## 1.1 – Non-Formal Mathematics

How can we define the concept of non-formal learning/education?

The term ‘non-formal education’ goes way back, to 1974, where Coombs and Ahmed used the term for the first time. Coombs and Ahmed, determined that learning and education could be equated, despite how, where, and when the learning process occurred (Mok, 2011). **Non-formal learning can be defined as a form of learning which occurs outside the classroom, separate from the formal school system**. In other words, outside the parameters of traditional learning institutions and structures. Thus, an educator together with a student, ‘hold’ their activities and learning outside the formal system. The terms community learning, adult education, lifelong learning can be used interchangeably in non-formal education (Khasnabis et al., 2010). Non-formal learning is about, recognizing how important learning, education, and training outside the standard educational foundations, are (*What Is Non-Formal Education?* 2015). Moreover, non-formal education is used in the procedure of lifelong education of individuals, as an addition, an alternative approach, or as a complementary learning method to formal education. It is not used as a replacement for formal approaches. Non-formal learning guarantees providing access to education to all individuals of all ages (*Non-Formal Education*, 2020). Thus, non-formal education and learning refer to a relatively methodical type of learning, which is not necessarily pre-planned. Similarly to formal learning, non-formal has as a target for learners and teachers to achieve specific learning tasks (Mok, 2011).

As we all know, there are different methods when teaching different learners, and different topics that can be taught in a different setting with a different approach. By introducing non-formal learning, teachers and students become equal. In other words, there is no need for students to call the teacher ‘Sir’ or ‘Miss’ and a student’s schedule is as important as the teacher’s schedule. Non-formal learning focuses on the empowerment of a learner in achieving more and challenges both the student and the educator to come up with a variety of ideas, to listen as well as argue with each other (Spiteri, 2016). It is important for all of us, to recognize non-formal learning and education as an indispensable part of the process of education and acknowledge the influence non-formal learning can make in educational organizations. Non-formal learning can be considered as an essential part of the concept of lifelong learning and can ensure that learners preserve the skills and abilities which are needed in adapting to a repeatedly changing environment. The collection of non-formal teaching tools and different learning structures can be seen as an innovative and creative alternative to the classic and traditional teaching schemes. The benefits of introducing non-formal approaches include :getting a chance to experiment and take responsibilities through engaging in non-formal education; being able to develop enthusiasm and curiosity towards the learning process; learning to work together in a team, and build decision-making skills. Furthermore, an educational process based on non-formal learning can help in the development of personal and social skills, through the experimental process (*Non-Formal Education*, 1999). Through the development of the personal and social skills of each individual, teachers can help the children boost their self-esteem. A healthy critical attitude of the surroundings can be developed by each individual’s learning and ‘discovering’ capacity (“What Is Non-Formal Education and Why It Is Important,” 2018).

## 1.2 – Mathematics in Kindergarten

Kindergarten math has as a main goal to prepare preschool children for the mathematics they will come across in first grade. Preparing children for this next step requires much more than just handing children worksheets and books. Young children will start to recognize and understand abstract concepts and symbols after they experience ideas. These experiences will integrate their senses by experimenting and making observations which allows them to examine a topic even further. Children learn math by grasping the concepts at their own pace. It is suggested that children return to previous tasks and try to solve them using a distinct way. Moreover, in order for preschool children to understand new math concepts and abstract ideas, they need to practice using concrete objects, such as blocks, sticks, counters, etc. Teachers should ensure that before being able to use the mathematical methods for guided math activities, children take a sufficient amount of time. Using mathematically based activities and games is a good opportunity for children to build a math vocabulary as well as connect mathematics to their everyday experiences (“How to Teach Kindergarten and Preschool Math,” 2019)

### 1.2.1 – The playful approach to math Modern approaches in teaching mathematics:

The concept of play is usually considered as a less academic activity and is frequently limited to young children and students when learning mathematics. On the other side, mathematics is known as a disciplined, logical, and boring subject–as considered by students most of the time. However, since learning through play is an acceptable pedagogical approach in kindergarten, educators should set up their teaching on this learning approach. While playing, children may reach a state called “flow”–an indefinable state of mind where time seems like disappearing when being deeply focused on what they are doing. To get to this optimal state, children’s mind requires freedom of play but also, a reaction from teachers to children’s ideas, and guidance through notions such as numbers and counting. Offering direction while creating freedom allows productive play which helps children open their minds and understand better the more difficult mathematical concepts. We can conclude that the amount of play in mathematics, a “serious” abstract subject, is inversely proportional to the age of the children/students. This means that, the older the children the less the amount of play in the process of learning mathematics. Nonetheless, this does not have to be the case (Oldridge, 2019)

### 1.2.2 – Improving the teaching of mathematics in kindergarten:

Teachers may use as a starting point for their preparation and decision making, their students capabilities. Resources that can be used as a starting point include mathematical reasoning, language, listening and reading skills ,as well as being able to cope with complex concepts. For children to develop context-related strategies, they need to envision the situation/events where the problem is set in. In this case, they can use their knowledge and experiences as the key basis for the aforementioned development of strategies. Instead of rejecting an ‘alternative interpretation of mathematical ideas, and categorizing it as “wrong thinking”, teachers can alternatively view them as an ordinary and necessary step in the learner’s development of concept formation. Teachers can provide several opportunities for them to learn from their errors. For instance, by formulating a discussion where children’s attention focuses on the difficulties that have appeared or by asking the children to share their thinking, their understandings, or solution strategies to compare and re-examine their solution. Kindergarten children develop ideas about mathematical concepts by engaging with math-related tasks, which helps them discover the scope of making sense of mathematics. Furthermore, learning experiences allowing original thinking, encourage students to be capable learners of important mathematical concepts. Instead of encouraging single-minded tasks, teachers should offer opportunities to children to struggle with concepts so that a variety of increasingly sophisticated mathematical processes is developed. Children should be encouraged and supported by teachers in creating connections between the distinct ways of problem-solving, between mathematical concepts and representations, and between mathematical concepts and everyday experiences(Anthony & Walshaw, n.d.).

It is necessary for educators to understand how the learning of mathematics is endorsed by young children’s engagement in play, as well a show children can support best this learning. For example, a child’s learning can be maximized if adults help them reflect and represent on their everyday experiences. Moreover, ‘learning through play’ is considered as a fundamentally good pedagogy in the learning of mathematics for young children. A “good” mathematics pedagogy includes a math-talk promotion, a productive disposition development, mathematical modeling emphasis, the use of tasks that are cognitively challenging, and a formative assessment. A good mathematics pedagogy can be endorsed when teachers engage and encourage children in activities across distinct areas of learning through a variety of mathematically-related activities. The activities should be child-initiated, in other words, rise from a child’s interest, concerns, questions, and everyday experiences. The features describing a good pedagogy require a deep understanding and should advise the distinct ways on how educators can engage kindergarten children in such mathematically-related activities such as play, project work, story reading, physical, and art education. When teachers focus on a child’s mathematical sense-making, can best realize the potential of such engaging activities for the development of mathematical proficiency. Furthermore, one of recreaMATHS aims is to maximize opportunities that engage children by providing a range of tools, such as digital tools to help in facilitating learning (Dooley et al., n.d.).

The foundation of a child’s very first mathematical experiences is a child’s play and its interests. Most children’s preschool mathematics learning takes place during play or playful activities that deliver the key contexts of mathematics. During the children’s free play, they may spontaneously engage in several mathematical concepts, which some of them can be quite advanced at some levels. Children may even play with math itself. The idea of the play depicts a context where children are able to reflect on their former experiences, connect experiences together, represent the experiences in distinct ways, explore different possibilities and create meaning out of them. Mathematical thinking has a strong connection with these procedures and may be inspired by children’s experiences. Mathematical language and notions are encouraged through the context of the play. Through this context of play, children can discover discrete mathematical ideas while teachers can be provided with a context that develops and supports the children’s ideas. Teachers–adults in general–have a critical role with the children they interact with. Their role is to help children reflect, as well as help them talk about their experiences while playing, to help them maximize their learning potential. This promotes and encourages children to think mathematically including mathematical learning. From this point of view, sensitive structuring of a child’s play as well as learning through play can be both seen as an important element to a good mathematical pedagogy of young children (Dooley et al., n.d.)

Key points summarised:

- For educators to comprehend the way the learning of mathematics is promoted through the engagement of young children while playing, as well as how teachers can best support this specific learning.
- Good mathematical pedagogy features can be acknowledged with reference to strong principles which relate people and relationships, and a learner’s environment with the learner
- The features along with the principles of a good mathematical pedagogy for kindergarten children (aged 4-7 years) relate to a variety of early educational settings which, on their side, are significant in the promotion of continuity in the pedagogical methods across all distinct settings (Dooley et al., n.d.)

### 1.2.3 – An example: The perception of mathematics in a Swedish preschool:

In 1998 the first curriculum-based iteration was produced, and since then the Swedish preschool curriculum has had clearly specified objectives about mathematics. Nonetheless, the aims are not objectives to be reached by children, but rather to be strived by the preschool itself. The emphasis of mathematical concepts was for teachers and preschool staff in general, to notice and consider mathematics in the everyday circumstances of preschool. In other words, to better comprehend their own views about mathematical concepts and to assess the contribution offered by the preschool on children’s mathematical development (Johansson, 2015)

Examples of mathematical concepts taught in a Swedish preschool–a curriculum change :

The main aims of the preschool are to ensure that each of the children:

- Develops an understanding of concepts such as shapes, space, direction, and location, as well as the basic properties of quantity, order, sets, and number and lastly notions such as direction, change, and time;
- Develops the ability to investigate using mathematical concepts and reflect over, as well as test, the different solutions of problems which were raised by themselves or by other children;
- Develops the ability of expressing, examining, distinguishing, and using mathematical concepts and their interrelationships;
- Develops math – related skills ‘in putting forward and the following reasoning

In order to develop further the debate of ‘what mathematics for young children in preschool should be’, Johansson continues with linking social practice to cultural practice. The auth or discusses that mathematical activity and mathematical practice are both cultural. Thus, the mathematics that takes form in this culture can be derived by dealing with quantities and the spatial understanding of the environment. A cultural understanding of mathematical concepts can be connected to the children’s perspective in developing a mathematical understanding (Johansson, 2015).

### 1.2.4 – Mathematics in a cultural context:

We can say that the learning of mathematics, explicitly, is formed by the shared understandings of one’s culture. In other words, for a child to comprehend mathematics and know how to express the knowledge gained in the classroom, needs to explore all distinct paths. Moreover, language and culture can play a tremendous role in the way a child learns to count ( Making the Connection between Culture and Mathematics Northwestern University | School of Education & Social Policy , n.d.) .

What are Art and Culture and how can it be linked with mathematics?

Culture can be defined as a set of ideas, customs, and social behavior as well as the characteristics and knowledge of a specific group of people ( What Is Culture? Definition, Meaning and Examples | Live Science , n.d.). Most of the educators teaching subjects such as mathematics, assume that mathematics is a non – cultural subject. However, mathematics is not a culture – free pedagogy. In some way, mathematics can be considered as an essential component of all cultural contexts (d’Entremont, 2015) . In this case we will use as an example of culture, art – defined as a huge subdivision of culture, broken down into several creative activities and disciplines (“Culture and the arts”, 2019) . Dooley et al, examine the learning of mathematics through the arts, which as mentioned above, is a vast subdivision of culture. They discuss specific ways in which links between mathematics and the arts (visual arts, music, and drama) can be established. Some examples include the following. Teachers can use the rich context of culture in music to develop a child’s mathematical concepts and language. Through the classification of sounds and movement, children can enhance their mathematical skills and understanding. Moreover, a strong link between timing, order, rhythm, and beat of the music, and the concepts of mathematics such as sequencing, counting, can be identified. By engaging children to music, helps them in the development of other attitudes and skills important to mathematics. This includes concentration, perseverance, creativity, sensitivity and self – confidence towards other individuals

Moving on to visual arts and mathematics. Both shapes and patterns are important features of mathematics and visual arts. A significant aim of the visual arts curriculum is the development of a child’s awareness of, enjoyment to and sensitivity of visual, tactile, aural and spatial environments, while also important is the awareness of the spatial and visual qualities in the environment for the mathematical understanding. Likewise, the enhancement of a child’s ability to apply mathematical knowledge in real 13 life and in the environment is as significant (Dooley et al., n.d.). For instance, in France, there is an increasing tendency to have distinct disciplines interact on the same subject [e.g., teaching Math, Physics, and Technology while creating/building a weather station]

Mathematical notions based on cultural perspective as well as art allow children to appreciate and reflect on their own culture while also, appreciating the traditions and culture of others. To benefit from these rich cultural experiences implies that students are exposed to several experiences as well as cultural resources. Kindergartens could set as an aim, to help children learn about their culture and the culture of others through activities which determine the connection between mathematics and culture (d’Entremont, 2015) . To expand a child’s, an educator’s and the parents’ views about preschool math, is not a straightforward procedure and should take time to accomplish this process (Johansson, 2015).

Mathematics is a way of thinking and understanding our livesand our world. It is a setof tools, a pair of glasses that we can use

Hyde & Bizar(1989)

There is a continual increase in the kindergarten mathematical education as well as an increase in the early childhood education in general. Several research findings confirm that the aching mathematical concepts in kindergarten education can facilitate the transition from non-formal kindergarten mathematics to formal elementary mathematics by providing cognitive foundations in children’s capability to become skilled in the systematic teaching of “real” mathematical concepts in advanced educational stages. The low performance of children internationally in the subject of mathematics, reflects the need for a distinct method to teaching the mathematical notions, being different from the traditional method to learning and teaching mathematics. Kindergarten children come to primary school with knowledge based on informal numeracy which can be extended, developed and enhanced through appropriately designed learning activities. Subsequently, kindergarten educators can make the teaching of mathematics more interesting by emphasizing in creating an innovative and different learning environment (Papadakis et al., 2016).