A case study
This is a case study related to street children who used to live in the street. The objectives of this study are; to identify the basic mathematical concepts of street children, and to investigate the process and sources of construction of these mathematical concepts by street children.
Over all there are a lots of street children in Nepal, among them the researcher selected only seven street children of Kanlaki, with different prospective. This research is limited into the city’s environment which cannot generalize all over the Nepal, due to their life style and sources of gaining knowledge of the city. The researcher collected the name of total street children of Kalanki camping and list out by identifying each member of camping then selected them on the basis of their work, caste, and occupation. For this research the researcher used the observation note and interview schedule to collect the data. For the data collection the researcher visited the street children of Kalanki camping regularly 30 days, to adapt their opinion and behavior, learning sources towards the mathematical knowledge. For the data analysis the researcher categorized the respondence response, coding them with mutual response, meaning making, and attaching with theory to interpret the data. The data were analyzed and interpreted for the following issues under different headings.
· Basic mathematical concepts of street children which refers to the concept of addition, subtraction, multiplication, and division operation used by them in their daily life.
· 
Process of construction of basic mathematical concepts by street children which is defined as the learning process of such mathematical concepts of them without formal school education system of Nepal.
Process of construction of basic mathematical concepts by street children which is defined as the learning process of such mathematical concepts of them without formal school education system of Nepal. So far, the researcher has not explained the case respondent’s background in details. Thus, in the following paragraph, they are introduced in terms of their socio economic and cultural background in the society and how they join in the Kalanki campaign. Some street children whom the researcher observed and interviewed are associated with a shelter and thus they were from different cultural background that were leading their lives, and learning mathematics in different ways according to the context. In this regard, the researcher had tried to discuss their individual’s stories obtained. After the observation and discussion with them, the researcher found that they have learnt mathematics by different things and their ways of learning have both similarities and dissimilarities in many respects. The researcher briefly tried to discuss their individual stories by using the information that was provided to researcher by these children as below.
Respondent A of this research was a Chamar. The caste Chamar has considered a so-called untouchable caste in our society. This is not really true in recent days, a disadvantaged caste of Nepal, and living all over the country. No written document was found where they are originated from but they are doing the same profession as “Lohar” who used to sharper the knife and other cutting metallic equipments. From the ancient period and some of them are engaged in farming. Chamar is dominated by other so-called higher caste in the society but they are trying to uplift themselves from the various suppressions of the society. They are living in close contact with Brahmans and Chhetri and, following the similar traditions and customs. They were more alcoholic previously now changing their habit. As their main occupation is – Lohar, their child also learn the skill from the very young age. Their parents teach them by giving smaller job like making different shapes as half circle, circle of this iron. When children get free time to play with hot iron then they made such circle simply to play the games by knowing that the circle moves in the road with the help of sticks. Children learn these skill by observing the players of this moving games and with the help of parents.
Respondent A was born in Sarlahi district, And Living in Kalanki camping till now. He has two brother and two sisters. His father work in farm field near to his village and mother works as housewife. He is the elder son of his parents. Due to the poor economic condition his parents has small house with four Aana field. He had interested to study in school. However, the poor economic condition of his parents, he moved to Kathmandu to do something, that’s why he works in Boutique centre. He earns Rs 100 per day when he moved to work; one of the secret things is that; he is here due to pay Rs 18,000 loan of his parents. He has a little bit knowledge of mathematics. He calculates with his own ideas. The researcher described his calculation method in details after the short history of selected street children.
A Second respondent named B was Pariyar. Let’s moved to the caste Pariyar- is also one of so-called untouchable and disadvantage caste of Nepal. The profession of the Pariyar is especially- tailoring- from the ancient period. They are also alcoholic earlier, now they changed their habit, slowly begun to joint their children at school, some are found in government office and teaching profession. The government and other project are lunched their programs to uplift Pariyar as well as other disadvantaged group. Pariyar are famous to play drums in the social and religious ceremonies such as marriage, naming and other religious rituals. Pariyar, whom the researcher has taken for the research purpose, does not have idea to play such instrument and sewing skill too. He was born in 2060 BS in Haripurba, 9 chamar tol, Sunsari. His father works in Panjab. His mother also engage in house work, similar as others, who dead after the birth of four children in 2067 BS. After the death of his mother, his father got married with other woman, that’s the reason he got of step mother. Pariyar is a healthy boy. His brother is engaged to sell “ pani puri”. His parents have only two Aana field in village area. They lived in hut. Due to the domestic violence he moved to Kathmandu by truck in the mid night, and then he joined with street children in Balkhu. He told to the researcher that he did not get any food till 6 pm in Balkhu of his first day of Kathmandu, and then he was lying in the street and drunk the water near to small café, and beg maize near to maize seller and got a relief. Then he started his own business, by collecting the leaf of Gaja near to the river and then made it dry to the purpose of smoke and sold it the people with a cheapest price. With this job he collected a little bit money only for survive, and got the knowledge of calculation.
In this research two respondents are from Magar community. Respondents C and D both are Magar, The caste Magar is one of the oldest tribe of Nepal. And Magar’s are scattering across the length of the country and also some part of India. The habitant of these people is the Terai region, where the hills commence and the small lower hills of the Mahabharat lekh, and some places in the high altitude mountainous area. The Magar tribe has been divided into basically seven sub castes: Thapa, Aale, Rana, Budhathoki, Roka, Gharti, Puri but here some differences must be made. Magar are mainly dependent on agriculture, in addition to agriculture they are found keeping goats and sheep’s. Magar people have their own language and they live in joint families. Respondent C , whom the researcher had selected for his study, only had idea about his caste Thapa Magar.
First, Thapa Magar is 14 years old young child. His hometown is Bhairahawa, Rupandehi district. He has one brother who lives in Butwal. But he moved to Kathmandu due to loneliness in home. His father sells woods, to earn some money by bringing from the jungle and his mother work as caregiver ( dhai aama) for the other family.
First, Magar moved in Kathmandu by stolen 1000 rupees from his father pocket along with bicycle, and he sold this bicycle in Rs 500 in Chitwan. And then he took a truck to come in city Kathmandu. When he arrived in Kathmandu he started to work in hotel as a dish washer, after some days he moved to street to rubbish the bicycle, t- shirt, helmet etc and sold it in the cheapest price. Not only this bur also he started to collect the rag in the street, by selling the goods he learns a little bit knowledge over the mathematics.
Respondent D was second street children of Magar community. He is 13 years old. He read in class one. His father and mother works together in Hatbazzar by selling drum stick “Neuro saag”, which is known as vegetable in Nepali. When he was in school he learnt 1,2,3 up to 100. He can say but cannot write even a single word “one”. Previously he worked in Palpa purano Buspark as a vehicle washer by leaving his study. His parents has small house in Batase dada in Palpa. One day he planned to go Butwal simply to visit with well known bus driver. When he reached in Butwal he ran away from their near to old market (purano tarkari bazzar), in Butwal where he felt too cold. Near to him someone is taking about sunsine in Kathmandu (Kathmandu ma gham lagechha). Then he planned to go Kathmandu and moved from there and came into Ratrapark. After finishing his money, he got an idea to sell the water bottle in street and asked some money in street and started his job by selling water in the street. He can recognize only the money not the numbers written in the money.
The fifth respondent named E was Khadka from Dolakha. Especially Khadka is known as a Chhetri. They could be found in all over the Nepal. Some of them are settling in the city area. After the retirement from the military service and foreign country job, most of them depend on agriculture. In addition they are in security forces of different countries like India, Britain, and Singapore as well as in Nepal.
Most of the Khadka family is interested in security force. So male of these family are encouraged to make their body physically fit rather than education. Girl child is taught for household activities, to make woolen clothes. They learnt such skills by observing their parent. Respondent E’s father and mother live in jadibuti in Bhaktapur. They work in street as a grocery (nanglo pasal). He is a single child of his parents, even he is single and cannot get care from parents due to economic condition.
The Khadka, whom the researcher selected in his study, does not have idea about his culture and other ethnicity. Just he was found him in the street that’s why he is well known about the street life. He has great ides to pick pocketing, when he saw a crowded of people, he used to run there for pick pocketing. One day he was catched and got a big slap then he left this job and moved to new buspark with one of the elder to do work as kulli and started to work in new buspark as a kulli to earn money not only this but also he started to take the drug ( den gum). With his work experience he got the knowledge of weight and distance of mathematics.
Respondent F was Lama, One of the so-called supreme castes of Gurungs. They have their own language and their physical structure is like mongoloid nature. The majority of lamas are in close contact with Brahmins and Chhetri.
Respondent F was 16 years old, with heavy body structure. He didn’t know about his father and mother and can’t say that where he was born. When he knows himself, he was in the street in Kalanki. Due to his heavy body structure, other street children obey the rules of him He was the boss of his gang and all the children gave him money for buy drug and collect the money from other street children and he calculate his own way to share equally, when he did mistake to share then he learn from other street children. In this way he was well known about the money up to hundred and then he planned to work himself in vehicle helper (khalasi) in micro bus.
Respondent G was Gautam, known as Brahmins, and they scatted all over the Nepal with highest population. Brahmins are very forward in the society. They mostly engaged in farming and government job. They are so- called worshipping class in our society. Since Brahmins are known as so-called preserver and conserver of Hindu religion, and they have followed this religion and worship Shiva, Vishnu, Ram, Krishna and thousands of other deities of the Hindu pantheon.
Respondent G is 8 years old. He only knows about his parents, when he was 6 years old in Bhanjyan. He left his parents in crowed of fair (mela) in Bhanjyan. Then he took the way to go to his home, but that way was wrong and he became to Dharke in the evening. He thought to beg the food in hotel and said his name to the hotel owner. Due to Brahmins caste, he got food by believing hotel owner Brahmins are god and our priest and also got some coins.
Respondent G was a Brahmin, and then he got food and money from the hotel owner. when he reached in Kathmandu after the long time on foot he felt tired and slept in the street without taking food, then next day he started to collect the money by begging the people who walk in the footpath. Later on, he joined to rag collecting and so on. Children learn these things by imitating and practicing with their senior members.
Communicating with these seven street children, the researcher was inspired a lot. Being the street children is really heart touchable things. No one wants to spend such a tragic life, but there is no any other way to survive in this challenging world. These seven street children are become in the street, not by their own interest. The similarities among these street children are; all the street children have the problem of economic, their parents are not so rich that’s the reason they can’t fulfill their children’s interest as well as care too and Parents thought themselves to survive. In this critical situation they haven’t any other way to full fill their duties towards their children. All the street children started their own job while they are in the street and all these respondents live in the street in the night.
On the other hand, dissimilarities of these seven street children are; they are from different society and castes. And all of them are not educated from the school because when the researcher gave them some mathematical problem related to count, add, subtract they weren’t able to solve and they were even don’t know that what is school means? Only one child got chance to read in grade one. These seven street children are from places as the result their acquiring process of knowledge is also different, some of them learns from elders, some of them from school and remaining of them are from shopkeeper as well as from the society where they lives.
The above discussion of personal stories of each respondent shows that they were from different parts of Nepal with cultural background. Moreover, they have some mathematical knowledge learned from society when interaction with elder members of society and friends of peers groups, in our so-called caste system found in our society has dominated to deprived people which compels children to be street children.
This research is encompassed with in basic mathematical knowledge of street children and how they learnt / construct basic mathematical knowledge even in street without formal education of school. These issues are thus analyzed and interpreted under the different heading and follows:
· Concept of addition and it’s learning process.
· Concept of subtraction and it’s learning process.
· Concept of multiplication and it’s learning process.
· Concept of division and it’s learning process.
Concept of Addition and it’s Learning Process
During the discussion and interaction with the street children and information provided by them, the researcher came here to describe the concept of addition , in short term it is known as “ add”, which is familiar word for all and most necessary concept for the human life, which gives the meaning of “ to joint so as increase”. In this research most of street children were familiar to the addition, for this, respondent A and respondent B said addition means “thetotal number”. Whereas the third respondent said “all” and the other respondent argue the meaning “the total number”. But the seventh respondent of this research said the term addition means “to joint so as to increase”. The concept toward the addition of street children gave the actual meaning which is familiar to all over the world in the subject of mathematics. These above mentioned replied were the similar to the concept of addition which we read in our classes of formal school. The total number means the all together with the sum of given calculation. In the calculation toward addition the street children were more familiar than the researcher expected. The respondents of this research said meaningful meaning of addition as to join so as increase. These concepts were more capable to share in the school children whom we teach in the real classroom teaching. Previously they were engage in any kinds of work like as dish washer, rag picking, and play to join the different parts of the poster of hero which was attached in the wall. This poster they used over the body to protect from cold in the winter, thus the mentioned concepts were learnt by their daily bases life in the street. For this concept Vygotsky said that the child cannot handle alone but to accomplish with the help of adults and more skilled peers. As children engage in co-operative dialogs with more mature partners, they take the language of these dialogue make it part of their private speech and use this speech organize their independent efforts in the same way.
For the concept of addition they also learnt by their elders, peers when they work in their daily life. They were more familiar with the money; in the practice with the money they got the concept of addition with the co-operation dialogues with more mature parents.
For the process of addition respondent A can add by counting orally without any doubt only for the two digits. For the describing the ways of his addition, when the researcher gave “GOTTI” to him in two different places, left and right hands 8 and 5 respectively then he started to calculating it by pairing method as below: the researcher asked to calculate it how many gotti do you have now in your both hands he replied as:
2 2
2 2
2 1
2 08 5
12 + 1 = 13
When he adds he used pairing method he made 4 pair of 8 and 2 pair of 5 and remaining as 1. Then again he adds 6 pair as 12 and remains 1 and said 13. For this addition, he learnt when he was at work. To know the his process the researcher asked how did you get this idea, he replied as
“I worked in boutique centre, in the boutique centre when I attached the “sinning metals in the saries I used to keep one pair in one place, that means two lighting metal at one place. With this process I can add easily how many these sinning metals I used in one row or in column”. For the addition of two digits up to 20 I used similar process. Over than 20 I cannot capable to add. Some times when I is in dilemmas then I used to add my fingers properly by counting 1,2,……13.
The above response shows that the addition process is learnt by street children in the work, when interact with pair groups. Thus this view is also supported interaction method to learn. In this regard Vygotsky (1930-1935) believed that child behavior is always guided by culture and environment of society. The society’s environment creates child to learn.
Thus, it is concluded that, he learns pair and became familiar with pairing method of addition. With the active participation he learns about pair with the help of his work place which creates the environment by adults.
Similarly the third respondent add for the 8 and 5 he used his fingers and cannot solve the large amount of addition as 20+38. He was familiar to the number 5 when he adds 10+15 then he used 10+5+5+5 by saying 10,15,20,25. He added with the help of his fingers to remember how many times he used 5.when the researcher asked him how did you familiar with the number five, he replied as
“previously I work in the street as water seller at that time shopkeeper gave me a bottle of water when I give 3 fives and I sold it in 4 fives which means 20 rupees, the shop keeper showed me sold it if they give a big red note which is similar to this. With this practice I became familiar with the number 5,10,15”.
The above response shows that the addition process learns by street children in the workplace when interact with the adults. In this regard vygotsky said reflecting upon the case, knowledge creates with the help of scaffolding, which means with the help of elders as shopkeeper he gain the knowledge of 5,10,15 With this daily work.
Among them respondent G, said his previous story to became a faster and forward up to do add for one digit, with the help of his finger rather than other children. He said as
“I used to play flute in my hometown so in playing process, fingers play most important role that’s the reason I was familiar with my finger and I can easily remember that how many finger did I used to close the hole of flute and how many remain to close. With this process I was well-known about the finger games”.
With the practice of playing instrument the learner’s knowledge structure is in facilitating thinking in the content field is to measure of learning. Here, Private speech indicates as with his own idea he could create the knowledge in his environment.
Respondent B read in class one so he can add by writing in his own copy. He used lining method which we read in our formal education in class room, as 8+5 =////////+/////=13. For this case the theory of vygotsky said a child can learn by the scaffolding, in this case he used lining method which is the concept we learn in formal school with the help of teacher. This process is similar to the all of children who learns in the school. Almost of the respondent did the addition by thinking it is the total process with the help of binary and lining method. Some of the respondent A,C and G are more familiar to the term addition but not to the sign of add. They could add the number in their daily activities mostly the addition was related to the money. They could perform addition up to two digits without knowing the carries over. They had not learnt addition from the institution and did not have the wider concept of it.
Similarly the others respondent learn the addition as the above mentioned ways, by using the fingers and help of the peers, with the help of elders and shopkeepers, passenger as well.
From the description and analysis the researcher knows that learning occurs through the active participation of the learner with the help of adults, peers and other society.
Concept of subtraction and it’s learning process
The street children were not familiar with the term “subtraction”. When the researcher moved to the concept, over all he got the result as subtraction indicates as reduce from big to small from the selected respondents.
Subtraction is the process to make it less from the large value by reducing from big to small or from small value, reach up to large value by adding necessary value, these added values is said to be the subtraction. (Street children)
Actually subtraction is the process where we define as the difference between the two numbers; where street children also gave the similar sense as reduce from big to small.
To investigate the process about the subtraction the researcher gave 20 marbles to each and took 8 from each then he asked how many marbles do you have now? As a result four respondent gave the right answer other were getting confused still to the last time. To analysis the process, the researcher divide in to two groups as x and y. in the group x there were respondent B,D,F and G who gave the right answer and in the group y there were respondent A,C and E. The researcher made two group because in the group x have the similar process where as in the group y they cannot well performed.
Then the entire member in x group said 12. For the process, all of them, of group x have similar way to do it. When the researcher asked one member of group x he replied as
“ at first I took all twenty marbles and removed eight from these twenty marbles then I start to count from 9,10,11,12,13,14,15,,16,17,18,19 and 20 with the help of fingers. As a result I took 9,10,11 as the nominally and count 9 as 1, 10 as 2 similarly 20 as 12.
The above response shows that the subtraction process is learnt by street children by copying adults and also with the previous knowledge of addition.
For the group x , they only know that to do subtract at first we have to take small number than add some more to reach the large number that is a little bit difference than we learnt in the school. In the formal school the teacher taught as 20-8=////////////////////=12 means keep the large number at first then reduce the small number from the large number. Where as in the street children are familiar as to take small at first and reach up to large number. Street children also used similar method when they go to buy the things to the shopkeeper, when they buy chocolates of 3 rupees with the note of 5 rupees they subtract as at first took small number as 3 and reach up to 5 by counting 4,5 with their fingers. These 4,5 are the just the number to reach the note 5 rupees. These 4,5 are the nominal when they count 4, 5 then they used their finger 4 as 1 and 5 as 2 and found the result as they took back 2 rupees from the shopkeepers. As a result they can do subtract with the result correspondence to their hands finger. Over the finger they were confused.
These all of group x had done similar manner, for the sources of this process they used their previous knowledge of addition 20-8=? They used to reach 20, add some number as 8+1+1+1+1+1+1+1+1+1+1+1+1= 12, when they played with the betting something they subtract as above system as additionally, which is different from the formal school.
In this regard vygotsky believed that looking the matter from the opposite perspective , could one suppose that a child behavior is always guided by meaning that a preschooler behavior’s is so arid that he never behaves spontaneously simple because he think he should behave otherwise? This strict subordination to rules is quite impossible in life, but in play it does and it is always possible. Thus play creates a knowledge vygotsky further writes “in play” a child is always above his average age, above in his daily behavior. In this case of group x they were in the simple counting process for the concept of subtraction. Vygotsky also said that child sees one thing and act differently in relation what they saw. Group x mastered learning through playing by emerging rules and more rigid they demand greater in his application, it can be regarded as a means of developing abstract mathematical thought.
From the above analysis it is concluded that learning mathematical skills developing through games and previous knowledge of addition, superficially play bears little resemble to the complex mediated from of thought and volition it leads to change in the behavior.
Concept of multiplication and it’s learning process.
Multiplication is known in short term as multiply. Multiply means to increase in number or quantity and to add (a number) to itself a stated number of times. When the researcher asked to the selected children in the concept of multiply most of them are unknown , in fact street children are unable to solve the problem of multiply directly because they don’t have chance to learn from formal education but street children had the creative mind which creates lots of knowledge. In the case of multiply even they don’t know the table but it has been found that they had the knowledge because when the researcher gave to add 8+5 in that case the respondents used to make it 4 pair of 8 which shows 2*4=8 which is known as multiply in mathematics. Thus the street children learn by their own way without knowing this is multiply. In another case , the respondent used 5+5+5 and said 5,10,15 which also refer that table of 5, which is the part of multiply, thus it can be conclude that street children know the multiply in their own ways not similar to school. Whereas in the case of multiply respondent F has a little bit idea about it, he was the vehicle helper, the researcher asked to the respondent when you asked the vehicle fair to the passenger, if they give three persons at once with fifty rupees then how did you do this? He replied as
“When the passenger gave vehicle fair of three person by one of them then I used addition method to multiply. That means when the passenger gave 50 rupees for 3 persons with actual 10 rupee fair to each of them I calculate as 3*10=10+10+10=30 in this way I calculate 30rupee and return back them 20 rupee. In this way I calculate a little bit knowledge over the multiply”.
From the learning process of subtraction he learnt during the time, when he worked in vehicle. The above response shows that the multiplication process is learnt by street children in the work place in vehicle as well as by previous knowledge of addition interacting with previous knowledge.
In this regard Vygotsky said that private speech, which is self directed speech that children used to guide their thinking and behavior. It is as the foundation of memorization and recall, categorization and self reflection. In this case respondent F also learnt this multiply with the help of addition which is his own self directed speech.
From the above analysis it is found that they are unknown about the multiplication operation and its sign. From the discussion and interaction with them it had found that they used this multiply concept on the basis of addition. They took it as long addition. They learned multiplication from their own practice during the long experience in money calculation. They internalized it by their individual practice, self guided and self reflection. This concept is drawn from the games which they play.
Concept of division and it’s learning process.
The street children whom the researcher selected were unknown about the concept of division and its sign. But they could perform it only for the small number with the help of their fingers but not clearly and easily. That’s why the researcher applied the games to get the concept and process about the division. One day the researcher decided to bring the “SELROTI” for their Tiffin. And he took 60 roti from the bakery café and asked to the selected respondent to divide it equally then the respondent started to talk with each other to share it equally but no one came at first to share then the researcher immediately takeout 20 coins from his pocket had asked to share equally. Then the respondent A started to give each of them by one- to-one method and gave one rupee coin to each. Then respondent came and asked to the researcher some coin to divide to some children then the researcher gave 10 coins to share only for the five children , who were in the last row of the class he also share similar method as above. Then the researcher asked who can share equally rather than this method then found that no one can share except this ways. So lastly the respondent G came and took 20 dish for the Tiffin and started to share them by keeping one-to-one methods to each dish, and repeated 3 times and said 3 roti to each of us, which is the division of 60/20=3. This is the same way of above mentioned. Even street children hadn’t ides about divide but they can share equally in the small portion of everything. In the formal education, student used to divide by using table, but the street children had not idea about the table so they used one-to-one method which gives the result as divide. In this regard Vygotsky’s socio culture theory said children are the active seekers of knowledge. In this theory also said that knowledge is being constructed in social situation of negotiation, rather than being the reflection of the object reality. Vygotsky argues that the child’s development cannot understand by studying that it needs to examine the external world. Children are socially associated which is present from the beginning arriving into the complex world of the social relationship and the culture. The culture itself has historical development .socio-culture theory. Thus this method of sharing also related to the cultural phenomena, for example when mother came from the market she took chocolates to the her children and she share one to one to the children in the home town that the reason that they acquire the knowledge from the cultural phenomena, which method is differ than the formal education system. From the above analysis, it can easily said that learning mathematics occur from socio-culturally as well. A street child who is working as a lobour on the street, is also learn the mathematics not only through memorization of formulas, practicing long time from given examples only. It occurs and develops through active participation in the culture and help from the mature parents as well as private speech.
From the above, it is observed that street children have mathematical knowledge which they need in their daily life. Even they had the knowledge of either to add or multiply but they cannot do as properly. No one encourage them to solve the problem related to the basic mathematics but they used to solve by their own way which gives real output but in process it is different. Their mathematical skills developed during the staying period on the road, with the help of scaffolding and socio-cultural system of Vygotsky theory.
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