HANDBOOK FOR TEACHERS AND PARENTS
• This handbook will help parents and teachers develop understanding of mathematical concepts with preschool children and prepare them for school. • The book contains a variety of examples and activities that will help pre-schoolers adopt basic mathematical ideas. • There are special illustrated activities intended for young, middle and older kindergarten-age children within each chapter (sets, natural numbers, halves and quarters, spatial relations, geometric shapes, mass, temporal orientation and measuring). • This handbook also contains instructions for ten mathematical games, ideas for various maths-related educational tools, as well as activities for encouraging the mathematically gifted.
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* Награда Европског удружења издавача уџбеника – Сребрно признање за најбољи европски уџбеник у категорији књига за предшколце у 2012. години
Mirko Dejić
ОЛАКШАЈТЕ ДЕЦИ УЛАЗАК У СВЕТ МАТЕМАТИКЕ!
Pre-schooler in the World of Mathematics
HANDBOOK FOR TEACHERS AND PARENTS
4 M i r k o D ejić
5
6
Pre-schooler in the World of
MATHEMATICS Activities ren for child r aged fou d to six an a half
About the Author Mirko Dejić, Ph.D. is a professor at the Teacher Education Faculty in Belgrade. His scientific work deals with building foundations and developing mathematics methodology at preschool and elementary school level. He authored several university and elementary school textbooks, as well as a number of popular mathematics books for both children and adults. He published around 150 scientific papers. In 2012, his books Matematika kao igra 1 and 2 were awarded by the European Educational Publishers Group as the Best European Schoolbook in the category of books for pre-schoolers.
Contents Foreword
4
Sets. Basic quantitative concepts
7
Young age
7
Middle age
16
Older age
19
Natural numbers
23
Young age
25
Middle age
27
Older age
35
Halves. Quarters
45
Older age
45
Spatial relations – position Young age
49
Middle age
52
Older age
54
Spatial relations – size
2
49
58
Young age
58
Middle age
64
Older age
67
Understanding geometric concepts
71
Young age
71
Middle age
74
Older age
77
Understanding the concept of mass
83
Young age
84
Middle age
85
Older age
87
Temporal orientation
89
Young age
90
Middle age
91
Older age
92
Concept of measuring Older age
95 96
Games for understating mathematical concepts
107
Activities for developing mathematical giftedness and skills
117
Educational tools
125
References 131
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Foreword Sense of mathematics is present in children from a very young age. Even at five months old, a child can differentiate between sets containing up to three elements, as well as perceive addition and subtraction of elements. Even though at preschool level children do not learn mathematics systematically, within a curriculum, they nevertheless encounter mathematics in everyday activities. Maths is an unavoidable part of everyday life for children. Understanding of mathematical concepts is developed to the level of intuitive image. Children learn spontaneously what is up, and what is down, what is behind, and what is in front of. They encounter terms such as left, right, in, on, outside, they perceive how much of something there is, they learn counting, series, measuring, they observe that something resembles a two-dimensional geometric shape (square, rectangle, circle, etc.) or a three-dimensional geometric shape (cube, cuboid, sphere, etc.). Likewise, they are able to single out objects according to their properties and compare them based on their differences, divide in equal parts and note where there is more and less of something, etc. Even though this is still basic mathematics, it creates the need for more organised activities related to developing mathematical concepts at a preschool level. This should under no means be understood as imposing mathematical knowledge on children. Parents and teachers should follow children’s interests and always work in a way that makes sense to children. When an opportunity arises for a child to count, measure or divide something, teachers and parents should guide them so that the appropriate mathematical concept could be explained to them in a clear way – it should leave an image in their mind based on which they will later be able to recognise the same concept in other situations. Nurturing comprehension of mathematical concepts in children is enabled by their continuous cognitive development. One should keep in mind that children younger than six cannot solve logical problems unless they are remarkably gifted. Interest in maths should be encouraged from the age of six to seven by giving the children more logical problems to solve. This is the way to facilitate further understanding of mathematical concepts. The content of this handbook relies in great part on the information provided in Methodology of Developing Basic Mathematical Concepts (“Metodika razvijanja početnih matematičkih pojmova”), published by the Teacher Education Faculty in Belgrade.1 The handbook in front you has been 1 Dejić, M. (2009). Metodika razvijanja početnih matematičkih pojmova. Belgrade: Praktikum. 4
written in simple language, so other than being a handy guide for handson work related to preschool mathematics for teachers, it can also useful to parents and other adults that are involved in raising children. With the aim of facilitating the communication with the readers, this book exclusively uses the term educator, which implies parents or any other adults who are involved in working with children. This handbook contains a number of activities that you can do together with children. They are marked with this symbol: If the activities are intended for kindergarten setting, parents can adapt them to their environment. The contents of the handbook have been structured so that the answers on what and how to work with children to develop mathematical concepts are provided in accordance with age: • Young age (three to four years old) • Middle age (four to five years old) • Older age (over five years old – preschool period from five and a half to six and a half years old) This structure should not be taken literally and abilities of each individual child should be respected. If the text states that a child can do something at age four, that does not mean that some children will not be able to do the same at an earlier age, and others at a later age. The handbook primarily illustrates the order of development stages.
Author
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One, many, few
Sets. Basic quantitative concepts Young age
– three to four years old
Grouping objects, classification and seriation
P
racticing how to group objects based on their properties enables children to learn how to carefully observe the world around them, to analyse and compare attributes of different objects. This is how they form the logical operation of classification.
Children aged two to four should be given various objects to hold and explore, to note their attributes, then to perceive those same attributes on other objects and note their differences and similarities. Observing similarities in objects helps children group objects, i.e. their classification. At the age of three, children single out colouring pencils from a group of objects, dolls from the toy basket, they put smaller balls in one basket, bigger ones in the other, etc. A child should always verbalise their actions and explain which criteria they are using to single out objects. Between the age of two and four, children do not yet possess the ability to abstract, i.e. they cannot separate relevant information from irrelevant. So instead of grouping objects based on their similarities, they group them based on a certain sense of belonging. So they will place balls next to figurines of football players because they are the ones who kick the ball; they will place their doll’s accessories next to their doll, etc. This level of group-
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ing is called figural collection. At this age, children should be encouraged to group objects based on their properties: same shape, same colour, i.e. to form non-figural collections, then to note the same property on other objects. Around the age of three, children are already able to group objects based on one criteria (shape, colour, etc.). Around age four, they can classify based on two criteria. One should keep in mind that some children will be able to classify based on more than two criteria, and they should be allowed to develop within their own abilities.
For example, one productive activity is to have a child single out all red circles (colour and shape) out of an attribute blocks box, or all small figures of the same colour (size and colour), or only large triangles (size and shape). If we want the children to classify objects based on three criteria, we can ask them to single out all the big yellow circles (size, colour and shape).
Attribute blocks
Children can also perform seriation based on a certain property: they can assort truck toys based on their size (in ascending or descending order), line up pencils based on their length, colour or shade, etc. The number of objects for children at this age should not exceed three. Seriation represents the basis for understanding the concept of set, and later the concept of number.
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One and many, many and few
T
he period when children are between three and four years old can be called pre-numeric. In order to develop the basic quantitative concepts, it is necessary to implement practical use of objects that are similar and varied. The initial understanding of the concept of one and many begins to form in children at a very early age. A child adopts the term one as early as one year old. When you tell them:“Show me one finger” or “Throw me one ball”, they know exactly what they are being asked to do. At this stage, a child exclusively associates the term one with objects. Furthermore, during the second year, a child can differentiate between concepts of singular and plural. They are able to understand when we tell them “Show me the doll” or “Show me the dolls”; “Place the block” or “Place the blocks”, etc. Exercises in which children are supposed to gather objects in a group and single out certain objects from that group can help them perceive the individual and the whole, which is composed of those individual parts. This will later help children to understand the concept of number as a set of ones. Concept of quantity is introduced through games in which children are asked to compile a group of objects with similar properties or disassemble a group into its individual parts. During these exercises, children should come to conclusion that a group (set) is composed of individual objects (elements) and learn how to single out individual elements from a group and determine how one element is a part of a whole. For example, an activity in which a child can perceive the relation between one and many, using the words: many, one, by one, none, more, less can be organised in the following way: Place various toys on the table, for example a hen and chicks, and cover them with something. Have children gather around the table. Uncover the toys and begin talking to the children:”What is this?” (show the toy – hen). “How many hens are there? Who else is here?” (show toys chicks). Emphasise how there is one hen and many chicks. Take one chick and ask the children:”How many chicks do I have in my hand?” (“One chick.”) “How many chicks does the hen have?” (“Many.”) The children are expected to provide correct answers and the educator should stress the words one and many.
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The game continues by having the children take the chicks one by one, until there are none left on the table. The educator is emphasising the words one by one and none. It is emphasised that children took all of the chicks one by one, and that there were none left on the table. Then they talk about how there is going to be many chicks on the table once again (there will be many after each child returns the chicks one by one). While the children are returning the chicks one by one, talk about how, for example, Ben put one chick on the table, Lucy put another one, Alex another one and now there is more and more chicks. Finally, everyone comes to the conclusion that, once they put chicks back on the table one by one, there are now many chicks. “How many hens are there?” (“One.”), “And chicks?” (“Many.”), “What is there more of?” (“There are many chicks and one hen.”) “Many is greater than one, and one is less than many.” You can extend the activity by having several children take a chick each, then asking:”How many chicks did Lucy (Ben, Alex) take?”, “How many chick do the other kids have?”, “How many chicks are there on the table?”, “Is there more chick’s on the table or in Ben’s hand?”, etc. Once a child adopts the terms one and many, we can teach them how to make a difference between two groups of objects with a few and many objects. For example: one group has three object, the other has five, ten or fifteen. We will say how there is a few objects in the first group and many object in the second group. In order for a child to clearly differentiate between the quantitative concepts of one, many and a few, adults should ask them to pass one sticker, to give many stickers, to take a few stickers, etc. The following activities can be useful as well: Place as many toys in front of you as there are children in the group. The toys should be the same. Give one toy to each of the children, then ask them to return them. The focus should be on showing the children that a group of objects becomes smaller if objects are taken out and bigger if objects are added to it. You can tell the children to take a toy one by one. When there is only one toy left, you can ask: “How many toys are left?” (“One toy.”) “How many toys did we take?” (“Many.”) When the last toy is taken from the group, you can follow it up with a question:“How many toys are left?” (“None.”) Afterwards, when some children return
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their toys one by one, you can ask questions such as:“How many toys have been returned?” (“A few.”), “How many toys still haven’t been returned?” (“Many.”)
Grouping objects with same properties can be practiced through the following activity: There are two differently coloured hoops and several teddy bears in the room. The children are asked to place one toy in the blue hoop and many (the rest of the teddy bears) in the red one. That way, children are partitioning the group made up of similar objects, which will lead to a more abstract understanding of the concept of set. When children intuitively come to the conclusion that a group consists of certain elements, you should find groups of similar objects in their surroundings and point to individual objects (one) and a whole (many). For example, a child can notice that one hand has many fingers, that room has many chairs, etc. You can also observe objects in the room and draw conclusions that there is one apple, many dolls (not more than four), etc. Other than questions related to quantity, the educator can also ask the child about size, colour, amount and other properties. Developing basic concepts on quantity continues with disassembling sets into a few and many objects. For example, a group of cards can be split, so a few cards are placed in one basket and many cards are places in another. Exercises that are related to terms one, many and a few can also be continued in other variations:”Put many chips in the blue basket”, “What has a few trucks and many dolls?”, “Bring me one pen and a few erasers”, etc.
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HANDBOOK FOR TEACHERS AND PARENTS
4
5
6
* Награда Европског удружења издавача уџбеника – Сребрно признање за најбољи европски уџбеник у категорији књига за предшколце у 2012. години
Mirko Dejić
• This handbook will help parents and teachers develop understanding of mathematical concepts with preschool children and prepare them for school. • The book contains a variety of examples and activities that will help pre-schoolers adopt basic mathematical ideas. • There are special illustrated activities intended for young, middle and older kindergarten-age children within each chapter (sets, natural numbers, halves and quarters, spatial relations, geometric shapes, mass, temporal orientation and measuring). • This handbook also contains instructions for ten mathematical games, ideas for various maths-related educational tools, as well as activities for encouraging the mathematically gifted.
Pre-schooler in the World of Mathematics
HANDBOOK FOR TEACHERS AND PARENTS
4 M i r k o D ejić
5
6
Pre-schooler in the World of
MATHEMATICS Activities ren for child r aged fou d to six an a half
About the Author Mirko Dejić, Ph.D. is a professor at the Teacher Education Faculty in Belgrade. His scientific work deals with building foundations and developing mathematics methodology at preschool and elementary school level. He authored several university and elementary school textbooks, as well as a number of popular mathematics books for both children and adults. He published around 150 scientific papers. In 2012, his books Matematika kao igra 1 and 2 were awarded by the European Educational Publishers Group as the Best European Schoolbook in the category of books for pre-schoolers.