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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.

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.

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

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.

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

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.

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.

National and international surveys highlight a strong disparity in males and females results in mathematics: boys outperform female in almost all the countries and in all school levels. In the international context, Italy appears to be one of the countries in which the results related to gender differences in mathematics are more alarming and INVALSI tests’ data confirm a very marked gender gap in favour of males at all school levels. In this paper we present the international situation regarding gender differences in mathematics starting from large scale assessment and paying particular attention to the Italian situation. Hence we present a detailed overview of the causes underlying the gender gap in mathematics, analysing the main studies on this issue and trying to provide a complete, though complex, interpretation of this phenomenon.

In this thesis, we present three topics concerning the study of Hilbert function. In the first topic, we obtained a partial solution to Froberg's conjecture. In the second topic, we gave the bounds for the second Hilbert coefficients of filtered modules. In the third part, we study the preservation of Hilbert function under small perturbations.

Several researches in STEM education research highlight the advantages of an inte- grated approach to these disciplines that relates knowledge and know-how, design and implementation, theoretical and practical problems [5, 4, 6]. In some researches, the effectiveness of these approaches on students conceptual understanding and motivation and has been studied through the use of quantitative analysis tools such as cluster analysis (CLA) [1, 7]. Through CLA it is possible to characterize students analyzing the strategies they deploy to tackle, for example, questionnaires built so as to investigate the lines of reasoning implemented by them when they are proposed with problematic situations. In particular, it is possible to characterize the students in terms of a limited number, m, of typical ways of answering the questionnaire questions [2]. Each student is therefore identified by a binary vector (each component can be 1 or 0) with m dimensions. Cluster Analysis techniques allow students to be grouped into homogeneous groups based on com- mon characteristics and the representation of these groups is ideally referred to an m-sized space. However, for reasons of simplicity and clarity, it is often preferred to perform the representation of groups in three or two dimensional spaces. One of the techniques used for this purpose is Multidimensional Scaling [3] (MDS). It allows the researcher to move from the m-sized space to a space with a smaller number of dimensions that is a function of the initial m-dimensional representation, preserving the global distances between the group elements. However, the application of the MDS methodologies depends strongly on the typology of the initial data and a non-thorough knowledge of the mathematical details at their base can lead to obtaining results that are not reliable and / or of little significance. In this paper we will study a MDS methodology based on Principal Component Analysis, with particular reference to a set of binary data, highlighting how the results obtained through this methodology can be reliable and significant for the researcher in education.

National and international large-scale assessments of mathematics show that, in most nations, males achieve better results than females and Italy is one of the countries with a greater gap. Many research studies in mathematics education have analysed this issue, using both quantitative and qualitative methods to understand the sources and characteristics of this gap. This study focuses on a specific Grade 10 task that requires algebraic manipulations of powers with the same base. Item-level analysis enables the study of gender differences on specific content, before using the lenses of mathematics education theories to interpret macro-phenomena emergeing from standardized assessment results. The quantitative analysis, carried out using the Rasch statistical model, highlights a gender gap in favour of males in this task and, furthermore, a variance in choices of incorrect options between males and females; the interviews conducted provide a key to understanding this phenomenon in terms of didactic contract.

In this paper, we examine the evolution of a specific mathematical problem, i.e. the nine-point conic, a generalisation of the nine-point circle due to Steiner. We will follow this evolution from Steiner to the Neapolitan school (Trudi and Battaglini) and finally to the contribution of Beltrami that closed this journey, at least from a mathematical point of view (scholars of elementary geometry, in fact, will continue to resume the problem from the second half of the 19th to the beginning of the 20th century). We believe that such evolution may indicate the steady development of the mathematical methods from Euclidean metric to projective, and finally, with Beltrami, with the use of quadratic transformations. In this sense, the work of Beltrami appears similar to the recent (after the anticipations of Magnus and Steiner) results of Schiaparelli and Cremona. Moreover, Beltrami's methods are closely related to the study of birational transformations, which in the same period were becoming one of the main topics of algebraic geometry. Finally, our work emphasises the role played by the nine-point conic problem in the studies of young Beltrami who, under Cremona's guidance, was then developing his mathematical skills. To this end, we make considerable use of the unedited correspondence Beltrami – Cremona, preserved in the Istituto Mazziniano, Genoa.

Research, at both national and international levels, has started to focus on the gender gap regarding student performances in technological/scientific topics. Fervent ideological and epistemological debates have sprung up, particularly concerning the differences in mathematics test results between males and females (with males outshining females in most countries). In this paper, we analyse some questions taken from the INVALSI mathematics tests administered at varying educational stages; emergent didactic phenomena are investigated via one of the key concepts of mathematics education, the didactic contract. From a quantitative analysis, based on the percentage of answers given and Rasch Model results, it is possible to identify the presence of significant gender differences between the performances which can be traced back to the effects of the didactic contract. Furthermore, study of the features which accompany these gender gaps in the case studies allows us to investigate the gender gap in mathematics via a new interpretative method which also takes into account student ability levels.

The paper presents some reflections and activities developed by researchers and teachers involved in teacher education programs on cultural transposition. The construct of cultural transposition is presented as a condition for decentralizing the didactic practice of a specific cultural context through contact with other didactic practices of different cultural contexts. We discuss the background theoretical issues of this approach and also give an analysis of two examples of cultural transposition experiences carried out in Italy. In particular, by means of qualitative analysis of some excerpts, discussions, and interviews, we show that the contact with different perspectives coming from China and Russia fostered educational practices and reflections on whole number arithmetic education.