ࡱ > 5@ bjbj22 ; X X $ ' F
F
F
F
X X X 8 X d RZ |
\ \ ( ] ] ] ^ ^ s^ ^ e g g g g g g $ R P! ~ 9
l ^ ^ l l F
F
] ] f f f l F
R ]
] C " f l e f f b w t
"
] \ P~4. X q 6
V ,
0
A " { "
F
F
F
F
"
4 ^ Ab f d f ^ ^ ^
D; F b
F A CHILDS PERSPECTIVE ON BEING IN DIFFICULTY IN MATHEMATICS
Troels Lange
Aalborg University & VIA University College
trla@viauc.dk
This paper is part of a study that explores learning difficulties in mathematics from the childrens point of view. An interview with a group of ten to eleven year old students is analysed with respect to their making sense of and ascribing meaning to their learning or non-learning of school mathematics. The analysis uses a three level procedure for analysing interviews adopted from (Kvale, 1984) that is coherent with the methodology and conducive to sensitivity towards the notion of childrens perspectives as an analytical construct. The students sense making seamlessly integrated into coherent wholes their immediate experiences in their mathematics classroom with the prospect of their future lives. It was also found that children in difficulties with learning mathematics can be reflective about the norms and expectations at play in school mathematics.
Framing childrens perspectives
In a previous paper (Lange, 2007), I explored methodological aspects of researching learning difficulties in mathematics from childrens point of view. In this paper, I report on research following these methodological considerations. It is shown young children can be interviewed about their experiences with school mathematics and their making sense of and ascribing meaning to school mathematics. A three level interpretation procedure inspired by (Kvale, 1984) is conducive to the construction of childrens perspectives. Finally, as anticipated in (Lange, 2007), children at the edge, e.g. performing poorly in mathematics, can be quite reflective about the norms and expectations at play in school mathematics.
In (Lange, 2007) learning difficulties in mathematics were seen as a social construction within the social practice of school mathematics education (Valero, 2002) and therefore closely related to the socio-cultural significance attributed to mathematics in Western societies. Consequently, the learning or non-learning of mathematics seriously affected childrens perceptions of themselves and therefore their construction of identity.
Children should be recognised, not just as objects of socialisation, but also as actors in their life with their own ways of constructing meaning and interpreting their world (James, Jenks, & Prout, 1997). As agents, children are co-constructors of the social practice of school mathematics teaching and learning because of their own sense-making, meaning ascription and identity formation. The recognition of childrens agency makes their construction of identity and meaning a unique and valuable source of knowledge on mathematics education and learning difficulties in mathematics (Lange, 2007).
It was anticipated that childrens identities and ascription of meaning to mathematics education would be expressed in narrative form. As narratives are made up from stories floating around (Sfard & Prusak, 2005), they connect individual agency and the social and cultural structure. Hence, childrens narratives about their learning or non-learning of school mathematics would reflect their individual meaning making and agency, as well as the social and cultural structure embedding the practices of mathematics teaching and learning.
The notion of childrens perspectives was claimed to be a theoretical construct of the researcher as opposed to a natural given (Lange, 2008). It was defined as meaning constructions: the meaning that children ascribe to their actions in the field of school mathematics learning. This definition referred to Skovsmoses conception of students foreground and background as resources for their meaning constructions (Skovsmose, 2005a; Skovsmose, 2005b; Skovsmose, 1994). Foreground and background of a child are the childs interpretation of the socio-political context. As children exert agency in interpreting the socio-political context and in ascribing meaning to mathematics education, children's perspectives express children's agency as well as embody the socio-political context.
Given that identity and meaning were considered narratives, it was imperative that research methods should be adopted that invited children to tell narratives. Hence, with reference to life history research (Goodson & Sikes, 2001) interviewing children seemed to be a method coherent with the aims of the research. The style of interviewing can be categorised as semi-structured life world interviews, the form of research interview defined as an interview whose purpose is to obtain descriptions of the life world of the interviewee with respect to interpretation the meaning of the described phenomena (Kvale, 1996, p. 5f).
In the next section, the study is described and the framework for analysing interview data introduced. In the second section, excerpts from a group interview are analysed. The final section concludes the analysis and reflects upon what can said about difficulties in learning mathematics.
Researching childrens perspectives
The research reported on in this paper is part of a larger study. The empirical material consists of interviews with children aged 10 or 11 years and observations of their mathematics classes. The children were students in a Year 4 class in a Danish Folkeskole (public primary and lower secondary school). I explained my presence in their classroom by saying that I would like to learn from them what it was like be in Year-4, learn mathematics and sometimes find it difficult, something in which they were experts.
The mathematics lessons were observed for almost a whole school year on a more or less weekly basis. Three rounds of interviews were conducted. In the first all students but one were interviewed in groups of six or seven students. In the second approximately half of the students were interviewed in pairs or alone. Half of the students were also interviewed in pairs in the third round. Some students took part in both second and third round. The interviews lasted from 30 to 45 minutes and were audio recorded; the group interviews were video recorded as well.
In this paper, I interpret key excerpts from the first group interview, which took place six weeks into the school year (September 2006). Following Kvale (1984), the excerpts are interpreted on three levels. The first level is a summary that the interviewees would recognise as a fair rendering of their statements in a language accessible to them and within their horizon of understanding. The second level of interpretation may transcend the interviewees understanding but remains within a common sense context of understanding. It can include general knowledge about the interviewees statements, address the form of the statement, the way it is expressed, and read between the lines. At the third level of interpretation, statements are interpreted within a theoretical framework or perspective. The interpretation is likely to transcend the interviewees self-understanding and a common sense understanding. Here, the theoretical frame is the notion of childrens perspectives as described above. Thus, I will be looking for how the children make sense of their experiences with school mathematics learning and what meaning they ascribe to school mathematics in their life world.
To some extent, the extracts and interpretations focus on one student, Kalila, while letting the other students in the group interviewed provide the context. The first reason for this is that the paper is exploring the possibilities provided through a particular methodology and conceiving of child perspective as a theoretical notion. Looking at one child in one interview context would constitute a simple case for trying out the methodology. The second reason is that observations and other interviews pointed to Kalila as a student who was particularly articulate. In the context of the group interview, Kalila could be seen as acting as a spokesperson for the students in the group in that she often reiterated and extended what the other students were saying. Sometimes they actively endorsed her statements, but generally, they neither contradicted nor challenged her. Hence, there are no reasons to think that her perspective was very particular or idiosyncratic. Even if her perspective was not coinciding with all of the other students, it outlined the sort of landscape within which students perspectives are to be constructed.
Transcripts and translation
The extracts are quoted in some length, and the original Danish transcript is translated in English. There is a difference between the researchers voice in a summary of an interview and the interviewees own voice (although filtered through a transcription). Goodson and Sikes (2001; ch. 3) discuss how in some cases the difference can be dramatic as to the impression the reader gets of the interviewees and their stories. As children of the age in question express themselves differently from adults, linguistically, grammatically and from a different perspective, it is important in the present context to render their ways of expressing themselves as a starting point of the interpretation.
The Danish transcript is provided so that readers familiar with Scandinavian languages have an opportunity to read the material that is analysed. The transcript is close to the wording of the recordings. A translation in written English that conveys the subtleties of (a transcript of) childrens spoken Danish is not always possible. When having to compromise, a rather literal translation has been chosen at the expense of what might be considered good English.
Background
There were twenty students in the class with equal numbers of boys and girls. The children also distinguished themselves as Arabs or Danes. In this terminology, half of them were referred to as Arabs and the other half as Danes. All of the children were born in Denmark and spoke the same regional dialect of Danish. The difference was that the Arabs were descendents of parents emigrated from the Middle East. For the group interviews, an even distribution of girls and boys as well as of children of Danish and non-Danish descent was sought.
When I began my observations in the beginning of the school year, the class had just become Year-4, moved from green corridor of the beginners level (Year 0 to 3) to their new classroom in blue corridor of the middle level (Year 4 to 6). From being the older among the youngest students, they were now the younger ones in the middle group of students. Moving into the middle level also meant having new teachers, most importantly a new Danish teacher and a new mathematics teacher. The Danish teacher was also class teacher and took the classes in English and Religious Knowledge. The mathematics teacher took the classes in music and science. These changes seemed to cause some unrest in the class dynamics and made the children unsettled in varying degrees. Kalila, for example, had many conflicts with her class mates and the mathematics teacher in the first months.
The mathematics teacher began the year by focusing on the multiplication tables. For each of the tables, she let the children produce a set of cards with all the q u e s t i o n s a n d a n s w e r s b e l o n g i n g t o a t a b l e , e . g . t h e q u e s t i o n s 3 "1 , 3 "2 , & , 3 "1 0 a n d t h e a n s w e r s 3 , 6 , & , 3 0 . T h e s t u d e n t p l a y e d g a m e s w i t h t h e c a r d s , a n d t h e y c o u l d t a k e t h e m h o m e t o a s s i s t t h e m i n p r a c t i s i n g t h e t a b l e s . T h e t e a c h e r l e t e a c h s t u d e n t c hose one table, sometimes more, for homework and checked their knowledge of the table afterwards. These activities took place in the weeks preceding the interview.
Constructing Kalilas perspective
This section deals with four excerpts from a 30 minute group interview with Kalila, Bahia, Isabella, Simon, Ishak, and Hussein. Each excerpt relates to the dialogue following one of the main question that structured the interview. The dialogues are analysed according to the three levels of interpretation. At the first two levels, the childrens understanding is summarised and a common-sense interpretation is suggested. The third level focuses on Kalilas contributions letting the other students contributions serve as background with the aim of constructing Kalilas perspective, i.e. her ascription of meaning to her experiences with school mathematics education.
In the transcript comma (,) is used to ease the reading by marking a new beginning of a sentence and repetitions; brackets around words ( ) means that the transcript is uncertain; underscore (_) signals that a few words are unintelligible; hyphen (-) signals a pause; text in sharp brackets [ ] gives the reader information that would be evident in the actual context of the interview. In the full transcript, the statements were numbered consecutively. In the interpretation, these numbers are referred to in brackets.
Can you tell me about something you have learned in mathematics?
The dialogue reproduced in the transcript began 12:40 minutes into the interview and lasted 4:20 minutes. Not all what was said in the period related to the question; this part has been omitted.
Transcript 1. Extract from 12:40-17:00 mm:ss of group interview 1
351Troels Kan I fortlle mig om noget I har lrt i matematik?Can you tell me about something you have learned in mathematics?354Simon PlusPlus355Hussein Vi har lrt at regne plus minusWe have learned to do plus minus356Ishak TabellerTables357Hussein Tabeller og minus og gange og dividereTables and minus and times and divide358Kalila Alts ved du hvad (der) er godt i fjerde klasse? Det er at (hun) [lreren] giver nogen tabel for og s siger hun _ fem gange tre og s skal man jo sige detDo you know what is good in year 4? It is that (she) [the teacher] sets some table[s] [for homework] and then she says _ five times three and then you must say it359Isabella Ja det kan jeg ogs godt lideYes I like that too360Kalila _ Og det sdan, det lrer man jo sdan lidt mere_ And that like, that you learn like a little more361-365Hussein bekrfter og Isabella og Kalila genbekrfter at de kan lide at lre tabellerHussein confirms and Isabella and Kalila reconfirm that they like learning tables.366Troels Hvorfor, hvad er det sjove ved det?Why, what is the fun about it?369Kalila Det er sdan at nogle, alts hun siger for eksempel sdan at vi flger med i tavle og s siger [lreren til] mig Tre gange tre? Og s, og s er det sjovt. JaIt is like that somebody, like she says for example that we follow whats happening in [the] board and then says [the teacher to] me Three times three? And then, and then it is fun. Yes380Kalila Ved du hvad jeg godt (kan lide)? Hun stter krydser hvis man kan. Til sidst for eksempel i gr Kalila du kan jo ni-tabellen for eksempel. _ S skal du lige have [et kryds]Do you know what I (like)? She puts crosses if you know. At the end for example yesterday Kalila you know the nine [times] table for example. _ Then you must have [a cross]381Troels Hvad er det gode ved at hun stter krydser?What is the good about that shes putting crosses?382Kalila Det er at s ved man jo hvad man kan. Hvis hun nu stter krydser bare ja det kan du godt s kan man jo altid sige Jeg kan fem seks syv otte ni ti og videre videre videre ogs sdan nr man ikke kan demIt is that then you know what you can. If she just puts the crosses yes you know that then you can always say I know five six seven eight nine ten and on on on also when you do not know them383Isabella forklarer at lreren noterer ved at stte enten en bagudvendt skrstreg [\] eller en fremadvendt skrstreg [/]. S ved hun detIsabella explains that the teacher keeps track by putting either a back slash [\] or a forward slash [/] respectively. Then she knows it384Troels OkOk385Kalila Alts s ved hun det. S er hun sikker p nr, frst hvis man ikke kan det s stter hun en prik. Hvis man kan det sdan midt imellem s stter hun en streg. Hvis det er helt korrekt s et kryds. Det er sdan man lrer megetLike then she knows it. Then she is sure when, first if you do not know it then she puts a dot. If you know it like in the middle then she puts a line. If it is completely correct then a cross. That is how you learn much389-396Simon kan lide at lre tabeller, men kun lidt. Ishak kan lide det fordi det er ligesom syvtabellen. Syv fjorten, enogtyve og s videre. Isabella er enig i detteSimon only likes learning the tables a little. Ishak likes it because it is like the seven times table. Seven fourteen twenty one and so on. Isabella agrees to this397Kalila Det der er godt ved det er at man fr en uddannelseWhat is good about it is that you get an education398Troels Er det godt at f en uddannelse?Is it good to get an education?399Kalila Ja det er rigtig godt fordiYes that is really good cos400Isabella Det tror jegI think so401Kalila Ligesom mig jeg vil godt vre en designerLike me I would like to be a designerSummary of the students understanding (1)
The students have learned plus, minus, times, divide and the times tables. Apart from Simon, they really like the way they work with the times tables: tables are set for homework and then the teacher ask them multiplication questions that they have to answer. The teacher makes notes about how well they know the tables. Kalila thinks this tells you what you know and that she learns well this way. That is good because then you get an education and may become a designer.
Common-sense interpretation (1)
Simon, Hussein and Ishaks answers to my question about what they had learned in mathematics dealt with mathematical topics (354-357). Kalila focused on how they worked with the multiplication tables. She highlighted that the teacher set tables for homework (358), that she tested the students table knowledge in class (358, 369), and that she kept a record (380), the details of which Kalila and Isabella reported in minute detail (383, 385). It was important that the teacher was serious in the recording (385) because the teachers record guaranteed to Kalila and Isabella that they knew the tables (381-5). The students liked this kind of teaching (358, 361-5, 389-96). For Kalila it was because it facilitated her learning (360, 385) and gave her an education (397) that pointed towards a future of her choice (398-401). She described her experience as being fun (369). What seemed to be fun was being asked questions from a times table you had practiced and being able to answer - and if not, to find it manageable to practice more for next lesson (369).
Kalilas perspective (1)
The question now is what may be said about Kalilas perspective on learning mathematics. How did she make sense of and what meaning did she ascribe to learning mathematics?
The students mentioned the four basic operations and the multiplications tables as examples of what they had learned. They did not mention examples of what is often called practical applications of mathematics, like converting a recipe for one number of persons to another number; working on this and similar problems had also been a substantial part of their lessons. Thus, the students ascribe the meaning to mathematics that the subject primarily comprises the basic rules and the times tables. This meaning ascription could be a making sense of their mathematics education or picked up from the popular conception of school mathematics. If the latter was the case, it can be inferred that their mathematics education either had not sought to challenge this conception or had been unsuccessful in doing so.
To Kalila (and Isabella) the authority to judge her learning seemed to reside solely with the teacher. The ticks in the teachers notes physically manifested that Kalila knew a table. It was not Kalilas subjective experience. This indicates that to Kalila learning the multiplication tables and in a wider sense also mathematics was not a way of systematising number relationships, but a question of getting it right, that is come up with the expected answer. The reasons for why this answer was expected or the logical connections in which in was embedded were not part of the business.
Thus, Kalila and the other students did not seem to ascribe the sort of meaning to mathematics that the curriculum talked about (Undervisningsministeriet, 2003). In this light, it was not accidental that Kalila answered a question about what with an answer about how. The key word in the how is fun. The experience of fun seems to link two dimensions of time. In the horizontal dimension of time, the immediate moment, the here-and-now, fun was derived from the liking of being able to comply with the requirements and expectations of the moment. When the teacher ask a times question, I can answer.
In what could correspondingly be called the vertical time perspective, the past, the present and the future, Kalila linked fun to like, learn, education, and job of own choice. The emotional experience of the moment (fun, like) informs and is formed by a future perspective (education, job). Becoming a designer takes education, which takes mathematics, which takes multiplication tables, which takes teachers ticks. This chain linking the school mathematics practice to her foreground (Skovsmose, 2005a; Skovsmose, 1994) constituted her perspective.
How is it when mathematics is easy and how is it when mathematics is difficult?
The next dialogue followed immediately after the first and was rather short.
Transcript 2. Extract from 17:00-17:41 mm:ss of group interview 1
443Troels Hvordan er det nr matematik er let og hvordan er det nr matematik er svrt?How is it when mathematics is easy and how is it when mathematics is difficult?444Bahia Matematik nr det er nemt s kan man lave, hvis man fr et ark s kan man lave det p to eller 10 minutter. Hvis det er svrt s sidder man og tnker og s begynder man mere at regne. Og hvis man slet ikke kan det s begynder man bare med at kede sig eller ogs s springer man over det. Mathematics when it is easy then you can do, if you get a [work] sheet then you can do it in two or ten minutes. If it is difficult then you sit and think and then you begin more to calculate. And if you cannot at all then you just begin to be bored or you skip it445Kalila Hvis det er svrt og man virkelig ikke kan det s gider man ikke det. Og man har prvet at regne det, ik , og man ikke kan. S sidder man sdan [albuen p bordet og hagen p hnden] S sidder man og snakker og render rundt og. Mske render man ikke lige rundt men s skal man ligeIf it is difficult and you really cannot then you dont feel like it. And you have tried to do it, havent you, and you cannot. Then you sit like this [elbow on table and chin in the hand] Then you sit and talk and run around and. Perhaps you not exactly run around but then you just have to448Isabella Spidse sin blyantSharpen you pencilSummary of the students understanding (2)
Bahia says that when mathematics is easy you can do a work sheet in a few minutes. When it is difficult, you have to think and calculate. When you do not know how to do it, you get bored or skip it. Kalila adds that when you cannot you do not feel like doing mathematics and then you start doing things you are not supposed to do.
Common-sense interpretation (2)
Bahia gave a clear answer to the question. Doing mathematics is (often) doing work sheets. These are either easy and quickly done, or they are difficult and requires thinking and calculation, or they are impossible to do and you get bored and skip them (444). This link between not being able to do what mathematics teaching requires of you and being bored, was elaborated on by Kalila. She stressed the experience of not succeeding despite trying hard, and how this undermined her will and stamina. She clearly perceived how this unpleasant situation raised unrest that was reacted out in bodily expressions like talking and running around (445) or just doing something different as Isabellas suggestion of sharpening her pencil (448).
Kalilas perspective (2)
Bahia expressed a common understanding among students that mathematics tasks should be quickly solvable (Schoenfeld, 1989). If this is not the case, then something is wrong either with the students ability or with the tasks. Kalila did not object and probably held the same view.
In the analysis of the previous extract it was shown how the experience of being able to started the chain of fun like learn education job. In the present extract, the opposite experience of not being able to is merged by Kalila into a cluster of cannot, being bored, dislike (not feeling like), and bodily unrest. Whereas the chain of fun like learn education job above links the experience of the moment with the future, its counterpart only mostly deals with the here-and-now experience. It logical continuation, not learn no education no job is not expressed. However, a closer look indicates that it nonetheless could be active.
Kalila described her feelings of dislike and of being bored when she could not do the mathematics that was set for her however hard she tried. She also described how this caused her to talk and walk around. I take this as a sign of stress. Consequently, it is suggested that failing causes stress. Now, what is stress? When the body is threatened, it becomes alerted, it wants to fight or flee. If none of these opportunities are available, the adrenalin cannot be transformed into appropriate action, and the result is stress. Hence, if Kalilas bodily unrest is taken as a sign of stress, this indicates that her body was in an alerted state with no available possibility of relevant action, which means that she was threatened. She had fought, tried hard without succeeding. She did not feel like doing more, she entered a state of dislike, her energy seeped out of her. She could not flee. She was caught in a no-way-out situation. If this analysis is accepted, it follows that not succeeding with mathematics is threatening. A plausible reason could be that then the continuation of cannot, being bored, dislike with not learn no education no job even if unsaid is active in her foreground. A future job of her liking depends on succeeding in mathematics. Not succeeding equals the opposite. In her perspective on mathematics, her future was at stake.
Why do you think the adults have decided that children should learn mathematics?
The dialogue following this question was rather focused as may be seen from the small number of omitted lines in the transcript.
Transcript 3. Extract from 18:58-22:00 mm:ss of group interview 1
476Troels Jeg kunne tnke mig at sprge jer om hvorfor tror I de voksne har bestemt at brn skal lre matematik i skolen?I would like to ask you why do you think the adults have decided that children should learn mathematics in school?...478Kalila Hvis man arbejder i en slikbutikIf you work in a sweet shop484Hussein Man kan ikke f en uddannelse hvis man ikke lrer matematik og sdan noget. Fordi hvis man skal arbejde i et tr_ og man laver et bord og det ikke er lige langt p begge sider s vil det vre et problem fordi man ikke kunne regne. Og man kan ikke f en uddannelse hvis man ikke kan lse og skrive og hvis man ikke kan regne og noget som helst s kan man ikke f en uddannelseYou cannot get an education if you do not learn mathematics and such. Cos if you should work in a wood_ and you make a table and it is not the same length on both sides then it would be a problem because you could not calculate. And you cannot get an education if you cannot read and write and if you cannot calculate and nothing then you cannot get an education491Bahia Hvis jeg nu arbejdede i en butik og du kbte den der lille, et ur for en krone og du s kom og gav mig en hund, en hundredekroneseddel. Hvad skal man s give tilbage og man ikke kan matematik s ved man jo ikke noget. S derfor skal man jo ikke bare plusse det hele. S ved jeg det. S er det nemlig nioghalvfems kroner. S skal jeg give dig tilbage.If I work in a shop and you bought that little, a watch for one krone and you then gave me a hundred kroner note. What should you then give back and you cannot do mathematics then you do not know anything. So therefore you should not just plus all of it. So I know. It is ninety nine kroner you see. So I shall give you change492Isabella Tror du, et ur koster ikke en kroneDo you think, a watch is not one krone493Prisen p et ur diskuteres i baggrunden mens Kalila taler:The price of a watch is discussed in the background while Kalila is talking:494Kalila Og det er ligesom i benzin_ Jeg ved ikke _ . Det har jeg bare hrt at der er nogen som kom til at stte benzin ned til to og et eller andet halljsa _ . Og s har de mistet, s har de mistet et eller andet med femten hundredeIn is like in petrol_. I dont know _. I have just heard that there is someone who happened to put petrol down to two and something _. And then they have lost, then they have lost something like fifteen hundred495Troels ok. S det var fordi de ikke kunne regne de kom til at?Ok. So it was because they could not calculate that they happened to?496Kalila Naj det er ikke derfor. De kunne jo godt regne. De kom til detNo that is not why. They could calculate. They happened to do it497Troels De kom til at stte prisen for langt ned?They put the price too far down?498Kalila JaYes503Troels Hvorfor tror du at brn skal lre matematik i skolen Simon?Why do you think that children should learn mathematics in school Simon?504Simon For s ved man hvor meget benzin man skal hlde p en crosserCos then you know how much petrol to put on a crosser [motocross bike]520Ishak Hvis man nu arbejder i en butik og der er en der kber mange ting og det koster hundrede kroner og s en han snyder ham med halvtreds s ved han ved han det _If you work in a shop and someone buys many things and it costs hundred kroner and then one he cheats him with a fifty then he knows it _527Troels Hvorfor tror du [Isabella] at brn skal lre matematik i skolen?Why do you [Isabella] think that children should learn mathematics in school?...529Isabella Ellers kan de jo ikke regne og s kan de ikke f en uddannelseOtherwise they cannot calculate and then they cannot get an education530Troels Ellers kan de ikke regne og s kan de ikke f en uddannelse?Otherwise they cannot calculate and then they cannot get an education?531Isabella Ja man skal jo kunne f en uddannelse fr man kan detYes you should be able to get an education before you know itSummary of the students understanding (3)
You must learn mathematics in order to get an education and a job. In order to work in a shop you must be able to calculate so that you can give the correct change, and you must know the notes to avoid being cheated.
Common-sense interpretation (3)
The students give three types of reasons for why they must learn mathematics at school that relate to either education, job or leisure time activities. Some of their reasons are explicitly or implicitly justified or contextualised by a mathematical topic, either money or measurements. The reasons and the mathematical topics are summarised and organised in table 1.
Table 1. Reasons given for school mathematics and topics used for exemplifying
Reason
Math topicEducationJobLeisureMoneyKalila: shop assistant in sweet shop, petrol station (478, 494-8)
Bahia: shop assistant, give change (491)
Ishak: shop assistant, not cheated with money (520)
Isabella: price of watch (492)Measurement (length, volume)Hussein: wood industry (484)Simon: petrol on motocross bike (504)UnspecifiedHussein (484) and
Isabella (529): no education without being able to calculateIsabella: education necessary to get a job? (531)
Kalila: in order to become a designer (397 401intranscript 1)
The main reasons were having a job and getting an education. Working in a shop and having to deal with money transactions was the dominant job example. No mention was made of education preceding becoming a job assistant or being a shop owner. Money was the most often mentioned mathematical topic, measurements being the other. However, the examples seem to have more to do with recognising that numbers are related to certain phenomena than actually involving mathematical operations. Hussein knew that numbers gave the lengths of the sides of tables. Simon knew the same went for volume of petrol. Isabella knew what number of kroner could be a reasonable price for a watch. Ishak knew the difference between a fifty kroner note and a hundred kroner note. Kalila knew that two kroner per litre was a low price for petrol, she thought that fifteen hundred kroner was a big loss, and she knew that the small litre price was connected to the total loss. Only Bahias example about giving the right amount of change involved a mathematical operation.
Hussein, Isabella, and Kalila saw mathematics as necessary for getting an education. Kalila and Isabella (perhaps) saw education as necessary for getting a job either as such (Isabella) or a specific job (Kalila). None of them expressed ideas about why or how mathematics was necessary.
Kalilas perspective (3)
The students largely gave reasons for learning mathematics at school related to their future, that is to their foregrounds. They thought that mathematics, in line with reading and writing, was necessary to get an education and a job. Apart from this gate keeping function, the role of mathematics in relation to education and jobs following education was unspecified. They saw clear connections between numbers, basic mathematical operations, and money-handling jobs. Possibly, the childrens backgrounds also are at play. The students, who gave money examples, Kalila, Bahia, and Ishak, are all descendents of emigrants from the Middle East. These immigrants often run shops, so these children have shop keeping as part of their environmental reference. Kalilas father was a shopkeeper.
In the analysis of the former transcript a negative cluster was found consisting of cannot, being bored, dislike, and bodily unrest. The hypothesised extension of this cluster into a chain with not learn, no education, no job is clearly stated in this transcript when the ability to calculate and use numbers in everyday and workplace situations on par with reading and writing are seen as prerequisites for getting an education and/or a job.
What is the most important thing you have learned in mathematics?
This question was the last in the interview. The students were getting tired and their concentration was waning. The two sets of statements prompted by the question are given in the transcript.
Transcript 4. Extract from 24:41-28:34 mm:ss of group interview 1
596Troels Hvad I synes har vret mest interessant eller mest spndende eller mest vigtigt [af alt det I har lrt i matematik mens I har get i skole]What do you think has been the most interesting or the most exciting or the most important [of all that you have learned in mathematics while you have gone to school]600Hussein Det mest spndende det var dengang vi lrte om gange og minus og plus og dividere.The most exciting was when we learned about times and minus and plus and divide602Kalila Nr man kan s er det jo rigtig rigtig sjovt i matematik ik. Hvis man kan alt, gange fem hundrede og fem s er man jo hurtig ik. S er det jo sjovt hvis man kan, hvis man nu fr et kopiarkWhen you know then it is really really fun in mathematics isnt it? If you know everything, times five hundred and five then you are quick arent you? Then it is fun you see if you can, if you get a copy [work] sheet603Troels S det er sjovt at vre god til detSo it is fun to be good at it604Isabella JaYes605Kalila Ja nr man er god s er det ogs sjovt at lave det. Men nr man ikke kan s er det kedeligt nr man ikke laver detYes when you are good then it is also fun to do it. But when you cannot then it is boring when you dont do it606Isabella Hvis man er drlig til det s er det kedeligtIf you are bad at it then it is boring607Kalila _ nr man ikke ved det_ when you do not know628Bahia jeg vil lige sige noget. Det der er godt ved matematik det er nr man kan det I want to say something. What is good about mathematics it is when you know it629Troels Hvordan kommer man til at kunne det? How do you get to know it?630Kalila Det er nr man hrer efter i timerneIt is when you listen in the lessons631Simon Og ver sigAnd practice632Bahia Hvis man kan tabellerne s kan man alt i matematik faktisk.If you know the tables then you know how to do everything in mathematics actuallySummary of the students understanding (4)
To learn how to calculate with times, minus, plus and divide has been the most exciting to Hussein. Kalila, Isabella and Bahia think mathematics is fun when you are good at it and know how to do it. If you are bad at maths and do not know how to do it, then it is boring. You learn mathematics by listening in the lessons and practice, especially the times tables.
Common-sense interpretation (4)
Hussein said that the most exciting was to learn the four basic rules (601). Kalila, supported by Isabella, said that mathematics was fun when you know and are quick and good at it (602-5). When you cannot, you are bad at it, and it is boring and (605-7). So while Hussein reacted to my what question by talking about mathematical topics and the accompanying emotion, Kalila (and Isabella) only talked about how, i.e. their experiences with learning mathematics. They phrased them in terms of competence/ability (can/cannot or know/not know), emotion (fun/boring) and identity (good/bad at maths). Bahias succinct statement What is good about mathematics it is when you know it (628) is similar to Kalilas and Isabellas in the absence of mathematical content as reasons to their perception of mathematics.
Kalilas perspective (4)
The dialogue repeated and extended the chains discussed in the previous sections. To the chain fun like learn education job is added excited, know, quick and good at maths. Its negative counterpart cannot bored dislike unrest not learn no education no job is supplemented with boring and bad at maths. With her statement What is good about mathematics it is when you know it, Bahia seemed to say that school mathematics is about competence as such and not about competence in something, mathematical topics for instance, and possible benefits from such competence. Contrary to her, Hussein gave mathematical subject matter, the basic rules, as sources of his excitement. He indicated no clues to why he found them exciting, and the order in which he listed them was different from the ones that would reflect their mathematical connections (plus - minus, times - divide or plus times, minus divide). So, maybe Hussein ascribed the same meaning to school mathematics as Bahia, namely that school mathematics is about competence or mastery of what is considered to be school mathematics. The circularity of this perception begs the question why school mathematics is worthwhile. An answer may be found in Kalilas (and Isabellas) statement. She went a step further than Bahia in pointing to a double meaning of good. Kalila seemed to say, What is good about mathematics is when you know it because then you are good at mathematics, or even more condensed What is good about mathematics is when you are good at mathematics. As seen in the analysis above, Kalila and the other students were well aware of the function of school mathematics as gatekeeper to her foreground of education and job. Being good at school mathematics promised passing through the gate.
The importance ascribed to the teachers ticks as a guarantee of Kalilas learning was highlighted in the analysis of transcript 1. This learning theory is elaborated in the present transcript when she explained that she would get to know mathematics if she listened in the lessons. In saying so, she paraphrased what the teacher often said. The implication of listening was remembering and following the instructions given, like practicing the multiplication tables. Her understanding, the sense she made, seemed to be that the teacher on behalf of mathematics has the authority to judge what is right or wrong. You become good at mathematics by getting it right. You get it right by listening to the teacher and doing what you are told. In this coherent understanding of mathematics learning students are ascribed a rather passive role and little space is left for them as active participants.
Relating Kalilas perspective on being in difficulties with mathematics
In this section, I will reflect upon the methodological ideas in this paper, that is interviewing a group of ten to eleven year old students, analysing the interview using the three level interpretation process, and constructing a childs perspective. Finally, I discuss what may be learnt from children in difficulties with learning school mathematics.
The interview with the students was a semi-structured life world interview (Kvale, 1996) with the purpose of obtaining descriptions of the childrens experiences with school mathematics learning that would allow interpretations of the meaning that the children ascribed this part of their life. At an interview technical level this meant that each main question was followed up by questions or reactions aimed at inviting and supporting the children to tell more, explore the issue further, or test my understanding of what they had said. The transcripts and the analysis show that it was possible to conduct this form of interview with ten-eleven year old students. The dialog let them form and express their point of view about their experiences with learning school mathematics.
The interview was analysed using a rather strict, almost pedantic, interpretation procedure in three levels. This was meant to discipline the interpretation process and make it transparent. The discipline was needed in order to avoid an unknowing conflation of my adult, mathematics teacher, researcher perspective, that is my meaning ascriptions and sense-making with those of the children. In the phrasing of the ethnographic research tradition it was an attempt to objectify myself as researcher (see for instance Eisenhart, 1988; Reed-Danahay, 2005; Prieur, 2002).
At the first level of interpretation, the interview showed that the students made sense of their school mathematics in a way that connected multiple dimensions. Summarising their understanding, the students had learned about plus, minus, times, divide and the times tables in mathematics. They felt they learned well when they had times tables of their own choice set for homework, and then had the teacher ask questions from the tables. It was important that the teacher made careful notes of their table knowledge because that testified to them that they had learned mathematics. The way they learned mathematics was by listening in the lessons and through practice. They found mathematics easy when work sheets could be done in a few minutes and difficult if they had to think. When they knew how to do it, they felt good at mathematics and that it was fun. If they could not, they felt it was boring and did not like doing mathematics. They thought that they must learn mathematics in order to get an education and a job.
As seen from the summary, the students sense making addressed a comprehensive range of questions on what school mathematics is about, what productive ways of teaching are, how they learn, what signs tell them that they are learning, what it is like to learn mathematics, and why they should learn mathematics. Their sense making apparently seamlessly connected their immediate experiences in the classroom with the prospect of their future life into a coherent whole.
The second level of interpretation, added to the children's understanding by pointing to that especially Kalila, sometimes supported by Isabella, addressed the emotional experience of learning mathematics. Also, these two students were particularly observant of the teachers note practices.
At the third and theoretical level of interpretation Kalilas perspective on school mathematics was constructed. This interpretation presumed that the notion of a childs perspective is an analytical construct. It is not a given, not even in an archaeological sense of dug out bits and pieces from pottery or a dinosaur skeleton that I have put together in an as-sensible-as-possible way because that would assume that an original whole once existed and thus also a right way to put the pieces together. It is my making sense of the interview from the assumption that sense and meaning of school mathematics are not imprinted on a passive child, but the child's active attempt to come to terms with her experiences with school mathematics and integrate them into a coherent identity and meaningful life world.
What about difficulties a perspective from a child at the edge
I envisaged that children, who performed poorly in mathematics, could be quite reflective about the norms and expectations at play in school mathematics (Lange, 2007). Because of the high social valorisation of mathematics, school mathematics constitutes an important social norm to students. Children who do not meet the expectations put to them by school mathematics could be expected to become acutely aware of the existence and nature of the norm in a way that children who fulfil the expectations do not. Having their belonging to a highly charged normality like school mathematics questioned would spur their reflective activity concerning their identity and sense making of school mathematics.
Judged from my observations and talks with her in the classes, Kalila struggled with learning mathematics. She did not find it easy and was happy when she succeeded, as was the case with the multiplication tables. Being able to learn mathematics as it was presented to her, was not a given. She could not rest confidently in a feeling of been a sufficiently good mathematics learner.
Kalila was not positioned as disadvantaged in the classroom (Valero, 2007). The school did not categorise her as having special educational needs in mathematics; in fact, the school rarely provided special needs teaching or assistance in mathematics as it did in reading. The new mathematics teacher only later in the school year recognised Kalila as low performing, and to my eyes she conducted her teaching without any public ranking of the children according to her perception of their mathematical performances. Still, Kalila was sensitively aware of the consequences of not succeeding with mathematics. In the interview, she was the first to draw attention to the link between learning mathematics and her dreams for her future. A number of times, she was the one who talked about how learning mathematics was experienced. So even without specific public labelling, Kalila could be seen to consider herself as being at the edge. Possibly this precarious position contributed to her being especially reflective and articulate.
Referring to Mellin-Olsens distinction between instrumental and social rationales for learning mathematics (Mellin-Olsen, 1987), Kalilas perspective seemed to include only instrumental rationales and be disconnected from topic related reasons or social rationales for her learning of school mathematics with money as a possible exception. Kalila and the other student displayed instrumental rationales for learning when they saw school mathematics learning as prerequisite for education and job later in life. The immediate experience of succeeding with the expectations presented by school mathematics, whether times tables to master or work sheets to complete quickly, seemed only to be related to the future, the vertical time perspective. To succeed was an experience of being good at mathematics and was felt to be fun; not succeeding was being bad at mathematics and felt to be boring. No other meaning seemed to be ascribed here-and-now, in the horizontal time perspective. School mathematics apparently was about being good at school mathematics. This perspective was coherent with Kalilas learning theory with no inherent meaning ascribed, how could she conceive of learning mathematics in any other way than listening and doing what she was told.
Her perspective from the edge highlights that much more than cognitive issues may be at play for children who struggle with learning mathematics. For Kalila a future of her choice was at stake.
Her instrumental rationales for learning school mathematics realistically reflect the socio-political context of schooling and valorisation of mathematics. However, the absence of social rationales is probably not conducive to her learning because it leaves her with no here-and-now meaning and no way of conceiving how her life world, her active imagination and thinking could contribute to her learning of school mathematics, and the other way round. One might speculate if children at the edge are especially prone to develop only instrumental rationales and hence that emphasising the importance of mathematics is not helpful to such students. If this is the case, then it is even more important for students at the edge that mathematics education counterbalance the strong socio-political incentives to instrumental rationales by encouraging and facilitating the formation of social rationales for learning school mathematics.
Acknowledgement
I want to thank Naomi Ingram, Tamsin Meaney, and Paola Valero for comments and suggestions for improving earlier versions of this paper and for linguistic assistance.
References
Eisenhart, M. A. (1988). The ethnographic research tradition and mathematics education research. Journal for Research in Mathematics Education, 19 (2), 99-114
Goodson, I. F. & Sikes, P. J. (2001). Life history research in educational settings. Buckingham: Open University.
James, A., Jenks, C., & Prout, A. (1997). Theorizing childhood. Cambridge: Polity Press.
Kvale, S. (1984). Om tolkning af kvalitative forskningsinterviews. Tidskrift fr Nordisk Frening fr Pedagogisk Forskning, 4 (3/4), 55-66
Kvale, S. (1996). InterViews. An introduction to qualitative research interviewing. Thousand Oaks - London - New Delhi: Sage Publications.
Lange, T. (2007). Students' perspectives on learning difficulties in mathematics. In L. . Johansen (Ed.), Proceedings of the 3rd Nordic Research Conference on Special Needs Education in Mathematics, (pp. 171-182). Aalborg: Adgangskursus, Aalborg Universitet.
Lange, T. (2008). The notion of children's perspectives. In D. Pitta-Pantazi & G. Philippou (Eds.), Proceedings of the Fifth Congress of the European Society for Research in Mathematics Education, (pp. 268-277). Department of Education University of Cyprus.
Mellin-Olsen, S. (1987). The politics of mathematics education. Dordrecht: D. Reidel.
Prieur, A. (2002). At stte sig i en andens sted. En diskussion af nrhed, afstand og feltarbejde. In M.H.Jacobsen, S. Kristiansen, & A. Prieur (Eds.), Liv, fortlling, tekst. Strejftog i kvalitativ sociologi (pp. 135-155). Aalborg: Aalborg Universitetsforlag.
Reed-Danahay, D. (2005). Locating Bourdieu. Bloomington: Indiana University Press.
Schoenfeld, A. H. (1989). Explorations of students' mathematical beliefs and behavior. Journal for Research in Mathematics Education, 20 (1), 338-355
Sfard, A. & Prusak, A. (2005). Telling identities: In search of an analytic tool for investigating learning as a culturally shaped activity. Educational Researcher, 34 (4), 14-22 25 Oct. 2005
Skovsmose, O. (1994). Towards a philosophy of critical mathematics education. Dordrecht: Kluwer Academic Publishers.
Skovsmose, O. (2005a). Foreground and politics of learning obstacles. For the Learning of Mathematics, 25 (1), 4-10
Skovsmose, O. (2005b). Meaning in mathematics education. In J.Kilpatrick, C. Hoyles, & O. Skovsmose (Eds.), Meaning in mathematics education (pp. 83-100). New York: Springer Science.
Undervisningsministeriet (2003). Flles ml - matematik. [Kbh.]: Undervisningsministeriet, Omrde for Grundskolen.
Valero, P. (2002). Reform, democracy, and mathematics education. Towards a socio-political frame for understanding change in the organization of secondary school mathematics. Copenhagen: Department of Curriculum Research, The Danish University of Education,
; < = J u v a
o
м}rg\g\g\gQgQFg; h* hy CJ aJ h* hF} CJ aJ h* h CJ aJ h* h7h CJ aJ h* h4 CJ aJ h9~) h* mHsH hy mHsH (h* ha CJ OJ QJ ^J aJ mHsH h* h.QT CJ OJ QJ ^J aJ h* ha CJ OJ QJ ^J aJ &h* h.QT 56CJ OJ QJ ^J aJ h* CJ OJ QJ ^J aJ h* ht CJ OJ QJ ^J aJ h* h* CJ OJ QJ ^J aJ < = J v $a$gd* $a$gd* $a$gd* $a$gd* gd* gd* gd* o
F c * + , 5 6 Q
"
Q
T
U
V
\
c
p
z
2 ? B i o p ȽȮȮz hyk h7h hyk h_' hI h-h h_' h%' hy hs h_' hz hvR h3- hpa h3- h7h hU h3- h3- hk h3- ha h h* hk CJ aJ h* h7h CJ aJ h* hy CJ aJ h* h4 CJ aJ 0p w } 2 l m { 1 : m r s < Z \ ] e f g мииЭКЏ}} h2- hIb h2- h$5 h|d h4 h4 hJL hpa hJL h10 h d hIb hhe hpa hB hB h h7h hTV@ hs6 h hpa hpa h*; hvR hvR h*; hvR hpa hI hu h_' hyk h_' 1g p s 9 : > ? H P Q ' ( ) d ; ÿαα{tp hwb hwb hwb hwb hK hwb hM hM hNp 6h9~) hK hM hfl hNp h2 *h{ hZJ hZJ h7h hZJ hZJ hu h2- hu hQ h7h h h2- h$5 h2- h` h2- he h2- hJ h2- hIb h2- h6: * " 7 < F Q R S j { ! * R l u
* - . 6 8 < h hT hG@ h3" h3" h3" hNp hNp ht~ hJm h.
hNp h{ h{ hZJ h6~S h9 h` hU h4 h=k ht hI h
h{] h2 hu hwb h$ 8< B I e m * 6 @ b z | $ - . ] n ;
h m r t Žŵ٩ɩѡѝɝɕ hDpE h+ hP7 hq# hKQ h6: h h< hz ht h7x h7x 6h hDFC h+ h3 hdk hz1 hNp hJm h~q hQ h hpa hkq hh h' h.
hL8 7
$ $ #) ?) + + - - / / ]3 ^3 6 6 e9 f9 t; ; x< y< < % $$a$gd* $a$gd* $a$gd* $a$gd* t P X
! ! # # # # # # # # # # # # # # # 2$ I$ $ $ $ $ 8% K% M% o% s% % % % % & 3& =& >& ?& ûҴݬݨݤ h- h@v h h: h: 6h: h*i h+; h*i hOp hn hQz h2 hf{ hDpE h h! 6h h C hf{ h hf{ h! hr hfm hVk h! h! hM hYY h+ hz1 hKQ 2?& F& G& K& _& j& l& q& v& & & & & & & & & & & & & & ' ' ' ' ' ' ' ' ' ' `( a( ( ( ) ) ") #) .) >) ?) P) U) V) ) ) ) ) ) ) ) ) ) ) * * * A* 鷳 h$ h" hf5 h? hQU h& h$9 hyk *h- h hs\
h NH hY h" h* hv hix hYY h h1A h hw_ h*i h@v h@v h ;A* C* J* T* U* b* c* k* n* o* r* * * * * * * * * * * * * w+ z+ }+ + + + + + , , , ", 8, D, E, O, c, k, , , , , , , , , , - /- 6- =- G- `- a- ÿ hf h
1K hB h h$ hM h& hv0 h_A h2L hIa hJ h hJ hJ hY| h& hj[ hS hf5 h9 hQU hi+ hVk hVk hi+ hVk hQU h" 8a- - - !. ". O. T. X. ]. g. s. . . . . . . / / / "/ / / 0 0 K0 Q0 e0 s0 0 0 1 12 22 :2 z2 {2 |2 2 2 2 2 2 \3 3 3 3 3 4 4 4 4 4 4 05 25 65 >5 H5 ^5 5 5 ල h> h; h? h; h % hBv= h- h6~S hn{ h6~S h- h@ h; 6h|E h6 h6: h2L hQ hn{ h h2L h; 6hB h; h" hv0 =5 6 6 ,6 66 86 a6 b6 e6 g6 6 6 6 6 6 6 6 6 6 7 7 u7 x7 y7 7 7 7 7 7 7 7 7 7 8 /8 =8 ^8 k8 q8 }8 8 8 8 8 8 d9 e9 f9 9 9 9 9 9 9 J: : ; 9; H; [; c; r; s; t; Ĺĵĵ쵱 h4Ad h& hO h
1K h > ø◞u◞# < hMs mHsH hZ< h# mHsH hZ< hMs 6 hMs hMs hMs h; hMs hs6 h; hMs hl7 hZ< h mHsH h; h h| h3 ht5 mH nH uh3 h3 h3 h]3 h]3 h hf h$ h6 h .< < < ;= <= @= K= P= d kd $$If F F # l
t $6 4 4
l a " $$If a$gd* $$If a$gd* P= Q= U= }= = " $$If a$gd* $$If a$gd* d kdp $$If F F # l
t $6 4 4
l a = = = = = " $$If a$gd* $$If a$gd* d kd $$If F F # l
t $6 4 4
l a = = = = > " $$If a$gd* $$If a$gd* d kdP $$If F F # l
t $6 4 4
l a > > > > > _> i> > > > > >
? ? ? ? ? ? ? ? _? c? k? ? ? ? ? ? @ @ @ l@ m@ @ @ @ @ @ @ @ @ @ A A A A A A A A A ŽŵŽԊ hZ< hrp mHsH hZ< h 6mHsHhZ< h6: 6mHsHhZ< h 6hZ< h 6hl7 hT h hZ< hl7 mHsH hZ< h mHsH hZ< h# mHsH h; h hZ< hMs 6 hMs hMs 2> > > > ^? " $$If a$gd* $$If a$gd* d kd $$If F F # l
t $6 4 4
l a ^? _? c? ? ? " $$If a$gd* $$If a$gd* d kd0 $$If F F # l
t $6 4 4
l a ? ? ? ? @ " $$If a$gd* $$If a$gd* d kd $$If F F # l
t $6 4 4
l a @ @ @ m@ @ " $$If a$gd* $$If a$gd* d kd $$If F F # l
t $6 4 4
l a @ @ @ @ A " $$If a$gd* $$If a$gd* d kd $$If F F # l
t $6 4 4
l a A A A A A " $$If a$gd* $$If a$gd* d kd $$If F F # l
t $6 4 4
l a A A A A B " $$If a$gd* $$If a$gd* d kd` $$If F F # l
t $6 4 4
l a A A A A A A A B B B B AB GB HB IB JB [B ]B |B B B B B C !C AC tC xC C C C C C C C C *D AD FD ^D bD hD 7E E E E E E F جإءإ؝ؕ hZ< h6: 6mHsHhZ< h 6hZ< h 6h3 hl7 h; h ht?t hZ< h2Z mHsH hZ< h# mHsH h4Ad h h4Ad hT hrp h hrp h@ hrp h hZ< h mHsH hZ< hT mHsH0B B B B B " $$If a$gd* $$If a$gd* d kd $$If F F # l
t $6 4 4
l a B B B AC C " $$If a$gd* $$If a$gd* d kd@ $$If F F # l
t $6 4 4
l a C C C *D ]D " $$If a$gd* $$If a$gd* d kd $$If F F # l
t $6 4 4
l a ]D ^D bD 7E E " $$If a$gd* $$If a$gd* d kd $$If F F # l
t $6 4 4
l a E E E F G " $$If a$gd* $$If a$gd* d kd $$If F F # l
t $6 4 4
l a F F F F F F F F F F F G G G #G 'G +G 1G H KH OH H H H H H I I I I I I I J J ;J =J BJ DJ HJ NJ Ƚ϶xphh붫 hZ< h2Z 6hZ< hG9! 6hZ< h# 6hZ< h2Z 6mHsHhZ< h8U 6ht?t hf]% h hZ< h mHsH hZ< h# mHsH h; h hZ< h mHsH h; h h hZ< h3 6hZ< hf]% 6hZ< hrp 6hZ< h 6hZ< h 6mHsH (G G G #G &G " $$If a$gd* $$If a$gd* d kd $$If F F # l
t $6 4 4
l a &G 'G +G H H " $$If a$gd* $$If a$gd* d kdp $$If F F # l
t $6 4 4
l a H H H I CJ " $$If a$gd* $$If a$gd* d kd $$If F F # l
t $6 4 4
l a CJ DJ HJ J J " $$If a$gd* $$If a$gd* d kdP $$If F F # l
t $6 4 4
l a NJ J J J J K K K +K IK MK UK cK oK sK yK zK {K K K K K K L L 7M jM kM M M M M M M M M M M N N N N N N N 'N +N 1N 4N KN LN PN QN WN kN lN sN Ę hE h hY,[ h6 h h4 6hWK h# h4 h`_ h3 ht?t hY ht hs h hf]% hZ< h2Z mHsH hZ< h# mHsH h; h h hZ< h mHsH8J J J J K " $$If a$gd* $$If a$gd* d kd $$If F F # l
t $6 4 4
l a K K K +K HK " $$If a$gd* $$If a$gd* d kd0 $$If F F # l
t $6 4 4
l a HK IK MK cK nK " $$If a$gd* $$If a$gd* d kd $$If F F # l
t $6 4 4
l a nK oK sK K K " $$If a$gd* $$If a$gd* d kd
$$If F F # l
t $6 4 4
l a K K K M M R .R R R V V XX YX Z Z \ $a$gd* $a$gd* d kd
$$If F F # l
t $6 4 4
l a sN tN |N N N N N N N N N N N O O O %O .O 9O ;O @O JO LO VO [O gO hO O O O O O O O O O O O O O O O O O P P P 0P 8P 9P ;P
Ұ ʰ հ ۰ [ ݰ ܰ \ \ \ \ \ 0 ڱ ʱ ] ] ] ] ] ^ ^ _ _ ÿÿûǝ|dz m < hm mHsH hZ< h# mHsH hZ< h mHsH h; h h5[c h ht5 mH nH u h hf h h h> h hG{7 h"P hJc; hp h15 hZO h hZO h hp 6 h h! h hp /\ 0] }] ~] ] ] ^ i^ j^ n^ _ ` j d kd
$$If F F # 7
t $6 4 4
l a " $$If a$gd* $$If a$gd* % $$a$gd* $a$gd* $a$gd* _ _ _ ` ` F` N` ` ` ` a b b b b b b c $c Nc Oc Tc c c c c c c c c c d d ;d Ad d d d d d d d e &e @e Ie _e e e e e e e e f $f 1f Hf If ~f f f f f f Ͽǫ hX hg q hUY hE hlt h2M hXU< h`_ h15 h% h# h hZ< h mHsH hZ< h# mHsH h; h h.{ h:w h @` ` ` a b " $$If a$gd* $$If a$gd* d kd` $$If F F # 7
t $6 4 4
l a b b b b b " $$If a$gd* $$If a$gd* d kd $$If F F # 7
t $6 4 4
l a b b b c #c " $$If a$gd* $$If a$gd* d kd@ $$If F F # 7
t $6 4 4
l a #c $c Oc d d g g h h _k `k p q q q $a$gd* $a$gd* $a$gd* d kd $$If F F # 7
t $6 4 4
l a f f f f f f f f g g ,g ;g Ag eg gg sg g g g g g g g h h h h h i )i ?i Bi Ei Ii Li Qi Ti ]i `i ci i i i i i i i i i i i i i i j j )j ,j /j 3j 6j ;j ȸȸȸȸȸȰȨȨȨȠȠȨȸȸȸ hw@ 6hn] hw@ 6h hw@ 6h
1K hw@ 6hp hw@ 6h.
L hw@ 6hw@ h.
L h7 h): hYz hlt h4
hf h5~ h!= h2M hX hUY h$k =;j >j Gj Jj Mj j j j j k k Nk Sk Vk `k gk ok pk qk k k k k k k l l =l ?l @l Pl \l ^l sl l l l l l l l l l l l l m ?m Am Bm Nm Vm Zm pm um ym zm m m m m m m m m m m
n n ļи临ммȼȴhBb hh hr
h&" hv=o hi h'T hisg h,!L hJ8 h.d hf hFt hTM h|z hm h*M6 hw@ 6hp hw@ 6hw@ Dn +n .n /n 5n 9n ;n An Bn Kn On Qn Vn hn vn {n n n n n n n n n n o o 7o Uo Yo do fo o o o o o o o o o o o o o o o o p p tp up p p p q q q q q h[ h-xA h-M hdU h-M h h> h 4 h 4 hK h 4 h*M6 hk 6hn] hk 6h hk 6hk h>ag he3 h)u h.d h'T h&" hisg hBb ;q q q q q q q q Cr [r ]r r r r r
s s s s s s t ru |u u u u u v v
v v v v Ow Vw uw x x x x x x x x y y ٵٱٱٱٱٓٱٱ hZ< h 6h:[ h5g hZ< heS mHsH hH heS hZ< h# mHsH h hZ< h.Z mHsH hX hX hX h hZ< hX mHsH hZ< h mHsH h; h hJ h[ h[ ht5 mH nH u .q q q Cr r r r r r z d kd
$$If F F # 7
t $6 4 4
l a " $$If a$gd* $$If a$gd* % $$a$gd* r r r r s " $$If a$gd* $$If a$gd* d kd
$$If F F # 7
t $6 4 4
l a s s s s
s " $$If a$gd* $$If a$gd* d kd $$If F F # 7
t $6 4 4
l a
s s s t v " $$If a$gd* $$If a$gd* d kdp $$If F F # 7
t $6 4 4
l a v v v
v v " $$If a$gd* $$If a$gd* d kd $$If F F # 7
t $6 4 4
l a v v v uw x " $$If a$gd* $$If a$gd* d kdP $$If F F # 7
t $6 4 4
l a x x x x y " $$If a$gd* $$If a$gd* d kd $$If F F # 7
t $6 4 4
l a y y y Uy y " $$If a$gd* $$If a$gd* d kd0 $$If F F # 7
t $6 4 4
l a y y Gy My Ty Uy y y y Qz qz z { { d{ h{ { { { { { { 8| :| <| z| ~| | | | | | | | | | | | )} .} 0} z} ~} } } } } } } } ~
~ ~ ̶̶̶̛̅Ӫ̶ hZ< h-M mHsH hZ< hs6 mHsH hZ< h9" mHsH h9" h hL mHsH hZ< h mHsH hZ< h# mHsH h; h hZ< h 6hZ< h# 6mHsHhZ< h 6mHsHhZ< h9" 6mHsH 4y y y z c{ " $$If a$gd* $$If a$gd* d kd $$If F F # 7
t $6 4 4
l a c{ d{ h{ { { " $$If a$gd* $$If a$gd* d kd $$If F F # 7
t $6 4 4
l a { { { 8| y| " $$If a$gd* $$If a$gd* d kd $$If F F # 7
t $6 4 4
l a y| z| ~| | | " $$If a$gd* $$If a$gd* d kd $$If F F # 7
t $6 4 4
l a | | | | | " $$If a$gd* $$If a$gd* d kd` $$If F F # 7
t $6 4 4
l a | | | | | " $$If a$gd* $$If a$gd* d kd $$If F F # 7
t $6 4 4
l a | | | 0} y} " $$If a$gd* $$If a$gd* d kd@ $$If F F # 7
t $6 4 4
l a y} z} ~} }
~ " $$If a$gd* $$If a$gd* d kd $$If F F # 7
t $6 4 4
l a
~ ~
~ ~ ~ " $$If a$gd* $$If a$gd* d kd $$If F F # 7
t $6 4 4
l a ~ ~ ~ ~ C " $$If a$gd* $$If a$gd* d kd $$If F F # 7
t $6 4 4
l a ~ ~ ~ ~ ~ E F G I M T U e m 7 | ~ ƀ ܀ Q " 7 L ^ _ d Ƃ ҂
hC hbS hS h|- h`_ hisg h9" h[ h> h heS hdU h hdU hZ< h9" mHsH hZ< h mHsH hZ< h# mHsH h; h h >C D F G H " $$If a$gd* $$If a$gd* d kd $$If F F # 7
t $6 4 4
l a H I M " $$If a$gd* $$If a$gd* d kdp $$If F F # 7
t $6 4 4
l a " $$If a$gd* $$If a$gd* d kd $$If F F # 7
t $6 4 4
l a 7 } " $$If a$gd* $$If a$gd* d kdP $$If F F # 7
t $6 4 4
l a } ~ ƀ
" $$If a$gd* $$If a$gd* d kd $$If F F # 7
t $6 4 4
l a
Q " $$If a$gd* $$If a$gd* d kd0 $$If F F # 7
t $6 4 4
l a # $ s z o X o $
&$ &`#$/ If a$gd* $$ &`#$/ If a$gd* % $$a$gd* $a$gd* $a$gd* d kd $$If F F # 7
t $6 4 4
l a
" # * + s ׄ
0 @ B Z ` a b c n o ą Ņ х ۅ ݅ ޅ Q | } # - / 7 8 A ǿǿϬǬǬ hM3 h4M hbS h8 hc h hZ< h 5hZ< h8 5hc h` h8 h h9# hZ< hc 5hZ< hC 5ht5 mH nH uhc) hl hC ? : kd $$If F \ Tl# \
o
t
6` 0 6 4 4
l / a $$ &`#$/ If a$gd* B a b c F kd $$If F \ Tl# \
o
t
6` 0 6 4 4
l a $$ &`#$/ If a$gd* ą Ņ х R F kd $$If F \ Tl# \
o
t
6` 0 6 4 4
l a $$ &`#$/ If a$gd* Y Q Q Q Q I Q Q $a$gd* $a$gd* kd $$If F \ Tl# \
o
t
6` 0 6 4 4
l a ƈ Lj ҈ ֈ
B C d j ~ Ӊ ؉ - / Ί Њ # % j o F I \ _ z č ō ! ' 1 2 3 7 h_e hL h:[ hC6 he3 h4M hCL hc) hCL h< hM3 hc) R J Q W Y ^ _ d f m s ȏ ʏ ֏ ؏ ޏ V Y x { א ݐ = E o ̑ ͑ ϑ ÿû h; h h|i h[ ht5 mH nH uh[ h[ h[ hL hL h+ hrx h hi h hC6 hU
L h h 6h*M6 hspC 6h#* hn] h8 6h h8 6hspC h8 h&( 6 A B D E F j d kdV $$If F F # h
t $6 4 4
l a " $$If a$gd* $$If a$gd* % $$a$gd* $a$gd* $a$gd* ^ ͒ ֒ @ C D E G K R T Q m Ô ̔ Д ֔ ٔ X Y [ q ̕
檦柘 h h h h h hi h-g h
hZ< h mHsH h_ h hZ< h# mHsH h; h h>> [ < hE% mHsH hZ< h mHsH hi h_) ; ğ ʟ ϟ ۟ ܟ ݟ ' ( ) / 6 9 = C U X p u N R S \ . / 6 7 8 > C K { h n p ~ ϣ ϣϣϣϻ h\ h[d hL hcf h% h^ h/C hu~2 h( hm h]5 h t h<- hC h hCW) hm^ h[* hU
L hu 6h2I hu 6hh hu 6hu =ϣ ӣ ף L [ ^ d Ȥ ɤ ʤ N R S U t u | ~ ˥ ͥ ԥ ֥ 4 7 B \ o
" B o ŽŹ൱ h:[ h-P h-P 6hB hC h0
h-P hp h9O hd9 h]5 h$ hp h$2 h h h h hL hn hG hcm hX hcf h[d h\ h\ 6h\ 7o 6 < I
( < = ? @ f g z ? c v { ~ Ȫ ɪ ʪ ž} hl h>Yg hw h2< hAg hZ h2< h h2< h'/ h2< h>Yg h2< h z h2< hO h2< hu h2< hs h2< hw h= hk, h hag h# hH$ h> h1 hf hp hB hS'! h 0 # q x ث $
" % I e r { ح ٭ ڭ 6 @ F M Ů Ȯ Ѯ G L Y a ͯ ί ѭh] h h] h,w h+c hCs h4/ h h~ hz h|- hnC hu h=" h z hAg hE6M h7 h2< h h h]n hl hw hv > > < = / ? [ $ a$gd* $a$gd* $a$gd* $a$gd* ; = C G d h n p } ' ( J ^ h i n v ³ س 1 ? @ c n y ȵ ʵ ֵ ٵ ۵ M X ÿᳯ htvR hm hP@ h h4/ hw h ho ) hu8 h hU$ hU$ h=" hA h9R} hU$ h h h h h=" E I t u ˷ η з # ^ t w Ϲ 2 < = > U V k v ~ hv h h$
hyk hy> X g< h] h] hQ ho hh8 ho hr[ ho ha' ho h] hm hRT h" h$ h]) h45 hn4 h ^ h= h7 hMm h
htvR hy> 2 ź % & ' ռ ۼ ; < X z Ž
' + ҿ ֿ 1 < = D [ v y " 2 3 8 : F I ¾β hlc hB hf hO h hs6 h# h0/ h6 hv hf&d h+ hhr h. hUvh hi[B h hg< h$
hy> hyk hy> hyk hv > ' 0 [ h j x 8 L " # G I z | t u " οڻ hsR hq h.p hsR h7L hsR hU htH h z hf&d hf h z hbPi h z h z hU h
h1|0 h1|0 h1|0 h$I h hlc hZ&: h hf hB hf hf hB hB 6hB hf 2" M X & 1 J L c ~ $ + e { . / ? O [ \ ] p } 䫧U hK h9~) h` hq h= h hq hq hq h4 h= hsR hsR h\hH hsR h{td hsR hV hsR hB4 hsR hj-< hsR h n hsR hT* hsR hZHE hsR h# g hsR hR hsR h8 4[ q u w % { d $
@h ^h`a$gd* $ a$gd* Valero, P. (2007). A socio-political look at equity in the school organization of mathematics education. ZDM Mathematics Education, 39, 225-233
PAGE 1
PAGE 1
hy,
h* 0J mH nH uh>[ hy 0J mH nH u
h>[ 0J j h>[ 0J Uh wl j h wl U h` h` j# hrL h wl <CJ UaJ j! hrL h wl <CJ UaJ h9~) hK h9~) $
@h ^h`a$gd* h]hgdt5 hh]h`hgdf &`#$gd 9 0 &P 1h:p* . A!"n#n$k% n $$If !v h5 l55
#v l#v#v
:V F
t $65 l55
n $$If !v h5 l55
#v l#v#v
:V F
t $65 l55
n $$If !v h5 l55
#v l#v#v
:V F
t $65 l55
n $$If !v h5 l55
#v l#v#v
:V F
t $65 l55
n $$If !v h5 l55
#v l#v#v
:V F
t $65 l55
n $$If !v h5 l55
#v l#v#v
:V F
t $65 l55
n $$If !v h5 l55
#v l#v#v
:V F
t $65 l55
n $$If !v h5 l55
#v l#v#v
:V F
t $65 l55
n $$If !v h5 l55
#v l#v#v
:V F
t $65 l55
n $$If !v h5 l55
#v l#v#v
:V F
t $65 l55
n $$If !v h5 l55
#v l#v#v
:V F
t $65 l55
n $$If !v h5 l55
#v l#v#v
:V F
t $65 l55
n $$If !v h5 l55
#v l#v#v
:V F
t $65 l55
n $$If !v h5 l55
#v l#v#v
:V F
t $65 l55
n $$If !v h5 l55
#v l#v#v
:V F
t $65 l55
n $$If !v h5 l55
#v l#v#v
:V F
t $65 l55
n $$If !v h5 l55
#v l#v#v
:V F
t $65 l55
n $$If !v h5 l55
#v l#v#v
:V F
t $65 l55
n $$If !v h5 l55
#v l#v#v
:V F
t $65 l55
n $$If !v h5 l55
#v l#v#v
:V F
t $65 l55
n $$If !v h5 l55
#v l#v#v
:V F
t $65 l55
n $$If !v h5 l55
#v l#v#v
:V F
t $65 l55
n $$If !v h5 l55
#v l#v#v
:V F
t $65 l55
n $$If !v h5 l55
#v l#v#v
:V F
t $65 l55
n $$If !v h5 l55
#v l#v#v
:V F
t $65 l55
n $$If !v h5 755
#v 7#v#v
:V F
t $65 755
n $$If !v h5 755
#v 7#v#v
:V F
t $65 755
n $$If !v h5 755
#v 7#v#v
:V F
t $65 755
n $$If !v h5 755
#v 7#v#v
:V F
t $65 755
n $$If !v h5 755
#v 7#v#v
:V F
t $65 755
n $$If !v h5 755
#v 7#v#v
:V F
t $65 755
n $$If !v h5 755
#v 7#v#v
:V F
t $65 755
n $$If !v h5 755
#v 7#v#v
:V F
t $65 755
n $$If !v h5 755
#v 7#v#v
:V F
t $65 755
n $$If !v h5 755
#v 7#v#v
:V F
t $65 755
n $$If !v h5 755
#v 7#v#v
:V F
t $65 755
n $$If !v h5 755
#v 7#v#v
:V F
t $65 755
n $$If !v h5 755
#v 7#v#v
:V F
t $65 755
n $$If !v h5 755
#v 7#v#v
:V F
t $65 755
n $$If !v h5 755
#v 7#v#v
:V F
t $65 755
n $$If !v h5 755
#v 7#v#v
:V F
t $65 755
n $$If !v h5 755
#v 7#v#v
:V F
t $65 755
n $$If !v h5 755
#v 7#v#v
:V F
t $65 755
n $$If !v h5 755
#v 7#v#v
:V F
t $65 755
n $$If !v h5 755
#v 7#v#v
:V F
t $65 755
n $$If !v h5 755
#v 7#v#v
:V F
t $65 755
n $$If !v h5 755
#v 7#v#v
:V F
t $65 755
n $$If !v h5 755
#v 7#v#v
:V F
t $65 755
n $$If !v h5 755
#v 7#v#v
:V F
t $65 755
n $$If !v h5 755
#v 7#v#v
:V F
t $65 755
n $$If !v h5 755
#v 7#v#v
:V F
t $65 755
n $$If !v h5 755
#v 7#v#v
:V F
t $65 755
n $$If !v h5 755
#v 7#v#v
:V F
t $65 755
n $$If !v h5 755
#v 7#v#v
:V F
t $65 755
n $$If !v h5 755
#v 7#v#v
:V F
t $65 755
$$If !v h5 55\
5o#v #v#v\
#vo:V F
t
6` 0 6 5 55\
5o/ / $$If !v h5 55\
5o#v #v#v\
#vo:V F
t
6` 0 6 5 55\
5o $$If !v h5 55\
5o#v #v#v\
#vo:V F
t
6` 0 6 5 55\
5o $$If !v h5 55\
5o#v #v#v\
#vo:V F
t
6` 0 6 5 55\
5on $$If !v h5 h55
#v h#v#v
:V F
t $65 h55
n $$If !v h5 h55
#v h#v#v
:V F
t $65 h55
n $$If !v h5 h55
#v h#v#v
:V F
t $65 h55
n $$If !v h5 h55
#v h#v#v
:V F
t $65 h55
n $$If !v h5 h55
#v h#v#v
:V F
t $65 h55
n $$If !v h5 h55
#v h#v#v
:V F
t $65 h55
n $$If !v h5 h55
#v h#v#v
:V F
t $65 h55
n $$If !v h5 h55
#v h#v#v
:V F
t $65 h55
n $$If !v h5 h55
#v h#v#v
:V F
t $65 h55
n $$If !v h5 h55
#v h#v#v
:V F
t $65 h55
n $$If !v h5 h55
#v h#v#v
:V F
t $65 h55
n $$If !v h5 h55
#v h#v#v
:V F
t $65 h55
n $$If !v h5 h55
#v h#v#v
:V F
t $65 h55
n $$If !v h5 h55
#v h#v#v
:V F
t $65 h55
n $$If !v h5 h55
#v h#v#v
:V F
t $65 h55
D d \
C * A B D 1 5 0 5 9 _ " @ @ b U =1M*|
ߨ 1 *" s n ) =1M*|
ߨPNG
IHDR a `PLTE!!!ZZZkkk > tRNS JO cmPPJCmp0712 H s DIDATWmI lAb8E&Et`&KޣHCU(0&-o x~ IENDB` D d \
C * A B D 1 0 2 6 3 _ " @ @ b P @]p8Uw /$ s n P @]p8UwPNG
IHDR ex sRGB PLTE';b RQxx,+yy;bqpbbuvZZLLxwjjNNA@zzmm^_ywyxxwRR,*++X6 cmPPJCmp0712 tRNS 0J IDAT]= P Ԙc
FF??!*pkjjp $i(
w?ѹ|L Յ{Qd:$~9lwVİCSLQE|18RK篽CPגZd?۵*$` IENDB` , @ @ @ z N o r m a l CJ _HaJ mH sH tH R @ R z H e a d i n g 1 $ < @& 5CJ KH \^J aJ R @ R z H e a d i n g 2 $ < @& 5CJ \]^J aJ J @ J z H e a d i n g 3 $ < @& 6\^J aJ D A@ D z D e f a u l t P a r a g r a p h F o n t R i@ R T a b l e N o r m a l 4
l 4 a ( k ( z N o L i s t < O < z
Q u o t a t i o n s
^ 6 D O D z
R e f e r e n c e s kx ^k` 4 @ 4 z F o o t e r
% . )@ !. z P a g e N u m b e r 4 @ 24 z H e a d e r
% H O H z P a p e r t i t l e $a$ 5CJ( aJ( tH L O L z
A u t h o r ' s n a m e $a$ 6CJ aJ tH P O P z A u t h o r ' s a f f i l i a t i o n s $a$ tH J O J z A u t h o r ' s e m a i l a d r e s s tH 4 O 4 z A b s t r a c t CJ tH 2 O 2 z B o d y t e x t tH d O d z N u m b e r e d t r a n s c r i p t
& F
S < 7$ 8$ CJ aJ l O l ' z C o m m e n t p a r a g r a p h x x -D` M
m$ B* OJ QJ ph 6 U@ 6 a H y p e r l i n k >*B*ph @ + @ z E n d n o t e T e x t x CJ aJ > * > < E n d n o t e R e f e r e n c e H* Z Y Z <,- D o c u m e n t M a p -D M
CJ OJ QJ ^J aJ j j z
T a b l e G r i d 7 :V 0 J O J z T r a n s c r i p t n u m b e r e d !
& F @ O "@ z
T r a n s c r i p t " n^n` > 2>
F o o t n o t e T e x t # CJ aJ @ & A@ F o o t n o t e R e f e r e n c e H* 8 "@ 8 C a p t i o n % 5CJ \aJ F V@ aF ! F o l l o w e d H y p e r l i n k >*B* ph n O qn h{ C o m m e n t p a r a g r a p h T e g n ) B* CJ OJ QJ _HaJ mH ph sH tH H H B a l l o o n T e x t ( CJ OJ QJ ^J aJ B ' B 6: C o m m e n t R e f e r e n c e CJ aJ < < 6: C o m m e n t T e x t * CJ aJ @ j @ 6: C o m m e n t S u b j e c t + 5\ O O < = J v
#! ?! # # % % ' ' ]+ ^+ - - e0 f0 t2 2 x3 y3 3 3 3 ;4 <4 @4 K4 P4 Q4 U4 }4 4 4 4 4 4 4 4 4 5 5 5 5 ^6 _6 c6 6 6 6 6 6 7 7 7 m7 7 7 7 7 8 8 8 8 8 8 8 8 9 9 9 9 9 9 9 A: : : : *; ]; ^; b; 7< < < < = > > > #> &> '> +> ? ? ? ? @ CA DA HA A A A A A B B B +B HB IB MB cB nB oB sB B B B B D D I .I I I M M XO YO Q Q S 0T }T ~T T T U iU jU nU V W W W X Y Y Y Y Y Y Y Z #Z $Z OZ [ [ ^ ^ _ _ _b `b g h h h h h Ci i i i i i i i i j j j j
j j j k m m m
m m m m un o o o o p p p Up p p p q cr dr hr r r r r 8s ys zs ~s s s s s s s s s s s s s 0t yt zt ~t t
u u
u u u u u u Cv Dv Fv Gv Hv Iv Mv v v v v v v v v 7w }w ~w w w
x x x Qx x x x y y #{ ${ s{ z{ { { { { { { { { | B| a| b| c| | | | | | | | } R} } } } } A B D E F G K Ë ̌ ~ ) P Q U s ; < @ n
L ʐ 7 W g ɡ ʡ ڤ ۤ > < = ˹ ̹ ¾ þ / ? [ q u w % { d ! $ & ) , / ; < = H I J K L M P 0 0 0 p 0 0 0 0 p 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 (! 0 (! 0 (! 0 0 0 % 0 % 0 % 0 0 0 - 0 - 0 0 0 - 0 0 0 |2 %0 |2 0 |2 "0 |2 0 |2 0 |2 0 |2 "0 |2 0 |2 0 |2 0 |2 "0 |2 0 |2 0 |2 0 |2 "0 |2 0 |2 0 |2 0 |2 "0 |2 0 |2 0 |2 0 |2 "0 |2 0 |2 0 |2 0 |2 "0 |2 0 |2 0 |2 0 |2 "0 |2 0 |2 0 |2 0 |2 "0 |2 0 |2 0 |2 0 |2 "0 |2 0 |2 0 |2 0 |2 "0 |2 0 |2 P 0 |2 0 |2 "0 |2 0 |2 0 |2 0 |2 "0 |2 0 |2 0 |2 0 |2 "0 |2 0 |2 0 |2 0 |2 "0 |2 0 |2 0 |2 0 |2 "0 |2 0 |2 0 |2 0 |2 "0 |2 0 |2 0 |2 0 |2 "0 |2 0 |2 0 |2 0 |2 "0 |2 0 |2 0 |2 0 |2 "0 |2 0 |2 0 |2 0 |2 "0 |2 0 |2 0 |2 0 |2 "0 |2 0 |2 0 |2 0 |2 "0 |2 0 |2 0 |2 0 |2 "0 |2 0 |2 0 |2 0 |2 "0 |2 0 |2 0 |2 ( 0 |2 0 B ( 0 |2 0 @ ( 0 |2 0 H 0 (I 0 H 0 (I 0 H 0 (I 0 (I 0 (I 0 (I 0 - 0 0 S %0 S 0 S "0 S 0 S 0 S 0 S "0 S 0 S 0 S 0 S "0 S 0 S 0 S 0 S "0 S 0 S 0 S 0 S "0 S 0 S 0 S ( 0 S 0 ~ 0 0 0 ;4 <4 P4 Q4 4 4 4 4 5 5 ^6 _6 6 6 7 7 7 7 8 8 8 8 9 9 9 9 : : ]; ^; < < > > &> '> ? ? CA DA A A B B HB IB nB oB B B iU jU W W Y Y Y Y #Z $Z i i i i j j
j j m m m m o o p p p p cr dr r r ys zs s s s s s s yt zt
u u u u Cv Dv Hv Iv v v v v }w ~w
x x x x { { b| c| | | } } A B F G P Q ; < P M90 M90 M90 8M90 M90 pM90 M90 M90 M90 M90 M90
CM90 M90 CM90 M90 dCM90 M90 8dCM90 M90 CM90 M90 CM90 M90 CM90 M90 eCM90 M90 fCM90 M90 PfCM90 M90 )CM90 M90 *CM90 M90" X*CM90 M90$ |]CM90 M90& ]CM90 M90( ]CM90 M90* $^CM90 M90, \^CM90 M90. qM90 M900 DqM90 M902 |qM90 M904 qM90 M906 qM90 M908 {M90 M90: {M90 M90< 0|M90 M90> 90 90 |90 90 |90 90 90 90 < M90 M90H t M90 M90J M90 M90L M90 M90N
M90 M90P T
M90 M90R
M90 M90T
M90 M90V
M90 M90X LIM90 M90Z IM90 M90\ IM90 M90^ IM90 M90` ,JM90 M90b dJM90 M90d JM90 M90f JM90 M90h KM90 M90j DKM90 M90l |KM90 M90n TyM90 M90p yM90 M90r yM90 M90t yM90 M90v 4zM90 M90x lzM90 M90z zM90 M90| zM90 M90~ {M90 M90 L{M90 M90 {M90 M90 {M90 M90 {M90 M90 ,|M90 M90 M90 M90 8M90 M90 pM90 M90 M90 M90 M90 M90 & & & ) o
p g < t ?& A* a- 5 t; > A F NJ sN #Q SW 6[ _ f ;j n q y ~
N ϣ o " l o p q r s u v w x y z < P= = = > ^? ? @ @ A A B B C ]D E G &G H CJ J K HK nK K \ ` b b #c q r s
s v v x y y c{ { y| | | | y}
~ ~ C H }
F P ; [ m t { | } ~ n " ) !! l , R $ g
fsT @ 0 (
B
S ? O
_ R e f 1 9 8 6 2 6 6 9 0
_ R e f 1 9 8 6 2 7 2 7 8 _ P i c t u r e B u l l e t s I ^ ! P -I ^ # P B Y Y T Y l6 Y ,? Y ,7 Y ,3 Y < Y l4 Y l3 Y 4 Y 4 Y ,6 Y 2 Y 7 Y ,8 Y ( Y ,. Y ,1 Y 2 Y 4 Y ,4 Y l, Y ) Y ,/ Y + Y - Y . Y / Y 0 Y * Y 3 Y 4 Y / Y l. Y l1 Y 0 Y l8 Y , Y 0 Y l1 Y l2 Y l( Y ,( Y ,) Y ( Y ( Y ) Y l) Y . Y ,. Y l+ Y l. Y 2 Y ,0 Y 0 Y / Y ,2 Y ,, Y / Y ) Y ,/ Y ,1 Y l/ Y l Y ,
Y
= J J R c c n v & & . 2 3 3 < A T OZ [ ^ h { S ! ] C ) ] B B m : : J J S S H ` a a o A n = = p p w P
! " # $ % & ' ( ) * , + - . / 0 1 2 3 4 6 5 7 8 9 : ; < = > ? & & . 2 3 3 < A T TZ [ ^ h { X & h H . j K K o G G P P \ \ O g k u u O z G G v P
! " # $ % & ' ( ) * , + - . / 0 1 2 3 4 6 5 7 8 9 : ; < = > @ ? A
C : *urn:schemas-microsoft-com:office:smarttagsmetricconverter _ 8 *urn:schemas-microsoft-com:office:smarttagscountry-region http://www.5iantlavalamp.com/8 6 *urn:schemas-microsoft-com:office:smarttagstime 9 *urn:schemas-microsoft-com:office:smarttagsState h ! *urn:schemas-microsoft-com:office:smarttagsCity0http://www.5iamas-microsoft-com:office:smarttags> B *urn:schemas-microsoft-com:office:smarttags
PersonName 8
*urn:schemas-microsoft-com:office:smarttagsdate V A *urn:schemas-microsoft-com:office:smarttagsplace http://www.5iantlavalamp.com/Z @ *urn:schemas-microsoft-com:office:smarttags PlaceName http://www.5iantlavalamp.com/Z ? *urn:schemas-microsoft-com:office:smarttags PlaceType http://www.5iantlavalamp.com/ 010121417182020052125440585f9DayHourMinuteMonth ProductIDYearB A @ ? A ? ? B :
A 8 A 6 6
6 6 6 6 A A A 6 A A A A A A A A B A A ! A A ! A ! A ! A A A ? @ A ! A A A A ! A @ ?
A ! A A ! A @ ? = C v N S \" ^" - - w4 |4 4 4 7 l7 7 7 |9 9 9 9 9 9 9 : : A: < 7< < < < < < =
? ? ? @ HA NA OA A I I
M M X X ^ ^ `b fb ln tn n n p p p Mp Np Tp r r Mx Px ւ ; m n Ë
~ o r ( 0 ĵ p x y { | Q ^ h s
" # . 3 ? V \ e h j t v { } r z < B C F I R U X v | ! # $ $ & & ' ' ) * , - / 0 J K P v * + Y \
)
4
] V d ' < # L! V! $ $ % % & & & ' + ++ K, O, - - . . _/ k/ 2 "2 W2 c2 ,3 43 y3 3 3 3 3 3 U4 |4 4 4 5 5 c6 6 6 6 7 l7 7 7 8 8 ^9 f9 |9 9 9 9 9 9 9 @: : ); F; K; b; 6< < = > "> +> ? ? ᴡ A A A * ѵ B ı ұ " L +> ? ? DA HA A A B B IB sB ~T T T T jU nU W W Y $Z h h h h i i j j m m Wn tn o o p p p p dr hr r r )s 8s zs ~s s s zt ~t u u Dv Mv v v ~w w x x ${ -{ s{ { B| | | | ψ B K Q U < @
! # $ $ & & ' ' ) * , - / 0 J K P ; < =
"! "! ' ' - - s2 s2 w3 w3 |4 |4 7 7 7 7 7 7 9 9 < < ? ? ? ? BA BA I I I I M M WO WO |T |T X X ^ ^ _ _ ^b ^b tn tn p p r r u u Px Px "{ "{ | | Q} Q} ͂ ͂ r r ; ; ʹ ʹ . . t t v v ! # $ $ & & ' ' ) * , - / 0 J K P P a u l E r n e s t ( `bbl Y~ i~} =1 #! AH ۂJ ^#DT d&%. D'|ZO $p)+! _a0 ! 4$= Y8z[Y 9qL '< f k Qz < $ - / PB R C @ @
! Mm t m* / 3
y,
0
N
.# % B[ d i+ i
jm
r
5 B Y f m s w z o0 L U dU D I > 5 h # $5 f5 8 Np h r M2 s6 S cm :w 9 > l ~ $ M 4 5 6: >[ L 5g t q{ 8 L oS kd h l " = J l r % ( M W ^ pa a 6" # * $2 7 K P _u 6: ? C at i N 3[ p 9/ 45 Q w ! S'! G9! b;! i! &" =" M" jj" 9# aG# 8\# q# J $ z?$ H$ K$ N$ U$ % )% >/% I8% E% f]% g% & ]& :' A' _' a' b' D( 1 ( %( &( =( 3x( ) o ) CW) ]) _) 9~) #* 6* [* V{* + C+ 5 q5 6 P/6 *M6 ,7 P7 l7 G{7 8 L8 |/8 )48 e8 h8 9 $9
*9 c9 f9 Ig9 j : Z&: 6: +; <; B; V; ]; Jc; q; < < j-< 2< 9< :< XU< g< h< = != Bv= #x= $'> O> AV>
h> y> 7? ? u? 4 ʰ ճհ n w 1 -洡 ! ڵ q / =6 ٹ B ܱᷡ ٱ跡 ұ緡 | > . E M TM XM N[M jM ON RN lN vN 9O P "P KQ QQ "@R kR tvR S .S =S rS TzS 6~S ))T t*T HT .QT pT U #U ,%U 8U uU V gLV >1W bIW .X BX 1^X |X }"Y X%Y DY EY HY UY YY Z Z Z Z )Z .Z Y,[ yP[ r[ s\ ]\ d\ k\ K] [] n] u] {] ^ ^
^ T^ m^ _ b+_ `_ Ia_ ` '` 9` Sb` :i` z` Ib wb 'c +c V/c Sc Wc
d f&d .d 4Ad Kd {td |d e
e 8e _e ]ee Qze |e f f Yg ag >ag Smg qg isg -h h 7h 7h Mh Uvh wi *i %>i bPi s]i ^ei Haj $k mk k p k =k Hk Vk dk yk yk 9l Wnl wl Ym jm n Un Nn Xn ]n v=o Hp BMp FVp rp g q ?q Eq Rq /Vq ~q hr r ?0r 2r cr s !s s /s Ms t?t Ft lt
u bu u )u Vu tu Xv Yv w jw w rx >>x {x y Iy /;z eEz Jz Yz n{ O| V| Y| )} 8} ;} D} F} 9R} ~ Q ~ ~ 5~ =~ t~ . f y ! # < ? C F :P [T { T{ 4
G 7 z 0/ 0 5 M wt { @ ]3 ] t' = -S y
* _- = > Bb 4 ( x } C ^ _ b { O k {4 O S 2Z M k L /* H f ! & 12 ? a yw > wI I ]5 p s 4 i 8 " Y2 J \ ] !i x R, :[ Op Z a p u fm 1 f o( 2 t; ; ? ZO V Z q [" m; o f{ I t 8 G { J f F* * B w :T :^ ,
K w_ Jn X b` m Z v T *) 2 @ F J F ( ܦ - : IJԧ n o * <. fh y - d c) '* 4M r 7 9" ? f h M PV Wt
H K ) + > e j (x } ' + j m s Cv ,w @}
I e
@ |m | a X MY " % J8 c + + 0 _ { ! '; B U \ n |z 3- 15 ~G i f VI T 2 Q ~ ` -P QU fl . v z h{ $
~! $ V& + I np z
9 D .G Aw f 2- . Q tl z f 5 # X |Y :g v5 M U 5] | i & ZJ
, U 9 s6 H8 1F OS t {
: O ;F n > G@ HP ] .
" $ & [. A I yQ - t5 bS f U $ S p $ - y M `x # ~) ^ ` u T* B J nR [ h - / nC C
. \ Cs * 9+ _. > Jm m v L / 3 vR $ C lI H m 'n * > eS p t 4/ @ q ]
.3 N Q i l 7x M 'T Y ;a he cf B 7 2: ~H (t )
L: [ P p w \ Z< 4X X ] h t t ! 93 XC K b Fm ,
O lc s@ RT i + ): ,i F L k .{ ! v0 6 W ] # = X _ + d9 VC O sR 0 u U @ Q W c > ~ . ! 8 XO mQ j[ \ S e R # - v & zI W 1 m! m kq 8~ A
# %' : z s ! 3" $ 4 CL .$ + . ; G JL Ia
P 4 Q
o " R ` h ={ 4 [d f #h .p ' = 3 3 3 ;4 <4 @4 K4 P4 Q4 U4 }4 4 4 4 4 4 4 4 4 5 5 5 5 ^6 _6 c6 6 6 6 6 6 7 7 7 m7 7 7 7 7 8 8 8 8 8 8 8 8 9 9 9 9 9 9 9 A: : : : *; ]; ^; b; 7< < < < = > > > #> &> '> +> ? ? ? ? @ CA DA HA A A A A A B B B +B HB IB MB cB nB oB sB B B B D eP T T U iU jU nU V W W W X Y Y Y Y Y Y Y Z #Z $Z [ ^ Qg h h Ci i i i i i i i i j j j j
j j j k m m m
m m m m un o o o o p p p Up p p p q cr dr hr r r r r 8s ys zs ~s s s s s s s s s s s s s 0t yt zt ~t t
u u
u u u u u u Cv Dv Fv Gv Hv Iv Mv v v v v v v v v 7w }w ~w w w
x x x Qx x x y s{ { { { { { { { a| b| c| | | | | | | } } } } څ A B D E F G K Ë ̌ ~ ) P Q U s ; < @ n
L W ? ! " # P aJ %6~ R E F M G R . I n s t a n t F o r m a t 8K
R E F M G R . L a y o u t O R E F M G R . L i b r a r i e s Os < I n s t a n t F o r m a t > < E n a b l e d > 1 < / E n a b l e d > < S c a n U n f o r m a t t e d > 1 < / S c a n U n f o r m a t t e d > < S c a n C h a n g e s > 1 < / S c a n C h a n g e s > < / I n s t a n t F o r m a t > < L a y o u t > < S t a r t i n g R e f n u m > C : \ D o c u m e n t s a n d S e t t i n g s \ t l \ D o k u m e n t e r \ P H D \ P h d L i t t D a t a \ R e f e r e n c e M a n e g e r \ S t y l e s \ A m e r i c a n P s y c h o l o g i c a l A s s o c i a t i o n 5 t h e d m o d . o s < / S t a r t i n g R e f n u m > < F o n t N a m e > T i m e s N e w R o m a n < / F o n t N a m e > < F o n t S i z e > 1 1 < / F o n t S i z e > < R e f l i s t T i t l e > < / R e f l i s t T i t l e > < S p a c e A f t e r > 1 < / S p a c e A f t e r > < R e f l i s t O r d e r > 1 < / R e f l i s t O r d e r > < C i t a t i o n O r d e r > 0 < / C i t a t i o n O r d e r > < N u m b e r R e f e r e n c e s > 0 < / N u m b e r R e f e r e n c e s > < F i r s t L i n e I n d e n t > 0 < / F i r s t L i n e I n d e n t > < H a n g i n g I n d e n t > 3 6 2 < / H a n g i n g I n d e n t > < L i n e S p a c i n g > 0 < / L i n e S p a c i n g > < S h o w R e p r i n t > 1 < / S h o w R e p r i n t > < S h o w N o t e s > 0 < / S h o w N o t e s > < S h o w K e y w o r d s > 0 < / S h o w K e y w o r d s > < S h o r t F o r m F i e l d s > 0 < / S h o r t F o r m F i e l d s > < S h o w R e c o r d I D > 0 < / S h o w R e c o r d I D > < S h o w A b s t r a c t > 0 < / S h o w A b s t r a c t > < / L a y o u t > B < D a t a b a s e s > < L i b r a r i e s > < i t e m > P h d L i t t < / i t e m > < / L i b r a r i e s > < / D a t a b a s e s > @ 4 , - O @ 4 l @ @ U n k n o w n S z T i m e s N e w R o m a n D o t u m 5 S y m b o l 3& z A r i a l CF C o m i c S a n s M S 5& z a T a h o m a ?5 z C o u r i e r N e w ; W i n g d i n g s ; A g e n c y F B " 1Hh ({f({fz : i w : i w # n 4 d 2q HP 7 ! .QT T C : \ D o c u m e n t s a n d S e t t i n g s \ t l \ A p p l i c a t i o n D a t a \ M i c r o s o f t \ S k a b e l o n e r \ N R C S N E M V a a s a . d o t ? A c h i l d ' s p e r s p e c t i v e o n b e i n g i n d i f f i c u l t i e s w i t h m a t h e m a t i c s T r o e l s L a n g e P a u l E r n e s t (
! " # $ % & ' Oh +'0 T
( 4 < D L @ A child's perspective on being in difficulties with mathematics ch
Troels Lange spe roe Final version including codes of article to 4th Nordic Research Conference on Special Needs Education in Mathematics in Vaasa November 2007 NRCSNEM Vaasa.dot ud Paul Ernest 2 ul Microsoft Word 10.0 @ @ 8@ (b.@ (b. : ՜. +,D ՜. +, T
p x
4 Version semifinal l bo Akademi w i A
@ A child's perspective on being in difficulties with mathematics Title h H ( @ \ _NewReviewCycle _AdHocReviewCycleID _EmailSubject
_AuthorEmail _AuthorEmailDisplayName _ReviewingToolsShownOnce Rev U Contribution to journal TRLA@VIAUC.DK o Troels Lange (TRLA) roe
! " # $ % & ' ( ) * + , - . / 0 1 2 3 4 5 6 7 8 9 : ; < = > ? @ A B C D E F G H I J K L M N O P Q R S T U V W X Y Z [ \ ] ^ _ ` a b c d e f g h i j k l m n o p q r s t u v w x y z { | } ~
! " # $ % & ' ( ) * + , - . / 0 1 2 3 4 5 6 7 8 9 : ; < = > ? @ A B C D E F G H I J K L M N O P Q R S T U V W X Y Z [ \ ] ^ _ ` a b c d e f g h i j k l m n o p q r s t u v w x y z { | } ~ R o o t E n t r y F G. D a t a
& 1 T a b l e z# W o r d D o c u m e n t ; S u m m a r y I n f o r m a t i o n ( D o c u m e n t S u m m a r y I n f o r m a t i o n 8 C o m p O b j j
F Microsoft Word Document
MSWordDoc Word.Document.8 9q