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

1 Ingenious and fun games of Maths

1. ITALY MEETING Project presentation

Progetto E.Te.Mat. “Effective Teaching of Mathematics”, programma UE Erasmus+, Key Action 2 - Partenariati Strategici-

V A S T O

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 SANA- MENTE 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 PARTICO- LARI 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

2 Ingenious and fun games of Maths

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 IN- QUADRARE L‟OPERATO DIDATTICO NEL CONTESTO VOLUTO DALLE INDICA- ZIONI, 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 im- postazione 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.

3 Ingenious and fun games of Maths

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 imposta- zioni 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

4 Ingenious and fun games of Maths

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 COSTITUZION- ALI.

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 INTERPRETA- ZIONE, 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 AT- TESE

5 Ingenious and fun games of Maths

O B I E T T I V I D I A P P R E N D I M E N T O

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:

Ingenious and fun games of Maths

1) PERSEGUIRE LE FINALITA’ EDUCATIVE

Elaborare il senso della propria esperienza Promuovere la cittadinanza attiva

a t t r a v e r s o

2) L` ACQUISIZIONE DEGLI ALFABETI DI BASE DELLA CULTURA

sapere (conoscenze) saper fare (abilità) a v e n d o c u r a d i

3) MIRARE AL PIENO POSSESSO DELLE COMPETENZE

LE COMPETENZE INDICATE NELLA < INDICAZIONI > SONO STRETTAMENTE DISCIPLINARI; MANCANO SUGGERIMENTI PER COMPETENZE COMPIUTAMEN- TE `TRASVERSALI`

L A M A T E M A T I C A D A I 6 A N N I I N P O I

.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 . .



Ingenious and fun games of Maths

I N V A L S I

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, individ- uandone 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, pro- prietà, strutture...); 2. conoscere e utilizzare algoritmi e procedure (in ambito aritmetico, geometrico, …);

8 Ingenious and fun games of Maths

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, argomen- tare, 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 . . . .

9 Ingenious and fun games of Maths

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

- L`alunno si muove con sicurezza nel calcolo scritto e mentale con i numeri naturali e sa valutare l`opportunità 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

10 Ingenious and fun games of Maths

-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 L`AZIONE EDUCATIVA ALLO SVILUPPO INTE- GRALE DELL‟ALLIEVO”

 COSTITUISCONO CRITERI PER LA VALUTAZIONE DELLE COMPETENZE AT- TESE

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 . . . .

Elaborare il senso della propria esperienza

FINALITA’ EDUCATIVE

Promuovere la cittadinanza attiva

AI FINI DELLO SVILUPPO INTEGRALE DELL‟ALUNNO ?

* * * * * * * * * * * * * * * * * * * * * * * * * * * *

Prof. Paolo ROTONDO

FEBBRAIO 2015

11 Ingenious and fun games of Maths

Ingenious and fun games of Maths

COUNTRY PRESENTATION

ITALY

www.nuovadirezionedidatticavasto.gov.it

WE ARE HERE

Ingenious and fun games of Maths

Monument to the bather

Punta Aderci

The castle

Punta Penna : the lighthouse

Ingenious and fun games of Maths

Fish soup is the typical of Vasto’s plate

ORGANIGRAMMA ORGANIZZAZIONE ORARIA

Dirigente Scolastico Prof.ssa Nicoletta DEL RE Consiglio di Giunta SCUOLA DELL’INFANZIA 7.45/8.00 – 16.00 Circolo Esecutiva

SCUOLA PRIMARIA 8.10 – 13.10 (LUNEDI –VENERDI) Direttore dei Collaboratori del 8.10- 12.10 (SABATO) S.G.A. Dirigente Funzioni Collegio TEMPO PIENO 8.10 -16.10 (SABATO LIBERO) Maria Giacinta Ins. Barbara Strumentali Docenti PICCONE GASPARI ITALIANO Ins. Paola MELIS 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

Assistenti Collaboratori Referenti di Referenti di Commissioni Coordinatori di Amministrativi Scolastici Plesso Primaria Plesso Infanzia Dipartimenti

Consigli Di Interclasse

STUDENTS TEACHERS COLLABORATOR

AnielloPolsi 151 14 3 STUDENTS TEACHERS COLLABORATOR Incoronata 100 10 2 Incoronata 94 13 2 S.Lorenzo 23 2 1 Peluzzo 221 24 4 Smerilli 50 6 1 Ritucci Chinni 232 23 3 S.Michele 106 15 2 S.Antonio 76 9 1 Pagliarelli 18 2 1 Vasto Marina 65 6 2 TOTALE 623 69 10

TOTALE 513 55 12

Ingenious and fun games of Maths

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

Ingenious and fun games of Maths

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. •LETTURA “LIBR…IAMOCI” P.O.F. PAROLE CHIAVE I PROGETTI D’ISTITUTO •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

17 Ingenious and fun games of Maths

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

18 Ingenious and fun games of Maths

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 Krakow

19 Ingenious and fun games of Maths

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)

Krakow Wawel – the Royal Castle The old capital of Poland

Jagiellonian University - the oldest in Poland (1364) Wawel Cathedral

20 Ingenious and fun games of Maths

Wroclaw – Our City Gdansk – ”A trace of Holland in Poland”

The „long” Market The Old Port

Poznan The Centre of Katowice

21 Ingenious and fun games of Maths

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 Delicious Food for Fat Thursday Season

In some regions of Poland the gifts are given to the The Thursday before Ash children only on December 6th - since St. Nicolaus Wednesday is celebrated as called also Santa Claus is a patron of this day. But in Fat Thursday - Tlusty the majority of houses children (and adults) can expect Czwartek . On this is the gifts twice- on December 6th and also on Christmas day when you forget about Eve. The atmosphere of this feast is different than the your diet and eat mountains atmosphere of Christmas eve since December 6th is a of donuts (paczki) and all normal working day. Whereas Christmas is usually the other things fat, greasy, celebrated as a family feast. sweet, full of cholesterol, generally unhealthy, and mmmmm.... delicious.

22 Ingenious and fun games of Maths

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 Winters in Poland were long and unforgiving. Therefore people are longing for spring. Catholic Church the willow (Polish: wierzba) symbolizes One of the ancient and pagan habits that supposedly was helping to get rid of winter was the resurrection and the immortality of the soul. "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 The Easter eggs are symbols of fertility and beginning of the new life. children love it! The baskets Some of the eggs were painted in traditional Polish folk patterns. These are then blessed by a priest. 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.

23 Ingenious and fun games of Maths

A long Mayday weekend in Poland St. John's Night

At the end of June, at the time of Summer Solstice, when night is shortest and May 1st - International Workers' Day Nature bursts with blossoms and growth, we May 2nd - Flag Day, it is also celebrated as a day of Polish celebrate the Holiday of immigration or Poles abroad, so called POLONIA DAY. Fire and Water, also May 3rd - The oldest feast is a feast of May 3rd which is called Noc Kupaly. devoted to the day of constitution, since the famous People gather at a fire, Constitution of the 3rd May was established on that day. jumping through the fire, sing songs, dance and Many people go on the outdoor trips during long Mayday having lots of fun. 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, Miners' Day (St. Barbara Day) November 1st

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, Miners are dressed in the special uniforms during Barbórka. The uniform candles and votive lights into the cemeteries where graves of family, consists of black suit and hat with a feather. The color of the feather friends or national heroes are. It is worth to mention that the cemeteries depends on the rank of the miner. in Poland are different than in any country. Graves and tombs are big and very individualized.

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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 FRANCE Heritage site and turned into a national park.

A little piece of France in the Indian Also "Maloya" music and dance of the slave has recently declared Ocean. 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 Water contains many geological curiosities: the Sugar cane fields 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 Circles Volcans landscapes.

The islanders are trying not to forget their roots. Muslims, Catholics and Hindus blend seamlessly and The isolation, the diversity of natural habitats and micro climates religious practices are very present in the lives of a have led many indigenous species present before the arrival of majority of residents. If the atmosphere is permeated man, to differentiate themselves over the millennia and become endemic species. Reunion passion, of course lovers of beauty with the dominant Catholic faith, it is however the vegetable, botanists as garden lovers. In the midst of the vast Hindu community that gives the island its most striking Indian Ocean, Reunion home to a unique flora. It is developed customs. Hinduism exhibits his thousand colors on the both on the coast and in the mountain forests. facades of temples that bloom throughout the island.

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 population mixing Sauveur”with Fabienne Couchat our teacher.

land of interbreeding

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ROMANIA

ROMANIA... THE CARPATHIEN GARDEN OF EUROPE

A big and beautiful country situated in the On its surface one can visit SOUTH-EAST of Central Europe The Danube Delta

ONE OF The biggest rivers in Europe: The Danube

We have an ancient history, starting with the dacic war Transilvania′s Highland and between 101-102 b.C. The Black Sea Ulpia Traiana Sarmizegetusa Dracula′s Castle

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Other monuments we are The Merry Churcyard and Corvinilor′s Castle proud of Monasteries from Northen Bucovina Peles Castle

UNESCO monuments in Romania Sighisoara Stronghold and

Densus Church in Hunedoara county wooden churches from Maramures Brancoveanu′s Monastery 1696 sec. XIII

You may have heard about ROMANIA just But you must know that names as: Eugen Ionesco was born in GICA HAGI 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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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 colletti- ve. 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 risolver-

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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 ne- cessarie per eseguire le prime misurazioni di lunghezze, pesi e altre quantità.  Individua posizioni di oggetti e persone nello spazio………..

Dai «Traguardi per lo sviluppo della competenza» di scuola primaria  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

Importante! La costruzione del pensiero matematico è un processo lungo e progressivo nel quale concetti, abilità, competenze e atteggiamenti vengono ritrovati, intrecciati, consolidati e svilup- pati 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’e- sperienza e di riflessione critica, non di addestramento “meccanico” o di apprendimento mne- monico.

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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, re- sponsabile, 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 sup- porto 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 cam- biano 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 ulteri- ori 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 man- tenendone 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 far- lo 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 ma- teria 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 sig- nificato 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 rista- bilire 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 de- crescente, 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 avreb- be 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

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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 con- quista 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)

I Maya (1.500 a.c. - 317 d.c.) - Il terzo 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 Ingenious and fun games of Maths

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 fri- jolito e palito).

I Greci (600 - 300 a.c.) - Neppure i 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 Ingenious and fun games of Maths

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 per- ché trovato descritto in un frammento attribuito ad Erodiano), di base 10, e con uno schema it- erativo 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., pre- vedeva l’associazione di ogni numero a una lettera dell’alfabeto; siccome però l’alfabeto classi- co conteneva solo 24 lettere, furono aggiunti altri tre simboli, per un totale di 27 simboli neces- sari 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 ac- cento 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 occi- dentale 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 C D M

54 Ingenious and fun games of Maths

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 rap- porti 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, Ger- berto 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 as- sumono 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 Ingenious and fun games of Maths

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”.

questo termine significa “vuoto” ma nelle traduzioniرفص (: ) Gli arabi chiamavano lo zero sifr 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 Fil- ippo 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 Ingenious and fun games of Maths

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 Ingenious and fun games of Maths

2. TURKEY MEETING Survey results (about parents-students-teachers) Questionnaire for students

1. Do you like Maths?

70% 60% 50% Poland Turkey 40% France 30% Italy Spain 20% Romania 10% 0% DEFINITELY YES RATHER YES RATHER NO DEFINITELY NO

2. Are you good at Maths?

60%

50% Poland 40% Turkey France 30% Italy 20% Spain Romania 10%

0% DEFINITELY YES RATHER YES RATHER NO DEFINITELY NO

3. Would you like doing more Maths? 60%

50% Poland 40% Turkey France 30% Italy 20% Spain Romania 10%

0% DEFINITELY YES RATHER YES RATHER NO

58 Ingenious and fun games of Maths

4. Which working method do you prefer?

60%

50% Poland 40% Turkey France 30% Italy

20% Spain Romania 10%

0% Individual work In couples Team work

5. What's your favourite topic in Maths?

60%

50% Poland 40% Turkey France 30% Italy

20% Spain Romania 10%

0% Arithmetic Geometry Logic No one

6. In which topic do you find more difficulties?

70%

60%

50% Poland Turkey 40% France Italy 30% Spain 20% Romania

10%

0% Arithmetic Geometry Logic No one

59 Ingenious and fun games of Maths

7. Do you like Maths?

70% 60% . 50% Poland 40% 30% Turkey 20% France 10% 0% Italy

Spain

Other

Solve

Mental

Decimal Decimal

time

table

numbers

problems

Money

arithmetic

1000

Fractions

Geometry

Memorize

calculations

Comparing Comparing

multiplication multiplication

Calendar and and Calendar numbers up to to upnumbers

8. You have much more troubles in maths when:

50% 45% 40% Poland 35% Turkey 30% France 25% Italy 20% Spain 15% 10% Romania 5% 0% You didn't pay The content was You didn't do You haven't got attention too difficult enough practise troubles

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 DEFINITELY BAD

60 Ingenious and fun games of Maths

10. You usually do your homework:

100% 90% 80% Poland 70% Turkey 60% France 50% Italy 40% Spain 30% Romania 20% 10% 0% On your own With a family member With an outside teacher

11. Which tools would you like to use in learning Maths?

70% 60% Poland 50% Turkey 40% France 30% Italy 20% Spain 10% Romania 0%

App Other

Books

tools

CD Rom CD

Interactive

technology Multimedia

Information Whiteboard

12. Do you think that cooperating with foreign countries for a Maths project could improve your skills?

90% 80% 70% Poland 60% Turkey 50% France 40% Italy 30% Spain 20% Romania 10% 0% DEFINITELY YES RATHER YES RATHER NO DEFINITELY NO

61 Ingenious and fun games of Maths

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 I DON'T YES NO 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% DEFINITELY RATHER YES RATHER NO DEFINITELY I DON'T YES NO KNOW

3. Which kind of method do you think it's more suitable for your child?

60%

50% Poland 40% Turkey France 30% Italy 20% Spain Romania 10%

0% Individual work In couples Team work

62 Ingenious and fun games of Maths

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 his/her own With a family member With an outside teacher

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% DEFINITELY RATHER YES RATHER NO DEFINITELY I DON'T YES NO KNOW

63 Ingenious and fun games of Maths

7. Do you feel able to help your child in maths?

60%

50% Poland 40% Turkey France 30% Italy

20% Spain Romania 10%

0% DEFINITELY RATHER YES RATHER NO DEFINITELY I DON'T YES NO 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% DEFINITELY RATHER YES RATHER NO DEFINITELY I DON'T YES NO 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 RATHER RATHER DEFINITELY I DON'T GOOD GOOD BAD BAD KNOW

64 Ingenious and fun games of Maths

10. In your opinion your child’s possible maths problems result from:

90% 80% Poland 70% 60% Turkey 50% France 40% Italy 30% Spain 20% Romania 10%

A lackof A

attention

motivation

The lackTheof

absences

from schoolfrom The frequentThe

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 10%

Solve

Mental

time

problems

to 1000 to

arithmetic

Comparing

numbersup Calendarand

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 DEFINITELY NO I DON'T KNOW

65 Ingenious and fun games of Maths

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% Individual work In couples Team work

3. In which activity do students show more difficulties in Maths?

100% 90% 80% Poland 70% Turkey 60% France 50% Italy 40% Spain 30% Romania 20% 10% 0% Arithmetic Geometry Logic No one

66 Ingenious and fun games of Maths

4. Do you do Maths activities outside the classroom?

60%

50% Poland 40% Turkey France 30% Italy 20% Spain Romania 10%

0% DEFINITELY RATHER YES RATHER NO DEFINITELY no answer YES NO

5. Do you think it’s important giving homework?

120%

100% Poland 80% Turkey France 60% Italy

40% Spain Romania 20%

0% DEFINITELY RATHER YES RATHER NO DEFINITELY NO YES

6. How do you evaluate students’ books of Maths?

120%

100% Poland 80% Turkey France 60% Italy 40% Spain Romania 20%

0% DEFINITELY GOOD RATHER GOOD RATHER BAD DEFINITELY BAD

67 Ingenious and fun games of Maths

7. In your opinion your child’s possible Maths problems result from:

100% 90% Poland 80% 70% Turkey 60% France 50% 40% Italy 30% Spain 20% Romania 10%

school

A lack of A

attention

A lackof A

Frequent

absences

motivation

Variouskinds of dysfunctionsof

8. Which tools do you consider more effective in teaching Maths?

100% 90% 80% Poland 70% Turkey 60% France 50% Italy 40% 30% Spain 20% Romania 10%

App

Other

Books

tools

CD Rom CD

Interactive

technology

Information

Multimedial Whiteboard

9. Do you think E.Te.Mat Project will improve our students’ skills?

120%

100% Poland 80% Turkey France 60% Italy 40% Spain Romania 20%

0% DEFINITELY RATHER RATHER NO DEFINITELY no answer YES YES NO

68 Ingenious and fun games of Maths

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% DEFINITELY RATHER YES RATHER NO DEFINITELY no answer YES NO

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% Poland 80% 70% Turkey 60% France 50% 40% Italy 30% Spain 20% Romania 10%

study

courses

Project

Training

E.Te.Mat E.Te.Mat

Individual

teachers

withinside

collegues

Conferences Comparison to Comparisonto

69 Ingenious and fun games of Maths

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

80% Turkey France 60% Italy

40% Spain

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% Poland 70% Turkey 60% France 50% Italy 40% 30% Spain 20% Romania 10% 0% Arithmetic Geometry Set problems Logic Reality tasks

70 Ingenious and fun games of Maths

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 Guidelines for the Curriculum : Successes for the Development of Skills at the end of primary school.

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 Ingenious and fun games of Maths

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

Perceiving their position in space and to estimate distances and volumes from their bo- dies. 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). Run- ning 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. Mea- suring 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. Per- form the four operations with security, considering if resorting to the mental or written calculation or to the calculator, depending on the situation . Car- rying out division with remainder of natural numbers; identifying multiple and partitions of a number . Estimating the result of an operation. Opera- ting with fractions and recognizing equivalent fractions. Use decimals, fractions and percentages to describe everyday situations . Interpreting negative integers in concrete contexts . Repre- senting 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 Ingenious and fun games of Maths

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 procee- dings. 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 Ingenious and fun games of Maths

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 Ingenious and fun games of Maths

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 Ingenious and fun games of Maths

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, qua- druple, 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 Ingenious and fun games of Maths

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 Ingenious and fun games of Maths

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.

Only 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 Ingenious and fun games of Maths

competencies Progress proposed by expected Progress proposed by the Progress proposed by the the To tier 2 O.B for CE2 O.B for CM1 the Common O.B for le CM2 Base

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 - Recognize, des- rectangle, diamond, vocabulary aligned points, square) and to draw cribe and name the figures and triangle. right, perpendicular lines, parallel lines. customary solid - Check the nature of a parallel lines, segment, me- - Check the nature of a

- Use the rule, plane figure using the ruler dium angle axis of symmetry, figure through the use and the square. center of a circle, radius, of instruments. square and com- - Build a circle with a com- diameter. - Build a height of a pass to check the pass. - Check the nature of a plane triangle.

- Use vocabulary situation: figure simple using the scale, - Reproduce a triangle nature of common side top angle setting. the square and compass. using instruments. plane figures and - Recognize that a figure - Describe a figure to identify build with care has one or more lines of it among other figures or to symmetry by folding or breed and precision 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 and - Recognize, describe name: name the solid rights: cube, and name the solid building 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-

ment a law firm patron.

79 Ingenious and fun games of Maths

Progress proposed by the Progress proposed by the competencies expected O.B for CM1 O.B for le CM2 To tier 2 Progress proposed by the the Common Base O.B for CE2

Reproductive problems, construction Reproductive problems, construction Reproductive problems, construction - Reproduce the figures (on plain paper, Complete a by axial symmetry. - Draw a figure (on plain paper, checkered - Ability to organize digital 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. 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.

Progress proposed by Progress proposed by competencies expected the the To tier 2 Progress proposed by the the Common Base O.B for CE2 O.B for CM1 O.B for le CM2

Reproductive problems, construction Reproductive problems, Reproductive problems, - Reproduce the figures (on plain paper, construction construction

- Ability to organize digital checkered or dotted), from a model. Complete a by axial symmetry. - 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 likelihood of a result. the instructions. 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.

80 Ingenious and fun games of Maths

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 Ingenious and fun games of Maths

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 Ingenious and fun games of Maths

The content of teaching- strands

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

The content of teaching- strands 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 Ingenious and fun games of Maths

The content of teaching - strands

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)

The content of teaching- strands

Calculating area, length, and other geometric properties The student:

1. calculates the perimeter a polygon 2. calculates the area of the square, rhombus, parallelogram, triangle and trapezium presented on a drawing and in practical situations 3. estimates and measures length using appropriate metric units 4. calculates area using acres and hectares 5. estimates and measures capacity using appropriate metric units 6. calculates angle measures

84 Ingenious and fun games of Maths

The content of teaching- strands

Practical calculations The student:

1. develops an understanding of simple percentages and relate them to fractions and decimals 2. solves problems involving operations with whole numbers, fractions, decimals and simple percentages 3. solves and completes practical tasks and problems involving times and dates 4. reads the temperature on the Celsius scale 5. renames the measures of weight and length 6. identifies given scale and draws items to a larger or smaller scale. 7. in the practical situation calculates the length of the road at a given speed and given time and so forth

The content of teaching- strands

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 Ingenious and fun games of Maths

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 Ingenious and fun games of Maths

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 Ingenious and fun games of Maths

2.nd CLASS 3-rd CLASS 4-th CLASS

Natural Numbers Natural Numbers Natural Numbers . Reading, position . Composing, reading, . Composing, reading, value and composition writting, comparison, writting, comparison, of numbers 0 -1000; ordering and rounding classes (units, . Comparison of between 0- 1 milion thousands, millions), numbers between 0- . Additions and ordering and rounding 1000; subtractions between 0 between 0- 1 milion -10000, grouping or . Positional numeral . Ordering and system: writing rounding 0-1000; counting . Using terms and numbers in . Additions and mathematical symbols decimal form (sum subtractions between 0 like sum , total, of products by a -1000,grouping or difference, minuend, factor 10, 100, 1000, counting ; subtrahend, even less, etc.);multiplication with 10, 100, . Multiplications and even more etc. 1.000. divisions till 100; . Finding an unknown number within a .Using terms and . Roman numbering relationship type ? + mathematical symbols system. a = b like sum , total, difference, minuend, .Highlighting some of subtrahend etc. the Assembly properties (commutativity, associativity, identity element) using objects and representations, without the use of terminology

88 Ingenious and fun games of Maths

2.nd CLASS 3-rd CLASS 4-th CLASS

. Measurements using non- . Measurements using con- . The use of unconvention- standard standards ventional standards: use al measures for determin- . Length measuring units : appropriate measuring ing and comparing the multiples, submultiples of tools: tape measure, ruler length; the metre graduated scales, scales, clock . The use of units of meas- . Measuring Unit capacity: urement for the determina- litre, multiples, submulti- . Length measuring units: tion of lengths, comparing ples . meter, multiples, submulti- and ordering various . Measuring Mass Units: ples, transformation via mul- events; kilogram, multiples, sub- tiplication and Division with multiples. 10, 100 and 1000; . Upon the completion of equivalent value exchanges . Units of measurement for . Measuring units capacity: through conventional rep- time: the hour, the minute, litre, multiples, submulti- resentation standard and day, week, month, year; ples, transformation via nonstandard using money . Coins and banknotes, in- multiplication and Division matters-simple game-type cluding those of the Europe- with 10, 100 and 1000; income-expenses, with an; . Measuring units: kilogram, multiples, submulti- numbers from 0-1000; . Using appropriate meas- ples, transformation via uring instruments: ruler, . Identification and use of multiplication and Division tape measure, calibrated customary units of meas- with 10, 100 and 100; ure for length, capacity, scales, balance Units of measurement for mass . time: the hour, the minute, second, day, week, month, year, decade, century, Mil- lennium, . Coins and banknotes

89 Ingenious and fun games of Maths

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 Ingenious and fun games of Maths

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 0,5 0,5 0,5 1,5 1,5 1,5 language/Support Break 2,5 2,5 2,5 2,5 2,5 2,5 TOTAL 25 25 25 25 25 25

CONTENTS

3º 4º 5º

Natural Numbers Natural Numbers Natural Numbers • Numbers up to five •Numbers up to the •Decimal numbers system. figures. Reading, position million. Reading, position Reading, position value N value and decomposition. value and decomposition. and decomposition. U •Comparison, ordering •Comparison, ordering •Comparison, order and M and rounding. and rounding. rounding. B •Ordinal numbers until the •Roman numbering •Operations: addition, E thirtieth. system. subtraction, R •Roman Numbering •Addition and subtraction multiplication, division. S system. with led in tenth and •Multiples and divisors. •Addition and subtraction hundredth. Properties. •Prime numbers and with led in tenth and •Multiplication by several compound numbers. hundredth. Properties. figures. Associative, •Divisibility criterion.: •Multiplication with commutative and 2,3,5 and 10. several figures with and distributive properties. Fractions . Concept. without led. Proper fraction. Improper fraction.

91 Ingenious and fun games of Maths

•Associative, commutative •Accurate and inaccurate •Fraction of an properties. division between two or amount.Equivalent •Multiplication by ten, one three figures. Division fractions. hundred, one thoussand.. test. •Comparing fractions. •Accurate and inaccurate Fractions. Concept. Addition and subtraction of

division. Division test. Reading, comparison and fractions. Fraction as exact Fractions. Concept, representation of division and not exact . reading,comparison and fractions. Mixed number. representation of fractions. •Fraction of an amount. •Decimal numbers. •Addition and subtraction •Application , reading and of fractions with the same writing od decimal denominator. numbers: tenth, hundredth Decimal numbers. and thousandth. Application, reading and •Comparison, ordering and writing of deccimal estimation decimal numbers: unit, tenth and numbers. hundredth. •Operations with decimal

•Comparing, ordering and numbers: addition, rounding decimals . subtraction, multiplication •Comparing numbers: division. natural numbers , fractions •Comparison numbers: and deecimals. natural fractions and decimal 3º 4º 5º

M Measurement of length, •Measurement of length, •Measurement of length , E capacity and mass capacity and mass. •Units capacity and mass. •Ways of S • UnitsU of mesurement. for measuring length, expressing measures. •ExpresingR measures units. capacity and mass. Diferent •Operations with measures,, •Time.E Units bigger than ways of expresing addition, subtraction, day:M day, week, month and measures. •Measures multiplication, division. year.E •Units smaller than operations: addition and •Measuring instruments. day:N hour, minute, second. subtraction. •Measuring •Area. Units of area. •CoinsT and notes. Euros instruments. •Area, Area •Addition and subtraction and cents. Addition and units. Addition and measures of area. •Time. subtraction with them. subtraction measurements Units higher than day: week, of area. •Time. Bigger Units month, year, lustrum, than day: day, week, decade, century. •Units month, year, lustrum, lower than day: hour, decade. •Smaller units than minute, second. Complex day: hour, minute, second. shape and not complex. Complex and not complex form. Addition and subtraction 92 Ingenious and fun games of Maths

3º 4º 5º

•Addition and •Addition and subtraction time subtraction time data. •Coins and data. Sexagesimal notes. •Euros and system. •Money, cents. coins and notes. Operations. •Euro and cents. Operations.

93 Ingenious and fun games of Maths

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 Ingenious and fun games of Maths

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, season- month, 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 Ingenious and fun games of Maths

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;

Students will be able to;

- 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 Ingenious and fun games of Maths

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;

Students will be able to;

-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 Ingenious and fun games of Maths

4th Grades Mathematic Curriculum(10 yeras old pupils)

Unit 1 The Geometry Around Us

Objectives:

Students will be able to ;

-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:

Students will be able to ;

-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 Ingenious and fun games of Maths

UNIT 3 LETS MEASURE,WEIGH,and GET TO REALITY

Objectives:

Students will be able to;

-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:

Students will be able to;

-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 Ingenious and fun games of Maths

UNIT 5 DECIMAL FRACTIONS&MEASURING LENGTH

Objectives:

Students will be able to;

-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:

Students will be able to;

-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 Ingenious and fun games of Maths

UNIT 5 DECIMAL FRACTIONS&MEASURING LENGTH

Objectives:

Students will be able to;

-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:

Students will be able to;

-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 Ingenious and fun games of Maths

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 Ingenious and fun games of Maths

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 .

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A story : Bark and the Incandescent Dragon .

ICEBREAKING SCRUM BRAINSTORMING (unplugged) (unplugged) (unplugged) Games heating From rugby : when 1. Designs inspired 1. Eyes closed all players are by the cards of pointing in the 2. Symbols and Stories same direction to controls : up, down, 2. Storm on take possession left, right characters of the ball. SCRUMBOARD STORY MOUNTAIN (unplugged/on - line) (unpluggedon - line) SPRINT PLANNING The board that helps (unplugged/on - line) The structure of the planning : story: the initial Planning of useful 1. to do; situation; that kicks work to give life to off the adventure; the story. 2. during construction; climax; remedial action; end. 3. already done.

FINAL EVALUATION OF THE " GOOD PRACTICES "

MATH CLASS INVENTED STORY (Task of reality) (Coding)

 The group elected facilitates The group formed by the teacher can collaboration and creative phase. work situations more balanced, especially for the enhancement of skills  This type of task stimulates the and roles of each boy. 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 Mathematics and storytelling “play” difficulties that may be encountered. togheter in games of different languages and fantasy.

 The students are using a precise language and appropriate (in Children learn to organize and plan the study time activities . mathematical terms). They learn how using creatively LIM, PC,  The students listened with tablet . seriousness and respect fellow presenting the lesson.

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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. Other teaching aids 4. Games 5. Maths competitions 6. 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 .

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Mathematical cards – multiplication table Students learn and improve multiplication table eagerly during games

Card games which help Mathematical cards Boxes for the game children to memorize "Multiplication table" "Olympic players" the multiplication table

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”

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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.

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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)

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competencies expected Progress proposed by the Progress proposed by the Progress proposed by the 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 - Recognize, describe and 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 the rule, square and - 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 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, Space Space Space - Recognize, describe and name: - Recognize, describe and name the solid - Recognize, describe and name the solid building 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.

Progress proposed by the Progress proposed by the competencies expected To tier 2 Progress proposed by the O.B for CM1 O.B for le CM2 the Common Base O.B for CE2 Reproductive problems, construction Reproductive problems, construction Reproductive problems, construction - Reproduce the figures (on plain paper, checkered Complete a by axial symmetry. - Draw a figure (on plain paper, checkered or - Ability to organize digital or dotted), from a model. - Draw a simple figure from a construction dotted), from a information or geometric, justify - 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.

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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.

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

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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 They can work in small groups. individual events. Various activities

In the end a diploma and a Chinese puzzle was given to each student of the winning class.

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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.

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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 Objectives: 1000 • practicing calculation in different contexts • to solve multiplication exercises using • getting used to other calculation methods; different methods; • identifying numbers and using them in • to solve different life situations using calculations appropriate operations; • interdisciplinary correlation of contents • to use their multiplication knowledge in a • identifying and correcting possible errors variety of contexts; • checking calculations in different ways • identifying everyday life situations in which we use such calculations: planning and organizing a party.

Resources: Types of activities: • student’s guide, • frontal , individual multimedia guide, and pair activities computer

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.

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Different methods to do multiplications .

Using the fingers Multiplication with lines Using a rectangle Let’s say we want to multiply this: 26 x 2 = ? 35 X 3. (only for multiplication by 9): We use a rectangle divided in two 1. Number each finger in your mind. 1. Represent the first number with lines. parts because we have a two-digit Group 3 lines for tens and 5 lines for Don’t forget the number you have number. given each finger. units.

Each part is divided by an oblique

line. We write the two-digit number above and the one digit number on 2. Cut both groups of lines with 3 lines the lateral. (from number 3) 2. Bend the finger which represents the number you want to multiply by 9. In turns we multiply the two digits with 2 and we write the results in the

rectangle . 3. Count the points where the lines meet in

Eg.: for 2 x 9 bend finger number 2. each group. For units there is 5.

There are 10 tens. (from 9 + 1) On the left side there is one finger, on We add the numbers in diagonal. The result is105 the right there are 8 fingers. 2 x 9 = The result: 52 (35 X 3 = 105). Check it! 18

FRIENDS OF NATURE Targeted competencies Objectives: • Validating the results • to identify in a text relevant information to solve a • Recording the data problem; observed in the • to drop out irrelevant information from a text to environment and facilitate understanding ; representing them • to identify contradictory information ; • Solving problems with the • to reword information for better understanding. studied concepts Types of activities Suggested activities • individual activities • improving analysis abilities of a text in order to search Resources: • frontal , select and use relevant information • student’s guide, activities(explanations,che- • appreciating information in terms of : useful/useless, multimedia guide, cking) complete/incomplete, corellated/ computer, contradictory, redundant, etc. information/images • rewording certain information in oder to clear up and about animals better understand • identifying the appropriate behaviour to protect environment.

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Suggestions for class activities: What have we got to do? How shall we monitor the activity? • give feedback to the students’ answers;

• come with additional explanations if needed ; • this unit has many texts and is based on • observe systematically the students’ involvement in the reading and analysing them in order to figure out activity; the useful information; • do not insist on the solving speed but on the correct • the texts are approached from a searching for solving; and selecting information perspective ; • encourage cooperation and mutual help ; encourage students to read the texts several • use a monitoring sheet to record the correctness of the times and to come back to the texts whenever answers for each student; necessary; • encourage self- assessment and evaluation in pairs/ • guide your students into a “mathematic” groups ; reading , helping them to make the difference • make sure all the mistakes have been corrected. between the answers they can explicitly find in the text and the ones which need correlating information from the text ; Reflection moment • lead them into choosing the information which • What happened during the activities? need to be correlated; • Why did this happen? • guide students in their activity of filling in with • Which part of the activity was the most certain data; emphasize the importance of using interesting for the students? real data, particularly when it comes to animals’, • Where have I encountered difficulties? plants’ objects’ phenomena’s characteristics, • Which activities shall I do again in other which might require a former documentation; 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? Write the answers to the following Observe, think, solve! - how many children brought photos questions : 1. How many children brought photos? How can we find this? What project did the students take part in? Can you find the answer to this question in We divide the total numbers of What grade were the participants in the the text? photos with 4 (because each child project? Is there any information in the text which has brought 4 photos): How many photos did each child bring for could help you find the answer ? Think and answer: the exhibition? Read the statement: 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 make? photos, each child displaying 4 photos of equal with the number of the Where did the children bring out the books? his tree. children involved in the project? Could you answer these questions? What What do we know? Explain your answer. Use helped you? - the number of photos information from the text. - How many photos each child has brought

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TURKEY good practices

MEASURING THE LIQUIDS

We are sharing one litre equally into two half litres.

116 Ingenious and fun games of Maths

We’re weghing 1litre of water with 1kilo.

Students practice on the board with projector…

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Drawing the geometric solids on isometric paper...

Cube

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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 ab- sence 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 .

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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. Se- condary 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 rehabilita- tion

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. Dif- ficulty 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.

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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

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Analog

No matter knowing the code!

LOGIC ANALOG The child needs to reason The child examines reality before “understanding to reason reality”

ANALOGICAL APPROACH (non conceptual)

 Abacus  Analogic tools

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 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).

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POLAND

12 + 29 = ? a) 38

b) 41 b) 42 b) 51

Good! You have saved 1 friendly alien!

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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

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Great! You have saved 3 friendly aliens!

31 - 7 = ?

a) 24 b) 25 b) 26 b) 27

Great! You have saved 4 friendly aliens!

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REUNION ISLAND

By Fabienne Couchat and Alexandre Schneider Maths teaching in a specialt class

activities: elements request to supplement the chickens

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Each student must complete their pool by exchanging 1 against number of counters defined by the master.

complexity of the task

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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.

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ROMANIA

Didactic ideas applicable to math lessons for students with special needs

Learning based on special needs

DIDACTIC IDEAS

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Effective communication ,

help in need

Form ing natural numbers with tens and units

Understanding passing to the next ten

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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

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Measuring capacity – practical application

Comparing measure units

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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.

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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.

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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.

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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

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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.

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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

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140 Ingenious and fun games of Maths

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

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 Turkey Spain France

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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

3th grade - Task 4

1.00

0.90

0.80

0.70

0.60

0.50

0.40

0.30

0.20

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3th grade - Task 5

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142 Ingenious and fun games of Maths

3th grade - Task 6

0.80

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3th grade - Task 7

0.80

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3th grade - Task 8

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143 Ingenious and fun games of Maths

3th grade - Task 9

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3th grade - Task 10

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144 Ingenious and fun games of Maths

4th grades Coun- try\Task 1 2 3 4 5 6 7 8 9 10 Total 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

4th grade - Task 1

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4th grade - Task 2

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145 Ingenious and fun games of Maths

4th grade - Task 3

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4th grade - Task 4

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4th grade - Task 5

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146 Ingenious and fun games of Maths

4th grade - Task 6

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4th grade - Task 7

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4th grade - Task 8

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147 Ingenious and fun games of Maths

4th grade - Task 9

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4th grade - Task 10

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5th grades Coun- try\Task 1 2 3 4 5 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 Ingenious and fun games of Maths

5th grade - Task 1

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5th grade - Task 2

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5th grade - Task 3

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149 Ingenious and fun games of Maths

5th grade - Task 4

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5th grade - Task 5

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5th grade - Task 6

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150 Ingenious and fun games of Maths

5th grade - Task 7

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5th grade - Task 8

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5th grade - Task 9

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151 Ingenious and fun games of Maths

5th grade - Task 10

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152

Edited 2016