# Qualitatively Compare The Problem Solving Behavior Education Essay

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The purpose of this study is to describe and to qualitatively compare the problem solving behavior of young schooled vendors in informal and formal settings. Ten vendors were systematically selected from a purposive population of 25 vendors in two open markets in Beirut. Vendors in the sample varied in schooling, age, and vending experiences. Ethnographic case study was the general methodological approach for this study. Four methods of collecting data were used: Participant observation, interviewing, collection of artifacts, and analysis of document. Interviews ranged from informal conversations, to semi-structured interviews, to formal interviews Two weeks after the semi-structured interviews and based on transactions executed by the subjects in the informal setting, a formal test was administered. Items of the formal test were presented as either computation exercises or as word problems. Upon completion of the formal test, each subject was asked to explain procedures used in problem solving. All interviews were tape-recorded and transcribed for analysis. The procedure used for data analysis was analytic induction which involved scanning the data for categories and for relationships among these categories. Upon comparing the problem solving behaviors of vendors across informal and formal settings, two findings emerged. First, vendors employed computational strategies in the informal setting which are different from those used when solving computation exercises in the formal setting. Second, the intuitive computational strategies that subjects used in the informal setting were identical to those employed when solving word problems and were associated with a higher success rate than computational strategies used when solving computation exercises in the formal setting. The results were discussed and interpreted using Vergnaud's model and cognition in practice theory. The results were similar to findings of a number of relevant empirical research studies. Implications and recommendations for education were presented along with suggestions for further research.

## CONTEXT

Achievement in schools has been decreasing steadily in many countries. In particular, the U.S.A and some European countries have shown in the last thirty years a decline in school achievement in mathematics (Millroy, 1992). In Lebanon there is a concern about the damaging effects of exam-driven instruction and particularly that of mathematical problem solving (Osta, 1997). Failing as well as not being able to cover the expenses are major causes of dropping out- of school. With no other source of support, students have to work to support themselves and their families thus work in what has been called the " informal sector of the economy".

In his book, The Other Path, the Peruvian economist, Hernando de Soto, gives a fascinating account of how Peru's informal economy was created by illiterate peasants who were excluded from participating in the formal economy. He describes how the informals responded by creating markets to support themselves with only limited resources. By organizing themselves and voluntarily obeying their own rules and norms, they created a subculture that socially and economically salient.

In most countries where the phenomenon of informal economy prevails, Street vending is considered as one of the most popular professions that children practice. In many developed and developing countries, the phenomenon of street vending or market children has been wide spreading. In this study, we are mainly interested in considering the case of Lebanon and India.

Street Children in India

India is the seventh largest country in the world with the largest population of street children. They work as porters on bus and railway stations, mechanics in auto repair shops, vendors of tea, food or handmade goods, tailors, ragpickers who pick usable items from garbage. According to the Civil Society forum report, it has a large and rapidly growing population of 1.027 billion of which 40% are under 18 (1/3 of the total population are under age15). In 2001, the rate of urbanization was 28.77%. The accelerated pace of industrialization and urbanization in the country has disrupted the family life and has compelled tribal and rural people to migrate to big cities. Migration from rural to urban areas (in search of employment) has resulted in the rapid growth of the urban population and nearly 29% of the total population lives in urban areas.

There are some negative consequences of the urban boom. One of the negative consequences is the existence of a large proportion of the urban poor living in slums and jhopad-patties or thatched huts (Phillips, 1994). An average of 50% of the urban population lives in conditions of extreme deprivation - compounded by lack of access to basic services, legal housing and poor urban governance. In addition, Agrawal (1999) found that almost 90 percent of the employment in the country is in unorganized and informal sectors.

Literacy levels are still low. Accessibility and facilities for education and social infrastructure is rather inadequate to meet the demands of a growing population. "Even now 2.6 percent of the children in the urban areas and 3.5 percent in rural areas have never attended school" (Agrawal, 1999, p.24). As the result, the number of street children in India is swelling. According to UNICEF's estimation, there are about 11 million street children in India (1994). These figures are considered to be conservative. An estimated 100,000-125,000 street children live in Mumbai, Kolkata and Delhi, with 45,000 in Bangalore.

According to previous studies about street children in India, majority of the street children who are of school-going age and even over school-going-age are children who have never been to schools. The increasing number of street children may have an impact on India's economy. Arbind Singh, coordinator, National Alliance of Street Vendors of India, outlined the contribution of street vendors to the local economy.

Street Children in Lebanon

After World War II and the creation of Israel state in 1948, thousands of Palestinian refugees entered Lebanon, many settling in Beirut. Seventeen refugee camps are spread all over Lebanon , the most densely populated are those found in Beirut. In 1964 and recently in 1994, the Lebanese government has passed two decrees which outlined the conditions of work for foreigners living in Lebanon. As foreigner refugees, the Palestinians are barred from working in over 70 professions. This lack of employment opportunity for the Palestinian refugees in Lebanon has created a devastating economic condition. (ØÙ„Ù‚ØÙ†ÙˆÙ† ØÙ„Ù„Ø¨Ù†ØÙ†ÙŠ)

In 1978, and after the Israeli occupation to Southern Lebanon, many Lebanese fled to the capital Beirut and settled next to the Palestinian refugee camps. Through out the refugees' camps, more than 60% of both Lebanese and Palestinians live below the poverty line. Children suffer greatly- born into camps as refugees, they have lived no other way. In many cases, neither have their parents. Life without adequate schools, health care, nutrition or shelter becomes the norm. Palestinians cannot join any professional associations- relegated to the unskilled and informal labour markets, they compete with 50,000 Egyptian and one million Syrian. In addition to employment and property restrictions, government bars them from enrolling their children in Public schools. ) United Nations Human Rights System, 2002)

For some time, educators who have studied school achievement in rural and urban communities have recognized that children do well in their daily life and indeed grow as successful citizens, in spite of their poor performance in school mathematics (D'ambrosio, 1992). For example, Saxe (1988) showed that Brazilian candy sellers with little or no schooling, can develop in the selling experience arithmetic practices that differ from the arithmetic taught in schools and that are associated with a high success rate. Increasingly, educators have found the cultural surroundings of children to be a factor affecting their achievement in school mathematics (Dawe, 1988), providing support to the hypothesis that cognitive power, learning capabilities, and attitudes towards learning are closely related to cultural background (D'ambrosio, 1992) to which Millroy (1992), adds a socio-political dimension that may create learning barriers affecting particularly children from disadvantaged groups. Outside the school environment, the performance of low-achieving children and adults in schools is often successful. Both children and adults perform "mathematically" well in their out-of-school environment: counting, measuring, solving problems and drawing conclusions using techniques of explaining, understanding and coping with their environment that they have learned in their cultural setting (D'ambrosio, 1992). These practices have been generated or learned by their ancestors, transmitted through generations, modified through a process of cultural dynamics and learned in a more casual and less formal way than school mathematics. It is the patrimonial knowledge of the groups. It is the "ethnomathematics". (D'ambrosio, 1992)

Ethnomathematics develops mostly when there is a discrepancy between people's need for problem solving and the amount of mathematics they have learned in school i.e. when people become involved in tasks requiring problem solving skills that are not learned in school (Nunes, Schliemann & Carraher, 1993). It has been suggested that there are informal ways of doing arithmetic calculations that have little to do with the procedures taught in school (Carraher & Carraher, 1985). Also studies have documented differences across groups as a function of their level of schooling. However, it is quite possible that the same differences between "street" and school arithmetic could exist within individuals. In other words, it might be the case that the same person could solve problems sometimes in formal, and at other times, in informal ways. This seems particularly likely with children who often have to do mathematical calculations outside school that may be beyond the level of their knowledge of school algorithms. It seems quite possible that these children might have difficulty with routines learned at school and yet at the same time are able to solve, by more effective ways, the problems for which these routines were devised. One way to explore this idea is to look at children who have to make frequent and quite complex calculations outside school. The children who sell things in street markets in Beirut form one such group.

## AIMS

While the short term aim of the present case study is to investigate the uses of math by a sample of young schooled vendors in the streets of Beirut who use math in their jobs, its long term aim is to be transferred and replicated in India. Specifically, the purpose of this study is to:

1. Describe the problem solving behaviour of a sample of ten young street vendors in informal and formal settings in Beirut.

2. Compare qualitatively the problem solving behaviour of the sample in informal and formal settings in Beirut.

## RATIONALE

Our intention is to study the mathematical practices and strategies that develop out of street vendors daily activities, to acknowledge their strengths and to see their weaknesses, as opportunities to negotiate broader understandings of what counts a mathematics . Millroy (1992) has stated that "an acknowledgment of these factors [the social, cultural and political aspects of math] would encourage a broader conceptualization of math and may begin a process whereby math could be seen as an active experience, accessible to all people". ( p.50)

Second, the results of this study may contribute to the growing body of research in "everyday cognition" or "cognition in practice" by studying the problem solving behavior of the same group in two distinct settings. Very few studies investigated the ways in which the arithmetic knowledge is learned outside school. In studying the arithmetic of Liberian tailors, Lave (1988) proposed that there were two qualitatively different modes of doing arithmetic. The unschooled tailors used a "manipulation of quantities" approach, an oral context-based way of working with numbers in contrast to the "manipulation of symbols" approach employed by their schooled counter parts. It is possible that such different modes of doing arithmetic may be found within the same individuals especially if they use math in every day work settings (Nunes et al., 1993). If so, it may be useful to describe and compare the uses of math by the same group in the context-based (informal) and school-based (formal) settings.

Third, the comparison of informal and formal procedures in arithmetic, that is the way people manipulate numbers in solving addition, subtraction, multiplication and division problems is a natural starting point for research for several reasons. D'ambrosio (1992) claims that arithmetic is a very simple aspect of math. Another reason is that reasoning about numbers is part of everyday experience as well as part of the formal discipline of math (Nunes et al., 1993). On the other hand, Lave et al. (1990) state that one of the several reasons for focusing on arithmetic was that "arithmetic activity has formal properties which make it identifiable in the flow of experience in many different situations" (cited in Millroy, 1992, p.6) and Lave (1988) states that "it (arithmetic) has a highly structured and incorrigible lexicon, easily recognizable in the course of ongoing activity". (p.5)

## SIGNIFICANCE FOR EDUCATION

The present study is significant for three main reasons. First, it represents the first attempt in Lebanon to analyze the mathematical problem solving behaviour of children outside the confines of the classroom using a qualitative approach. Second, it studies the performance of schooled children across two different contexts. Third, it contributes to the growing body of research on learning in terms of "Apprenticeship" model of instruction. Through gathering evidence that could be seen as a challenge to the conventional definition of math, mathematical activity can be seen as interwoven with everyday practice outside the academic formal settings. This, in turn, could open new perspectives for further research into other models of teaching and learning since "for years, math educators and researchers in math education have focused on the classroom as the primary setting in which math learning takes place" (Nunes et al, 1993, p. 557).

Another contribution from this work concerns teachers. The detailed description and comparison of problem solving behaviour of schooled vendors in work and school settings may provide insights for teachers into their students' level of mathematical understanding. By creating opportunities for students' problem solving activities in practical contexts, teachers might generate dilemmas to stimulate students' invention, discovery, and understanding in patterns of activity. For, problem solving that relies heavily on the learning of rules can be often "plagued" with bugged (consistent error) algorithms. If students can come to understand the rules through imagining situational contexts, they may be able to strengthen their understanding of these rules.

A further practical value of this study is the proposal it offers to curriculum developers on how to present mathematical concepts. In a school context, a mathematical concept is usually described and explained by invoking the standard algorithm for its calculation. The analysis of the problem solving behaviours of vendors in work contexts may provide curriculum developers with alternative and more effective ways of presenting mathematical concepts.

## LITERATURE REVIEW

A good deal of interest has been generated recently by evidence that unschooled persons solve everyday math problems successfully using invented strategies and that many schooled persons solve every day math problems using strategies different from those learned in school (Carraher et al., 1985; Saxe, 1991). For many years, math education researchers have questioned the math that is generated and used outside of institutions of learning (Millroy, 1992). This is the math that allows unschooled and sometimes illiterate people to practice crafts and trades, conduct business transactions and make their livings in a variety of ways. This mathematical activity has been called "informal" math (Ginsburg, 1988) or "everyday" math (Lave, 1988) or "ethnomath" (D'ambrosio, 1992), or even "street" math (Nunes et al., 1993).

Several contributions to the literature on informal math can be grouped into two classes of studies: (a) work that aims at describing informal math used in Western cultures and (b) work that aims at describing non-Western indigenous forms of math existing in cultures, where no systematic transmission in school prevails (Nunes et al., 1993).

A good portion of the work on informal math in Western cultures focuses on young children and elementary arithmetic. Several important contributions to our knowledge of elementary arithmetic in preschool years were made by Ginsburg (1988) who demonstrates that when children learn a numeration system and understand it well, they can then invent ways of using it to solve arithmetic problems through counting and decomposition. A second group of studies on informal math in Western cultures focuses on math used outside school by adults, not by children. This line of investigation has shown that it is one thing to learn formal math in school and quite another to solve math problems intertwined in everyday activities "Whether it is inventory taking at work or shopping or calculating calories in cooking, school math does not play a very important role"( Nunes et al., 1993, p. 3). Hence, the idea prevails that informal math has its own forms that are adaptations to the goals and conditions of the activities.

On the other hand, work on non-Western math showed that several groups of people who learn numeracy without schooling, use their indigenous counting systems to solve arithmetic problems through counting, decomposition, and regrouping (Gay & Cole, 1967; Ginsburg, 1988). For example, Gay and Cole (1967) report that the Kpelle people of Liberia used stones as support in solving arithmetic problems and could solve addition and subtraction problems using numbers up to 30 or 40 with accuracy. Beyond that, their method became tedious, and people tended to guess the number rather than give an exact answer.

Several studies (Carraher et al., 1985; Ginsburg, 1988) seem to indicate that school-learned algorithms may not be people's preferred ways for solving numerical problems outside the classroom. This observation seems to be true of children with varying degrees of schooling (Carraher et al., 1985), adults with an elementary and secondary education and children up to fifth grade in both the United States and the Ivory Coast (Ginsburg, 1988). Carraher et al. (1985) have suggested that the situation in which arithmetic problems are solved may have an important role in eliciting different types of strategies; school situations tend to elicit school-taught procedures, and out-of-school situations are more likely to give rise to informal procedures. In their study, five children, aged 9 to 15 years and with various levels of schooling (first to eight grade), were asked to solve arithmetic problems in the course of their work as market or street-vendors and in a school-like setting. Their performance in the natural situation was significantly better than their performance in the school-like setting. Moreover, their approaches to problem solving varied across situations; school-like problems were more likely to be solved through recourse to the school algorithms whereas the natural situation gave rise to a variety of informal procedures that were highly unlikely to have been learned at school.

These results have motivated further investigation of the effect of the situation on the problem-solving procedures since many differences exist between the settings under consideration. Several possible explanations for the differences in performance observed in the informal and formal tests were suggested. In particular, Nunes et al. (1993) present two types of theory that could explain these results. One stressing the social-interaction aspects of the situation and a second stressing the social-cognitive aspects.

Informal math has often been treated in the literature as "lesser" math involving "idiosyncratic, intuitive, child-like procedures, techniques that did not allow for generalization and should thus be eliminated in the classroom through carefully designed instruction." (Nunes et al., 1993, p.19). However, there are many calls that legitimize the forms of knowledge associated with out-of-school practices.

## METHODOLOGY

## Population and Sample

The population of this case study consists of young schooled vendors in two open markets in Beirut who had at least three years of schooling and three months of vending experience.

The method used for selecting the sample is purposive sampling. The reason for choosing this method was simply because particular vendors, whose characteristics were known and dictated by the study before sampling, were deliberately chosen in order to match and facilitate the study. Ten vendors were purposively chosen from two market settings in Beirut, namely: Haret Hreik and Sabra.

Vendors in the sample varied in years of schooling (three to seven years), in age (10 to 16 years), and vending experience (one to eight years). Four of the vendors worked alone while the other six helped their fathers or neighbors. Only three were totally responsible for purchasing the produce at wholesale market and pricing it for selling.

Since competition was usually high in these open markets, the vendors would constantly be obliged to revise and change their selling prices unexpectedly even during the same day. Of the ten subjects, six had complete freedom in changing the prices of the produce they were selling, while constantly revising their profit and loss. Vendors devoted long time for their work: Seven subjects worked from six to seven days per week with a mean of ten hours per day; whereas, the other three subjects, still attending school, worked after school and during vacations.

Failure was the basic reason for subjects dropping out from school. Seven subjects were out-of-school during the time of the study, six had dropped school because they simply had failed and repeated classes and only one had to quit and work to support his family.

During the course of their daily work, the subjects were involved in transactions that required them to mentally solve a large number of mathematical problems without the use of calculators or even paper and pencil.

## DESIGN

An ethnographic case study approach was adopted as the main methodology.

The bounded unit being the problem solving behavior of young street vendors in two open markets: Sabra and Haret Hreik. These two markets are located in relatively densely populated neighborhoods in Beirut. The two areas attract a large number of migrant workers who live at the nearby camps. These workers come from a low socio-economic background where family members, including children, normally work to support the family. Both are open markets for selling fruits and vegetables in fixed kiosks whose roofs are basically covered with corrugated sheets of iron, weighted with blocks of stones and held by thin wooden and iron supports. The architecture of this roof helps to shade and protect the vendors and their produce from rain and direct sunlight. Inside the markets, vendors have wooden tables, each at his own spot, on which fruits and vegetables are exhibited. Other vendors who stand on the borders of the market have their own carriages, each shaded by an umbrella. Photographs of the vendors and the two markets are provided and are used as data sources (Merriam, 1998). (See Appendix A).

A mix of qualitative and quantitative methods is undertaken. The general methodological approach in the informal setting was to conduct naturalistic observation of the subjects at work in both markets and to note their problem solving behavior on the arithmetic tasks encountered during their daily practice as vendors. In the formal setting, a formal test was administered and the problem solving behavior of subjects was studied from worksheets and transcribed audio-taped interviews.

## DATA COLLECTION TECHNIQUES

In an attempt to strengthen reliability of findings( Merriam, 1998, Yin, 2003), data was triangulated using four methods of collecting evidence from multiple sources: participant observation, interviewing, analysis of documents, and Collecting artifacts.

## Participant Observation

To get a rather emic perspective on the phenomenon of street vending, the researcher posed as customer asked questions on the prices of fruits and vegetables for a purchase or a possible purchase. During observations, interactions with the vendors as well as vendors' interactions with other customers were recorded.

## Interviewing and Testing

Interviews ranged from informal conversations, to semi-structured, to formal-structured interviews which were preceded by a formal test.

Informal conversation. These conversations took place the first two weeks of the study. They consisted, essentially, of general and open-ended questions that would make the subject start talking about his life. The second type involved rather specific questions, a script of which is provided in Appendix B. The main purpose of these conversations was to get to know the subjects better, to obtain information about their age, level of schooling, nationality, and residency.

Semi-structured interviews. The semi-structured interviews were administered in Arabic, the native language of the subjects and the verbal responses were taped-recorded along with subjects' explanations of the procedures used for obtaining the answer. A script of the semi-structured interviews is provided in Appendix C. It is worth mentioning here that though questions posed in these interviews were relatively formulated following a general guideline, they were also generated in the natural setting and were not identified prior to interviewing.

Formal test . Upon transcribing data from the semi-structured interviews, conversations with the subjects were separated from transactions. The items of the formal test were thus extracted from the transactions executed by subjects in an attempt to achieve a sell or a possible sell. In this way, each operation performed by a subject in the semi-structured interviews was chosen as an item to be included in the formal test taken by that subject. Problems were presented as either computation exercises or as word problems.

After transforming the transactions into mathematical operations exercises, items were chosen randomly for each subject to be presented as word problems. Problems involved different contexts such as transactions with different currencies, $ and L.L, measurements and weights. A script for word problems is provided in Appendix E.

The formal test was administered a couple of weeks after the semi-structured interviews, formal-structured interviews were scheduled. The formal test took place in the market or at the subjects' homes. It is formal in the sense that it took place in a formal, school-like setting where subjects were given papers and pencils and were asked to perform a school-like task while sitting at a table.

Formal-structured interviews. Upon completion of every test item in the formal test, each subject was interviewed and oral explanations of the procedures used in problem solving were tape-recorded.

## Collecting artifacts

This method involved collecting anything a community makes and uses which reflects their experiences and practices. The artifacts gathered consisted of photos of subjects at work picturing the way these subjects exhibited their products and the weights and scales used, in order to show the natural situation that provided meaning for their problem solving behavior. Also, specimen of papers on which subjects wrote their calculations was collected. (See Appendix D)

## Analysis of documents

Statistical national and international records from international organizations (UNICEF and UN) as well as official and legal documents from the Lebanese government were examined.

## ANALYSIS

Data consisting of descriptive and reflective field notes, transcribed taped interviews as well as problem solutions were read and reread several times. The main purpose for scanning the data was to ensure its completeness and to record significant observations that helped in launching the analysis process. Careful scanning of the data resulted in outlining a general and preliminary framework for sorting these data. This classification was primarily based on the calculations carried out by subjects in observable fashions in both settings during problem solving and their explanations for responses.

As an initial step in the process of analysis, Eisenhart (1988) emphasized the establishment of "meaningful" units of analysis according to which observed phenomena were divided and patterns and regularities evolved in the vendors' problem solving behavior. Similarities and differences between patterns of behavior were delineated and eventually major categories emerged emphasizing broad outlines of vendors' problem solving behavior. Relevant chunks of data were assembled to fit these categories and additional categories were formed to include "negative" instances which did not fit the general framework. Finally, by comparing and matching these categories and subcategories and referring to field notes, "coherent intact schemes" for classifying and categorizing problem solving behavior of vendors in both settings, started to emerge. At this point, data were categorized and results were produced.

## SUMMARY OF RESULTS

Upon analyzing the problem solving behavior of street vendors in formal and informal settings, three major findings emerged. First, when solving the three types of problems: problems in the informal work setting; computation exercises; and word problems, three heuristics, three computational strategies, and eleven computational substrategies were used by the vendors. These heuristics, computational strategies and substrategies involved a combination of standard school-taught algorithms and nonstandard procedures invented by the vendors. Vendors in the informal setting solved proportion problems through building-up heuristic which constituted 66% of the heuristics employed and was associated with a high success rate namely 92%. Also, vendors attempted addition, multiplication, and subtraction problems using informal, intuitive computational strategies, the most frequent of which was decomposition which represented 62% of the computational strategies employed and which elicited high percentage of correct responses, namely 89%.

Second, vendors in the formal setting used formal computational strategies (combination of traditional and idiosyncratic algorithms) for solving computation exercises that were different from the informal computational strategies used for solving word problems. For 81% of vendors' computational strategies when solving computation exercises were formal whereas 78% of the computational strategies used for solving word problems were informal. Informal computational strategies were associated with a high success rate on both types of problems; 85% for computation exercises and 82% when solving word problems. However, using formal computational strategies, this success rate decreased considerably when solving computation exercises (46%) and increased when solving word problems (91%). Third, vendors employed computational strategies in the informal setting that were identical to those used when solving word problems but were qualitatively different from the computational strategies used for solving computation exercises. For, the informal, intuitive computational strategies were exclusively used by the vendors in the informal setting and represented 78% of the computational strategies in word problems, whereas 81% of vendors computational strategies when solving computation exercises were formal (combination of traditional and idiosyncratic algorithms). Also, informal, intuitive computational strategies were associated with a high success rate across settings whereas the formal computational strategies elicited high success rate, 91%, only on word problems. One of the implications drawn was that applied problems were much easier and meaningful than pure computation exercises. Also, the presence of real objects could not by any reason reduce the complexity of the mathematical problems posed and thus contribute to this relative success in the market, since performance on word problems was considerably high.

## INTERPRETATION OF RESULTS

Theoretical frameworks that were proposed by cognitive developmental theorists, specifically the works of Vygotsky and Piaget, may, to a large extent, explain within and across individual differences in performance in the informal and formal settings. Vergnaud (1988) has developed a theoretical model of concepts which may explain the use of heuristics as well as differences in computational strategies within and across groups and settings. Vergnaud's model is based upon the idea that concepts always involve three aspects: invariants, representations, and situations. A possible interpretation for this difference in computational strategies usage could be the differential impact of the situations that elicited such computational strategies. The informal computational strategies that were employed in meaningful vending situations required understanding and their use by the subjects developed understanding. It was an understanding of numbers and number system developed within a larger context, a context of meaningful and reasonable relationships. But the formal strategies were rather more symbolic, restricted only to meaningless representations that messed up the subjects' performance and led to uncertainty and confusion.

## IMPLICATIONS FOR EDUCATION

The most important implication that can be extracted from this study is the new conception about what counts as math in general and arithmetic in particular. Math is intuitive, realistic, subjective, and can be used as a tool for accomplishing purposive activities. In this respect, the results of this study confirm the view that math, specifically arithmetic, is not an abstract body of rules but rather can be invented by the people.

## Implications for Teaching

This study has provided evidence that children can invent problem solving strategies for solving addition, subtraction, multiplication, and simple proportion problems which may not have been taught to them in school. Teachers could facilitate more meaningful learning by establishing links between children's intuitive strategies and the traditional algorithms. Also, Students can best learn a concept when they have experienced for themselves manifestations of that concept. A third implication for teaching is the fact that students' errors can be valuable part of the learning process because they can provide information about students' understandings

## Implication for Curriculum Developers

One direct implication of this study to curriculum development is the designing of curriculum around primary concepts and presenting it in a whole-part approach as suggested by constructivists ( Brooks & Brooks, 1993). The vendors' informal computational strategies were holistic in that they dealt with complete numbers rather than individual digits and they worked from left to right, preserving the meaning and place value of numbers. Presenting mathematical content and structuring problems around "big" ideas can provide opportunities for students as well as teachers to acquire component skills, gather more information, and thus build mathematical concepts for, "with curriculum activities clustered around broad concepts, students can select their own unique problem solving approaches and use them as spring boards for the construction of new understandings" (Brooks & Brooks, 1993, p.47).

The results of this study have generated a number of questions that are worth considering for further research. Perhaps, the most significant question is the way in which school learning interacts with the kinds of understandings children generate through their participation in every day cultural practices. Despite the importance of this question, we have little empirical research in this area. Also, describing and comparing the problem solving behavior of vendors in informal and formal settings have triggered the enduring questions about what a mathematical concept is and what it means to solve a problem in nonacademic settings. It may be interesting to replicate this study on different mathematical concepts and with a different group of apprentices and to compare the problem solving behaviors across contexts.

Further research in support of the idea of people's practical theorems, or Vergnaud's theorems-in-action should be conducted. We probably need to develop skillful ways for describing different sorts of implicit knowledge and determine the scope of intuitive problem solving behavior.

## POSSIBILITIES FOR REPLICATION IN INDIA

While our main focus in this case study was to examine the problem solving behaviour of street children in Beirut, we are interested in extending it to India. However, we are aware of certain challenges including those pertinent to language as different languages are spoken by children in various cities in India. Also, the gender role differences will be present. Girls are required to marry early and boys remain on the streets longer. Begging by families is common too. The laws do not permit children to put up small boxes to sell their wares so they run when they see police coming. There is a surcharge to be paid to the government to set up small booths to sell their wares. Also, there are specific areas that these children can sell their goods. Most times they are selling and setting up their boxes where it is illegal to do so. So, as a researcher you may have to wait days for your subject to return from jail etc.

## Appendix A

A participant weighing

The architecture of Sabra's market

Selling and exchanging money

Negotiating the price

## Appendix B

## Script of Informal Conversations

## Adapted from Millroy(1992)

A. General, open-ended questions to make the subject talk about his life.

B. More specific questions

1. What is your name?

2. How old are you?

3. Where are you from?

4. At which class have you dropped school?

5. How many years have you studied?

6. Where do you live?

7. How old were you when you dropped school?

8. Why did you drop school?

9. For how many years have you been working in the market?

10. At what time do you come to the market and when do you leave?

11. How many are you at home?

12. Does your father work?

13.Have you taken addition, subtraction, and multiplication at school?

14.Do you know how to compute? Do you use paper and pencil or a calculator?

15. What do you sell?

16. Do you sell alone or somebody helps you?

17. Do you make wholesale purchases?

18. Who makes the pricing on the produce?

19. Can you change the prices, make discounts or increase the price?

20. Do you compute profit and loss?

21. Can you give a change to a dollar bill?

22. Do you use the things you have learned in school while working in the

market?

23. Do you like working in the market?

24. Do you like your brothers to work in the market?

25. Is it profitable to work in the market?

26. When have a problem do you ask for help from anybody?

27. Do you consider going back to school?

28. What does it take to be a good vendor?

## Appendix C

## Script for semi-structured interviews

Questions posed were drawn from the subjects' natural setting, from the type of

transactions used and the questions they may face in their work.

1. I am going to take X kilos of this produce. How much is that? How do you know?

2. I will take X kilos. I am going to give you z L.L bill, what do I get back?

How did you get it ?

3.You are selling X kilos for y L.L but I want z kilos, how much do I have to

pay? Why?

4.I want to buy X kilos of this and y kilos of that. How much do I have to pay? How?

5. I have X L.L. I want to take y kilos from this produce, how much will I

have left? How did you find out?

I have X L.L How many kilos can I buy with it from this produce?

How did you know?

7. You are selling X kilos of this produce for y L.L, but I only want one kilo.

How much does one kilo cost? How did you get the answer?

8. Have you changed your prices today ? By how much ? Why?

9. I want X kilos from this produce. I will pay you with a y $ bill. How much

is the change in $? In L.L? How?

10. Can you estimate how much the leftovers from this produce weigh?How?

11. From the leftovers can you possibly guess how much have you sold?

How do you know?

12. How much have you sold today? Can you determine your profit? How?

## Appendix D

Papers on which the vendors wrote their solutions of arithmetic problems