# What is the educational value of mathematics

## All children are math researchers

### Numeracy in kindergarten

Analogous to LiteracyLiteracy ||||| Literacy in early childhood and in the transition to school is a
Collective term for children's experiences and skills related to book,
The narrative, rhyming and written culture approach, which paves the way for storytelling, writing and reading skills in a child, is the so far lesser known numeracy approach. It stimulates children to experience numbers, quantities, forms and solutions and supports the natural curiosity for mathematical understanding of the world that every child carries within themselves.

Mathematics shapes our everyday life. Both boys and girls are fascinated by mathematical relationships long before school starts. They are enthusiastic about exploring numbers, shapes and patterns.

Four-year-old Anna, for example, hardly misses a chance to count trees, cars or points on ladybirds and can already count safely past 50. She is particularly fond of floral designs. Max, five years old, sorts the vehicles and building blocks in the group room according to colors, shapes and other details with enormous perseverance and knows almost all traffic signs. He knows exactly whether they look like a triangle, square or hexagon. He's not very good at counting, so he avoids using numbers. Fynn, on the other hand, already enjoyed arithmetic tremendously at the age of four. His favorite activities are sudokus, number puzzles and questions such as "How many different tasks can you find for the number 20?" It is easy for him to use all four arithmetic operations.

Just like Anna, Max and Fynn, every child has very special strengths and a wide range of mathematical skills that they confidently demonstrate. All children are autonomous inquisitive learners from birth and actively acquire learning objects (including mathematical ones). In addition to the child's independence, this constructivist perspective also emphasizes his or her natural curiosity and curiosity.

Early math education includes opportunities to:
• One-to-one mapping
• Counting and counting
• Recognizing the immutability of quantities
• Forming sequences
• spatial orientation
• Compare, sort and organize
• Discover numbers in the environment
• indirect computing
• Recognize patterns
• Capturing and perceiving
• Sizes (length, weight, time, volume, money, area)

### Mathematics in the daycare?

Mathematics in elementary school does not mean "teaching" children numbers or arithmetic before school. Early mathematics education is all about acquiring basic mathematical skills that enable children to discover mathematics in their world. This means that children reconcile processes in their everyday world with their previous mathematical experience. Mathematics in the elementary area means accompanying and developing the mathematical thinking of the children.

This is what it takes
• a professional pedagogical specialist,
• a stimulating learning environment,
• meaningful activities and interaction processes
• as well as a holistic and complex understanding of mathematics.

Mathematics is not a statically closed system of definitions, formulas and proofs, but a dynamically developing science in which problem-solving processes, mathematical activity and discovery-based learning play an essential role.

Mathematics is more than just dealing with numbers, counting and calculating problems. On the one hand, it includes content areas such as space and form; Number and structure; Measures, time and money as well as data, chance and probabilities and, on the other hand, includes essential process goals: being creative and solving problems; Communicating and arguing; Justify and check, as well as organize and use patterns. In addition, there are mathematical ways of thinking and acting, such as classifying and comparing. Mathematical learning in kindergarten should take exactly this complex and holistic understanding of mathematics into account.

### The numeracy approach

»Numeracy« can generally be translated as everyday mathematics and describes competencies that are required in order to react appropriately to mathematical requirements in everyday situations. This means that children study math in ways that suit the needs of their present and future lives. This of course also includes later challenges in mathematics lessons, but not primarily. The children gradually recognize and understand the role and importance of mathematics in their world.

Everyday mathematics is mainly acquired through play, imitation and personal activity. Numeracy competence is shown in the understanding handling of mathematics and in the ability to use mathematical terms as »tools« in a variety of contexts. This understanding of mathematics corresponds to the »numeracy approach« known in the English-speaking world and can be compared with the »literacy approach«, which has been discussed for a long time in the field of language and is increasingly accepted in practice.

The design and support of early childhood mathematical education that follows such an understanding should accordingly encompass three areas:
• Mathematics in everyday life,
• Math in the game and
• Mathematics within openly designed learning opportunities.

In the following, all three areas will be explained as examples.

### Mathematics in everyday life

Observations in day-to-day life in day-care centers impressively show that children often do mathematics in a natural way. Mathematics in everyday life can be found e.g. in various rituals in the daily routine, namely:
• count the children present in the morning circle,
• setting the table at meals,
• handing out sweets to birthday parties,
• go shopping and pay with money before cooking together,
• measure and weigh the ingredients when baking cakes,
• keep the hourglass time when brushing your teeth,
• determine the date and day of the week in the morning,
• open a door in the advent calendar,
• organize and sort toys when tidying up,
• Use space-position relationships when looking at hidden objects (next to, below, to the right of ...)
• watch yourself in the mirror while taking care of your body,
• Recognize the shapes of traffic signs while walking or collect, count and sort natural materials.

So the situations arise in everyday life. In the close connection with observation and documentation, it is now important as a pedagogical specialist to react appropriately and specifically for mathematics. The learning guides should e.g.

• develop a feeling for mathematical moments in order to be able to recognize certain situations as mathematically significant,
• accompany these moments in the everyday life of children,
• classify the children's mathematical ideas and recognize mathematically significant content in the children's utterances,
• ask mathematical questions and react productively,
• enter into a dialogue with children about mathematical issues and moderate co-constructive educational processes among the children.

Accordingly, children can be sensitized to mathematics in their everyday life in a variety of ways and be encouraged to mathematically "understand" their environment.

Materials with mathematical potential:
• Building blocks of different shapes and colors
• The same material in large quantities, e.g. 1000 ice cream cups, ice cream spoons, small wooden cubes, 1 cent coins ... (see K. Lee, 2010)
• Muggle stones
• Geo boards, tangram, pentominos
• Collectively collected buttons, clothespins, toilet paper rolls, yoghurt cups, screw caps from tetra packs, paper clips ...
• Laying tiles in different shapes and colors (triangles, squares, circles)
• Packaging materials
• Natural materials (nuts, chestnuts, stones, shells, cones ...)
• Game dice in different designs

Tools for research and discovery:
• various measuring instruments (scales, measuring cups, measuring tapes, ...)
• Drawing instruments (rulers, templates, compasses ...)
• mirror
• Pens (pencils, crayons ...)
• Material for exploring number spaces and for estimating
• calculator
• math games
• Building materials (cardboard, boxes, tubes ...)
• Reference works and non-fiction books
• Children's search engines on the Internet

### Math in the game

In the elementary sector, informal and non-formal, playful forms of learning predominate. Mathematical education becomes possible when the children - especially in free play - become mathematically active on their own, e.g. when they
• use numbers and sizes in role-playing games, such as »grocery store«,
• act with shapes and building blocks in building and placement games,
• roll dice and place pieces in board games,
• orientate oneself in movement and hide-and-seek games in space,
• use numbers in counting rhymes,
• create patterns or mandalas while painting,
• cut and fold something while playing,
• hop on tiles or run through (maize) mazes,
• use the seesaw or run a race,
• Pour sand or water into different containers.

Activities with a mathematical reference should first be perceived by the learning companions and, in due course (so as not to disturb or prematurely interrupt play activities), stimulate further thought and action processes through clever impulses or stimulating open questions. The children should always be given the opportunity to follow their own paths of thought and to deal with mathematical phenomena in greater depth. To do this, they need interactions with a person "who works with them to develop the mathematical principle behind their work" (Fthenakis et al. 2009, p. 47). Such support requires both professionalism and sensitivity.

Mathematical learning opportunities do not always arise (which they do not have to) and not every everyday or game situation is equally mathematically productive. Furthermore, differences between the children can of course also be determined.

Some children find numerous stimuli in their environment and become involved in mathematics on their own, without adults having to interfere. Other children, however, need more impulses, encouragement or suggestions to gain mathematical experience. Accordingly, child-oriented mathematics education should also include consciously initiated open mathematical learning situations.

### Mathematics within open learning opportunities

This in no way refers to course-oriented support and training programs in which, for the most part, a "schooling" can be observed that is deficit-oriented and sees the children as passive consumers. This does not correspond to the numeracy approach represented here. Rather, it is based on open work, learning within projects as well as in (learning) workshops or on special settings such as "Children invent mathematics" (cf. Lee, 2010) or the use of open mathematical games and learning fields (cf. Fuchs, 2015).

In addition to their professional orientation, they are all characterized by an orientation towards the child and its learning processes, which means that each participating child can contribute to working on a selected general topic according to their individual level of development and learning as well as their special interests. Such a perspective focuses on mathematical activities that allow the children to construct mathematical concepts, to reflect on them and to exchange ideas in a communicative manner. On the one hand, however, you can confront learning guides with enormous technical challenges and, on the other hand, demand constant reflection on their own learning experiences.

Open learning offers are differentiated from free play situations in that learning guides stimulate problem-oriented activities on a topic with previously deliberately selected material. Ideally, the children can choose between various open play and learning fields (also from other open offers). The topic itself should be characterized by a certain complexity and offer opportunities for cross-educational learning. Every child has enough time to deal with the material individually and with their own questions.

While mathematical learning processes develop spontaneously in free play phases from the actions and ideas of the children, they are initiated in open play and learning fields on the basis of observations through non-restrictive suggestions, impulses or problems. Such open forms of learning are characterized by a balance between learning on one's own path (self-education, self-construction) and learning from and with one another (social interaction, co-construction). The selection of suitable materials with a certain mathematical potential and a high incentive character for research, discovery and experimentation is of particular importance here (see box: Materials with mathematical potential). Open playing and learning fields should have the following characteristics:

Openness to diverse ideas and approaches
The children have the opportunity to work on the topic or the mathematical problem in different ways, e.g. to approach them intuitively, to try persistently or to proceed systematically (see problem-solving styles according to Fuchs, 2006). They can be active with material (= enactive), paint pictures for visualization (= iconic) or operate with numbers, shapes or figures in their heads (= formal-symbolic, cf. the three levels of representation according to Brunner et al., 1971 ).

Openness regarding creativity and the variety of possible discoveries
The subject, the material or stimulating impulses from the specialist should support or challenge the creativity of the children and thus enable diverse (mathematical) discoveries in connection with this. This requires an open and neutral attitude on the part of the educational specialist
ahead of children's ideas. The specialist is challenged to endure unexpected situations with regard to the children's original ideas (initially) and to question them at a suitable time or to reflect on them themselves and, ideally, to take up and expand them.

Openness to the choice of tools
The children should have various materials or aids available to them to work on the topic, which they can but do not have to use individually depending on the approach or ideas (see box: Tools for research and discovery).

An openness regarding the documentation and presentation of results
Depending on their age and development, the children should receive various suggestions for documenting and presenting their discoveries, e.g. painting or drawing, taking photos, entering them in tables, in the researcher's diary or world discovery book, setting up exhibitions, making posters, etc.

An openness regarding communication
The professional should encourage the children to communicate and interact with one another. This is useful during the research phase but also in the presentation and evaluation phase. Depending on their age, level of development and interest, the children can be encouraged to compare, organize, justify, argue and examine (cf. Rathgeb-Schnierer 2012).

An openness regarding the participation and length of stay of the childrenr
The children should often have the opportunity to decide for themselves whether and for how long to participate.

Despite the six aspects mentioned regarding the openness of mathematical playing and learning fields, it is advisable to adhere to certain framework conditions. This includes structure-giving and recurring processes (rituals) that offer the children the necessary security on the one hand and the greatest possible freedom on the other. For this, a high degree of flexibility and spontaneity as well as sensitive and appropriate time management are required from the learning guides. For the specific process of open play and learning fields, a three-part sequence of steps has therefore proven its worth: an introductory phase, a research phase and a presentation and evaluation phase.

Don't be afraid of math!

The attitudes and attitudes of the learning guides towards mathematics are formative for the mathematics education of children. In order to be able to accompany the mathematical learning processes of the children appropriately and authentically in everyday life, the educational specialists should
• don't be afraid of math yourself,
• show willingness to (re) engage with mathematics,
• Ideally, you have had your own positive experience with mathematical problems or be open to new mathematical experiences,
• Being able to experience joy, at best enthusiasm, in doing math and
• Take the time to feel the fascination that mathematics emanates in the children and perhaps in yourself.

### Conclusion

All children are math researchers in the sense of the numeracy approach described here. They should be given many opportunities to discover and experience mathematics in their own way and in exchange with others.It makes sense to establish an individually shaped relationship between the use of everyday and game situations that contain mathematics and the creation of open mathematical learning opportunities. The latter should - following the constructivist learning approach - be primarily child-oriented.

Early math education depends on both the professionalism of the educational staff and their attitudes and attitudes towards the child and mathematics. In order for all children to be able to demonstrate and develop mathematical skills, they therefore need professional educational specialists, a stimulating learning environment, rich activities and interactive processes that challenge them to solve problems with the help of mathematics and to explore the world.

Guiding principles of an early mathematical education based on the numeracy approach

Math play and learning activities should
• encourage children to discover and explore mathematics in their world and in their everyday lives,
• Maintain and strengthen children's curiosity and enthusiasm for mathematics,
• give the children opportunities to be active, creative and cooperative at all times,
• take place within stimulating mathematical learning environments,
• Strengthen the children's learning methodology and problem-solving skills,
• are always understood as self-directed constructive processes that enable the children to
• to realize your own ideas and procedures,
• include the individual previous experiences and interests of the children,
• encourage children to think and act mathematically,
• challenge children to communicate about math,
• be based on the basic mathematical ideas,
• be characterized by a balanced balance between the use of everyday and game situations containing mathematics and the creation of open mathematical learning opportunities,
• take basic mathematical skills into account in a variety of ways and
• help to develop and promote the potentials and talents of all children, including small math aces.

The book "All children are math researchers" contains numerous examples for the practical implementation of open play and learning fields with authentic in-house productions by children as well as more detailed explanations on the topic.
Mandy Fuchs: All children are math researchers. 224 pages, Kallmeyer 978-3-78004800-4 Euro 29.90

### literature

• Bruner J. S .; Oliver, R. & Greenfield, P. M. (1971): Studies on Cognitive Development. Stuttgart: Velcro Cotta
• Fthenakis, Wassilios E., Schmitt, Annette; Daut, Marike; Eitel, Andreas; Wendel, Astrid (2009): Creating knowledge of nature. Volume 2: Early Math Education. Troisdorf
• Fuchs, Mandy (2006): Approaches of mathematically gifted third and fourth graders in problem solving - Empirical investigations for the typification of specific problem handling styles - Talent research. Series of publications by the ICBF Münster, Vol. 4
• Fuchs, Mandy (2015): All children are math researchers - early childhood talent development in heterogeneous groups. Seelze
• Lee, Kerensa (2010): Children invent mathematics - creative activity with the same material in large quantities. Weimar et al.
• Rathgeb-Schnierer, Elisabeth (2012): Mathematical Education. In: Kucharz, Diemut u.a .: Elementarbildung. Pp. 50-85. Weinheim / Basel

First published in Betrifft Kinder 08-09 / 2015, pp. 22-27. Transfer with the kind permission of the publisher Das Netz

Last edited on: Friday, November 20, 2015 11:39 AM by Karsten Herrmann