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Building Bridges to Algebra Through a Constructionist Learning Environment”

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Building Bridges to Algebra Through a Constructionist Learning Environment” TOC The Autopoiesis of Social Systems Constructionism and its Criticisms Introduction The Autopoiesis of Social Systems From self-reference to autopoiesis of social systems Applications of autopoiesis to social systems Building Bridges to Algebra through Biological criticisms Sociological criticisms Conclusion a Constructionist Learning Environment Open Peer Commentaries What Is Sociology? Eirini Geraniou • University College London, UK • e.geraniou/at/ioe.ac.uk Towards a Consistent Constructivist General Systems Theory Manolis Mavrikis • University College London, UK • m.mavrikis/at/ioe.ac.uk Does Social Systems Theory Need Autopoiesis? Does Social Systems Theory Need a General Theory of Autopoiesis? Missing: The Socio-Political Dimension > Context • In the digital era, it is important to investigate the potential impact of digital technologies in education Autopoiesis Applies to Social Systems Only and how such tools can be successfully integrated into the mathematics classroom. Similarly to many others in the Explaining Social Systems without Humans constructionism community, we have been inspired by the idea set out originally by Papert of providing students Communication is Meaning-based Autopoiesis with appropriate “vehicles” for developing “Mathematical Ways of Thinking.” > Problem • A crucial issue regarding the The Concept of Autopoiesis: Its Relevance and Consequences for Sociology design of digital tools as vehicles is that of “transfer” or “bridging” i.e., what mathematical knowledge is transferred The Concept of Autopoiesis from students’ interactions with such tools to other activities such as when they are doing “paper-and-pencil” Authors’ Response mathematics, undertaking traditional exam papers or in other formal and informal settings. > Method • Through the On the Criticisms against the Autopoiesis of Social Systems lens of a framework for algebraic ways of thinking, this article analyses data gathered as part of the MiGen project Authors’ Response from studies aiming at investigating ways to build bridges to formal algebra. > Results • The analysis supports the Combined References need for and benefit of bridging activities that make the connections to algebra explicit and for frequent reflection and consolidation tasks. > Implications • Task and digital environment designers should consider designing bridging activities that consolidate, support and sustain students’ mathematical ways of thinking beyond their digital experience. > Constructivist content • Our more general aim is to support the implementation of digital technologies, especially constructionist learning environments, in the mathematics classroom. > Key words • Algebraic generalisation and language, transition, exploratory learning, microworlds, bridging tasks. Introduction designed to help them with. Therefore the ment (eXpresser) specially designed to sup- tool cannot immediately become an efficient port and address students’ difficulties with « 1 » There is a growing concern that mathematical instrument for them (Artigue learning algebra (see Mavrikis et al. 2013) despite the increased availability of dig- 2002). Consequently, teachers can be hesi- – to paper-and-pencil (PaP) activities. ital technologies designed for mathematics tant to use such tools as it is hard for them « 2 » We are not the only ones con- learning, students rarely use ideas, concepts to be convinced of the tools’ value and they cerned with the transition to formal algebra or strategies that they could acquire through are reluctant to dedicate time to learning (e.g., Radford 2014), and the literature is their interaction with such technologies in how to use them effectively and incorporate replete with examples of student difficulties other contexts (cf. EACEA Eurydice Report them into mathematics teaching practice (e.g., Stacey & Macgregor 2002). Our view 321 2011). As such, the full or intended po- (Clark-Wilson, Robutti & Sinclair 2014). is similar to that of Luis Radford (2014), tential of such technologies is diminished. However, a growing awareness of digital who claimed that there is a need for spe- One of the earliest (and most articulate) technologies’ potential and limitations can cially designed classroom activity to support examples of these concerns is discussed in support teachers using these technologies students’ developmental path to formal al- Jean-Luc Gurtner (1992). Referring to the in the classroom (Abboud-Blanchard 2014). gebra, and to Gurtner (1992) and Stephen Logo environment, Gurtner demonstrated In this article, we discuss our approach to Godwin and Rob Beswetherick (2003), who that the tool’s features, which are designed supporting students’ transition in moving suggested presenting structured tasks, using to support students when faced with com- back and forth from paper-and-pencil to appropriate digital tools and making explicit plex mathematical problems, may impede interacting with digital tools. We therefore interventions during students’ interactions. them from making connections between consider ways of facilitating the integration We claim that a digital tool specially de- their work in Logo and any mathematical of digital technologies into the mathematics signed to support the development of alge- or geometrical ideas they are already famil- classroom in an effort to shed some light on braic ways of thinking (AWOT), together iar with and use when problems seem less this issue that did not have the attention it with carefully designed bridging activities, complex. One of the reasons behind this deserves, as yet. In particular, our focus is should scaffold the transition to formal al- is that students might know how to use a on the transition to formal algebra and how gebra. Besides “learning” the tool and devel- digital tool procedurally, but can fail to un- students “transfer” their knowledge from oping expertise in using it, students should derstand conceptually the mathematical their interactions with an algebraic micro- be supported in making the connections to concepts and procedures that the tool was world – a constructionist learning environ- the maths. http://www.univie.ac.at/constructivism/journal/10/3/321.geraniou « 3 » Considering the issues discussed heart, root and purpose of algebra” (Mason & Kieran 1988). Moreover, students tend above, the research carried out in the Mi- 2005b: 2). to ignore interesting perspectives on math- Gen project1 has offered major gains in « 5 » In the digital era, where digital ematics while interacting and gaining more the understanding of students’ develop- technologies are increasingly making their and more experience with digital tools (e.g., ment of AWOT through their interaction appearance in the mathematics classroom, Gurtner 1992; Godwin & Beswetherick with exploratory learning environments. In students are faced with another transition, 2003). Saying that though, it is worth re- this article, we present data gathered from that of moving back and forth from PaP visiting the arguments in Seymour Papert 11–14-year-old students who worked on to interacting with digital tools. It is im- (1980) that students who interact with Logo bridging activities carefully designed to sup- perative, therefore, to investigate how and can visit mathematically rich areas that they port their transition from interacting with whether students “transfer” their knowl- would not have approached otherwise. Us- the MiGen constructionist learning envi- edge from their interactions with digital ing digital tools that are specially designed M ronment, namely eXpresser, to traditional tools to PaP activities. We put transfer in to support students’ difficulties and possi- S I PaP algebraic generalisation tasks. General quotes because it refers to different con- ble misconceptions on the topic of algebra, ON concluding remarks based on the initial structs for different communities. There for example, should scaffold the transition I T stages of our data analysis are discussed on is of course a lot of research on “transfer” C to formal algebra without rendering it im- U this transition to PaP tasks, as well as to al- (e.g., diSessa & Wagner 2005). Our view is possible for them to reach the mathematical R gebra in general. Some research outcomes aligned with King Beach (2003) who has bank of algebra. Besides “learning” the tool are shared regarding the successful integra- argued that the metaphor should be viewed and becoming experts in using it, students tion of the eXpresser microworld, as well as as transition instead of transfer, as crossing should then be able to make the connec- N CONST I the successful integration of similar digital boundaries from one location to another is tions to the mathematics behind their digit- tools, into the mathematics classroom. in fact a process of transition and he consid- al interactions. The challenge is to find ways NTS ers that people are the ones who move and to support students to make these connec- not knowledge or learning. Other authors tions. Theoretical background claim that transfer “entails re-use of knowl- « 7 » In the case of Logo, Gurtner PERIME edge, demonstrated and / or acquired in one (1992: 247) considered “the type of con- EX H « 4 » Major transitions in a learner’s situation (or class of situations), in a ‘new’ nections generally expected, and very sel- life are much studied: the transition from situation (or class of situations)” (diSessa dom observed, between Logo practice and EARC counting to number (Cobb 1987), from & Wagner 2005). Similarly, Robert
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