milieu interaction combines the virtual with paper and pencil within a computing environment for human learning.
- 10.
To help the reader, it is best to remember that 32 768 = 2 (8 762 + 7 622) = 215, 268 435 456 = 228 and 1 073 741 824 = 230.
- 11.
Like a linguistic wink, digital artefact translates into French as «artéfact numérique», the word digital referring only to fingers, as in formal language in English. See also digital multiplication in Fig. 1.
- 12.
According to the Jean Piaget Foundation (2021): «The very abstract notion of an epistemic subject characterises this part of cognitive function, which is common to all subjects of a certain level of development, and which constitutes notions used to organise, transform or explain reality. It is the centre of cognitive function rather than the cognitive system as a whole, abstractly detached from those elements of this function that are not properly cognitive».
- 13.
At the time, the sharing platform Intergeo (2021) brought together the systems relative to interactive geometry in one place in order to make it easier to access the resources provided by different systems, traditions and regions (file formats, didactic documents, assessment tools, study programme content, etc.). Today it is no longer active in open access. Information on it can be found in the Sésamath journals (2021) and in The Advanced Open Source Enterprise Wiki (2021).
- 14.
As an anecdote, first worksheet was uploaded in April 2011, but first official public use was with the release of GeoGebra 4.0 on 20 October 2011, although it had been available in beta before that. The first worksheet uploaded by someone who was not on the GeoGebra team was by the Catalan Carlos Giménez Esteban on 30 April (2011).
- 15.
In this activity, the blue figure is an instrumented geometric figure for which there is no equivalent traditional figure, unless it is considered as a collection of known figures (point, segment, triangle, trapezoid, rectangle). This figure capable of metamorphosis plays the role of reality (or model situation) that is in relation to the model in mathematical analysis. However, the graphic representation of the function is also an instrumented representation when the red curve is plotted with the «trace» tool for point P activated, as, and to the extent that, point M moves. So then it is possible to speak of an instrumented mathematical model that is revealed within the play between the two windows.
- 16.
WIRIS (2021) is a suite of tools for the teaching of mathematics that integrates perfectly with Moodle (2021), an online learning platform that allows for the creation of learning communities gathering around pedagogical content and activities.
- 17.
In its general characteristics, the system allows the student to simply expressions, solve equations, factor polynomials, represent mathematical expressions graphically and write functions and draw geometric objects by hand. Nevertheless, unlike calculators like those of the Photomath type (2021), which are available in portable devices equipped with a camera, it does not recognise handwritten mathematical equations or external views and does not provide step-by-step directions on the solving process.
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Appendix 1: Evolution of Instrumented Mathematical Work with Digital Artefacts in the MWS Community
Appendix 1: Evolution of Instrumented Mathematical Work with Digital Artefacts in the MWS Community
Here below, we refer to the work done in the MWS symposiums in a succinct manner. So, in ETM3 we focus on works that show semiotic and instrumental genesis. While in ETM4 and based on what was addressed in the previous symposium, in addition to the other two geneses, discursive genesis is introduced in a genesis connection. Then in ETM5, the research focused on mathematical work with a certain independence from ETM. Finally, in ETM6, the digital artefact gives a new drive to instrumental genesis, like discursive genesis (articulation of meanings + control structure par excellence). So instrumental genesis is linked in a particular manner to digital artefacts which, in turn, are linked to semiotic genesis since it can be experienced representational registers such as figures and graphs.
In this sense, at the third symposium on MWS, ETM3, the research was linked to the system of signs and semiotic genesis. The research of Blossier and Richard (2012) was aimed at accounting for the mathematical work generated by a problem that asked for the completion of the construction of a figure using the GeoGebra 3D digital artefact. The potentialities of the artefact are identified, which then allow the optimisation of the construction of an idoneous MWS. Likewise, the work of Tessier-Baillargeon et al. (2012) showed the influence of geometric working spaces on the design and validation of the geogebraTUTOR tutorial system, created for the development of geometric thinking. The design of this digital artefact consolidates the interpretation of an idoneous Geometric Working Space (GWS) in which the student is successful in solving a deductive geometry problem. For her part, Coutat (2012) presents the use of dynamic geometry software in a geometry task to analyse, in mathematical work, the different geneses that emerge and explain how the instrumental knowledge acquired by working with the software are linked to the conceptual knowledge acquired in the paper and pencil environment. Furthermore, the research of Gómez-Chacón (2012) used GeoGebra to solve problem on geometric locus and analysed how figural and instrumental geneses are articulated, proposing the need for a balance between the non-iconic visualisation and instrumental genesis. On the other hand, Carrión & Pluvinage (2012) analyse the predominance of the digital artefact in mathematical work when tasks are solved regarding real variable functions. They conclude that the use of digital technology exerts a strong influence on the solving process and conceptual handling.
In the edition of the symposium, ETM4, the research presented enriched the works presented in the ETM 3 symposium with regard to signs, genesis and artefacts. Santos-Trigo & Camacho-Machín (2014) analysed the extent that coordinated use of digital artefacts (YouTube videos, WolframAlpha and GeoGebra) favour the solving of problems that involve representation, exploration and communication of results. Their work presents a task that consists in folding a sheet of paper, which allows them to analyse and discuss how the coordinated use of various digital technologies provides opportunities to represent and explore tasks from different angles and perspectives. The research of Rivera et al. (2014) also evaluates the use of an interactive digital whiteboard to highlight its utility when addressing matters of calculating differentials. It showed that when the digital whiteboard is used, it can be a mediator of knowledge as compared to the traditional blackboard. For their part, Páez Murillo & Vivier (2014) studied the diversity of semiotic representations that arise when working on tasks regarding linear inequalities and interacting with the GeoGebra digital artefact. They showed that the use of signs and digital artefacts as vehicles of knowledge connect naturally with mathematical work and semiotic representations. While at the same time, Lagrange (2014) questioned the notion of function based on the research done in the Casyopée project, where functions were incorporated and used for work in the same functional dependency in four different settings, based on the representations of the functions. This led to the reconsideration of the notion of function and research into the potential of linked functional working spaces to design and analyse situations in technological environments. Furthermore, Abánades et al. (2014) analysed and compared the effectivity of the “Locus”, “LocusEquation” and “Locus GC” GeoGebra tools in order to solve a given task in the hopes of identifying the personal and idoneous MWS and analysed the instrumental geneses that emerge. Drouard and Kuzniak (2014) posed questions regarding the role of tools and instruments in mathematical work and introduced a multidimensional point of view on tools and instruments that resonates with the research on the domain of algebra and geometry. Finally, they explained how the different functions and use of tools and instruments in the MWS should allow for a detailed study of mathematical work.
The advances in the fifth edition of the symposium, ETM5, can be split into two groups. In the first group, the articles analysed the interaction of students with digital artefacts and, in the second group, they instead analysed the design of digital artefacts and the potential MWS they could activate. The first group included the work of Lagrange (2016), who used Casyopée as an artefact and which was developed in the Casyopée group. It presented students with a task to model a suspension bridge, in which the digital artefact allowed them to work with elements of algorithmic geometric working space, with functional working space and to connect them. It should be pointed out that unlike the other dynamic geometry software programs, Casyopée’s connection between the geometry window and the algebra window is bidirectional. This means by modifying (moving/dragging) elements of the geometric window, the parameters of the algebraic expressions are modified as well.
The second group includes the work of Leduc, Tessier-Baillargeon, Corbeil, Richard & Gagnon (2016), who presented a tutorial system called QED-Tutrix—which is both an adaption of GeoGebra and a revision of geogebraTUTOR—that helped students work a proof within a dynamic geometry environment. According to the authors, QED-Tutrix can be considered a Didactic MWS, where the artefact is an adidactic milieu. Different geneses are considered in the design of a tutorial system’s architecture, which implies more precise guidance of the students’ work. In that same group, there is the work of Gaona (2016), which analysed the design of a set of tasks in an online assessment environment with Moodle and Wiris and compared them with the paper and pencil assessments. He concludes that the idoneous MWS proposed in the platform is reduced as compared to the idoneous MWS of the platform. Nevertheless, the data presented only allow conjecture regarding these differences and the reasons behind them.
With regard to ETM6, it also saw works that analysed from the perspective of the use of digital artefacts as mediators and the design of digital artefact. In relation to the use of digital artefacts, Páez and Pluvinage (2018) used a guided examination of the interactive environment to study how the zoom command in the GeoGebra software favoured the visualisation and activation of the [Sem-Ins] and [Ins-Dis] planes in engineering students. Furthermore, Salazar and Carrillo (2018) analysed the mathematical work of students and the activation of the [Sem-Ins] and [Ins-Dis] planes based on a task on piecewise functions using a GeoGebra applet. Likewise, López (2018) studied the role of visualisation of students’ mathematical work in relation to parametric curves when the GeoGebra digital artefact was used. The work of Cyr, Font, Gagnon, Leduc, Richard and Tessier-Baillargeon (2018) and of Gaona (2018) reported on design in continuations of the work they presented at ETM 5. One of the convergent elements of these different works is related to the automatisation of a validation process of the students’ answers. In the case of Cyr and his collaborators, they addressed the small-scale automatisation of a test process, while Gaona looked at the automatisation of graph reading questions. Both of them observed that there is a concern in the design of a certain openness of the tasks so that they permit partially correct answers and as well regarding the representation of objects. These elements configure and influence the sense of the mathematical objects involved and in the potential MWS of the tasks mediated by digital artefacts.∎
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Flores Salazar, J.V., Gaona, J., Richard, P.R. (2022). Mathematical Work in the Digital Age. Variety of Tools and the Role of Geneses. In: Kuzniak, A., Montoya-Delgadillo, E., Richard, P.R. (eds) Mathematical Work in Educational Context. Mathematics Education in the Digital Era, vol 18. Springer, Cham. https://doi.org/10.1007/978-3-030-90850-8_8
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