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The Influence and Shaping of Digital Technologies on the Learning – and Learning Trajectories – of Mathematical Concepts

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Mathematics Education and Technology-Rethinking the Terrain

Part of the book series: New ICMI Study Series ((NISS,volume 13))

Abstract

The significant development and use of digital technologies has opened up diverse routes for learners to construct and comprehend mathematical knowledge and to solve problems. This implies a revision of the pedagogical landscape in terms of the ways in which students engage in learning, and how understandings emerge. In this chapter we consider how the availability of digital technologies has allowed intended learning trajectories to be structured in particular forms and how these, coupled with the affordances of engaging mathematical tasks through digital pedagogical media, might shape the actual learning trajectories. The evolution of hypothetical learning trajectories is examined, while the transitions learners make when traversing these pathways are also considered. Particular instances are illustrated with examples in several settings.

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Notes

  1. 1.

    1 In theDual Process Theory cognition is seen as operating in two quite different modes calledSystem 1 and System 2.

  2. 2.

    2 Nuñez (1993) explains that this confusion arises when there are several competing components (processes) present; that is, when two types of iterations of perhaps different nature (cardinality vs. measure) are confused: the process itself and the divergent process of adding terms to a sequence.

  3. 3.

    3 For example, during the explorations of sequences of the type {(1/k)n}, some students discovered that the corresponding series:\({\sum}_{n=1}^\infty 1/k^n\), where the integer k > 1, converge to 1/(k - 1). They then tested the validity of their conjecture using all the available tools in the microworld, in order to “prove” it.

  4. 4.

    4 Computer Intensive Algebra is a beginning algebra curriculum that introduces students to algebra in the context of mathematical modeling computer explorations, that provide access to multiple representations and assist in reasoning about algebraic expressions (Heid 1996).

  5. 5.

    In this study, a complete learning sequence (theVisual Math curriculum) is prepared in order to observe learning processes throughout a longitudinal period of 3 years (grades 7–9) using an alternate approach (a functional approach) to algebra teaching. One of the findings was that when using the alternate treatment, changes expected – for example in conception of functional variation and the rate of change – took a fair amount of time (Yerushalmy2000).

  6. 6.

    6 In this project, algebra is introduced to pupils at the beginning of primary school. Its approach is based on a Russian framework created by the melding of multiple theories (e.g. theories by psychologists like Davidov and Vygotsky). Pupils begin with generalizations rather than with specific instances, so that they can see the concepts in action rather than trying to build the bigger picture from a variety of specific examples. Symbolism is naturally integrated to children’s tasks as well as the notion of relationships between and among quantities (Dougherty 2001).

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Acknowledgments

We acknowledge the collaboration of Hee-chan Lew and James Nicholson in the discussions leading to the writing of this chapter.

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Authors and Affiliations

  1. Departamento de Matemática Educativa, Centro de Investigacióny de Estudios Avanzados del IPN(Cinvestav), Mexico, Mexico

    Ana Isabel Sacristán, Teresa Rojano & Manuel Santos-Trigo

  2. Mathematics, Science and Technology Education, University of Waikato at Tauranga, Tauranga, New Zealand

    Nigel Calder

  3. Department of Science Teaching, The Weizmann Institute of Science, Rehovot, Israel

    Alex Friedlander

  4. University of Muenster, Muenster, Germany

    Hartwig Meissner

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  1. Ana Isabel Sacristán
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  2. Nigel Calder
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  3. Teresa Rojano
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  4. Manuel Santos-Trigo
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  5. Alex Friedlander
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  6. Hartwig Meissner
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Editors and Affiliations

  1. Inst. Education, University College London, Bedford Way 20 , London, WC1H 0AL, United Kingdom

    Celia Hoyles

  2. IUFM de Reims, rue Gabriele Voisin 11, Reims, 51100, France

    Jean-Baptiste Lagrange

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Sacristán, A.I. et al. (2009). The Influence and Shaping of Digital Technologies on the Learning – and Learning Trajectories – of Mathematical Concepts. In: Hoyles, C., Lagrange, JB. (eds) Mathematics Education and Technology-Rethinking the Terrain. New ICMI Study Series, vol 13. Springer, Boston, MA. https://doi.org/10.1007/978-1-4419-0146-0_9

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