wims at the university of milano-bicocca · 2010-05-28 · university of milano-bicocca – dip. di...

University of Milano-Bicocca – Dip. di Matematica e Applicazioni WIMS at Milano-Bicocca – pagina 1

WIMSat the University of Milano-Bicocca

Marina CazzolaDipartimento di Matematica e Applicazioni

Facolta di Scienze della Formazione

29 may 2010

WIMS


Petit historique IT-WIMS


September 2000: first Italian translation of WIMS

In a short, in the whole system, you should translateeach file or subdirectory terminating with an .en or a .frinto an additional file/subdirectory terminating with an.it. [. . . ]

To know what in the file is language-dependent (soshould be translated) from what is not (for example thecommands), you can compare visitor.phtml.en withvisitor.phtml.fr, to locate the places where the two filesare different.


WIMS• Petit historiqueIT-WIMS

Teacher training

Testing students’proficiency

E-learning


December 2000: first use with “real” students

Linear Algebra for students in computer science.

Use of the ready available exercises on the topic(mostly by Xiao Gang), in English (very fewtranslation into Italian).

Used both for self study and (“selectively”) forexams; 145 registered users.



Teacher training


E-learning


October 2002: wims.matapp.unimib.it listed as aWIMS mirror site

Sporadic work on the translation, as teaching to noclass suitable for the use of WIMS (no more linearalgebra course) and no time/not enought experienceto write down my own exercises.



Teacher training


E-learning


Novembre 2004: colleagues in Florence start usingWIMS.

Milano-Bicocca takes care of the translation of thecore/adm modules, while Florence starts totranslate the exercise modules.

Teacher training

WIMS

Teacher training


E-learning



WIMS

Teacher training


E-learning


October 2003: I was enrolled by the Facolta diScienze della Formazione

Teaching activity mainly in the pre-service primaryschool teacher training programme.

Courses not suitable for the use of WIMS

• no computer labs available for the students

• pedagogical concerns


WIMS

Teacher training


E-learning


Future teachers needs to acquire abilities that gobeyond computational skills: they need to be able toformulate abstract mathematical definitions orstatements and to write simple proofs.


WIMS

Teacher training


E-learning


Future teachers needs to acquire abilities that gobeyond computational skills: they need to be able toformulate abstract mathematical definitions orstatements and to write simple proofs.

In the academic year 2005/06 computer labs weremade available, so we could really start using WIMSwith our students. The use of WIMS was alsostimulated by the will of the faculty to startexperimenting “e-learning”.


WIMS

Teacher training


• Maths for primaryschool teachers• Basic computationalskills

• Manipulatives

• . . . contratto didattico

• must know• you do not reallyknow if you cannot tell

• Geometry

E-learning


Maths for primary school teachers

WIMS

Teacher training



• Manipulatives



• Geometry

E-learning


The pre-service teacher training university degreecourse is a four years program. Future teacher havethe following math courses

• “Istituzioni di matematiche I” (Elements ofmathematics, first part): arithmetic (first year)

• “Istituzioni di matematiche II”: geometry (secondyear)

• “Didattica della matematica 1B” (mathematicseducation): probability and example of goodteaching practice (third year)

• “Didattica della matematica 2”: more teachingpractice (fourth year)

Basic computational skills

WIMS

Teacher training



• Manipulatives



• Geometry

E-learning


http://wims.matapp.unimib.it/wims/wims.cgi?lang=it&+module=adm/class/classes&+type=authsupervisor&+class=8480673


WIMS

Teacher training



• Manipulatives



• Geometry

E-learning


The easiest thing to do is to use WIMScomputational capabilities in order to test students’basic computational skills at the very beginning oftheir course of study.

All the first year students (about 400/450 every year)has to take the WIMS test.

10% of the students fail the test

(Academic years: 2006/07, 2007/08, 2008/09 and2009/10).



WIMS

Teacher training



• Manipulatives



• Geometry

E-learning


The easiest thing to do is to use WIMScomputational capabilities in order to test students’basic computational skills at the very beginning oftheir course of study.

All the first year students (about 400/450 every year)has to take the WIMS test.

10% of the students fail the test

(Academic years: 2006/07, 2007/08, 2008/09 and2009/10).

But with students in their second and third year, weneed to go beyond testing basic computationalskills.


Manipulatives

WIMS

Teacher training



• Manipulatives



• Geometry

E-learning


. . . contratto didattico


. . . contratto didattico


When mathematics is concerned

• a few things are a must know

◦ being able to do the computations (with integers and withfractions),

◦ being able to convert between square meters and squarecentimeters, cube meters and cube centimeters,

◦ being able to tell a square from an hexagon

• but future teachers also need to be able to “talk” about maths

must know

WIMS

Teacher training



• Manipulatives



• Geometry

E-learning


you do not really know if you cannot tell


In my “Didattica della matematica 1B” course (third year), theexams consist in a WIMS test, a written test and an oralexamination.The written test partly consist of giving a full explanation of oneof the question already answered in the WIMS test.

Rispondere al seguente quesito WIMS, dando unapiena giustificazione (NOTA BENE : in questo esamesara valutata la capacita di argomentare e dare unagiustificazione e non semplicemente il raggiungimentodi un risultato corretto in quanto quest’ultimo aspetto egia stato verificato con la prova preliminareinformatizzata)



you do not really know if you cannot tell

WIMS

Teacher training



• Manipulatives



• Geometry

E-learning


Geometry

WIMS

Teacher training



• Manipulatives



• Geometry

E-learning


During this academic year for the first time we areexperimenting WIMS activities for the “Istituzioni dimatematiche II” course (second year).

• big effort in collecting (translating + creating)suitable exercises

• still think part of the final exam should consist ofexercises in which the studens have to “giveexplanations” (i.e. still need for a written exam)


E-learning

WIMS

Teacher training


E-learning

• E-learning project atBicocca• How to teach in ane-learning course?

• Active learning

• Problems andmathematics• Problem-basedlearning

• The “problem”

• Role of the teacher

• Guessing

• Explaining andteaching

• Success?


E-learning project at Bicocca


The university funds a projectwith the aim of experimentinge-learning on a vaste scale.

In particular the Facolta diScienze della Formazione(Faculty of education)(e.g. in the last academic yearthe faculty has 37 coursestaught in e-learning)

How to teach in an e-learning course?


Which tools and which activities are suitable for e-learning?

How to teach in an e-learning course?


Which tools and which activities are suitable for e-learning?

• Didattica della matematica (master)

• Didattica della matematica 1B (2005/06)

• Didattica della matematica 1B(rec-2006/07)

• Didattica della matematica 1B (2008/09)

Manipulatives

WIMS

Teacher training


E-learning


• Active learning




• Guessing


• Success?


Active learning

WIMS

Teacher training


E-learning


• Active learning




• Guessing


• Success?


Active learning

WIMS

Teacher training


E-learning


• Active learning




• Guessing


• Success?


What the teachers says in the classroom isnot unimportant, but what the studentsthink is a thousand times more important.The idea should be born in the students’mind and the teacher should act only as amidwife.1

1George Polya, Mathematical Discovery: On Understanding, Learning,and Teaching Problem Solving, volume Combined edition, John Wiley andSons, 1962

Problems and mathematics

WIMS

Teacher training


E-learning


• Active learning




• Guessing


• Success?


. . . there is a grain of discovery in thesolution of any problem. Your problem maybe modest; but if it challenges yourcuriosity and brings into play your inventivefaculties, and if you solve it by your ownmeans, you may experience the tensionand enjoy the triumph of discovery. Suchexperience at a susceptible age maycreate a taste for mental work and leavetheir imprint on mind and character for alifetime.2

2George Polya, How to solve it: a new aspect of mathematical method,Princeton University Press, 1945

Problem-based learning

WIMS

Teacher training


E-learning


• Active learning




• Guessing


• Success?


Problem-based learning (PBL) is a constructivistlearner-centred instructional approach based on theanalysis, resolution and discussion of a givenproblem. It can be applied to any subject, indeed itis especially useful for the teaching of mathematics.

PBL “is an instructional (and curricular)learner-centered approach that empowers learnersto conduct research, integrate theory and practice,and apply knowledge and skills to develop a viablesolution to a defined problem”.3

3John R. Savery, “Overview of problem-based learning: Definitionand distinctions”, The Interdisciplinary Journal of Problem-based Learning,1(1):9–20, 2006

Problem-based learning


Typically a PBL session follows these steps 4:

• pupils are given a problem they have never seen before;

• they discuss the problem and/or work on the problem insmall groups, collecting information useful to solve theproblem;

• all the pupils gather together to compare findings and/ordiscuss conclusions; new problems could arise from thisdiscussion, in this case

• pupils go back to work on the new problems, and the cyclestarts again.

4Marina Cazzola, “Problem-based learning and teacher training in mathematics”, preprint,2008

The “problem”


The “problem”


Doc

ente

Avendo completato la procedura di iscrizioni, siamopronti a cominciare.Chiedo a tutti voi una partecipazione attiva fin da ora(“non si impara matematica se non si fa matematica”).Il corso comincia con una revisione del primo capitolodel testo “La lotteria a Babilonia” di Giuliano Spirito, chee il testo fondamentale per la parte di calcolo dellaprobabilita.Come primo “compito” vi chiedo di iniziare la lettura inparticolare dei paragrafi “Che cosa e unacorrispondenza” e “Una definizione elegante dicorrispondenza”.

The “problem”


Doc

ente

Nella revisione del testo vi chiedo in particolar modotener conto dei seguenti aspetti:

• che cosa e una corrispondenza?

• saper dare esempi di corrispondenze (oltre a quellicontenuti nel testo)

• data una corrispondenza, saperla rappresentare indiversi modi

The “problem”

WIMS

Teacher training


E-learning


• Active learning




• Guessing


• Success?


Doc

ente

Infine vi chiedo di leggere con attenzione i“Quesiti”, che costituiscono una ricca fonte diesempi, e di scrivere qui di seguito glieventuali dubbi che vi sono sorti e che nonsiete riusciti a risolvere nel lavoro di gruppo.

The “problem”

WIMS

Teacher training


E-learning


• Active learning




• Guessing


• Success?


• Read the textbook

• try to understand definition, explanetions,examples&exercises given in the textbook

• discuss about them with your collegues in yourgroup using the forum

• practice with WIMS exercises completing theassigned worksheets

• write down an explanation of your findings for theteacher

The “problem”

WIMS

Teacher training


E-learning


• Active learning




• Guessing


• Success?


The assignement emphasized the need of “activeparticipation” by all of the students: only after goingthrough the steps above the students could ask forhelp from the teacher.

One of the main themes of the course wascombinatorics, with respect to “countingcorresondance, permutations and functions”, so itwas easy to implement WIMS exercisescorresponding to the exercises given in thetextbook

Note : in the spirit of “discovery” the students wereasked to find out the formulas by themselves.

http://wims.matapp.unimib.it/wims/wims.cgi?lang=it&+cmd=new&+module=local/probability/oefprob.it&+exo=funzioni&+qnum=5&+qcmlevel=6&+scoredelay=

Role of the teacher

WIMS

Teacher training


E-learning


• Active learning




• Guessing


• Success?


Role of the teacher

WIMS

Teacher training


E-learning


• Active learning




• Guessing


• Success?


As the aim of the course was indeed “teach toteach”, the actions of the teacher where ment tostimulate metacognitive reflections

Doc

ente Secondo voi perche gli esercizi [del test di

autovalutazione] vi costringevano a lavorarecon numeri anche piuttosto alti?

(Why do you think WIMS poses exercises with quitebig numbers?)

Role of the teacher

WIMS

Teacher training


E-learning


• Active learning




• Guessing


• Success?


It was good to discover that some students did getthe point

mi viene da pensare che usare i numeri alti estata una strategia per “costringerci” atrovare una generalizzazione e forse ancheuna giustificazione, che, a pensarci bene,con numeri piccoli non e necessaria.

Stu

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(I sort of think that big numbers force us to find out ageneralization ad a justification of our procedure,justification we do not feel the need for with lowernumbers)

Students and WIMS


http://wims.matapp.unimib.it/wims/wims.cgi?lang=it&+module=local/probability/oefprob.it&exo=funzioni&qnum=1&qcmlevel=3&scoredelay=&special_parm2=&special_parm4=

Students and WIMS


io non conoscevo le formule e dopo aver tentato invanodi trovare qualche chiarimento sui libri ho provato adoperare da “profana” cioe senza conoscenze (o poche)su questi argomenti [. . . ] ho iniziato a provare eriprovare il test segnando ogni volta quale fosse larisposta corretta ai quesiti e anche qui ho costruito unatabella Subito mi sembrava che non ci fosse alcunarelazione tra il risultato ottenuto e il numero deglielementi degli insiemi. Ho comunque iniziato a provare:di nuovo scomponendo i numeri ottenuti come risultato.Ed ho scoperto che. . .Secondo voi questa costituisce una giustificazionesoddisfacente? Perche?

Stu

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Students and WIMS


I did not know any formula and after looking for somehint in the books I tried to act as a “profane” that is withalmost no knowledge about these subjects[. . . ]I started trying and trying again the exercise, andwriting down the correct answers filling up a table. At afirst glance I could not find any relation among thosenumbers. Then I tried to factorize the result. And Ifound out that. . .Do you think this is good enought as an explanation?Why?

Stu

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Guessing

WIMS

Teacher training


E-learning


• Active learning




• Guessing


• Success?


Let me recommend here just one littlepractical trick. Before the students do aproblem, let the guess the result, or a partof the result. The boy who expresses anopinion commits himself; his prestige andself-esteem depend a little on the outcome,he is impatient to know whether his guesswill turn out right or not, and so he will beactively interested in his task and in thework of the class—he will not fall asleep ormisbehave.

Explaining and teaching


Ho risolto un dubbio, ma con esercizi diversi io nonriesco a ragionare e sbaglio...Se lancio il dado 5 volte ed esce sempre lo stessorisultato la probabilita non dovrebbe essere di 1/6 allaquinta?

Stu

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e1

http://wims.matapp.unimib.it/wims/wims.cgi?+lang=it&+cmd=new&+module=local/probability/bordin.it&+exo=dadistep&+qnum=1&+qcmlevel=3&+scoredelay=&+intro_qcmpresent=4&+intro_presentsol=1&+intro_check=1&+intro_check=2&+intro_check=3&+intro_check=4&+intro_expert=yes



I solved one of the exercises, but with a different one Icannot think and end up making mistakes . . .If I throw a dice 5 times the probability I always get thesame result shouldn’t be 1/6 to the fifth? S

tude

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tude

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attenta al testo. . . ti dice stesso risultato, non “sempre ilnumero 6” (per cui e giusto quel che dici tu. . . e vabene come prima parte di risoluzione di questoesrcizio, ma poi cosa succede?)Dicendoti lo stesso risultato vuol dire non deviragionare su un numero preciso. . . ma su. . .

Stu

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tude

nte

1

read carefully. . . it says same result, does not say“always number 6” (what you say is right. . . it is the firststep, but then what happens?)Saying the same result means you do not have to thinkof a precise number. . . but . . .

Stu

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Come sempre non capisco...Al primo lancio io ho la possibilita che mi esca qualsiasinumero giusto? Quindi 6/6. E’ da qui in poi che devostare attenta. Al 2o lancio che possibilita ho che miesca lo stesso numero del lancio precedente? Io direi1/6, ma non ne sono sicura.Facendo questo ragionamento io direi di avereprobabilita 6/6x1/6x1/6x1/6, ma sara sbagliato.

Stu

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As usual I do not understand. . .At the first throw I can get any number, right? So theprobability is 6/6. From now on I have to be careful. Atthe second throw what is the possibility that I get thesame number as in the previous throw? I would say1/6, but I am not sure.Going on like this, I would say the result is6/6x1/6x1/6x1/6, but it is certainly going to be wrong.

Stu

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As usual I do not understand. . .At the first throw I can get any number, right? So theprobability is 6/6. From now on I have to be careful. Atthe second throw what is the possibility that I get thesame number as in the previous throw? I would say1/6, but I am not sure.Going on like this, I would say the result is6/6x1/6x1/6x1/6, but it is certainly going to be wrong.

Stu

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Aspetta, andiamo per gradi, prova prima a risolverequesto:“Lacio un dado 5 volte. qual e la probabilita che escesempre 4?”risolto questo andiamo avanti. . .

Stu

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Io direi 1/6 alla quinta, ma ormai non ho piu certezze.

Stu

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Io direi 1/6 alla quinta, ma ormai non ho piu certezze.

Stu

dent

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e qui ci siamo, quindi potremmo dire che per calcolarela probabilita di avere sempre lo stesso numero faccio(con questi numeri) (1/6)alla 5.Se nel teso del mio esercizio, pero non ho piu unnumero preciso a cui fare riferimento cosa posso fare?Per prima cosa calcolo la probabilita dell’evento comese avessi un numero in particolare, ma questo non mibasta piu... 1/6 alla quinta (vale solo per un numero allavolta)...

Stu

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cosa posso fare se la mia richiesta e generica e puoriguardare qualunque numero (ovviamente tra quellipresenti sul dado)?

Stu

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Stu

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Non lo so... 1/6 alla quinta vale per ogni numero... nonso... non so...

Stu

dent

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Stu

dent

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Non lo so... 1/6 alla quinta vale per ogni numero... nonso... non so...

Stu

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1/6 alla 5 vale per ogni numero preso singolarmente...ma se io non so a quale numero fa riferimentol’esercizio, che mi dice che potrebbe essere unoqualsiasi, nell’incertezza (diciamo cosı), li devoconsiderare tutti... proviamo a scrivere i casi per esteso

Stu

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“Lancio un dado 5 volte. Qual e la probabilita che escasempre lo stesso risultato?”1. Scomponiamo l’esercizio, cosa puo accadere?- lancio il dado 5 volte ed esce sempre 1 (che calcolofaccio qui?)- lancio il dado 5 volte ed esce sempre 2 (e qui?)- lancio il dado 5 volte ed esce sempre 3 (e qui?)(continua tu...)come lego tutto cio per trovare la probabilita?

Stu

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Ogni volta per ogni numeo io dovrei fare 1/6 alla quinta(almeno credo), e questo per 6 volte perche i numerisono 6, ma non penso che poi possa moltiplicare 1/6alla quinta per 6... S

tude

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tude

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giusto, non posso moltiplicare per sei, posso... (pensaai tipo di relazione che posso avere...)

Stu

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tude

nte

1

giusto, non posso moltiplicare per sei, posso... (pensaai tipo di relazione che posso avere...)

Stu

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Ma non posso considerare l’evento “escono tutti numeriuguali” come il complementare di “escono tutti numeridiversi”?Pero in questo caso 1/6 alla quinta non mi serve piu. S

tude

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1



infatti non e neppure il complemetare, a questo puntomanca un ultima operazione da considerare...(per poi fare i conti in questo ultimo passaggio risolvi1/6 alla quinta, che diventa 1/7776) S

tude

nte

2




tude

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2

Ma parli della somma?

Stu

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tude

nte

2

Ma parli della somma?

Stu

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esatto, per esclusione e rimasta solo quella, qundi ladomanda e cosa sommo e quante volte lo sommo...

Stu

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Io sommerei 1/7776 + 1/7776 + 1/7776 + 1/7776 +1/7776 + 1/7776pero l’ho fatto perche non rimanevano alternative, nonperche abbia capito perche prendo la somma. S

tude

nte

1



Io sommerei 1/7776 + 1/7776 + 1/7776 + 1/7776 +1/7776 + 1/7776pero l’ho fatto perche non rimanevano alternative, nonperche abbia capito perche prendo la somma. S

tude

nte

1

fondamente prendo la somma per il solito motivo, a mebasca che accada almeno uno dei casi elencati in unodei messaggi precedenti, che il numero che esce siasempre 1, 2, 3,... poco importa, basta che sia semprequello (almeno uno dei miei sei casi)

Stu

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Ti ringrazio anche per questo, ma come ti dicevo prima,ogni esercizio per me e a se...Non riesco a ragionare con le probabilita, quindi anchegli esercizi in cui mi chiede la probabilita che escaesattamente 2 volte un numero o che ogni numero siadiverso da quello prcedente mi trovano in difficolta...Vorrei riuscire a fare quel salto che mi permetta dientrare nella logica delle probabilita, ma non riesco,quindi lo step del test 3 mi rimane ancora da superare enon lo superero se ogni volta mi areno in un eserciziodiverso.

Stu

dent

e1

Success?

WIMS

Teacher training


E-learning


• Active learning




• Guessing


• Success?


Success?

WIMS

Teacher training


E-learning


• Active learning




• Guessing


• Success?


ciao antonella l’esame e andato bene midispiace che non sei potuta venire anche tu[. . . ] ti consiglio di riguardare bene gliesercizi del forum che la prof ha messo comeesempi d’esame. [. . . ] Guarda bene cosa tichiede l’esercizio e vai tranquilla che il lavorodel forum ti aiuta tantissimo!!! Devo dire laverita all’inizio sembrava faticoso trovarsitutte le sere ma ne e valsa la pena perchel’esame l’abbiamo superato grazie al lavorofatto qui!!!! In bocca al lupo!

Stu

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wims at the university of milano-bicocca · 2010-05-28 · university of milano-bicocca – dip. di...

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