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We present a replication study of one of the most renowned seminal works in the field of mathematics education, i.e., the Fischbein et al. (1985) study on implicit models of multiplication and division in solving word problems. The original instrument was administered to a sample of 902 grade 7 students, proposing some methodological variations. The discussion focuses on four items involving partitive division of integer numbers, and results are compared to the original study: the main results of the original study are confirmed, but we contribute further with considerations on the statistical significance of differences observed in item difficulty. Finally, through our new methodological approach, we broaden the results of the original study by analysing students’ behaviour on the basis of their general arithmetical problem-solving ability. Students with medium-level ability are more influenced by the primitive model than those with lower ability.

Luigi Cremona is considered the founder of the Italian school of algebraic geometry. He formed a group of students of great value, very active in scientific research. Examining the letters from Eugenio Bertini, Ettore Caporali, and Riccardo De Paolis to Cremona preserved in the archive of the Istituto Mazziniano in Genoa, we have reconstructed their biographies, careers, studies, and relationships with their teacher. They had the merit of cultivating the scientific innovations of the period and passing them on to the subsequent generations.

Il fenomeno didattico della multiculturalità è un tema in molti casi discusso, vissuto però sovente in ambito scolastico come problema, non come risorsa. Parecchi sono infatti gli insegnanti che, sperimentando quotidianamente la realtà multiculturale nelle loro classi, manifestano un disagio, spesso associato ad una successiva richiesta di formazione; in molti casi disattesa o non adeguata ai loro bisogni. La ricerca nazionale in campo educativo, rivolta allo studio di questi fenomeni, è infatti a uno stadio quasi embrionale: non molte sono le riflessioni pedagogico-didattiche dirette ad insegnanti e formatori che operano in contesti complessi come quelli qui presi in esame, ancor meno le indagini teorico/sperimentali più strettamente disciplinari. Il presente contributo, attraverso l’esplicitazione del quadro teorico della Trasposizione Culturale (Mellone et al. 2019), si propone di offrire al lettore un possibile approfondimento su queste tematiche attraverso una lente storico- didattico focalizzata, nello specifico, sull'insegnamento/apprendimento della Matematica. Uno degli obiettivi che il contributo intende raggiungere è favorire nel lettore una riflessione autonoma sulla propria strategia di insegnamento e formazione di stampo interculturale, specie nel particolare ambito disciplinare discusso, anche grazie a testimonianze di esperienze laboratoriali, qui raccontate.

Il presente lavoro di ricerca e sperimentazione riguarda l’introduzione del concetto di numero naturale e punta alla costruzione di una prima idea dei suoi diversi aspetti, in una prospettiva di curricolo verticale. Attraverso la sinergia di tre artefatti e con il gioco si è favorita la costruzione del ‘senso’ di numero naturale, nei suoi tre aspetti cardinale, ordinale e ricorsivo. Il presente percorso, progettato sulla base della teoria della Mediazione Semiotica e della sinergia tra artefatti, ha avuto come obiettivo principale quello di costruire il concetto in senso globale di numero naturale attraverso la realizzazione della linea dei numeri naturali passando da un’idea intuitiva legata alle quantità alla concettualizzazione rigorosa del numero come ‘qualità’. This research and experimental project concerns the introduction of the concept of natural number and aims to construct an initial idea of its various aspects, in a vertical curriculum perspective. Through the synergy of three artifacts and using the methodology of play, the construction of the ‘sense’ of natural number, in its three aspects cardinal, ordinal and recursive, was fostered. The main objective of this path of research and experimentation, designed on the basis of the theory of Semiotic Mediation and the synergy between artifacts, is the construction of global sense of natural number through the creation of the natural number line, moving from an intuitive idea linked to quantities to the rigorous conceptualization of number as ‘quality’.

In this contribution, we discuss phenomenological research related to a pilot study carried out by the Consortium of the MaTeK Horizon 2020 project during the 2020–21 academic year. The research aims to analyse the effects of the Covid-19 pandemic on the use of teaching resources in mathematics in five coun- tries. A questionnaire made of seven questions was administered to a data sample made of teachers of all grades. The answers coming from the questionnaire were quantitatively and qualitatively analysed. Closed-ended questions were analysed by using a clustering methodology called k-means. Open-ended questions were qualitatively analysed. The results show that almost all the teachers are aware of the impact of the Covid-19 pandemic on education. However, their perception of how the pandemic situation changed the use of educational resources mostly appears linked to cultural factors, age, teaching experiences and school grades of the teachers.

The issue of students’ difficulties in processing operations with percentages has been addressed in several international research studies from a qualitative perspective. In this study, we analyze students’ difficulties on this topic, focusing on the transition from middle school to high school with a mixed methods research design. We focus on students’ responses in a specific task belonging to the Italian large-scale assessment analyzed through the Rasch model, and we deepen the task analysis thanks to interviews, which enlightened image schemas and metaphors underlying students’ reasoning. From the qualitative point of view, the Rasch model shows that students’ difficulties in dealing with percentages is a macrophenomenon that involves the higher levels of competences. From the qualitative point of view, the metaphoric approach outlines the image schemas that foster the correct conceptualization of percentage and those that hinder their correct learning and can be one of the possible causes of the emerging aforementioned macrophenomenon.

We present an external close replication of the 1985 Fischbein et al. study about intuitive models of multiplication and division. We administered two batteries of mathematics items developed in the original study, via a spiralling process, to a quota sample of 903 students attending grade 7. Compared with the analytic strategy based on the count of correct answers employed in the original research, our study goes a step further as we propose a methodological approach that guarantees measurement invariance, thus allowing for the direct comparison of different groups of students and/or items. The advantages of Rasch estimates compared to percentages of correct answers over the total are critically discussed to show why the former should be considered as more robust than the latter.

For a category B with finite products, we first characterize pseudofunctors from B to Cat whose corresponding opfibration is cartesian monoidal. Among those, we then characterize the ones which extend to pseudofunctors from internal groups to 2-groups. If B is additive, this is the case precisely when the corresponding opfibration has groupoidal fibres.

I corsi di matematica del primo anno in architettura mirano a fornire agli studenti il ​​linguaggio scientifico, aumentare il pensiero spaziale, creativo, insieme alla capacità di riconoscere e creare forme e consentire un uso consapevole dei software di progettazione. Tuttavia, i futuri architetti italiani considerano spesso questi corsi marginali nella loro formazione (Pagano & Tedeschini Lalli, 2005). Per superare questa criticità, abbiamo sviluppato un’officina sperimentale di 4 ore seguendo la metodologia DBR (Brown ,1992; Barab & Squire, 2004). Il contenuto didattico è la parabola, familiare agli studenti del primo anno, riscoperta con attività volte a svelare il legame tra forma geometrica e descrizione analitica. Gli studenti piegano (su carta) l'inviluppo di una parabola, ne verificano la proprietà di riflessione, "scoprendo" così la descrizione algebrica della curva e, infine, la applicano ad un problema di luminosità in architettura. Dopo la fase di progettazione, il laboratorio è stato sperimentato con due gruppi di 75 studenti del primo anno del Politecnico di Torino e dell'Università Roma Tre nell'a.a. 2021-22. Il confronto delle nostre note sul campo con l'analisi comparativa delle risposte di un questionario finale ci ha fornito risultati incoraggianti sull'apprendimento concettuale e sul coinvolgimento, con un impatto sulla matematica al di là dell'esempio specifico considerato nell’officina., Annali online della Didattica e della Formazione Docente, V. 14 N. 24 (2022): Processi di insegnamento-apprendimento della Matematica: aspetti significativi

Il tema dell’influenza delle idee e delle metodologie bourbakiste su discipline che trascendono l’ambito matematico è stato frequentemente affrontato e, in particolare, la riflessione sul rapporto tra Bourbaki e Oulipo ha animato un appassionato dibattito in seno alla critica letteraria. Lo scopo di questo contributo è quello di investigare le relazioni tra Le Lionnais, Queneau e Roubaud - e più in generale l’Oulipo - e Bourbaki. Attraverso un esame comparativo tra i due gruppi si è evidenziata l’esistenza di un innegabile carisma del “matematico policefalo” sul movimento artistico-letterario. The theme of the influence of Bourbaki's ideas and methodologies on disciplines that transcend the mathematical field has been frequently addressed and, in particular, the reflection on the relationship between Bourbaki and Oulipo has animated a passionate debate within literary criticism. The purpose of this contribution is to investigate the relations between Le Lionnais, Queneau and Roubaud - and more generally Oulipo - and Bourbaki. Through a comparative examination between the two groups, the existence of an undeniable charisma of the "polycephalic mathematician" on the artistic-literary movement was highlighted.

What could the Venice of the future look like? A project for an ideal city can take off from an allegory: arts and sciences inside a seed of Lodoicea maldivica, whose bipartition reminds us of a human brain. From the seeds left by the past, we derive the vision of the future. The ideal city certainly needs brains able to conceptualize images and develop ideas, and bridges to strengthen connections and in- teractions. A well-working brain needs “bridges” as connections between ideas and techniques. My vision for a future city contains a livable and stimulating space enhancing at one time creativity, enthusiasm, and scientific development. To this aim, I use the image and the metaphor of cerebral hemispheres, specialized in activities of different typologies yet interconnected. The image of the brain is one of the symbols of cognition. The right hemisphere deals with artistic expression, while the left hemisphere refers to logic thinking, mathematics, scientific attitude toward the world. Different cerebral areas, which contribute to the complex and rich life of an individual, are a metaphor for different places in a city, contributing to the completeness of a community life. Separation between hemispheres, also as a homage to Venice, is seen as a sort of Canal Grande, and connections are represented by bridges. The whole brain is seen as a giant seed of Lodoicea maldivica: the city of the future needs to develop from seeds, that is, knowledge inherited from the past and new ideas from our minds and thoughts. Thoughts that, in turn, are fed upon the Science of Complexity.

Some research studies concerning the didactics of Algebra discuss how learning to solve problems using symbolic algebraic language can be hard for students (Bohlmann et al., 2014; Palm, 2009). Students, in fact, have often difficulty to learn the ways in which the sym- bols should be manipulated to argue or to prove an assertion in order to reach a problem solu- tions. Although many studies conducted by mathematics educators discussed important contri- butions to this subject (e.g. Arzarello, Robutti & Bazzini, 2005; Boero, 2001; Carraher, Schliemann, Brizuela & Earnest, 2006; Lins & Kaput, 2004; Ursini & Trigueros, 2001), not many analysis were conducted on the role of the School textbook in this complex teaching /learning context. The paper, focusing on this aspect, discuses some preliminary MaTeK (En- hancement of Research Excellence in Mathematics Teacher Knowledge) Horizon 2020 project results, related to the analysis of some algebraic reasoning and proving tasks, discussed in 8th grade school mathematics textbooks in Italy.

Mathematical memes are an Internet phenomenon with an epistemic potential noteworthy for the teaching and learning of mathematics. The aim of this paper is to conceptualize this phenomenon on an em...

White light can be decomposed into different colors, and a complex sound wave can be decomposed into its partials. While the physics behind transverse and longitudinal waves is quite different and several theories have been developed to investigate the complexity of colors and timbres, we can try to model their structural similarities through the language of categories. Then, we consider color mixing and color transition in painting, comparing them with timbre superposition and timbre morphing in orchestration and computer music in light of bicategories and bigroupoids. Colors and timbres can be a probe to investigate some relevant aspects of visual and auditory perception jointly with their connections. Thus, the use of categories proposed here aims to investigate color/timbre perception, influencing the computer science developments in this area.

Mathematics education literature involves studies that sought a way of investigating the mode of reasoning in mathematics textbooks because textbooks are the main resource for teachers in planning their mathematics lessons. In this vein, this study aimed to analyze the ways of reasoning in mathematics textbooks that are currently used in five countries: Slovakia, Czech Republic, Italy, Norway, and Turkiye, as a part of a Horizon 2020 Project. We initially started with a framework that aimed to examine the effect of teachers’ participation in the lesson study on the improvement of students’ mathematical reasoning (Project LESSAM). However, as the textbook analysis of different countries proceeded, we realized that this framework would not solely be sufficient to address the types of reasoning with alternative tasks presented in those textbooks. Thus, our framework development followed four main steps: (1) starting with a proposed framework that focused on different types of reasoning (2) identification of reasoning and proof tasks among worked examples, (3) categorization of ways of reasoning through the proposed framework, and (4) developing new categories based on a review of existing frameworks, to cover and differentiate all types of reasoning in the worked examples. Hence, in this study, we aim to present the integrated framework that we developed to analyze the ways of reasoning in worked examples (the problems with explained solutions) in the above-mentioned countries’ textbooks. We also discuss future research agenda for analyzing mathematics textbooks with this integrated framework.

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In Italia, come in molte altre nazioni, il divario nei risultati ottenuti da maschi e femmine nelle prove standardizzate di matematica risulta essere marcato e a favore dei primi. Se si considerano però i risultati in termini di voti scolastici questo divario non risulta o è addirittura a favore delle femmine. In questo contributo rifletteremo sui principali fattori che possono influenzare le differenti performance di maschi e femmine nelle prove standardizzate, tra questi prenderemo in considerazione anche aspetti metacognitivi e fattori legati alle convinzioni di insegnanti, genitori e studenti rispetto alla matematica. I risultati di un questionario proposto a oltre 500 docenti di diversi gradi scolastici ha fatto emergere il ruolo delle convinzioni dei docenti in relazione alle differenze di genere e ci ha permesso di interpretare il divario presente in termini di valutazioni scolastiche anche in termini di convinzioni., Annali online della Didattica e della Formazione Docente, V. 14 N. 24 (2022): Processi di insegnamento-apprendimento della Matematica: aspetti significativi

If we study the history of societies, we can find several pandemic events such as smallpox, cholera, plague, and SARS [1, 2]. Each one of these pandemics events af fected human life in many aspects from health to the economic sphere [3], education is one of these aspects. The SARS-CoV-2 pandemic (Covid-19) has had a massive impact on Education: many students of different countries have been affected by school and university closures due to the Covid-19. Italy was the first Western coun try to suffer a coronavirus emergency

The first part of the paper presents some theoretical reflections and some methodological notes about a Professional Development (PD) path worked out during the last two years by Italian researchers for prospective primary teachers. The theoretical construct of Cultural Transposition defines the framework of the PD path’s activities and the related research. It was used to define an interesting cultural lens to delineate possible new approaches for effective pre-service teacher education programs in particular for the primary level. The defined methodology was based on the possibility to reflect on the he decentralization of didactic practices based on a specific cultural context through one or more contacts with other “realities” coming out from different selected cultural contexts. In the second part of the paper, we particularly focus on one of the stimuli proposed in the PD’s path: the Chinese “variation” as a systematic method adopted in Chinese textbooks. In this context, we concentrate our analysis on the use of three counting art artifacts as the Abacus, the Straws, and the Chinese abacus, the Suanpan (算盘).

In this era of great uncertainty, imagining the future may be challenging, especially for young people. In science education, the interest in future-oriented education is now emerging, research needs, however, to keep eyes on youngsters’ future perceptions and on the development of a future literacy. In this article, starting from a sample of individual students’ narratives about their future daily life in 2040, we aim to delineate which ways of grappling with the future can be observed in the essays and which methodological tools are suited to operationalize their identification and characterization. The analysis led to the definition of “polarization” and “complexification” attitudes that represent the ways in which the students’ narratives are positioned with respect to a bunch of dichotomies: personal–societal, functional–aesthetics oriented, good–bad, natural–artificial, and certain–uncertain. Moreover, with this study, we provide a contribution to the methodological reflection that deals with the collection and analysis of data, when students’ future perceptions need to be investigated. Discussing the limits of the current data collection tool, we introduce the design of a SenseMaker® questionnaire which contributed to feeding a collaboration with #OurFutures project, recently launched by the European Commission to collect future narratives all around Europe.

The difficulties encountered during the COVID19 emergency have led to reflection on teaching methods and the introduction of new digital tools. In this paper we highlight how the use of a digital platform can support mathematical discussion, playing a fundamental role in the construction of meaning of new mathematical objects during laboratory activities. We qualitatively analyse the results of a teaching experiment involving a discussion based on comparison of different solutions to the same problem, which is recognised as a powerful pedagogical activity but also a challenge for both teachers and learners.

Differences in performance of native and foreign students in reading comprehension and mathematics emerge in all standardised assessments, both at national and international levels. This study is the continuation of a previous research project in which we examined citizenship-based differences in Italian mathematics standardised assessments, both at item level and in connection with students’ competence in text comprehension. We present a deeper analysis of items which presented a significant Differential Item Functioning (DIF) in favour of native students, and interpret these differences in terms of linguistic difficulties. On the basis of this analysis, we then present a classification of possible item characteristics causing disadvantage to foreign students, going one step further classification of items in terms of reading demand. The results obtained are then supported by further analysis of items similar to those considered in the first part of the present study, but administered in different years: all the results in terms of DIF analysis are confirmed also in the similar items’ analysis.

This paper discusses some theoretical/methodological observation and some qualitative results coming from a Cultural Transposition experience, implemented in the Italian school context (grade 8), according to the methodology of variation, as one of the most significant problem solving approach in Chinese schools. The framework of the Cultural Transposition and the methodology of variation are presented as an important condition for “decentralizing” the didactic practice from a specific social and cultural context. We argue that looking at different teaching/learning mathematics strategies coming from East-Asia cultures can favor some cultural contaminations at school and allow students to a significant and unusual thought about “inclusion” and “diversity” in mathematics. Our variation problems are designed on 3D Geometry and are aimed to guide students in discovering the relationship between pyramid and cone areas/volumes.

The research described in this paper focused on the issue of describing and understanding how mathematical discussion develops in a hybrid learning environment, and how students participate in it. The experimental plan involved several classes working in parallel, with pupils and teachers interacting both in their real classrooms and in a digital environment with other pupils and teachers. The research was based on a rich set of data collected from the M@t.abel 2020 project, which was developed in Italy during the Covid health crisis. Based on Complementary Accounts Methodology, the data analysis presented in this paper involved specialists from the fields of mathematics education and inclusive education. In the study we considered the complexity of learning and the different elements that have an impact on students’ activity and participation, when they are engaged in mathematical discussions within the multilevel-digital environment that emerged due to the pandemic. These parallel analyses showed that ‘mathematical discussion in the classroom’ is a complex (and sometimes chaotic) phenomenon wherein different factors interweave. A complementary approach assists in developing a global vision for this dynamic phenomenon and in highlighting local episodes that are crucial in this interplay of factors. It is precisely in these episodes that the role of the teacher is fundamental: these episodes appear as catalysts for the different variables, with the teacher acting as mediator.

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In July 2021, the ICME 14 took place in Shanghai, China, as a hybrid event; a thematic focus of this international conference was the TSG 36 Research on Classroom Practice at Primary Level under the leadership of Shuhua An. The aim of this TSG is to share and discuss, at an international level, a wide variety of experiences of teaching and learning mathematics in the classroom (An et al., 2022). According to this aim the TSG 36 explored state-of-the-art strategies and approaches to address the concerns and problems, and advance the research on classroom practice with an ultimate goal of supporting it.