Ingenious and Fun Games of Maths - Nuova Direzione Didattica Vasto

Transcript

Ingenious and Fun Games
of Maths
Strategies for an Effective
Teaching of Maths
Final product of
Ingenious and fun games of Maths
Summary
1. Italy
 Project presentation
 Countries and schools presentation
 Project LOGO
 Maths convention
2. Turkey
 Survey results (about parents-students-teachers)
3. Romania
 Countries curriculum
 Good practices
4. Reunion Island
 Test for 3th - 4th-5th grade first (version by Poland)
 Special needs games
5. Spain
 Games
 Test results first version
6. Poland
 Guide lines
 Tests for 3th -4th-5th grade (second version by
Turkey)
 International competition
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1. ITALY MEETING
Project presentation
Progetto E.Te.Mat. “Effective Teaching of Mathematics”, programma UE Erasmus+,
Key Action 2 - Partenariati StrategiciVASTO
Prof. Paolo ROTONDO
L’ORGANIZZAZIONE DEL CURRICOLO
TRA COMPETENZE E OBIETTIVI
DI APPRENDIMENTO
IL CASO DELLA MATEMATICA
18 FEBBRAIO 2015
1. LE < INDICAZIONI PER IL CURRICOLO > D. M. 254 / 16 NOV. 2012
2. LA SCUOLA DEL PRIMO CICLO
3. L‟ORGANIZZAZIONE DEL CURRICOLO
4. LE COMPETENZE
5. LE INDICAZIONI INVALSI
6. LA MATEMATICA NEL PRIMO CICLO
ALCUNI SPUNTI DI RIFLESSIONE
a) PER LE SUE SUGGESTIONI FORMATIVE E CULTURALI (ANCHE SE A VOLTE SANAMENTE UTOPISTICHE), LA PREMESSA (CULTURA – SCUOLA – PERSONA + FINALITA’
GENERALI) APPARE LA PARTE PIU‟ IMPORTANTE DEL DOCUMENTO DEL 4 / 09 / 2012,
ANCHE SE LA GENERALITA‟ DEGLI ENUNCIATI VA POI CONCRETIZZATA IN PARTICOLARI PERCORSI DIDATTICI.
b) LA SUCCESSIVA PANORAMICA DELLE DISCIPLINE, ARTICOLATE IN OBIETTIVI DI
APPRENDIMENTO (PER LA 3.a E LA 5.a PRIMARIA E PER LA 3.a MEDIA) E TRAGUARDI
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b) LA SUCCESSIVA PANORAMICA DELLE DISCIPLINE, ARTICOLATE IN
OBIETTIVI DI APPRENDIMENTO (PER LA 3.a E LA 5.a PRIMARIA E PER LA 3.a
MEDIA) E TRAGUARDI PER LO SVILUPPO DELLE COMPETENZE, AL TERMINE
DEI DUE CICLI, E`RICCA DI SPUNTI CHE VANNO INTERPRETATI E CONDIVISI SIA
A LIVELLO DI CURRICOLO COMPLESSIVO (L‟INSIEME DEGLI INSEGNANTI CHE
OPERANO CON I MEDESIMI ALUNNI), SIA TRA DOCENTI DI UNA MEDESIMA
DISCILINA IN UN ISTITUTO (GRUPPO DISCIPLINARE), AFFINCHE‟ SI POSSA INQUADRARE L‟OPERATO DIDATTICO NEL CONTESTO VOLUTO DALLE INDICAZIONI, CHE COMUNQUE NON APPAIONO UNA RIVOLUZIONE RISPETTO ALLA
PRECEDENTE “RIFORMA MORATTI” ( D. 59 / 2004) O ALLE PRECEDENTI
“INDICAZIONI 2007”, E NEMMENO RISPETTO AI PRECEDENTI “PROGRAMMI”
DELLA SCUOLA ELEMENTARE (1985) E DELLA SCUOLA MEDIA (1979).
LA SCUOLA DEL PRIMO CICLO
L’alfabetizzazione culturale di base

La scuola primaria mira all‟acquisizione degli apprendimenti di base, come primo esercizio dei diritti costituzionali. Ai bambini e alle bambine che la frequentano va offerta
l‟opportunità di sviluppare le dimensioni cognitive, emotive, affettive, sociali, corporee,
etiche e religiose, e di acquisire i saperi irrinunciabili. Si pone come scuola formativa che,
attraverso gli alfabeti delle discipline, permette di esercitare differenti potenzialità di pensiero, ponendo così le premesse per lo sviluppo del pensiero riflessivo e critico. Per questa
via si formano cittadini consapevoli e responsabili a tutti i livelli, da quello locale a quello
europeo.

La scuola secondaria di primo grado rappresenta la fase in cui si realizza l‟accesso alle
discipline come punti di vista sulla realtà e come modalità di interpretazione, simbolizzazione e rappresentazione del mondo.
La valorizzazione delle discipline avviene pienamente quando si evitano due rischi: sul
piano culturale, quello della frammentazione dei saperi; sul piano didattico, quello della impostazione trasmissiva. Rispetto al primo, le discipline non vanno presentate come territori da
proteggere definendo confini rigidi, ma come chiavi interpretative. I problemi complessi
richiedono, per essere esplorati, che i diversi punti di vista disciplinari interessati dialoghino e
che si presti attenzione alle zone di confine e di cerniera fra discipline.
L’ambiente di apprendimento

Il primo ciclo, nella sua articolazione di scuola primaria e secondaria di primo grado,
persegue efficacemente le finalità che le sono assegnate nella misura in cui si costituisce
come un contesto idoneo a promuovere apprendimenti significativi e a garantire successo
formativo per tutti gli alunni.
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come un contesto idoneo a promuovere apprendimenti significativi e a garantire successo
formativo per tutti gli alunni.
A tal fine è possibile individuare, nel rispetto della libertà di insegnamento, alcune impostazioni metodologiche di fondo.
- Valorizzare l’esperienza e le conoscenze degli alunni
- Attuare interventi adeguati nei riguardi delle diversità
- Favorire l’esplorazione e la scoperta
- Incoraggiare l’apprendimento collaborativo
- Promuovere la consapevolezza del proprio modo di apprendere

Realizzare percorsi in forma di laboratorio
PROPOSTA: GLI INVARIANTI DELLA DIDATTICA
1. LA CONTINUITA
2. LA COMPRENSIONE DI UN TESTO SCRITTO
3. ORGANIZZARE E RAPPRESENTARE LE INFORMAZIONI
4. LE MAPPE CONCETTUALI
5. LE CONCEZIONI DELLA MATEMATICA E DEL SUO APPRENDIMENTO
L‟O R G A N I Z Z A Z I O N E D E L C U R R I C O L O
< Ogni scuola predispone il curricolo all’interno del Piano dell’offerta formativa con
riferimento al profilo dello studente al termine del primo ciclo di istruzione, ai traguardi
per lo sviluppo delle competenze, agli obiettivi di apprendimento specifici per ogni disciplina. >
FINALITA‟ EDUCATIVE (Profilo dello studente)
TRAGUARDI PER LE COMPETENZE
OBIETTIVI DI APPRENDIMENTO
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F I N A L I T A` E D U C A T I V E
LA FINALITA‟ DEL PRIMO CICLO E‟ LA PROMOZIONE
DEL PIENO SVILUPPO DELLA PERSONA
Elaborare il senso della propria esperienza
Promuovere la pratica
consapevole della cittadinanza

LA SCUOLA PRIMARIA MIRA ALL‟ACQUISIZIONE DEGLI
APPRENDIMENTI DI BASE, COME PRIMO ESERCIZIO DEI DIRITTI COSTITUZIONALI.
LA SCUOLA SECONDARIA DI PRIMO GRADO
RAPPRESENTA LA FASE IN CUI SI REALIZZA L‟ACCESSO ALLE DISCIPLINE
COME PUNTI DI VISTA SULLA REALTÀ E COME MODALITÀ DI INTERPRETAZIONE, SIMBOLIZZAZIONE E RAPPRESENTAZIONE DEL MONDO.
I TRAGUARDI PER LE COMPETENZE
SONO INDICATI AL TERMINE DI CIASCUN CICLO:
FINE QUINTA ELEMENTARE
FINE TERZA MEDIA

RAPPRESENTANO “RIFERIMENTI PER GLI INSEGNANTI”

INDICANO “PISTE CULTURALI E DIDATTICHE DA PERCORRERE”
AIUTANO A “FINALIZZARE L‟AZIONE EDUCATIVA ALLO SVILUPPO
INTEGRALE DELL‟ALLIEVO”

COSTITUISCONO CRITERI PER LA VALUTAZIONE DELLE COMPETENZE ATTESE
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OBIETTIVI DI APPRENDIMENTO
ACQUISIZIONE DEGLI ALFABETI DI BASE DELLA CULTURA
VENGONO POSTI COME SAPERE / SAPER FARE ED INDICATI:
Per la SCUOLA PRIMARIA
AL TERMINE DELLA TERZA E DELLA QUINTA CLASSE
Per la SCUOLA SECONDARIA DI I GRADO
AL TERMINE DELLA TERZA CLASSE
ACQUISIZIONE DEGLI ALFABETI DI BASE DELLA CULTURA
VENGONO POSTI COME SAPERE / SAPER FARE ED INDICATI:
Per la SCUOLA PRIMARIA
AL TERMINE DELLA TERZA E DELLA QUINTA CLASSE
Per la SCUOLA SECONDARIA DI I GRADO
AL TERMINE DELLA TERZA CLASSE
I TRAGUARDI PER LE COMPETENZE RAPPRESENTANO – PROBABILMENTE – A
META‟ STRADA TRA FINALITA’
E OBIETTIVI, UNA SORTA DI COLLEGAMENTO
TRA L’EDUCAZIONE (PRODOTTA DALLE FINALITA‟)
E L’ ISTRUZIONE (PRODOTTA DAGLI
OBIETTIVI DI APPRENDIMENTO)
EDUCARE ISTRUENDO POTREBBE PERTANTO VOLER DIRE:
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1) PERSEGUIRE LE FINALITA’ EDUCATIVE
Promuovere la cittadinanza attiva
attraverso
2) L` ACQUISIZIONE DEGLI ALFABETI DI BASE DELLA CULTURA
sapere (conoscenze)
saper fare (abilità)
avendo cura di
3) MIRARE AL PIENO POSSESSO DELLE COMPETENZE
LE COMPETENZE INDICATE NELLA < INDICAZIONI > SONO STRETTAMENTE
DISCIPLINARI; MANCANO SUGGERIMENTI PER COMPETENZE COMPIUTAMENTE `TRASVERSALI`
LA MATEMATICA DAI 6ANNI IN POI
dalle INDICAZIONI PER IL CURRICOLO
SUGGERIMENTI GENERALI PER LA MATEMATICA

In Matematica è elemento fondamentale il laboratorio . . .

Caratteristica della pratica matematica è la risoluzione di problemi . . . .

Nella scuola secondaria di primo grado si svilupperà un’attività più propriamente
di tematizzazione, formalizzazione, generalizzazione.

Un’attenzione particolare andrà dedicata allo sviluppo della capacità di esporre e
di discutere con i compagni le soluzioni e i procedimenti seguiti.

L’uso consapevole e motivato di calcolatrici e del computer deve essere incoraggiato opportunamente fin dai primi anni della scuola primaria . . .

Di estrema importanza è lo sviluppo di un’adeguata visione della matematica . . .
riconosciuta e apprezzata come contesto per affrontare e porsi problemi significativi . .

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INVALSI
QUADRO DI RIFERIMENTO
PRIMO CICLO DI ISTRUZIONE / PROVA DI MATEMATICA
Le due dimensioni della valutazione
Le prove INVALSI di matematica per il primo ciclo scolastico sono volte a valutare le conoscenze e le abilità matematiche acquisite dagli studenti in entrata e in uscita del ciclo
d‟istruzione (classe II della scuola primaria; classe V della scuola primaria; classe I della
scuola secondaria di primo grado; classe III della scuola secondaria di primo grado).
Le domande di matematica sono costruite in relazione a due dimensioni:
- i contenuti matematici coinvolti, organizzati nei quattro ambiti (Numeri, Spazio e figure,
Dati e previsioni, Relazioni e funzioni);
- i processi coinvolti nella risoluzione.
Questa bi-dimensionalità della valutazione è utilizzata in quasi tutte le indagini internazionali
ed è indispensabile per fotografare correttamente gli apprendimenti dello studente, individuandone le componenti strutturali.
È importante sottolineare il fatto che (in matematica) non è possibile in generale stabilire una
corrispondenza univoca tra il singolo quesito e un unico contenuto (conoscenza o abilità) il
cui possesso venga verificato in esclusiva mediante quello stesso quesito.
Infatti, in generale, la risposta a ciascuna domanda coinvolge diversi livelli di conoscenze di
vario tipo e richiede contemporaneamente il possesso di diverse abilità.
È questa una conseguenza della natura stessa del pensiero matematico, che non consiste solo
in convenzioni o procedure di calcolo, ma in ragionamenti complessi, fatti di rappresentazioni, congetture, argomentazioni, deduzioni.
Ogni quesito delle prove del Servizio Nazionale di Valutazione viene quindi riferito a un ambito di contenuti e a un singolo processo, ma va sempre inteso che quelli indicati sono l'ambito e il processo prevalenti.
I processi utilizzati per costruire le domande e analizzare i risultati sono i seguenti:
1. conoscere e padroneggiare i contenuti specifici della matematica (oggetti matematici, proprietà, strutture...);
2. conoscere e utilizzare algoritmi e procedure (in ambito aritmetico, geometrico, …);
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2. conoscere e utilizzare algoritmi e procedure (in ambito aritmetico, geometrico, …);
3. conoscere diverse forme di rappresentazione e passare da una all'altra (verbale, numerica,
simbolica, grafica, ...);
4. risolvere problemi utilizzando strategie in ambiti diversi – numerico, geometrico, algebrico –
(individuare e collegare le informazioni utili, individuare e utilizzare procedure risolutive, confrontare strategie di soluzione, descrivere e rappresentare il procedimento risolutivo,…);
5. riconoscere in contesti diversi il carattere misurabile di oggetti e fenomeni, utilizzare strumenti di misura, misurare grandezze, stimare misure di grandezze (individuare l'unità o lo strumento di misura più adatto in un dato contesto,stimare una misura,…);
6. acquisire progressivamente forme tipiche del pensiero matematico (congetturare, argomentare, verificare, definire, generalizzare, ...);
7. utilizzare strumenti, modelli e rappresentazioni nel trattamento quantitativo dell'informazione
in ambito scientifico, tecnologico, economico e sociale (descrivere un fenomeno in termini
quantitativi, utilizzare modelli matematici per descrivere e interpretare situazioni e fenomeni,
interpretare una descrizione di un fenomeno in termini quantitativi con strumenti statistici o
funzioni ...).
8. riconoscere le forme nello spazio e utilizzarle per la risoluzione di problemi geometrici o di
modellizzazione (riconoscere forme in diverse rappresentazioni, individuare relazioni tra
forme, immagini o rappresentazioni visive, visualizzare oggetti tridimensionali a partire da una
rappresentazione bidimensionale e, viceversa, rappresentare sul piano una figura solida, saper
cogliere le proprietà degli oggetti e le loro relative posizioni, …).
POSSIBILI CURRICOLI VERTICALI OMOGENEI
Poiché gli “Obiettivi di apprendimento” al termine di 3.a elementare, 5.a elementare e 3.a media sono declinati omogeneamente in termini di:

NUMERI

SPAZIO E FIGURE

RELAZIONI, DATI E PREVISIONI
sembra possibile e naturale costruire curricoli verticali (da 6 a 14 anni) disciplinari e condivisi
tra i docenti dei due ordini di Scuola, ed espressi soprattutto in termini di “saper fare”, più
spesso che di “sapere”.

Qualche osservazione . . . .
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1. I concetti
- al termine della classe 5.a, tra gli “Obiettivi di apprendimento” si scrive:
< Utilizzare e distinguere tra loro i concetti di perpendicolarità,
parallelismo, orizzontalità, verticalità >
2. L’ATTENZIONE AL LINGUAGGIO

Dagli “Obiettivi di apprendimento >
a) TERZA ELEMENTERE
Eseguire un semplice percorso partendo dalla descrizione verbale o dal disegno, descrivere
un percorso che si sta facendo e dare le istruzioni a qualcuno perché compia un percorso
desiderato.
b) QUINTA ELEMENTARE
- Riprodurre una figura in base a una descrizione, utilizzando gli strumenti opportuni
(carta a quadretti, riga e compasso, squadre, software di geometria).
3. LA TRADIZIONE
a) TERZA ELEMENTARE
- Riconoscere, denominare e descrivere figure geometriche.
b) QUINTA ELEMENTARE
- Determinare il perimetro di una figura.
- Determinare l‟area di rettangoli e triangoli e di altre figure per scomposizione.
I TRAGUARDI PER LO SVILUPPO DELLE COMPETENZE
al termine della scuola primaria
- Làlunno si muove con sicurezza nel calcolo scritto e mentale con i numeri naturali
e sa valutare lòpportunità di ricorrere a una calcolatrice.
- Descrive, denomina e classifica figure in base a caratteristiche geometriche, ne
determina misure, progetta e costruisce modelli concreti di vario tipo.
- Riesce a risolvere facili problemi in tutti gli ambiti di contenuto, mantenendo il
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-Riesce a risolvere facili problemi in tutti gli ambiti di contenuto, mantenendo il
controllo sia sul processo risolutivo, sia sui risultati.
OSSERVAZIONI
RICORDIAMO CHE I < TRAGUARDI PER LE COMPETENZE >

RAPPRESENTANO “RIFERIMENTI PER GLI INSEGNANTI”

INDICANO “PISTE CULTURALI E DIDATTICHE DA PERCORRERE”

AIUTANO A “FINALIZZARE LÀZIONE EDUCATIVA ALLO SVILUPPO INTEGRALE DELL‟ALLIEVO”

COSTITUISCONO CRITERI PER LA VALUTAZIONE DELLE COMPETENZE ATTESE
SI PROVI ALLORA A:
1. SPECIFICARE – PER QUALCUNO DEI “TRAGUARDI” SOPRA
ESAMINATI – QUALI CONCRETE PISTE DA PERCORRERE SIANO
DA ATTUARE;
2. COME VALUTARE IL GRADO DI RAGGIUNGIMENTO DI
TALI TRAGUARDI;
3. COME CONNETTERE GLI OBIETTIVI DI APPRENDIMENTO CON
QUESTI TRAGUARDI;
4. COME MIRARE DAI TRAGUARDI SUGGERITI ALLE . . . .
FINALITA’ EDUCATIVE
Promuovere la cittadinanza attiva
AI FINI DELLO SVILUPPO INTEGRALE DELL‟ALUNNO ?
****************************
Prof. Paolo ROTONDO
FEBBRAIO 2015
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COUNTRY PRESENTATION
ITALY
www.nuovadirezionedidatticavasto.gov.it
WE ARE HERE
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Monument to the bather
Punta Aderci
The castle
Punta Penna : the lighthouse
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Fish soup is the typical of Vasto’s plate
ORGANIGRAMMA
ORGANIZZAZIONE ORARIA
Dirigente Scolastico
Prof.ssa Nicoletta DEL RE
Consiglio di
Circolo
SCUOLA DELL’INFANZIA
Giunta
Esecutiva
7.45/8.00 – 16.00
SCUOLA PRIMARIA
Direttore dei
S.G.A.
Collaboratori del
Dirigente
Maria Giacinta
PICCONE
ITALIANO
Ins. Barbara
GASPARI
Ins. Paola MELIS
Assistenti
Amministrativi
Collaboratori
Scolastici
Referenti di
Plesso Primaria
Referenti di
Plesso Infanzia
Funzioni
Strumentali
8.10 – 13.10 (LUNEDI –VENERDI)
8.10- 12.10 (SABATO)
TEMPO PIENO 8.10 -16.10 (SABATO LIBERO)
Collegio
Docenti
Inizio a.s. 11 sett. 2014 chiusura 9 giugno 2015
Vacanze di Natale dal 23 dicembre al 6 gennaio
Vacanze di Pasqua dal 2 all’ 8 aprile
Commissioni
Coordinatori di
Dipartimenti
Consigli Di
Interclasse
STUDENTS
TEACHERS
COLLABORATOR
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10
2
6
15
2
6
3
2
Incoronata
94
13
2
1
Peluzzo
221
24
4
1
2
Ritucci Chinni
S.Antonio
232
76
23
9
3
1
Vasto Marina
151
100
23
50
106
18
65
TOTALE
623
69
10
TOTALE
513
55
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Aniello Polsi
Incoronata
S.Lorenzo
Smerilli
S.Michele
Pagliarelli
STUDENTS
1
2
15
TEACHERS
COLLABORATOR
Scuola dell’infanzia Aniello Polsi
Scuola dell’infanzia e Scuola Primaria Incoronata
Scuola dell’infanzia S.Lorenzo
Scuola dell’infanzia Smerilli
Scuola dell’infanzia S.Michele e Scuola Primaria Peluzzo
Scuola dell’infanzia Pagliarelli
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Scuola Primaria Ritucci Chinni
Scuola dell’infanzia Vasto Marina
Scuola Primaria S. Antonio
PIANO DELL'OFFERTA FORMATIVA (POF)
UNA SCUOLA PER LA VITA
P.O.F.
P.O.F.
I PROGETTI D’ISTITUTO
PAROLE CHIAVE
•LETTURA “LIBR…IAMOCI”
•SOLIDARIETA’
•RECUPERO/POTENZIAMENTO
•OSPITALITA’
•ORTOLIAMO
•AMBIENTE DI APPRENDIMENTO
•CONTINUITA’
•DIDATTICA LABORATORIALE
•AREA A RISCHIO-IMMIGRAZIONE
•COMPETENZE
•INTERPRETO I SEGNI DEL TEMPO
•CONTINUITA’
•COMENIUS “LEARN TO READ”
•UNA SCUOLA PER TUTTI E PER CIASCUNO
•ERASMUS PLUS “E.TE.MAT.”
•SCUOLA A DOMICILIO
•L.I.M. IN CLASSE
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POLAND
The national emblem and the flag
POLAND
area: 312,685 square km
population: 38,2 million
European Union memeber since 2004
The river Wisla and the biggest cities
Gdansk
Poznan
Warszawa
POLAND
Lodz
WROCŁAW
Wroclaw
Katowice
Krakow
Mazury – lake district
The Baltic Sea coast – Hel peninsula
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The Carpathians – Tatra Mountains
RYSY - the highest peak of Poland (2,499 m)
(the south of Poland)
Zakopane
Bieszczady
The magic forest of Bialowieza
The European Bison – a unique animal
Gdansk
Poznan
Warszawa
Lodz
Wroclaw
Katowice
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Krakow
WARSZAWA (Warsaw) Population: 1,7 million
the capital and the largest city of Poland
The Old Town in Warsaw (rebuilt after total destruction
of the city in World War II)
Wawel – the Royal Castle
Krakow
The old capital of Poland
Jagiellonian University - the oldest in Poland (1364)
Wawel Cathedral
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Gdansk –
Wroclaw – Our City
”A trace of Holland in Poland”
The „long” Market
The Old Port
Poznan
The Centre of Katowice
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Mikolaj Kopernik, Copernicus
Fryderyk Chopin
(astronomer)
(composer, pianist)
Lech Walesa
Karol Wojtyla – John Paul II, the Pope
Legendary leader of „Solidarity”. Former president of Poland.
Who and When brings Gifts to
Polish Children during Christmas
Season
Delicious Food for Fat Thursday
The Thursday before Ash
Wednesday is celebrated as
Fat Thursday - Tlusty
Czwartek . On this is the
day when you forget about
your diet and eat mountains
of donuts (paczki) and all
the other things fat, greasy,
sweet, full of cholesterol,
generally unhealthy, and
mmmmm.... delicious.
In some regions of Poland the gifts are given to the
children only on December 6th - since St. Nicolaus
called also Santa Claus is a patron of this day. But in
the majority of houses children (and adults) can expect
gifts twice- on December 6th and also on Christmas
Eve. The atmosphere of this feast is different than the
atmosphere of Christmas eve since December 6th is a
normal working day. Whereas Christmas is usually
celebrated as a family feast.
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Sinking of Marzanna
Palm Sunday Traditions
Palm Sunday niedziela palmowa is called also The Sunday
of the Lord's Passion. Here we will focus mainly on the
tradition of Polish palms
The most popular palms that people usually carry to the
church are made of blooming pussy willows branches
called bazie or kotki decorated with branches of birch,
raspberry, currant and also some boxwood bukszpan, dry
flowers and grass, ribbons and other decorations. In the
Catholic Church the willow (Polish: wierzba) symbolizes
the resurrection and the immortality of the soul.
Winters in Poland were long and unforgiving. Therefore people are longing for spring.
One of the ancient and pagan habits that supposedly was helping to get rid of winter was
"sinking of Marzanna". Kids made a doll from old grass and tree branches and take it to
the river. They burn the doll and throw her into the river. The symbolic meaning of this
ceremony is to get rid of winter therefore it is performed in early spring.
Art of Coloring Easter Eggs
Easter Saturday in Poland
Easter Saturday in Poland is
a busy day. Every Polish
family visits a church with a
basket full of food products
(a piece of bread, salt,
sausage, egg - usually
painted etc). Especially
children love it! The baskets
are then blessed by a priest.
The Easter eggs are symbols of fertility and beginning of the new life.
Some of the eggs were painted in traditional Polish folk patterns. These
eggs were called "pisanki". Word "pisanki" comes from the root-word
meaning "to write". Painting eggs is a multi-layered process of writing
on an egg with hot beeswax, dying the egg, then finally melting and
rubbing off the egg for a finished product.
Wet Monday
Smigus Dyngus (shming-oos-ding-oos) is an unusual
tradition of Easter Monday. This day (Monday after
Easter Sunday) is called also in Polish "Wet Monday",
in Polish: "Mokry Poniedzialek" or "Lany
Poniedzialek". Easter Monday is also a holiday in
Poland. It was traditionally the day when boys tried to
drench girls with squirt guns or buckets of water.
The atmosphere of All Saints' Day is unique. In the evening cemeteries
are decorated in glowing and flickering colorful lights of countless
candles.
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St. John's Night
A long Mayday weekend in Poland
At the end of June, at the
time of Summer Solstice,
when night is shortest and
Nature bursts with
blossoms and growth, we
celebrate the Holiday of
Fire and Water, also
called Noc Kupaly.
People gather at a fire,
jumping through the fire,
sing songs, dance and
having lots of fun.
May 1st - International Workers' Day
May 2nd - Flag Day, it is also celebrated as a day of Polish
immigration or Poles abroad, so called POLONIA DAY.
May 3rd - The oldest feast is a feast of May 3rd which is
devoted to the day of constitution, since the famous
Constitution of the 3rd May was established on that day.
Many people go on the outdoor trips during long Mayday
weekend.
St. Andrew's Night
There is a long tradition of fortune telling
especially for non-married girls on the
November 30th in Poland. The main purpose
of Andrzejki celebrations is to predict the
future of unmarried girl, especially her
prospects for a good marriage.
Presently people do not take seriously the
fortune-telling during st. Andrew Day but
this day is still celebrated because it is lots of
fun
Girls wore wreaths of flowers on their heads. If the burning wreath was thrown in
the river and then pulled by a single man it might mean they are engaged.
All Saints' Day in Poland,
November 1st
Miners' Day (St. Barbara Day)
One of the most celebrated days associated with workers group is St.
Barbara's Day on December 4th. St. Barbara is a patron of coal miners.
Poles take flowers (especially fall flowers like chrysanthemum), wreaths,
candles and votive lights into the cemeteries where graves of family,
friends or national heroes are. It is worth to mention that the cemeteries
in Poland are different than in any country. Graves and tombs are big and
very individualized.
Miners are dressed in the special uniforms during Barbórka. The uniform
consists of black suit and hat with a feather. The color of the feather
depends on the rank of the miner.
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REUNION ISLAND (RANCE)
REUNION ISLAND
What about ?
Five years ago, 40 percent of Reunion, Island
Reunion, was named a UNESCO World
Heritage site and turned into a national park.
FRANCE
A little piece
of France
in the Indian
Ocean.
Also "Maloya" music and dance of the slave has recently declared
UNESCO heritage
Put the dates in the ascending order:
1545.- Discovered by Pedro Mascarenhas. (Portuguese)
1848 - Slavery abolished.
1642 - Arrival of French. Early settlement.
1810-1815 - British interlude.
1946 - Reunion becomes a Department Overseas (DOM).
1869 - Start of economic decline.
English…and a stop for pirates and their treasures .
provides close to the shores. It thus is visited by many browsers, Arabic, Portuguese and
century. She is a stopover on the trade route, popular because of the abundance of fresh water it
The Island of the Meeting today- remains uninhabited until the middle of the seventeenth
• Born of a volcano out of the bowels of the earth there are three million years, Bourbon Island
Representation with geometric shapes of the discovery of Reunion Island
Answers :
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Ideal destination for hikers, Reunion
contains many geological curiosities: the
Piton of the Fournaise of course, but also
circuses or extraordinary reefs, such as the
blower spitting foam sprays. On a small
area, the island offers a range of remarkable
landscapes.
Water
Sugar cane fields
Circles
Volcans
The islanders are trying not to forget their roots.
Muslims, Catholics and Hindus blend seamlessly and
religious practices are very present in the lives of a
majority of residents. If the atmosphere is permeated
with the dominant Catholic faith, it is however the
Hindu community that gives the island its most striking
customs. Hinduism exhibits his thousand colors on the
facades of temples that bloom throughout the island.
The isolation, the diversity of natural habitats and micro climates
have led many indigenous species present before the arrival of
man, to differentiate themselves over the millennia and become
endemic species. Reunion passion, of course lovers of beauty
vegetable, botanists as garden lovers. In the midst of the vast
Indian Ocean, Reunion home to a unique flora. It is developed
both on the coast and in the mountain forests.
843 617
Reunion's cuisine is as mixed as its people. No dish
has remained in its original flavor, all have been
enriched and embellished by the generous
inspiration of Bourbonnais stoves and influences
from elsewhere: French, Indian, Chinese ...
800 000 + 43 000 + 600 + 17
The local specialty is curry, fragrant stew of meat,
fish or crustaceans, simmered with garlic, onions,
ginger, cloves, turmeric and other local spices.
843 617: it’s just the numbers of poeple of Island Reunion (2013)
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We are pleased to participate in the ERASMUS project
mathematic, and look forward to receive you at home.
Appointment to our town (The Tampon), in our school “Just
Sauveur”with Fabienne Couchat our teacher.
population mixing
land of interbreeding
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ROMANIA
ROMANIA...
THE CARPATHIEN GARDEN OF EUROPE
A big and beautiful country situated in the
SOUTH-EAST of Central Europe
On its surface one can visit
The Danube Delta
ONE OF The biggest rivers in Europe: The
Danube
We have an ancient history, starting with the dacic war
between 101-102 b.C.
Transilvania′s Highland and
The Black Sea
Ulpia Traiana Sarmizegetusa
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Dracula′s Castle
Other monuments we are
proud of
The Merry Churcyard and Corvinilor′s Castle
Monasteries from
Northen Bucovina
Peles Castle
Sighisoara Stronghold and
wooden churches from Maramures
UNESCO monuments in Romania
Brancoveanu′s Monastery 1696
Densus Church in Hunedoara county
sec. XIII
You may have heard about ROMANIA just
names as:
But you must know that
GICA HAGI
Eugen Ionesco was born in
Romania (La Cantatrice Chauve)
NADIA COMANECI
Constantin Brancusi and his
sculptures are made in Romania
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SPAIN
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TURKEY
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38
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Project LOGO
Proposals countries for LOGO
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AND THE WINNER PROJECT
LOGO
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MATHS CONVENTION
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La matematica
Nelle
«Indicazioni Nazionali»
per il 1° ciclo
•L’apprendimento della matematica è una componente fondamentale nell’educazione e nella
crescita della persona
•Nella società attuale la matematica è nel cuore del trattamento quantitativo dell’informazione,
nella scienza, nella tecnologia, nelle attività economiche e nel lavoro
•La competenza matematica è un fattore fondamentale nella consapevolezza delfuturo cittadino
e nella sua riuscita nel mondo professionale
Struttura delle Indicazioni Nazionali
•Campi di esperienza, aree disciplinari e discipline
•Traguardi per lo sviluppo delle competenze (indicano piste culturali e didattiche da percorrere e aiutano a finalizzare l’azione educativa allo sviluppo integrale dell’allievo).
•Obiettivi di apprendimento e Nuclei tematici (individuano campi del sapere, conoscenze e
abilità ritenuti indispensabili al fine di raggiungere i traguardi per lo sviluppo delle competenze).
Perché sono solo "indicazioni" e non un vero curricolo?
1. perché mancano i collegamenti fra i traguardi e i corrispondenti obiettivi;
2. perché gli obiettivi di apprendimento sono poco dettagliati;
3. perché non ci sono le suddivisioni per anno ma solo per periodi di due o tre anni.
N. B. Il curricolo lo deve costruire ciascuna singola scuola.
La Mission della matematica
La formazione culturale delle persone e delle comunità
Sviluppando le capacità di mettere in stretto rapporto il «pensare» e il «fare»
Offrendo strumenti adatti a percepire, interpretare e collegare tra loro fenomeni naturali,
concetti, artefatti ed eventi quotidiani.
Quattro cose interessanti
1.Il Laboratorio di matematica inteso sia come luogo fisico, sia come momento in cui l’alunno
è attivo, formula le proprie ipotesi e ne controlla le conseguenze, progetta e sperimenta, discute
e argomenta le proprie scelte, impara a raccogliere dati, negozia e costruisce significati, porta a
conclusioni temporanee e a nuove aperture la costruzione delle conoscenze personali e collettive.
2. Risolvere problemiche devono essere intesi come questioni autentiche e significative, legate
alla vita quotidiana, e non solo esercizi a carattere ripetitivo o quesiti ai quali si risponde semplicemente ricordando una definizione o una regola.
N.B. I problemi, oltre a risolverli, bisogna saperli inventare.
N.B. I problemi, oltre a risolver43
3. Gli strumenti di calcolo
L’uso consapevole e motivato di calcolatrici e del computer deve essere incoraggiato opportunamente fin dai primi anni della scuola primaria, ad esempio per verificare la correttezza di calcoli mentali e scritti e per esplorare il mondo dei numeri e delle forme.
4. Sviluppo di un’adeguata visione della matematica non ridotta ad un insieme di regole da
memorizzare e applicare, ma riconosciuta ed applicata come contesto per affrontare e porsi problemi significativi e per esplorare e percepire relazioni e strutture presenti in natura e nelle creazioni dell’uomo.
Dai «Traguardi per lo sviluppo della competenza» di scuola infanzia

Il bambino raggruppa e ordina oggetti e materiali secondo criteri diversi, ne identifica alcune proprietà, confronta e valuta quantità; utilizza simboli per registrarle; esegue misurazioni
usando strumenti alla sua portata.
Ha familiarità sia con le strategie del contare e dell’operare con i numeri sia con quelle necessarie per eseguire le prime misurazioni di lunghezze, pesi e altre quantità.
Individua posizioni di oggetti e persone nello spazio………..
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Sicurezza nel calcolo scritto e mentale (uso calcolatrice)
Riconoscimento di forme del piano e dello spazio
Capacità di classificazione
Disegno geometrico
Rappresentazione e lettura (tabelle e grafici)
Comprensione e soluzione di problemi


Dai «Traguardi per lo sviluppo della competenza» di scuola primaria
Importante!
La costruzione del pensiero matematico è un processo lungo e progressivo nel quale
concetti, abilità, competenze e atteggiamenti vengono ritrovati, intrecciati, consolidati e sviluppati a più riprese; è un processo che comporta anche difficoltà linguistiche e che richiede un’acquisizione graduale del linguaggio matematico.
Due considerazioni … importanti
•Verticalità:
Sforzo di costruire un curricolo verticale, in continuità tra i diversi ordini di scuola
•Coerenza tra Documenti ministeriali e non:
In questi ultimi anni, documenti diversi come struttura e finalità cominciano a parlarsi tra loro
(es. Ind.Naz. Con il Questionario di Rilevazione per la matem. Invalsi)
Cosa fa l’Invalsi?
Ha il compito di sondare se le conoscenze che la scuola stimola e trasmette, sono ben
ancorate ad un insieme di concetti fondamentali di base e di conoscenze stabili (almeno sui livelli essenziali). Se, cioè, si tratta di conoscenza concettuale, frutto di interiorizzazione dell’esperienza e di riflessione critica, non di addestramento “meccanico” o di apprendimento mnemonico.
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Migliorare l’apprendimento in “spazio e figure”
Diana Cipressi
Al termine della scuola primaria sono fissati - nelle Indicazioni Nazionali per il curricolo 2012 i traguardi per lo sviluppo delle competenze della matematica, disciplina “non ridotta ad un insieme di regole da memorizzare e applicare, ma riconosciuta e apprezzata come contesto per
affrontare e porsi problemi significativi e per esplorare e percepire relazioni e strutture che si
ritrovano e ricorrono in natura e nelle creazioni dell’uomo”.
A questo proposito B. D’Amore suggerisce una riflessione nell’articolo Che problema i
problemi! pubblicato sul sito http://www.dm.unibo.it/rsddm/it “È vero che, in prima istanza, chi
risolve tenta di applicare regole (norme, esperienze,…) o procedimenti (meglio se vincenti)
precedentemente esperiti con successo; ma è anche vero che, se la situazione problematica è
opportuna, il soggetto potrebbe non trovare una situazione analoga o identica ad una precedente. Egli può invece trovare una particolare combinazione di regole (norme, esperienze,…)
del tutto nuova e che andrà ad arricchire il campo delle esperienze cui far ricorso in futuro.
Insomma: risolvendo il problema, il soggetto ha appreso”.
Il compito della scuola è quindi quello di promuovere occasioni di apprendimento caratterizzate
non da esercizi ripetitivi e meccanici ma da situazioni problematiche concrete, la cui risoluzione
porta l’alunno alla scoperta di un concetto o di una regola.
Diciamo che un problema è significativo se è costruito in modo realistico e strutturato, aperto a
più risposte, e l’approccio verso la risoluzione, genera curiosità o motivazione, sviluppa processi più che prodotti, stimola formulazione di ipotesi e creatività, favorisce un apprendimento sociale e condiviso.
L’alunno d’altra parte sa riconoscere una situazione-problema: egli osserva che si presenta
come un problema atipico, “un problema diverso dagli altri”, “un problema impossibile da risolvere” e che l’aiuto dei compagni è prezioso.
Un problema significativo da proporre sarà quello dove l’allievo ha un ruolo produttivo, responsabile, dove il docente assume il ruolo di guida dell’alunno che apprende, dove il sapere è
costruito attraverso esperienze concrete e dinamiche.
Nel laboratorio matematico l’alunno potrà commettere errori, riflettere su di essi, ragionare e
discutere con i compagni.
Fissiamo ora l’attenzione su alcuni aspetti della geometria.
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1. La percezione. “La Geometria può essere significativa solo se esprime le sue relazioni con lo
spazio dell’esperienza […] essa è una delle migliori opportunità per matematizzare la
realtà” (Freudenthal, Mathematics as an Educational Task).
Il pensiero geometrico va quindi ricercato – a partire dalla scuola d’infanzia - in una molteplicità di
esperienze, nelle quali l’alunno vedendo, toccando e organizzando le forme nello spazio ricava le
informazioni legate alla forma, alla grandezza, alla posizione e alla trasformazione di quegli oggetti.
Durante l’approccio esplorativo l’alunno impara ad osservare, a descrivere gli oggetti e le forme che
li rappresentano, per esempio a riconoscere un quadrato da un rettangolo, a capire che il cilindro
rotola più facilmente di un parallelepipedo.
2. Il linguaggio.
a) Il linguaggio naturale utilizzato dall’alunno ogni giorno può diventare una sorgente di difficoltà
nel processo di apprendimento. Ad esempio
l’angolo della strada, l’angolo del tavolo, l’angolo-cottura, ecc. sono espressioni diverse che
provocano una distorsione dell’immagine dell’angolo di un poligono.
I termini orizzontale, verticale, obliquo, ecc. non sono specifici dell’ambito matematico e possono produrre rappresentazioni stereotipate delle forme geometriche.
b) Il linguaggio specifico gioca nella matematica un ruolo fondamentale. La discussione in classe
sarà efficace per individuare termini, definizioni e concetti e la riflessione condivisa permetterà di
correggere gli errori, semplificare le formulazioni e ricercare un linguaggio chiaro e univoco.
c) Il processo di comprensione di un problema è estremamente complesso, in quanto richiede competenze linguistiche relative ai significati diversi di una parola, ai termini espliciti ed impliciti,
all’ordine delle informazioni, ecc.
Ad esempio uno stesso oggetto può essere designato con nomi diversi: il “segmento” diventa “lato”
di un poligono oppure “altezza” di un triangolo ecc.; l’alunno è disorientamento.
3. Il disegno di una figura geometrica. Il disegno è un’immagine statica, e non favorisce
l’osservazione e l’intuizione dell’alunno, come osserva E. Castenuovo nel 1965: “ il disegno è insufficiente; se io traccio una figura alla lavagna o se il bambino fa egli stesso il disegno, la sua attenzione si ferma sul tratto disegnato, cioè sul contorno della figura, non sull’interno. Per lui il triangolo è il contorno del triangolo, per lui l’angolo è l’insieme di quelle due semirette: l’interno della
figura è vuoto, perché il bambino non ha l’educazione necessaria per un’interpretazione più generale.”
Disegniamo un angolo con un archetto. L’alunno può allora identificare l’angolo con la coppia di
semirette, oppure con l’archetto, oppure con la parte limitata dall’archetto, … e non con una
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superficie illimitata.
S. Sbaragli - in L’apprendimento della matematica- fa notare che: “Se l’insegnante mostrerà all’allievo sempre la stessa rappresentazione del concetto, senza pensare alle conseguenze che questa
sua scelta potrebbe comportare, si potrebbero verificare ostacoli di tipo didattico per il futuro apprendimento.”
Posizioniamo un quadrato in una posizione diversa da quella classica, con il lati non paralleli al
pavimento ad esempio. Gli alunni erroneamente non riconoscono un quadrato ma un rombo.
Il disegno dunque è insufficiente per
poter dare alla geometria un carattere costruttivo: limita le possibilità manipolative, richiede competenze grafiche specifiche, fornisce
un’immagine statica della realtà.
4. Materiali didattici.
a) L’uso di modelli concreti realizzati con carta, bastoncini, elastici ecc. possono offrire un supporto efficace all’intuizione nella costruzione dinamica della geometria.
Uno strumento didattico articolato e mobile è utile per attirare la curiosità del bambino, il quale
attraverso la manipolazione dell’oggetto può osservare le sue trasformazioni, analizzare i casi possibili, e essere condotto dal concreto verso l’astratto.
La piegatura della carta e la riflessione nello specchio sono delle attività formative in cui l’alunno ricerca o verifica le regolarità di una figura.
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Quattro strisce uguali unite tra loro possono servire per vedere come alcune proprietà cambiano e altre restano costanti, per vedere quindi che il quadrato appartiene alla famiglia del
rombo.

Uno spago è teso tra le quattro dita a mo’ di rettangolo. Spostando le dita si ottengono rettangoli che hanno tutti lo stesso contorno, cioè sono isoperimetrici. L’area invece cambia: se
la sua altezza viene ridotta a zero il rettangolo ha area nulla! Se la sua altezza è uguale alla
base l’area è massima. L’area viene percepita nel suo divenire come una funzione matematica.

L’uso di tessere in cartone o legno, a forma di poligono regolare, possono essere accostate per
costruire figure, per confrontare perimetro e area, per ricercare figure con il perimetro minimo,
ecc.
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
Mentre un modello di poliedro pieno concentra l’attenzione sul numero delle facce, un
modello di poliedro scheletrato mette in risalto il numero di vertici, di spigoli.
b) La tecnologia fa parte integrante dell’apprendimento della matematica, in quanto offre ulteriori forme di rappresentazioni. GeoGebra è uno strumento che può migliorare le pratiche didattiche, è un software "open source" dinamico, adatto all'insegnamento e all'apprendimento della
matematica a tutti i livelli di istruzione. Alla LIM l’alunno può riprodurre una figura in base ad
una sua descrizione, ma può anche manipolarla, trascinarla, ruotarla, ingrandirla, ecc mantenendone inalterate le proprietà.
Ad esempio nella costruzione di un parallelogrammo, a partire dalle diagonali, l’alunno esprime
la successione di operazioni da effettuare:
“Disegno un punto A; un segmento AC di lunghezza 3cm; il punto medio M di AC; un segmento
MB di lunghezza 4 cm; il punto D simmetrico di B rispetto ad M; il segmento MD”.
Spostando la figura con il cursore, l’alunno scoprirà quali quadrilateri hanno le diagonali che si
dividono a metà.
Un’azione didattica promossa
con oggetti e modelli
favorisce senz’altro l’apprendimento dell’alunno, ma è anche vero che il modello non rappresenta il concetto in modo esaustivo. Per tale motivo l’insegnante avrà cura nel procedere non
sempre dal concreto all’astratto e nel guidare l’alunno verso un ragionamento che motivi ciò
che egli vede.
5. Problemi autentici.
Le sperimentazioni possono coinvolgere alcuni degli aspetti geometrici riscontrabili nella vita
quotidiana attraverso le varie discipline. Tanti sono i percorsi possibili: la storia di alcune figure
geometriche, ad es. la stella a cinque punte e i Pitagorici; la piegatura della carta e le simmetrie
negli origami; una pavimentazione con poligoni regolari e lo studio delle regolarità; le strutture
architettoniche della città; un gioco strategico; ecc.
Le varie dimensioni, storiche, tecniche, linguistiche ecc., contribuiscono a dare vita agli oggetti
matematici e, contestualizzandole in una dimensione culturale e sociale, concorrono tutte allo
sviluppo delle competenze.
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Laboratorio di Geometria dinamica
Geometria Dinamica significa che gli oggetti geometrici, punti, segmenti, rette e poligoni, non sono statici, come quelli disegnati sul quaderno o sulla lavagna, ma si muovono. Per farlo hanno bisogno di particolari strumenti, come il software GeoGebra o le Tassellazioni, che si
utilizzeranno nel percorso che, con l’insegnante Marina Iovacchini, stiamo effettuando nella
quinta classe del Plesso Incoronata, studiando le isometrie (traslazioni, simmetrie e rotazioni).
Nella relazione odierna racconteremo il percorso effettuato in un’altra scuola primaria
di Vasto, dove per quattro anni i bambini (dalla seconda alla quinta), hanno lavorato su quattro
percorsi: Cornicette, Origami, Geogebra, Tassellazioni.
I Laboratori di geometria dinamica permettono agli alunni di scoprire quanta geometria
e più in generale quanta matematica sia presente nella realtà e quindi di non considerare la materia come qualcosa che appartiene solo all’ambito scolastico ma che è utile e necessaria nella
vita di tutti i giorni.
Marina Gallo
Vasto, 17 Febbraio 2015
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Dalla “Numerazione Unaria” alla “Numerazione Binaria”
Evoluzione dei sistemi di calcolo ed elaborazione dati
Perché lo zero? Lo zero è forse una delle invenzioni più geniali della Storia, e tuttavia, com’è
avvenuto con la maggior parte delle scoperte umane più significative, esso viene ampiamente
utilizzato senza che se ne riconosca l’importanza, sfruttato e svuotato della propria natura. E poi
è la cifra più curiosa che esista: è un numero oppure no? E ha senso parlare di zero o la questione è del tutto irrilevante ed è sufficiente considerarlo solo quel semplice circolino che siamo
abituati a vedere fin dai nostri primi calcoli in colonna?
Possiamo dire che lo zero equivale ad una mancanza, ad una assenza, ad un buco: insomma,
equivale al nulla. E che cos’è questo nulla di cui gli antichi Greci avevano tanto orrore, che ha
terrorizzato la letteratura cristiana medioevale e ha assillato artisti e filosofi alle prese con il significato dell’esistenza umana? Il nulla c’è, non possiamo negarne l’esistenza; il cosmo ha origine dal nulla, la vita ha origine dal nulla e, come l’universo, ad esso ritorna. Ma come esprimere
un qualcosa che non c’è, come rappresentarlo visivamente e renderlo tangibile?
Dagli Egizi ai Sumeri: primi passi verso lo zero. Nell’antichità gli Egizi erano notoriamente
definiti veri maestri di geometria. Plutarco narra che la insegnarono a Talete e a Pitagora. I papiri ritrovati testimoniano conoscenze piuttosto elaborate: essi sapevano misurare terreni e ristabilire i confini dei campi dopo le inondazioni del Nilo, conoscevano formule per calcolare l’area di figure piane e il volume di solidi come il tronco di piramide. Eppure nei papiri non vi è
alcuna traccia dello zero, il primo e il più ambiguo dei numeri, così come non si trova nella matematica greca, che ampliò considerevolmente le conoscenze degli Egizi e con la creazione della logica costituì le basi di tutta la matematica moderna. La mancanza dello zero non si fece infatti sentire fino a quando si usarono sistemi additivi di rappresentazione numerica. La numerazione egizia ricorreva alla ripetizione di una sequenza di simboli corrispondenti ad uno, dieci,
cento, mille, diecimila, centomila e un milione; i segni comparivano in ordine di grandezza decrescente, ma soltanto per una questione stilistica: le posizioni relative dei simboli dei numerali
non fornivano alcuna informazione numerica, cosicché non vi era la necessità di un simbolo per
lo zero; se i numeri possono stare in qualsiasi posizione senza modificare la quantità totale che
rappresentano, non c’è possibilità di un “posto” vuoto e un segno della sua presenza non avrebbe senso. E nel caso in cui non ci fosse nulla da contare, semplicemente non si scriveva alcun
simbolo. Al vantaggio del sistema additivo, e cioè l’indipendenza dall’ordine degli addendi, si
opponevano però sostanziali svantaggi: da un lato la teorica necessità di infiniti simboli per le
infinite potenze della base, dall’altro la pesantezza della rappresentazione, che richiedeva troppe
ripetizioni. I Sumeri tentarono di ovviare al problema introducendo una nuova caratteristica: il
loro sistema di numerazione non era puramente decimale, in quanto si serviva
51
della base dieci per individuare le grandezze, ma introduceva anche il numero sessanta come
seconda base; i simboli individuavano i numeri uno, dieci, sessanta, seicento, tremilaseicento e
trentaseimilaseicento.
I segni che rappresentavano i numerali sumeri
I Babilonesi (3000 a.C. - 200 d.C.) utilizzarono verso il 200 o 300 a.C., ai tempi della conquista di Alessandro il Grande, un segno speciale consistente in due piccoli cunei disposti
obliquamente, segno che era stato introdotto perché servisse come indicatore di spazio dove
mancava una cifra.
Attorno al 300 a.C. i babilonesi iniziarono a usare un semplice sistema di numerazione in cui
impiegavano due cunei inclinati per marcare uno spazio vuoto. Questo simbolo tuttavia non
aveva una vera funzione oltre a quella di segnaposto, né tantomeno veniva considerato un numero.
(Notazione Cuneiforme)
317 d.c.) - Il terzo
I Maya (1.500 a.c. sistema posizionale
della storia della
matematica mondiale in ordine cronologico venne ideato dai Maya. Il loro sistema di numerazione si fondava su una base venti e i numeri erano composti da combinazioni di punti, ciascuno equivalente a uno, e di aste, equivalenti a cinque. I primi diciannove numeri erano costruiti
con punti e linee secondo uno schema additivo, derivato probabilmente da un sistema di numerazione anteriore basato sulle dita delle mani e dei piedi.
52
Quando si dovevano scrivere numeri maggiori di 20 si creava una sorta di torre di simboli, il cui
piano terreno indicava i multipli di uno, mentre il primo piano conteneva multipli di 20; al secondo piano, poi, non vi erano multipli di 20 x 20, ma di 360, in maniera tale che ogni livello
rappresentasse multipli di 20 volte maggiori di quelli del livello precedente, leggendo il numero
dall’alto verso il basso. Il sistema posizionale maya era integrato da un simbolo per lo zero a
indicare l’assenza di moltiplicatore a uno dei livelli della “torre”; il simbolo assomigliava ad
una conchiglia, o secondo altre interpretazioni, ad un occhio. I Maya usavano lo zero sia in
posizione intermedia, sia in posizione finale nelle loro sequenze di simboli. Tuttavia, nel nostro
sistema decimale ciascun livello è correlato al precedente tramite potenze della base dieci e ciò
permette di “quantificare” l’effetto dello zero, dato che aggiungerlo alla destra di un numero
comporta sempre la moltiplicazione per il valore della base; il sistema dei Maya, invece, manca
di questa proprietà a causa delle distanze diseguali tra un livello e l’altro.
Come strumenti per contare i Maya utilizzavano fagioli o chicchi di mais e legnetti (detti frijolito e palito).
a.c.) - Neppure i
I Greci (600 - 300
Greci, i più grandi
matematici della
storia, concepirono lo zero come numero: i loro numeri partivano da due, dato che per loro il
numero era molteplicità; perciò uno non era un numero e zero men che meno.
Motivi ispiratori della matematica greca - Nell’atmosfera del razionalismo ionico nacque la
matematica moderna, che non solo risponde alla domanda "come?" ma anche alla domanda che
caratterizza la scienza moderna:"perché?". Tradizionalmente padre della matematica greca é
Talete, mercante di Mileto: egli simbolizza le circostanze in cui si stabilirono i fondamenti, non
solo della matematica moderna, ma anche della scienza e della filosofia moderne.
Agli inizi: epoca omerica - La prima notazione numerica utilizzata dai Greci provenne senza
dubbio dall’influenza micenea: essa era decimale e additiva, e attribuiva segni grafici particolari
53
solo all’unità e a ognuna delle prime potenze della sua base. All’epoca di Omero (IX-VIII sec.
AC) si rappresentava l’unità con un punto, un piccolo arco di cerchio o un tratto verticale. La
decina, invece, con un tratto orizzontale, o con un cerchietto. Questo sistema presentava però lo
svantaggio della troppa semplicità, in quanto per scrivere cifre molto elevate era necessario ricorrere ad una eccessiva ripetizione di segni uguali.
Sistema erodiniaco - A partire dal VI sec. AC nacque il sistema erodiniaco (così chiamato perché trovato descritto in un frammento attribuito ad Erodiano), di base 10, e con uno schema iterativo alquanto semplice. L’unità era rappresentata con un trattino verticale, e così fino al 4: 1
= I; 2 = II; 3 = III; 4 = IIII. Furono introdotte cifre speciali per rappresentare il 5, il 50, il 500,
ottenute dalle iniziali dei nomi dei corrispondenti numeri (principio dell’acrofonia). I numeri
dal 6 al 9 erano rappresentati aggiungendo al simbolo del 5 i trattini indicanti le unità, in modo additivo. Anche le potenze intere positive della base erano rappresentate con le lettere iniziali delle corrispondenti parole numeriche. A partire dall’età alessandrina (III sec. AC) il sistema erodiniaco fu sostituito da quello ionico.
Sistema ionico - Il sistema ionico, di tipo additivo, adottato in Grecia dal III sec. a. C., prevedeva l’associazione di ogni numero a una lettera dell’alfabeto; siccome però l’alfabeto classico conteneva solo 24 lettere, furono aggiunti altri tre simboli, per un totale di 27 simboli necessari alla numerazione. Nacquero però fondamentalmente due problemi: il primo, come distinguere numeri e parole, fu risolto tracciando delle linee sopra ai numeri o aggiungendo un accento alla fine. Il secondo problema, come scrivere simboli per numeri maggiori di 999, fu risolto in modi diversi. Una virgola davanti alla cifra la moltiplicava per 1000. Per numeri ancora
più grandi si utilizzò la M del sistema erodiniaco, sopra cui si scriveva l’altro fattore della
moltiplicazione.
Talete di Mileto (624-546 a.c.) è comunemente considerato il primo filosofo della storia occidentale e tra i Greci fu il primo scopritore della geometria, l’osservatore sicurissimo della natura, lo studioso dottissimo delle stelle.
Pitagora di Samo (582-507 a.c.) è stato un matematico, legislatore e filosofo greco antico.
I Romani (753 a.c. - 476 d.c.) - Nel sistema additivo romano i numeri possono stare in
qualsiasi posizione senza modificare la quantità totale che rappresentano.
ALFABETO
I
V
X
L
54
C
D
M
ESEMPIO
10
10
10
X
X
X
DIFFERENZA CON LA NOSTRA
Centinaia
Decine
Unità
7
7
7
GLI INDIANI (200 - 1200 d.c.) - GLI ARABI (700 - 1400 d.c.) - Gli Arabi, in stretti rapporti commerciali con l’India, vennero a contatto con gli efficienti metodo di calcolo elaborati
e iniziarono a tradurre molte opere matematiche provenienti dalla valle dell’Indo. Baghdad
divenne un centro di smistamento culturale di primaria importanza; agli inizi del IX secolo il
grande matematico arabo Al-Khuwarizmi illustrò la notazione indiana nel proprio trattato di
aritmetica, gettandone le basi. La diffusione del sistema indo-arabo in Europa è da attribuire a
Leonardo da Pisa, più noto come Fibonacci (Pisa 1170-1250) e a uno studioso francese, Gerberto d’Aurillac, futuro Papa Silvestro II (999), che ne venne a conoscenza durante lunghi
soggiorni in Andalusia. L’aspetto più interessante è stato quello di usare un numero limitato
di simboli con cui scrivere tutti i numeri. Gli indiani hanno iniziato ad utilizzare solo i primi 9
simboli del sistema decimale in caratteri Brahmi, in uso dal III secolo a.C. Questi simboli assumono forme leggermente diverse secondo le località e il periodo temporale, ma sono comunque questi che gli arabi più tardi copiarono e che, in seguito sono passati in Europa fino
alla forma definitiva standardizzata dalla stampa nel XV secolo.
I matematici indiani mutarono il ruolo dello zero, da mero segnaposto in un numero in piena
regola.
55
ESEMPI
migliaia
centinaia
decine
unità
7
5
4
2
migliaia
centinaia
decine
unità
7
0
4
2
Nel XIII secolo Leonardo da Pisa, più noto come Fibonacci, tentò di mostrare la ragion pratica
di quel numero, svuotandolo di ogni pericoloso riferimento: battezzò lo zero arabo zephirum, o
cephirum, da cui poi deriverà zefiro, zefro o severo, infine abbreviata in dialetto veneziano in
zero. “Gli indiani - scrive Fibonacci nel suo Liber abaci - usano nove figure: 9, 8, 7, 6, 5, 4, 3,
2, 1 e con queste, assieme al segno 0, scrivono qualsiasi numero. [...] et dovete sapere chel
zeuero per se solo non significa nulla, ma è potentia di fare significare... Et decina o centinaia o
migliaia non si puote scrivere senza questo segno 0”.
Gli arabi chiamavano lo zero sifr ( :) ‫رفص‬questo termine significa “vuoto” ma nelle traduzioni
latine veniva indicato con “cephirum”. Fibonacci tradusse SIFR in ZEPHIRUM. Da questo si
ebbe il veneziano ZEVERO e quindi l’italiano ZERO.
Bisognerà peraltro attendere il 1491 e il testo stampato a Firenze, Aritmetica Opusculum di Filippo Calandri, per veder considerato lo zero alla stregua di un qualsiasi altro numero.
Oggi, non è del tutto vero che lo zero conti solo come numero e non sia presente nella nostra
vita di tutti i giorni:
 sulla bilancia, in assenza di oggetti sul piatto, ci si aspetta di vedere apparire lo zero;
56
il termometro segna zero gradi, e allora non è che non accade nulla, ma avviene la fusione del
ghiaccio;
 il tasso zero, vendita di un prodotto a tasso zero, zero interessi, può essere un buon acquisto;
 dai capelli a zero alla crescita zero in economia e in demografia;
 Renato Zero, il gruppo musicale degli “Zero assoluto”, la trasmissione televisiva “Anno zero”, Senza poi dire di Ground Zero, espressione inglese per indicare un territorio toccato da una
terribile deflagrazione.
Infine, anche il nostro Trilussa ha utilizzato lo zero in un’amarognola poesia moraleggiante del
1944 che si rifà a un’antica diatriba tra il numero uno e il numero zero.
Nummeri
Numeri
Conterò poco, è vero:
Conterò poco, è vero:
diceva l’Uno ar Zero -
diceva l’uno allo zero -
ma tu che vali? Gnente: propio gnente.
ma tu che vali? Niente, proprio niente.
Sia ne l’azzione come ner pensiero
Sia nell’azione che nel pensiero
rimani un coso voto e inconcrudente.
resti una cosa vuota e inconcludente.
lo, invece, se me metto a capofila
Io, invece, se mi metto a capofila
de cinque zeri tale e quale a te,
di cinque zeri uguali a te,
lo sai quanto divento? Centomila.
sai quanto divento? Centomila.
È questione de nummeri. A un dipresso
È questione di numeri. Più o meno
è quello che succede ar dittatore
è quanto succede a un dittatore
che cresce de potenza e de valore
che cresce di potenza e di valore
più so’ li zeri che je vanno appresso.
più sono gli zeri che lo seguono.
Rocco Di Scipio
57
2. TURKEY MEETING
Survey results (about parents-students-teachers)
Questionnaire for students
1. Do you like Maths?
70%
60%
Poland
Turkey
France
Italy
Spain
Romania
50%
40%
30%
20%
10%
0%
DEFINITELY YES
RATHER YES
RATHER NO
DEFINITELY NO
2. Are you good at Maths?
60%
50%
Poland
Turkey
France
Italy
Spain
Romania
40%
30%
20%
10%
0%
DEFINITELY YES
RATHER YES
RATHER NO
DEFINITELY NO
3. Would you like doing
more Maths?
60%
50%
Poland
Turkey
France
Italy
Spain
Romania
40%
30%
20%
10%
0%
DEFINITELY YES
RATHER YES
58
RATHER NO
4. Which working method
do you prefer?
60%
50%
Poland
40%
Turkey
France
30%
Italy
Spain
20%
Romania
10%
0%
In d ividual wo rk
In co up les
Team wo rk
5. What's your favourite topic
in Maths?
60%
50%
Poland
40%
Turkey
France
30%
Italy
Spain
20%
Romania
10%
0%
Arith metic
Geo metry
Lo g ic
No o n e
6. In which topic do you find more difficulties?
70%
60%
Poland
50%
Turkey
40%
France
Italy
30%
Spain
20%
Romania
10%
0%
Arith metic
Geo metry
Lo g ic
59
No o n e
7. Do you like Maths?
.
Poland
Turkey
France
Other
Decimal
numbers
Fractions
Solve
problems
Geometry
Calendar and
time
Money
calculations
Memorize
multiplication
table
Comparing
numbers up to
1000
Italy
Mental
arithmetic
70%
60%
50%
40%
30%
20%
10%
0%
Spain
8. You have much more troubles
in maths when:
50%
45%
40%
35%
30%
25%
20%
15%
10%
5%
0%
Poland
Turkey
France
Italy
Spain
Romania
Yo u d id n't p ay
atten tio n
Th e co n tent was
Yo u d id n't d o
to o d ifficult
en o ug h practise
Yo u h aven 't g ot
tro ubles
9. What do you think about the book of Maths?
80%
70%
60%
Poland
50%
Turkey
France
40%
Italy
30%
Spain
20%
Romania
10%
0%
DEFINITELY GOOD
RATHER GOOD
RATHER BAD
60
DEFINITELY BAD
10. You usually do your homework:
100%
90%
80%
Poland
70%
Turkey
60%
France
50%
Italy
40%
Spain
30%
Romania
20%
10%
0%
On yo ur o wn
With a family member
With an o utside
teach er
11. Which tools would you like to use in learning
Maths?
70%
60%
Poland
50%
Turkey
40%
France
30%
Italy
20%
Spain
Romania
Other
CD Rom
App
Multimedia
Interactive
Whiteboard
Information
technology
tools
0%
Books
10%
12. Do you think that cooperating with foreign countries for a Maths
project could improve your skills?
90%
80%
70%
60%
50%
40%
30%
20%
10%
0%
Poland
Turkey
France
Italy
Spain
Romania
DEFINITELY YES
RATHER YES
RATHER NO
61
DEFINITELY NO
Questionnaire for parents
1. Does your child like Mathematics?
70%
60%
Poland
50%
Turkey
40%
France
30%
Italy
Spain
20%
Romania
10%
0%
DEFINITELY RATHER YES RATHER NO DEFINITELY
YES
NO
I DON'T
KNOW
2. Would he/she like doing more Maths?
50%
45%
40%
Poland
35%
Turkey
30%
France
25%
Italy
20%
Spain
15%
Romania
10%
5%
0%
YES
NO
I DON'T
KNOW
3. Which kind of method do you think it's more suitable for your child?
60%
50%
Poland
40%
Turkey
France
30%
Italy
Spain
20%
Romania
10%
0%
Individual work
In couples
62
Team work
4. How does he/she do homework?
100%
90%
80%
Poland
70%
Turkey
60%
France
50%
Italy
40%
Spain
30%
Romania
20%
10%
0%
On h is/her o wn
With a family member
With an o utside
teach er
5. Homework is:
100%
90%
80%
Poland
70%
Turkey
60%
France
50%
Italy
40%
Spain
30%
Romania
20%
10%
0%
TOO MUCH
RIGHT
SO LITTLE
6. Has your child got any difficulties in Maths?
70%
60%
Poland
50%
Turkey
40%
France
30%
Italy
Spain
20%
Romania
10%
0%
YES
NO
63
I DON'T
KNOW
7. Do you feel able to help your child in maths?
60%
50%
Poland
40%
Turkey
France
30%
Italy
Spain
20%
Romania
10%
0%
YES
NO
I DON'T
KNOW
8. Can your child use maths contents in everyday
life?
90%
80%
70%
Poland
60%
Turkey
50%
France
40%
Italy
30%
Spain
20%
Romania
10%
0%
YES
NO
I DON'T
KNOW
9. What is your opinion about the maths book? It is...
80%
70%
Poland
60%
Turkey
50%
France
40%
Italy
30%
Spain
20%
Romania
10%
0%
DEFINITELY
GOOD
RATHER
GOOD
RATHER
BAD
64
DEFINITELY
BAD
I DON'T
KNOW
10. In your opinion your child’s possible maths
problems result from:
Poland
Turkey
France
Italy
Spain
The lack of
motivation
The frequent
absences
from school
Romania
A lack of
attention
90%
80%
70%
60%
50%
40%
30%
20%
10%
0%
11. From the topics provided choose the one your
child has learned the best:
80%
70%
Poland
60%
Turkey
50%
France
40%
Italy
30%
Spain
20%
Romania
Solve
problems
Calendar and
time
Comparing
numbers up
to 1000
0%
Mental
arithmetic
10%
12. Do you think that cooperating with foreign countries for a math
project could improve your child skills?
70%
60%
Poland
50%
Turkey
40%
France
30%
Italy
20%
Spain
Romania
10%
0%
DEFINITELY YES
RATHER YES
RATHER NO
65
DEFINITELY NO
I DON'T KNOW
Questionnaire for teachers
1. In your opinion, is the number of hours of maths
per week adequate?
90%
80%
70%
Poland
60%
Turkey
50%
France
40%
Italy
30%
Spain
20%
Romania
10%
0%
DEFINITELY YES
RATHER YES
RATHER NO
DEFINITELY NO
2. In your opinion, which working method is more
effective?
90%
80%
70%
Poland
60%
Turkey
50%
France
40%
Italy
30%
Spain
20%
Romania
10%
0%
In d ividual wo rk
In co up les
Team wo rk
3. In which activity do students show more difficulties in Maths?
100%
90%
80%
70%
60%
50%
40%
30%
20%
10%
0%
Poland
Turkey
France
Italy
Spain
Romania
Arithmetic
Geometry
Logic
66
No one
4. Do you do Maths activities outside the classroom?
60%
50%
Poland
Turkey
40%
France
30%
Italy
Spain
20%
Romania
10%
0%
YES
NO
n o an swer
5. Do you think it’s important giving homework?
120%
100%
Poland
80%
Turkey
France
60%
Italy
Spain
40%
Romania
20%
0%
DEFINITELY
YES
RATHER YES
RATHER NO
DEFINITELY NO
6. How do you evaluate students’ books
of Maths?
120%
100%
Poland
Turkey
80%
France
60%
Italy
Spain
40%
Romania
20%
0%
DEFINITELY GOOD
RATHER GOOD
RATHER BAD
67
DEFINITELY BAD
7. In your opinion your child’s possible Maths
problems result from:
Poland
Turkey
France
Italy
Spain
Frequent
school
absences
A lack of
motivation
Various kinds
of dysfunctions
Romania
A lack of
attention
100%
90%
80%
70%
60%
50%
40%
30%
20%
10%
0%
8. Which tools do you consider more effective in
teaching Maths?
Poland
Turkey
France
Italy
Spain
Other
CD Rom
App
Multimedial
Interactive
Whiteboard
Information
technology
tools
Romania
Books
100%
90%
80%
70%
60%
50%
40%
30%
20%
10%
0%
9. Do you think E.Te.Mat Project will improve our
students’ skills?
120%
100%
Poland
Turkey
80%
France
60%
Italy
Spain
40%
Romania
20%
0%
DEFINITELY
YES
RATHER
YES
RATHER NO DEFINITELY
NO
68
n o an swer
10. Do you think E.Te.Mat Project will increase your
motivation as a teacher?
100%
90%
80%
Poland
70%
Turkey
60%
France
50%
Italy
40%
Spain
30%
Romania
20%
10%
0%
YES
NO
n o an swer
11. Which results do your students reach in percentage?
(average taken from answers)
80%
70%
Poland
60%
Turkey
50%
France
40%
Italy
30%
Spain
20%
Romania
10%
0%
Unsatisfactory
Satisfactory
Good
Very good
12. What do you think it’s useful to improve Maths teaching/learning
process?
100%
90%
80%
70%
60%
50%
40%
30%
20%
10%
0%
Poland
Turkey
France
Italy
Spain
69
Comparison to
collegues
Individual
study
Conferences
with inside
teachers
Training
courses
E.Te.Mat
Project
Romania
13. Do you use to let your students practise logic exercises?
90%
80%
70%
Poland
60%
Turkey
50%
France
40%
Italy
30%
Spain
20%
Romania
10%
0%
DEFINITELY YES
RATHER YES
RATHER NO
DEFINITELY NO
no answer
14. Do you think it’s important to propose your students everyday life tasks?
120%
100%
Poland
Turkey
80%
France
60%
Italy
Spain
40%
Romania
20%
0%
DEFINITELY YES
RATHER YES
RATHER NO
DEFINITELY NO
15. Which kind of difficulties do you meet the most in teaching?
100%
90%
80%
70%
60%
50%
40%
30%
20%
10%
0%
Poland
Turkey
France
Italy
Spain
Romania
Arithmetic
Geometry
Set problems
70
Logic
Reality tasks
3. ROMANIA MEETING
Countries curriculum
ITALY curriculum
National Guidelines for the Curriculum in kindergarten and the first cycle of' Education
(2012 September)
Mathematics contributes to the cultural formation of individuals and communities, developing the ability to put in close relationship "thinking " and "doing" and offering tools to
perceive, interpret, and linking natural phenomena, concepts and artifacts built by man, everyday events.
In particular,
mathematics gives tools for the scientific description of the world and for a useful application
in everyday life; it helps to develop the ability to communicate and discuss, to argue properly,
to understand the views and arguments of others.
The
National
Successes
for
at the end of primary school.
Guidelines
the
for
the
Development
Curriculum
:
of
Skills
Students move with confidence in written and mental calculations with whole numbers and they are able to evaluate the opportunity to use a calculator.
Students recognize the forms in space, relationships and structures found in nature or
created by man.
They describe, they denominate and classify figures based on geometric characteristics, they determine measures, they design and manufacture concrete models of various types.
They
use tools for geometric design (line, compass...) and the most common measuring instruments
(meters, protractor...).
They
are
able to research data to obtain informations and to construct representations (tables and
graphs). They also extract informations from data presented in charts and graphs.
They
read and comprehend texts that involve logical and mathematical aspects.
They
are able to solve difficult problems in all areas of content while maintaining control on both the
solution process and the results.
They
describe the procedure followed and recognizes solution strategies than their own.
They build
reasoning with assumptions, supporting their ideas and confronting the views of others.
They recognize and use different representations of mathematical objects ( decimals, fractions, percentages, scale reduction...).
They develop a positive attitude to mathematics, through meaningful experiences, that let them understand how the mathematical tools - that they learned to use - are useful to operate in reality.
Learning goals at the end of third class of primary school (9 years old)
Numbers Counting objects or events, verbally and mentally, in the sense progressive
and regressive, for jumps of two, three...
Reading and write natural numbers in decimal notation, having awareness of positional
notation; compare and order them, even representing them on the line.
Knowing with certainty the tables of multiplication of numbers
up to 10. Performing operations with natural numbers with the
71
Knowing with certainty the tables of multiplication of numbers up to 10. Performing
operations with natural numbers with the usual algorithms written.
Reading, writing, comparing decimal numbers and representing them on the straight;
performing simple additions and subtractions, also with reference to the coins or the result of
simple measures.
Space and figures
dies.
Perceiving their position in space and to estimate distances and volumes from their bo-
Communicating the position of objects in physical space, both compared to the subject
and both compared to other people or objects, using appropriate terms (up/down, front/back,
left/right, in/out).
Running a simple path from the verbal description or by a drawing, describing a path that you are
doing and giving instructions to someone because he performs a desired path.
Recognizing,
denominating and describing geometrical figures.
Drawing geometric shapes and materials to build models in space.
Reports, data and forecasts
Sorting numbers, figures, objects based on one or more properties, using appropriate
rappresentations, depending on the contexts and purposes.
Arguing on the criteria that were used to create classifications and regulations assigned.
Reading and representing data and reports with charts, diagrams and tables.
Measuring sizes (lengths, time, etc.) , using arbitrary units and conventional units and instruments
(meter, clock, etc.).
Learning goals at the end of the fifth class of primary school
Numbers
Reading, writing, comparing decimal numbers.
Perform the four operations with security, considering if resorting to the mental or written
calculation or to the calculator, depending on the situation .
Carrying out division with remainder of natural numbers; identifying multiple and partitions
of a number .
Estimating the result of an operation.
Operating with fractions and recognizing equivalent fractions.
Use
decimals, fractions and percentages to describe everyday situations .
Interpreting negative integers in concrete contexts .
Representing the numbers known on line and use scales in meaningful contexts for science and
technology .Knowing notation systems of the numbers that are or have been in use in places, times and other cultures .
72
Space and figures
Describing, denominating and classifying geometric figures, identifying significant elements and symmetry, also in order to reproduce them by themselves.Playing a figure based
on a description, using the appropriate instruments (squared paper, ruler and compass, teams,
geometry software).
Using
the Cartesian plane to locate points.
Building materials and using models in space and plan in order to support a first display capabilities.
Recognizing figures
rotated, translated and reflected.
Relations, data and forecasts
Representing public relations and data and, in important situations, using representations to obtain information to formulate an opinion and to take decisions.
Using notions of frequency, trend and arithmetic means, if appropriate to the type of data given.
Representing problems with tables and charts that express their structure.
Using the Main Unit for lengths, angles, areas, volumes/capacity, time intervals, masse,
to make weights and measures estimates.
Switching from one unit to another one, limited to common units more used, also in the
context of Monetary System.
In concrete situations, of a pair of events guessing and beginning to argue what is more
likely, giving a first quantification in more simple cases, or recognizing if they are events
equally likely.
Recognizing and describing regularity in a sequence of code numbers or figure.
Comparing and measuring angles using properties and tools.
Using and distinguishing between the concepts of squareness, parallelism, horizontality,
verticality.
Playing in scale a figure assigned (using, for example, the squared paper).
Determining the perimeter of a figure using the most common formulas or other proceedings.
Determining the area of rectangles and triangles and other shapes for breakdown or
using the most common formulas.
Recognizing flat representations of three-dimensional objects, identifying points of several view of the same object (top, front, etc).
73
REUNION ISLAND (FRANCE) curriculum
Extracts from official instructions 2008 or french primary school concerning
Mathématiques
Extract from the preamble valid for three cycles
of primary school:
"National primary school programs define for each area of education the knowledge and
skills to be achieved in the context of cycles; they indicate the annual benchmarks to organize
progressive learning in French and mathematics. However, they give free choice of methods
and approaches, demonstrating trust in teachers for implementation tailored to students. "
Cycle program Extracts 3 on mathematics
In continuation of the first years of primary school, mastering the French language as
well as the main elements of mathematics are the priority objectives of CE2 and CM.
The teachings of French and mathematics are subject to increases by grade, attached to
this program.
The practice of mathematics develops a taste for research and reasoning, imagination
and the capacity for abstraction, rigor and accuracy.
CE2 to CM2 in the four areas of the program, students enrich their knowledge, acquire
new tools, and continues to learn how to solve problems. It strengthens mental math skills. It
acquires new automation. The acquisition of mathematics mechanisms is always associated
with an intelligence of their meaning.
The mastery of the main elements using mathematics to act in everyday life and prepare
further studies in college.
Proportionality is approached from situations involving the percentage of notions of
scale, conversion, enlargement or reduction of figures. For this, several procedures (particularly
that of the so-called "rule of three") are used.
74
Competency 3 : The main elements of mathematics and scientific and technological culture
The main elements of mathematics
The student is able to
- Write, naming, comparing and using whole numbers, decimals (up to hundredths) and some
simple fractions;
- Restoration of the addition tables and obtained when 2 to 9;
- Use surgical techniques of the four operations on whole numbers and decimals (for the division, the divisor is a whole number);
Mentally calculate using the four operations;

Estimate the magnitude of a result;


Use a calculator;

Recognize, describe and name the usual figures and solids;

Use the rule, square and compass to verify the nature of common plane figures and build
with care and precision;

Use common units of measurement; use measuring instruments; perform conversions;

Solve problems involving the four operations, proportionality, and involving different mathematical objects: numbers, measurements, "rule of three", geometric figures, diagrams;

Learn organize digital or geometric information, justify and assess the likelihood of a result;

Read, interpret and construct some simple representations: tables, graphs.
75
a - Nombres et calcul
They study organized numbers continued until billion, but higher numbers may be encountered.
The natural numbers:
- Principles of decimal numeration position: value depending digits of their position in writing numbers;
- Oral designation and write numbers and letters;
- Comparison and storage of numbers on a number line identification, use of signs
> And <;
- Arithmetic relationships between commonly used numbers: double, half, quadruple, quarter, triple, third ... The concept of multiple.
Decimal numbers and fractions:
- Simple and decimal fractions: writing, mentoring between two consecutive integers, writing as sum of an integer and a fraction less than 1, the sum of two decimal fractions or two fractions of the same denominator;
- Decimal numbers: oral descriptions and figures scriptures, value of numbers based on their position, passage of scripture to write a fractional point and vice versa, comparison and storage, tracking on a number line; approximate value of a decimal to the
nearest unit to the nearest tenth, to the nearest hundredth.
The calculation:
- Mental: addition and multiplication tables. Daily training in mental calculation
on the four operations promotes appropriation of numbers and their properties.
- Placed: mastering a surgical technique for each of the four operations is essential.
- The calculator: Calculator been wise use depending on the computational complexity faced by students.
The resolution of problems related to everyday life helps to deepen the knowledge
of the numbers studied, strengthen the control of meaning and practice of operations,
develop a taste for rigor and reasoni
76
b – Geometry
The main objective of teaching geometry CE2 to CM2 is to allow students to move progressively from a perceptual object recognition to a study based on the use of instruments and
measurement plot.
Relationships and geometric properties: alignment, squareness, parallelism, equality of
lengths, axial symmetry, the middle of a segment.
The use of instruments and techniques: ruler, square, compass, tracing paper, graph paper, dotted paper folding.
The plane figures: square, rectangle, diamond, parallelogram, triangle and its particular
case, the circle:
- Description, reproduction, building;
- Specific vocabulary related to these figures: side top angle diagonal symmetry axis, center,
radius, diameter;
- Enlargement and reduction of plane figures, in connection with proportionality.
Conventional solids: cube, cuboid, cylinder, prisms, pyramids.
- Recognition of these solids and study of some patterns;
- Specific vocabulary concerning these solid: vertex, edge, face.
The problems of reproduction or construction of various geometric configurations
mobilizing knowledge of the usual figures. They are an opportunity to make good use of
the specific vocabulary and the steps of measuring and layout.
77
c - Sizes and measurements
Lengths, weights, volumes : measurement, estimation, Legal metric units, the calculation variables, conversions, perimeter of a polygon form the perimeter of squares and rectangles, the length of the circle, the volume of cuboid.
The areas: comparison of surfaces according to their areas, common units, conversions;
formula for the area of a rectangle and a triangle.
The angles: comparison, use a jig and the square; right angle, acute, obtuse.
The identification time: Reading the time and calendar.
Durations: measurement units durations, calculating the elapsed time between two
given moments.
Money
Solving concrete problems helps to consolidate the knowledge and skills relating to
quantities and their extent, and give them meaning. On this occasion custom estimates can be
provided and validated.
d - Organization and data management
The capacities of organization and data management develop by solving problems of
everyday life or from other teachings. This is gradually learn to sort data, to classify, to read or
to produce tables, graphs and analysis.
Proportionality is approached from situations involving the percentage of notions of
scale, conversion, enlargement or reduction of figures. For this, several procedures (particularly
that of the so-called "rule of three") are used.
ning.
The following tables provide benchmarks for teaching teams to organize escalation learOnly new knowledge and skills are mentioned in each column.
For each level, the knowledge and skills learned in the previous class are consolidated.
Problem solving plays an essential role in mathematical activity.
It is present in all areas and is exercised at all stages of learning.
78
competencies
expected
To tier 2
the Common
Base
- Recognize, describe and name
the figures and
customary solid
- Use the rule,
square and compass to check the
nature of common
plane figures and
build with care
and precision
Progress proposed by the
O.B for CE2
O.B for CM1
Progress proposed by
the
O.B for le CM2
In the plane
In the plane
In the plane
- Recognize, describe,
- Recognize that the lines are
- Use instruments to
name and reproduce, draw
parallel.
check the parallelism of
geometric shapes: square,
- Use experiencing geometric
two lines (rule and
rectangle, diamond,
vocabulary aligned points,
square) and to draw
triangle.
right, perpendicular lines,
parallel lines.
- Check the nature of a
parallel lines, segment, me-
- Check the nature of a
plane figure using the ruler
dium angle axis of symmetry,
figure through the use
and the square.
center of a circle, radius,
of instruments.
- Build a circle with a com-
diameter.
- Build a height of a
pass.
- Check the nature of a plane
triangle.
- Use vocabulary situation:
figure simple using the scale,
- Reproduce a triangle
side top angle setting.
the square and compass.
using instruments.
- Recognize that a figure
- Describe a figure to identify
has one or more lines of
it among other figures or to
symmetry by folding or
breed
using tracing paper.
- Perceive and
- Draw on graph paper, the
recognize parallel
symmetrical figure of a
figure given in relation to a
and Perpendicular
given line.
- Solve reproduc-
Space
Space
Space
tive problems,
- Recognize, describe and
- Recognize, describe
name:
name the solid rights: cube,
and name the solid
a cube, a cuboid.
paved prism.
rights: cube, pad, cylin-
- Use vocabulary situation,
- Recognize or complete a
der, prism.
face, edge, summit.
pattern of cube or tile.
- Recognize or supple-
building
ment a law firm patron.
79
O.B for CM1
O.B for le CM2
Reproductive problems, construction
- Reproduce the figures (on plain paper,
Complete a by axial symmetry.
- Draw a figure (on plain paper, checkered
information or geometric,
checkered or dotted), from a model.
- Draw a simple figure from a construction
or dotted), from a
justify and assess the
- Build a square or a rectangle of given
program or by following the instructions.
construction program or a freehand
likelihood of a result.
dimensions.
competencies expected
To tier 2
the Common Base
- Ability to organize digital
O.B for CE2
drawing (with indications regarding the
properties and dimensions).
Angles
Angles
- Compare the angles of a figure using a
- Reproduce a given angle using a
template.
template.
- Estimate and check by using the square, a
right angle is acute or obtuse.
The practice of mathematics develops a taste for research and reasoning, imagination and the
capacity for abstraction, rigor and accuracy.
CE2 to CM2, students enrich their knowledge, acquire new tools, and continues to learn how to solve
problems. It strengthens mental math skills. It acquires new automation. In mathematics, the
acquisition mechanisms is always associated with understanding.
the
O.B for CM1
the
O.B for le CM2
Reproductive problems,
Reproductive problems,
- Reproduce the figures (on plain paper,
construction
construction
checkered or dotted), from a model.
- Draw a figure (on plain paper,
information or geometric,
- Build a square or a rectangle of given
- Draw a simple figure from a
checkered or dotted), from a
justify and assess the
dimensions.
construction program or by following
construction program or a freehand
the instructions.
drawing (with indications regarding the
To tier 2
the Common Base
O.B for CE2
likelihood of a result.
properties and dimensions).
Angles
Angles
- Compare the angles of a figure using a
- Reproduce a given angle using a
template.
template.
- Estimate and check by using the square,
a right angle is acute or obtuse.
The practice of mathematics develops a taste for research and reasoning,
imagination and the capacity for abstraction, rigor and accuracy.
CE2 to CM2, students enrich their knowledge, acquire new tools, and continues
to learn how to solve problems. It strengthens mental math skills. It acquires
new automation. In mathematics, the acquisition mechanisms is always
associated with understanding.
80
Tier 2 CM2 /
Competence validated on
1. Mastering the French language
2. Practice a foreign language
3. Key elements of mathematics Scientific and technological culture
4. Mastery of common information technology and communication
5. humanistic culture
6. Social and civic competences
7. Autonomy and initiative
81
POLAND curriculum
Aims Broad objectives:
Calculating skills
Students make simple memory operations: addition, substraction, multiplication,division
using the positive integers (whole numbers), integers and fractions. They know and use algorithms to make calculations. Students can use acquired concepts, skills and processes in real
problem-solving situations.
Making and using mathematical information
Students interpret text, number and graphical information. They know the basic mathematical concepts and are able to explain the meaning of them. Students formulate answers and
write the results in a correct way.
Mathematical modelling
Students adjust correct mathematical formulae to simple situations. Students explore, perceive, use and appreciate mathematical patterns in order to convert the text of the exercise into
simple, arithmetic equation.
Understanding and making of strategies
Students use acquired simple concepts, establish the sequence of doings (including calculations) in the process of problem-solving. Students plan, monitor and evaluate solutions.They can also draw conclusions using different kind of information and facts provided or
learnt.
The content of teaching- strands
The positive integers in decimal system.
The student :
reads and writes the multi-digit integers (whole numbers)
interprets the integers on a number line
compares the integers
round whole numbers e.g round whole numbers to nearest ten, hundred, thousand
changes decimals up to 30 into Roman numerals and vice versa
Whole number calculations
The student:
explores and identifies place value in whole numbers
adds and substracts multi-digit numbers and solves simple problems
knows and recalls addition and subtraction facts
solves word problems involving addition and subtraction
develops an understanding of multiplication as repeated addition and vice versa.
adds and subtracts whole numbers without and with a calculator
multiplies and divides integres by other whole numbers, without and with a calculator
identifies whole numbers divided by 2, 3, 5, 9, 10, 100
identifies and explores square and cube roots
divides two-digit numbers into prime factors
82
Directed numbers
The student:
1.
identifies positive and negative numbers in context
2.
identifies positive and negative numbers on the number line
3.
calculates the absolute value
4.
compares directed numbers
5.
makes simple memory calculations usind directed numbers
Fractions and decimal fractions
The student:
1.
calculates a unit fraction of a number and calculates a number, given a unit fraction of the number
2.
reduces or simplifies the fractions
3.
finds common denominator to fractions
4.
expresses improper fractions as mixed numbers and vice versa
5.
rounds decimal fractions
6.
adds and subtracts simple fractions and simple mixed numbers
7.
compares and orders fractions and decimals
8.
multiplies a fraction by a whole number and a fraction by a fraction
9.
expresses tenths, hundredths and thousandths in both fractional and decimal form
Fraction and decimal fraction calculations
The student:
1.
adds and subtracts whole numbers and decimals without and with a calculator
2.
multiplies and divides a decimal by a whole number, without and with a calculator
3.
makes simple calculations using fractions and decimal fractions
4.
estimates the results of calculations
5.
compares fractions differentially
6.
identifies and explores square roots and cube roots of fractions
7.
computes the integer fraction
Straight lines and sections (lenghts)
The student:
1.
identifies, describes figures: point, straight ,ray, length
2.
identifies, describes and classifies vertical, horizontal and parallel lines
3.
identifies, describes and classifies oblique and perpendicular lines
4.
measures the length with an accuracy of 1 millimeter
5.
knows if he wants to find the distance from a point to a line he must find the
length of the perpendicular line
Angles
The student:
1.
recognises, classifies and describes angles, their rays and vertex
2.
measures angles less than 180 degrees with accuracy of 1 degree
3.
draws an angle less than 180 degrees
4.
classifies angles as acute, obtuse and right angles
5.
estimates angle sizes
6.
recognises and uses features of apex and adjacent angles
83
2-D shapes
The student:
1. classifies and describes triangles and quadrilaterals
2. identifies, describes and classify 2-D shapes: equilateral, isosceles and
scalene triangle, rectangle, parallelogram, rhombus, trapesoid
3. explores, describes and compares the properties (sides, angles, parallel
and non-parallel lines) of 2-D shapes
4. identifies the properties of the circle: diameter, radius and chord circle
5. uses the assertion of the sum of angles in a triangle
3-D shapes
The student:
1. identifies, describes and classifies 3-D shapes, including cube, cuboid,
cylinder, cone, sphere, triangular prism, pyramid
2. recognises the nets of prisms and pyramids
3. draws the nets of simple 3-D shapes (prisms)
Calculating area, length, and other geometric properties
The student:
1.
2.
3.
4.
5.
6.
calculates the perimeter a polygon
calculates the area of the square, rhombus, parallelogram, triangle and
trapezium presented on a drawing and in practical situations
estimates and measures length using appropriate metric units
calculates area using acres and hectares
estimates and measures capacity using appropriate metric units
calculates angle measures
84
Practical calculations
The student:
1.
2.
3.
4.
5.
6.
7.
develops an understanding of simple percentages and relate them to
fractions and decimals
solves problems involving operations with whole numbers, fractions,
decimals and simple percentages
solves and completes practical tasks and problems involving times and
dates
reads the temperature on the Celsius scale
renames the measures of weight and length
identifies given scale and draws items to a larger or smaller scale.
in the practical situation calculates the length of the road at a given
speed and given time and so forth
Elements of mathematical statistics
The student:
1.
compiles and uses simple data sets
2.
uses charts, graphs and tables to read and interpret data
Problem-solving activities
The student:
1.
reads with understanding a simple text which includes number information
2.
can extract important information and data from the activity, makes auxiliary drafts
before solving the operation
3.
notices the relationship between the information and data provided
4.
selects appropriate concepts, methods and techniques to apply to mathematical
problems.
5.
makes connections and begins to reason deductively in geometry, number and
algebra, including using geometrical constructions
6.
verifies the outcome of the execise, judging its meaningfulness
85
ROMANIA curriculum
Basic aquisitions cycle (kindergarten – grade II) having as main objective
accomodation with the demands of the scholar system and initial literacy;
Cultivation/ building-up cycle (grade III – grade VI) when the main aim is building
basic skills necessary to carry forward their studies.
In our country , The National Curriculum is built on the following seven curricular
areas:
Language and communication
Mathematics şi Science
Man and society
Arts
Physical Education and sport
Technology
Counselling and guidance
For Mathematics , and for the other subjects as well, the number of hours intended for
compulsory activities for all students, which is meant to ensure equal chances for all of them, is
established in the plan frame.
Being a dynamic tool, the Romanian curriculum is undergoing a reform.
Differences :
The traditional field of view, where the curricular areas included mono disciplinary
approached subjects was given up and it turned towards a multi or interdisciplinary approach ,
packing together more school subjects.
The actualul curriculum aims at building functional educational competencies,
absolutely necessary for students to get into the work market.
86
In the school year 2014-2015 grade 2, is the first generation who started primary school at
the age of 6.The new curriculum is working out. The syllabus keeps to the curricular projection
model ,focused on skills.
The compulsory number of hours:
2 Grade
3 Grade
4 Grade
5 hours
4 hours
4 hours
87
2.nd CLASS
3-rd CLASS
4-th CLASS
Natural Numbers
Natural Numbers
Natural Numbers
. Reading, position
value and composition
of numbers 0 -1000;
. Composing, reading,
writting, comparison,
ordering and rounding
between 0- 1 milion
. Composing, reading,
writting, comparison,
classes
(units,
thousands, millions),
ordering and rounding
between 0- 1 milion
. Positional numeral
system: writing
numbers
in
decimal form (sum
of products by a
factor 10, 100, 1000,
etc.);multiplication
with 10, 100,
1.000.
. Comparison of
numbers between 01000;
. Ordering and
rounding 0-1000;
. Additions and
subtractions between 0
-1000,grouping or
counting ;
. Multiplications and
divisions till 100;
.Using terms and
mathematical symbols
like sum , total,
difference, minuend,
subtrahend etc.
.
Additions
and
subtractions between 0
-10000, grouping or
counting
. Using terms and
mathematical symbols
like sum , total,
difference, minuend,
subtrahend, even less,
even more etc.
. Finding an unknown
number within a
relationship type
?+
a=b
.Highlighting some of
the Assembly
properties
(commutativity,
associativity, identity
element) using objects
and representations,
without the use of
terminology
88
. Roman numbering
system.
2.nd CLASS
. The use of unconventional measures for determining and comparing the
length;
. The use of units of measurement for the determination of lengths, comparing
and ordering various
events;
. Upon the completion of
equivalent value exchanges
through conventional representation standard and
nonstandard using money
matters-simple game-type
income-expenses, with
numbers from 0-1000;
. Identification and use of
customary units of measure for length, capacity,
mass .
3-rd CLASS
4-th CLASS
. Measurements using nonstandard standards
. Measurements using conventional standards: use
appropriate measuring
tools: tape measure, ruler
graduated scales, scales,
clock
. Length measuring units :
multiples, submultiples of
the metre
. Measuring Unit capacity:
litre, multiples, submultiples .
. Measuring Mass Units:
kilogram, multiples, submultiples.
. Units of measurement for
time: the hour, the minute,
day, week, month, year;
. Coins and banknotes, including those of the European;
. Using appropriate measuring instruments: ruler,
tape measure, calibrated
scales, balance
. Length measuring units:
meter, multiples, submultiples, transformation via multiplication and Division with
10, 100 and 1000;
. Measuring units capacity:
litre, multiples, submultiples, transformation via
multiplication and Division
with 10, 100 and 1000;
. Measuring units: kilogram, multiples, submultiples, transformation via
multiplication and Division
with 10, 100 and 100;
Units of measurement for
time: the hour, the minute,
second, day, week, month,
year, decade, century, Millennium,
. Coins and banknotes
89
SPAIN curriculum
LOMCE 8/2013
Ley Orgánica para la Mejora de la Calidad Educativa
The Curriculum for Primary:
Objectives. To reach at the end of the Primary Education.
Competences:
1. Linguistic competence
2. Mathematics competence
3. Digital competence
4. Learning to learn competence.
5. Social competence
6. Entrepeneur actitud competence
7. Cultural awareness competence.
Contents
Standars of evaluation
Methodology
The government establishes: six levels ( 1º- 6º) or 6-12 years old.
Areas: Natural Science
Social Science
Language and literature
Mathematics
Fisrt foreing language
Phisical Education
Religion (optional)
Specific areas ( autonomy of centres and regions):
Artistic Education
Second foreing language
Social and Civic values
90
Weekly Timetable
AREAS
1º
2º
3º
4º
5º
6º
Language and literature
4,5
4,5
4,5
4,5
4,5
4,5
Mathematics
4,5
4,5
4,5
4,5
4,5
4,5
Natural Science
1,5
1,5
2
2
2
2
Social Science
2
2
2
2
2
2,5
First foreing language
2,5
2,5
2,5
3
3
3
Phisical Education
3
3
3
2,5
2,5
2
Religion/ Socilal Civic Values
2
2
2
1
1
1
Artistic Eduaction
2
2
1,5
1,5
1,5
1,5
Second foreing
language/Support
0,5
0,5
0,5
1,5
1,5
1,5
Break
2,5
2,5
2,5
2,5
2,5
2,5
TOTAL
25
25
25
25
25
25
CONTENTS
3º
Natural Numbers
• Numbers up to five
figures. Reading, position
N value and decomposition.
U •Comparison, ordering
M and rounding.
B •Ordinal numbers until the
E thirtieth.
R •Roman Numbering
S system.
•Addition and subtraction
with led in tenth and
hundredth. Properties.
•Multiplication with
several figures with and
without led.
4º
5º
Natural Numbers
•Numbers up to the
million. Reading, position
value and decomposition.
•Comparison, ordering
and rounding.
•Roman numbering
system.
with led in tenth and
hundredth. Properties.
•Multiplication by several
figures. Associative,
commutative and
distributive properties.
Natural Numbers
•Decimal numbers system.
Reading, position value
and decomposition.
•Comparison, order and
rounding.
•Operations: addition,
subtraction,
multiplication, division.
•Multiples and divisors.
•Prime numbers and
compound numbers.
•Divisibility criterion.:
2,3,5 and 10.
Fractions . Concept.
Proper fraction. Improper
fraction.
91
•Associative, commutative
properties.
•Multiplication by ten, one
hundred, one thoussand..
•Accurate and inaccurate
division. Division test.
Fractions. Concept,
reading,comparison and
representation of fractions.
3º
M
Measurement
of length,
E
capacity
and mass
S
•Units
of
mesurement.
U
•Expresing
measures units.
R
•Time.
E Units bigger than
day:M
day, week, month and
year.E •Units smaller than
day:Nhour, minute, second.
T and notes. Euros
•Coins
and cents. Addition and
subtraction with them.
•Accurate and inaccurate
division between two or
three figures. Division
test.
Fractions. Concept.
Reading, comparison and
representation of
fractions.
•Fraction of an amount.
of fractions with the same
denominator.
Decimal numbers.
Application, reading and
writing of deccimal
numbers: unit, tenth and
hundredth.
•Comparing, ordering and
rounding decimals.
•Comparing numbers:
natural numbers, fractions
and deecimals.
•Fraction of an
amount.Equivalent
fractions.
•Comparing fractions.
Addition and subtraction of
fractions. Fraction as exact
division and not exact .
Mixed number.
•Decimal numbers.
•Application , reading and
writing od decimal
numbers: tenth, hundredth
and thousandth.
•Comparison, ordering and
estimation decimal
numbers.
•Operations with decimal
numbers: addition,
subtraction, multiplication
division.
•Comparison numbers:
natural fractions and
decimal
4º
5º
•Measurement of length,
capacity and mass. •Units
for measuring length,
capacity and mass. Diferent
ways of expresing
measures. •Measures
operations: addition and
subtraction. •Measuring
instruments. •Area, Area
units. Addition and
subtraction measurements
of area. •Time. Bigger Units
than day: day, week,
month, year, lustrum,
decade. •Smaller units than
day: hour, minute, second.
Complex and not complex
form. Addition and
subtraction 92
•Measurement of length ,
capacity and mass. •Ways of
expressing measures.
•Operations with measures,,
addition, subtraction,
multiplication, division.
•Measuring instruments.
•Area. Units of area.
measures of area. •Time.
Units higher than day: week,
month, year, lustrum,
decade, century. •Units
lower than day: hour,
minute, second. Complex
shape and not complex.
3º
4º
•Addition and
subtraction time
data. •Coins and
notes. •Euros and
cents.
Operations.
93
5º
•Addition and
subtraction time
data. Sexagesimal
system. •Money,
coins and notes.
•Euro and cents.
Operations.
TURKEY curriculum
CIRRICULUM FOR THE 2nd GRADES
UNIT 1: GEOMETRIC SOLIDS & NUMBERS
Objectives 1: Ss will be able to show the surfaces,vertex &edges
2 :Ss will be able to measure the lengths using both the standard &non-standard
measures.
3: Ss will be able to explain dozen with examples
Activities :Classroom objects,ropes, meter ,pitures ,toys
UNIT 2: NUMBER HUNTING
Objectives 1: Ss will be able to tell the time
2: They will be able to explain the relation between the ‘whole’,’half’ and
‘quarter’.
3: They will be able to put the numbers smaller than 100 into order.
4: …explain the symmetry with models & use the relation in a new
one with different materials.
Activities : 1-Shoe sizes
2-Making a necklace
3-What time is it ?
4-Fold & cut
94
UNIT 3: ADDITION TIME
Objectives 1: ……add the numbers below 100with carry &without carry
2: ….recognize the banknotes & coins
3: …settle & solve addition problems with natural numbers
Activities :1-How many baskets of apples are there ?
2: Game: How much Money have you got ?
3: How long is it (objects ) ?
UNIT 4 : SUBSTRACTION & MULTIPLICATION
Objectives 1: ……to explain the operation with models
2:….. to do multiplication &substraction with numbers below 100.
3: …to explain the ,’0’ (zero) & ‘1’ ( one ) effect in multiplication
Activities : 1- Let’s count the toothpicks & buttons
2-Let’s calculate mentally
UNIT :5 DIVISION & MEASURING
Objectives
1: …. to create number patterns
2:… to divide maximum 20 objects into 1,2,3,4,5 groups & tell the number
3: … to explain the relation between hour-day,week-day,month-day, seasonmonth, year-week & year-month
Activities :
1- Let’s share the pencils &erasers equally
2-Shopping at the greengrocer’s
3-Let’s make a class birthday calendar
95
3rd GRADES CURRICULUM (9 Years old pupils)
UNIT 1 THE SHAPES and THE NUMBERS
Objectives:
Students will be able to;
-İllustrate ‘dot’ by themselves.
-name and show easily ‘dot’ from our daily lives.
-describe prism ,triangular, cone, triangular, cylinder, geometrical cone and orb.
- to read and write Romanian numbers.
-to identify and to classify the odd and the even numbers.
-able to use origami.
UNIT 2 ADDICTION and THE WORLD OF THE SHAPES.
Objectives:
Students will be able
- to solve addiction problems mentally.
-solve money problems.
-measure the range of shapes’ faces with non-standart metric.
-use patterns by drawing different geometrical shapes.
Fold papers to form symmetricalness.
UNIT 3 SUBSTRACTION, ANGLES and SHAPES
Objectives;
- form and solve problems with addiction and substraction.
- solve substraction problems mentally .
-draw angles using ruler or mitre.
Classify acute angle, right angle,obtuse angle and straight angle.
96
UNIT 4 MULTIPLICATION and MEASURING THE LENGTHS
Students will be able to ;
-explain the relation between centimetre an metre.
-predict the lengths of authentic objectsaround themselves and compare their presuppositions
with real results.
-form and solve the problems with meter and centimeter units.
UNIT 5 DIVISION and MEASURING
Objectives;
-form and solve the problems with at least two operations.One of them issupposed to be
division.
-tell the time and show the time by themselves.
-create their own clocks.
-weigh the authentic materials
97
4th Grades Mathematic Curriculum(10 yeras old pupils)
Unit 1 The Geometry Around Us
Objectives:
-name square,rectangle, triangle.
-define the edge & angles and show with symbols
Classify the triangles according to their sizes.
ACTIVITIES
1.Letsmake different angles with strings/ropes.
2.We can draw triangles
3.Lets examine square rectangle triangle
Unit 2Data &Operation with numbers
Objectives:
-make a column chart
-write and read the numbers with 3,4,5 and 6 digits.
-explain the relation between a year,a
month,a week and a day
ACTIVITIES
1.We can substract, add, and multiply
2. We can read and write the numbers correctly.
3.We are making a classroom calender.
98
UNIT 3 LETS MEASURE,WEIGH,and GET TO REALITY
Objectives:
-recognize ton,kilogram,gram,and miligram and convert one to another.
-to settle and solve problems about litre and mililitre
-to use the words about possibility in appropriate sentences
Activities:
1.Lets add,substract mentally.
2. How many litres is it?
3.Lets use the words that express possibility .
UNIT 4 FRACTIONS and AREAS
Objectives:
-do addition&substraction with fractions with equal denominators.
-guess an area using non standart area measures to check it out using units.
-compare the fractions.
-to show the fractions on the number line.
Activities:
1.Lets share it
2.Lets put the fractions into order
3.How can we measure the the area?
4. Lets add substract the fractions
99
UNIT 5 DECIMAL FRACTIONS&MEASURING LENGTH
Objectives:
-tell that it’s a decimal fraction when a whole is divided into 10&100 equal parts.
-to write the decimal fractions using ‘‘comma’’
-to compare 2 decimal fractions
-show with ‘‘< , > or =’
-to express certain lengths with different units/lengths
Activities:
1.From fractions to decimal fractions
2.Which one is bigger?
3.We can classify big and small lengths
4.The circumference length of geometric shapes
UNIT 6 OPERATIONS WITH NUMBERS&TIME
Objectives:
-multiply numbers mostly with 2 and 3 digits
- 5 ,25,50 mentally
10,100 and 1000 mentally
-to make a connection between a pattern and numbers to complete the missing part.
To do the operations with two steps.
Activities;
1)We can multiply mentally .
2)Lets form number patterns.
3)Operations with paranthesis.
100
UNIT 5 DECIMAL FRACTIONS&MEASURING LENGTH
Objectives:
-tell that it’s a decimal fraction when a whole is divided into 10&100 equal parts.
-to write the decimal fractions using ‘‘comma’’
-to compare 2 decimal fractions
-show with ‘‘< , > or =’
-to express certain lengths with different units/lengths
Activities:
1.From fractions to decimal fractions
2.Which one is bigger?
3.We can classify big and small lengths
4.The circumference length of geometric shapes
UNIT 6 OPERATIONS WITH NUMBERS&TIME
Objectives:
-multiply numbers mostly with 2 and 3 digits
- 5 ,25,50 mentally
10,100 and 1000 mentally
-to make a connection between a pattern and numbers to complete the missing part.
To do the operations with two steps.
Activities;
1)We can multiply mentally .
2)Lets form number patterns.
3)Operations with paranthesis.
101
GOOD PRACTICES
ITALY good practices
First experience – A REALITY TASK:
design and implement a math lesson for student of another
class.
EXPECTED RESULTS

Developing
logical
skills
and
organizational
to
achieve a goal.

Knowing how to work in a team work, respecting
their own commitments, their own role and the
commitments and the roles of others.

Reworking and refining their own knowledge, then
share them with others as comprehensively as
possible.

Considering how it is possible to facilitate the
learning of a mathematical concept.

Overcoming
the
embarrassment
of
speaking
in
public and logically organize their own speech.

Making experience of fun and entertain with math.
STRATEGIE OPERATIVE





Creating five working groups of elective, to favor as much as
possible the cooperation and the achievement of objectives.
Choosing interesting topics, that require the knowledge of
some mathematical concepts, but which also give the
possibility to acquire a new one during the work of the
group.
Providing guidance on where to find the necessary materials
( books, websites ).
Doing enough lessons to work in groups and encourage,
where possible, at least an afternoon meeting in the care of
families .
Before starting the lesson, it is necessary to review the work
of each group and to reflect together on the procedures .
102
Second experience - CODING :
an introduction to computational thinking
EXPECTED RESULTS

Stimulating creativity and problem solving skills.

Knowing how to define the aim of a project.



Identifying their own role in a team, working in a
cooperative atmosphere.
Learn to operate following rules already validated, but
that can be improved.
Developing a system of controls to verify progress.
OPERATIONAL STRATEGIES

Teaching using the laboratory (learning by doing ).

Cooperative learning/peer education .



Eagerness of the students in the choice of the free
and creative content of the project, the definition of
the objective and the identification of their own role
in the working group.
Using motivational techniques work: icebreaking,
brainstorming, scrum, story mountain, sprint
planning and scrumboard .
Alternating moments of work "unplugged " (analog
mode, without Network ) and other on - line .
103
A story : Bark and the Incandescent Dragon .
ICEBREAKING
(unplugged)
Games heating
1. Eyes closed
SCRUM
BRAINSTORMING
(unplugged)
(unplugged)
From rugby : when
all players are
pointing in the
same direction to
take possession
of the ball.
1. Designs inspired
by the cards of
Stories
2. Symbols and
controls : up, down,
2. Storm on
left, right
characters
SCRUMBOARD
STORY MOUNTAIN
(unpluggedon - line)
The structure of the
story: the initial
situation; that kicks
off the adventure;
climax; remedial
action; end.
SPRINT PLANNING
(unplugged/on - line)
Planning of useful
work to give life to
the story.
(unplugged/on - line)
The board that helps
planning :
1. to do;
2. during construction;
3. already done.
FINAL EVALUATION OF THE " GOOD PRACTICES "





MATH CLASS
INVENTED STORY
(Task of reality)
(Coding)
The group elected facilitates
collaboration and creative phase.
The group formed by the teacher can
work situations more balanced,
especially for the enhancement of skills
and roles of each boy.
This type of task stimulates the
metacognitive skills of the boys , as
well as the recovery of knowledge.
The CODING urges both logical skills that
creative ones.
Gaming activities on mathematical
release tension compared to the
difficulties that may be encountered.
Mathematics and storytelling “play”
togheter in games of different
languages and fantasy.
The students are using a precise
language and appropriate (in
mathematical terms).
Children learn to organize and plan the
study time activities .
They learn how using creatively LIM, PC,
tablet .
The students listened with
seriousness and respect fellow
presenting the lesson.
104
POLAND good practices
The examples of good practice in
teaching Mathematics
Written by Krystyna Ceszkiel
Translated by Małgorzata Zubalewicz
The list of content
1. Tips for teachers
2. Teaching aids provided by GWO (Gdańsk
educational publishing house)
3.
4.
5.
6.
Other teaching aids
Games
Maths competitions
What do students expect from teachers?
Motivational stickers
Motivational stickers:
e.g.
•Super-setsquare,
•The Lord of Fractions,
•The Real Pythagoras,
•The Conqueror of Equations,
•Not Bad Denominator,
•Squared Congratulations
.
105
Mathematical cards – multiplication table
Students learn and improve multiplication table
eagerly during games
Mathematical cards
"Multiplication table"
Card games which help
children to memorize
the multiplication table
Boxes for the game
"Olympic players"
The bingo game contest in brief
•
•
•
•
•
3 to 7 people take part in the game (the best number of
participants is 4)
Everyone is given 9 cards with ready results and puts them in
front.
When the teacher shows the card with the mathematical
activity, the student shouts „ I`ve got it” and points his/her
card.
If he/she shows the right card, the teacher gives him/her the
paired card.
After having coverd all cards the pupil shouts „BINGO” and
then, he/she wins the game.
Celebrating the First Day of Spring in our school
using mathematical cards in the tournament
„The mathematician- hockey player”
106
Other games used during lessons
• Extra Mathematics (GWO magazine) – pdf file
• Mathematical domino: revising the knowledge
of angles and triangles (GWO magazine)
• Jigsaw puzzles : the surface area of polygons
(it comes from the teacher`s guide written by
Małgorzata Paszyńska)
• A game „Crazy Shopping” (GWO)
Extra Maths - the instruction
•
•
The player who throws the biggest number begins the game. Players move their pawns
ahead as many squares as the dice shows.
Description of the squares :
? The player draws one question card. If he answers correctly he can throw the dice one
more time. If his answer is wrong he misses his turn.
•
← The player draws one question card. If he answers correctly he can „take a shortcut”
•
! The player draws one question card. If he answers correctly he moves 5 squares ahead. If
his answer is wrong he goes 5 squares back.
•
!! The player draws two question cards. If he answers correctly twice he can throw the dice
two times more. If he answers one question correctly he doesn`t move forward. If his both
answers are wrong he goes back at the BEGGINING.
•
•
The player who finnishes the first is the winner.
The teacher is the one who checks if the answers are correct. In the end, the winners of
particular games can play the final game to gain the title of „The Master of Extra Maths”
Have fun!
•
•
Competitions:
•
•
•
are essential for the students who are „hungry
for more”
help the teacher to recognise the most
talented ones
motivate students to practise harder
Every year our students take part in many
mathematical competitions, where they solve
(so they say) very interesting maths problems.
107
REUNION ISLAND (FRANCE) good practices
Draw me a tangram!
A rally-math! Motivate to solve!
Activity geometry and problem solving
with students aged 7 to 10 years.
Just Sauveur school By Fabienne Couchat,
School teacher
District TAMPON 1, Academy Reunion
France
The geometry and the implementation of the
Common Skills Base.
The 2008 programs:
"The main objective of teaching geometry from CE1 to CM2 is to allow students to
move progressively from a perceptual object recognition to a study based on the
use of instruments tracing and measuring"
The common base :
Définition :
The common base is "the set of knowledge and skills that are essential to master to
successfully complete their education, pursue training, build their personal and
professional future and successful life in society"
(Act of 23 April 2005)
108
To tier 2
the Common Base
O.B for CE2
O.B for CM1
O.B for le CM2
In the plane
In the plane
In the plane
- Recognize, describe, name and reproduce, draw
- Recognize that the lines are parallel.
- Use instruments to check the parallelism of
geometric shapes: square, rectangle, diamond,
- Use experiencing geometric vocabulary
two lines (rule and square) and to draw parallel
triangle.
aligned points, right, perpendicular lines,
lines.
name the figures and
- Check the nature of a plane figure using the ruler
parallel lines, segment, medium angle axis of
- Check the nature of a figure through the use of
customary solid
and the square.
symmetry, center of a circle, radius, diameter.
instruments.
- Build a circle with a compass.
- Check the nature of a plane figure simple
- Build a height of a triangle.
- Use vocabulary situation:
using the scale, the square and compass.
- Reproduce a triangle using instruments.
compass to check the nature of
side top angle setting.
- Describe a figure to identify it among other
common plane figures and
- Recognize that a figure has one or more lines of
figures or to breed
- Use the rule, square and
symmetry by folding or using tracing paper.
build with care and precision
- Draw on graph paper, the symmetrical figure of a
- Perceive and recognize
figure given in relation to a given line.
parallel and Perpendicular
- Solve reproductive problems,
building
Space
Space
Space
- Recognize, describe and name:
- Recognize, describe and name the solid
- Recognize, describe and name the solid
a cube, a cuboid.
rights: cube, paved prism.
rights: cube, pad, cylinder, prism.
- Use vocabulary situation, face, edge, summit.
- Recognize or complete a pattern of cube or
- Recognize or supplement a law firm patron.
tile.
To tier 2
the Common Base
information or geometric, justify
O.B for CM1
O.B for CE2
O.B for le CM2
- Reproduce the figures (on plain paper, checkered
- Draw a figure (on plain paper, checkered or
or dotted), from a model.
- Draw a simple figure from a construction
dotted), from a
- Build a square or a rectangle of given dimensions.
program or by following the instructions.
construction program or a freehand drawing
and assess the likelihood of a
(with indications regarding the properties and
result.
dimensions).
Angles
Angles
- Compare the angles of a figure using a template.
- Reproduce a given angle using a template.
- Estimate and check by using the square, a right
angle is acute or obtuse.
The practice of mathematics develops a taste for research and reasoning, imagination and the capacity for
abstraction, rigor and accuracy.
CE2 to CM2, students enrich their knowledge, acquire new tools, and continues to learn how to solve problems. It
strengthens mental math skills. It acquires new automation. In mathematics, the acquisition mechanisms is always
associated with understanding.
109
A rally-math ! Motivate to solve !
PROBLEM SOLVING
Problem solving is a highly complex task that requires the successive implementation and possibly reiterated skills within different fields
and have been grouped under the following headings:
a. to search and organize information;
b. initiate a process, reason, argue, demonstrate;
c. calculate, measure, apply instructions;
d. communicate using a mathematical language adapted.
It is therefore useful to take the information, thinking and performing processing of information, and communicate results.
Problem solving plays an essential role in mathematical activity.
It is present in all areas and is exercised at all stages of learning.
The practice of mathematics develops a taste for research and reasoning,
imagination and the capacity for abstraction, rigor and accuracy.
CE2 to CM2 in the four areas of the program, students enrich their knowledge,
acquire new tools, and continues to learn how to solve problems.
It strengthens mental math skills. It acquires new automation.
The acquisition of mathematics mechanisms is always associated with an intelligence of their meaning.
The mastery of the main elements using mathematics to act in everyday life and prepare further studies in college.
COMMON BASE / SECOND LEVEL FOR THE CONTROL OF THE JOINT BASE : SKILLS EXPECTED AT THE END OF CM2
Competency 6:The social and civic competences.
Capacities The student is able to:
- Take part in a dialogue to address the others, listen to others, make and defend a point of view;
- Cooperate with one or more classmates.
-Communicate And teamwork, which involves listening, to express his point of view, negotiate, seek a consensus, carry out its work
according to the rules group
-Evaluate The consequences of his actions: to recognize and name emotions, impressions, to assert constructively
-Know Build his personal opinion and be able to challenge the shade (for awareness on the part of affection, influence of prejudice,
stereotypes).
Attitudes
-Respect Self and others
-Need For solidarity: taking into account the needs of people in difficulty
(Physically and economically) in France and around the world.
-Conscience Of his rights and duties
-Volonté To participate in civic activities
Competency 7:The autonomy and initiative.
Capacities The student is able to:
- Follow simple instructions independently;
- Show some perseverance in all activities;
- Get involved in an individual or group project.
-S'appuyer On working methods (organizing time and plan their work, take notes, prepare a dossier)
-Take The opinion of others, exchange, inform
Attitudes
-Volonté To take charge personally
-Conscience The influence of others on their values and choices
-Motivation And determination in achieving goals
110
Organization of a meeting several times a year :
- Students are grouped in small heterogeneous groups.
- 1 test is distributed by group.
- Each group has 15 minutes to find one or more answers, and find one or more solutions.
- On an answer sheet, the pupils of the group must offer an answer, after discussing and following
consultation.
- Rotation of the tests.
- In one hour students will meet 4 puzzles.
- The teacher refers to changes.
- At the end of each session is proposed answers and the correct answers are valid.
- A collective correction can then be proposée.- For each correct answer you can give one or more points.
at the end of several sessions, each group made the point total.
- ability to reward the winning group
Students have
individual events.
They can work in small groups.
Various activities
In the end a diploma and a Chinese puzzle was given to
each student of the winning class.
111
What interest ?
For students : craze, increased autonomy and self-esteem, differentiation of tasks and
methods beneficial to pupils, change in relation to math, better mobilization of knowledge
(benefits provided to successfully transfer skills built during the rally at the other meetings
of math and forms of work, including individual).
For Teachers : Another look at the student and class (highlighted relational dynamics and
learning modalities)
Providing analysis of student productions elements (in some rallies)
Accountability and socialization of students (civics / debate, respect differing opinions)
Reinvestment decontextualized and more fun math concepts already discussed.
Constitution of a bank problems allowing the teacher to use it wisely, knowing what
mathematical concepts requires resolution
Three deviations to avoid :
The Maths Rally must not be a disconnected contest classroom work (other meetings of
mathematics).
The Maths rally should not become the only opportunity to do math
The importance of classification must be undervalued if one wants the fun aspect
predominates.
Finally the Project Etemath allowed me to change and
improve my classroom practice.
Exchanges with other partners is very positive for me
and for the students.
The many situations observed in other countries
stimulate me to seek to offer innovative situations and
make them want to do math.
A very rich experience to share!
Thank you for your attention.
112
ROMANIA good practices
THE CHILDREN’S PARTY
Targeted competencies:
 Computing with natural numbers in different contexts
 Validating computing results
 Solving problems based on the studied concepts
Targeted activities:
• multiplication of natural digits smaller than
1000
• practicing calculation in different contexts
• getting used to other calculation methods;
• identifying numbers and using them in
calculations
• interdisciplinary correlation of contents
• identifying and correcting possible errors
• checking calculations in different ways
• identifying everyday life situations in which we
use such calculations: planning and organizing a
party.
Objectives:
• to solve multiplication exercises using
different methods;
• to solve different life situations using
appropriate operations;
• to use their multiplication knowledge in a
variety of contexts;
Resources:
• student’s guide,
multimedia guide,
computer
Types of activities:
• frontal , individual
and pair activities
Suggestions for class activities:
What have we got to do?
• getting into the unit according to the social standard of the class; the teacher will gradate the term “party” so as
to meet reality;
• more atypical methods to solve the exercises are introduced in the unit; do not insist on learning them, just
introduce them as curiosities which might be useful; they might play a motivating role in getting involved in
mathematical activities;
• allow enough time to check the correctness of the calculations, encourage cooperation and mutual help;
• insist on questions which might facilitate understanding: Are there enough…? Why? How many should there be
to….. ? Is it possible that ….? Why? ; allow enough time for reasoning;
• support your students in making up a budget; encourage them to repeat for other situations ( in class or family);
• understanding each step should be one of the main concerns; use frontal activity whenever you feel that certain
tasks are above the students’ level of understanding;
• encourage the finding of more solutions and personal approach ;
• allow time for interdisciplinary correlation; encourage students to use their own experience or to look for new
information;
• throw a party in the classroom getting the students involved in planning and organization.
113
Different methods to do multiplications .
Using the fingers
Using a rectangle
(only for multiplication by 9):
1. Number each finger in your mind.
Don’t forget the number you have
given each finger.
2. Bend the finger which represents
the number you want to multiply by 9.
26 x 2 = ?
We use a rectangle divided in two
parts because we have a two-digit
number.
Each part is divided by an oblique
line. We write the two-digit number
above and the one digit number on
the lateral.
2. Cut both groups of lines with 3 lines
(from number 3)
In turns we multiply the two digits
with 2 and we write the results in the
rectangle .
Eg.: for 2 x 9 bend finger number 2.
On the left side there is one finger, on
the right there are 8 fingers. 2 x 9 =
18
Multiplication with lines
Let’s say we want to multiply this:
35 X 3.
1. Represent the first number with lines.
Group 3 lines for tens and 5 lines for
units.
We add the numbers in diagonal.
The result: 52
3. Count the points where the lines meet in
each group.
For units there is 5.
There are 10 tens.
(from 9 + 1)
The result is105
(35 X 3 = 105). Check it!
FRIENDS OF NATURE
Targeted
competencies
Objectives:
• Validating the results
• Recording the data
observed in the
environment and
representing them
• Solving problems with the
studied concepts
Resources:
• student’s guide,
multimedia guide,
computer,
information/images
about animals
• to identify in a text relevant information to solve a
problem;
• to drop out irrelevant information from a text to
facilitate understanding ;
• to identify contradictory information ;
• to reword information for better understanding.
Types of activities
Suggested activities
• individual activities
• frontal
activities(explanations,checking)
• improving analysis abilities of a text in order to search
, select and use relevant information
• appreciating information in terms of : useful/useless,
complete/incomplete, corellated/
contradictory, redundant, etc.
• rewording certain information in oder to clear up and
better understand
• identifying the appropriate behaviour to protect
environment.
114
Suggestions for class activities:
What have we got to do?
How shall we monitor the activity?
• this unit has many texts and is based on
reading and analysing them in order to figure out
the useful information;
• the texts are approached from a searching for
and selecting information perspective ;
encourage students to read the texts several
times and to come back to the texts whenever
necessary;
• guide your students into a “mathematic”
reading , helping them to make the difference
between the answers they can explicitly find in
the text and the ones which need correlating
information from the text ;
• lead them into choosing the information which
need to be correlated;
• guide students in their activity of filling in with
certain data; emphasize the importance of using
real data, particularly when it comes to animals’,
plants’ objects’ phenomena’s characteristics,
which might require a former documentation;
• give feedback to the students’ answers;
• come with additional explanations if needed ;
• observe systematically the students’ involvement in the
activity;
• do not insist on the solving speed but on the correct
solving;
• encourage cooperation and mutual help ;
• use a monitoring sheet to record the correctness of the
answers for each student;
• encourage self- assessment and evaluation in pairs/
groups ;
• make sure all the mistakes have been corrected.
Reflection moment
• What happened during the activities?
• Why did this happen?
• Which part of the activity was the most
interesting for the students?
• Where have I encountered difficulties?
• Which activities shall I do again in other
occasions?
My tree !
The III rd grades A, B and C students took part in the project "Adopt a tree".
Each student has chosen a tree in the surroundings or in a park and has observed its evolution all through the school
year , recording the findings in a worksheet. Students have taken photos of the tree in different seasons and aspects :
when in bud , blooming, leafy, partly or completely bare. They organized an exhibition with 312 photos, each child
displaying 4 photos of his tree. All the students involved in the project brought photos for the exhibition. They have also
made up poems and stories about their trees and put them in a book of the class. Each student made a page in the book.
The book of the students in III C had 28 pages , while the books of those in III A and B had an equal number of pages. The
students brought out the books at the school year end ceremony .
What do we want to find out?
Observe, think, solve!
Write the answers to the following
- how many children brought photos
1. How many children brought photos?
questions :
How can we find this?
Can you find the answer to this question in We divide the total numbers of
What project did the students take part in?
the text?
What grade were the participants in the
photos with 4 (because each child
Is there any information in the text which has brought 4 photos):
project?
could help you find the answer ?
How many photos did each child bring for
Think and answer:
Read the statement:
the exhibition?
Is the number of the children who
How many pages in the book did each child They organized an exhibition with 312
brought photos for the exhibition
photos, each child displaying 4 photos of
make?
equal with the number of the
Where did the children bring out the books? his tree.
children involved in the project?
What do we know?
Could you answer these questions? What
Explain your answer. Use
- the number of photos
helped you?
information from the text.
- How many photos each child has brought
115
TURKEY good practices
MEASURING THE LIQUIDS
We are sharing one litre equally into two half litres.
116
We’re weghing 1litre of water with 1kilo.
Students practice on the board with projector…
117
Drawing the geometric solids on isometric paper...
Cube
118
4. REUNION ISLAND MEETING
Special needs game
ITALY
Dr. Rachele Giammario
Psychologist - pedagogist, psychomotrist,
Therapist of the neuro and psycomotricity of the childhood
Teacher at L’Aquila University
5 children per class
Have difficulty with
calculation

5-7 children per class
Have difficulty with problem
solving

+ 20% OF POPULATION
Difficulties of Children with numbers and maths in general .
It’s a disorder related to learning numbers and calculation.
It’s often associated to dyslexia.
The diagnosis of dyscalculia can be given in the 3rd grade .
Dyscalculia is a specific disorder related to numbers and calculation system in absence of neurological lesions and cognitive problems.
There can be dyscalculia even with a normal education, a right intelligence, a
good culture and a nice family atmosphere .
119
Dyscalculia :
Such disorder concerns the acquisition of easy skills, e.g.:
Writing
numbers
Reading
numbers
Calculation system (like memorising calculation tables, executing calculations etc.).
Dyscalculia is divided in primary and secondary
Primary dyscalculia is a disorder about numerical and arithmetic skills.
Secondary dyscalculia is associated to other learning problems, such as dyslexia, la
dysgraphia, etc. In these situations we will deal above all with the dyslexia and its rehabilitation
Dyscalculia:
Children with a dyscalculia desorder frequently make the following mistakes:
They don’t recognize numbers when reading or writing , in particular if
they have got many figures.
They can’t recognize the figures that make a number .
They can’t recognize relations between figures inside a number.
Difficulty in grasping mathematical links.
Difficulty in associating a quantity corresponding number.
Difficulty in learning the meaning of signs (plus, minus, times and divided for) - Difficulty
in analysing and recognizing data that can give a problem solution.
Difficulty
in learning the rules of calculations (loan, reporting, queuing, etc.) Difficulty in learning easy
operations like calculation tables, the results of which are got automatically, without making
difficult calculations.
Difficulty in space-time and
look-space organization.
Difficulty in motor coordination, above
all handy.
Difficulty in making works in sequences.
Maths has got a fundamental role in compulsory school:
It tries to develop concepts, methods and ability to order, quantify and measure reality
facts and phenomena and to give the necessary ability to critically interpret and to knowingly
operate on it.
120
Use of fingers
Counting by the use of fingers
(so useful mechanism to learn
the ability to count and
automate correspondences,
stable order and cardinality)
Operations
with abacus
Digital
I must know the code to decode time
121
Analog
No matter knowing the code!
LOGIC
The child needs to reason
before “understanding
reality”
ANALOG
The child examines reality
to reason
ANALOGICAL
APPROACH
(non conceptual)

Abacus

122
Analogic tools


From a motivational point of view, it is
necessary to find out participation strategies, to
make students active and participating.
Studying the multiplication tables, the playing
activity allows to come from a learning based
on the connection “stimulus-verbal response”
(S-R) to an holistic one, more complete and
rewarding for the student, that is expressed by
the formula “Stimulus-Personality-Response”
(S-P-R).
123
POLAND
12 + 29 = ?
a) 38
b) 41
b) 42
b) 51
Good! You have saved
1 friendly alien!
124
17 - 9 = ?
a) 6
b) 7
b) 8
b) 9
Good! You have saved
2 friendly aliens!
23 - 11 = ?
a) 11
b) 12
b) 21
b) 22
125
Great! You have saved
3 friendly aliens!
31 - 7 = ?
a) 24
b) 25
b) 26
b) 27
Great! You have saved
4 friendly aliens!
126
REUNION ISLAND
By Fabienne Couchat and
Alexandre Schneider
Maths teaching in a
special
class
t
activities:
elements request to supplement the chickens
127
Each student must complete their pool by
exchanging 1 against number of counters defined
by the master.
complexity of the task
128
Organisation :
• Students are in small groups, 2 in autonomy and directed 1.
• 2 adult help those who are independently on activities of counting on
a digital file or Lotto cards. Students must associate a correspondence
between a number and its different representations (points,
constellations, calculations...)
• Directed group works with the master. It offers a problem situation
simple at first then harder.
Activity led by the master:
The master presents the material and asks students what he see. This is not difficult
because the master a choose a hen that is an animal known by all students.
He noted that 1 hen consists of 1 head, a body and 2 legs. Each student then has
chips depending on the number of travel to reconstitute 1 hen. (It is possible to give
1 number of tokens = to the numbers of the components missing to begin the
activity)
When every child has understood he must leave his table and order missing parts
by swapping them with the same number of chips. Caution it is entitled to 1 single
trip.
Then when all the students have understood, the master will distribute 2 body and
asked to do the same thing. For many it is a difficult situation: either they lose parts
of the body or they are wrong in the quantity demanded. But soon they will in look
because they do not have enough chips.
129
ROMANIA
Didactic ideas
applicable
to math lessons
for
students with special
needs
Learning
based on
special
needs
DIDACTIC IDEAS
130
Effective communication ,
help in need
Forming natural numbers
with tens and units
Understanding passing to the next
ten
131
True / false
represented
through
colours
Students
calculate and
find the
correpondence
between the
result and the
symbol given.
Klammerkarte ZR 30, Subtraktion Andrea Haunold
http://vs-material.wegerer.at/mathe/m.htm
132
Measuring capacity – practical application
Comparing
measure units
133
SPAIN
Introduction
GMG is a pupil in our school who is seven years old. He has
got specific and permanent educational needs due to a
total developmental delay caused by Down Syndrome.
134
Numbering activities: We have written numbers from
one to twenty on bottles caps.
He then orders them in ascending order.
With the number line, we perform a number dictation.
I say the number outloud and he places a piece of clay
on it.
135
The following activity consists on relating numbers with
quantity: We will use a circle made of cardboard and
wooden clothespins. (The quantities are in the cards and
the numbers are in the clothespins). This activity is
designed to improve his mobility.
Finally, we watch a short story in the computer. He likes
this activity and consequently it is a reward for working
hard. This activity is very important to improve listening
and speaking skills.
136
TURKEY
Education of Special Needs Children
It is the education that is held in environments that are appropriate for the individuals
with special needs for their disabilities and characteristics with the help of the professional staff
who use specially developed tecniques.
Individual With Special Needs is defined as the individual whose personel
characteristics and educational sufficiency differs from his peers because of any reason.
Each individual in the society, has different characteristics individual sufficiency. That’s
why contemporary & democratic mind requires the sensitivity and the need to have the kind of
education that is accurate for each individual.
Recently important steps have been taken for the benefit of individual with special
needs.
The 42nd article of the constitution says, ’’ The State takes measures that the people
with special needs will make use of in their daily life.’’
Kind Of Disability and
Characteristics
Mentally Retarted Individuals
Normal Function IQ=100
.Slight, IQ=55-70 (can be educated)
.Medium , IQ=40-55 (can be taught)
.Heavy , IQ=25-40 (some can be taught)
.Heavier IQ= lower 25 (need total care)
Partially Sighted
Light Loss
Total Loss
137
WHERE IS INCLUSIVE
EDUCATION APPLIED?






Ordinary school
Source Room
Separate
Classroom
Separate School
Boarding School
Home/Hospital
Individualized Education Plan
A plan that shows the actions that
the person has to take according to
his /her needs and how and with
whom the secondary steps will be
taken .
 Is compulsory according to the 573 rule
BEP(IEP) Individualized Education
Programs.

138
Individualized Education Unit at
Schools





HEADMASTER
SCHOOL COUNSELOR
CLASS TEACHER
BRANCH TEACHERS
STUDENT’S PARENTS AND THE
STUDENT HIMSELF/HERSELF.
BROAD EVALUATION FORMS
Long Term Goal : First Reading-Writing
Short Term Goal: Develops the coordination and muscular force with hand –finger exercises.
EDUCATIONAL GOALS:
1. Stretches arms forward and opens and closes the right hand faster and faster gradually.
2. Does the same with the left hand.
3. Stretches a rubber,etc. With two hands .
4. Makes a fist on his/her chest level his/her thumbs free,ties to spin the thumbs around their own axis.
5. Thumbs free,other fingers adjoınt, spins the thumbs into different directions.
6. Thumbs free, opens and closes the other fingers to and from the palm.
7. All fingers separated, touches the thumb with each finger.
8. Presses with each finger on a smooth surface.
9. Making a fist, raises all fingers in turns, starting from the little finger.
10. Puts his hands open on the desk, raises all fingers in turns ,starting from the thumb.
11. Placing a small object between any two fingers, tries to move the fingers.
12. Without using the thumb lift and put down some objects with the other fingers.
13. Turns pages.
14. Gives forms to clay etc. With
139
140
Test for 3th - 4th-5th grade first (version by Poland)
3th grades
Country\Task
1
2
3
4
5
6
7
8
9
10
Total
Poland
0.87
0.77
0.81
0.92
0.97
0.60
0.62
0.79
0.72
0.78
0.81
Italy
0.92
0.80
0.80
0.56
0.53
0.48
0.32
0.86
0.74
0.65
0.74
Romania
0.93
0.87
0.84
0.93
0.96
0.68
0.72
0.91
0.92
0.86
0.88
Turkey
0.79
0.58
0.73
0.63
0.63
0.56
0.55
0.59
0.79
0.61
0.69
Spain
0.94
0.74
0.86
0.85
0.82
0.23
0.32
0.93
0.57
0.84
0.80
France
0.68
0.46
0.51
0.68
0.32
0.34
0.31
0.45
0.49
0.46
0.47
3th grade - Task 1
1.00
0.90
0.80
0.70
0.60
0.50
0.40
0.30
0.20
0.10
0.00
Poland
Italy
Romania
Turkey
Spain
France
Spain
France
3th grade - Task 2
1.00
0.90
0.80
0.70
0.60
0.50
0.40
0.30
0.20
0.10
0.00
Poland
Italy
Romania
141
Turkey
3th grade - Task 3
1.00
0.90
0.80
0.70
0.60
0.50
0.40
0.30
0.20
0.10
0.00
Poland
Italy
Romania
Turkey
Spain
France
Spain
France
Spain
France
3th grade - Task 4
1.00
0.90
0.80
0.70
0.60
0.50
0.40
0.30
0.20
0.10
0.00
Poland
Italy
Romania
Turkey
3th grade - Task 5
1.20
1.00
0.80
0.60
0.40
0.20
0.00
Poland
Italy
Romania
142
Turkey
3th grade - Task 6
0.80
0.70
0.60
0.50
0.40
0.30
0.20
0.10
0.00
Poland
Italy
Romania
Turkey
Spain
France
Spain
France
Spain
France
3th grade - Task 7
0.80
0.70
0.60
0.50
0.40
0.30
0.20
0.10
0.00
Poland
Italy
Romania
Turkey
3th grade - Task 8
1.00
0.90
0.80
0.70
0.60
0.50
0.40
0.30
0.20
0.10
0.00
Poland
Italy
Romania
143
Turkey
3th grade - Task 9
1.00
0.90
0.80
0.70
0.60
0.50
0.40
0.30
0.20
0.10
0.00
Poland
Italy
Romania
Turkey
Spain
France
Spain
France
3th grade - Task 10
1.00
0.90
0.80
0.70
0.60
0.50
0.40
0.30
0.20
0.10
0.00
Poland
Italy
Romania
144
Turkey
4th grades
Country\Task
1
2
3
4
5
6
7
8
9
Poland
0.73
0.81
0.66
0.47
0.65
0.80
0.48
0.72
0.31
0.55
0.63
Italy
0.86
0.80
0.82
0.79
0.84
0.86
0.62
0.62
0.69
0.71
0.78
Romania
0.95
0.91
0.87
0.87
0.95
0.96
0.89
0.89
0.92
0.86
0.91
Turkey
0.83
0.88
0.77
0.62
0.54
0.70
0.67
0.81
0.53
0.50
0.70
Spain
0.83
0.92
0.82
0.86
0.78
0.95
0.59
0.73
0.33
0.58
0.75
France
0.42
0.72
0.36
0.33
0.36
0.56
0.33
0.36
0.39
0.22
0.40
10
Total
4th grade - Task 1
1.00
0.90
0.80
0.70
0.60
0.50
0.40
0.30
0.20
0.10
0.00
Poland
Italy
Romania
Turkey
Spain
France
Spain
France
4th grade - Task 2
1.00
0.90
0.80
0.70
0.60
0.50
0.40
0.30
0.20
0.10
0.00
Poland
Italy
Romania
145
Turkey
4th grade - Task 3
1.00
0.90
0.80
0.70
0.60
0.50
0.40
0.30
0.20
0.10
0.00
Poland
Italy
Romania
Turkey
Spain
France
Spain
France
Spain
France
4th grade - Task 4
1.00
0.90
0.80
0.70
0.60
0.50
0.40
0.30
0.20
0.10
0.00
Poland
Italy
Romania
Turkey
4th grade - Task 5
1.20
1.00
0.80
0.60
0.40
0.20
0.00
Poland
Italy
Romania
146
Turkey
4th grade - Task 6
1.20
1.00
0.80
0.60
0.40
0.20
0.00
Poland
Italy
Romania
Turkey
Spain
France
Spain
France
Spain
France
4th grade - Task 7
1.00
0.90
0.80
0.70
0.60
0.50
0.40
0.30
0.20
0.10
0.00
Poland
Italy
Romania
Turkey
4th grade - Task 8
1.00
0.90
0.80
0.70
0.60
0.50
0.40
0.30
0.20
0.10
0.00
Poland
Italy
Romania
147
Turkey
4th grade - Task 9
1.00
0.90
0.80
0.70
0.60
0.50
0.40
0.30
0.20
0.10
0.00
Poland
Italy
Romania
Turkey
Spain
France
Spain
France
4th grade - Task 10
0.90
0.80
0.70
0.60
0.50
0.40
0.30
0.20
0.10
0.00
Poland
5th grades
Country\Task
1
Italy
2
3
Romania
4
5
Turkey
6
7
8
9
10
Total
Poland
0.72
0.57
0.31
0.44
0.66
0.53
0.45
0.63
0.42
0.46
0.54
Italy
0.82
0.95
0.96
0.87
0.97
0.90
0.96
0.44
0.86
0.84
0.84
Romania
0.97
0.84
0.98
0.89
1.00
1.00
0.77
0.47
0.95
0.96
0.88
Turkey
0.53
0.53
0.32
0.39
0.61
0.50
0.41
0.50
0.40
0.38
0.46
Spain
0.70
0.67
0.33
0.22
0.58
0.13
0.50
0.78
0.62
0.68
0.56
France
0.78
0.53
0.49
0.31
0.61
0.53
0.61
0.69
0.28
0.33
0.53
148
5th grade - Task 1
1.20
1.00
0.80
0.60
0.40
0.20
0.00
Poland
Italy
Romania
Turkey
Spain
France
Spain
France
Spain
France
5th grade - Task 2
1.00
0.90
0.80
0.70
0.60
0.50
0.40
0.30
0.20
0.10
0.00
Poland
Italy
Romania
Turkey
5th grade - Task 3
1.20
1.00
0.80
0.60
0.40
0.20
0.00
Poland
Italy
Romania
149
Turkey
5th grade - Task 4
1.00
0.90
0.80
0.70
0.60
0.50
0.40
0.30
0.20
0.10
0.00
Poland
Italy
Romania
Turkey
Spain
France
Spain
France
Spain
France
5th grade - Task 5
1.20
1.00
0.80
0.60
0.40
0.20
0.00
Poland
Italy
Romania
Turkey
5th grade - Task 6
1.20
1.00
0.80
0.60
0.40
0.20
0.00
Poland
Italy
Romania
150
Turkey
5th grade - Task 7
1.20
1.00
0.80
0.60
0.40
0.20
0.00
Poland
Italy
Romania
Turkey
Spain
France
Spain
France
Spain
France
5th grade - Task 8
0.90
0.80
0.70
0.60
0.50
0.40
0.30
0.20
0.10
0.00
Poland
Italy
Romania
Turkey
5th grade - Task 9
1.20
1.00
0.80
0.60
0.40
0.20
0.00
Poland
Italy
Romania
151
Turkey
5th grade - Task 10
1.20
1.00
0.80
0.60
0.40
0.20
0.00
Poland
Italy
Romania
152
Turkey
Spain
France
Edited 2016

Ingenious and Fun Games of Maths - Nuova Direzione Didattica Vasto

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