The career of measurement

Measuring Beyond the Standard: Informal Measurement Systems as Cognitive Technologies

This paper explores the role of measurement as a cognitive technology across human history, emphasizing the coexistence of formal and informal measurement systems. While standardized systems dominate contemporary culture and are well documented across large-scale societies of the past, this manuscript highlights the less explored domain of informal measurement practices that have been integral to daily life from the past to the present. Through the examination of body-based measurement systems and proportional heuristics, we demonstrate how these informal strategies were not merely precursors to formal standards but essential adaptive tools for solving everyday problems. Often, these informal solutions come with practical advantages. This manuscript calls for a broader recognition of their significance in cultural and technological evolution.

Creating ad hoc graphical representations of number

The ability to communicate about exact number is critical to many modern human practices spanning science, industry, and politics. Although some early numeral systems used 1-to-1 correspondence (e.g., 'IIII' to represent 4), most systems provide compact representations via more arbitrary conventions (e.g., '7' and 'VII'). When people are unable to rely on conventional numerals, however, what strategies do they initially use to communicate number? Across three experiments, participants used pictures to communicate about visual arrays of objects containing 1-16 items, either by producing freehand drawings or combining sets of visual tokens. We analyzed how the pictures they produced varied as a function of communicative need (Experiment 1), spatial regularities in the arrays (Experiment 2), and visual properties of tokens (Experiment 3). In Experiment 1, we found that participants often expressed number in the form of 1-to-1 representations, but sometimes also exploited the configuration of sets. In Experiment 2, this strategy of using configural cues was exaggerated when sets were especially large, and when the cues were predictably correlated with number. Finally, in Experiment 3, participants readily adopted salient numerical features of objects (e.g., four-leaf clover) and generally combined them in a cumulative-additive manner. Taken together, these findings corroborate historical evidence that humans exploit correlates of number in the external environment - such as shape, configural cues, or 1-to-1 correspondence - as the basis for innovating more abstract number representations.

Unlocking the Power of Gesture: Using Movement-Based Instruction to Improve First Grade Children's Spatial Unit Misconceptions

Gestures are hand movements that are produced simultaneously with spoken language and can supplement it by representing semantic information, emphasizing important points, or showing spatial locations and relations. Gestures' specific features make them a promising tool to improve spatial thinking. Yet, there is recent work showing that not all learners benefit equally from gesture instruction and that this may be driven, in part, by children's difficulty understanding what an instructor's gesture is intended to represent. The current study directly compares instruction with gestures to instruction with plastic unit chips (Action) in a linear measurement learning paradigm aimed at teaching children the concept of spatial units. Some children performed only one type of movement, and some children performed both: Action-then-Gesture [AG] or Gesture-then-Action [GA]. Children learned most from the Gesture-then-Action [GA] and Action only [A] training conditions. After controlling for initial differences in learning, the gesture-then-action condition outperformed all three other training conditions on a transfer task. While gesture is cognitively challenging for some learners, that challenge may be desirable-immediately following gesture with a concrete representation to clarify that gesture's meaning is an especially effective way to unlock the power of this spatial tool and lead to deep, generalizable learning.

Embodying measurement

Measuring with body parts is a handy and persistent cross-cultural phenomenon.

Body-based units of measure in cultural evolution

Measurement systems are important drivers of cultural and technological evolution. However, the evolution of measurement is still insufficiently understood. Many early standardized measurement systems evolved from body-based units of measure, such as the cubit and fathom, but researchers have rarely studied how or why body-based measurement has been used. We documented body-based units of measure in 186 cultures, illustrating how body-based measurement is an activity common to cultures around the world. Here, we describe the cultural and technological domains these units are used in. We argue that body-based units have had, and may still have, advantages over standardized systems, such as in the design of ergonomic technologies. This helps explain the persistence of body-based measurement centuries after the first standardized measurement systems emerged.

The puzzle of ideography

An ideography is a general-purpose code made of pictures that do not encode language, which can be used autonomously - not just as a mnemonic prop - to encode information on a broad range of topics. Why are viable ideographies so hard to find? I contend that self-sufficient graphic codes need to be narrowly specialized. Writing systems are only an apparent exception: At their core, they are notations of a spoken language. Even if they also encode nonlinguistic information, they are useless to someone who lacks linguistic competence in the encoded language or a related one. The versatility of writing is thus vicarious: Writing borrows it from spoken language. Why is it so difficult to build a fully generalist graphic code? The most widespread answer points to a learnability problem. We possess specialized cognitive resources for learning spoken language, but lack them for graphic codes. I argue in favor of a different account: What is difficult about graphic codes is not so much learning or teaching them as getting every user to learn and teach the same code. This standardization problem does not affect spoken or signed languages as much. Those are based on cheap and transient signals, allowing for easy online repairing of miscommunication, and require face-to-face interactions where the advantages of common ground are maximized. Graphic codes lack these advantages, which makes them smaller in size and more specialized.

Exact Number Concepts Are Limited to the Verbal Count Range

Previous findings suggest that mentally representing exact numbers larger than four depends on a verbal count routine (e.g., "one, two, three . . ."). However, these findings are controversial because they rely on comparisons across radically different languages and cultures. We tested the role of language in number concepts within a single population-the Tsimane' of Bolivia-in which knowledge of number words varies across individual adults. We used a novel data-analysis model to quantify the point at which participants (N = 30) switched from exact to approximate number representations during a simple numerical matching task. The results show that these behavioral switch points were bounded by participants' verbal count ranges; their representations of exact cardinalities were limited to the number words they knew. Beyond that range, they resorted to numerical approximation. These results resolve competing accounts of previous findings and provide unambiguous evidence that large exact number concepts are enabled by language.

Metaphor and the Philosophical Implications of Embodied Mathematics

Embodied approaches to cognition see abstract thought and language as grounded in interactions between mind, body, and world. A particularly important challenge for embodied approaches to cognition is mathematics, perhaps the most abstract domain of human knowledge. Conceptual metaphor theory, a branch of cognitive linguistics, describes how abstract mathematical concepts are grounded in concrete physical representations. In this paper, we consider the implications of this research for the metaphysics and epistemology of mathematics. In the case of metaphysics, we argue that embodied mathematics is neutral in the sense of being compatible with all existing accounts of what mathematical entities really are. However, embodied mathematics may be able to revive an older position known as psychologism and overcome the difficulties it faces. In the case of epistemology, we argue that the evidence collected in the embodied mathematics literature is inconclusive: It does not show that abstract mathematical thinking is constituted by metaphor; it may simply show that abstract thinking is facilitated by metaphor. Our arguments suggest that closer interaction between the philosophy and cognitive science of mathematics could yield a more precise, empirically informed account of what mathematics is and how we come to have knowledge of it.