Abstract:

In the Italian lower secondary school context, the International System of Units (SI) is often introduced as a static set of rules rather than a dynamic human invention. Students learn to convert units by mechanically shifting a decimal point along a scale. This "algorithmic shortcut" creates a significant pedagogical gap: students learn the how (the procedure) without ever grasping the why (the multiplicative structure of the metric system). Consequently, measurement is perceived as a "magical" manipulation of symbols rather than a physical and proportional reality.

This paper presents a teaching pathway designed to re-root the concept of measurement in its geometric and physical origins. Drawing on the principles of mathematical laboratory and enactive learning, the project aims to: deconstruct the "magic" of decimal shifts by making the multiplicative relationships between units explicit; utilize geometric theorems (specifically Thales’ Theorem) as practical tools for unit construction; provide a multisensory scaffold for learners to visualize the scaling process.

The pathway is structured into three iterative phases that mirror the historical development of measurement:
1. The Problem of the Remainder. Students begin by measuring classroom objects using a non-standard unit (a drinking straw). They inevitably encounter the "remainder" problem—objects that do not align with integer straw-lengths. This creates a "productive frustration" that motivates the conceptual need for submultiples.
2. Geometry in Action. To divide the straw into equal parts, students are introduced to a hands-on application of Thales’ Theorem. By constructing a device consisting of an auxiliary graduated scale and parallel lines, students can divide their straw into an arbitrary number of segments (e.g., 2, 5, or 10). This phase is crucial: it transforms Thales’ Theorem from an abstract geometric statement into a functional engineering tool. Choosing to divide the unit into ten parts then becomes a discussed choice of computational convenience, linking measurement to the base-10 number system.
3. The Equicalculator. In the final phase, students assemble the "Equicalculator," a paper-based manipulative device. Unlike a static conversion table, this tool requires students to physically move between scales, making the ratio between units a tangible experience. The device acts as a cognitive bridge, translating physical length into decimal notation.

The pathway was piloted in ten lower secondary classes with diverse learner profiles. Qualitative feedback from teachers and systematic observation revealed a marked increase in "metrological awareness." Notably, the project proved highly inclusive: the manipulative nature of the activities allowed students with SEN to participate fully, as the physical tools bypassed the cognitive load often associated with abstract numerical calculations.

Keywords:

Mathematics Education, Measurement, Thales’ Theorem, Manipulative Didactics, Lower Secondary School.