ENG_Matematica-con-90-Minilibri_DEF by Edizioni Centro Studi Erickson

The MINIlibrary of the World

Desirèe Rossi

Mathematics with

90 Minibooks First Grade

BUILD, ETE COMPL ! OLOUR AN D C

Workshops and activities for primary school

INDEX

Activating learning with Minibooks

Prerequisites

Counting and numbers up to 20

Addition and subtraction up to 20

107

Numbers over 20

115

Problem solving

133

Geometry

142

How did it go? I observe myself

143

The minilibrary

Activating learning with Minibooks

A child, a teacher, a book and a pen can change the world. Malala Yousafzai

When enabled to do, boys and girls become active participants in the processes we seek to stimulate, in every context. When doing and thinking are connected, motivation to learn is obtained, which is a drive to pursue an activity despite cognitive effort and the inevitable mistakes. The minibooks are small thematic booklets of eight pages, each dedicated to a specific learning content. Children can build them using creativity and fine motor skills: folding, cutting precisely, mentally rotating and predicting the result, experimenting, making mistakes and learning from mistakes, comparing, learning through modelling or peer education, etc. Inside the book there is a page with illustrated instructions on how to assemble the minibooks and a QR code to access the video tutorial. With minibooks, first the hands are ‘set in motion’ and then the cognitive part is stimulated, although there is still a certain need for children to remain focused on coordination and stroke management to make the most out of the minipage space. Book after book, children enrich the ‘shelves’ of their minilibrary, represented in the final page of the book. It is advisable to photocopy it, preferably in A3 format, and then invite the pupils to colour in the backs of the minibooks as they complete them. In the final part of the book, there are also self-assessment sheets with an emoticon Likert scale where children express the degree of commitment, difficulty and emotions they felt during the activity.

Mathematics with 90 Minibooks – First Grade

Each volume in the series ‘The World’s Minilibrary’ focuses attention and learning content on a subject area. This volume, dedicated to the development of logical-mathematical prerequisites and counting skills, introduces the activities in an extremely gradual way, starting with pre-mathematical and visual processes such as perceptual quantification and progressing to counting, knowledge of numbers, grouping and addition and subtraction operations functional to problem solving. The 90 minibooks are divided into six sections: – Prerequisites (logical-mathematical prerequisites; sets and relationships); – Counting and numbers up to 20 (counting; numbers up to 20; major/ minor; groupings in tens; friends of 10); – Addition and subtraction up to 20; – Numbers over 20; – Problem solving; – Geometry. At the beginning of each section, there are theoretical indications of the processes involved. The importance of using active, diversified and interdisciplinary teaching methods has out of necessity determined the inclusion of a workshop part that anticipates the minibooks and becomes complementary for an integral work on the proposed skills. The workshops are closely linked to the contents of the minibooks and address the need to deal with the same subject area with different skills and perspectives. Minibooks can be used to: – introduce a topic – explore a content that has already been developed in class

– revise a content – stimulate a certain subject content through a different input. Each volume has a funny lead character who guides the entire group of minibooks, stimulates and encourages the children to act, and is a real tutor who, through the use of symbolic images, is able to show which skill will be most required in the process of completing the minibook and thus anticipates the delivery written on the pages. At the beginning of the volume there is a minibook dedicated precisely to the character, here the caterpillar Lello, aimed at illustrating the basic words of the tasks: observe, listen, pronounce, read, write, colour, circle, mark with an x, etc. In this way, the child looking at the picture of Lello the caterpillar already knows which activity will characterise the minibook he or she is about to do. There is also some sort of ‘intuitive challenge’ regarding what is asked in the minibooks because during the building phase, which precedes the completing one, children have the opportunity to look at the different pages and try to imagine what they will have to do. For the sake of completeness, the minipages include, in most cases, written assignments, which can be read by the teacher or the pupil depending on their skills, precisely with the aim of recognising different abilities even among peers and of acknowledging this difference in approach to a task rather than seeing it as a limitation. The use of minibooks in the classroom involves, in addition to individual work, work in pairs or small groups, to encourage self-correction between classmates and monitoring how one is working (in some minibooks this step is explicitly requested).

The illustrations are presented in black and white to allow each child to customise their own cards, colouring them as they wish. The minibook consists of eight pages with activities designed to start and end within a lesson time; it is a task with a precise timeframe that must have a beginning and an end, to avoid the effect of ‘effort’ in starting, stopping and starting an activity again. It is desirable for the teacher to use the classmate resource to ‘check in’ on how each child performed the activity, as in a kind of inclusive peer learning and self-correction. The special horizontal layout is designed to facilitate photocopying, preferably in A3 format, for an optimal yield and to facilitate children in using the page space. Once assembled and completed, the minibooks can be attached to the notebook, collected in a holder or in a display case, hung on a poster board or stored in a small box acting as a travelling minibookshop. All it takes is the curiosity to look at the minibook proposals and discover their different potentials, leaving the children free to experiment independently and to get involved in the realisation of the minibraries. Their creativity could lead them to create their own special minibooks, which perhaps have little to do with learning content, but which show how to get an ‘idea’ or a tool ‘away and use it in other areas is a valuable learning goal.

Activating learning with Minibooks

FOLD THE SHEETS LIKE THIS TO CREATE THE MINIBOOKS:

1 Lay the sheet in front of you.

The printed side is facing downwards. Fold the sheet in half. To do so, place the right-hand edge over the lefthand edge.

2 Fold in half a second time.

3 And fold in half again.

6 Re-open the sheet and fold it in half, in order to see pages 4, 5, 6 and 7.

5 Fold the sheet in half again and cut along the hatch between pages 2 and 5.

4 Re-open the sheet with the white part facing upwards.

7 Finally, fold the sheet as in the illustration. Your minibook is ready!

Watch the tutorial on how to build minibooks!

Logical-mathematical prerequisites

Studies and psychological research have shown not only that we are born with a predisposition to numerical intelligence as well as verbal intelligence, but also that numerical intuition and the ability to identify/visualise and differentiate between small quantities (subitizing) is innate; it gets perfected towards complexity thanks to the stimulation and cultural tools that the growing environment provides the child. From the ages of 2 and 3, calculating skills are already being developed: through discriminating numerosity between sets, children, although they do not yet possess counting skills, are able to perform simple addition and subtraction visually presented through concrete objects. Then, it is important to stimulate and develop these basic skills through formal education in order to improve the mathematical skills leading to increasingly abstract processes. But what is numerical intelligence or quantity intelligence? It is the specific ability to understand, interpret, reason through the cognitive system of numbers and quantities. Gardner, in his multiple intelligences’ formulation, defines logical-mathematical intelligence, associated with scientific thinking and brought back to direct confrontation with the real world, as the ability to think and process information in numerical and abstract relation terms. Underlying numerical competence there are skills such as comparing, ordering and classifying objects, estimating quantities, creating and solving problems, analysing situational and variable components, using abstract symbols, discovering and using algorithms and logical sequences. It is undeniable that numerical competence is closely linked to the development of language: in fact, every quantity has a name that is expressed through verbal language; in this regard, it is enough to consider how language

Mathematics with 90 Minibooks - First Grade

influences quantification, for example in the use of the plural (one is different from many as dog is different from dogs). As indicated by Piaget, who dealt with the link between cognitive skills in relation to the number concept processing, a child›s ability to produce a correct verbal sequence of numbers does not indicate an ability to count according to quantity; it is important that in the child›s mind, or better still in his or her practical experience, each word that identifies a number corresponds to an object and that therefore each number corresponds to a certain quantity. According to the psychologist, for children to be able to use numbers, they need to learn to master their logical operations of classification and seriation. The development of numerical skills therefore involves a mental representation of analogical quantities and assumes the existence of specific skills such as subitizing (estimating between 4 and 6 elements) and estimating quantities (greater than 6-7 elements). Based on studies regarding the nature of difficulties in mathematics conducted by the University of Padua (Lucangeli, Poli and Molin, 2003) and those concerning the development of numerical competence, it is possible to infer what processes underlie the construction of the number concept and, therefore, the learning of calculation. These processes are estimated to be preverbal in nature, precisely because they appear when children are still very young and can be defined as precursors of numerical ability since they anticipate calculation skills and represent their fundamental basis. They have been summarised as follows: – semantic processes or of ‘quantitative understanding’ and/or ‘number sense’; – counting processes (counting skills);

– lexical processes (“verbal labelling” i.e. the naming of the number); – pre-syntactic processes (related to the structure of the numerical system). The transition from preverbal skills to the actual skill of counting requires the ability to associate number-concepts with number-words, deducing among many logically possible meanings the correct meaning of the number-words. There are two different theoretical positions concerning the transition from preverbal skills to the acquisition of number-words (Lucangeli, Poli and Molin, 2003). The first is The counting theory developed by Gelman and Gallistel (1978), for whom the innate concept of number is present in children and evolves into calculation procedures through the following steps: – the principle of one-to-one correspondence, according to which each element of the counted set must correspond to a single number-word; – the principle of stable order, which refers to the ability to arrange number-words according to a fixed sequence that repeats in order the elements to be counted; – the principle of cardinality, according to which the last number-word used in a counting represents the quantity of elements belonging to the group in question. According to the two researchers, number-words are included in the innate list of mental number-labels, with the prevailing role of preverbal competence over the verbal one. The second theory that explains the transition between number-concept and word-number is the theory of different contexts by Fuson (1991). It confirms the importance of innate competences, but from the perspective of a reciprocal relationship of interaction and support, and places the same value on competences learned through practice and imitation. Fuson therefore recognises the innate basis of the processes indicated by Gelman and Gallistel (correspondence, stable order and cardinality), but considers that, in order to develop them and learn to use them competently, it is important to have

several moments of learning and interaction with the stimulating environment, as these processes take time to consolidate. Hence, interaction and stimulation of the environment becomes a key factor in the development of more complex numerical skills. For Fuson, the integration of lexical, semantic and functional aspects of calculation as counting occurs, firstly, when the child recognises that each number-word refers to the total of the units that precede it (including itself) and, secondly, when the child understands that the position of any unit in the number series acquires the value ‘plus one’ in relation to the preceding unit and ‘minus one’ in relation to the following one. The three principles underlying counting lie in the three closely related number concepts, which are: the number sequence, the one-to-one correspondence between number-words and counted elements, and the cardinal value of numbers (Lucangeli, Poli and Molin, 2003). And how does the transition from word-number to writing and reading of the Arabic numeral develop? Although there is no single theory on the development of written number competence, some experimental and empirical studies reveal a classification depending on the represented graphic mode: – idiosyncratic (there are no understandable notations compatible with quantity, they are incomprehensible from the numerical point of view to the outside eye looking at them); – pictographic (figurative reproduction of the elements that are part of the group); – iconic (graphic signs, rods or symbols that have a biunivocal correspondence with objects); – symbolic (presence of Arabic numeral writing, even if not always in the correct position). This research shows us that numerical competence does not develop from primary school onwards, when children are taught the correspondence between Arabic numeral and numerical quantity. The prerequisite skills that are present in children already at a very early age are developed in the first and second childhood through the presence of meaningful experiences in the

Prerequisites

field of quantitative analysis during interaction and immersion in their living environment. This will allow progress along the way with a certain degree of confidence; the transition that primary school makes is the reinforcement of prerequisites and the consolidation of quantification skills in the first cycle, still relying heavily on the concreteness of quantification with a gradual transition to the written symbolic system and the ability to work through contents that are gradually more abstract. In the early years of primary school, counting strategies are still fundamental, while in the following years they tend to be abandoned because calculation strategies are based on mnemonic retrieval. Bibliography Fuson K.C. (1991), Relations entre comptage et cardinalité chez les enfants de 2 à 8 ans, «Les chemins du nombre», pp. 159-179.

Mathematics with 90 Minibooks - First Grade

Gelman R. e Gallistel C.R. (1978), The child’s understanding of number, Cambridge, Harvard University Press. Lucangeli D., Poli S. e Molin A. (2003), L’intelligenza numerica (3 volumi), Trento, Erickson. Wynn K. (1990), Children’s understanding of counting, «Cognition», vol. 36, pp. 155-193. Wynn K. (1992), Addition and subtraction by human infants, «Nature», vol. 358(6389), pp. 749-750.

WORKSHOPS IN THE CLASSROOM…

KEEP AN EYE ON THE DISGUISE Goals Stimulating attention, recognition, and visual memory.

Materials a Clothes to dress up with (trousers, skirts, sweaters, shirts, hats, carnival dresses, wigs). a A box to store the costumes.

a Accessories (glasses, handbags, earrings, ties, hats, beanies, scarves).

Unfolding Set the chairs of the class group in a circle and divide the children into two groups (corresponding to the two semicircles). All the children sit down, except one, who has to go dress up, helped by a male or female classmate, using not only the clothes, but also the accessories available. When he/she is ready, the child in disguise places him/herself in the centre of the circle. The classmates observe the child and can ask questions about his or her disguise, such as where he or she comes from, what his or her name is, what he or she is doing in such a dress. After answering and after the classmates have observed him/her carefully, the child in disguise comes out of the circle and, with the help of his/

her chosen classmate, quickly changes three elements of the disguise: he/she can replace clothes and accessories, he/she can move accessories, he/she can change shape of a garment (e.g. he/she can tie a shirt at the waist or leg instead of wearing it) The child then returns inside the circle. The two teams observe him in silence for up to one minute and, after a brief consultation between the members, try to spot the differences. The team that guesses the greatest number of differences wins the round. The game continues with a new disguised classmate and a new helper.

Prerequisites

COLOUR THE CELLS WITH NUMBERS ONLY.

DRAW THE PEOPLE WHO ARE IMPORTANT TO YOU.

PEOPLE WHO MATTER... 2

TODAY’S DATE IS...

THE CASE OF NUMBERS

CIRCLE THE NUMBERS OF THE DATE OF YOUR BIRTHDAY.

COPY FROM THE BLACKBOARD THE DATE MADE ONLY OF NUMBERS.

2 3 4 5 6 7 9 10 11 12 13 15 16 17 18 19 21 22 23 24 25 27 28 29 30 31 1 2 3 4 5 6 7 8 9 10 11 12

WRITE DOWN THE NUMBERS YOU KNOW.

1 8 14 20 26

IN THE WORLD OF QUANTITIES

MY FACE

I’M __ YEARS OLD

DRAW YOUR PORTRAIT AND COLOUR.

DRAW ON THE CAKE AS MANY CANDLES AS YOUR YEARS.

THE NUMBERS OF MY LIFE...

Counting and numerical acquisition

The ability to count visually and discriminate small quantities occurs from a very young age. If for quantities up to 6 the perception of quantity is immediate (subitizing), when quantities increase it is necessary to develop counting skills through practice and the verbal-quantity association. When counting, attention must be directed, by moving the eyes from one object to another, towards the elements that are part of the quantification. This skill requires the acquisition of the principles of one-to-one correspondence, stable order, and cardinality. These counting principles in turn, thanks to exercise and immersion in experiential contexts of stimulation, are based on semantic, lexical and syntactic processes. Semantic processes refer to the awareness of the quantities that a certain number defines, with the ability to discriminate where in two sets, if asked, the larger quantity is present (Lucangeli, 2012). Lexical processes involve the ability to correctly name a number through a bidirectional encoding between verbal/written code and Arabic code (Lucangeli, 2012). In short, it is a matter of giving a conventional name to the mathematical object. Syntactic processes, on the other hand, involve the ability to organise quantity into different orders of size and to attribute operating rules ‘to the mathematical object’ (positional order of digits, groupings into tens and units). These processes can act in synergy with each other in the activities that the teacher prepares everyday or be selected/checked individually by arranging targeted activities aimed at their reinforcement. The teacher is responsible for recognising the level at which his or her pupils are in relation to these processes. Here, the following workshop proposals and those in the minibooks aim precisely

Mathematics with 90 Minibooks - First Grade

at strengthening these processes to transform them into secure skills on which to build the abstract and symbolic framework characterising numerical skills. The development of calculation skills and memorisation with quick access to the solution are part of algorithmic and executive practices and include the ability to automate numerical facts (fast and correct retrieval); for example, if I work with the number friends of 5, the numbers that compose it can be 2 and 3, 3 and 2, 4 and 1, 1 and 4, 5 and 0, 0 and 5. This quick breakdown of the parts that compose the number is a fundamental stage for working with oral calculation in the transition to the ten and takes place through repeated activities and acquired mainly during the first cycle of primary school. The exercise on comparing quantities, first in a concrete way and then accompanied by a comparison on the number line, requires an understanding of, for example, the mathematical vocabulary, what the words “greater than”, “less than”, “equal to” mean and then a correspondence of these words with the graphic symbols >, <, =, which, if they are not fixed with their sense and meaning, turn out to be pure labels that are easily mistaken. The same thing happens with the mathematical terms ‘preceding’ and ‘following’, which first require an understanding of the meaning and then become very simple mechanisms with which to identify the number that comes before and after a given number. The need to semantically understand these contents is also found in children of recent 0-language immigration who need these steps in order to be able to develop competences, since this area is not ‘disconnected’ from the linguistic sphere, as some may think (see introduction), but on the contrary is closely connected in the development of competences.

In the logical-mathematical area of quantification and knowledge of number, the National Curriculum Directions indicate that in the first cycle of primary school, boys and girls: – gain confidence and expand their counting skills; – know, name and write the numbers from 0 to 20 and become aware of the recursiveness of numbers; – acquire a basic understanding of mathematical relationships and learn to work with numbers up to 20, and then replicate the same mechanisms with higher quantities; – compose and decompose quantities.

especially at the graphic/written level. The workshop proposals, together with the number and counting minibooks that follow in this section, aim precisely at supporting these two integrated aspects of number learning. Bibliography Lucangeli D., Poli S. e Molin A. (2012), L’intelligenza numerica Abilità cognitive e metacognitive nella costruzione della conoscenza numerica dai 6 agli 8 anni, vol. 2, Trento, Erickson.

Children must therefore be stimulated, without falling into pure exercise mechanics but keeping their motivation high, to carry out experiences of quantifying, comparing, composing and decomposing numbers using various methods that are as active and involving as possible. The part of exercise, order and recursiveness is present for a systematisation of the knowledge and skills learnt

Counting and numbers within 20

RIDDLES WITH NUMBERS Goals a Being able to identify on an image a certain amount of elements possessed by a certain animal. a Thinking and trying to answer according to the given clues.

a Asking questions using low quantities to identify a living being.

Materials a Images on cardboard/card of different types of animals with recognisable numerical characteristics (e.g. number of legs, tentacles, wings, antennae, etc.).

Unfolding Arrange the children in a circle and choose one child to start the game. Ask him/her to pick a card, look at it closely without showing it to the others and describe the animal represented to his/her classmates. He/ she can say, for example: ‘Guess what... This animal has eight legs and a hairy head’. The classmates answer in turn, trying to guess the mystery animal. Whoever guesses wrong is eliminated from the game, while

Mathematics with 90 Minibooks - First Grade

the one who correctly guesses takes the place of the game leader and picks a new card. BEWARE! It is important, on penalty of exclusion from the game, that the riddle is always and only about numerical characteristics associated with other qualitative features (e.g. 6 legs, 2 wings, 8 tentacles, etc.).

COLOUR 16 ELEMENTS.

COLOUR NUMBER 16.

WRITE NUMBER 16.

COLOUR 16 FINGERS.

CONNECT THE NUMBERS FROM 1 TO 16.

COLOUR THE DOTS AND FIND…

16 ON THE DICE. COMPLETE THE LAST SERIES.

COLOUR THE GROUP WITH 16 ELEMENTS.

NUMBER SIXTEEN 8

Addition and subtraction operations

Within the competences of numerical intelligence, the mechanism of calculation under stimulus and training operated in different contexts, but particularly in school, also evolves. At first, the child implements counting strategies that are based on very concrete and visible elements (use of hands, fingers, objects, symbolic representations, verbal aloud counting without the need for fingers, etc.). Subsequently, these strategies are abandoned in favour of others that are based on the mnemonic retrieval of counting results and typical procedures. Siegler and MitChell (in Lucangeli et al., 2003) highlight that the choice adopted by the child in performing the first calculations is an automatic process based on a level of ‘confidence’, considered as the threshold below which the child does not feel safe to give the answer to the arithmetic question. With experience and training, strategies are abandoned whereby, for example, one depicts the addends in a visible way by lifting the fingers or through a mental image. This is what happens when, for instance, children, making their first counts within ten, first raise their first hand and say 5, then raise their second hand and add another 5: at first a single counting of the fingers takes place, then at some point an automatic mnemonic retrieval occurs which makes you say 10 without raising your hands. Through exercise, the activation power of the retrieval process increases, and this type of strategy becomes the predominant and effective one. In calculating operations, there are different simplification strategies; while at the beginning of primary school the child counts all the time and individually in an addition by adding in the order in which the elements are presented, he/ she later starts counting from the largest addend in order to make less effort. It is important to combine the practical and experiential acts with the symbolic

Mathematics with 90 Minibooks - First Grade

part, both written and verbal: when the teacher adds elements, he says plus and writes the symbol + on the blackboard, when he takes them away, he says minus and writes the symbol -. Addition is followed by words such as ‘put together’, ‘join’, ‘add’. Subtraction, on the other hand, is followed by words such as ‘take away’ or ‘eliminate’, introducing, depending on the type of experience performed and always by means of example, subtraction as remainder or difference. Children learn calculation strategies by experimenting with them at school; the continuous comparison and verbalisation/demonstration of effective calculation strategies speeds up these processes and makes them consciously available for use. Therefore, in the first counting activities, whether in addition or subtraction, it is a good idea to use visual aids, materials, more or less active number lines. When children have reached the confidence level on basic calculations (e.g. number friends of 10) they will automatically retrieve the results from memory and can be quick and effective. Given this possibility in the stimulation of oral calculation, first with low quantities and gradually higher, from an educational-didactic point of view, research shows the possibility of developing mathematical skills precociously and of being incisive on the acquisition of these skills through an appropriate programme that integrates the concrete experiential part with the visual and verbal mnemonic part. Bibliography Lucangeli D., Poli S. e Molin A. (2003), L’intelligenza numerica – Volume 1, Trento, Erickson.

WORKSHOPS IN THE CLASSROOM…

AT THE GARDEN CENTRE Goal Introducing the notion of addition as an aggregation of quantities through an experience/reality task.

Materials and Resources a Sheets, pencils, colours, price tags and a black permanent marker.. a 1 euro per child or fake construction of coins. a Enlarged photocopy of 1 euro coin (front and back).

a Small piece of school garden or large box/pot with potting soil to make a mini school garden.

Unfolding Offer the children to make a small school garden and then experience going to a garden centre to buy seeds, plants and/or bulbs. If possible, ask them to bring one euro each and count the total amount. Introduce the euro as a basic monetary system: observe the one-euro coin with the children and make one out of paper, cutting out a greatly enlarged photocopy of the coin, adding colour or missing elements and joining the front to the back of the coin. Hypothesise together with the children what can be bought with the achieved sum and what is worth buying for the vegetable garden in the ground or for the vegetable garden in the pot.

Take the children out to the garden centre. Ask them to write down the prices of the products of interest on a sheet of paper and add them up one by one in the garden centre. Check that the total is not greater than the starting budget. Buy the products, sow or plant them in the school garden and follow the growth phase of the plants. You can also have the same experience in the autumn months with tulip bulbs. If the trip to the garden centre is not possible, the objective can be modified by buying other things or by involving the pupils in the buying experience with an online garden centre.

Additions and subtractions within 20

4 + 3 = ____ 4

2 + 7 = ____

5 + 4 = ____

THE HOPPING FROG

6 + 4 = ____ 8

1 LOOK AT WHERE THE FROG STARTS AND HOW MANY JUMPS IT MAKES FORWARD. SCORE THE RESULT. 60

1 + 3 = ____ 2

3 + 2 = ____ 3

Numbers over 20

Counting is one of the innate pre-verbal skills and has a very important function, because it enables a visual quantity to be associated with a verbal label that conventionally establishes a specific quantity. As children learn to count and visually quantify numbers, first up to 10 and then up to 20, a ‘picture’ is created in their minds of where the numbers that repeat in the units and tens after 20 are placed, allowing them to read and identify them immediately. When they are helped through an ordered arrangement of symbols (squares, balls or stars, etc.) perceptually grouped by ten, counting becomes intuitive. Once assimilated for the first 20, this process is like a repeating format that allows them to go on in counting. Their motivation is sparked and ready to progress further. The recognition of numbers occurs through their position and topological location. So why, if children are ready to count over 20, that is, to acquire a greater concept of visual quantity than the ‘syllabus’, should we hold back and extinguish the enthusiasm of this special learning period? At the basis of the verbal encoding of a number is the fact that each digit, depending on its position, takes on a name: the units; the teens, which contain the subcategory of ‘-teen’ (11, 12, 13, etc.); the tens (30, 40, 50, etc.).

108 Mathematics with 90 Minibooks - First Grade

Every element is characterised not only by the class to which it belongs, but also by the position it occupies. For example, the six has the sixth position at the level of units, in sixteen the sixth position at the level of teens and in sixty the sixth position at the level of tens. As we could notice from this example, there is a consistency in guessing the position of a number and associating its verbal label. Mathematics is permeated with challenging elements that lead children to go further and try to build solutions to new problem situations with their knowledge. It is an area that illustrates well Vygotsky’s famous Zone of Proximal Development. Addressing counting with a practical, experiential and visual approach encourages participation, engagement and the success of the activity itself.

WORKSHOPS IN THE CLASSROOM…

CELEBRATING THE FIRST 100 DAYS OF SCHOOL Goals Use a special occasion to start counting over 20. Work in an interdisciplinary way by introducing coordinates, the euro monetary system, learning about how popcorn are made and which vegetable they come

from, listening to, watching and understanding a movie, cooperative learning by doing

Materials and Resources a Cardboards. a 100 coloured paper circles of the same size. a Stationery material. a Corn seeds for making popcorn. a White envelopes.

a Colour photocopies of one and two euro coins and of five and ten euro notes. a Box cardboard. a Movies to be screened or watched on the Lim (e.g. One Hundred and One Dalmatians).

Unfolding In a first instance, have the children reﬂect in a circle about the school days already gone by and then prepare a poster on which to stick a dot for each school day that has been spent. Agree with the children whether to count the actual school days (excluding Saturdays and Sundays) or the days since school started. Arrange the circles in rows of 10 (if you wish, write a date inside each circle). If the 100 days have not yet passed, take a few minutes

each day with the class to count the days remaining, making possible connections with numbers already seen and perceived, and their location. Ask the pupils to organise a celebration for ‘the first 100 days of school’ and brainstorm any ideas that may emerge.

Numbers over 20

109

At this point perform a reality task in helping the teachers organise the party: first decide which activities to do. One proposal could be ‘cinema at school’. Now, if the children agree (“raise your hand” voting), you start to involve them by assigning roles: one group of girls and boys will take care of the writing/graphic part of the entrance ticket, which must be numbered and created following the example of real tickets. Another small group will take care of the classroom set-up and together with the adults decide how to arrange the chairs, which can be organised as in the cinema hall, ordered by letter and number. In this case, the coordinates of the place will have to be written on the ticket, and the children will have to be able to read them. Another group will take care of the scenic set-up of the box office: a frame made out of a box and placed on a counter bearing the name of the cinema chosen by the children. A group will work on cutting out the money with which to pay for the ticket. The same group will randomly prepare envelopes (wallets) with

110

Mathematics with 90 Minibooks - First Grade

figures (max. 10 euros) which will be distributed arbitrarily to the classmates in order to ‘pay for the cinema entrance’. Decide together which film to watch, voting on different proposals made by the teacher. Considering the theme of the party, you could propose One Hundred and One Dalmatians. Given the importance of the 100 days celebration, the children will prepare a simple guest letter for another class in the school. There is no cinema without popcorn: recreate, using digital visual sequences, the corn’s journey from plant to seed for popcorn. Ask the cooks to help you make the popcorn, or ask the children to bring them already made from home. The whole experience represents an important learning moment. On the day of the first 100 days of school, the whole class will be involved in some role (cashier, ticket taker, pop-corn distributor, chaperone to the right chair, greeter at the entrance of the cinema).

DO AS IN THE PREVIOUS PAGE.

NUMBERS OVER 20

OBSERVE THE QUANTITIES, THEN WRITE DOWN THE NUMBER. HOW MANY ARE THERE?

DO AS IN THE PREVIOUS PAGE.

HOW DID YOU FEEL?

HAVE YOU SUCCEEDED IN COUNTING THE NUMBERS OVER 20? CHECK YOUR WORK WITH A PARTNER.

BEWARE: ONE LINE CONSISTS OF 10 SQUARES, THERE IS NO NEED TO COUNT THEM EVERY TIME. 70

Problem solving competence

Problem solving within the skills related to mathematical thinking is identified as one of the eight key competences of the Recommendations of the European Parliament and of the Council of 18 December 2006 (2006/962/EC). In this document, there is reference to the ability to develop and apply mathematical thinking to solve problematic situations, authentic and meaningful questions related to everyday life: using mathematical models of thinking (logical and spatial) and presentation (diagrams, graphs, tables, etc.). The same conception is present in the National Curriculum Directions where it is specified that mathematical competence closely links doing with thinking. The pupil analyses situations to translate them into mathematical terms, recognises recurring patterns, establishes analogies with known models, chooses the actions to be performed (operations, geometric constructions, graphs...) and connects them in order to produce a resolution to the problem (National Directions, 2012).

But what exactly is meant by problem solving? The Pisa-Ocse Document defines it as a complex competence that is developed through a suitably organised collaborative methodology (division of roles, organisation and strategies) that contains identifiable key steps within which the teacher has key roles that change according to the phase we are in: – provide the input of a problem situation; – provide the information needed to start the search for a solution or ensure that the pupils have this information; – support the formation of a hypothesis and transformation into action;

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– check the effectiveness of the solution found by the pupils by analysing the error and, if necessary, reformulate the hypothesis. Another component that is involved in solving a problem situation is the use of creative thinking (which encourages people to find unexpected solutions) and critical thinking (which helps to evaluate the advantages and disadvantages in each solution hypothesis). In problem solving, the knowledge possessed by the pupil helps him/her to learn new ones (learning to learn). Problem solving is a reflective and critical attitude that confronts us with various everyday problem situations. What cognitive and metacognitive skills underlie problem solving? – Understanding the problem situation: the identification of key information, the cognitive representation of each piece of information and the integration of verbal and arithmetical information into a unified representation. This level implies a general understanding of the verbal language and of the specific mathematical vocabulary (each, all, etc.). – The representation: the connection and relationship between all the information in the crucial problem, to guide the subject towards the correct problem-solving strategy. – The categorisation of the structure of the problem: the recognition of similarities and differences in the problem-solving patterns that allow one to identify as similar problems that belong to the same category and which are solved in the same way.

- The planning of operation procedures: the metacognitive competence of sequencing strategically controlled algorithms, which enables the choice of the sequence of solution strategies. - Monitoring and final assessment (Lucangeli et al., 2002). Each resolution thus leads the pupil to build up his/her own competence and experience in the mathematical approach to problem solving, which then also has important effects in everyday life. At an early stage of learning to solve problems, it is important for children to focus on the problem situation in a concrete and controllable way (proposing problems with numbers that are accessible to them in terms of the operations they can perform). There are methodologies that facilitate this approach and can be summarised as: – Problem-situation teaching: working in groups with specific and precise tasks aimed at solving a real or realistic problem. The teacher prepares the materials, sets the tasks, assigns roles within the groups and starts the discussion through brainstorming in which ideas are shared. The final phase consists of comparing different solution strategies and implementing plans thanks to the teacher’s mediation. – Project Based Learning: an interdisciplinary activity related to educational co-design, in which students and teachers work together; it is usually a longterm activity, for which the teachers’ ability to work in teams emerges, each according to their own discipline and the links they can interweave with the others. – Ateliers or creative workshops: projects in in which creativity, manual skills and a playful approach are encouraged, including experimentation with technology in order to create concrete or digital objects that can be used in

everyday life. The setting up of the atelier implies the inclusion in the class of an external professional able to bring a different viewpoint to the offering of workshop-type experiences. – Authentic tasks: activities and proposals in which the pupil really gets involved and carries out useful and functional activities with classmates. The authentic task involves the realisation of something that can be material, organisational or digital (e.g. organising the school trip or a sales market for fundraising, etc.). It is worth specifying that the activation of the problem-solving process does not only occur in the mathematical disciplines, but is a way of thinking about the contents within any subject area that stimulates thinking with creativity and curiosity. Therefore, it should be an interdisciplinary input with which to address minor and major everyday questions within our classrooms. The competence of problem solving is complex, but the skills that compose it should be stimulated in primary school from the first cycle, to allow the skills to consolidate over time and during the workshop experiences offered at school. Bibliography and sitography Lucangeli D., Ianelli M., Franceschini E., Bommassar G. e Marchi S. (2002), Laboratorio logica, Trento, Erickson. https://www.metodologiedidattiche.it/problem-solving/

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THE TREASURE AT SCHOOL Goals a Reading/listening to and understanding simple indications concerning the location of certain clues in school spaces. a Activation of problem solving. a Knowing and finding one’s way around school spaces.

a Understand problem vocabulary (one, none, each, etc.). a Following a sequence of information and the clues they lead to. a Stimulate motivation to participate in a group activity, to collaborate and to learn.

Materials and Resources a Cards where clues can be written down. a A ‘treasure’ (a book or a game and some sweets, etc.).

a Teaching team. a Third grade pupils.

Unfolding This activity is carried out by a class of pupils from first year together with some children from the third grade who will accompany them on their discovery of the school spaces and who, like tutors, will help them in reading the clues. Prepare three different itineraries with some clues in common. The location of the treasure must be the same. Plan the clues according to the school spaces and always include maths words (one, none, each, all, some, etc.) linked to elements (so the children will have to count). Divide the class into three groups (one for each itinerary) and assign two

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third-class tutors to each one; they have the task of stimulating the children, supporting them in reading the clues and helping them in in case of difficulties.. Start the game and leave the children free to experiment and solve the problem situations. Intervene as little as possible and let the tutors help their peers. The group that first arrives at the treasure wins. Appreciate everyone’s effort by ending the game with a convivial moment of celebration.

... GOING TO THE BEACH?

... GOING TO SLEEP?

... GOING FOR A RUN?

... EATING DINNER?

... CYCLING?

WHAT DO YOU DO BEFORE...?

... EATING A SANDWICH?

... GOING TO SCHOOL?

CHOOSE AND COLOUR THE ACTION THAT IS DONE BEFORE THE ONE INDICATED. 8

Learning geometry

Not many studies in psychology have dealt specifically with how children recognise geometric shapes and develop learning in this particular area. It was not until Piaget and Inhelder (1979) that research began attempting to study the development of geometric cognition within the aspects of topological, projective and Euclidean relations. What has emerged is that the development of geometric cognition is related to immersion in specific contexts and stimulation, to education and to the experiences made in this field, rather than to the children’s chronological age. In the development of geometric cognition, visual-spatial skills play a special role and assume a different function depending on the used process: In visualisation, the child recognises figures on the basis of their similarity to known objects. “It is a square like the picture on the wall”: from this image-object the child derives the familiar visual elements that characterise the figure. The next step involves descriptive-analytical analysis, where the features of a figure allow its attribution to a certain category or concept (“It has four sides, it is a quadrilateral”; Hershkowitz, 1989). Visual-spatial ability incorporates the ability to interpret information about geometric figures and to manipulate visual representations of these figures themselves. In order to develop geometric skills, it is important to work on the visual aspects at a practical-manipulative level from the start and to leave verbal language

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in the background with the definitions that will come later. In the teaching of geometry, pupils must be stimulated to do and reason in relation to the information received through the visual approach in order to manipulate these images. Pupils are stimulated to work through comparisons and reflections on the visual information they obtain from the work proposals. The active methodology thus first passes through the experiential approach (e.g. folding, drawing, moving, searching for shapes in reality, etc.), and then leads to the acquisition of the regularities that characterise geometric shapes. The visual approach and visual analysis are functions that have already been stimulated by the workshop proposals and minibooks in the ‘Prerequisites’ section. Working in depth and focused on these aspects represents a fundamental starting point for the geometric area. Bibliography Hershkowitz R. (1989), Visualization in geometry: Two sides of the coin, «Focus on Learning Problem in Mathematics», vol. 11, pp. 61-76. Piaget J. e Inhelder B. (1979), La rappresentazione dello spazio nel bambino, Firenze, Giunti e Barbera.

WORKSHOPS IN THE CLASSROOM…

LINES HUNTING Goal Recognising the main geometric lines: curved, broken, mixed, straight, open, closed, dashed.

Materials and Resources a Paper tape. a 14 cards printed with the image of different types of lines.

a Pen, permanent markers or wax crayons. a Digital camera.

Unfolding Preparation: Create and print cards, each with the drawing of a type of line and the request to search for it. For example: “Look for a broken open line”, “Look for a mixed closed line”, etc. Then locate a wide area (for example, the foyer or corridor) and draw a series of different types of lines (like those used in the cards) with paper tape. It is important that they are large and do not cross each other, so that they are clearly visible and distinguishable. Then take a photo of each line to record the work in the exercise book and for self-correction at the IWB. Finally, prepare a digital sheet with photos of the lines and the corresponding definitions.

The activity: Divide the class into pairs and give each one a card. The children should read the handout, compare themselves with their classmates and identify their own line on the floor. Once they have found it, ask them to go over it with a felt-tip pen or wax crayon, following the line on the tape without taking their hand off. When everyone has finished, return to the classroom and display the digital sheet on the interactive whiteboard with the pictures and line definitions. Each pair will check their association. Finally, move on to the reworking of the activity in the exercise book using photos.

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• THE CURVING BRAIDED LINE THE SHORTEST PATH IS:

• THE CURVED LINE

• THE BRAIDED BROKEN LINE

• THE CURVING BRAIDED LINE • THE CURVED LINE

THE SHORTEST PATH IS:

• THE STRAIGHT LINE • THE BROKEN LINE THE SHORTEST PATH IS:

THE SHORTEST PATH

THE SHORTEST PATH IS: • BRAIDED CURVED LINE • DASHED STRAIGHT LINE 8

OBSERVE THE PATHS CONNECTING THE TWO CHARACTERS AND RETRACE THE SHORTEST ONE. WHICH LINE IS IT? CIRCLE THE CORRECT ANSWER. 87

THE SHORTEST PATH IS:

• THE CURVED LINE

• THE MIXED LINE

• THE BROKEN LINE

• THE CURVED LINE

HOW DID IT GO? I OBSERVE MYSELF Fill in the table at the end of each section: did the minibooks seem difficult to you? How hard did you have to work? How did you feel when doing the exercises?

How hard I worked from 1 to 5. Colour the box.

How hard did I work/was I in difficulty? Indicate on the bar from 0 (no difficulty) to 5 (very in difficulty)

Prerequisites (minibooks 2-10)

Counting, sets and relationships (minibooks 11-57)

Additions and subtractions within 20 (minibooks 58-69)

Numbers over 20 (minibooks 70-73)

Problem solving (minibooks 74-85)

Geometry (minibooks 86-90)

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Which emotions did I feel when performing these minibooks?

This volume offers 90 minibooks to be used in class to learn numbers up to 20 and the first operations in a fun, engaging and motivating way. Boys and girls will be able to build their own minibook independently from an A4 photocopy. It’s very easy: you choose a template and with a few simple folds it turns into a book. Each minibook includes exercises or pictures to be coloured in order to practise and learn; thus, while pupils build their own pocket library, they practise the didactic topics of the first grade. The book also contains methodological guidelines for classroom work, workshops for shared use of the minibooks, self-assessment sheets and the library card to mark the minibooks completed.

BUILD THE MINIBOOK: photocopy the page, fold and cut!

90 MINIBOOKS ON: • Prerequisites • Counting and numbers within 20

(Counting; Numbers within 20; Major/ minor; Groupings in tens; Friends of numbers)

• Addition and subtraction within 20 • Problem solving • Geometry.

Enter the QR code and discover how to build minibooks!