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Description
Studies with first-semester mathematics students indicate that after leaving highschool, they understand the subject primarily as a set of calculation rules, i.e. as a ‘toolbox’ (Törner & Grigutsch, 1994; Girnat & Hascher, 2021). However, mathematics as taught at university, as well as any scientific approach to mathematics, should rather be understood as an interconnected system of concepts expressed linguistically in a rich terminology. Learning mathematics therefore involves learning mathematical language and understanding how mathematical concepts are interrelated (cf. Riccomini et al., 2015).
Students are thus in need of lexical/terminological tools that support them in two ways: (a) to understand basic mathematical concepts and their relations (cognitive needs, in terms of the lexicographic function theory, cf. Tarp, 2008; Fuertes-Olivera & Tarp, 2014); and (b) to get acquainted with the phraseology needed for understanding and producing texts about mathematical topics (communicative needs: receptive and productive).
We analysed a number of sample entries from sources students may typically resort to: Wikipedia or other (online) encyclopedias, AI generated summaries of answers to keyword-based questions as well as (online) learning material for first years. We checked, a.o., the German terms 'Gleichung' (equation), 'Äquivalenz' (equivalence), 'Lösung' (solution) and 'Lösungsmenge' (solution set), all of which appear early in the first semester introductory lecture.
Wikipedia and similar tools provide substantial conceptual knowledge, but they require a considerable effort to extract relational concept data from the rather complex article texts (cf. Kruse, 2025, pp. 81-83). The analysed online material either contains wikipedia-like mini-essays or it focuses on computing examples. None of these make phraseology explicit.
We assume that formulating meaning explanations and conceptual relations explicitly may help students to master (and to memorize) mathematical concepts (a sort of 'learning by doing'). In an experimental course (cf. Kruse et al., 2024) administered since 2023, we thus invited participants of the first year introductory lectures to mathematics to restructure elements of the lecture contents by designing their own dictionary entries for the mathematical terms learned in the lecture; we also asked them to produce small ontological networks. To ensure that they would adhere to a general guideline, we gave a lexicographical mini-introduction to the microstructural items we are interested in, e.g. definitions, possibly paraphrases of definitions for school children, synonymy, hyp(er)onymy, related concepts that are part of the subdomain in question as well as collocations. We equally introduced the students to concept maps and ways to represent relations.
An analysis of a subset of the students' submissions (96 submissions) shows that their dictionary entries contain more types of microstructural items than their networks. All dictionary entries and almost 80 % of the networks contain definitions. Synonyms and Hyponyms are present in ca. 80 % of both types of submissions, while collocations only show up in dictionary entries and mathematical notations (symbols etc.) almost exclusively in the network graphs.
Overall, the students are quite successful in the exercise: they provide more conceptual relations than most online learning material. It seems, however, that they should need to be made more aware of mathematical phraseology. A combination of particularized information in the style of dictionary entries and of small network-like representations may be a good way to provide them with their own learning material. In the medium term, a study of the effectiveness of the approach would be needed.