Abstract: We offer a response to three themes arising from the commentators’ inquiries and critiques: (a) The epistemological compatibility of enactivism and conceptual metaphor theory; (b) the way enactive metaphorization works in the teaching and learning of mathematics, particularly in problem-posing and problem-solving activities; and (c) the nature of mathematical abstraction and its relation with enactive metaphorizing.

For over three-quarters of a century, the implicit learning theory underlying the curriculum and pedagogy of career and technical education has been behaviorism, but the emerging theory of constructivism may have implications for career and technical education practice in the future. Preparation of workers for entry into and advancement in the workplace of the next decade requires an educational program that provides not only job skills, as career and technical education did throughout the 1900s, but also higher order thinking, problem solving, and collaborative work skills. Classical behaviorist theory does not adequately address the latter kinds of learning, but constructivist theory may. Constructivist principles are examined in light of the fundamental requirements of career and technical education as we move into the new century with a new name for a redesigned profession. Of the three basic types of constructivism discussed, cognitive constructivism is most compatible with career and technical education. The authors recommend a more thorough examination of the relative efficacy of behaviorism and cognitive constructivism to serve as the learning theory on which to base career and technical education in the future. To embrace such a foundational change, leaders in the profession must re-think many of the fundamental assumptions underlying the mission, curriculum, and pedagogy of career and technical education. Perhaps such a rethinking is due.

In this study we compared the effects of small-group constructivist and explicit mathematics instruction in basic multiplication on low-achieving students’ performance and motivation. A total of 265 students (aged 8–11 years) from 13 general and 11 special elementary schools for students with learning and/or behavior disorders participated in the study. The experimental groups received 30 minutes of constructivist or explicit instruction in groups of 5 students twice weekly for 5 months. Pre- and posttests were conducted to compare the effects on students’ automaticity, problem-solving, strategy use, and motivation to the performance of a control group who followed the regular curriculum. Results showed that the math performance of students in the explicit instruction condition improved significantly more than that of students in the constructivist condition, and the performance of students in both experimental conditions improved significantly more than that of students in the control condition. Only a few effects on motivation were found. We therefore concluded that recent reforms in mathematics instruction requiring students to construct their own knowledge may not be effective for low-achieving students.

This paper deals with organisational complexity, seen from the perspective of its unfolding from global to local concerns. Historically, this unfolding has produced rigid social systems, where those in power positions have forced unfair constraints over the majorities at the local level, and often excluded them. There is a need to move towards flexible, fair, social systems, inclusive in character. This transformation requires an increasing appreciation of communication problems in society and the embodiment of effective social systems. This transformation is presented as a problem‐solving paradigm which requires social systems with capacity to create and produce their own meanings, with capacity to manage necessary structural couplings among existing social systems, thus making this management a heuristic to produce necessary social differentiation to overcome communication failures among existing self‐producing, operationally closed, social systems. A key construct used in this paper to practically produce this management is the viable system model, developed by Stafford Beer.

Key words: cybernetics, complexity theory, communications, social interaction.

Open peer commentary on the article “Examining the Role of Re-Presentation in Mathematical Problem Solving: An Application of Ernst von Glasersfeld’s Conceptual Analysis” by Victor V. Cifarelli & Volkan Sevim. Upshot: The value of Cifarelli & Sevim’s target article lies in the analysis of how reflective abstraction contributes to the description of mathematical learning through problem solving. The additional value of the article lies in its emphasis of some aspects of the learning process that goes beyond radical constructivist learning theory. I will look for common ground between the humanist philosophy of mathematics and radical constructivism. By doing so, I want to stress two converging elements: (i) the move away from traditionalist ontological positions and (ii) the central role of the students’ activity in the learning process.

Open peer commentary on the article “From Problem Solving to Problem Posing, and from Strategies to Laying Down a Path in Solving: Taking Varela’s Ideas to Mathematics Education Research” by Jérôme Proulx & Jean-François Maheux. Upshot: Problem solving should not be reduced to situated or localized activity since cognizers also draw on non-local resources that are not actually experienced but nevertheless impart on their situated cognition. A Varelian-inspired epistemology neglects this non-locality, which is a vital trait of human life.

Context: Mathematical problem solving is considered important in learning and teaching mathematics. In a recent study, Proulx and Maheux presented mathematical problem solving as a continuous dialectical process of small problem posing and solving instances in which the problem is continuously transformed, which they call problematizing. This problematizing conceptualization questions many current assumptions about students’ problem solving, for example, the use of heuristics and strategies. Problem: We address two aspects of this conceptualization: (a) how does problematizing evolve over time, and (b) how do the students’ problematizations interact? Method: In this study, we apply and further develop Proulx and Maheux’s enactivist perspective on problem solving. We answer our questions by applying micro-analysis to the mathematical problematizing of a group of students and, using Ingold’s pathways and meshwork as our framework, illustrate the lived practice of a group of students engaged in mathematical problem solving. Results: Our analysis illustrates how mathematical problematizing can be viewed as a complex, enmeshed and wayfaring journey, rather than a step-by-step process: in this enactive journey, smaller problems co-emerge from students’ interactions with one another and their environment. Implications: This research moves the focus on students’ mathematical problem solving to their actions, rather than strategies or direct links from problems to solutions, and provides a way to investigate, observe and value the lived practice of students’ mathematical problem solving. Constructivist content: Our work further strengthens the understanding of mathematical activities from an enactivist perspective where mathematical knowledge emerges from interaction between individual and environment.

Key words: Problem solving, problematizing, journey, heuristics, emergent, enactive, movement, rhythm, traces, artefacts.

Open peer commentary on the article “From Problem Solving to Problem Posing, and from Strategies to Laying Down a Path in Solving: Taking Varela’s Ideas to Mathematics Education Research” by Jérôme Proulx & Jean-François Maheux. Upshot: The target article prioritises the emergence of pupils’ mathematical ideas. Other constructivist approaches have focussed on how teachers might act to facilitate pupils’ mathematical activity. How might teachers be helped to use Varela’s insights into the uncontrollable emergence of ideas while teaching in the context of dominant intentional problem-solving approaches?

The purpose of this study was to learn whether Science Curriculum Improvement Study (SCIS) or more recent interpretations of the learning cycle could be used by teachers to engage students in social constructivist learning. To accomplish this purpose, two university researchers and six science teachers planned, implemented, and reflected upon instruction based on the reciprocal use of language and action within the learning cycle framework. The study examined teachers’ changing beliefs and practices as well as issues and problems that emerged. Discrepant case analysis was used to analyze the data, which included transcriptions of instruction, reflection sessions, and teacher and student interviews as well as copies of teachers’ written plans and instructional materials. In this paper, we present a case study of one teacher and profiles of five others. The case is organized chronologically and describes Martha, a high school physics teacher, in terms of her instruction and concerns at the beginning, middle, and end of the school year. Analysis revealed that several of Martha’s beliefs and practices gradually changed across the year. Martha initially expressed the positivistic view that the goal of science instmction was for students to arrive at scientifically acceptable conclusions. As Martha explored social constructivist teaching, she gave her students increasingly more opportunities to test and discuss their ideas during problem solving. Along with this change in practice, Martha experienced a tension between her efforts to give her students opportunities to develop their own understandings and her efforts to present scientific information. As Martha’ perspective changed, she became dissatisfied with her existing grading system. Like Martha, each of the other five teachers gave their students more opportunities to explore their own ideas and each experienced tensions in the process. We interpreted these findings within a social constructivist theoretical framework to suggest changes in the learning cycle.

Open peer commentary on the article “Examining the Role of Re-Presentation in Mathematical Problem Solving: An Application of Ernst von Glasersfeld’s Conceptual Analysis” by Victor V. Cifarelli & Volkan Sevim. Upshot: The key philosophical premise of von Glasersfeld’s radical constructivism is not necessary to the insightful conceptual analysis presented by Cifarelli and Sevim, which could benefit from abandoning it.