The first part of this paper examines the differences between Piaget’s constructivism, what Papert refers to as“constructionism,” and the socio-constructivist approach as portrayed by Vygotsky. All these views are developmental, and they share the notion that people actively contribute to the construction of their knowledge, by transforming their world. Yet the views also differ, each highlighting on some aspects of how children learn and grow, while leaving other questions unanswered. Attempts at integrating these views [learning through experience, through media, and through others] helps shed light on how people of different ages and venues come to make sense of their experience, and find their place – and voice – in the world. Tools, media, and cutural artifacts are the tangible forms, or mediational means, through which we make sense of our world and negociate meaning with others. In the second part of this paper, I speak to the articulations between make-believe activities and creative symbol-use as a guiding connection to rethink the aims of representations. Simulacrum and simulation, I show, play a key role besides language in helping children ground and mediate their experience in new ways. From computer-based microworlds for constructive learning (Papert’s turtle geometry, TERC’s body-syntonic graphing), to social virtual environments (MUDing). In each case, I discuss the roles of symbolic recreation, and imaginary projection (people’s abilities to build and dwell in their creations) as two powerful heuristic to keep in touch with situations, to bring what’s unknown to mind’s reach, and to explore risky ideas on safe grounds. I draw implications for education.

Context: There seem to be relatively few sustained implementations of microworlds in mathematics instruction. Problem: We explore the roles of and demands on university instructors to create an environment that supports students’ constructionist learning experiences as they design, program, and use interactive environments (i.e., microworlds) for doing mathematics. Method: We draw on the experiences of instructors in programming-based courses implemented since 2001 at Brock University, Canada, as a case study, and use Ruthven’s model on the professional adaptation of classroom practice with technology to guide our analysis of these experiences. Results: We describe how, in adapting to a design of empowering students to engage in programming for authentic mathematical explorations, instructors adopt characteristics of constructionist teaching that, nevertheless, demand expertise, a shift in traditional roles, and time from instructors. Implications: The results contribute to our understanding of roles of and demands on “ordinary” instructors in classrooms, who aim to create rich environments for supporting students’ constructionist learning experiences of computational thinking for mathematics. Constructivist content: The teaching approach aligns with Papert’s constructionism: a constructivist learning theory, but also a pedagogical paradigm. However, the approach presented has two salient characteristics: it is a university-level constructionist implementation, and it is a sustained long-term authentic classroom implementation. The focus is on the roles of and demands on instructors in that kind of implementation. Through the analysis using Ruthven’s work, we enrich our understanding of constructionist teaching features.

Key words: Computational thinking, constructionism, mathematics, programming, microworlds, university, teachers.

Context: Computational thinking denotes the thinking processes needed to solve problems in the way computer scientists would. It is seen as an ability that is important for everybody in a society that is rapidly changing due to applications of computational technologies. More and more countries are integrating computational thinking into their school curricula. Problem: There is a need for more effective learning environments and learning methods to teach computational thinking principles to children of all ages. The constructionist approach seems to be promising since it focuses on developing thinking skills. Method: We extract and discuss insights from the target articles. Results: There are several learning initiatives and curricula that successfully apply constructionist learning to acquiring computational thinking skills. Implications: Computational thinking as a subject at school presents a chance to bring more constructionist learning to schools.

Key words: Computational thinking, concept building, constructionist learning, deconstructionism, learning environment, makerspace, microworlds, short tasks

Context: In the digital era, it is important to investigate the potential impact of digital technologies in education and how such tools can be successfully integrated into the mathematics classroom. Similarly to many others in the constructionism community, we have been inspired by the idea set out originally by Papert of providing students with appropriate “vehicles” for developing “Mathematical Ways of Thinking.” Problem: A crucial issue regarding the design of digital tools as vehicles is that of “transfer” or “bridging” i.e., what mathematical knowledge is transferred from students’ interactions with such tools to other activities such as when they are doing “paper-and-pencil” mathematics, undertaking traditional exam papers or in other formal and informal settings. Method: Through the lens of a framework for algebraic ways of thinking, this article analyses data gathered as part of the MiGen project from studies aiming at investigating ways to build bridges to formal algebra. Results: The analysis supports the 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.

Open peer commentary on the article “Problem Posing and Creativity in Elementary-School Mathematics” by E. Paul Goldenberg. Abstract: Supporting the inborn curiosity of children is the motivation for our involvement in developing novel curricula, textbooks and microworlds. Our main goal of implementing the constructionism as a fundamental educational strategy is to keep the students “as question marks,” i.e., to encourage them to pose questions, to make experiments, to invent their own problems. We strongly support the ideas behind Goldenberg’s experience in learning environments, generating curiosity and creative engagement (§15. As an extension of the ideas in §54 we propose a metaphor to visualize how programming can be “repurposed” to wrap the math in an attractive, yet educationally effective way.

In recent years, the social interaction involved in the construction of the mathematics of children has been brought into the foreground in order to sptcify its constructive aspects (Bauersfeld, 1988; Yackel, etc., 1990). One of the basic goals in our current teaching experiment is to analyse such social interaction in the context of children working on fractions in computer microworlds. In our analyses, however, we have found that social interaction does not provide a full account of children’s mathematical interaction. Children’s mathematical interaction also includes enactment or potential enactment of their operative mathematical schc:mes. Therefore, we conduct our analyses of children’s social interaction in the context of their mathematical interaction in our computer microworlds. We interpret and contrast the children’s mathematical interaction from the points of view of radical constructivism and of Soviet activity theory. We challenge what we believe is a common interpretation of learning in radical constructivism by those who approach learning from a social-cultural point of view. Renshaw (1992), for example, states that ‘In promulgating an active. constructive and creative view of learning,… the constructivists painted the learner in close-up as a solo-pia yer, a lone scientist, a solitary observer, a me:ming maker in a vacuum.’ In Renshaw’s interpretation, learning is viewed as being synonymous with construction in the absence of social interaction with other human beings. To those in mathematics education who use the teaching experiment methodology, this view of learning has always seemed strange because we emphasize social interaction as a primary means of engendering learning and of building models of children’s mathematical knowledge (Cobb and Steffe, 1983; Steffe, 1993).