ࡱ>  SbjbjUqUq "77ӫl"""8ZnD$  :y|""0MATHEMATICAL IDENTITIES OF BOYS IN SPECIAL EDUCATION: A CASE STUDY Melinda Browne mb282(at)exeter.ac.uk INTRODUCTION In the United States boys are more likely than girls to be placed in special education programs for reasons such as learning disabilities and emotional and behavioral disorders. To elucidate this complex problem as it pertains to mathematics education, I consider boys' mathematical identities in the context of Special Education, wherein they may spend the majority of their educational careers. Mathematical identity refers to just one aspect of self, which is shaped by the learner's experiences (Ernest, 2004b). My investigation explores case studies of two high school students, Rick and Sam (pseudonyms), who are eligible to receive specifically designed instruction across the curriculum, including in mathematics, with a view to measuring the effectiveness of social constructivism and/or to provide insights to serve the needs of struggling students. The boys' Special Education programs complied with The Individuals with Disabilities Act (IDEA) of 1975, which requires public schools to actively identify students with disabilities in order to make certain that they get proper assistance. Protecting students with special needs, the IDEA may be viewed as an important section of United States civil rights legislation (Kopp, 2001). Rick and Sam qualified because they were each found to have a disability that adversely affected their educational performance. This outcome was the result of a multidisciplinary evaluation and identification process, which involved a battery of testing. School officials, parents, regular education and special education teachers negotiated to determine what accommodations the school should provide. The process concluded with a consensus between all parties, and the approval of an Individual Educational Plan (IEP) for each boy. Not only did the content of their IEP describe the services that they would receive, but it also projected the future outcomes of those services. Developing the plans and performing annual evaluations was a time consuming, multi-step process. In the United States, special education services offered by public schools are at no cost to the parents. Even placements outside the student's regular school may be incurred at public expense for circumstances such as a court order as it was in Sam's case. However, in the case of both Rick and Sam, parents eventually went beyond the public school system to utilize private interventions. Behrens and Satterfield (2006) found that these programs predominantly contain white, upper-middle to upper-class teenagers who are below average academically. Parents are almost always responsible for placing their children in these programs and paying the cost of services. For Rick, the shift from public to private placement occurred because his single mother could no longer control his defiant behavior, which attracted negative attention in the suburban community where they lived. After leaving public school, Rick attended a therapeutic wilderness program followed by a non-traditional boarding school for boys, which is the setting for my investigation. Rick is diagnosed with Attention Deficit Hyperactivity Disorder (ADHD) as well as slow processing. His Full Scale IQ on the Wechsler Adult Intelligence Scale Third Edition (WAIS-III) is high average, and based on my experience with him, reflects his academic potential. According to Rick, the special education system at his public high school allowed him to coast through the classes he disliked such as mathematics, and advance in subjects he enjoyed such as history. It also entitled him to one hundred percent extended time on the Scholastic Aptitude Test (SAT), which he took during the time of this investigation. However, Rick's circuitous educational path created difficulties for him, especially in mathematics. Going into the test, he knew that receiving extended time on the Math section of the SAT may not compensate for his lack of continuity. And even if he did achieve a high score, rather than celebrate his mathematical prowess, he may instead attribute his success to his accommodations. This no-win scenario characterizes the conflicting messages that students may receive as a result of receiving help from Special Education. And it could suggest possible implications for Rick's developing mathematical identity. REVIEW OF LITERATURE Mathematical Identity To set up a framework to analyze the problem, my investigation begins with the postmodernist assumption that students construct mathematical identities that influence their learning of mathematics (Ernest, 2004b). Mathematical identity is only one of many possible identities. Ernest (2004b) summarizes this relatively new research perspective: A critical reconsideration of the learning subject, i.e., the learner in school and in mathematics lessons, thus involves acknowledging the development of a fragmented postmodern self -- multiple selves in different social contexts -- that change over time. (page 77) This means that a single research subject can assume different identities depending upon the social situation he/she experiences. Rick's identity was a function of what setting he was in. When he returned home for vacations he was always at risk for reliving the negative behavior patterns that landed him in trouble to begin with. In stark contrast to Rick's public school image as a special-ed, troublemaker, at boarding school he was a studious, high-achieving student. Ernest (2004b) acknowledges that it is impossible to draw a line between what happens inside and outside the school house. Brown, Jones and Bibby (2004) assert that identity is motivated by an individual's need to belong socially. Some of Rick's negative behavior was prompted by his desire to fit in with a certain group of friends. Removed from his friends' influence, he was free to form a new identity, which allowed him to succeed academically. At boarding school, the supportive classroom environment may have encouraged Rick to overcome his feelings of failure, and such an experience would likely facilitate his learning of mathematics (2004c). As Ernest (2004b) points out, identities may evolve over time. Rick had the opportunity to construct a new identity and improve his academic performance. But the special-ed designation follows him. Even as a private school student, outside the public system, he is still entitled to receive extended time on the SAT. In the past, Rick has used the system to his best advantage, such as when he utilized his accommodations for mathematics the most, if only because he disliked mathematics the most. Garibaldi (2006) refers to similar behavior when he calls the special-ed classification "the bane of the modern boy." A conversation with a tenth grade boy in Garibaldi's (2006) classroom illustrates the "game" of Special Education: Brandon has been on the special-ed track since he was nine. He knows his legal rights as well as his caseworkers do. And he plays them ruthlessly. In every debate I have with him about his low performance, Brandon delicately threads his response with the very sinews that bind him. (page 5) Garibaldi (2006) describes the case of a boy who has assumed a special-ed identity, and will commit to doing no more work than to satisfy the requirements of his IEP. Recent literature has highlighted the problems associated with the "underachieving boy" identity in schools. Sommers (2000) blames boys' poor school performance on the so called feminization of the classroom, which creates a culture that is inhospitable to boys. Pollack (1999) thinks the problem lies with boys' self-esteem. Rather than deal with their emotions boys hide behind a "tough-guy" faade and act out their insecurity through bad behavior. Pollack (1999) also cites the unrealistic images of men portrayed by the media that add to the boys' difficulties. When boys act out for whatever reason, a special-ed designation is an easy fix, removing the problem from the traditional classroom. And once the diagnosis is made and the educational goals are laid out in the formal way, which I described earlier, the boy may have little opportunity to change his brand name. Conflicting Messages Taking Ernest's (2004b) position that mathematics is socially constructed; adopting this special-ed identity will have an impact on that student's learning. Social constructivism takes into account the student's entire social context of learning (Ernest, 1998). Besides the mathematics itself, otherwise minor details such as the language used in the lesson may contribute to the learner's social experience. Scanning a typical secondary algebra textbook, one finds many verbs in the active tense. Problems are often written as succinct commands: evaluate, simplify, solve etc. The words appear to direct the learner to adopt a self-reliant approach. To encourage learners' active involvement, the familiar sentence "mathematics is not a spectator sport" appears on many math-classroom walls. For the special-ed student the expectation of individual competence is sometimes in direct opposition to his/her accommodations. These accommodations may include extra help, supervised study time, and low level classes. This contradicts the status placed in mathematics education on precocity and acceleration (Thurston, 1990). Thurston (1990) brings to light the emphasis in secondary-mathematics on efficiency and speed, rather than depth and insight. Related to this is the tendency to think of mathematics as a sport like a foot race. High schools in the United States cultivate the competitive nature of mathematics by offering inter-school math leagues. The active language combined with likening math to a sport should appeal to young men in general. In schools, sports programs are the primary masculinity-making institutions (Connell, 2000). When considering the pride associated with the construction of the hegemonic male identity, a boy who is at once subjected to the competitive, sports-like culture of mathematics is likely to cringe at the special handling afforded to them through Special Education (Connell, 2000). Competition is a primary driving force behind achievement in a meritorious society. Bateson (2000) observed that competition and displays of independence are a common phenomenon in the United States; and this "symmetric" behavior may not lead the populace to the peace and harmony of what he called the "steady state." Bateson (2000) defined symmetric behavior as people reacting to others' activity by doing a comparable thing themselves. The problem with this tendency towards rivalry is that it often escalates. In an ethnographic study, Lareau (2003) found that middle class children were raised in accordance with "concerted cultivation," in which they experienced endless competition through extracurricular activities. The middle class parents in her study were willing to invest significant economic resources and the majority of their own free time to promote their children's success. And Lareau (2000) also established that social class was a factor in how much time parents invested in assisting their children with school-related work. By far it was the upper-middle-class parents who demanded a more individualized approach to their children's education. This socio-economic group was often willing to seek out private programs beyond the public school at their own expense (Behrens & Satterfield, 2006; Lareau, 2000). However, a high level of parental involvement may place enormous performance pressure on children. Rosenfeld and Wise (2000) offer a critique of what they call "hyper-parenting," and Tofler, Knapp and Drell (1999) raise questions about the mental health implications for children subjected to the intense talent development, which may be part of this lifestyle. For Special Education students, parents can utilize their child's disabled identity to the learner's advantage, such as when taking the SAT (Rosenfeld & Wise, 2000). Playing the disability card, entitles students to extra time, supposedly making it more likely that they will achieve a higher score, and become a winner in the super-competitive, college admissions game. If the goal of learning mathematics is reduced to obtaining a number on a scale, like one's math score on the SAT, it may become more important to learn how to win points rather than to find meaning in learning. In this scenario, conventional success, not the love of knowledge is the variable maximized. Rosenfeld and Wise (2000) make the point that in the United States everyone may love a winner, whether he or she feels like one or not. The Paradox of Special Education In my discussion thus far, I have argued that Special Education is a politically sanctioned system that may at once lock an individual into a disabled identity, while simultaneously attempting to propel him/her towards a new, improved future identity. According to Heller (1999) this sort of paradox combines elements of the technological mindset, with issues of social allocation and social justice. Such a conflict between opposing viewpoints is typical in postmodern society. In the case of Special Education, what may result is a tug of war between the bureaucratic-machine set up to identify and prescribe accommodations for a learner's disabilities, and the learner's own desire to break free of these imposed binds. In the United States, the process is justified by political power, spelled out in the legal language of IDEA. Applying Heller's (1999) theory, technological thinking in Special Education embraces the belief that things will be better tomorrow as a result of the planning we do today. If we follow a learner's IEP it will result in that student being able to assume a more successful future identity. From this vantage point the possibilities appear limitless. The technologically sound prescriptions are based on research by people with credentials who have published in educational journals (Heller, 1999). As more special needs are identified, greater and greater measures are taken to remedy them. Pushed to the limit, the result may be that assessing children to quantify their lack of ability makes students problems to solve rather than people. With all the required documentation, a student like Rick or Sam could end up with a file two inches thick. Internationally, as well as in the United States, success with mathematics is often measured by how well students perform on State and National standardized tests. William, Bartholomew, and Reay (2004) observed eleven-year-old students in an English school who adopted mathematical identities based on the assessment results they earned: From thinking of themselves as students who might get a particular level, the students changed to talking about themselves as being a level three, four, five or six. The causes of this shift are, of course, complex, but there can be little doubt that a major influence was the culture of the school which had embraced the need to improve its test scores irrespective of the consequence for the students' achievement in wider terms. Students were increasingly valued not for their personal qualities, but rather for what they could contribute to the targets set for the school by the school district. For many of the students in the class, the results of these assessments came to be bound up with not just what kinds of careers might be open to them, but who they were now, who they could be, and even their moral worth. (page 58) Performance targets may also provide an incentive for schools to liberally designate students as "special-ed." Rosenfeld and Wise (2000) describe how Special Education services were used in a well-funded school district of an affluent community to bolster the performance of mediocre students on State-mandated tests. However, students in remedial classes learn less mathematics and often have fewer opportunities for personal exploration (Ernest, 2004c). Once on this track, students find it difficult to get off, since they are behind the usual progression of studies (Brahier, 2000). For example to spend more time reinforcing the fundamentals, the curriculum may stretch one academic year of algebra over two years, making it unlikely that students who take these classes will ever be introduced to rich topics like trigonometry. The backlash of all this emphasis on quantified results at the expense of humanistic considerations is sometimes to reject the system that produces the numbers altogether (Ritzer, 2004). Glendinning (1995) echoes Max Weber's work in her harsh criticism of technological society: What I am describing is a human-constructed, technology-centered social system built on principles of standardization, efficiency, linearity, and fragmentation, like an assembly line that fulfills production quotas but cares nothing for the people who operate it. (page 45) Glendinning (1995) goes on to argue that a technological world lacks the ingredients for human satisfaction. Furthermore, according to her view, learning mathematics prepares us to manage our splintered lives in technological society: The hallmark of technological education is to learn mathematics to quantify reality, and to master fragmented thinking to function in a mechanistic world. (Glendinning, 1995, page 50) The children William, Bartholomew, and Reay (2004) observed appeared to quantify their reality likewise. Portrayed in a negative light, mathematics is associated with dehumanizing, "disconnected" values (Ernest, 2005). Disconnected learning would store knowledge in locked compartments, preventing flow from one social context to another (Ernest, 2004b). Glendinning (1995) argues that life experienced "piecemeal" irrevocably changes a person's nature-based and human self. Rather than advocate for one side or the other, I look to Heller's (1999) model of the two trains of thought headed in opposite directions. In her view, both are necessary; future-oriented technology must be balanced with past-oriented, humanistic, ideology to preserve self-determination. Heller (1999) believed that education plays an important role in the allocation of social position. For example, higher-mathematics is sometimes a gateway to a professional career, an achievement that may well depend upon building bridges (Ernest, 1991, 2004b). Again relying on Heller (1999), the biggest challenge in helping boys (or girls) in Special Education attain such a future-identity may lie in spanning the void between the private skills that would give them a life, and the social capacity to find pleasure and meaning in work that would give them a world. Summary Boys in Special Education programs may receive conflicting messages, which could cause difficulties for their developing mathematical identity (Ernest, 2004b). Sports programs and extracurricular activities outside the classroom encourage middle-class boys to be sturdy and ready for action (Connell, 2000; Lareau, 2003; Pollack, 1999). Schools' approach to mathematics education often echoes the competitive spirit (Thurston, 1990). In any case, "symmetric" behavior patterns are widespread in the United States (Bateson, 2000). Winning with help has less value to boys than winning on their own (Connell, 2000; Pollack, 1999). However, schools, parents and the students themselves, sometimes play the Special Education "game" to their best advantage (Garibaldi, 2006; Rosenfeld & Wise, 2000). In addition to the accommodations assured by IDEA, higher socio-economic parents are more likely to rely on youth-programs outside of the public realm (Behrens & Satterfield, 2006; Lareau, 2000). Regardless of the well-intended efforts to help them, boys may get locked into a disabled identity, and a low-level academic program, which makes it less likely that they can study advanced mathematics in high school (Brahier, 2000; Ernest, 2004c). Looking out for the best interests of Special Education students, negotiating the conflict between technological and humanistic-ideological objectives may well necessitate building bridges between various social contexts (Ernest, 2004b; Heller, 1999). METHODOLOGY Background for the Investigation Social Constructivism assumes that mathematical knowledge is liquid rather than solid, flowing from conversation (Ernest, 2004a). Indeed, Ernest (2004a) writes that mathematical identity is shaped by conversation. And as I have previously discussed, mathematical identity evolves over time (Ernest, 2004b). Because learners bring their entire selves into mathematics class, their mathematical identities are based in part on their personal experience (Ernest, 2004b). Accordingly, my investigation takes into consideration the personal backgrounds of two boys, who are eligible for public Special Education services, but are currently in a private program. Rick and Sam, grew up in the same geographic area of the United States, and are the oldest of two siblings in single-parent households. One of their parents is deceased. Rick's father died two years ago; and Sam's mother died less than one year ago from this writing. (In both cases, losing a parent occurred after a divorce.) The diagnosis that led to their placement in Special Education occurred many years before these tragic events. Needless to say, Rick and Sam have taken steps, including individual therapy, to deal with their troubled family situations. The primary learning diagnosis for each boy is ADHD and slow processing. Despite their hardships, neither boy is visibly depressed. In fact, as mathematics students, I usually found them upbeat and engaged. At the time of my investigation, Rick was seventeen years old and a senior; Sam was sixteen and one-half years old and a junior. Data Gathering Methods It is difficult to broadly depict a learner's mathematical identity in little more than a snapshot. In the context of Special Education, each boy brings his own psychological composition and emotional sensitivity, making his experience infinitely complex (Ernest, 2004b). Another limitation is the idea of mathematical identity itself. Critics would proclaim it is a nebulous concept; one that is heavily subjective by nature. Nevertheless, this form of analysis may unveil useful patterns (Ernest, 2004a, 2004b). My research method intended to reveal each boy's mathematical identity by interviewing him. I also asked the boys to write a short essay about why they liked or did not like mathematics. I scanned their personal files for useful details. During the interviews, my questioning was exploratory, but certain specific questions provided focus; What has been your past experience with mathematics? Why do you or do you not like mathematics? I aimed at determining how the boys felt about their current performance in mathematics, and whether or not they saw themselves continuing to study the discipline in the future. I directed some of my questioning towards Special Education at the boys' former schools. If they received accommodations on a standardized test, I asked them to evaluate their experience. I made digital audio recordings of the interviews and saved the data to my computer. Scanning the recordings at a later date, I transcribed useful comments to word processing. (I did the same thing with their hand-written essays.) Making sense of empirical observations is the intention of interpretive social science (Thomas, 1993). Since I am the boys' mathematics teacher, I reported relevant details concerning my curricular choices for them. To gather pertinent background information, I utilized each boy's academic file, which may contain his IEP, a psycho-educational evaluation, transcripts from previous schools, standardized test results, teacher reports and current grades, and parent communications. Their participation in my study was voluntary. I made an effort to protect the boys' privacy by changing their names and not revealing the geographic location of their homes. RESULTS OF THE INVESTIGATION PART I: CASE STUDIES Rick After Rick's six-month hiatus, an email from his mother sent to my school confirms that life at home is still a problem: I have determined that I absolutely cannot allow Rick to come home this summer, even for a weekend. (Rick's mother) She then gave a detailed account of the fiasco that accompanied Rick's first visit home. Acknowledging his good behavior at boarding-school, she refers to him as "Jekyll and Hyde." It appears that in spite of her aggravation, Rick's mother was very involved with his education over his entire academic career (Rick's file). She approved his IEP every three years, and saw to it that he had accommodations on the SAT. Rick sat for the College Board exam during my investigation. After taking advantage of one-hundred percent extended time on the SAT, he talks about his experience: (Rick's score on the Reading "Verbal" section was significantly higher than his score on the Math section.) [In an individual interview] M.B. Do you think it (100% extended time) was helpful? Rick I do not think that the extra time the full double time was helpful. I think that it was maybe a little bit long. And I just went in and started taking the test with, like, a bad attitude M.B. If you had done really well on the Math section, how would you have felt about it? Rick I would have been a lot happier. I would have been excited that I had done well, but I just knew that I wasn't going to do well, and that is why I skipped entire sections, because it's, like, I just don't want to sit here and take this test. If I'm going to sit here, I'm just going to sit here and not do math, because I hate math The English I guess I could say was more enjoyable to do. M.B. Is there any particular reason why you like the Verbal (section)? Rick I don't know, I guess it's because I just never liked math to begin with, and I'm not very good in math, and I'm better at the verbal M.B. Was that always the case? Rick Yes I guess it's because my mom is a librarian and my dad majored in English I just don't like math. M.B. What types of special services were you offered to help you with your math? Rick I was in low math. M.B. Did you do better? Rick I did okay in middle school, but in high school I just didn't do well because I knew for a pretty long time that I wasn't going to have a career that used it, so I just stopped caring about it. It's the wrong attitude M.B. If you went to take the SAT again would you want one hundred percent extended time? Rick Not double (time) I would probably want fifty percent (extended time). According to his transcript from public school, Rick studied remedial math and Advanced Placement (AP) history in his previous year of high school. Attempting to build on his strong verbal skills, I directed Rick to read Levitt and Dubner's (2005) Freakonomics: A Rogue Economist Explores the Hidden Side of Everything. After completing the assignment, I asked him to answer the question: What does reading this book have to do with math? Reading Freakonomics has everything to do with math. Math is not just numbers and multiples and algebraic equations. Math is about solving problems using the tools given to you. And that is what Freakonomics does; it takes problems in the world or questions and answers them using mathematical techniques whether it be four plus four or why the crime rate has gone up. (Rick's essay) Sam Sam was identified early on as eligible for Special Education services. He did well in elementary school, but when he got to high school he lacked the concentration to complete college preparatory classes. As a result, he was placed in remedial classes even though he claimed that he was "good" at math. According to Sam, he spent the majority of his time preparing for State mandated assessments. I verified on his public school transcript that he was indeed studying mathematics at a low level: [In an individual interview] M.B. How was it that you were placed in such a low math? Sam I kind of wanted to take low math. It was more important for me to get an "A." [In another individual interview] M.B Do you consider yourself good at math? Sam Yes. (pause) I consider myself average. But, I don't know it's one of my stronger points. M.B. What was the reason that you were in those (low level) classes? Sam I just didn't do any homework. Last year, when Sam was a sophomore at a public school in an elite-executive community, he was caught selling marijuana. After a brief stay at a juvenile correction center, he was placed in an alternative school run by a university located in a nearby city. Sam reported that most of the other students were from working-class neighborhoods and lived near the school, while he continued to reside in his affluent suburb, which meant an hour commute both ways. Since his placement at the alternative school was made by his Special Education Department, Sam's tuition was paid by his public school district. Sam called the math taught at the alternative school "a joke." According to his transcript, he received a grade of "A" for his first quarter (remedial) work. Dissatisfied with the academic program at the alternative school, and worried about negative peer influences, Sam's father hired an educational consultant to find another school for him. This process brought Sam to my school where he is receiving a tutorial education at his father's expense. At the end of his first term, Sam earned a grade of "A" in college-preparatory geometry. [In an individual interview] M.B. You've really done well this term in math, any thoughts on that? Are you doing anything different? Sam I'm not smoking "pot." M.B. That makes a difference? Sam No, I'm just doing my workI've just been doing a lot of work I guessI haven't been getting away with half doing everything M.B. You don't think that you can do that here? Sam Not really. It's harder. I can't blend in with the chair or anything (He demonstrates how to do this by slouching over and putting his head down on the table.) Sam has a growing interest in marine biology and diving, sparked by a sailing trip in the Caribbean, where he encountered coral reefs and numerous varieties of fish. That experience also gave Sam the opportunity to take diving lessons: [In an individual interview] M.B. Do you see yourself as building on your math background? Sam Of course, there's a lot of math during marine biology. Plus it improves your chances for a better career in marine biology. M.B. Do you know how to scuba dive? Sam Yes. M.B. Do you use calculations? Sam Yes. You go down a depthsay you are going down 55 feet. The chart says that if you stay at 55 feet, you'll only have 3000 psi, and you can only go down for say 33 minutes. At 20 feet you can stay there for about two hours because you have enough air. If you go down like 150 feet you only have 2 minutes. You need more air supply because air goes much faster down there. M.B. Do you actually make a calculation or do you look it up on a chart? Sam It depends. You can make calculations saying you are going 20 feet, then you are dropping down to 40 feet, then you are coming back to 20 (feet) to be down for enough time to do what you want I asked Sam to write a short essay about why he likes (or does not like) mathematics: I enjoy math because everything is very precise. You can't have a debate about what the answer is because you either get it or you don't. That is kind of how people feel about math; they either love it or hate it. Numbers always went better through my head than facts or names. So the bottom line is I love it because I am good at it and it comes easy to me. It has a lot to do with common sense. (Sam's essay) PART II: ANALYSIS Rick Rick's negative comments about mathematics characterize his mathematical identity (Ernest, 2004b). Furthermore, when I interviewed him he did not think that he would use math in his future career, a choice based at least in part on his poor test performance, past and present (William, Bartholomew, and Reay, 2004). Rick has been able to slip through the system in low level classes due to his special-ed designation (Garibaldi, 2006). Before Rick arrived at my school, his mother had taken the necessary steps to qualify him for one-hundred percent extended time on the SAT (Rosenfeld & Wise, 2000). Allowing him additional time to read the passages may have helped him on the Verbal section. But on the Math section in which he was ill-prepared, having more time to flounder frustrated him. Even with accommodations Rick felt like he could not win, which may have prompted him to act tough, strongly emphasizing his dislike of math during our interview (Pollack, 1999; Connell, 2000). Rick is, however, confident about his verbal abilities. His record from his previous school indicates that he excels in history and English. He justifies his success, in part, because both his mother and his father (now deceased) were strong in this area (Ernest, 2005). As Rick's mathematics teacher, I attempted to build a bridge between school-mathematics, and mathematics in a verbal context, to strengthen his developing mathematical identity (Ernest, 2004b). I did this by asking him to read the popular book Freakonomics: A Rogue Economist Explores the Hidden Side of Everything (Levitt & Dubner, 2005). Based on his essay, Rick appeared to find this assignment meaningful. Sam Sam clearly enjoyed discussing the practical mathematics of diving. He appeared to have a no-nonsense, business-like pattern to his thinking, such as when he referred to math as "common sense" in his essay. Nevertheless, he shows some restraint in assessing his own ability. His reticence may be due to underlying issues with his self-esteem (Pollack, 1999). Or it may be that he has quantified his mathematical identity in the same way that William, Bartholomew, and Reay (2004) discussed. Sam attended a public school in a very affluent community. For that school, Sam was at a remedial level for his age. A high percentage of his peers were college bound, with many attending elite colleges and universities. Whether it was due to ADHD or the distraction of a difficult home-life, this school system placed him in low level classes to increase his chances of success. It is reasonable to assume that the school district benefited from his placement there as well (Rosenfeld & Wise, 2000). And Sam revealed his incentive to remain at a low-level when he confirmed that in those classes he was able to get an "A." Not challenging himself allowed him to easily satisfy the requirements of his IEP (Garibaldi, 2006). According to Sam, at his next school, the alternative school, the math classes were even less challenging. However, staying on the low-math track lessens Sam's chances of ever having a scientific career, a personal goal that he articulated. Maintaining his low-achieving identity could block his progress towards this more accomplished future-identity. Fortunately for Sam, in the setting of a small, college-preparatory school, he can form a new identity based on his interest in science and mathematics. Sam's concern for the mathematics of diving could become a bridge from his remedial studies to more advanced work (Ernest, 2004b). Validity Though I compared information from multiple sources to ensure a triangulation of the data, which was a means to validity, certain factors created obstacles for generalizing my results (Carspecken, 1996). For one, my research subjects were troubled boys who have personal issues that may affect their learning of mathematics. And the steps each boy has taken to deal with their problems resulted in broken continuity with their mathematics' studies. Also, when I interviewed them, they may have sounded more positive about mathematics while attending a small, supportive, school than they would have in a large, competitive academic setting. As a mathematics teacher, my independent school offers me the autonomy to build upon learners' interests. This opportunity may not be available everywhere. CONCLUSION A small scale study limits the extent to which I may draw any conclusions, but I think that Rick and Sam had spent the majority of their high-school careers working below their potential in mathematics. In my opinion, the well-intended efforts to help them succeed academically, as required by IDEA quite likely yielded mixed results, which I have already discussed. Along Heller's (1999) line of thinking, my intent of making such a statement is not to find a party to blame, but rather to recognize that complex forces are at work concerning each boy's developing mathematical identity. Understanding how these competing interests come into play may provide an insightful way to serve the needs of struggling students. Each boy in my study expressed a desire to pursue a college education and a career. However, Rick also voiced anger and frustration with mathematics, and at the beginning of my investigation had pretty much given up on the discipline. Albeit Sam knew he was capable of more, he enjoyed receiving easy "A's" in his low level math-classes, not seriously considering the long-term implications of continuing on this track (Brahier, 2000; Ernest, 2004c). Ironically, it was more the result of Rick's and Sam's bad behavior, not academic disenchantment, which led their parents to remove them from the public education system. In the setting of a small boarding school, both boys were able to move up to a college-preparatory level in mathematics. New mathematical identities could be sparked by building bridges to topics that interested them (Ernest, 2004b). These two boys were fortunate to attend a private school where they could receive a tutorial education. To improve my study, I would expand the discussion surrounding my methodology, and how certain research subjects could affect the outcome. Lower socio-economic, underachieving boys would probably not have the economic resources to elect the option of forgoing publicly-funded support (Behrens & Satterfield, 2006). Nevertheless, my experience with Rick and Sam illustrates the power of social constructivism; it was through conversation that I was able to discover math-related topics that engaged them (Ernest, 2004a). And this opens up the idea that if we aggressively look for a learner's problems, then we must equally as assertively look for his/her strengths. With regard to Special Education, I advocate for turning technology in the direction of finding what learners are good at, as well as identifying their special needs. A more balanced version of a learner's IEP might even inform building a bridge. Within the process of quantifying performance, it is possible to forget that people need to be treated with dignity and respect (William, Bartholomew, & Reay, 2004). REFERENCES Bateson, G. (2000). Morale and National Character. In Gregory Bateson (new forward by Mary Catherine Bateson), Steps to an Ecology of Mind (pp. 88-127). Chicago, IL: The University of Chicago Press. Behrens, E. & Satterfield, K. (2006). Report of Findings from a Multi-Center of Youth Outcomes in Private Residential Treatment, Presented at the 114th Annual Convention of the American Psychological Association at New Orleans, LA, August, 2006. Retrieved October 28, 2006 from natsap.org/Behrens.doc, NATSAP. Brahier, D. J. (2000). Teaching Secondary and Middle School Mathematics. Needham Heights, MA: Allyn & Bacon. Brown, T., Jones, L. & Bibby, T. (2004). Identifying with Mathematics in Initial Teacher Training. In Walshaw, Mathematics Education within the Postmodern (pp. 161-179). Greenwich, CT: Information Age Publishing, Inc. Carspecken, P. F. (1996). Critical Ethnography in Educational Research: A Theoretical and Practical Guide. New York, NY: Routledge. Connell, R. W. (2000). The Men and the Boys. Berkeley, CA: University of California Press. Ernest, P. (1991). The Philosophy of Mathematics Education. Abingdon, OX: RoutledgeFalmer. Ernest, P. (1998). Social Constructivism as a Philosophy of Mathematics. Albany, NY: State University of New York Press. Ernest, P. (2004a). Postmodernism and the Subject of Mathematics. In Walshaw, Mathematics Education within the Postmodern (pp. 15-33). Greenwich, CT: Information Age Publishing, Inc. Ernest, P. (2004b). Postmodernity and Social Research in Mathematics Education. In Valero & Zevenbergen, Researching the Socio-Political Dimensions of Mathematics Education: Issues of Power in Theory and Methodology (pp. 65-84). Norwell, MA: Kluwer Academic Publishers. Ernest, P. (2004c). Advanced Course Module: Mathematics and Special Educational Needs. Exeter, UK: , School of Education. Ernest, P. (2005). Advanced Course Module: Mathematics and Gender. Exeter, UK: , School of Education. Garibaldi, G. (2006). How Schools Shortchange Boys. City Journal Summer 2006, 16(3), Retrieved October 11, 2006 from www.city-journal.org, Manhattan Institute. Glendinning, C. (1995). Technology, trauma, and the wild. In T. Roszak, M. E. Gomes & A. D. Kanner (Eds.), Ecopsychology: restoring the earth, healing the mind (pp. 55-67). San Francisco, CA: Sierra Club Books. Heller, A. (1999). A Theory of Modernity. Malden, MA: Blackwell Publishers, Inc. Kopp, D. A. (2001). Significant Cases in Maine School Law. Portland, ME: Drummond Woodsum & MacMahon. Lareau, A. (2000). Home Advantage: Social Class and Parental Intervention in Elementary Education. Lanham, MD: Rowman & Littlefield Publishers, Inc. Lareau, A. (2003). Unequal Childhoods: Class, Race, and Family Life. Berkeley and Los Angeles, CA: University of California Press. Levitt, S. D. & Dubner, S. J. (2005). Freakonomics: A Rogue Economist Explores the Hidden Side of Everything. New York, NY: HarperCollins. Pollack, W. (1999). Real Boys: Rescuing Our Sons from the Myths of Boyhood. New York: NY: Henry Holt and Company. Ritzer, G. (2004). The McDonaldization of Society, Revised New Century Edition. Thousand Oaks, CA: Pine Forge Press. Rosenfeld, A. & Wise, N. (2000). The Over-Scheduled Child: Avoiding the Hyper-Parenting Trap. New York, NY: St. Martin's Press. Sommers, C. H. (2000). The War Against Boys: How Misguided Feminism is Harming Our Young Men. New York, NY: Simon & Schuster Paperbacks. Thomas, J. (1993). Doing Critical Ethnography (Qualitative Research Methods: Vol. 26). Newbury Park, CA: Sage Publications, Inc. Thurston, W.P. (1990). Mathematical Education. Notices of the AMS, 37, 844-850. Tofler, I.R., Knapp, P.& Drell, M.J. (1999). The "Achievement by Proxy" Spectrum: Recognition and Clinical Response to Pressured and High Achieving Children and Adolescents. The Journal of the American Academy of Child and Adolescent Psychiatry, 38(2), 213-216. William, D., Bartholomew, H., & Reay, D. (2004). Assessment, Learning and Identity. In Valero & Zevenbergen, Researching the Socio-Political Dimensions of Mathematics Education: Issues of Power in Theory and Methodology (pp. 43-61). Norwell, MA: Kluwer Academic Publishers. 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