Maria G. Bartolini Bussi

Full professor of mathematics education

Università di Modena e Reggio Emilia

Italie

Curricula

I have been asked to give a short introduction to the problem of curricula all over the world. It is really very difficult and I shall limit myself to some observations about the Italian one (my country) and the Chinese one (a country I have been studying for some years).

In general, I could say that ICMI has recently launched a DATABASE PROJECT whose ultimate goal is to build and update a database of the mathematics curricula all over the world.

http://www1.mathunion.org/icmi/other-activities/database-project/introduction/

This database, with documents in local languages or English (most are in English), would allow comparative studies between different national standards. Both France and Italy are present (and US too, but not China yet). This is a first step to have collected in only one place relevant information. The documents are about “approved” standards. The study of curricula is really much more complex as it would require to refer not only to the standards but also to the variety of materials for teachers and for schools and similar.

Comparative analysis of standards (and of curricula even more) is not an easy matter at all, especially when very different cultural traditions are into play. But there is a much subtler reason. Usually a researcher who is embedded in one culture share a lot of implicit norms and systems of values, that make it difficult to understand the traces of them in the written documents from his/her own country. Everybody is strongly centered in his/her culture (up to the “naive” belief that it is the only GOOD model, that seems to be shared especially by teachers). A very interesting comparative book is:

Xie X, Caspecken P. F. (2008), Philosophy, Learning and the Mathematics Curriculum: Dialectical materialism and pragmatism related to Chinese and US mathematics Curriculums. Sensepublisher.

In the above book naïveté is avoided by having two authors, one from each culture. In that book, starting from the NCTM Standards (US) and from the Chinese governmental Standards, the analysis is made also on two very popular textbook series (one in US and one in China). The reader is accompanied to a trip that allows to recognize the traces of the background culture, philosophy, systems of values and so on in both the Standards and the exemplary textbooks. This model might be applied also to compare other pairs (Standard, curriculum) and to discover some differences between our own model and a model from a far culture.

Actually, another way of avoiding naiveté (I am choosing this one, as I shall argue below) is to try to detach myself from my Italian (and European) culture, taking a point of view from a complex and ancient culture that is as far as possible from mine, i. e. to look at Italy from a Chinese perspective. I started about 6 years ago to study Chinese curricula and textbooks to try to understand why Chinese children are better in Mathematics (e. g. arithmetics, problem solving) than Italian children in Italian school (in REGGIO EMILIA, the city of the Faculty of education where I train prospective pre-primary and primary school teachers, there is one of the largest Chinese community in Italy). This “impression” is confirmed by the international assessment like OECD PISA, where Shanghai was absolutely the best in 2009. As I discovered very soon, the answer was not easy and required to start a long trip inside Chinese culture and language. Some years later I found that my personal approach (realized with the help of family members and colleagues) was consistent with the approach of a French philosopher and sinologist (François Jullien) who studied Chinese culture to understand French (and European) culture. My first approach to Jullien was mediated by:

Chieng A. (2006), La pratique de la Chine, Editions Grasset & Fasquelle.

I studied carefully:

• ICMI study n. 13: Mathematics Education in Different Cultural Traditions- A Comparative Study of East Asia and the West;

• Chemla papers (and edition of the Nine Chapters) about the ancient Chinese mathematics.

I exploited some travels to Beijing and Taiwan, related to my activity as a member of the ICMI EC to have a first-hand contact with Chinese culture and school.

Now I have a small group of students on this “Chinese” project that is, in Italy, quite original: educators are usually very interested in studying the ways of teaching as soon as possible Italian as a second language to students, whilst we are more interested in exploiting in teacher education students from abroad (especially from China, who is the farthest situation from a cultural perspective) as a challenge and a resource to become aware of some hidden choices of our mathematics curriculum and to prompt changes in the teacher system of beliefs and, later, in the classroom practice.

I have already published something about this theme (in Italian) and I have made the first international presentation (in English) at SEMT 11 last August (see below). The focus in this case was “problems with variation”, a typical traditional approach to problem solving in China (see below).

An example from the “Chinese” study: problems with variation

The Chinese standards are very interesting to be read, although it is not always easy to find traces of the traditional Chinese culture. A very useful book is the book by Xie & Carspecken quoted above.

It is known also in the West now that the way of approaching problem solving in the Chinese school is quite different from the western way. This was well explained in the Chinese national presentation in Monterrey (ICME 11, 2008). The Chinese method is known as “problems with variation”. In short this means to be able to see, in the same situation (from everyday life or mathematics as well) different ways to pose and solve problems. I shall present now a small example taken from a first grade textbook (introducing the numbers 6 and 7) with a comment taken with small adaptation from Chinese curricula The problem mentioned in the Standards concerns 2 and 3 but the image is very ugly!).

Comment:

One can see 6 + 1 = 7 in different ways (from different perspective, in a more flexible way):

6 children and 1 teacher;

6 persons in classroom and 1 just entering

6 persons cleaning the classroom and 1 cleaning (or writing on) the blackboard.

But also:

6 desks and 1 teacher’s desk

6 chairs close to the desks and 1 still moving

What is also important is that in the same page the relationships are written in many ways:

6 + 1 = 7; 7 – 1 = 6; 7 - 6 = 1.

Whenever there is an addition there is a subtraction and viceversa. This links the inverse operation to each other in a very flexible way.

In the Chinese Standards, surprisingly, problems with variation are not mentioned explicitly, as if they were a stable part of the Chinese tradition. Chinese colleagues tell that in China there are thousands of journals for teachers where problems with variations are discussed. Below you find a beautiful example of a very complex set of problems with variations taken from a second grade textbook.

I have discussed this set of problems in:

Bartolini Bussi M. G., Canalini R. & Ferri F. (2011), Towards cultural analysis of content: problems with variation in primary school, Proc SEMT 11, Prague.1

In teaching experiments carried out in Italy, I have used this set of problems, with a very complex metacognitive task, to prompt a change in teachers’ system of beliefs. We have used similar tasks also for multiplicative structures.

One might think that in this case there is nothing new (from the research perspective) with respect to Vergnaud’s analysis of the conceptual field of additive structures (and multiplicative structure in the other case). This might be true, but the attitude is a different: rather than analyzing and classifying problems (a typical western attitude) the above Chinese problems are considered as a whole by the teachers and are discussed as a whole by 2nd graders. They appear in this way in a standard textbook.

I do not know how addition and subtraction are treated in problem solving in French textbooks. In Italy they are usually introduced separately (in spite of the indications of our curricula). There is a very famous series of teacher’s guide where there is a volume on addition problems and a volume of subtraction problems, written by two different authors with no connection with each other.

An example from Italian Standards: relationships between didactical research and institutional choices

In the document (the most recent Standards for grades k-8):

http://archivio.pubblica.istruzione.it/normativa/2007/allegati/dir_310707.pdf

there is a clear reference to the curricula prepared for the Ministry of Education by the Italian Mathematical Union (and the Italian Commission for Mathematical Instruction, chaired at that time by ARZARELLO). 3 volumes have been produced by a large group of didacticians (including dozens of teacher-researchers). The volumes have been published by the Ministry of Education and are available at:

http://www.umi-ciim.it/in_italia--28.html

with the names Matematica 2001, Mathematica 2003, Matematica 2004.

In those big volumes the most powerful methods and examples produced by the community of Italian didacticians have been exploited. They draw on international research but with a special perspective on the institutional feature (e.g. Italian teachers usually spend many years – 3/5 – with the same group of students, hence the importance of long term processes). For instance – without explicit quotation to any author - the idea of Laboratory of Mathematics (not so far from Inquiry Based Mathematics Education, but with a major focus on the historic-cultural roots of mathematics), the idea of Mathematical discussion orchestrated by the teacher as a major form of social interaction in the mathematics classroom, the research studies on mathematical instruments and ICTs and so on. These 3 volumes had a strong impact on all the subsequent official documents by the Ministry of Education (including the 2007 document for grades K-8 quoted above). I am not claiming that teachers are actually exploiting in a strong way the above ideas, but, nevertheless, some national big projects for in-service teacher education have been developed by the Ministry of Education itself, drawing on these curricula (e.g. M@tabel for secondary school teachers; PLS for high school teachers and similar). In a sense, for Italian researchers involved in teacher education, it is good to have extended document (implicitly quoted in the national standard) with a lot of methodological indications and examples about didactics of mathematics that exploit the most meaningful findings of Italian didactical research.

This experience suggests me some observations and questions about the French situation, although I have to confess that my knowledge of the French situation is quite limited.

I have read the document “le B.O.” (19 juin 2008). I was a bit surprised to have not found any mention to the famous tradition of didactical research in France. For instance I know (and use)

Vergnaud’s studies about conceptual fields (and problem solving);

Brousseau’s studies about the Theory of Didactical Situations

Chevallard studies about Anthropological theory

Artigue’s studies about didactical engineering.

I have not found any trace of them in the standards. Surely curriculum is defined not only by standards but also by textbooks, journals for teachers, teacher’s guides, training courses and so on. For instance I am familiar with a very good material for pre-primary school (DVD):

Apprentissage Mathématiqué en maternelle (Hatier) by Briand, Loubet, Salin.

In this multimedia (that I have exploited also for my students) there is reference to Brousseau TDS.

But I was not able to find any reference to this large body of research studies in the standards B. O.. Which are the relationships between didacticians and the institutions in charge for preparing standard or organizing teacher education?

An example of the links between fundamental research, classroom practice and teacher education in Italy

This example comes from my personal experience as a researcher, as a “theoretician” and as a teacher educator.

The theoretical framework of semiotic mediation after a Vygotskian approach.

After some decades of teaching experiments in classrooms at all school levels (from primary to secondary), Bartolini Bussi & Mariotti have published a paper to illustrate the theoretical framework, named Semiotic mediation after a Vygotskian approach.

Bartolini Bussi M. G. & Mariotti M. A. (2008), Semiotic Mediation in the Mathematics Classroom: Artefacts and Signs after a Vygotskian Perspective. In: L. English, M. Bartolini, G. Jones, R. Lesh, B. Sriraman, D. Tirosh. Handbook of International research in Mathematics education (2nd edition). p. 746-783, New York: Routledge Taylor & Francis Group.2

(this document is available; Mariotti held a course in the last école d’été).

In short, the focus is on artifacts (and on semiotic activity), ranging from classical artifacts (e. g. abacus, compass, curve drawing devices, pantographs, and so on) to ICTs (dynamic geometry softwares, softwares for symbolic calculus and so on). In this framework the instrumental approach by Rabardel is exploited together with Vygotsky’s semiotic approach (with the basic ideas of Zone of Proximal Development, genesis of higher psychological functions, genesis of consciousness and similar). The theoretical framework has been built in a dialectical and dialogical way together with teacher-researchers to interpret, design and analyze effective classroom practices where a major focus in on artifacts and on the teacher’s role in acting as a cultural guide.

Exploiting the theoretical framework in a complex program for teacher education developed at the regional level.

This theoretical framework has been exploited in a very complex project for in-service teacher education (secondary school) in the region Emilia-Romagna. 5 laboratories of mathematical instruments – mostly related to geometry - have been opened (2008/10) in 5 different provinces and a regional network of 80 secondary school teachers has been created to cope with didactics of mathematics in the laboratory. 3 more laboratories are on the way in 2012 in 3 more provinces with a similar organization. A complete report of the first two-year period is available in Italian

http://www.mmlab.unimore.it/site/home/progetto-regionale-emilia-romagna/risultati-del-progetto/libro-progetto-regionale.html (editor Martignone).

(other materials are available at: http://www.mmlab.unimore.it/site/home/progetto-regionale-emilia-romagna.html , Michela Maschietto presented the project in Lyon – 2010 - http://ife.ens-lyon.fr/editions/editions-electroniques/apprendre-enseigner-se-former-en-mathematiques).

(Short presentation in English are available (RF at PME 35 – 2011)).

The theoretical framework has allowed to produce a standard script of mathematical exploration of a mathematical instrument for teachers that can be summarized as follows (in parenthesis one can see the main reference for this particular question): in this case we imagine to apply it to the study of a Scheiner pantograph, the famous instrument to enlarge/reduce drawings.

1) How is this instrument (better artefact) made? (Rabardel instrumentalization)

2) What does the instrument (better artefact) do? What can you do with the instrument (better artefact) to solve a particular problem? (Rabardel instrumentation)

3) Why can the instrument (better artefact) do that? (Vygotsky mediation) – up to the construction of proofs.

4) What could happen if .....? (problem solving) – e.g. if some of the geometric features of the artefacts are changed?

The third question is related to the link between the artefact and mathematics by means of semiotic mediation.

The transfer from tasks for teachers to tasks of teacher (i. e. from tasks in teacher education to tasks in mathematics classroom).

This script is used in teacher education and is exploited by teachers in the construction of classroom tasks (classroom tasks are different from teachers’ tasks as they are to be modified by each teacher according to the classroom context).

This example from Italy is consistent with Italian curricula (see the above quoted Matematica 2001 / 2003 / 2004). It has been realized together with the stakeholders of the Region (regional and school authorities, pedagogists, policy makers and so on). All the involved teachers have made experiments in their classrooms (also from vocational schools, where the tasks were transformed in “physical” construction tasks to produce working copies of instruments [1]) and produced logbooks. Half of the teachers have produced scientific reports and an increasing number of teacher has produced paper for teacher journals. The activity is going on after one year of the “official” conclusion.

Transfer to other school levels

What is interesting in this approach is that we have applied it also to teacher education for pre-primary and primary school. For instance we have applied a similar script (with an additional “0-task”, see below) to explore a big size Slavonic abacus for pre-primary school and to explore a “pascaline” for primary school.

The 0-task is:

0) What is this? Can you imagine a story about it?

It opens a narrative space, quite useful for young students, and realize a context suitable to devolution (according to Brousseau theoretical construct).

This 0-task might be not suitable for elder students, because they are not accustomed to be interviewed in a very open way on a task that is perceived as not-mathematical. However in some case we use it, maybe later, to introduce the history of the artefact (as our artefacts are all reconstruction of historical instruments).

Research perspective

National seminar 2012

Now in Italy we are moving, as a group [2], towards research on mathematics teacher education. In the last national seminar (January, 26-28, 2012) a group coordinated by Arzarello (http://www.seminariodidama.unito.it/) has chronicled the development of activity about mathematics teacher education in the last 20 years (quoting Dumas’ Vingt ans après). They started from the so-called Research for Innovation

(ARZARELLO F., BARTOLINI BUSSI M.G. (1998). Italian Trends in Research in Mathematics Education: A National Case Study in the International Perspective,. In: KILPATRICK J., SIERPINSKA A.. Mathematics Education as a Research Domain : A Search for Identity. vol. 2, p. 243-262, DORDRECHT: Kluwer).

Then they used Chevallard’s Anthropological theory to study the development of praxeologies in the field of mathematics teacher education in Italy in the collaboration between didacticians, institutions, stakeholders.

This is a national research project we are launching just now to be funded by the Ministry of Education. The strict links between didacticians and institutions (embodied by Arzarello, who is both an outstanding researcher and the president (for nine years up to 2005) of the Italian Commission for Mathematics Instruction) allowed to have an institutional acknowledgment which left traces in the governmental documents and in some very large project for in-service teacher education launched by the Ministry of education.

I am very interested in knowing whether such links are present in France too and how you foster and increase them. I mean the links between fundamental research in didactics of mathematics, institutional development of Standards and curricula and classroom practice.

Notes:

[1] It is worthwhile to recall that Emile Borel (1904, Lecture at the Pedagogical museum in Paris) immagine a laboratori of mathematics as a carpenter workshop (menuiserie).

[2] The National group of researchers in didactics of mathematics has recently formed a scientific association similar to ARDM. This Association has a yearly meeting in the National seminar named after Giovanni Prodi, that a couple of weeks ago held the 29° session.