Therefore, the learning of mathematics
entails both the interpretation of mathematical signs and the construction of
mathematical meanings through communication with others. These interpretations
and these meanings are not constructed on the spot. Rather, they
evolve in a continuous manner, a manner resulting from the individual’s
exposure to a variety of closely interrelated experiences within different
mathematical, social, and physical contexts. In such experiences, multiple
semiotic systems combine (e.g., language, mathematical signs, and gestures)
to ground a continuous and evolving interpretation of mathematical
meanings. Since communication is possible only through signs (Peirce, CP
4.7), acts of communication are in themselves acts of interpretation. p. 183

Kanes observes
that natural language “atworst distorts the formation of ideas” and “at best it
makes the construction of concepts and communication possible” (p. 135).
By the same token, mathematical notations at worst distort mathematical
meanings if they are not interpreted in context and unfamiliar distinctions
are not taken into account; and at best they enable us to state explicitly what we want to state circumventing the ambiguities which the structure
of natural language is not equipped to avoid (Nagel, 1956). p. 184-5

Really? That’s the best they can hope for? This would seem to go against research into notations as tools for reasoning, which suggests that a good notation can actually significantly boost and form reasoning in a non-transparent way. See Muntersbjorn.

This is based on Peirce and sign-systems, but the author talks about intentions a lot. This is intriguing.

There is the Intentional Interpretant, which is a determination of the mind of the
utterer; the Effectual Interpretant, which is a determination of the mind of the
interpreter; and the Communicational Interpretant, or say the Cominterpretant,
which is a determination of that mind into which the minds of utterer and interpreter
have to be fused in order that any communication should take place. This mind may
be called the commens. [The commens] is all that is, and must be, well understood
between utterer and interpreter, at the outset, in order [for] the sign in question [to]
fulfill its function (Peirce, 1908, p. 478; emphasis in the original). p. 189

Peirce is, as ever, tricky to read, but the intentional interpretant seems to be based in the intentional actions of the utterer rather than any perceived intentions playing a part in interpretation. I ought to take more notice of this in my methodology, perhaps, in talking about the relationship between semiotics and pragmatics.

While mathematical concepts to be learned are static, interpreting processes
are dynamic. p. 194

Again – not sure about this – can they be said to be static? See Muntersbjorn.

For Wittgenstein (1991) language games are mediating processes for
meaning-making that impact how meanings are produced according to the
particular circumstances in which words and sentences are used in sociocultural
contexts. Language games go beyond the logic and structure of
the language, they account for the semantic aspects of linguistic expressions
(Harris, 1990). Likewise, Interpreting games are also mediating processes
of meaning-making that have an effect on how interim mathematical
meanings are produced according to the meaning-giving contexts used by
the teacher and interpreted by the students. Interpreting games account for
the semantic aspects of mathematical symbolizations. p. 199


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