Percorso di formazione ad un sapere riflessivo nella professione medica attraverso l’uso dei film
Mese: settembre 2009
Matematica e scuola
Problemi personali poco simpatici quanto ineludibili mi obbligano a limitarmi agli adempimenti.
Aspettanto che passi ‘a nuttata trovo consolazione in letture amene. È la volta di
Un libro delizioso, scritto da Stefano Beccastrini e Maria Paola Nannicini, che offre numerosi spunti per collocare storicamente il pensiero matematico nell’insegnamento della matematica nella scuola primaria.
Quella dell’insegnamento della matematica è una questione che mi ha sempre turbato e ogni tanto ci ritorno perché la matematica è fondamentale per la soluzione di problemi di ogni tipo nella vita privata e nel lavoro, perché è una formidabile palestra del pensiero e perché è bellissima. E invece, come dicono gli autori di questo libro, è la cenerentola delle materie.
La suddivisione dello scibile in materie ben distinte, e principalmente la suddivisione fra mondo umanistico e mondo scientifico, è probabilmente una delle cause principali, anche se non l’unica, dell’inadeguatezza della scuola a formare cittadini pensanti e non solo abili (più o meno) a fare un mestiere.
Non si dovrebbe parlare di materie bensì di prospettive. Non dovrebbe essere tanto importante cosa insegnare ma insegnare a vedere qualsiasi cosa dal maggior numero di prospettive possibili: la prospettiva storica, la prospettiva matematica, la prospettiva geografica, letteraria, sociale, politica e via dicendo. In altre parole, si dovrebbe insegnare che non è mai tempo perso quello speso a risalire una connessione insospettata con un mondo del tutto diverso. Non è tempo perso nemmeno quando a causa di tali diversioni si rischia di non finire il programma.
Per l’appunto, la matematica, alla fin fine, nient’altro che di questo è fatta: connessioni fra mondi apparentementi lontani. Connessioni perfettamente e minuziosamente definite e quindi spesso faticose da conquistare; connessioni che generano stupore, piacere estatico; connessioni senza le quali ciò che conosciamo come tecnologia non avrebbe mai visto la luce.
Ecco, a me sembra che la proposta di Stefano e Paola vada in questa direzione prendendo le mosse dal momento più importante, quello nel quale gli insegnanti accompagnano i bambini nei loro primi passi in quella che dovrebbe essere una passeggiata in un bosco incantato anziché in un labirinto pieno di mostri dispettosi.
Viareggino fra i lucchesi
Non riesco a metter mano a nessuna delle varie cose che mi pareva di voler fare, oppresso dagli adempimenti che in ottobre sempre montano e che quest’anno mi paiono più uggiosi che mai.
Così finisce che il più delle volte, espletati gli adempimenti, vado a farmi raccontare storie, in questo periodo da Mario Tobino.
Traggo dal racconto “Capitolo nero” in
I lucchesi pensano sempre al denaro, i viareggini non ci resistono, la loro mente si perde in vaneggiamenti, in sogni. I lucchesi hanno il cerchio delle mura, che è contornato da un cerchio di monti, da secoli lì ruminano monete, tentati dalla lussuria e spaventati dal peccato, ugualmente bramosi di afferrare e consapevoli dell’odiata morte che li farà vuoti di ogni possesso. I viareggini hanno il mare, la spiaggia è molle e aperta, non conosce né insenature né scogliere. I lucchesi sono delle volpi, il commercio è la loro massima passione; i viareggini ci si annoiano e non lo sanno esercitare, un astuto negoziatore è guardato con irritazione, con uggia, quasi un traditore, uno che tende brutti lacci all’abbandono giocoso della vita. I lucchesi sanno stare in silenzio, calcolando dentro la testa, non si mettono a chiaccherare, chi si confida è da loro considerato un baggiano da giustamente gabbare. I lucchesi non hanno amici, non credono esista l’amicizia, la simulano per trarne beneficio. I viareggini stanno volentierissimo insieme e all’aperto, la loro casa è la strada, le loro porte sono spalancate che ci entri l’aria, la luce, gli amici; i viareggini non sopportano di stare nel chiuso, al contrario dei lucchesi che si aggirano cauti nelle loro strette, buie, contorte strade, fitte e zeppe di muri e muraglie, a loro volta fasciate dalle alte mura delle città, dai bastioni. E intorno il cerchio, l’anello dei monti, quasi che quel posto sia l’unico stato della terra. l’unico vero governo, al di là delle mura e di quei monti, l’albergo del diavolo.
…
I lucchesi hanno una educazione, si comportano, specie nella loro città, con cautela; per le strade camminano in silenzio non scontrando né questo né quello, non guardano con spalvaderia né con sfida, dentro la loro testa attentissima l’ipocrisia e la prudenza, non sprecare le forze in esterni atteggiamenti, che non ottengono altro che danno. I lucchesi sono risparmiatori al massimo; in quel cerchio di monti che li serra, dentro la serratura delle mura non penetrano soffi di novità, siano queste nefaste o apportatrici di bene. Essi conservano tutto e sopra ogni altro l’esperienza, che dice di stare nella misura, anzi al di sotto di questa, fare il passo ancora più breve di quello che la gamba potrebbe compiere con facilità. I lucchesi non hanno cultura letteraria, umanistica, scientifica, né desiderano impadronirsene, sono soltanto educati di maniere, esatti nel loro mestiere, ricchi di consuetudini. Essi ignorano che esista il genio, non riconoscono il talento, non vogliono neppure sapere dell’estro. Stimano ed anzi adorano chi eccelle nei negoziati. Tutte le volte che in Lucca nasce un uomo di qualità, se ne deve andare poiché essi non lo lasciano respirare, gli tolgono l’ossigeno, ignorandolo. I sogni, le fantasie, i furori di grandezza, di gloria, di generosità, il programma di una felicità futura, sono dai lucchesi giudicati giochi di bambini o di dissennati o di fannuloni o di volgari imbroglioni. I lucchesi aborrono il teatro, odiano che sulle scene si rappresenti la vita come è, che il ladro lo si veda che rubi, il prete che non ci crede, l’avaro sia mostrato nella sua miseria. I lucchesi stanno rinchiusi in casa, dove non ricevono nessuno, neppure i conoscenti.
I viareggini sono ignoranti, non solo senza cultura ma anche maleducati, i loro modi sono troppo aperti, con molto dello spavaldo, del senza ritegno, dello sfaccendato, del fannullone che neppure si preoccupa dell’accuratezza del vestire, e, d’un tratto, invece è tutto dipinto da damerino settecentesco, teatrante e vanitoso. Il viareggino fa come la natura di Viareggio in primavera, dalla burrasca alla incantata trasparente notte di stelle.
CCK09: How to look at a blogroom (3)
(Versione italiana)
Abstract, A kind of introduction
Beginning to look at numbers
After the kind of introduction it would be natural to give some details about methods and tools but I prefer to further clarify what we are about first.
I’m reporting data relative to an online class of 21 students who were teachers of primary and secondary schools from all over Italy. The class was about using digital technologies in teaching activities and took part of a three-years graduation in an all-online university called Italian University Line based on the cooperation of five italian universities
- Università degli Studi di Milano-Bicocca
- Università degli Studi di Firenze
- Università degli Studi di Macerata
- Università degli Studi di Lumsa Roma
- Università degli Studi di Palermo
and a National Agency devoted to the research and development in education: Agenzia Nazionale per lo Sviluppo dell’autonomia Scolastica .
This is the class where the blogroom method was most successful. The students reported this experience as a very rewarding one and all had the feeling to have learned a lot of things in a meaningful way. Furthermore, they felt to be engaged in a community.
The impression of having learned a lot of things is remarkable because the course was organized in a very loose way. I did not declare a “program” as a list of contents but I suggested a number of objectives, partly to be determined in function of the experiences and the needs of students. I claimed that contents would emerge from activities and that the whole course had to be thought of as a path, not necessarily the same path for all the students.
The case of this class was advantaged by the small number of students with respect to the majority of my classes which may reach more than 200 students. However there were also strong adverse factors, such as the busy life of adult students and the 100% online nature of the course, that made the experience significant.
The thesis I’m anticipating here is that, the perception of meaningfulness claimed by the students has to be advocated to the emergence of a living community, very much akin to a community of practice, a context where the learning process is substantially improved.
The idea is therefore to use this class as a kind of reference to be compared with other classes, in order to understand the most significant factors affecting the learning experience.
The formal part of this course consisted in eight assignments where the students were required to write a short essay or to comment about a specific activity. Therefore, in the conventional perspective, each student was supposed to produce eight posts on its blog, meaning a total of 168 posts for all the students. Instead, the students, wrote 484 posts meaning that 316 posts have been written spontaneously. In other words, we can say that the students wrote about 23 posts each instead of eight.
One could wonder how much pertinent these extra 15 posts per student are. Here comes another reason I have chosen this class among many others. These students are in the range 30-50, therefore they usually have families duties and a job. That is to say that these are highly motivated people and, actually, almost all the posts concerned the topics of the course. Having said that, I do not mean that students are blamed for broadening the subjects of their posts, and very often their blogs are quite reach and colored, indeed. However, for what I’m trying to show here, it is good to know that the extra posts had a high probability of being “meaningful”.
So far, it appears that those 21 people did a great deal of work producing a 200% more of what it was expected. Even if this statement has some value and can be used for comparison with other classes, it is worthwhile to look at the results in more detail.
First of all, the average result may be a significant estimator of the global performance of a class but it is a very poor estimator of individual performances that are by no means homogeneous. Let us look at the distribution of number of posts per student. In reality, the figure shows the number of posts for each blog. In all cases this corresponds to the number of posts per student except in one case were two students shared one blog. The point is the first one at the left. Keeping this exception in mind, in what follows I will refer to the number of posts per student.
We see that there are very large differences among the students. As a matter of fact, we found that the 484 posts were distributed among 143 “due posts” and 341 “extra posts”. This is due to the fact that the five students at the right wrote less then eight posts for a number of reasons: some had still to finish at the time of data uptake and some others needed a reduced number of credits for this course. Fifteen students wrote more than the minimum of eight posts, and about half wrote more the double of the required posts.
The 70% figure of “extra posts” (341 over 484) can be thought of as a measure of the community effect, in some way. In fact, it is well known that when a population is just free to undertake something or to contribute actively, the so called 1-9-90 rule holds true. That is over 100 people, one starts contributing, 9 follow the first one and the remaining 90 just look at what’s going on. These last people are the so called lurkers in the Internet. According to such a rule, the plot should be quite different, showing perhaps one or two outstanding contributors and the rest with just eight posts. The plot should therefore be completely flat except for a very narrow peak at the extreme left, instead of being almost triangular.
However, even if the distribution of posts gives an idea of the contributions exceeding the required minimum, we still miss the most important side, which is represented by comments exchanges among the students. In order to get a visual feeling of such kind of activity it is useful to look at the sociogram of the blogroom.
The sociogram shows the connections among the participants as well as their role. The existence of a line between two nodes means that at least one of the participants relative to that nodes made a comment to the other but it may also mean that multiple comments in both directions were made. Red nodes represents students belonging to the class, blue nodes students from other classes, green nodes educators external to the class that got involved in the blogroom, the light blue one is me, the teacher. The layout of the sociogram was determined by means of an algorithm called “Multidimensional Scaling (MDS)”, which is capable to discriminate somewhat the role of different actors in the network solely on the base the connections.
Actually, the MDS algorithm did a pretty nice job by placing the most active actors to the right and the less active ones towards the left side. However, the most relevant point is the amount of connections among the nodes and, in this respect, it is worthwhile to compare this sociogram with the extreme simulations I showed in the introduction, were a conventional class and an almost all-connected classes were represented.
The star-like sociogram describing a conventional class is compatible with the plot of the number of posts. At most, we cold add arrows towards the teacher to say that the students produced a significant amount of contents and that this fact made a difference for the teacher.
This diagram represents a conventional class were the students produced a lot of content, beyond what was required by the teacher.
We could call this a kind of high-performance class with highly performant students but there isn’t any “network effect”, everybody is just working harder, both the students and the teacher. Instead, in our blogroom a great deal of communication took place among the members.
The sociogram of our blogroom gives a spatial perception of the web of connections but it is interesting to go further by quantifying those connections. We can do this by plotting the number of comments that have been made or have been received and, eventually, some measures of the role played by different members in the network.
The first twenty points at the left represents students belonging to the class and are the same of the precedent plot about the number of posts. The nine points at the right side are relative to students belonging to other classes or educators that engaged spontaneously in the blogroom. The last point at the right is the teacher. The number of posts written by these last members (black line) is not plotted.
The black line corresponds exactly to that of the precedent plot. The difference is that here are reported data for all the members of the blogroom and not only the students, such as students not belonging to this class or educators that became interested in the activities of this blogroom. For these new members the number of posts in their blogs is not reported here so that the black line drops to zero towards right.
The red line shows the number of comments received from other members by each student. In the jargon of social network analysis this is the “indegree” value of a node. The green line shows the comments made to other members blogs by each student, that is the “outdegree” value. Indegree and outdegree are measures of “centrality” of a node, that is the relative importance of that node in the network. In our case, a high indegree describes a prestigious actor and a high outdegree an outgoing one.
There is a rather loose correlation among the number of posts, the number of received comments and that of written comments. There are people that wrote quite a large number of posts but experienced few contacts with respect to others who wrote a comparable quantity of posts. Conversely, there are someone who preferred to comment others blog instead of writing on its own. Such particular behaviours are worthwhile of careful consideration because they may reveal interesting approaches as well as potential problems. It is difficult to follow all the interactions that are going on within the blogroom, thus this kind of representation may be useful detect relevant situations.
Even more interesting is the quantity plotted as a blue line: the so called betwenness, another measure of centrality which takes into account how much a node “is between” other nodes. The following picture taken from the article Centrality of Wikipedia (in this article you can find also the rigorous definition of betwenness centrality) gives an intuitive idea:
Color (from red=0 to blue=max) shows the node betweenness.
Together with the degree measures of centrality, betwenness is among the “classic” parameters of social network analysis since it was introduced in the seventies, however it is still considered a very effective measure of centrality. The difference with respect to the degree measures of centrality is that these are local, counting how many nodes are connected to a node, whereas betwenness measures how many paths among couple of nodes a node is intercepting, were the nodes may be also many steps apart.
That is, betwenness measures how much a node is connecting different parts of the network and not only neighbour nodes. Actually, I found that this centrality measure was very good at detecting the most active students.
A node with high betwenness acts as a sort of catalyst enabling the spread of a 1-9-90 kind of participation towards a broader distribution. Thus, the high-betwenness nodes may play a crucial role for the blogroom life. We can think to that nodes as “key nodes” to which it is worthwhile to pay attention in order to facilitate the rise of the community.
It is not to say that those students associated to “key nodes” necessarily will get high grades. This may happen but it is not the point. It is to say that the teacher should spend some time in facilitating the action of “key nodes” because this may foster the health of the whole network improving, in turn, the participation of all its members.
I foresee to use betwenness in much larger classes, where it is difficult to become aware of all potential “key nodes” in time. I would like to detect those nodes at an early stage so as to exploit their potential before the class is going to finish.
The instrumentarium of social network analysis is extremely reach and with the simple examples I have made we have only scratched the set of possibilities. Of course, abundance of possibilities does not mean certainty of success but it is worthwhile to explore them provided we stick to this objective: devising methods to nourish the community in the blogroom giving rise to an experience of meaningfulness in its members.
In successive posts I will tell about other ways to look at blogrooms and as well as about the methods I’m applying.
Risposta ad un amico su massa e individui
Un mio caro amico (ex studente) commentando il post Il mio primo sociogramma, chiede:
… poi voglio fare una domanda veloce schietta e banale: te arf iamarf arfs dici spesso che la massa è ganza e che bisogna star attenti alle dinamiche che emergono dall’interazione fra persone più che alle persone stesse ecc. ecc., come la storia di quei prof che dovrebbero star più attenti alle dinamiche della comunità-classe che alle singole persone
e tuttavia a me mi interessano anche – se non altrettanto – le singole persone.
quindi voglio chiedere: che valore hanno per te le singole persone che compongono l’insieme?
Le persone singole hanno per me interesse primario.
Nella classe tradizionale la persone singole vengono isolate le une dalle altre da un metodo che è sostanzialmente militare: tutti sincronizzati e ceffate a chi comunica trasversalmente.
Io sostengo che la classe avrebbe anche la possibilità di essere apprezzata e coltivata come essere vivente a se stante perché è così che funzionano gli esseri umani.
In particolare le donne che hanno sempre fatto “rete” più facilmente degli uomini. Il senso della comunità è un fattore di civiltà e di crescita per i singoli membri.
Io credo che fare maggiore attenzione alla comunità che può svilupparsi in una classe finisca col beneficiare assai i singoli, il cui benessere e la cui crescita ci deve interessare sopra a tutto il resto.
In principio la tecnologia non c’entra nulla con tutto questo. Mi sono state narrate storie di insegnanti che hanno applicato queste idee trent’anni fa. Mosche bianche.
La tecnologia può aiutare e basta.
P.S. A me capita di dire che la massa è ganza ma su larga scala, al livello dei milioni e anche centinaia di milioni di persone che finiscono per rappresentare un terreno sul quale possono germogliare cose straordinarie. Nell’ambito di classi scolastiche parlo di comunità e non di massa. È una cosa diversa. In comune ci può essere il fatto di imparare a riconoscere la vita quando e dove essa appare. E per rispondere anche al tuo P.P.S. l’interesse per le dinamiche emergenti nasce proprio dalla stupore di vedere emergere processi vitali su substrati considerati solitamente passivi. Mi ha aiutato molto Fritjof Capra.
CCK09: How to look at a blogroom (1)
A kind of introduction
Why I’m doing this work?
I have been applying the blogroom method during the past four semesters with about 800 students out of a total of about 1400 distributed in 20 different curricula, some of the Faculty of Medicine and some of the Faculty of Education Science.
I have being applying this method when teaching digital literacy or related subjects, so far.
Short description
To describe the method in a nutshell, there are only a couple of conventional lessons at the beginning of the course, where the students are told the basics of the method. Then, the teacher is available for some weeks in a computer room to help students when needed, to engage in discussions or to make seminars.
In the computer room the students are encouraged to help each other. The rest of the course life take place within the blog community. The teacher gives assignments in his or her blog and the students are supposed to write their essays as posts in their own blogs.
The assignments are not intended to “cover” a range of topics or to assess skills directly but are conceived as stimula to explore autonomously. We may say that assignments are chosen just to address some broad relevant topics.
In online courses where it is impossible to organize conventional lessons and other kind of meetings, the blogroom is supported with weekly virtual meetings.
Peculiarities of a blogroom
Besides the rough participation numbers and the distribution of grades, the teacher is left with a tangle of impressions and feelings, a very strong trace even if a subjective one.
Nonetheless, from such tangle of impressions some clear points emerge regularly:
- Something very different happens when the blogroom method “works”.
- The success of the method is very much peculiar to the curriculum. Even the degree of a priori acceptation – that is the number of students that decide to try the new method, since they have the freedom to choose between the new and the conventional approach – is variable, ranging from 10-20% to almost 100% depending on the curriculum.
- The most striking feature of a working blogroom is the emergence of an emotional component: a significant number of students get involved in a manner that is quite unusual in a conventional class.
- In a working blogroom a significant proportion of the students would like to continue and, as a matter of fact, the blogroom survives the term of the course.
In short, the blogroom appears to be a living being.
The problem
In the classroom the teacher follows each student separately and usually, the relationships among classmates are kept to a minimum.
On the contrary, in the blogroom, students are encouraged to engage in discussions with classmates and other people and not necessarily about the taught topics.
When students get the message, the blogroom comes to life in the shape of a network that is quite difficult to follow since a living network is a chaotic entity.
Of course, I remember well a lot of posts, comments and whole threads of discussions. I know very well many students and often I get a rather clear idea of their personalities. However, for sure a great deal of threads get lost and in my perceptions, the relative proportions may be distorted by my own feelings or preferences.
The idea
However, by its nature, the Internet has memory or, perhaps, is a kind of memory. It hosts life forms and often their traces impressed in the time course can be recovered. This happens to the blogroom, too. All the posts and the comment are retrievable from the feed aggregator and they can be analyzed by means of mathematical and graphical tools.
The idea is to try to construct ways to look at the blogroom in its entirety from different perspectives and having the possibility to focus on specific aspects.
We could put it in the following way. The blogroom is a living entity but its multidimensional nature may be quite complex and therefore we need instruments that allow us to look at it in a reliable way.
I believe that the idea could be interesting being aware that 1) the subjective perception of the teacher remain fundamental, 2) mathematical devices are not magic.
PS: I will be very grateful to anyone willing to tell me about similar works or papers on this subject.
Sociograms have been made with Stanet.
Andreas Formiconi 