Andrea MaffiaUniversity of Bologna | UNIBO · Department of Mathematics MAT
Andrea Maffia
Doctor of Mathematics Education
About
70
Publications
29,327
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135
Citations
Citations since 2017
Introduction
Andrea Maffia, PhD in Mathematics Education. His research mainly concerns teaching and learning of multiplication properties at primary level. Semiotics of mathematical
teaching/learning and networking of theories in maths ed. are other general interests.
He is currently researcher at the University of Bologna (in Italy).
Additional affiliations
November 2018 - present
September 2017 - September 2018
Position
- Lecturer
Description
- Lectures on Mathematics Foundations and Mathematics Education for pre-service primary teachers.
July 2014 - July 2015
Education
January 2013 - May 2017
Publications
Publications (70)
International literature is increasingly disclosing the relevance of cultural aspects within the processes of teaching and learning mathematics. Knowledge is inextricably linked to the activities in which the subjects engage, and must be considered in relation to the socio-cultural context wherein the activity takes place. Literature reveals the re...
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 o...
Sin dai primi anni di scolarizzazione, l’introduzione di nuovi oggetti di conoscenza è spesso accompagnata dall’accostamento degli stessi con colori: è frequente l’uso di colori per identificare lettere, numeri e figure. Alcuni autori sostengono che il daltonismo possa comportare difficoltà nel successo scolastico anche se le strategie compensative...
Design of inclusive educational contexts can be applied in unusual contexts such as prison, a multicomplexity contextaddressed by teachers whenever they have to design teaching materials and conduct classroom activities. Teaching inprison needs specific reflections on the context and the socioeconomic and cultural backgrounds of students, to imag...
This contribution presents the role of self-evaluation within the project "Numeri e Pedine" based on a community of inquire formed, in the last two years, by researchers and primary mathematics teachers from the whole country. After a short introduction about the features of the boardgames chosen for this mathematics education project, we analyze t...
With the aim of nourishing the discussion that has taken place during CERMEs in TWG13, we present a review of results presented in previous CERMEs about the possible roles of boardgames in developing young children's number sense. Several contributions in CERMEs have shown that difficulties related to teacher education and curriculum design emerge...
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. Th...
While the development of creativity, or creative thinking, in mathematics is considered important by many researchers, there are several difficulties in implementing creative tasks, especially before secondary school. Within the original context of a mathematical escape game, this paper reports two episodes exemplifying the difficulties met by sixt...
Parents can play a key role in their children's homework activities; this has been particularly true during the home-schooling caused by the pandemic. However, how parents affect students’ mathematical activity is scarcely researched, in particular in the case of primary school students.
Within the framework of activity theory, in this contribution...
The purpose of this paper is to analyse, following Sfard’s theoretical framework, the
students’ discourse about irreducible polynomials in order to understand their
difficulties. Through the students’ explanations, given during the interviews conducted
after carrying out the questionnaire, it is possible to study which processes were
triggered by t...
There is a small but still growing body of literature about the implementation of board games as means for teaching mathematics. Researchers have noticed the opportunities provided by specific board games in developing number sense. However, there are empirical results showing that primary teachers can struggle in understanding the potential of a g...
Attraverso due studi di caso, si propone una riflessione sull’attività matematica individuale che i bambini della scuola primaria svolgono a casa e su come l’interazione con i genitori influenzi tale attività. Dopo aver presentato una breve rassegna della letteratura sul tema del “fare scuola a casa” e del rapporto dei genitori con i compiti di mat...
Recognizing the relevance of the construction of knowledge by the learners, a complex educational problem concerns the movement from individual activities towards sharing and institutionalizing mathematical knowledge. Teacher’s role is crucial: how to manage the movement from a task to the evoked mathematics? We face this problem in the specific ca...
Thematic Working Group 13 focuses on the mathematics learning of 2–8-year-olds. We acknowledge that this learning may take place in kindergarten, preschool, primary school, and other less formal settings. We pay attention to how mathematics is approached implicitly and explicitly by children and teachers, and other responsible adults. A wide range...
Due to the pandemic situation in 2020 the ERME summer school (YESS10) was designed for an online format using a conference system. Our design choices were based on previous experiences with YESS and the use of the research pentagon as a tool to think about research. This paper elaborates theoretically and empirically the specification of the concep...
La rappresentazione simbolica di relazioni tra numeri viene spesso introdotta nella scuola media. Vari autori sostengono che l'adozione di un paradigma relazionale e l'uso di rap-presentazioni grafiche possono supportare gli studenti nel primo approccio all'algebra, tutta-via esempi di sequenze didattiche del genere non sono ancora largamente diffu...
L’uso dei giochi da tavolo nei contesti educativi e formativi non è certamente qualcosa di nuovo: tuttavia negli ultimi anni la riflessione circa le potenzialità di questo strumento nell’ambito della didattica scolastica è divenuta oggetto di attenzione della ricerca scientifica in modo sistematico.
Nel nostro contributo presentiamo alcune rifless...
Boardgames and tabletops are used in primary schools, but only as ludic tools and not as assessment practices. Preliminary results from a project for primary teachers' professional development are presented. The project is aimed at teachers' professional development in terms of their assessment practices in mathematics teaching and learning. The pr...
Molti studenti considerano un'uguaglianza non come una relazione di equivalenza, ma come un "operatore" che indica lo svolgimento delle operazioni. In questo lavoro si presentano i risultati relativi a un questionario mirato a far emergere la contrapposizione tra queste due concezioni dell'uguaglianza nel corso del primo ciclo di istruzione. La per...
Partendo dal presupposto che l’insegnamento degli algoritmi per il calcolo scritto è ancora un obiettivo importante dell’educazione matematica (quando ci si interroga su come e perché funzionano), presentiamo un esempio di attività laboratoriale centrata sull’analisi dell’algoritmo standard per la moltiplicazione in colonna. Mostriamo come l’attivi...
Presentiamo un’indagine sulle convinzioni di docenti e studenti sulle prove INVALSI per la classe quinta della scuola secondaria di secondo grado (grado 13). Basandoci sull’idea di competenza in matematica, analizziamo quanto le opinioni dei docenti sulla valutazione si avvicinano alla valutazione per competenze e come le opinioni degli studenti so...
In questo contributo vengono presentati i primi risultati di un progetto di Ricerca-Formazione mirato allo sviluppo professionale di docenti di scuola primaria in merito alle pratiche valutative relative all’insegnamento della matematica. Il progetto, centrato sull’uso del gioco da tavolo “Chiudi la scatola” prevede due diversi livelli di coinvolgi...
Aiming at unfolding possible cultural differences concerning pre-service teachers’ beliefs towards mathematics and its teaching, this paper presents a comparative study between nearby regions: Italy and Germany. The sample is composed by 460 pre-service teachers from three universities, one in Germany and two in Italy, one of them close to the Aust...
Prendendo ispirazione dalla letteratura sul pensiero algebrico prescolare, è possibile stimolare e riconoscere il pensiero relazionale a livello di scuola dell’infanzia. Si presenta una ricerca basata sull’analisi delle risposte dei bambini ad una consegna; lo scopo è sia sviluppare il loro pensiero relazionale sia ottenere indizi su come poterlo r...
Questo documento descrive l'analisi di un testo interdisciplinare, sulla matematica e la letteratura, secondo la teoria della comognizione di Anna Sfard. Questa indagine è inquadrata in un progetto più ampio: "Liceo Matematico". Uno degli obiettivi del progetto è il superamento delle barriere tra materie scolastiche e la creazione di una collaboraz...
The use of artifacts to introduce the distributive law of multiplication over addition in primary school is a diffused approach: it is possible to find pre-constructed learning trajectories in instructional materials. However, it is still unclear how the teacher might support his/her students in transitioning from concrete to symbolic representatio...
Literature has already shown that gestures play a relevant role in classroom interactions between students and teacher. We integrate the perspective of Theory of Semiotic Mediation with the notion of Semiotic Bundle to illustrate how gestures can be used as “pivot signs” in semiotic chains. This means that gestures can be performed by the teacher t...
The process of pairing a name with representations or peculiar properties permeates many mathematics classroom situations. In school, many practices go under the label 'definition', even though they can be very different from what mathematicians conceive as a formal definition , and in fact there are substantial differences between these different...
This paper sets out to explore the different uses made of Organisation for Economic Co-operation and Development Program for International Student Assessment (OECD-PISA) tests and data in mathematics education research. Through a comprehensive literature review of journals and conference papers, we show that although a large variety of topics is ad...
Since the last decades, there has been a lot of research on affect in teaching and learning mathematics. In this chapter, we show examples of different methodological approaches and outline recent issues arising from the discussions that took place during the MAVI25 conference. Based on examples both from the literature and from contributions in th...
In the last years, research has paid strong attention to pre-service primary teachers’ views of mathematics. Interviews and questionnaires to pre-service teachers during their academic studies are the mainly used tools for collecting data. Qualitative and quantitative approaches may give different insights. In this paper, after a review of the diff...
Lo studio dell'autoefficacia degli insegnanti e delle caratteristiche che ne influenzano lo sviluppo è un tema di grande rilevanza per la didattica della matematica. In questo contributo presentiamo un'indagine sulla relazione tra le emozioni verso la matematica e il suo insegnamento e il senso di autoefficacia nei futuri docenti. Lo studio è condo...
Classes about mathematics education within the degree in "Scienze della Formazione Primaria" must work both on the content knowledge and the pedagogical knowledge of content. Possible strategies to evaluate content knowledge have been deep studied in the past. It seems more difficult to find suitable data about the pedagogical knowledge of content....
This work aims at laying the base of a model for studying how the mathematical learning process triggered by the use of an artifact in the classroom can be affected by the use, even in different moments and for different purposes, of another artifact by the same students. Taking a Peircean framework, we define semiotic interference as an enchaining...
Integrating history of mathematics in classes could be a hard task with young pupils. Indeed, original historical sources have a language that is far from the modern one. Such texts represent
cultural artefacts that can give access to mathematical knowledge.
The teacher can exploit such potential acting as a mediator between the mathematical signs...
Taking inspiration from recent literature about algebraic thinking at early grades, we wonder if and how we can prompt and recognize relational thinking at the kindergarten level. Adopting a design-based research approach, we analyze students' answers to a task, both with the aim of developing its potential in prompting relational thinking and the...
La ricerca sul richiamo dei fatti aritmetici moltiplicativi (ovvero le tabelline) si è già interrogata molto sul modo in cui le informazioni sono immagazzinate nella nostra memoria. Poco ancora si sa sull'effetto di diverse modalità di immagazzinamento di tali informazioni nella memoria a lungo termine. Dopo una rassegna della letteratura esistente...
Multiplication can be presented to students through different models, each one with its pros and cons. In this contribution we focus on the repeated sum and the array model to investigate the relations between the two models and those between them and multiplication properties. Formal counterparts are presented. Taking both a mathematical and cogni...
Il potenziale didattico degli artefatti nell’apprendimento è stato ampiamente studiato, focalizzandosi soprattutto sul loro possibile uso da parte degli studenti e sui benefici che ne derivano. Vi è però la tendenza a sottostimare la complessità legata alla piena espressione di tale potenziale, e in particolare l’importanza del ruolo del docente ne...
The didactic potential of artefacts for learning have been extensively studied, with a main focus on their possible use by students and the subsequent benefits for them.
However, there has been the tendency to underestimate the complexity of exploiting this potential, and specifically the complexity of the teacher’s role orchestrating the teaching...
School mathematics is often related to a needing for memorization of a lot of information, times-tables are a paradigmatic example. A large amount of research on arithmetical facts has been implemented within cognitive psychology but rarely it is related to mathematics education research and quantitative methods are always used. In this paper it is...
Pre-service teachers’ education usually consists in a theoretical part,
then followed by some kind of practice in school. In our experience, it is
quite common to hear about the discrepancy between what is taught in
universities and what is put in practice. If you are the new young teacher
of your school, doing things differently than your more exp...
Integrating the history of mathematics in class could be a hard task with young pupils. Indeed, the language of historical sources often poses problems for the modern reader.. Such texts represent cultural artefacts that can give access to mathematical knowledge. The teacher can exploit such potential by acting as a mediator between the mathematica...
Il percorso della scuola secondaria di primo grado costituisce un tuttuno con quello della scuola primaria formando il cosiddetto primo ciclo di istruzione. Pertanto, la fine della classe terza della secondaria di primo grado coincide con il termine del primo ciclo. L’evento che caratterizza questo momento è l’esame di stato volto a certificare le...
Riassunto Il presente contributo si inserisce all'interno delle ricerche svolte nel progetto Comenius FAMT&L dell'Università di Bologna, volto a promuovere un corretto uso della valutazione formativa in matematica. Uno dei principali outcomes del progetto è un corso di formazione pilota svolto con insegnanti in formazione e in servizio, di scuole s...
Structure sense can be mobilized by pupils to compare and to transform arithmetical expressions, however sometimes it can lead to mathematical inconsistency that pupils might be not aware of. This paper provides evidence of this type of phenomenon. Through the analysis of an interview with a third grader, it is shown that the development of structu...
La letteratura mostra come l’uso della Latent Class Analysis permetta di individuare quesiti che presentano particolari
difficoltà per gli studenti “poveri di conoscenza”. A partire da ciò abbiamo portato avanti un’ulteriore riflessione sulle
prove di valutazione standardizzate nazionali allo scopo di fornire strumenti significativi utilizzabili ne...
It is widely shown in literature that gestures take part in classroom interactions. Adopting the perspective of Theory of Semiotic Mediation, we illustrate how they can act as “pivot signs” in semiotic chains, namely they can link signs referring to a situated context (such as the activity with an artifact) to the mathematical domain. More specific...
L'obiettivo del progetto è di costruire strumenti di analisi per selezionare, nelle prove di Valutazione Nazionale dei diversi livelli, catene di quesiti (ossia quesiti somministrati in livelli successivi che possono essere collegati attraverso l'intreccio di analisi qualitative e quantitative), che identifichino studenti che possono essere, divent...
La moltiplicazione viene presentata presto nella scuola primaria, ma le sue proprietà sono introdotte solo dopo che le cosiddette tabelline sono state memorizzate. Nell'articolo si presenta un teaching experiment volto a introdurre precocemente le proprietà della moltiplicazione per facilitare la memorizzazione di fatti moltiplicativi. L'esperiment...
In this paper, some of the main theoretical constructs of the
discursive approach presented in the book “Thinking as
communicating” are described. It is argued that the basic principles
of this approach are not so far from the paradigms of some Italian
theories. Particular attention is devoted to the notion of
‘commognitive conflict’ which is exemp...
Starting from a multiple-choice question taken from the INVALSI standardized national test in grade 5, we investigate the conception about rational numbers developed by Italian students. The aim of this research is to point out which invariant-operators emerge in students' answers and which one of them conducts more easily to right/wrong conclusion...
L’uso dell’indagine OCSE-PISA è estremamente limitato nella ricerca in didattica della matematica, nonostante
il forte interesse che i risultati di questi test hanno suscitato nell’opinione pubblica nei confronti dell’educazione
matematica (Sfard, 2005). Sono stati analizzati 107 articoli che fanno uso del PISA tratti dalle principali riviste inter...
Number line is a representation often used for the teaching of number sets. The usage of this representation is also suggested by national standards for all grades, so in INVALSI national tests there are questions specifically designed for the evaluation of students' ability in the management of number line. In 2014 such questions have been adminis...
Gli obiettivi delle Indicazioni Nazionali si riferiscono esplicitamente alla capacità di stima del risultato di un'operazione. Il termine "stimare" richiama l'approssimazione di un numero piuttosto che il suo calcolo esatto; tale numero è concepito come il risultato di un'operazione. Sono coinvolte quindi conoscenze relative alle operazioni e agli...
In this paper we analyse some peculiar students’ mistakes in a task selected from the Italian National
Mathematics Standardized Tests. In particular we show different types of errors identified in the solutions of
grade 6 students who faced an item that involved the management of the number line. We use both the
statistical results of the natio...
School mathematics is often related to a needing for memorization of several informations, times-tables are a
paradigmatic example. A large amount of research on arithmetical facts has been implemented within cognitive psychology but rarely it is related with mathematics education research. In this paper, ten interviews to adults are analyzed using...
Trovare consegne adatte alle esigenze dell’intera classe e di ogni singoli allievo non è un compito semplice per l’insegnante. Questo appare ancora più complesso quando si vogliono implementare attività in cui sia l’intuizione sia l’argomentazione da parte degli studenti giocano un ruolo importante. A disposizione dei docenti ci sono molte risorse...
This paper presents a longitudinal analysis of the outcomes of the Italian national standardized mathematics tests. By intertwining quantitative and qualitative methods, we selected and analysed a set of linked questions among the tests administered to the same cohort of students first in grade 6 and then in grade 8. In particular, we focus on poor...
An early introduction to arithmetical expressions is realized in a teaching experiment involving an artefact based on the rectangle model for multiplication. Children elaborate signs, strictly related to the activity with the artefact, which evolve to mathematical ones: the enchaining of different representations in many semiotic systems is describ...
The paper consists of a review of the book “The Language of Mathematics: Telling Mathematical Tales” by Bill Barton. Barton's studies concern about Ethnomathematics and, in the analyzed book, he joined the results about some of his main researches about the relationships between language and Mathematics Education. The review focuses on the three pa...
L'elaborazione della prova equipollente alla prova nazionale (INVALSI) dell'esame di stato a termini del primo ciclo per gli studenti con PEI è diventata pratica comune negli ultimi cinque anni, ma rimangono aperte molte problematiche relative alla stesura della prova e alle sue finalità. Per questo il 18/01/2013 si è svolto a Firenze un workshop m...
Questions
Question (1)
On the book "Early Algebrization" edited by Cai and Knuth (2011) there is a Chapter by Russel, Schifter and Bastable in which the idea of "Representation-Based Proof" is presented through some examples. Are there any more recent papers in which these authors (or others) develop this idea? I am particularly interested in examples about multiplication properties.
Thanks for any suggestion!
