In a recent book titled Beyond Constructivism: A Models & Modeling Perspective on Mathematics Problem Solving, Learning & Teaching (Lesh & Doerr, 2003a), the concluding chapter describes a number of specific ways that a models and modeling perspective moves significantly beyond the implications that can be drawn from constructivist theories in the context of issues that are priorities to address for teachers, curriculum developers, or program designers. In that chapter (Lesh & Doerr, 2003b), the following topics were treated as cross-cutting themes: (a) the nature of reality, (b) the nature of mathematical knowledge, (c) the nature of the development of children’s knowledge, (d) the mechanisms that drive that development, (e) the relationship of context and generalizability, (f) problem solving, and (g) teachers’ knowledge and the kinds of teaching and learning situations that contribute to the development of children’s knowledge. In this article, we organize our comments directly around the preceding topics and describe how a models and modeling perspective provides alternative ways of thinking about mathematics teaching and learning that enable teachers, researchers and others to produce useful and sharable conceptual tools that have powerful implications in the context of decision-making issues that are of priority to practitioners.

Open peer commentary on the article “A Cybernetic Approach to Contextual Teaching and Learning��� by Philip Baron. Upshot: The target article does an excellent job of describing the theoretical basis of a cybernetic approach to teaching at the university level. In addition, it also describes changes that must occur in the teacher’s perspective and attitude. Yet I am left wondering how any of this can actually happen.

My purpose in this paper is to illustrate the way in which an enactivist methodological approach guided me as I conducted a two-case longitudinal study where the learning of algebra was explored in different contexts throughout time. Three groups of students in two different schools in the city of Puebla, Mexico, were followed from the last year of primary school (Year 6) to the second year of their secondary education (Year 8). Learning was characterised as the ongoing structural change that allows individuals or groups to act effectively in a changing environment [Maturana (GAIA, a way of knowing: political implications of the new biology. Lindisfarne, New York, pp 65–82, 1987)]. An enactivist methodology, which revolves around the idea of research being a form of learning [Reid (Proceedings of the 20th conference of the international group for the psychology of mathematics education. PME, Valencia, pp 203–209, 1996)], implied that, as I carried out the study, what I was doing was learning about how people learn algebra. My initial questions arose from my experiences with the teaching and learning of mathematics, which gave me a sense of the complexity involved in the learning processes. Later, as I became immersed in the process of investigation of the teaching and learning of algebra, my conceptions continually evolved. The meaning of the phrase ‘algebraic learning’, which I used as a way of maintaining a wide perspective that allowed me to explore the events in the classroom in a complex way, arose as I engaged with enactivist ideas about learning, with the research literature on the learning of algebra, in conversations with people and in interactions with the participants of my project. I went through a process of continual development and change which I describe in this paper.

Key words: enactivism, algebraic learning

Excerpt: In a typical study of students’ conceptions and conceptual change, researchers analyze what a student does or says in a classroom or in an interview and recognizes ideas that match or do not match their own understanding of the topic. Attributing the perspective they recognize in the student, those studies support the idea that a conception is the way by means of which an individual intrinsically conceives (of) a given phenomenon. They then hypothesize the existence of some mental structures that can be theoretically and objectively re-constructed based on what is observed in a student’s performance. Thus, researchers studying conceptions commonly assume that the observer and the observed are separate entities. However, even in the most theoretical and hardest of all sciences, physics, the independence of the measured object and the measuring subject is not taken for granted: Light, for example, will present itself as waves or as particles depending on how we examine it. The artificial sense of separation from the object(s) of study found in many accounts on students’ conceptions makes irrelevant the relationship that exists between the observer and the observed: an interdependence and co-emergence of the observer and the observed. This tight relation exists because each participant not only reacts upon what others say but also acts upon the reactions that his/her own actions give rise to. With this situation come epistemological, practical, and ethical implications for those researching in mathematics and science education. Positing or questioning the existence of an objective reality mediates how we accept or reject another human being and the worldviews s/he develops. It provides a rationale that guides our actions. This is especially important when it comes to teaching and learning at a time where the ability to deal with the plurality and diversity of human culture have emerged as significant referents for our social behavior.

This paper offers a critique of, and an approach to, education that is heavily influenced by Heinz von Foerster, especially the idea that knowing takes place through our cognition organizing itself through cycles of perceiving and acting. Von Foerster’s approach was to make the manner of teaching embody the ideas which he wanted to teach. The author describes elements of constructivist second-order cybernetic thinking that can serve as tools for teaching and learning.

Open peer commentary on the article “A Cybernetic Approach to Contextual Teaching and Learning��� by Philip Baron. Upshot: I consider implications of Baron’s article on change in university education. In particular, I address the problem of why change happens with difficulty and how the principles and practices of second-order cybernetics that Baron discusses are applicable beyond South Africa to a wide range of situations.

Open peer commentary on the article “Heterarchical Reflexive Conversational Teaching and Learning as a Vehicle for Ethical Engineering Curriculum Design” by Philip Baron. Upshot: Philip Baron focuses on changing university curricula in South Africa to enable students to succeed who do not share the culture, expectations, and experience of their teachers. With increasing need and desire for more education worldwide, his article is relevant to university education in all countries, especially in those with underserved populations. This commentary focuses on the factors Baron describes which can be generalized to all university education.

Review of The Content of Science. A Constructivist Approach to its Teaching and Learning, edited by Peter Fensham, Richard Gunstone and Richard White, Farmer Press, London, 1994.

Open peer commentary on the article “Constructing Constructivism” by Hugh Gash. Upshot: Radical constructivism is explicitly discussed in Gash’s target article outlining “stages” or types of constructivism. The stages contextualize radical constructivism in a series of research phases involving a number of domains using a variety of approaches. The target article begs the query: “just how radical are many constructivist approaches in teaching and learning?”

The Next Generation Science Standards (NGSS) [Achieve, Inc. [2013]] represent a broad consensus that teaching and learning expectations must change. Rather than memorizing and reciting information, students are now expected to engage in science practices to develop a deep understanding of core science ideas. While we want to share in the optimism about NGSS, the standards are not a silver bullet for transforming science classrooms. They are, instead, another reform document designed to suggest opportunities for students to actively engage in knowledge construction themselves – to be doers of science, rather than receivers of facts. A foundational contradiction underlies these efforts – while we want students to do science, we seem to mean that students should mimic practices others have selected as important to learn, and content others have selected as foundational. As a result, students are rarely positioned with epistemic agency: the power to shape the knowledge production and practices of a community [Stroupe [2014] Science Education 98:487–516]. We argue that unless the field tackles significant questions around precisely how students can be active agents in knowledge construction, we will likely continue to implement learning environments that position students as receivers of scientific facts and practices, even as classrooms adopt NGSS. In this conceptual analysis article, we unpack the construct of “epistemic agency” and its relationship to the NGSS, using a vignette to illustrate how students are typically positioned in researcher-developed curricula. The vignette, which describes a seventh-grade class exploring which of two lakes is more at risk for invasion by the spiny water flea, provides an exemplar of what we take to be a loose consensus about learning environments consistent with the NGSS. However, when we look beneath the surface of the consensus, the vignette reveals contradictions and unresolved issues around epistemic agency.