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PAGE 1 1 DOMAIN KNOWLEDGE, ATTITUDES, SELFEFFICACY BELIEFS, AND ATTRIBUTIONS FOR ACHIEVEMENT WORK ING TOGETHER IN THE COMMUNITY COLLEGE REMEDIAL MATHEMATICS CLASSROOM By KENNETH SCOTT MURPHY A THESIS PRESENTED TO THE GRADUATE SCHOOL OF THE UNIVERSITY OF FLOR IDA IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF ARTS IN EDUCATION UNIVERSITY OF FLORIDA 2007 PAGE 2 2 2007 Kenneth Scott Murphy PAGE 3 3 To my hero, Kenneth Howard Murphy. PAGE 4 4 ACKNOWLEDGEMENTS I would like to acknowledge th e patience and vision of my colleagues at Santa Fe Community College. Without their continued su pport and encouragement, this document and the degree which accompanies it woul d never have come to fruition. Special thanks to: Carole Windsor, Mark Dicks, Lela Elmore, and Megan Swilley. PAGE 5 5 TABLE OF CONTENTS page ACKNOWLEDGEMENTS.............................................................................................................4 ABSTRACT....................................................................................................................... ..............6 CHAPTER 1 INTRODUCTION................................................................................................................... .7 Introduction................................................................................................................... ............7 Statement of the Problem....................................................................................................... ...7 Purpose........................................................................................................................ .............8 2 LITERATURE BACKGROUND............................................................................................9 Mathematical Domain Knowledge...........................................................................................9 Attitudes & SelfEfficacy Beliefs...........................................................................................10 Attributions for Achievement.................................................................................................13 3 FRAMEWORK AND DESIGN.............................................................................................16 Participants................................................................................................................... ..........17 Instruments.................................................................................................................... .........18 Methods........................................................................................................................ ..........19 4 DATA ANALYSIS................................................................................................................21 Knowledge Attitudes & Efficacy.....................................................................................25 Attitudes & Efficacy Attributions....................................................................................26 5 CONCLUSIONS....................................................................................................................29 APPENDIX A INTERVIEW QUESTIONS...................................................................................................35 B STUDENT SURVEY.............................................................................................................36 LIST OF REFERENCES............................................................................................................. ..37 BIOGRAPHICAL SKETCH.........................................................................................................39 PAGE 6 6 Abstract of Thesis Presen ted to the Graduate School of the University of Florida in Partial Fulfillment of the Requirements for the Degree of Ma ster of Arts in Education DOMAIN KNOWLEDGE, ATTITUDES, SELFEFFICACY BELIEFS, AND ATTRIBUTIONS FOR ACHIEVEMENT WORKING TOGETHER IN THE COMMUNITY COLLEGE REMEDIAL MATHEMATICS CLASSROOM By Kenneth Scott Murphy August 2007 Chair: Thomaseina Adams Major: Mathematics Education Community colleges are faced w ith increasing numbers of students who are unprepared to complete and receive passing grades for collegel evel coursework. Mathematics is one subject area in which many students often require remediation. Researchers have discussed remedial mathematics in community colleges as it relate s separately to mathematical domain knowledge. However, there are no studies which examine how the relationships between and among attitudes toward mathematics, selfefficacy beliefs, and at tributions for achievement influence students acquisition of mathematical domain knowledge. The findings of this study showed that attitudes toward mathematics, selfefficacy beliefs, a nd attributions for achievement influenced mathematical domain knowledge acquisition among four students. PAGE 7 7 CHAPTER 1 INTRODUCTION Introduction Mathematics is a troublesome area of study fo r many students. Many students either experienced difficulty with mathematics or simply have not successfully mastered a significant amount of mathematics. From an educational stan dpoint, the former is preferable to the latter. Because mathematics is an exact science, student s and teachers alike need not expect that the learning of mathematics will, or should, be without obstacles or a need for reteaching. Many factors affect the way that students relate to mathematics. These fact ors include, but are not limited to, mathematical domain knowledge, attitudes toward mathematics, selfefficacy beliefs, and attributions for achievement in mathematics. The researcher believes that knowledge affects attitude and efficacy beliefs, atti tude and efficacy beliefs affect at tribution, and viceversa in both cases. Through a description of stud ents previous experiences, this study explored situations in which attitude toward mathematics, mathema tical efficacy beliefs, and attributions for achievement in mathematics influenced the acqui sition of mathematical domain knowledge as it pertains to remedial algebra coursework. Statement of the Problem Community colleges and many universities in America have adopted policies and mission statements that reflect the now commonly held belief that higher education should be made available to all who would make a serious attempt at such an endeavor. This shift in focus has led to opendoor policies and relaxed admission re quirements, which have afforded opportunities for many to have access to education. Community colleges, for example, are not critically selective in their admissions processes. A st udent need not even possess a high school diploma because, in many cases, community colleges offe r programs of study leading to G.E.D. or PAGE 8 8 certificate of completion of high school courses. This has resulted in an influx of students who are not prepared for collegelevel coursework. Mathematics is one subject area in which many unprepared students require re mediation. A lack of speci fic domain knowledge, negative attitudes toward mathematics, lack of selfeffi cacy, and external attributions for mathematics achievement often lead to a pattern of severely limited success in collegelevel mathematics courses. Purpose As colleges and universities seek new ways to serve their stude nts needs, American higher education has witnessed the relo cation of college preparatory c ourses, from highs schools or preparatory institutions, to the campuses of colleges themselves. Although universities sometimes offer these remedial courses, commun ity colleges do so almost invariably. It has become common for students to attend one or more of these preparatory courses, usually not for credit, to ready themselves for th e collegelevel material required for their major fields of study. As evidenced by proliferation of remedial course s, mathematics is one subject in which students typically lack adequate knowledge. Since secondary schools in the United States presumably do not purposely furnish colleges with academically deficient students, one might wonder just how we have arrived at the point where we now stand. Despite how it has happene d, community colleges, in particular, must find ways to help students overcome their academic deficiencies and move on to collegelevel coursework. The primary purpose of this study is to identify and describe relationships between and among attitudes about the usef ulness of mathematics, selfeffi cacy beliefs, and attributions for achievement in mathematics and their infl uences on mathematical knowledge acquisition among community college students enrolled in a remedial algebra course. PAGE 9 9 CHAPTER 2 LITERATURE BACKGROUND Research exists, albeit not very current, wh ich addresses mathematical domain knowledge, attitudes and selfefficacy beliefs about mathem atics, and attributions for achievement in mathematics separately. There exists some litera ture which explores relationships between these areas; for example, Xin Ma (1997), relates atti tude toward mathematics to achievement in mathematics. Literature will be referenced below, particularly that which pertains to proposed or established relationships between mathematical domain knowledge, attitu des toward and selfefficacy beliefs about mathematics, and attri butions for achievement in mathematics. Mathematical Domain Knowledge Mathematical domain knowledge refers to know ledge that is specific to the domain of mathematics. For example, knowing that the qua dratic formula is requi red to factor general equations of the form Ax2+ bx + c = 0, requires mathematical domain knowledge. Mathematical domain knowledge consists of several types of knowing, including knowing that, knowing how, knowing why, and knowi ng to. According to Selden and Selden (1996), knowing that may simply refer to a stud ents ability to reprodu ce facts they learned. Knowing how usually refers to knowing the pr ocess required to so lve a problem. For example, a student who has obser ved the teacher solving quadratic equations using the quadratic formula ostensibly knows how to solve quadratic equations themselves. By knowing why, (it is) meant having various stories in ones head about why a mathematical result is so (Selden and Selden, 1996). It should be noted that know ing why and proving a thing are different; proofs require much more than just stori es in ones head. Knowing to refers to a students ability to act, and is arguably the most impor tant type of knowing. This t ype of knowing allows learners to shift their attention from thinking to doing. A ccording to Selden and Selden, knowing to is PAGE 10 10 the type of knowing that most of ten translates to achievement; it is this type of knowing that teachers should try to teach most. Specific domain knowledge in the area of ma thematics is essential to the successful teaching and learning of mathematics. Unlike many other knowledge domains, mathematics requires linear thinking and a pr ecise understanding of strategies that can be used to solve problems. That is not to say that teachers and st udents must always arrive at the correct answer to a mathematics problem in exactly the same wa y. In fact, employing seve ral strategies is an asset to the mathematics problem solver. Given that teachers and mathem atics students need to access the same canonical domain knowledge, we mi ght expect that students enter college equipped with a certain amount of basic knowle dge of mathematics. Basic knowledge of mathematics includes terminology along with knowledge of how to perform basic arithmetic and algebraic operations. This is not the case, in fact, It can no longer be assumed that the beginning college algebra student understands the most basic con cepts of mathematics (Chang, 2000, p. 17). Community colleges are uniquely situ ated to confront this problem because teaching is the primary function of faculty at community colleges. That is, behavior management is not an issue as it may be in high school. In addition, community college faculty do not have research and service interests competing with teaching duties for their time as is the case in larger colleges an d universities. Attitudes & SelfEfficacy Beliefs An important factor bearing on the forethought phase of problem solving in mathematics is selfefficacy. Selfefficacy is ones belief about thei r ability to be successful. Selfefficacious learners very often judge themselves able to solve problems a priori, and thus make choices about strategy and effort that often lead to succ essful problem solving. There is a related idea, that of selfconcept, which is distinguishable fr om selfefficacy in that selfefficacy is a context PAGE 11 11 specific assessment of competence to perform a specific task Selfconcept is not measured at that level of specificity and includes beliefs of selfworth associated with ones perceived competence. (Pajares & Miller, 1994) Learners may have a positive selfconcept in general, but lack selfefficacy in a particular area. Also, a relationship exists between ones perceived ability to solve a specific mathematics problem and ones actual efficacy in solving that problem. As the former increases, so does the latter. That is no t to say that selfefficacy alone can ensure success in mathematics problem solving, rather that wh en enhanced selfefficacy is combined with domain knowledge and proper use of strategies, a greater degree of succ ess may be attained. First, selfefficacy beliefs among learners of mathematics can be justifiably considered by instructors to be the single most important factor contributing to persistence in mathematical problem solving. It is not that selfefficacy is strictly more important than domain specific knowledge or achievement attributi on strategies; rather selfefficacy occupies a position which is somewhat psychologically predominant. Unles s people believe that they can produce desired effects by their actions, they have little incen tive to act. (Bandura, Barbaranelli, Caprara, & Pastorelli, 1996, p. 1206). Believing in their capabil ities is what motivates mathematical learners to persist in the learni ng of mathematics. Without some degree of selfefficacy, students are likely to pursue mathematical learning only becaus e it is required, and not for the purpose of mastery of the material. Mathematical self efficacy beliefs are rela ted to mathematical achievement in a number of ways. First, efficacy beliefs are formed and influenced by past attempts to achieve mathematical success. The results(indicate) that perceived selfefficacy is a significant contributorincluding past performance (Bandura & Locke, 2003, p. 90). Depending upon the level of success attained, students form beliefs about thei r future abilities to perform the same mathematical task or function. A string of unsuccessful attempts to perform PAGE 12 12 mathematically could contribute to a negative se t of mathematical selfe fficacy beliefs. Once these negative beliefs exist, they are difficu lt to overcome. Often, students recall that an outstanding teacher in their educational experien ce was instrumental in their understanding of a certain subject. In the area of mathematics, it is probable that such teachers overcome negative selfefficacy beliefs that the students already posse ss, foster positive selfefficacy beliefs that the students do not possess, or both. Second, according to Bandura (1996), efficacy be liefs are related to social beliefs of individuals. For example, a student may perceive that he or she is efficacious in mathematics based upon successes with math in their immediate family, or in their peer groups. Parental aspirations and perceived efficacy build children s sense of efficacy and academic aspirations (Bandura, Barbaranelli, Caprara, & Pastorelli 1996, p. 1213 ). When parents facilitate and encourage academic pursuits in their children, effi cacy beliefs in those children are likely to be high. That is not to say that those same childre n are always the high achie vers; rather that they are not hindered in their learning by the effects of low selfefficacy beliefs. Peer groups also can have an influence, positive or negative, on e fficacy beliefs. Perceived academic efficacy affect(s) their academic achievement both indepe ndently and through the mediated effects of peer relations (Bandu ra et. al., 1996, p. 1211). Third, efficacy beliefs have a reciprocal re lationship with motivation. People generally avoid activities with which they do not feel comf ortable. This is cert ainly true of college students, who tend to choose courses, lodging, and even friends based on comfort level and confidence. Mandatory mathematics requirements, however, force college students to complete a certain level of mathematics coursework. Since many students enroll in these mandatory courses with no other motivation th an to satisfy general educati on requirements, it is important PAGE 13 13 that instructors recognize the re lationship between efficacy beliefs and motivation. According to Bandura and Locke (2003), The evidenceis c onsistent in showing that efficacy beliefs contribute significantly to the level of motivation and performance. (p. 87) Another factor possibly cont ributing to the poor mathema tical achievement of many college students is attitude toward mathematic s. Negative attitudes toward mathematics are common among students today, and may be reinforced by parents and even teac hers. That is not to say that most parents and teachers share a ne gative view of mathematic s; rather that these groups evidently do not undertake to foster more positive perception of mathematics among students. Human activity has determined that previous mathematics performance and perceived ability are both key elements for success in mathematics (Kloosterman & Stage 1995, p. 296). Because rational consideration does not allow one to conclude that all students who are ill prepared for college mathematics courses are lazy and/or not very inte lligent, attitudes and beliefs toward mathematics must play a substant ial role in their poor performance. Certainly, there could be other factors at work as well, such as lack of exposure to mathematics and poor instruction. According to Mau (1991), researchers ha ve concluded that ne gative beliefs about learning and doing mathematics seem to be a key to many students inability to focus enough to survive mathematics courses (p. 25). Attributions for Achievement For the purposes of this study, the phrase attri butions for achievement will refer to what or to whom a person gives credit for mathema tical achievement. St udents might attribute success in the area of mathematic s to various influences. A popul ar attribution for achievement is studying. Students associate studying well with academic success, not only in their own experience, but in general. This is a desira ble attribution, but studying may be interpreted too broadly. Because of differences of opinion, it is difficult to meas ure some attributions made by PAGE 14 14 students. For example, what one student cons iders studying in depth, another might consider nothing more than reading over material. Neverthe less, attributions are routinely made, and can serve in a general way to inform mathematics inst ructors as to the mindset of their students. The types of attributions student s make fall into one of two cate gories. One is that of an external attribution, for example, The test was to o hard, so I failed it. Th e other is that of an internal attribution, for example, I did not pr epare properly for the exam so I failed it. The second type of attributio n, internal attribution, claims that the person was directly responsible for the event. (Butterfield, 1996, p. 4). Some examples of positive external attri butions are: I was just born with math intelligence; my brother is really good at math, I really like this teacher, my grade depended on it, the test was easy, etcetera. Ther e are, of course, also negative ex ternal attributions such as: I was just not born with any math intelligence, the test was too hard, the teacher is terrible, etcetera. These external attribut ions do not leave the student any recourse to perform better next time. With this mindset, if the external factors remain the same, the students achievement level will also remain the same. In the case of external attributions, the student places the responsibility for his/her performance on someone or something other than his/her own efforts. Some examples of positive internal attributions are: I studied hard for th at test, I utilized the proper strategies, I recalled th e algorithm for solving that probl em, etcetera. Here are a few examples of negative internal at tributions: I forgot to use th e proper problemsolving strategy, I am not the kind of person who spends any time st udying, I did not internalize the material in this unit, etcetera. These attributions give room for improvement, may lead to increased selfefficacy, and as such are more desirable than exte rnal attributions for achievement. According to Gredler (1992), attributions with an internal locus lead to feel ings of confidence. Internal PAGE 15 15 attributions tend to be habit forming. For exam ple, performing well on a mathematics exam after studying at a certain part of the library with another student might encourage the students to study together at the same place for the next exam. Over the course of the semester or year, it may become a habit for the students to meet at th at same place to study for mathematics exams. So, the student may believe that their mathemati cal achievement is linked to studying with that other student (external), or at the location in the library (exter nal), or the methods and strategies (internal) he or she used to study. Clearly, attributing mathema tical success to the methods and strategies used for studying is the desirable attr ibution because the othe r two rely upon factors out of the students control. In fact, attribut ing successes to external factors can undermine an existing habit (Butterfield 1996, p. 6), such as might occur in our scen ario above if all the attribution is given to the location and the st udy partner. The student may cease to employ the methods and strategies that caused the successes in the first place. Attributing successes to proper (mostly internal) factors reinforces good habits, thereby possibly increasing success in mathematics and mathematical selfefficacy beliefs PAGE 16 16 CHAPTER 3 FRAMEWORK AND DESIGN This study utilizes description to isolate and magnify the evident rela tionships between and among attitudes toward mathematics, selfefficacy beliefs, and attributions for achievement. While the positive influences that these ha ve on each other and on mathematical knowledge acquisition will be the focus, it is useful to understand the interplay in a practical way that includes the potential negative infl uences as well. The effects of domain knowledge on attitudes toward mathematics and selfefficacy beliefs can be positive, negative, or neutral. Increased mathematical knowledge may lead to better attitudes and selfeffi cacy beliefs; lack of such knowledge may lead to poor attitudes and selfeffi cacy beliefs, and it is possible that varying degrees of domain knowledge have no significan t effect on attitudes toward mathematics and selfefficacy beliefs. It should be noted that e ffects of increased knowledge on attitudes and selfefficacy beliefs could be differential. That is, increased domain knowledge could improve attitudes without having a signifi cant impact on selfefficacy. Fo r the purposes of this study, a positive effect on attitudes and selfefficacy shall mean on either instead of on both, meaning that improvements in attitude are not assumed to be intrinsically linked to improvements in selfefficacy beliefs and viceversa. The effects of attitudes and selfefficacy be liefs on achievement attributions can be positive, negative, or neutral in the same way as described above. An example of a positive effect on achievement attributions might be that improved attitudes and se lfefficacy beliefs led a student to attribute success on a mathematics asse ssment to studying or to st rategies rather than to luck or to natural ability. By the arrows in the diagram, it is meant that selfefficacy beliefs and attitudes toward mathematics directly affect both domain knowledge and achievement attributions, while domain knowledge and attributions directly affect attit ude/selfefficacy only. PAGE 17 17 This interpretation speaks to th e importance of attitudes and self efficacy beliefs in the area of mathematics. It would be difficult to overem phasize the positive effects of improvement in attitudes toward mathematics and selfefficacy be liefs as they pertain to domain knowledge and achievement attributions. Participants Four students at a community college in the sout heastern United States participated in this study. Each of these students was, at the time of the study, enrolled in a remedial algebra course at the community college. The course, MAT0020 (In tegrated Mathematics) is geared to begin with a brief review of arithmetic skills and move to basic algebra skills. The course culminates in a standardized state exit exam which requires mastery of algebra skills comparable to those required to pass Algebra II in a st ates high school. This level of mastery indicates that students possess the skills necessary to enroll in a coll egelevel algebra course. The Common Placement Test (CPT) is a knowledge based exam which community colleges in the state use to place students. The focus, then, was the breadth a nd depth of domain knowledge up to and including algebraic thinking. Each stude nt placed in MAT0020 demonstr ates a reasonable level of arithmetic knowledge, but fails to demonstrat e that same reasonabl e level of algebraic knowledge. Because each of the students in the study was enrolled in MAT0020, mathematical domain knowledge was uniformly less than desirabl e for college students. Participants shall hereafter be referred to as stude ntA, studentB, student C, and studentD, which approximates their selfreported achievement levels (gradewise) in MAT0020. Thes e reports were confirmed via the course instructor with written, signed pe rmission from the students. In each case, the students report of achievement level was a match with the grade reported by the MAT0020 instructor. StudentA and stude ntB occupy the high level of achievement range, studentC occupies the middle range, and studentD occupi es the low range. StudentA is a caucasian PAGE 18 18 female, 1820 years of age, enrolled in her second semester of college. StudentB is a black male, 1820 years of age, enrolled in his second semest er of college. StudentC is a caucasian male, 1820 years of age, enrolled in his second semester of college. St udentD is a hispanic male, 1820 years of age, enrolled in his third semester of college. Each of the participants reported a somewhat weak background in mathematics, which st ands to reason, as each of them is enrolled in a remedial algebra course. In order to be sure that achievement in terms of mastery and performance were uniform, the four participants were all chosen from the same section of MAT0020. Each of the participan ts took part in an impromptu end of class meeting attended by their instructor and the re searcher; at this time they all verbally agreed to participate in the study. It was announced that, in addition to their gr ades and mathematical background information, a time commitment would be necessary for surveying and interviewing. There was an exchange of email addresses and phone numbers in preparatio n for beginning the study. The participants all voluntarily signed an informed consent document, and all spoke to the researcher about their experiences with mathematics. Instruments The instruments utilized in this study may be viewed in the appendix and include the Student Survey and an interview guide designed by the researcher to re veal attributions for achievement in mathematics. The fifteen items in the Student Survey were separated for this purpose, with the first eleven indicating the st udents attitude toward mathematics, and the remaining four indicating the students level of se lfefficacy with respect to mathematics. This Student Survey has been piloted in an ongoing st udy of college students enrolled in a remedial algebra course (MAT0020 or MAT0024). Except fo r cases of total disinterest, the Student Survey gives an indication of either positive or negative general attitude toward mathematics, and of either generally positive or generally negative mathematical selfefficacy beliefs. The PAGE 19 19 personal interview was utilized to ascertain the level of mathematical domain knowledge possessed by each student. During the interviews, the students related grades, mastery level, and familiarity with various mathematical concepts The interview items were designed to evoke responses which would indicate the achievement attr ibutions of the student. That is, to what does the student attribute success (or failure) in th e area of mathematics? For example, a student might attribute success on a mathematics assignme nt to having the natural ability to do well in math, or Im just smart attribution. The fifteen item Student Survey has been piloted, in particular, on a group of over 100 remedial algebr a students at the community college, excluding the four research participants of this study. This made for ease of use a nd interpretation, as the items had been edited over time to elicit responses aimed directly at the issu es of attitudes toward mathematics and mathematical selfefficacy beliefs. Within the structure of the five mathrelated statements used in the personal interview, tendencies toward one type of attribution or another were determine d. Attributions for mathematical achievement were categorized according to their type. The two types of attributions were external attributions and internal attributi ons. The former indicates that the student attributes success or failure to some entity other than themselves, while the latter indicates that the student attr ibutes success to studying, reca lling examples, and generally working toward success in an incremental manner. Methods The three indicators of knowledge, efficacy/att itude, and attribution were, respectively, personal interview/ course grades, Student Survey responses, and verbal responses to openended mathematics statements in the interview instrument (appendix). Within two weeks of the initial meeting, each student made contact with the rese archer, and took time to respond to the fifteen item Student Survey (appendix). The responses to the Student Surv ey were analyzed to ascertain PAGE 20 20 a general idea of the students attitude toward mathematics and selfefficacy beliefs. At the time when the students responded to the Student Surv ey, a subsequent interv iew time was scheduled. The participant and researcher agreed to a fo rtyfive minute window to explore the students general mathematics background and to have th em respond to five openended mathematicsrelated statements (appendix). Interviews were taperecorded and conducted in a semiprivate manner, meaning that while the student and research er were in sight of ma ny other students, their conversations could not be overheard. The inte rviews began with an informal chat about mathematics, more or less an icebreaker, follo wed by a request of the st udent to relate their general mathematics background. This informa tion included memories of difficulty with mathematics, grades received in current or pa st mathematics courses, certain teachers that motivated the student (or not) to attempt to im prove their mathematical situation, and general comments from the student about their feelings and thoughts about mathem atics. After this informal beginning to the interview, each stud ent was asked to formally respond to five mathematicsrelated statements. Students responde d to each of the five items, and then were invited to make any closing statements they wi shed before concluding th e interview. Without exception, the students spoke at length about th eir relationship to mathematics and how that relationship had developed over time. PAGE 21 21 CHAPTER 4 DATA ANALYSIS Student D reported that his mathematics back ground was never good. He related that he has always received tutoring since early youth. He feels as though he will never need mathematics, and related that he becomes bored easily when doing mathematics. Student D expressed anxiety problems when d ealing with mathematics, and stated that uncertainty is the norm for him when attempting to solve mathematics pr oblems. He also stated that he has always had to play catch up in the area of mathematics, more often than not, having to work backwards from provided solutions to understand how to so lve problems. Student D sees no reason to do mathematics without a calculator, and believes he performs satisfactorily when allowed to use a calculator to solve mathematics problems. In general, student D believes he possesses unacceptably little mathematical domain knowledge, and that what little he does possess is not very clear. Although student Ds responses to the Student Survey were mixed, his general attitude toward mathematics is negative. This is best evidenced by his agreement with the statements: I often do not care whether I get math problems corre ct or incorrect, and Mathematics is more something I have to do than something I choose to do. Student Ds selfefficacy beliefs are high relative to his generally negative attitude toward mathematics. For example, even though he agreed with the statement: I often think of other things while attempting to solve math problems, he also agreed with the statement: Sol ving math problems is easy for me. It is also interesting to note that even after mentioni ng his mathematics anxiety problems, student D strongly disagreed with the statement: Perfo rming mathematical computations makes me anxious. In his interview, student D attribut ed success to eleventh hour studying, and included the possibility, though not for himself, that some pe ople are born with mathematical ability. PAGE 22 22 Student C reported that his mathematics background was okay up until high school. He indicated that the difficulty arose when he began to become interested more in extracurricular activities than in school itself. Teachers play a large role in the learning process of student C ; in fact, he claims to do little to no studying outside of class. He stated that in high school, his mathematics teachers did not care all that much, a nd that as a result, he did not care either. Student C related that he enjoye d no support structure, such as tutoring and the like, and still does not make use of such things even when th ey are available. Having already failed MAT0020 once, student C feels the need to perform this time, even though he is often bored with mathematics. He believes that, in general, he possesses a typical amount of mathematical domain knowledge, and that mathematic s is not all that difficult. Student C displays a generally positive attitude toward mathematics. He converses in an upbeat manner about the subject, and strongly disagreed with the statements: Mathematics is not very useful in the real world. and I ofte n do not care whether I ge t math problems correct or incorrect. Student C responded in agreem ent to the statements: Mathematics is more something I have to do than something I choose to do., and I often think of other things while attempting to solve math problems. This indicates that student C is not really interested in mathematics for reasons other than the general education requirement that he complete a certain amount of college level mathematics coursework. Student Cs responses to the Student Survey items 12 through 15 indicated that he felt efficaci ous in the area of mathematics. Student C attributed success in mathematics to paying atte ntion in class and to the influence of his instructor. He believes that seeing and hearing about a mathem atics concept once is usually enough to master that concept, and that it is not necessary to repeat edly practice a skill for mastery. Student C does indicate that he believes he will have to study, eventually. PAGE 23 23 Student B indicated that hi s dealings with mathematics were good until high school. Similar to student C, student B related that the difficulty he experienced then was of his own making. He stated that this difficulty has made college very challenging, and that he has been forced to work much harder to make up for the lack of knowledge. Student B comes from a large family, and related that th erein a support structure exists on which he regularly falls back. Having siblings who can help with school occasio nally increases student Bs will to succeed and enhances his selfefficacy beliefs. He believes that there are many people and support structures other than his family that students can take a dvantage of in the college setting; mathematics instructors and tutors were two that he menti oned. Student B possesses an acceptable amount of mathematical domain knowledge, but he does no t seem to be comfortable utilizing the knowledge. While student Bs expressions would not likely give it away, he has a decidedly negative attitude toward mathematics. His agreement w ith items 3, 4, 7, and 8 of the Student Survey, which all address attitude toward mathematics, along with his disagreement with the statement: I enjoy studying mathematics confirms this negative attitude. Stude nt B possesses low selfefficacy beliefs, as shown by his responses to items 12 through 15 of the Student Survey. He related that he experiences great difficulty in le arning mathematics and that he believes he will never be good at doing mathematics. In spite of student Bs negative attitude and low selfefficacy beliefs, he is performing well on exams in MAT0020. It is also noteworthy that he strongly agrees that mathematics is an important field of study. Student B attributes his success in MAT0020 to lots of behind the scenes work. In his interview he related that he spends a great deal of time mastering the concepts covered in the course. Student B re presents a bit of an PAGE 24 24 oddity because he is currently experiencing success with mathematics in spite of his low efficacy beliefs and his negative attitude to ward mathematics. Student A considers mathematics to be a very theoretical science, and she does not believe that her path in life will require her to make use of very much of the mathematics she is required to learn. She indicates that she has always memorized mathematical concepts, and does not usually see the big picture where mathematics is concerned. Student A believes that her middle school played a role in her struggle with mathematic s, stating that she was placed in a lower level course because of overcrowding at the school. As a result, student A believes, she was illprepared for high school mathematics courses and subsequently for college mathematics courses as well. Although mathematics has never come easily to her, she does not experience great difficulty as long as she puts in the necessary stu dy time. Student A describes her relationship to mathematics as a love/hate relationship. Student A exhibits a positive attitude toward mathematics, as evidenced by her strong disagreement with items 4, 5, 6, and 8 of the Student Survey, all of which address attitude toward mathematics. She is the lone student in this study who disagrees at all with the statement: Mathematics is more something I have to do th an something I choose to do. In conversation, student A disparages the study of mathematics, bu t not because she finds it difficult. She speaks of a feeling of accomplishment when doing mathem atics, but relates that washing her car might produce the same feeling. Her responses to item s 12 through 15 of the Student Survey indicate that student A possesses high selfefficacy beliefs with respect to mathematics. She reports mathematics anxiety, but still believes she can perform mathematical ope rations successfully. The attributions for mathematical achieve ment given by student A included existing PAGE 25 25 mathematical knowledge that she possesses, studying, and her current teachers ability to teach effectively. This study lends itself to descrip tive analysis of the data acqu ired from the Student Surveys and the interviews. The connections and implications made will be more focused on why and how they occurred than on quantitative measures such as to what extent they occurred. The data have been summarized in the preceding pa ges; next they will be analyzed. Knowledge Attitudes & Efficacy Mathematical domain knowledge is sometime s difficult to measure because summative assessments, which are the usual measuring tool sometimes do not give the best glimpse of domain content knowledge. This happens for a vari ety of reasons, including test anxiety, lack of feedback, and lack of motivati on. For the participants in this study, mathematical domain knowledge is lacking in general, hence the remedial algebra course (MAT0020). With this in mind, performance in MAT0020, in the traditional gr ading scale sense, will be used to measure differences in the students domain knowledge. Student A and student B are both performing near the high end of the grading scale, with student A slightly ahead. It will be assumed, then, that student A possesses slightly more mathema tical domain knowledge than student B. Recall that student A exhibits a positive attitude toward mathematics, and that she believes that she is efficacious in the area of mathematics. St udent B harbors a negative attitude toward mathematics, not outwardly, but in his candid res ponses to certain items of the Student Survey. He also possesses low selfefficacy beliefs with re spect to mathematics. Since students A and B possess very similar amounts of domain knowledge, this factor may not have contributed to the large difference in selfefficacy beliefs and attitu de. Rather, improvements in the attitude and efficacy beliefs of student B would serve to increase his domain knowledge. It is widely accepted that increased knowledge leads to increa sed efficacy, but perhaps the relationship is PAGE 26 26 more reciprocal than commonly believed to be. Allowing domain knowledge, efficacy beliefs, attitudes, and attributions for achievement to function in a positive feedback cycle is certainly desirable, perhaps even necessary for some lear ners. An example of a positive feedback cycle would be a learning environment where equal em phasis was placed on each of the four factors mentioned above. In this way, the learning needs of the student could be met in a timely manner which could prevent undesired outcomes. Student C is performing in the middle range of the grading scale for MAT0020. He possesses significantly less mathematical dom ain knowledge than student A and student B. Student C displays a positive attitude toward ma thematics and believes he is efficacious in the area of mathematics. To date, his positive attitude and efficacy beliefs have not greatly enhanced his mathematical domain knowledge. Student Cs upbeat personality and can do beliefs have likely prevented his lack of domain knowledge from having a negative im pact on his attitudes and beliefs. This is good, but not enough to enhance his performance in MAT0020. Student D is performing near the low end of the grading scale. He possesses minimal mathematical domain knowledge and displays low selfefficacy beliefs and a negative attitude toward mathematics. When speaking with him, it is clear that the reciprocal relationship between domain knowledge and atti tude/efficacy beliefs is working against student D. That is, his low efficacy beliefs and attitude actually se rve to limit the amount of mathematical domain knowledge he possesses and his lack of domain knowledge perpetuates his negative attitudes and beliefs about mathematics. In particular, student D has a good selfconcept, but fails to relate his troubles with mathematics to a specific context. Attitudes & Efficacy Attributions The relationship between selfefficacy beli efs/attitude and mathematical achievement attributions is clear. The potential positive effects of one on the other can be seen in any PAGE 27 27 mathematics classroom. For example, perf orming well on an assessment because of implementation of effective study habits leads to improved selfefficacy beliefs immediately and perhaps to an improved attitude toward mathem atics ultimately. Conve rsely, positive attitudes and efficacy beliefs about mathematics lead to pr oper attributions for achievement. That is, a student who displays good attit udes and beliefs is much less lik ely to attribute success with mathematics to luck or natural ability as they ar e to attribute that success to hard work or good strategies. Recall that attitudes and efficacy beliefs were me asured with the Student Survey, and that attributions were docu mented during interviews. Student D displays a negative attitude towa rd mathematics and possesses relatively low selfefficacy beliefs. His attributions for achieveme nt are less than desirable, indicating that his attitudes and beliefs about mathematics may be discouraging him from developing proper study habits and problemsolving strategi es. In addition, student Ds attributions for achievement, which included being born with mathematical ability and cramming for exams, are not allowing for improvement of his attitudes and beliefs. Af ter all, being born with whatever mathematical abilities one possesses does not allow for increases in those abilit ies. It is equally as disturbing to hear a student attribute failure on a mathematics assessment to lack of natural ability. This type of attribution seems to doom the stude nt to limited, if any, success in the area of mathematics. Students C B and A all attributed successes in mathematics to desirable causes. These included attending and paying atten tion in class, doing the assigne d homework, lots of practice, and strategic studying. In addition to these, st udent C reported that sometimes winging it, or hoping for the best, resulted in achievement gains. In the case of student A, there seems to be a positive reciprocal relationship between her attri butions and her attitudes and beliefs. For PAGE 28 28 student B, however, his desirable attributions do not seem to have had a positive influence on his attitudes and beliefs about mathematics. He attr ibutes achievement in ma thematics to hard work, etcetera, but does not believe th at he is good at doing mathematics even though he puts in the work necessary. Student C possesses strong e fficacy beliefs. Though his performance in MAT0020 is just average, he believes it would be mu ch better if he so desired. In the interview, he indicated that he puts forth the necessary effort to pass the course, but no more effort than that. Ironically, it is probably this same selfassuredness which prompted student C to attribute success to winging it. This is an example of a positive influence having a negative effect. When this happens, there are more than likely overridin g factors involved, such as student Cs nearly invincible selfconcept, which di sallows him to entertain the thought of failure. In general, the students experienced a more dynamic relationship between attitudes/efficacy and attributions than between knowledge and attitudes/efficacy. In ei ther case, attitudes and efficacy beliefs had the strongest influence over their counterparts. PAGE 29 29 CHAPTER 5 CONCLUSIONS Because some degree of mathematical domai n knowledge is necessary to perform any mathematical task, a rank ordering of domain knowledge, attitudes & belie fs, and attributions would place domain knowledge at the top. Researc h on expertise indicates that domainspecific knowledge is the most important single com ponent in effective le arning, outstrippingother components (Ericcson, 1996), (2004, p. 4). Attitude s & beliefs and attributions could be ranked in either order, depending upon the context. Interestingly, this ranking scheme limits the interplay between domain knowledge and othe r components because domain knowledge must exist, to a certain degree, a priori. There were three primary conclusions reached as a result of this study. First, there is a reciprocal relationship between mathematical do main knowledge and mathematical selfefficacy. By understanding this relationship, instructors can provide instruction in a way that increases both efficacy and knowledge. Since efficacy beli efs are specific (e.g. I can solve quadratic equations easily), students can build efficacy beli efs slowly, as specific areas of mathematics are addressed. Conversely, mathematical selfefficacy beliefs can increase opportunities for students to gain mathematical domain knowledge. To use the earlier example, a student who believes he or she is efficacious in the area of solving quadratic equations woul d be more likely to excel in the area of applying solutions of quadratic equations to realworld problems. Students generally have an aversion to word problems, but efficacious beliefs with respect to the skills necessary to solve word problems can overcome this aversi on. Repeated successes with solving word problems (i.e. displaying the knowledge), in this case applied quadratic equations, can lead to increased efficacy beliefs, thereby completing th e reciprocal loop between efficacy beliefs and knowledge. Instructors can use their understanding of the reci procal relationship between PAGE 30 30 efficacy beliefs and domain knowledge. However, instructors must also praise, support, and acknowledge student learning through a variety of feedback methods. Second, the relationships among attitudes toward mathematics, mathematical selfefficacy beliefs, and attributions for achievement in math ematics are certainly more evident, and perhaps more important than the relationship between knowledge and efficacy beliefs. By more important, the researcher means that the impact on learning mathematics is greater. That is, attitudes toward mathematics, efficacy beliefs, and attributions for achievement can affect learning via their interaction with each othe r, whereas domain knowledge and efficacy beliefs primarily affect learning directly. It is reasonable, then, to co nclude that the dynamics of the relationships among attitudes toward mathematic s, efficacy beliefs, and attributions for achievement are more important to increasi ng mathematical learning than is the dynamic between domain knowledge and efficacy beliefs Interaction with students and engaging students in the learning process is essential to the meaningful learning of mathematics. Many students display a negative attit ude toward mathematics, which is usually easy to detect. This makes for an uphill battle in the mathematics cla ssroom. When students make comments such as Im just no good at math, it exhibits very clearly that the student harbor s a negative attitude toward mathematics and that the student attributes his or her failures in mathematics to external factors. These negative attitude s and these external at tributions are undesirable as they lead to low selfefficacy beliefs among students. The clai m that these relationships are highly evident can be justified in almost any mathematics classroom on exam day. Invariably, there will be some pretest and posttest commentary by stude nts which displays attitude, attribution, and efficacy beliefs. This commentary can serve as the basis for classroom discussion and an opportunity to explore students be liefs, learning needs, and/or misperceptions. Clearly, simply PAGE 31 31 overhearing negative comments on ex am day and then mentioning it to the class will not suffice to accomplish this goal. Rather, some class time ma y be well spent in exploration of the origins of negative feelings about mathematics as well as exploration of reasons for poor performance on mathematics exams. It is difficult to predict th e likely outcomes of such exploration. Changes in instruction, using activities that lead to success repeatedly and over time, may improve attitudes toward mathematics and promote internal attributi on. In either case, the importance of efficacy beliefs, attitudes, and attributi ons is made clear in the sense that meaningful learning requires both responsibility for achievement leve l and acknowledgement of beliefs. Third, attitudes toward mathematics and ma thematical selfefficacy beliefs play an indispensable role in the learning of mathematic s, even when the motivation for a student to engage in mathematical learning is completely external. The participants in this study were freshman community college students enrolled in a remedial algebra course (MAT0020), which usually indicates that mathematics is not a fa vored subject of study for a student. General education requirements mandate that students comp lete two collegelevel mathematics courses, which cover material significantly more involved than that of a re medial college algebra course. This requirement represents an external motivation for students to enroll in mathematics courses. Having solely external motivation to learn mathematics may not be the most desirable situation, but it does not preclude the meaningful learni ng of mathematics. Indeed, students who are externally motivated, by requirements, to study mathematics may become efficacious in the area of mathematics and come to hold positive attitudes toward mathematics. Other factors, such as mode of instruction and feedback also may have an effect on attitudes toward mathematics. This is the challenge for, and hopefully the goal of mathematics instructors. Attributions which allow for positive change in efficacy beliefs and attitudes, namely internal attributions, minimize PAGE 32 32 external reasons for students to perform poorly in mathematics. In this way, students are prompted to take responsibility for their st udies, and can enjoy successes in mathematics by giving themselves credit for those successes. Th is behavior enhances se lfefficacy beliefs, and makes more sense to the student th an attributing successes to luck of the draw or natural ability. In addition to improving mathematical performance, proper attributions wh ich lead to increased efficacy beliefs and better attitude s can be brought to bear in other domains. Even the student who intends to complete exactly the required am ount of mathematics and no more can benefit from making desirable attribut ions and building selfefficacy in mathematics in ways that enhance their ability to do well in other courses of study. Implications: The conclusions reached in this study were of a ge neral nature and should be discussed here and elsewher e to explore their applicability to particular mathematics environments. To that end, let us apply the fi ndings to some specific situations, namely those surrounding students A, B, C, and D of this study. Since student A is performing exceedingly well in MAT0020, and attributes part of that success to her instructor, it would be interesti ng to track her through her future mathematics coursework. To see that the reciprocal re lationship between mathematical domain knowledge and mathematical selfefficacy beli efs can be utilized in the mathematics classroom in a way that increases both efficacy and knowledge, consider the situation of student B. Student B performs near the top of the grading scale for MAT0020, and possesses a good deal of mathematical domain knowledge. However, student B also po ssesses low selfefficacy beliefs and a negative attitude toward mathematics. His mathemati cal knowledge, which is substantial, could be utilized to foster better attitu des and efficacy beliefs about mathematics. Perhaps student B would benefit from journaling his beliefs and attit udes about mathematics. This could allow him PAGE 33 33 to give himself credit for the good performan ce he enjoys in MAT0020. For example, having student B write down the reasons for each step when solving problems might prompt him to recognize that his persistence and strategies are th e reasons he is successful with mathematics, and that he is actually fairly efficaci ous in the area of mathematics. To see that the relationship among attitudes toward mathematics, mathematical selfefficacy beliefs, and attributions for achievemen t in mathematics is more evident, and perhaps more important than the relationship between kno wledge and efficacy beliefs, consider student C. Student C, whose performance in MAT0020 is less than desirable, continues to display a positive attitude toward mathematics and exhibits high selfefficacy beliefs. His beliefs and attitudes influenced his internal attributions for achievement. This will allow student C to improve his performance, average though it is, because he realizes that success is mostly due to diligent studying and practice on his part. His att itudes, efficacy beliefs, and attributions being good could ultimately be more important for st udent C than his grade in MAT0020. Perhaps instruction could be geared to take advantag e of student Cs willing attitude, assigning him formative assessments in addition to the existing summative assessments in hopes that his mathematical domain knowledge will be increased. To see that attitudes toward mathematics a nd mathematical selfefficacy beliefs play an indispensable role in the learning of mathematic s, even when the motivation for a student to engage in mathematical learning is completely exte rnal, consider that all of the participants in this study were forced to enroll in a remedial algebra course. Despite this, most of them are learning mathematics, and doing so in a way that minimizes the fact that their motivations are almost certainly external. Of the four partic ipants, only student D, who is performing rather poorly in MAT0020, indicated that the course was simply a means to the end of satisfying PAGE 34 34 general education requirements. In his case, it is hoped that the negative effects of low efficacy beliefs, poor attitude toward mathematics, and limited domain knowledge can be reversed through careful instruction and al ternative assessment. Suggestions for future study or for advancement of this study include working with a larger population, follow ing these participants through their college mathematics courses to provide longevity, and perhaps studying the relationship that exists direct ly between mathematical domain knowledge and attributions for achievement. PAGE 35 35 APPENDIX A INTERVIEW QUESTIONS Interview questions desi gned to reveal attribu tions for successes and failures in mathematic s among remedial colle ge algebra students: 1. If I have performed well on a math ematics exam, it is because 2. Lack of ability in the area of mathematics can cause 3. Trying hard to solve mathematics problems will result in 4. Factors which are out of my control wher e mathematics is concerned are 5. Mathematicians most likely have success with mathematics because PAGE 36 36 APPENDIX B STUDENT SURVEY 1 Mathematics is an important field of study. ( )strongly agree ( )agree ( )disagree ( )strongly disagree 2 I enjoy studying mathematics. ( )strongly agree ( )agree ( )disagree ( )strongly disagree 3 A calculator is necessary when doing mathematics. ( )strongly agree ( )agree ( )disagree ( )strongly disagree 4 Many mathematical concepts are too difficult to understand. ( )strongly agree ( )agree ( )disagree ( )strongly disagree 5 Mathematics is not very useful in the real world. ( )strongly agree ( )agree ( )disagree ( )strongly disagree 6 I often do not care whether I get ma th problems correct or incorrect. ( )strongly agree ( )agree ( )disagree ( )strongly disagree 7 Mathematics is more something I have to do than something I choose to do. ( )strongly agree ( )agree ( )disagree ( )strongly disagree 8 Mathematics is best left to mathematicians. ( )strongly agree ( )agree ( )disagree ( )strongly disagree 9 I often think of other things while attempting to solve math problems. ( )strongly agree ( )agree ( )disagree ( )strongly disagree 10 Knowing why mathematics works is as important as knowing how it works. ( )strongly agree ( )agree ( )disagree ( )strongly disagree 11 Mathematics is a powerful tool. ( )strongly agree ( )agree ( )disagree ( )strongly disagree 12 Performing mathematical computations makes me anxious. ( )strongly agree ( )agree ( )disagree ( )strongly disagree 13 I experience great difficulty in learning mathematics. ( )strongly agree ( )agree ( )disagree ( )strongly disagree 14 Solving math problems is easy for me. ( )strongly agree ( )agree ( )disagree ( )strongly disagree 15 I believe I will never be good at doing mathematics. ( )strongly agree ( )agree ( )disagree ( )strongly disagree PAGE 37 37 LIST OF REFERENCES Ball, D. L., Hill, H. C., & Rowan, B. (2004). Effects of teachers mathematical knowledge for teaching on student achievement. Presented at the 2004 annual meeting of the American Educational Research As sociation, San Diego, CA. Bandura, A., Barbaranelli, C., Caprara, Gian V., & Pastorelli, C., (1996). Multifaceted impact of selfefficacy beliefs on academic functioning. Child Development 67 12061222. Bandura, A., & Locke, E. A., (2003). Negativ e selfefficacy and goal effects revisited. Journal of Applied Psychology 88 # 1, 8799. Bandy, I. G. (1985). 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Contemporary Educational Psychology 22, 363377. Mau, S. (1991). Belifs of collegelevelremedial mathematics students. Paper presented at the annual AERA meeting in Chicago, April 1991; 25. Pajares, F. & Miller, D., (1994). Role of selfefficacy and selfc oncept beliefs in mathematical problem solving: a path analysis. Selden, A.& J., (1996). Of what doe s mathematical knowledge consist? MAA Online Sorensen, C. T. & Hoyt, J. E. (1999). Promoting academic standards?: The link between remedial education in college and st udent preparation in high school. Department of Institutional Research and Management Studies 12,5,7. Zimmerman, B.J., & Paulsen, A.S. (1995). Self monitoring during colle giate studying: an invaluable tool for academic selfregulation. New diections in college teaching and learning: Understanding selfregulated learning, 63 1327. Zimmerman, B.J., & Kitsantas, A. (1997). Develo pmental phases in self regulation: Shifting from process to outcome goals. Journal of Educational Psychology 89 2936. Zimmerman, B.J. (1998). Developi ng SelfFulfilling Cycles of Academic Regulation: An Analysis of Exemplary Instru ctional Models. In D.H. Schunk & B.J. Zimmerman (eds.), Selfregulated learning: from te aching to selfreflective practice (pp.119). New York: Guilford Press. PAGE 39 39 BIOGRAPHICAL SKETCH Scott Murphy was born in Gainesville, Florida, in the late sixties. As a native son of the area, he realized that it does not get much better than this. Ha ving never entertained the idea of going elsewhere to college, Scott obtained a Bach elor of Science in mathematics in the early nineties and a Master of Arts in Education, speci alizing in mathematics, in August 2007. Scott is a mathematics instructor at Santa Fe Community College in Gainesville and has nearly ten years of service to SFCC. While Scotts real passi on in life is his family, which includes his wife Jessica and tenyearold son Jordyn, mathematics r uns a close second. Scott intends to remain in Gainesville, where his extended family also resi des and where Gator sports events are held. Scott is dedicated to helping students who have trouble with ma thematics overcome those issues and also enjoys teaching those who are motivated mathematicians. 