The language of INVALSI Mathematics tests of grade 8

INVALSI Mathematics tests are mainly "world problems," i.e., "a text (typically containing quantitative information) that describes a situation assumed familiar to the reader and poses a quantitative question, an answer to which can be derived by mathematical operations performed on the data provided in the text, or otherwise inferred" (Greer & al., 2002). World problems strive to make a mathematical question more relevant to the student (and they do not always manage to do so, given the ontological category of "[senseless] mathematical problems," cf. Zan 2017) but the addition of a natural language layer could introduce a further obstacle to comprehension if the language is not clear enough to the student. This is particular relevant from an inclusiveness perspective, since disadvantaged students will often not have Italian as mother tongue (be it a local dialect or a foreign language). On the other hand, it is clear the need that "an individual»" — and henceforth a student — "should be able to reason mathematically, understand mathematical proof and communicate in mathematical language." The language of mathematics, that is, has to be clear to students. Clearness of language has many different components, amongst which a relevant one is vocabulary: do students actually understand the meaning of all the words in the problem? De Mauro (2019) recognizes a "fundamental core" of 2000 Italian words, which any functional speaker of the language can be supposed to know; 3000 "high frequency" words, which occur often in the spoken or written language; 2500 "high availability" words, that albeit not appearing frequently are usually known to native speakers (e.g., "pepper"). Following De Mauro, we call these 7500 words the "basic" Italian vocabulary, to which we have to add 33,000 "common" words (and some 220,000 regional, poetical, specialistic or obsolete words). The Italian school system does not have an official set of words that have to be known at a given grade, but the introduction to De Mauro's fundamental core points out that it is not unreasonable to consider the basic 7500 word vocabulary as commonly known at grade 8. Toth (2021) analyses the word frequency of basic and common vocabulary in INVALSI Italian tests at grade 5, 8 and 10 and finds that word frequency does not explain the difficulty level: this is partially surprising, and Toth suggests that further inquiries are needed on students' vocabulary to understand the phenomenon. Finally, Ferrari (2021) hypothesizes that "part of learning difficulties in Mathematics are of linguistic origin". In this contribution, thus, we want to answer the question: does word fequency explain part of the difficulty level of Mathematics tests? We compare the top 5% and the bottom 5% of INVALSI mathematics items at grade 8 as ranked by number of correct answers to verify if there is a significant vocabulary difference. As source data we used the corpus of INVALSI Mathematics tests of grade 8, which we analysed in full in order to recognize the basic mathematical vocabulary. Collected and sorted words were processed by hand to remove proper names and mathematical non-words (e.g., ABCD in "the square ABCD") and later stemmed using the SnowBall (Porter 1980) library and finally post-processed by hand to merge stemmings. This shows that some technical words like "diagonal" or "perimeter" while not appearing in De Mauro "basic" vocabulary should be considered part of a "mathematical basic" vocabulary. Data show that 97% of the words used both in the top 5% and the bottom 5% belong to the basic vocabulary, with the fundamental core covering respectively 84% and 87%. This answers negatively the research question. Data show also that Mathematics tests make much more use of the basic vocabulary than the Italian tests: Toth (2021) reports a value around 60% for grade 8. We infer from these numbers that the Mathematics tests, at least in grade 8, have been built with a great linguistic care of avoiding words that could bothersome to some students. It remains to apply the same techniques to the other available tests to confirm this fact. An interesting offproducts of this work is the building of a "basic mathematical vocabulary" for each given grade.