Design heuristic (6 September 2026):
Don't begin by deciding which mathematical content cLogos should teach. Begin by asking what relations are present in the mathematical activity, how those relations can become objects of attention, and what the representational environment enables learners to do with them.

Introduction

Logo is a programming language that invites to ask: "What kind of thinking do we want this feature to cultivate?"

This page is intended as a place for proposals of educational activities, both on and off the computer, within a cLogos framework constituted by the above principle.

It rests on the insight that mathematics may be understood as a discipline that playfully yet systematically explores and examines the relations that constitute tangible and intangible patterns and structures by making sets of parts distinguishable from other sets: numbers are related; operations relate quantities; equations express relations; functions organise dependencies; geometrical structures express relations; algebra transforms relations while preserving others; proof establishes relations among propositions.

From this perspective, it appears appropriate that an educational intervention intended to offer learning opportunities with mathematical content should potentially afford relational insight alongside further intended learning targets, such as developing concepts of cardinality or ordinality. The focus is therefore not on preparatory activities intended to elevate arithmetical thinking to algebraic thinking. It is on a relational orientation to mathematical activity.

In Papert's constructionist position, learners build things that are meaningful to them, discovering and enacting powerful ideas through their activities. This foundational principle resonates with various educational and psychodevelopmental theories and approaches. Papert and his colleagues arrived, through their educational work with Turtle geometry, at three principles of "an appropriable mathematics" (Papert 1980, p. 54):

First, there was the continuity principle: The mathematics must be continuous with well-established personal knowledge from which it can inherit a sense of warmth and value as well as "cognitive" competence. Then there was the power principle: It must empower the learner to perform personally meaningful projects that could not be done without it. Finally there was a principle of cultural resonance: the topic must make sense in terms of a larger social context.

This also resonates with contemporary positions in mathematics education. The power principle helps to clarify Papert's relative term powerful idea: an idea becomes powerful for the learner insofar as it empowers them to build something meaningful. Here, meaningfulness or significance appears to be more than merely personal, in conjunction with the principle of cultural resonance. Yet the powerful idea also feels powerful while being discovered, explored, applied, and thereby appropriated. The memory associated with such an idea may sustain a warm feeling while the learner moves on to discover new powerful ideas within the continuity principle. At the same time, the experienced "magnitude" of the power of an appropriated idea may diminish. Here, particularly the Vygotskian notion of mutually travelling through the Zone of Proximal Development, relating an individual's level of actual development to its level of potential development, is reflected.

In cLogos, it is therefore posited that, beyond the potential to build things that are meaningful for particular individual reasons, a deeper and more concrete design issue is whether the environment allows learners to construct, inspect, alter, and reason about relations during this personal process of appropriation. Relations can thereby become things that learners encounter, make, vary, observe, compare, and eventually represent symbolically.

Through such activities, a foundational concern of mathematics as a discipline may become a coherent frame for possible individual discoveries. As a possible consequence, the gradual refinement of relational observations, mathematical analyses, and syntheses may foster generalisations across disciplines and contexts, connect real-world contexts with mathematical thinking, and contribute to an integrated image of mathematics itself.