The Algebra Project (AP) is a grassroots initiative organized to teach algebra to adolescents and developed by civil rights activist and educator Bob Moses (1935-2021). The AP defines mathematical literacy as the ability to wield words and symbols to solve math problems, that is, to speak the languages of mathematics fluently (www.algebra.org). AP pedagogy is based on a five-step learning progression that starts (1) from a shared experience that (2) students represent in drawings before (3) comparing their experiences and drawing with each other. From these conversations (People Talk), (4) students learn to speak more regimented discourses (Feature Talk), which make implicit mathematical features of their experiences explicit and precise. In the last step, (5) Feature Talk expressions are translated into mathematical symbols. According to Moses et al. (1989), "The purpose of the five steps is to avert student frustration in 'the game of signs,' or the misapprehension that mathematics is the manipulation of a collection of mysterious symbols and signs" (433). The enculturation theory of Menary (2015) and others suggests that dynamic group interactions with cognitive tools, including diagrams and symbols, transform how we think. When people are denied access to quality math education, they are prevented from developing those cognitive capacities the mathematically literate benefit from, but may take for granted, such as reasoning about functions and variables, as well as properties of functions and relations between variables, etc., using the representational tools and strategies of mathematics.

Addressing the issue of whether a scientific work, especially one handed down from an ancient past, uses an artificial language confronts us with the following question: do we truly see the original language created and utilized by the author in an edition of the work? In my presentation, I will discuss why the editors of a work play a crucial role in shaping our understanding of the language of the text. For this, I concentrate on different editions of Sea Mirror of Circle Measurements (測圓海鏡 Ce yuan hai jing, 1248), the first mathematical book known to us which reflects the use of the method of computing with unknowns to establish equations in ancient China. My study compares Li Rui’s 李銳 (1769–1817) collated edition (1798), which had a great influence on contemporary and later scholars and is the foundational text for future study, with other editions before his. By focusing on the differences in the representation of numbers and equations in these various editions, as a hint, my study reveals that different editions of a work reflect different artificial languages, each containing the author's/editor's own conception. Therefore, when studying artificial language, one must first conduct textual criticism in order to clarify the foundation on which our discussion of these artificial languages is based.

In this paper I will present the abacus representation as a model for representing numeral systems in different modalities. For example, systems of finger counting, verbal and symbolic numeral systems, can all be represented by the same abacus, differing only in the ways the rows and columns are interpreted. Thus, this model can serve to bridge seemingly incompatible systems of numeration by making explicit the structural commonalities and the individual differences between them.