Success Factors Of Calculus

SUCCESS FACTORS OF CALCULUS: HISPANICS VERSUS WHITE NON-HISPANICS by SHIRLEY ALLENE DAVIS, B.S.

A THESIS IN MATHEMATICS Submitted to the Graduate Faculty of Texas Tech University In Partial Fulfillment of the Requirements for the Degree of MASTER OF SCIENCE

Approved

Accepted

Dean of the Graduate School December, 2001

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^ •^

ACKNOWLEDGEMENTS

'f-C.

I would like to thank Dr. Jat.es Epperson for his tinte and effort . init.attng th.s

project. I also appreciate the patience and assistance I received from Dr. Benjamin Duran and Dr. Carl Seaquist. Above all, I want to thank the chair of my committee, Dr. Gary Harris who at an inconvenient time inadamantly took the responsibility and nurture of my thesis. If it was not for him, I would still be working on this thesis today. Thank you from the bottom of my heart. I also extend thanks to all of my friends, family, and, especially, classmates who have supported me through this graduate school experience. I, especially, appreciate the patience, understanding, and support from my closest friends, Leah, Traci, Richard, Ruby, and Amanda during the numerous times when I was certain there was no end in sight. Man> thanks also goes to Texas Tech's Ultimate Frisbee team for allowing me to use you all as an emotional punching bag. I also would like to thank Jim for re-introducing me to Christ. Somewhere along the way, I became spiritually lost wandering aimlessly with despair, and although it occurred at the tail end of my graduate school experience, I have found Christ again which has allowed me to find myself Thanks Jim. Finally, I thank God. I know through him, I have the ability to do anything.

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TABLE OF CONTEXTS

ACKNOWLEDGEMENTS

ii

ABSTRACT

v

CHAPTER I. INTRODUCTION

1

II. LITERARY REVIEW

3

IIL METHODOLOGY

11

r / . RESULTS

13

4.1

4.2

Interviews from Group A

13^

4.LI

Student A

13

4.1.2

Father A

22

4.1.3

Teacher A

26

Interviews from Group B

31

4.2.1

Student B

31

4.2.2

Father B

42

4.2.3

Teacher B

4"

V. CONCLUSIONS 5.1

3.2

54

Analysis of Interviews

54

5.1.1

Student A

54

5.1.2

Student B

56

Conclusion

58

111

REFERENCES

60

APPENDIX

62

A. QUESTIONS FOR STUDENTS

63

B. QUESTIONS FOR PARENTS

66

C. QUESTIONS FOR TEACHERS

68

IV

ABSTRACT In this study we are interested in those social and academic factors that might contribute to success in college level calculus for Hispanic students. In particular how do these factors compare with success factors for white non-Hispanic students. Via in-depth case studies we attempt to isolate and compare such factors for two students, one Hispanic and one white non-Hispanic, who successfully completed a one year calculus sequence at a large state comprehensive university.

CHAPTER I INTRODUCTION

Students who perform successfiiUy in calculus tend to have strong study habits, adequate math preparation prior to college matriculation, and positive influences such as mentors and/or supportive parents. Statistically, among these students, very few Hispanics appear with the vast majority being white non-Hispanic. This raises an obvious question: To what extent do the factors that contribute to success in calculus for Hispanic students differ from those that contribute to the success in calculus for white non-Hispanics? Prior research has explored the challenges and resources related to Hispanic culture, language, and economic status in attempts to examine the structural phenomena that contribute to Hispanic marginalization within the institutions and its potential effect on academic performance. The research identified faculty contact and mentoring, social and academic integration, parental and sibling influence, and, of course, study habits as the main factors that affect the achievement in calculus for Hispanic students, just as for non-Hispanic students. Among these selective factors, one of the more reoccurring and popular factors is faculty involvement. It is felt that faculty role modeling positively affects students' attainment and retention in college. Although high school faculty is seen as an influential factor in preparing students for college matriculation and math/science majoring, college faculty interaction is viewed as a primary factor for college retention. Additionally the research strongly suggests that social, as well as academic, integration into the university are necessary for students' persistence in college through

degree completion. In this context social integration involves faculty interaction, peer relations, and involvement in extracurricular activities. Again, are these factors of equal importance for Hispanic students as for white non-Hispanic students? In this study we are interested in those social and academic factors that might contribute to success in college level calculus for Hispanic students. In particular how do these factors compare with success factors for white non-Hispanic students? Via in-depth case studies we attempt to isolate and compare such factors for two students, one Hispanic and one white non-Hispanic, who successfully completed a one-year calculus sequence at a large state comprehensive university.

CHAPTER II LITERARY REVIEW

We summarize those studies from the literature which we feel ha\e been in our study. In a seminal paper, Treisman [1992] conducted a comparative smdy between 20 Chinese and 20 Blacks. He was interested in these two ethnic groups because the Chinese excelled academically in mathematics while the Blacks perpetually struggled in this area. He discovered that the amount of time the Chinese studied was nowhere near important as how they studied. The Chinese devoted just a couple more hours a week studying more than the Blacks but their most successful studying tool was that they spent some of that time stud\ ing in groups. This discovery explained the minorities' lack of success in calculus far better than the other preconceived beliefs such as inadequate preparation, motivation gap, lack of family support or understanding of higher education, and income. After realizing the effects of these studying skills, Treisman's efforts turned towards constructing an anti-remedial program for students providing not only tutoring, time management, and study skills but. also, a challenging, yet emotionally supportive academic environment. In his attempts to assist and help these students, he had to convince them that success in college required peer group working and studying so as to create for them a community based on shared intellectual interests and common professional aims. In their 1997 study, Terenzini and Pascarella [1997] studied freshman attrition for a random sample of 500 students in the College of Arts and Sciences at Syracuse University. In particular, they w ere interested in patterns of social and academic integration that contributed to students w ithdrawing from or remaining in their classes. Students remaining

in their classes were called "stayers." They found that stayers reported more informal contact with faculty members outside the classroom and their nonacademic li\ es w ere more demanding than that of leavers. Also stayers were more involved in the social system of the university. Their results tended to support Tinto's [1975] view that social and academic integration are approximately of equal importance in students' decisions to remain or withdraw. Mayo, Murguia, and Padilla [1995] examined the formal and informal social integration among 400 students from Hispanic, Black, and Native American groups and 400 White students. They found that relationships with faculty and ethnically matched role models are successful in contributing to academic achievement. They claim that formal social integration has a much greater significance on the effects of academic performance than informal. However, this is not necessarily so with Hispanics since their social life impedes with their studies negatively affecting their academics. Nevertheless, Hispanics do consider student organizations as an important necessity for their existence and success in college so Mayo, Murguia, and Padilla suggests that the development of well-organized ethnic-specific student groups would prove beneficial to the minority groups on campus. Gainen [1995] examined four factors that grant some students success and not others who possess similar ability and interest. She claims that the most limiting factor in a student's academic success is insufficient or inadequate preparation in mathematics. Her claim is supported when considering that of the HS graduates, only approximately one-fourth have the sufficient mathematical and science preparation to proceed into SME majors in college and among this one-fourth, only 11% of women compared to 14% of men have the appropriate background in mathematics upon HS graduation.

Students affiliated w ith peer culture that fa\'or academic pursuits are much more likely to succeed in challenging majors, however, in contrast, when peers are uninterested in academics, students find it difficuU to maintain momentum in the face of academic challenges of SME majors. Students often find themselves uncomfortable and alienated and yet are unable to determine the cause. Gainen attributes these feelings of the students to the professor's low expectations of minorities and their favoritism to the non-minorities. In addition to these o\erbearing obstacles, students also complained of unstimulating introductory SME courses, bad teaching, poor advising, and unhelpful faculty. Again. Gainen has shown that faculty plays an important role in the attraction and retainment of students in SME majors, and proclaims that success of these students depends at least in part on the willingness of faculty to resist cultural norms of competition. In the secondary analysis study of Hedges and Nowell [1995]. six large data sets were collected between 1960-1992. They found that although average sex differences are small, females perform better in reading and writing whereas the males perform better in science and math, and concluded, that the differences in ability arise because of differences in experiences and socialization and not necessarily due to differences in sexes. Rendon and Triana [1989] were interested in factors that impede Hispanic students progress in science and mathematics at the high school level. Among the factors they isolated are the following: 1. Low socio-economic status 2. Poor academic performance 3. Nonparticipation in HS academic programs

4. Academic related problems 5. Enrollment in remedial courses 6. Discriminatory practices 7. Mathematics fear, anxiety, and avoidance 8. Curricular deficiencies 9. Teacher shortages 10. Low transfer from 2-year colleges to 4-year colleges. Attinasi [1989] analyzed interviews with 18 Hispanic students from a single entering class of a large public southwestern university 8 to 11 months following the end of their freshman year. His intent was to identify, from the students' point-of-view, factors that affected their decisions to persist or not to persist in their university programs. He grouped these factors into two categories: factors involved with getting ready to go to college and factors involved with getting in. Getting ready involved initial expectation, fraternal influence, mentor influence, and direct and indirect simulation. Getting in involved physical, social, and academic concerns. Being able to deal with the latter three was shown as a positive relation to persistence. Bennett and Okinaka's [1989] study was a revision of a 1982 quesfionnaire administered to an ethnically mixed sample in 1986 so as to compare characteristics of persisters and nonpersisters and their attitudes and perceptions. Persisters pertain to students who remained in college upon degree completion whereas nonpersisters are those who withdrew before degree completion. They found that persisters, especially the Hispanics, experienced more interracial contact prior to college and had more positive attitudes tow ards their college instructors than nonpersisters, and that involvement in ethnic organizations.

accumulation of college friends, adjustment to college life, and openness to equity policies were shown as predictors of college satisfaction among the persisters. Of all these factors, students that felt the most positive about their college instructors were the most satisfied with the university's social environment, administration, and classes. In the case study paper, Galindo and Escamilla [1995] examined two Chicanos, one male and one female, through autobiographies, biographical sketches, and interviews in order to analyze their educational experiences and the sociocultural factors that influenced those experiences. The female, Linda contributes her good grades to her early foundation in learning established in her home and at church and reports that her family highly supported her efforts in education. Although she had the support of her family and church, she struggled with her ethnic identity in HS and college. Due to her different cultural background and social antagonism, she felt alienated in school and eventually in church. She contributes this alienation to the lack of role models who could have provided an example of how other minorities had negotiated the system. On the other hand, the male, Richard, was ridiculed in HS for being culturally different making him feel alienated and socially outcasted. He learned to overcome this in college and succeeded academically and socially. Although his father's occupation was not conducive to educational success, which alienated him from students and teachers, his parents supported his education by allowing time for school and homework. They also had all the children serve as tutors to one another including Richard. Besides the support of his family, Richard also attributed his educational success to the impact that his pre-college teachers had on his and his sibling's schooling. He mentions that these teachers were not just exceptional but were extraordinary since they were constantly going out of their way to help educate Richard and his family. 7

A sample of 100 Mexican-American undergraduate students that were divided into two divisions: lower (age 18-21) and upper (age 20-30) were examined by Lopez [1995] in this statistical paper. He is particularly concerned with the age and gender patterns in the perceptions of challenges and resources that stem from the family, peer relationships, and the university. Upper division males reported more challengesfi-omdomestic responsibilities, perceived more racial discrimination, and received more academic encouragement from siblings and from male friends than lower division students. There were no age-related differences found between the two divisions in their perceptions of challenges and resources. Hilton and Lee [1988] studied the data obtained from two studies: one which surveyed 23,500 high school seniors with follow up surveys every 2 years for 6 years, the second a similar study begirming with high school sophomores. They found that most students who choose to major in mathematics, science, or engineering (MSE) do so prior to entering college. And by the first year of college Hispanic students are less likely to be interested in majoring in MSE than either blacks or whites, while showing more interest in MSE as high school sophomores. Jay and D'Augelh's [1991] study focused on assessing patterns of social support of AA and White freshmen attending a predominantly White university and their adjustment to university life. Their participants included 81 White and 84 African American freshmen attending a 4-year institution. They found that predictors of adjustment were based on first year college GPA, psychological well-being, and physical well-being. Sax [1992] samples 15,050 students from 392 different 4-year universities. She was interested in mathematical self-concept development and its relation to institutional selectivity. In her discoveries, men exhibited greater initial mathematical self-confidence 8

than women. However, women who persist in the mathematically oriented disciplines and participate in tutoring other students tend to gain confidence in their mathematical abilities. On the other hand, interaction with faculty and attendance of a public college promotes a declinafion in mathematical self-confidence and intellectual self-esteem, respectively. They also determined that predictors of mathematical self-confidence are more evident in the precollege experiences, and therefore, are stronger influences on students' level of mathematical confidence. In Carter's meta-analysis [1987], Astin showed that minority students prefer certain academic disciplines, but the biological sciences, engineering, mathematics, and physical sciences are not included in their preferences of studies. An exception to this is Black students who are graduating with degrees in mathematics but from black institutions (Trent, 1984). Besides feeling excluded, discouraged, criticized, and disregarded, women also feel like they are betraying their family, friends, and relatives (Sandler & Hall, 1982). These feelings of alienation, personal isolation, and rejection by the instructors in the classroom are more apparent among the Black women. However, it is with these minority women that once they have established status and success, they develop a strong sense of self-confidence, perform effectively and sufficiently in their respective careers, and, most importantly, encourage all other women to consider studying the fields (Epstien, 1973; Scott, 1977; Scott & Horhn, 1975). When determining a career choice, it seems that prestige, nature of the work, and personal characteristics affect occupational preferences (Anderson & others, 1984; Axelson, 1974; Kanter, 1975; Knudson, 1982). Among the women who choose nontraditional careers, it was discovered that their success is attributed to having a high

achievement orientation, disregard of sex role stereotyping, a development of better stud) skills, and, mostly, encouragement in the exploration of nontradidonal interests. Anderson [1990] summarized the efforts of the National Research Council's Mathematical Sciences Education Board (MSEB) in eliminating the underachie\ement and underrepresentation of minorities in mathematics. She found that the le\'el of parental education is directly related to student performance, and poverty is directly related to academic achievement. Boli, Allen, and Payne [1983] looked at students in introductory mathematics and chemistry courses at Stanford University in the fall of 1997. They were interested in gender related factors that might contribute to success or failure in such course. They found no direct gender factors related to course performance; however, female students did appear to perform somewhat better if they had female mathematics instructors. Erekson [1989] studied 544 undergraduate students from a random sample of 100 students at Miami University to determine factors related to students grade point average (GPA). He found that library effort and course selection did not significantly affect GPA, but male students had significantly lower average GPA than did female students.

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CHAPTER III METHODOLOGY

In the fall semester of 1999 there were 443 students enrolled in the first semester calculus course. Of these students 88 received a grade of "B" or higher and maintained a grade point average of at least 2.5. The 88 students were comprised of the following (Table 3.1). Table 3.1

Ethnicity and Gender

Hispanic

White Non-Hispanic

Total

Males

6

61

67

Females

5

16

21

Total

11

77

88

To avoid questions regarding possible gender dependence of success factors in our comparison, we chose two male students, one Hispanic and one white non-Hispanic. We refer to the Hispanic student as Student A and to the non-Hispanic student as Student B. Students A and B were selected based on their similar academic credentials and prior mathematics preparation, with emphasis on major, calculus I grade, SAT/ACT scores, and HS size and rank. Both students had received the grade "B" in calculus I. Student A*s major was mechanical engineering and Student B's major was computer engineering. Student A's SAT score of 600 was comparable to Student's B ACT score of 28. Both students graduated from their high school highly ranked. Student A ranked 2 out of 527 placing him in the top 1% of his graduating high school class. Student B ranked 7 out of 457 placing him in the top 2% of his graduating high school class. 11

Once the two students were chosen, personal and phone interview s were conducted with the students, their fathers and their high school mathematics teachers. All interview s w ere audio recorded with the full knowledge and permission of each participant. The audio recordings were later transcribed for future analysis. The students were interviewed in person with both researchers participating in the questioning of each student. These interviews lasted approximately one hour each. The students were first asked for information regarding their HS experience and then for information about their college experience. Information such as time spent studying, methods of studying, and external and internal influences towards education was obtained. The student interview questions are included in Appendix A. The interviews with the fathers were conducted over the phone by one researcher only. They also lasted approximately one hour each and were audio recorded for later transcription. The fathers were asked to discuss their prior knowledge and/or observation of their son's studying habits and behaviors while in HS, their contribution to their son's education, and their attitude towards math as well as education. Both fathers elaborated on most of the questions and contributed additional comments outside the realm of our originally prepared questions. The interview questions for the fathers are included as Appendix B. Interviews with the teachers also were audio-recorded phone interview s conducted solely one researcher and lasting approximately one hour each. The information obtained from the teachers was almost identical to that from the fathers. Questions focused on the teachers' prior knowledge and observation of the their student's studying habits and behaviors while in HS, their contribution to the student's education, what factors they think lead to the success of that particular student and factors they believe contribute to the success of students in general. The teacher interview questions are included in Appendix C. 12

CHAPTER IV RESULTS

4.1 Interviews from Group A 4.1.1 Results from Interview: Student A

Student A attended a large high school in which a majorit\ of the students w ere Hispanic; he graduated ranked 2 out of his class of 527. He took Algebra I in the 8* grade followed by Algebra II. geometr}. Trigonometry, and AP (Honors) Calculus in high school. WTien asked to describe his high school preparation in mathematics, he responded: I went all the w ay through calculus I in high school. Basicalh. the math preparation I had was through UIL math. So starting in the 6 grade and through high school, I w as in the UIL math and science competition. That basically prepared me through high school. My calculus professor in high school was \er\ good preparing us for college and calculus courses. When asked if he was alw ays good at mathematics growing up, he responded with "Math has always been my strong point. I like that subject." One objective of the interviews with the students was to assess factors dealing with their motivations for stud>ing mathematics and factors contributing to their confidence levels with regard to mathematics. We consider two types of such factors, internal and external. Internal factors involve the students" personality types, learning styles, personal goals, etc. External factors relate to the influence of mentors, family, peer groups, etc. The following excerpts from the inter\iew with Student A relate primarih with internal factors. Personalit> traits are presented first followed by his learning and studying habits. '>

In regards to Student A's personality, the following quotes are presented. Question: If I asked you how would you describe your moti\'ation level in studying mathematics in high school, would you describe as "high," "moderate," or "below average"? Why? Response: Moderate in the amount that they wanted me to know but abo\ e average in the amount I wanted to know. I don't think that in the high schools that I have known or that I have been to or that I have competed that they actually demand much of you. But in the competition like UIL, you should know more. You are expected to know more. In just the mathematics itself, it is intriguing study, and I like it so I delved into it further.

Question: If I were to ask you about your motivation level and said, "Were you motivated by your teachers, by your family, internally, or externally?" Response: It was just basically just me. My motivation is self-induced.

Question: Would you describe your motivation level now in studying mathematics (in college) as "high," "moderate," or "below average"? Why? Response: As a few months ago, above average.

Question: So last year, you would have considered it above average? Why? Response: I was considered about moderate. For the past couple of months or several weeks, I would say above average. I just see the compelling need to do so. It's part of my field. That's what it is - mathematics, applied mathematics.

Question: So it is motivating you now because you want to understand it for your career? Response: Right.

Question: Would you describe yourself now as being internally moti\ated or externally motivated? Response: I say a little bit of both but more internal than anything else.

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Question: So you have a strong sense of competition? Response: My dad tells me I do.

Question: Did you have any ideas about what mathematics in college w ould be like? Response: No I hadn't really thought about it. When I was in the 6th grade, I never thought about taking calculus. I just saw a book and saw all these symbols and never thought I would get there. But I just took one course after another, and it just kind of snowballed, and I was just ready for the next course. When asked what it is about mathematics that intrigues him and make him persevere, he commented, "I think I can do it. I know I can do anyone of these subjects. I don't think it is beyond me to do anyone of these math problems. Calculus III now I understand it. I understand pretty well now. It should be ok. Linear algebra, I don't know. The way it is taught, it just seems to be so abstract so theoretical. It seems to be beyond the scope of the class of what it is suppose to be like." In regards to Student A's learning style, the following quotes are presented. Question: Did you feel any kind of pressure in high school to do well in mathematics that being from friends, teachers, or from parents, brothers, or sisters? Why or why not? Response: I just basically saw everything as a little bit of a competition throughout all the way we did it the valedictorian through like the 10th rank in our class. It was all very competitive. We all tried to outdo each other so we kept each other in check.

Question: Why do you say it was UIL that prepared more so than your high school courses or than your mathematic courses? Response: Because it was a competition. It was me against somebody else. 1 like to do better than whoever else I am competing against.

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Question: There is something that satisfies you to know that on the test you beat everybody else? Response: In the begirming, I take the test or whatever as it comes. I get one back, and it looks bad. I say that is not the grade I like, and it's me against the test now. It usually does better after that.

Question: You think of it as a competition with the material rather than a competition with others? Response: Yea.

Question: So, in other words, if you understand it than you are victorious? Response: Right

Question: And that allows you to explain it others and that makes you...? Response: Even better.

Question: Would you describe the exams and homework assignments in high school as challenging? Why or why not? Response: Not really. It just seemed like.... We didn't really put all our effort into anything. I think that most of the effort or like many just about 20 or 30%. It was adequate enough to go through all the tests. Through college so far, it has been the same way.

Question: Now, at the presence, do you feel some pressure to do well in mathematics from your teachers, or instructors, or your parents? Response: Again, from me and my grades.

Question: Did your confidence come from having done well or did it come from....? Response: I think it comes from having done well m the past and expecting myself to do well in the fiiture as well as the competitive nature of the class: It's me vs. the class. 16

Question: Do you have high expectafions of yourself? Response: I think I do a little bit but it's only at the point that I see it as a competition. If I don't see it as a competition then I don't put as much effort into it. Student A's studying style revolved around an individual effort with minor group involvement both in high school and college. While in college in his calculus class. Student A studied approximately 2 hours a week, however, this time was reserved only for exams. He did not consider homework part of studying. He committed an hour to each homework assignment and an hour and a half to group studying. His studying method involved working extra problems concentrating on the conceptual problems. He would also read the sections in the book and write formulas although he didn't particulariy appreciate the textbook due to the excessive errors found and its literal construction. He preferred working/studying with students who possessed the same intellectual ability as he or better, and he did not discriminate among these intellectuals. However, he also tutored the mathematically underdeveloped students because it reinforced his own knowledge and comprehension of the material. He considered tutoring a better studying skill than individual studying. When engaged in group studying, the group, which consisted of himself and his social group friends, would study in a relaxed and casual atmosphere. The following excerpts from the interview with Student A relate primarily with external factors. The following interchange expands on the initial reference to his high school calculus teacher: Question: Who was your favorite teacher and why in mathematics? Response: I would have to say my calculus professor .. .(Teacher A below, a male Hispanic teacher who began his teaching career as a football and 17

basketball coach).. .He would assign homework but never take it up. Only quizzes and tests count so it kind of forced you to do your homework. So that's basically hke college.

Question: If you had to name a particular teacher/person who influenced you to pursue mathematics at the college level, who would it be? Why? Response: .. .(Teacher A)... He just brings about the subject in the best possible way. He likes to show you more than what you need at the time. He encourages you to think. He also is one of those where everyone should do better than the past generation. He reinforced family ideas.

Question: Somehow he communicated to you, that he had this hope that you would go on do something more? Is that what you felt from him? Response: Right. I felt the same thing from him as my parents.

Question: What was it about him you liked other than that method? Response: You could ask him any question, and he would have the answer for you. It wouldn't matter what section, why, or the reasoning, proofs or whatever. He had the answer for everything. He explained it quite clearly.

Question: So .. .(Teacher A)... inspired you to continue to study math? Response: He did in a way he brought mathematics about. So for the most part, he made it a little bit more fim than any the other teachers have. He could answer questions relevant to the things I talked about. He furthered the knowledge.

Question: How do you think your high school teacher became so good at math? What theories did you have? Response: I never really thought about it. I just thought it was pretty cool that he knew so much math. I could ask him anything, and he would know the answer. My other teachers couldn't do that. Some of them if you took them out of the book, they got lost a httle bit but he could answer any question. I thought that was very interesting.

18

Question: Is your motivation level in college influenced by your instructors? Let's talk about last year. Your motivation last year. Why or why not? Response: I can't think of a professor in college who has motivated me anymore than I already am. It just seems to be icons just like giving out facts and figures.

Question: How is that different from high school? Response: In high school, they give practical applications and show you why you might need it. It just seems like all the courses here are theoretical, you might need this, you might not. We'll teach you this just because. It just seems like that. Student A was curious about his HS teacher's decision to study mathematics, but he chose not to ask his him since he felt it inappropriate. However, he was not at all curious about his college professors' decision to major in mathematics. It did not seem to matter to him nor make a difference. To him, they chose what they wanted as he has chosen what he wanted for his own reasons. The following interchange expands on the initial reference to his family: Question: What are your parent's wishes for your future career? Response: They always said whatever makes me happy so I just go with that. I always felt like I need to something good for me so I picked engineering.

Question: So they kind of told you that you can do what makes you happy and you feel like pursuing an engineer degree makes you happy right now? Response: Right.

Question: What role, if any, did your family play in supporting your decision to come to college? Response: They support my decision with everything they have.

19

Question: Did they encourage you to go on to college or say do whatever you want to do? Response: No. They definitely encouraged college.

Question: In what way did they encourage it? Response: It's basically like a culture type idea. It's like further culture in our family. I think it is good for all of us.

Question: Could you explain that to me about "culture idea"? Response: Like each generation should do better than the previous one so since the majority of my family members didn't go college. Some graduated, some have degrees; I should also have a degree and do better.

Question: Growing up did they ever say since maybe you were in 1^^ grade that you were going to college or did they say...? Response: They never did put that type of pressure. My dad said I put too much pressure on myself I think it is just me. I am naturally competitive.

Question: This confidence that you have academically, do you feel like it has always been or do you feel like it was something that was built up over the years from you like your parents rewarding, always telling you good job, and that you were always working hard and being competitive? Do you feel like it was always there, or it just got stronger through your course of life and academics? Response: I think it was always there. My parents never rewarded me for doing good. If you do your best then that is what you are suppose to do. If you earn a C, and it was hard fought then it is all right but if you could have gotten an A then you slacked off then you know what the problem is.

Question: What kinds of responsibilities did you have in high school? Did you have to work or did you have to take care of your sister? Response: No. I was just like a mentor to my sister and that's about it. I helped my other friends in their subjects. Other than that, I didn't have much more to do 20

The following interchange expands on the initial reference to his high school and college peers. Question: If I asked you did you study with others in high school, what would you say? Response: I basically studied by myself That's basically how it w orked.

Question: Describe your social group in high school like who did you hang around with? Response: I don't know if I had a certain chck. I pretty much just hung around with everyone. We had our own group but it was quite di\ erse.

Question: In what way was it diverse? Response: We had people that were more into like the liberal arts or English and type of stuff to math or just regular people. We just hung out with everyone.

Question: What was the common thread? Response: Just personality I suppose.

Question: You all kind of like to do the same things? Response: Yea, we all kind of did the same thing beside from studying.

Question: If you had to pinpoint a number of people in your social group say would it be less than 10 or less than 5? Response: It would be close to 20. We hung out quite frequently.

Question: Why were they from your circle of friends and not from outside your circle of friends? Response: My circle of friends usually included the brightest to the outspoken. That's how it worked in high school. We were all very' close. 21

Question: How important were your friends' opinions when you were choosing a place to go like a university ... when you were tr> ing to decide what university to attend, how important were your friends opinion about that? Response: I basically wanted to go to a college they weren t going to. I didn't want that social distraction. I didn't want to go to a place where I had all the same friends again. I wanted to interact with different people. I want to branch out a little bit. I didn't want the same competition from them. I wanted to see different people. I keep close fies with them but I like me own space too.

Question: Would you care to explain a little bit what you mean b\ that? Response: You always have the socialistic type of friends who go out and party or do something else other than studying.

4.1.2 Results from Student A's Father - Father A Father A is a self-motivated and self-driven educated man with a Master's degree in the social sciences. Bom into an uneducated and working class family that strongly values education, he chose "the military shortly after high school graduation (since) Vietnam was at its height" and he thought it better to serve first then pursue an education. He has a high regard for education and the hard sciences and encourages the pursuance of the mathematical disciplines although he is not very confident in his ability to perform and comprehend mathematics. After spending a lifetime working and raising a family, he still entertains the idea of returning to college for a PhD, if not in the social sciences then, alternatively, in law. Question: Are there other ways that have not been mentioned that you think t that your high school could better prepare students for college? Response: High school itself Oh yea, high school could a lot better job. A lot of people go out there. You shouldn't have asked me that question. In our high schools, a lot of educators and even the counselors, I am sorrv to say, I'm not sorry that's why I am saying it. Seriously, they go up there and they are 22

interested in doing what they do and that's it. They don't publicize enough. They don't tell the kids look if you are going to do this then you should do. They more or less let the kid and his parents, if he had the help at home, figure it out mosfly. They give you the degree plan for four \ears, and if you ha\ e questions, come ask. They don't. Here ask if you want but that's it. I think I would take a different approach to that. That's my opinion. They could do a lot better. I think a lot of kids are not prepared for college because somebody did not tell them this is what you need to do because we ha\e a jr. college here. I know some kids come back from universities because they w eren't prepared. That's to me, although the student has a responsibility, the school bears a lot of the responsibility. That is what they are there for. Yea, they could do a lot better.

Question: Would you encourage a degree/career in the mathematical disciplines compared to any other degree or career or is there not really a difference there; is it whatever the kid desires? Response: Yea but if you are going for a math major, that's fine and fantastic, but you almost have to want to teach. Either you want to be a teacher or... That's fine. There is nothing wrong with that. If that is what the person wants to do. If you don't want to be a teacher then you need to do something else. Once they tell me, I just try to put into perspective for him. That's all. You choose what you want to do. That's it. Easy!

Question: You said teachers did not encourage you? Response: No, no. Heck no! I am serious. I don't want to sound mean or anything. The teachers, I guess it is still the same—that's how I see it—. For example, when I was in high school, I didn't know that they were doing me a disservice by not encouraging me to take algebra. I thought fantastic. I don't have to go to algebra. I don't have to take algebra in order to graduate. So I didn't. Nobody would say that if you are going to go or if you are going to do this then ... nobody would say anything. We thought we were getting away with something. I went to college after, and I realized that I had to pass college algebra before I could graduate from any major university. I said well, that is where I am going to start because if I don't then I am not going to graduate. So I just started with the basic algebra. The very first one. No teacher.. .1 can honestly say they did not. I am not saying that I am upset because really it is a person's responsibility to find out. If you care then you are going to or if you have someone who cares then they are going to help you. Everybody else, sometimes, they have their own issues, their own thing to do.

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Question: You wished that you were a little bit more well informed? Response: I wish people, educators especially, do more than w hat they are paid to do. I think that would go a long ways in helping people sometimes who come from families who are not well educated or not in tuned w ith the system or the consequences. Educators sometime disappoint me. They don't share the informafion they have. I wish educators would take more interest in doing what they are supposed to do.

Question: So they valued education? Response: Oh yea, yea absolutely! I am tired of hearing people say that some people don't value education and stuff and this and that. I think that is bunk and uncool. I have never met any parent who did not value education. The problem is that people don't see that factors and things get in the way of education and sometimes people have to eat and it is more pressing the immediate. The immediate need is more pressing than going to school. That is why you have people who were migrants who leave and go off even to Lubbock and to the fields and stuff. It is not that they don't value education, it's because they got to eat. One time I had to I said the same thing. Teachers, for example, they get their undergraduate degree and then they go to work. If you ask them, I know a lot of educators, how come those.. .why haven't you got your Master's degree. Well, it's because I have bills to pay, I have this to pay. Well, that is the same thing with people who go out and work...to eat. The have priorities. The same reason why you are not getting your Master's degree immediately is the same reason they don't. They have bills to pay and they have to eat. It is not because they don't value education.

Question: You don't think that it could have been the teachers in college who turned you off? Response: Let me tell you this. This is my experience. I told Jose the same story. My experience in college was the teacher or professor or whatever he was .. .he would stand there and interact with the brightest. The questions would be directed at the brightest. Interaction was between that student and the teacher.

Question: So he discriminated against the non-brightest? Response: The non-brightest. The non-brightest were not challenged. They were not because the teacher liked 'this is what we did and how we did it 24

and.. .the quadratic equation.' That's it. If he had a question to share, he would direct at the brightest. He would not direct an\ thing at an>bod} else. I suppose. Now me. I am thinking. I am not sure. I couldn't say anything but I imagine that was because this guy doesn't know then Lll ha\ e to explain it again. Maybe I'll I have to lower how I explain this stuff We went through it pretty fast. The guys that were the brightest would of course be like this and this and this.. ..would you share your homework, how did you do this problem, and how did you do this, this, and that. Sometimes we would get in groups and get by that way.

Question: What factors contributed to your child's success in high school mathematics courses? (i.e. social group, amount of time studying, parental involvement, etc.) Response: He took the higher-level courses. The honor courses, the hardcore mathematic courses. He went through calculus. I had an influence only up to 7^^ grade. After that, he lost me. I helped him up to 7^^ maybe 8^^ but not much more than that. He's a pretty bright guy. I don't know how he got through. I helped him the early years but not in high school. Question: Would you describe your child's motivation level in studying (mathematics) as "high", "moderate", or "below average"? Why? Response: High because he is a self-starter and very competiti\ e. He likes to explore. You give him a task and the tougher it is, the more he likes it. I'm not sure. But he is self-driven.

Question: You said you didn't have to do too much; did you ever pressure Jose to do well academically? Response: In my opinion, no. The only thing I tell my kids is to do the absolute best and to seek and find.

Question: Do you reward them for a good performance? Response: No, I always told them that doing their best is what they are suppose to do. I go to work, my wife goes to work, we get paid for doing what we are suppose to do. I always tell them everybody has a job. Your job is to do well in school as well as you can. That's the most I can ask. That's it. I didn't holler and scream if they didn't bring an A home. I just tell them if that is the best you can do then that is the best you can do. If you need help, let me know, and we'll do something.

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Question: Do you feel that your child's high school adequately prepares students for college? Response: Yea but let me say this. Yes but it was mostly he. I don't want to say me but it was mosfly him. If there was anybody else, telling him anything, it was me. His teachers were there. He had very good math teachers in high school at the higher levels. But the school, if \ou decided to go the vocational way, they don't do anything or say an> thing. So it is up to you as the student to meander your way through whatever it is you want to do. He knew what he wanted to do. He did have all those courses. He was w ell prepared. I have to give him a lot of credit Jose himself

Question: Based on the inadequateness of the counseling in high school, would you say that you had a large amount of influence as far as helping Jose leam about the colleges and universities? Response: Yea, I think I may have contributed there because 1 alwa> s put the stuff in front of him. I always told him. The only concern I had in that regard was if he was not ready to go out, leave home. I didn't tell him that way. I made it plain to him that he could go to the jr. college and take his basics then precede. I made the offer if he wanted to do that before he went on. He was well aware about all the academic opportunities, and I left it up to him.

4.1.3 Results from Student A's Teacher - Teacher A Although Teacher A was a dedicated and devoted HS mathematics teacher for 33 years, teaching was unintentional career choice. His first career inclination was coaching baseball. Since a coaching job was contingent upon him teaching mathematics concurrentK. he chose the dual career, and as years passed, transferred his love from coaching to teaching. After the cessation of his coaching career, he sought UIL participation in mathematics in order to fill the void of competition. In an overview of his career, he comments, "If I had to do my life over again, I would still want to be a teacher.'' The following excerpts from the interview with Teacher A relate to his philosophy and teaching style. 26

Question: Did you encourage academic competition in your class among your students or academic cooperation? Response: I was teaching mainly honors. I didn't have to encourage it. In most cases, it was already there.

Question: Did you encourage academic cooperation or w as that alread>' there? Response: I did encourage that but see I was an UIL sponsor. Most of the kids were in UIL also. Especially because I wanted them to help each other not only for that but also for competition. But then there are some kids who don't want to do it. If they know how to do something then they don't work with nobody else. In regards to Teacher A's perspective of his own classes, he comments. "My classes were very competitive against each other.. .Everybody would compete against each other. 1 didn't make it that way. They made it." In Teacher A's interaction with his students, he never volunteered his rationalization for majoring in mathematics losing the opportunity to elaborate of his struggles and triumphs in mathematics because he assumed that they already knew everything about him since he had most of their parents and older siblings as students. Question: What was your role in Jose's performance in math? Response: I came from a poor family myself but I see these kids and try to help them as much as I could. I wouldn't trade it for any pay. I would spend hours after school. I would stay late until 5:00 in the e\ening. I would always tell them, 'If you can win here at the local level then you can win at regional and at the state.' That is something that you can put on your resume. Whenever you apply somewhere, people are going to look at you of a sudden. Not that you just one in district but you also went to the state level. To be perfectly honest, at our high school, I was the Calculator coach for fifteen years. Actually 16 but one year, I had cancer. Out of those 16, we won district 14 times in a row. In the Number Sense, we won half the time or more.

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Question: What qualities do you feel that students must possess in order to pursue a career in a mathematics-based discipline? Response: I feel like for a student to pursue this career, they must first ha\ e a love for math cause if you don't have a love for math then you are going to be bored. Next, I think you have to have a lot of patience to soh e problems. You can't give up too easily. And the next thing, you must be willing to work. If you don't have those qualities then you won't get an> where.

Question: How do you encourage/model these qualities? Response: I usually let them model themselves. I start off easy and let them work. I had only honors classes so they were fighting for grades for rank in the class so they would have to stay up. I didn't physically threaten any of them. I might once in while scare some of them. That was part of it. Question: If a student lacks theses qualities, what advice do you give the student? Response: I have never discouraged anybody. That is one thing I think a lot of math teachers do wrong. I've known a lot of math teachers to do that especially teachers at Jr. colleges. I teach at a Jr. College also. There are teachers where the students come and there and they put some problems on the board and they say, 'Oh, you can't work this problem cause you don't belong in this class.'

Question: So you think a lot of teachers... Response: I think that we can always bring a kid that is down low up to a certain level.

Question: Do you think that the high school teachers discourage students from majoring in math? Response: Yes

Question: What suggestions do you usually give your students on how to study for mathematics courses?

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Response: I ask them to keep a notebook. Something that they can w rite down the formulas in. I could get them to do anything because they like me. It didn't matter how hard I worked them, they enjoyed it cause they liked me. It took me about five years to become a good math teacher. If they like you then you can do anything with them. I think in my case that was the most successful thing I did. A lot of the things I taught them, I learned from other teachers. When asked if Teacher A encouraged cooperative learning, he responded, "Cooperative learning is great. I do feel so that cooperative learning in calculus is not as good as it is for developmental. In calculus, they need to think on their ow n. Somebody needs to lead them a litfle." Besides cooperative learning, he encourages students to take calculus because they may need it depending on their major. He responded with "I don't like to push any universities but I do tell them this, "Go to an university that has what you need.' An example I use. Let's say you get in the car with your friend and here you are thinking about getting some enchiladas. On the way over, you stop at McDonald's.... This is also what I tell them. What is the one thing that they can't take away from you? Money, car, .. .what is the one thing that they can't take away from you? Your education. No matter where they put you, they cannot take they away from you. I also tell them that that little piece of paper that you get opens a lot of doors. I won't tell them in class but if they ask and they start talking about it. I tell them that you don't have to belong to a fraternity. All you have to do is get the most education you can out of school. If your grades are good enough, somebody will find you. You don't need anybody else's. That is what I try to encourage. I try to encourage them to get the most they can out of this and that they need all the education you can get. That is the one thing that they cannot take away from you," when asked if he encouraged (Student A) to attend a particular university.

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The following excerpts from the interview with Teacher A relate to his perspecti\ e of Student A. Question: Describe Jose's performance in your mathematics course(s)? Response: I knew Jose before my mathematics classes. First course he took from me was either a trigonometry/analytic or a pre-calculus class. He likes to compete with other people.

Question: Did you encourage Jose to pursue more mathematical courses and possibly a mathematical career? Response: I didn't pursue him to do that. I think he made up his mind.

Question: Do you think there was any contributions from extra-curricular activities or social group? Response: I think the extra-curricular activities helped more than the social group. He was more of an individual. He didn't like to socialize that much. He wanted to socialize but then he didn't want to socialize. He would only have certain people hanging him. Question: What extra-curricular activities was he in? Response: French, Chem., Mathematics for a while, I don't know. Maybe, calculator.

Question: Did you think that maybe extracurricular actixities and the social group distracted Jose from his work in school at all? Response: Nothing distracted him from his work. He would be the type of student who would come in after school and do his work. You know after school in my classroom. If he couldn't do something, he would ask for help on it. He was very dedicated. When asked what role did the parents have in Student A's success, he commented, "I don't think his father really helped him

I don't think his father helped a whole lot

although I don't think his father discouraged him either."

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Question: Name (and rank) three factors that contributed to Jose's performance in your course(s)? (e.g. mofivation level, aptitude for mathematics, smdy habits, etc.) Response: I think probabh' the biggest part was that he wanted to be somebodv in that class.

Question: He wanted to make his presence known? Response: He wanted to beat other people although the\' were better people than he was. It was still there. It wouldn't go away for some of those people. Those people probably drove in more than I did. It was not a competition type thing. We were all going to go to the same contest.

Question: Name the factors that contribute to Jose's success? Response: He would do his work. He was not the lazy t\pe. He would alwavs ha\e his work.

Question: Would \ ou sa\ he his internally motivated? Response: Yes. I think so. He may not have had the best ability of all the kids that I had but intemalh he was motivated. Not only did he ha\ e that ability but also he overachieved. Teacher A's additional comment relating to Smdent A s success is the following. "The only thing that helped Smdent A is that he never became girl crazy. He ne\ er had any girlfriends."

4.2 Interviews from Group B 4.2T Results from the Interview: Student B Student B attended a 5A high school (HS) in Dallas. TX. With the completion of algebra I in junior high, he had the liberty to take geometrv. algebra II. pre-calculus, and

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calculus I in HS. He received A's in all his mathematical courses with the exception of geometry, in which he received a B. His father and older brother hold a degree in engineering and his mother and older sister have a degree in education. Student B chose to major in computer science. In regards to Student B's personality, the following quotes are presented. Question: Would you describe the exams and homework assignments in high school as challenging? Why or why not? Response: Exams were challenging; the homework was not because homework was straight out of the book. The book problems were not easy but they are not challenging either.

Question: How was it that the exam problems were challenging and the homework wasn't? Response: The exams had more of a conceptual focus other than just calculating problems whereas the homework problems mainly focused on calculation.

Question: Would you describe your motivation level in studying mathematics in high school as "high", "moderate", or "below average"? Why? Response: Below average in high school. I did the work but I really didn't strive to go beyond doing the work. I didn't try to leam anything on my own or anything like that.

Question: Would you describe your motivation level in studying mathematics in college as "high", "moderate", or "below average"? Why? Response: I would say it is very high right now. Probably because of the major I chose and to see how math plays an integral role in that. Also, just being more aware that math is a part of everyday life like my teachers told me in high school. I understood that but I didn't actually really look at how it goes whether you're counting money or pay somebody with money, you ha\ e to know how to count in order to know how much change you're going to get. I understood that but I really didn't understand how the upper level of math

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goes. 1 want to get a better understanding of that included in my major I chose and technology is really important as a foundation for my major.

Question: Would you describe yourself as internally motivated (motivated from within) or externally motivated (motivated from without)? Why or why not? Response: Internally motivated cause I have set high standards for myself and I like to succeed at anything I do, and I don't like to do anything that I'm not But I'm willing to do stuff I may not succeed at but if I do decide to do it, I'm going to try to succeed. Maybe, I can succeed at something I thought I couldn't do. I don't really know. I think it's just the type of personality that I have.

Question: In your college courses last year, do you feel like you gained any conceptual knowledge of calculus or its applications? Response: Yes very much so. I find it very easy to memorize formulas. I can look at a problem, and I may not understand the conceptual applications but I can solve it because in my leaming process, it is something very easy for me. So in high school that's what I did with calculus but it is something that is harder for me to do. I feel like with the classes in college I'm getting a better background.

Question: You said you were below average in high school, do you feel like in high school, there is a tendency to experience external pressure or internal? Response: It was part of it. I knew high in school that the teachers, if you are not motivated yourself then they will push you to do it. I think also a part of it was in high school if you didn't push yourself to do the work, generally there is going to be a way you can get out of it. If you don't perform as well as you like then there is going to be a way you can make it up. You can make better grades, or you can make up your grades. In high school, the big emphasis was grades. Where college, the professors care to a certain extent beyond that, it is your responsibility, and you push yourself in the class cause they are not going to push you. You have to push yourself in college whereas in high school.

Question: Did you leam to push yourself by being pushed by your teachers in high school?

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Response: Yes and no. I did leam a lot from my teachers in high school but 1 also knew in high school, I wasn't going to ha\e to push myself I think all along I knew I could motivate myself and have the motivation within me but I also knew in high school, I didn't have to use it.

Question: So in other words, high school could have been a little bit more challenging? Response: Yes, it could have been.

Question: It seems as though now, you have a lot of confidence in yourself based on what your parents that they have an occupation now but not dealing with their major, do you feel that the reason that you don't receive any pressure is because you think that given this like high school, if it doesn't work out, there is always that outlet? Response: That's a nice question! I think that can be a part of it. I think it can. Even with what I am interested in now, there is many different options I can take if engineering doesn't work, or I decide I don't want to be an engineer, there is many different things I can with that degree.

Question: Besides engineering, what other occupation would you be interested in? Response: I am very interested in teaching on the secondary teaching level. I enjoy teaching people things. I enjoy engineering, and the degree I am pursuing but I also do not necessarily enjoy a classical, typical engineer like the material I am leaming. I don't just want to work for a big corporation and design new things and stuff like that. So teaching is an option or a different career I have looked at.

Question: Are you competitive at all? Do you have a competitive nature? Response: Very much so. Question: When you take an exam, do you feel that you compete internally or do you compete with the rest of the class, or do you like to know what the average was? Response: I like to know what the average was to see where I stand against that but I think that is more for just my knowledge. My friends, I like to know how they did on the exam and then talk about why didn't you do this problem 34

right or what was missing, can you help out, and w hy 1 didn't do this one. I do like to know that but it's more for my knowledge. I do definitely compete within myself on each exam that I take. In regards to Student B's leaming style, the following quotes are presented. Question: When you were in high school did you need any help on the material and how did you get help? Response: After school hours, my math and all other teachers stayed at least 30 minutes after school so you could talk them more or you could set up a time where you could stay longer after school to get extra help.

Question: Did you do that very often? Response: More often in the later years, my junior and senior years. Sometimes it was to get a little extra help, sometimes it w as just to do the homework while the teacher was around. We had several students who w ould do the same.

Question: When you do work in groups with other people, who did you feel more comfortable with? (i.e. others of your ability le\ el. others from your social background, others from high schools similar to your own, etc.) Response: I feel more comfortable with people with the same ability le\ el. The people with the same ability level, somebody will have a different idea than I did. They're going to see a different way to soh e a problem than I did. But they are not very out of my league such that they are understanding everything, and I'm trying to play catch up. Somebody on the lower level, I find it hard to study with because they may not be understanding the concepts and rather than studying, I'd be going back and helping them understand how to do something. Which in a way is studying but I don't mind helping people because I generally leam something when I'm trying to help someone understand the same thing. But if it's actually studying for a test or quiz then I feel like I'm missing out in the time where I need to focus on what I need to work on so generally, somebody at the same ability level. Question: Describe your experiences in your college calculus course, (i.e. did you like the professor, the textbook, the structure of the class, etc.) Response: Calculus 1,1 like my professor, and I like the idea of projects but I felt the projects we were doing were a lot of busy work with not as much return on the actually leaming the concepts. I didn't really follow the way she taught as well. She didn't have a problem teaching; I just wasn't following 35

the way she taught. I like the way the class was set up in that we had quizzes and we had projects where you could work with other people in understanding the material and there wasn't a big focus on doing the homework. I understand the point of doing homework but I don't like to feel that I have to have it done every single time completely completed because sometimes life gets busy, and 1 don't get it done.

Question: When you did study for calculus classes, how did you study? Did you use the book? Response: I like to review the sections in the book that were covered, look at the homework especially if it had been graded to see what part of the sections I was missing the most problems on. If it wasn't graded then I would just review some of the problems that I had done to see if I remembered and to see if I could still do them. Right before the test, I would review the notes taken in class. Student B studied majority of the time alone with an occasional group study session. In college, he studied approximately 3-4 hours a week outside of class alone. For the entire semester, Student B spent close to 25% of the time studying with other people. The following exchanges expands on the initial references to his high school teacher: Question: If you had to name a particular teacher/person who influenced you to pursue mathematics at the college level, who would it be? Why? [e.g. Were they interested in mathematics and tell you about their own story of how they became interested in mathematics] Response: Algebra II / Calculus I teacher in high school. Her name was.. (Teacher B).. . She was my trainer my freshman/sophomore year. My junior/senior year, she wasn't a trainer but she still taught math; taught math all the time. She focused on her relationship with her students. She kept encouraging me to stay with math and some sort of field that would require me to continue to take math in college. Question: How did your teacher foster these relationships you talked about her getting to personally know the students; what did she do exactly? Response: She's the type of teacher in class who doesn't keep necessarily strict rules. She keeps the class lively and encourages us to talk about the topics we're leaming. She's just a real fun teacher. She has a reputation for being fun. I remember when I had her for algebra II as a sophomore, I w ould 36

see upperclassmen, and people who had graduated came back to see her just because she really is a very caring teacher. So we got that before we met her and then when we talked to her, she was a lot of fun and very helpful even outside of mathemadcs.

Question: How did she convey to you that she cared? Response: She's very open and would joke around with you. If you weren't an open person, she'd tried to get you to open up and enjoy the class and not just the mathematics that she was teaching but also just to enjoy life as it was.

Question: Did the teachers encourage you to get together outside of class or were they hands off? Response: My teacher was laid back about it. She realized it was helping us out so she made her classroom available after school. She suggested that we do it if we were having problems but she wasn't real pushy about it.

Question: Did you feel that your high school preparation in mathematics was adequate for college? Response: Yes, because the teachers were aware what we needed as far as what kind of mathematics or how to prepare us for the math classes in college. They not only taught us to the concepts that we needed for our high school classes but they also taught us how to leam it on our own, and taught us to understand what we were leaming and how it would be in college. They still did high school grading but talked about that in college you wouldn't have all these homework grades so focus more on the tests.

Question: Was your motivation level in high school influenced by your teachers? Why or why not? Response: Yes, my teachers pushed me to raise the mofivation level. I think that kept me still going in math. Question: How would they motivate you? Response: By encouraging me that I didn't just have to do the basics to get by with math. I could go further with that. They design some projects that had some challenges to them but were also fun which try to keep you interested. That was building a relafionship with their students to help keep people mofivated as well. 37

Question: Is your mofivafion level in college influenced by your instructors? Why or why not? Response: Yes it is. Some classes seem boring at times. I may revert back and highlight some classes in high school and realize I can leam just as much and I'll be just fine cause I'm understanding the stuff Most instmctors that I've had, I don't think I haven't had an instmctor who doesn't try to show you how much further you can go with the material. Once I see that that then I can go farther than I am letting myself It's just kind of natural for me to ingrain that. If I can go further, why not go further because I am paying for the class. If an instructor is willing to show me how muchfiartherI can go then I should take the chance and see what I can actually leam. The following interchange expands on the initial reference to his family. Question: Would you describe your motivafion level in studying mathematics in high school as "high", "moderate", or "below average"? Why? Response: Not really. My parents would encourage me to do well. The wanted me to do well and keep my standards high as far as achieving goals in math as well as all my classes. They didn't pressure me into math. That w as my choice.

Question: Are your parents' occupations related at all to your career choice/future occupation goals? If yes, why? If no, why not? Response: Yes and no, my mother has physical education degree for teaching elementary physical education. My father has an electrical engineering degree. They both were working within their degree right out of college but are now with a Christian organization. That is something, I am also looking at. Even though I am in the engineer degree, I'm not sure I will stick with it once I graduate.

Question: What are your parent's wishes for your future career? [may want to pursue this with why?] How do they compare with your own career goals? Response: Their main wish is I choose something I'm happy and that I enjoy doing for life. My dad knows about engineering so they don't wish for me to

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do engineering vs. something else. They just want me to choose something 1 enjoy doing.

Question: What role, if any, did your family play in supporting your decision to come to college? Response: Yes, I think they did in that they encouraged me to go to college. I really never had any thoughts about going to college just because, for one, it helps you choose a career eariy and prepares you for a career. If you don't go to college then it's harder to choose a career and, I really don't know what I'm that interested in so I think that's why my family encouraged me to go to college.

Question: What family and peer relationships empower you in your academic pursuits? How? Response: My relafionship with my enfire family empowers me in my academic pursuits. My entire family is very supportive whether what I choose to do. I'm the youngest in the family. I have my oldest brother and sister to look up to and see what they have chosen and be able to talk to them about why did you choose this, what if you didn't choose this. That helps me in my decision to be able to choose my own path that I want to follow. The following interchange expands on the initial reference to his high school and college peers: Question: So the other times you studied by yourself? Response: Yes

Question: How/What did you study with other people in calculus? Response: It was my senior year and was more fun to study with others. Less work on my part. Generally, most of us had the same types of questions so when we studied together, we could narrow it down to a few main questions of a concept we stmggled with then the teacher had to answer those questions we had.

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Question: Describe your social group in high school, (i.e. what interests did you have in common?, are they now attending college?, how many of your close friends were in your mathematics courses?, etc.) Response: Most of my friends were in math classes. Most of us were in marching band in high school so we had that as general interest. Some of m\ friends and I were interested in engineering and then just good ole goofing off and sports.

Question: How did ya'U meet each other either marching band or classes? Response: Both of those, largely due to marching band and then the mathemafics classes as a freshman being in geometry. Most of us within that group started algebra in middle school and then had the same math classes from geometry through calculus.

Question: If you studied with other people, how many of them were from your social group (i.e. friends)? Why were they the same? Or why were they different? Response: Studied with same people.

Question: How important were your friends' opinions when it came to choosing a university to attend? [Why were they important? Why weren't they important?] Response: My friends' opinions were something I definitely thought about but I knew even though we had been in high school, most of us for 4 years were together. We were close friends but we were probably going to choose different universities just because of our different interests. There was some merit to it but I knew I couldn't base my enfire decision off of that.

Question: What family and peer relationships empower you in your academic pursuits? How? Response: My relationship with my entire family empowers me in my academic pursuits. My entire family is very supportive whether what I choose to do. I'm the youngest in the family. I have my oldest brother and sister to look up to and see what they have chosen and be able to talk to them about why did you choose this, what if you didn't choose this. That helps me in my decision to be able to choose my own path that I want to follow.

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Question: Describe your (current) social group at college? (i.e. w hat are their majors?, do you study together?, etc.) Response: My social group doesn't include anybody I knew in high school. The majors range from all over. There really is not one major. I have many friends from engineering college at Tech. I ha\ e to study for those classes so 1 study with them but I also have many friends with majors ranging from pohfical science to pre-med majors. Just because I like engineering and 1 enjoy the friends I have. I also know there are a lot of other people who offer different views such as a polifical science major vs. engineering major.

Question: So you described that you study with people in your social group, so you tend to study with your friends in the engineer field? Response: Yes

Question: What kinds of information do you share (among you and your friends) about navigating the university (i.e. registration, financial aid, what professors to sign up for, what courses to take, emotional support, etc.)? Response: We share pretty much all of that You can always find out who's teaching what class, what section, who students recommend, who they don't recommend, what kind of test this professor gi\es so I think all that kind of information is shared very openly. In fact, I find out who's taken this class before me; who they suggested I take, who teaches it and in which way. The following interchange expands on the initial reference to his extracurricular activities: Question: How would you say those exams and homework assignments in college differ from those in high school? Response: Homework assignments in college generally aren't graded, and if they are, it's generally just certain problems and not the whole thing so complefion grades aren't that important. In high school, there w as a bigger focus on the homework problems that they were done. The exams (most of) in college have more of a conceptual compression and less calculation.

Question: Did you work while you attended high school?

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Response: I had a couple of jobs. They went into my school once classes started but I really didn't have a job that I kept the entire school year because I was in the marching band and was really hard to work during practice so generally I wouldn't have a job during the fall semester of the year. I didn't have a job until the end of the spring semester.

Question: So you worked at the end of the spring and in the summer? Response: Yes

Question: What kinds of responsibilities (housework, child care, job, etc.) did you have in high school? Do you think that these responsibilities enhanced or detracted from your educational experiences? Response: I think it enhanced it. When I had a job, I had to manage my time better because I ha set a schedule; I couldn't just goof off and play all the time because I had the responsibility of being somewhere at a certain time. It really helped me understand that with taking classes I had to work at it but I could still have fun. I had to make sure to manage my time to do the work and have fun. Holding a job helped me that way.

Question: How much do you think that helped you in your classes, having described as though you've had a pretty good background? So how would you say that helped you? Response: I thought it helped quite a bit especially that I had calculus in high school but I retook it my freshman year in college so just the fact that the background that I had in math was already a strong background. It was easier for me to get into, as a freshman, college classes and not to worry as much that I'd be struggling as much in the area of math. I knew that 1 already had a good understanding of it. That just kind of helped me just kind of take it and leam to sit in as a freshman and leam what was required of me in college.

4.2.2 Results from Student B's Father - Father B Father B received a BS in electrical engineering (EE) in college then retumed for his MS degree in EE for reasons of diversity and marketability. Once as an engineer and. presently, as a missionary, he has had the experience of changing his career along the w ay

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crediting his degree for the easy transifion between each profession. His parents had only a high school educafion, but they had strong views towards education. Missing their opportunity to attend college, they prepped and planned for both of their children to attend. With the influence of his high school math teacher, he chose the engineering route in college. As parents, he and his wife, educated as well, shared the responsibility of tutoring their children by dividing up the schoolwork. She handled the biology, the reading, and the history, and he took the math and the sciences excluding biology. His "mother worked in the local school district as a secretary (so he) was around ideas of education all (his) life." He "believes in making (mathemafics) fun and interesting and trying to apply it to real world problems." He prefers a mathemafics teaching instmction "based on rules (and not) think about it and try to figure what to do." He stresses the importance of mathemafics to his children in the following quotes: Question: Did you encourage your child to take mathemafics all four years of high school or just the high school requirement (of three years)? Response: We required all three children to take 4 years or 5 years.

Question: Why would you encourage them to do this? Response: I think it is very important no matter what course of study you go into. (Advanced level mathematics) happens to have a natural tendency towards math and science and physics. That was a natural thing. Even the other children who had chosen to do types of studies, all took fairly advanced math. I just think it's healthy. Father B's view of the adequateness of high school's preparation is exposed within the following quotes: Question: Do you feel that your child's high school adequately prepares students for college? Response: I definitely think so. 43

Question: Did it adequateh prepare your child, particularly, in mathematics? Response: Yes, I do. Again, I think the teachers were ver> enthusiastic. By that. I mean not just by teaching lessons in the book, how to soh e problems, or whate\ er but tying it to real life type stuff

Question: Are there other ways that have not been mentioned that you think that your (son's) high school could better prepare students for college? Response: No. I don't think so We actually switched Jonathan to a larger high school so he would ha\e the opportunity to take ad\anced classes in multiple subjects without a conflict. The high school did a real good job of making sure there was an opportunity for all that and for staffing it with people who w ere good at their job. Even though Father B encouraged ad\'anced mathematical courses, he was also ••leer\ of trying to speed up the process" of Student B's education. He elaborates about this idea w ith the following comment, "Universities put together their curriculum based on some assumptions that might not be tme. If you take those classes some other way. I never said no don't do that but I would have been a little leery of it because I wouldn't want him to get to college and start the 2"^^ semester of math and science courses." Question: In what ways, do you promote pursuing a higher education among your child(ren)? Response: Basically, it was a given. Non-negotiable. Formnately. we never got into a test of will over you'll do this. It wasn't that way. and it worked out smoothly. Both my w ife and I have degrees. We work in a Chrisfian organization where e\ ervbody there has a degree. I think that they just thought that the world is a much broader place if you have a degree. I think it was kind of osmosis, I guess, but it was non-negofiable. They were going to go away and at least try. It never came to a test of will of ordering that you were going to do it.

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Question: Do you have definite ideas about the career path your child should follow? Why or why not? (Do you encourage them to do w hatever they desire?) Response: No. I think they should choose what they want to do. With the experience of career change and geographically relocating. Father B strongly advises a degree in the mathemafical disciplines simply because it allows for marketability. The following excerpts reveal his reasoning: Question: Are your ideas about your child's career path the same as your child's goals? I'm not saying you would direct his career path but have you ever thought about him taking a certain career or mentioned it to him? Response: Jonathan was bom in Africa while we there as missionaries, and 1 have real strong feelings that there is a likelihood that he may want to go back and live overseas. He was a 3'^'^ culture kid. Children who grow up in a 3'^^ culture tend to want to move back into a 3"^^^ culture but not necessarily the same one. There is a real high probability of that happening. We have talked to them and said if you are going to want to have the option, you are going to have to have a degree that's, in American terminology, it's called a marketable skill. You have to have a degree that the country sees as reason to grant you a VISA to live there. That tends to mean something in teaching, medicine, or some type of degree that you can do where the country thinks it's valuable We were pleased, and I don't know how much pressure we exerted or how much of an influence we were but we were pleased that each of our children including Jonathan has chosen a degree that is marketable; one you can get a job with. It's not that I expect them to get a job but because I like to expect them to consider full-time Christian work but it is still a very important criteria in that department. As opposed to, picking on theologians, doing an undergraduate degree in religion education and seminary degree in theology and trying to go to another country, they are not going to look at that academic qualification say, "How are you going to help us?

Question: You would consider (a degree in mathematics) marketable? Response: Yes, it would be marketable in a sense of that it would be a basis for education or something. Probably, teaching not education. Father B's perspective of Student B is reflected in the following quotafions.

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Question: Would you describe your child's motivation level in studying (mathemafics) in high school as "high," "moderate," or "below average"? Why? Response: Above average

Question: Why do you think this is? Response: Probably, natural apfitude. Enthusiastic parents.

Question: Would you say Student B's motivation is within? Response: Yes, definitely.

Question: Why do you think this is? Do you think he inherited it from one of you or...? Response: Yea, it transcends math. He's an achievement-oriented person.

Question: Have you ever pressured your child to perform well academically? (i.e. punishment for poor performance, reward for good performance, etc.) Response: No, but, of course, he probably feels different. We are not trying to live out our lives through our children but don't settle less for what we know they are capable of giving.

Question: Did you reward for good performance? Response: No

Question: So did you punish for bad performance? Response: No, we didn't punish either but it depends on what you mean by reward. If you mean extemal reward then no. We feel like praise and cheering them on and going out having a meal and celebrating good grades; well, I guess, those are rewards too, but we did that each semester. It was not tied to monetary or a car or anything like that.

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Question: Besides helping and tutoring Student B to excel in math, how else did you assist him to excel in math? Did you provide other tutors or did he get help from teachers after school? Response: I'm not sure about talking to teachers. We didn't find other tutors. We done things like, again, because part of it is just my interest too, done things like tour Cape Canaveral, toured the Soviet Space Exhibit w hen it was in Fort Worth during his high school years. Kind of tied it to the real world of how people accomplish what they've done.

4.2.3 Results from Student B's Teacher - Teacher B Teacher B decided to become a HS mathemafics teacher after "teaching a miserable semester of Health and P.E." She realized that she abhorred teaching these two subjects and preferred to teach mathematics. An interesting situation when considering that she had not always liked mathematics until her experience with trigonometry. For some unknown rationale, the course just clicked for her. Thus, trigonometry convinced her to minor in mathematics since a major in mathematics seemed unattainable due to its demanding mental challenge. She thought that she was real good at mathematics unfil a couple of upper le\el mathemafics courses knocked the chip off her shoulder. Nevertheless, she simply loved the course, which led her to love mathematics so much that she wanted to share that feeling of goodness. The following excerpts from Teacher B's interview reflect her attitude towards education and her students. Question: Do you think your role in contribufing to Jonathan's performance in mathemafics a success or a failure? Describe your role a little? Response: I can only do so much as a teacher and that is what I tell the kids. I have you only for 45 minutes a day for 180 days out of the year, and that is not very much. If you look at your lifefime. the whole span of your lifefime. 45 minutes a day is nothing. That is why I want to have a good time with those kids. I like to cut with those kids. They are fun if you can get them 47

rolling on a topic but they can tell 1 enjoy. They tend to pay attention a little more and stuff I had a good time with the kids. They could tell if I was disappointed in them, or if I was very proud of them or if 1 was mad at them. When they could tell what my mood was, they knew w hether or not they w ere screw ing up. I taught the material.. .1 gave them what I think they needed to know out of algebra II and calculus at least to my abilities when it came to calculus. It just goes above and beyond when you really care about the kids. As far as the math part of Jonathan's success. I did what I was supposed to do as a teacher to the best of my ability.

Question: Did you ever inform them or did they ever ask you how you became involved in math or why you became a math teacher? Response: They asked me that several times. I told them that I started out in pre-vet, and I hated chemistry so I got out of pre-vet and went into Health and P.E. because I also liked athletics. When I got involved with athletic training. I did my student teaching. I didn't want to minor in driver's education, which was the big deal in those days when I was in college. Driver's ed. was the big minor. There was no way I was going to do that. There wasn't really anything else I liked as much as math so I minored in math. It turns out that I did my student teaching in Health and P.E., and I hated it. When 1 got my first teaching job, he asked me if I wanted to teach P.E. I said, "Not on your life! Math." I tell the kids that trigonometry is what made me love math. I didn't always like math. It was because for some reason, trig, just clicked. 1 had a really good time with it and that is part of the reason why I decided to minor in math. 1 liked it so much that I wanted to share that.

Question: Do you inform these students of your ability in math? Do you have to know how intelligent they are compared to you or how much more gifted and talented they maybe you? Response: It doesn't bother when a kid, especially in calculus, will ask me a question that I can't answer because I know very well that they w ill do a whole lot better than I did in college math. They are the ones that will go into engineering. I didn't have a problem with that. If I didn't know the answer then I would try and find out. If 1 couldn't find out the answer then, usually, either the kid would or he was going to find out through somebody else anyway or over the internet even.

Question: Do you feel that your mathematics courses prepare students w ell for college calculus? Why or why not?

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Response: That's very subjective. I hated calculus in college. I hated it. I minored in math, and I had to take two semesters of calculus, and 1 hated calculus because I just didn't get the calculus because I just didn't get the instmctor. When I finished them, I said never again. About 5 or 6 years, my principal asked me if I would teach calculus. I haven't seen calculus in sixteen years. I swear I would never touch it again and now you w ant me to teach it. The following quotes reveal Teacher B's perspective of educafion. Question: Do you encourage academic competition in your class among your students or academic cooperation? Response: I make the kids work together on those because some kid maybe a lot better on one topic in calculus than another. That goes for Algebra II also. As far as the compefifion, I don't have to promote competition with those kids because those gifted and talented kids compete against themselves, and I don't ever have to say a word.

Question: Do you feel that your mathematics courses prepare students well for college calculus? Why or why not? Response: .. ..The very first year I taught calculus, I don't feel like they were prepared. However, they came back to me, those kids who went into engineering, they came back and said, "As basic as a background you gave us, it got us at least through the first semester." I guess what we offered them in calculus and what I gave them in calculus.. .1 got better at it as I taught it...

Question: Do you think your tests and homework prepare them for the tests and homework in college mathematics? Response: My tests are very much geared towards what I taught I test over what I taught then I throw some stuff that they might haven't seen before but they have the background to get it from what I taught. When asked if Teacher B encouraged students to pursue more mathemafics in HS and, possibly, a mathematical career, she responded, ".. ..I expected them to go to Pre-Cal, and I expected them to go to Calculus...."

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Question: For those who didn't want to take more math, did you try to put fear in them about what college was like and staying out a year or two years without math...? Response: Very much so. I did very much so. If I had a chance to talk to the parents, I did with them to. I tell the kids all the time that I don't even like them to skip calculus, if they take Pre-Cal their junior and their planning on doing something or if their planning on going to a 4-year college then I try and talk them into at least go through calculus even you don't take the AP test at the end, at least keep some math under your beh because college algebra is going to kick your butt. I really, very much did not like it much if they stopped at Algebra II but those kids, I'm thinking of one in particular, aren't going to go into a math-related field, and they already know that. I still don't like the fact that they skip two years of math because, half the time, they ha\ e to go a lower math course then they even belong in. Teacher B expresses her opinion about success in educafion in the following quote, "1 tmly think that you have to come from a strong family background to be as successful as some of these kids are." Question: What qualities do you feel that students must possess in order to pursue a career in math or any mathematical-based discipline? Response: They got to have the smarts. If you don't get math then you just don't get math. That is going to be hard going into engineering if you are not good at math in the first place. Just the basic ability to do math in the first place is one quality; I think that is a lot of intelligence. Self-motivation for sure because those engineering courses are not easy, and those kids have to stay motivated. They have to stay motivated through themselves because mommy and daddy aren't there anymore. They have to have the support from somewhere because if they do screw up that support can come from a college professor, or it can come from mom and dad back home, or it can come from an email from a friend or anything. They have to have support in those tough courses.

Question: How do you encourage or model these qualities? Response: Some of those kids are more intelligent than me so that doesn't count. I have had a couple of kids that have tested me, and I promise you... .there is no way I can make a 5 on calculus AP test. I have some kids that are more intelligent than me so I do what I can. If they ask me a question that I can't answer, I will try and find out from somebody else, another 50

teacher, or someone else. As far as the support and stuff, I hope the kids knew when I was proud of them. Teaching gifted and talented kids, you w ill have a lot of kids that will go nafional merit in that sort of academic competition. 1 hope that I model.. ..I hope they can tell through me that there is somebody there that is proud of them. If it is me that they need when they go through college they know where to find me.

Question: If a student lacks these qualities, what ad\ise do you gi\e them? Response: I'm not one to discourage a kid from something they think they can do. If they lack those any of those quahfies, 1 tell them to just gi\e it a try. If you think you can do it, you go for it. They ha\ e a lot of kids in engineering who don't have the background that some of these gifted and talented kids have. That just means they will have to work a lot harder. Those kids are probably better off than these very intelligent kids in the first place. They have to work for that. I think that in itself makes for a better professional sometimes.

Question: That is all the questions that I have, however, is there anything that you would like to add? Response: I do to tell the kids, and I harp on this, "If you screw up the first semester in college, it will take you the next three years to pick up your GPA." I tell them that all the time, and that doesn't matter which class we are talking about. If you screw up your first semester, you are going to spend the rest of your semesters trying to pick up your GPA. You still ha\e to ha\ e a good time because that is part of what college is all about but keep your priorifies straight too. If you are having too much of a good time and your academics is slipping, you are messing yourself up for the rest of your college career. At 16, 17, and 18,1 am stupid. They will not listen to me when it comes to college. Those kids have to leam for themseh es. I harp on the first semester thing the most because that's when they get lost. They go away from home, they don't have mommy and daddy there to kick them in the butt, and give them a curfew, and ask them if they did their homework, they are on their own, and that is why I harp on the first semester stuff As far as anything else, if I think of something, I will tell them but these kids have to leam for themselves. They will not listen to anything we have to say when it comes to college. They have to find out for themselves, and that is not always a bad thing.

Question: If you explain to them how rough and tough some of those professors will be?

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Response: I do because they gripe at me about the homework I gi\ e them. They have so many things that they have to do that they just don't ha\e time to do it. I tell them, "Hey you think it is tough now w ait till you get to college. Those professors are not going to care what your schedule is so you better get a grip on things now. At least the teachers you ha\ e now care about what your schedules are. You are right here, and you are a much smaller group than most of the colleges you are going to pick. College professors see thousands of kids everyday, and they are not going to be able to pick you out, and point you out, and help you out, and hold your hand in every class. Grow up!" Teacher B's perspecfive of Student B can be seen in the following excerpts. Question: Could you name and rank three factors that contributed to Jonathan's performance in your courses? For instance, study habits, motivation, aptitude. Response: He was very self-motivated and very intelligent. His attitude.. .1 never saw the child down. He was always in a good mood but if you weren't, he would do what he could do to make. I think all of those had a lot to do with it. He was just a good kid all around.

Question: Describe Jonathan's performance in your mathematics course(s)? Response: Algebra II, he was always very much grade conscious. I never had to worry about him not doing what I ask him to do. He was a pretty good kid. He was usually on time with his homework and everything; and every once in a while, he would slip up but he was so involved in so many things that but he would come around in the end though. However, Teacher B responds with Student B's performance his senior as somewhat different in the quote, "He wasn't quite as worried about his grades as a senior..." Question: At the times he was slipping up in your class, did you encourage him to work harder or provide any type of verbal encouragement? Response: .. ..I always gave them a break and he knew that but I also told him when they were taking calculus if you slip up in calculus, it is going to be that much harder to catch up so you need to try and stay on top of things....

Question: To the best of you knowledge, what role (if any) did the following have in contribufing to Jonathan's success in mathematics such as parents.

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extra-curricular activifies, social group as far as friends &: peers, and things like that? Response: I think that all of that had a factor. I think that Jonathan had a lot of good teachers and whether or not the\ were renaissance teachers, whether or not they w ere gifted and talented teachers, or regular teachers, he had a lot of good teachers. And I think that had a lot to do with it but Jonathan was a leader. He wasn't a loud leader, and he wasn't realh a foul leader but kids looked up to him. I think he had a strong famih background. I think he had strong spirimal background. 1 think church w as important to him and extracurricular activities as far as band w ere \ er\ important to him. I think all of that combined made him \ ery successful. He was just a great kid all around. Teacher B's answer to what distractions did Student B experience w as "Girls w ere a distraction. Band was a distraction.... He was so involved in so many things. I can sa\ at any given time, any one of those things was a distraction including his other academic classes like his renaissance English class..." In conclusion. Teacher B's lack of confidence and knowledge of mathematics and her w illingness to open up w as the most effective aspects of her teaching style.

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CHAPTER V CONCLUSIONS

5.1 Analysis of Interviews 5.1.1 Analysis of Student A Student A is a self-motivated, confident student whose perseverance and endurance exceeds the average student and helps him overcome any lack of innate mathematical ability that might otherwise adversely affect his academic performance. As a practical person, he prefers applied mathematics over theoretical mathematics, and attains his understanding of mathematics one course at a time. Competitive in nature, he thrives on competition for motivation, excitement, and success. Although he competes both externally and internally, he perceives his intemal motivation to the dominant. Academically, his success is attributed to three major influential factors: competition, HS mathematics teacher, and his family's "culture idea." In high school, the competition stemmed from UIL participation, grades, and a particular collegetailored teaching style. UIL was more challenging than his high school mathematics courses because it demanded more attention, time, and effort; thus, allowing for more and better preparation in mathematics. Also, throughout high school he remained in constant grade competition with other students, especially, with his closest friends. He enjoyed the college-like teaching style that his math teacher used in high school

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because he viewed it as a compefifion between himself, the material, and the teaching method. In college, his grades and major provided his primary motivation. Again, he viewed his mathemafics courses as a competition, not just between himself and his fellow students, but also between himself and the subject. His major forced him to focus not on just leaming the concepts but understanding them well enough for application use. Among all the individuals in his life, his HS mathematics teacher was the most influential. He highly respected the man for his knowledge, teaching skills, and support for the "culture idea." Although he did not inquire more about his high school teacher outside of the classroom. Student A wanted to emulate his knowledge. In college, he was not affected or influenced by any professor, largely because he considered them more theoretical than practical. His family supports all of his educational aspirations because of their "culture idea." They expect him and each child to surpass them educafionally, and that success is defined by performing one's best. When studying mathematics. Student A preferred isolation and invested a small amount of fime each week. Occasionally he would study with a select group of intellectual peers or tutor mathematically weaker friends but most of the time he would study alone. Socially, Student A was not a gregarious person. In high school he preferred to socialize with friends he viewed as intellectuals, and in college he socializes within

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his immediate group of friends. He keeps the group small so as to avoid social distractions. In conclusion, it would be fair to describe Student A as an introv erted personality type, both socially and academically, with strong personal motivation to succeed.

5.1.2 Analysis of Student B Student B is a gregarious, confident, happy-go-lucky individual with a lust for social interaction and unencumbered by an overly exaggerated work ethic. Indeed, his academic abilities allowed him to succeed in high school with minimal effort. Although he claims to be internally motivated in college, his motivation in high school was definitely extemal, coming from parents, teachers, and peers. His competitive nature does not appear to have caused him any stress. Student B was reared in a Christian home, his faith and family are the most important factors in his life. He thrives on social interaction and challenging new tasks. Academically, his success is contributed to three major influential factors: family, HS mathematics teacher, and extracurricular activities. In high school, he depended upon extemal influences for motivation, particularly, his HS mathematics teacher. Through her persistent enforcement of deadlines, leadership, and leniency with respect to allowing make-up work, she kept Student B on pace with his mathematics. His parents were well educated and qualified to tutor him in the sciences, mathematics, and humanities. His friends and extra-curricular activities provided him with social entertainment and peer support. 56

In college, realizing the lack of support that his professors pro\ ide, he has had to discover other means of mofivafion. His major and grades act as motivational instmments. He competes mainly with himself but to some extent w ith other classmates as well. He challenges himself to do all that he can do and expects nothing less than his best effort. His father and brother have been role models for Student B. He has witnessed both choosing a career beyond the expected intentions of their college degree, and has considered the possibility of utilizing his degree in various occupafions as well. On the other hand, his HS mathemafics teacher initially inspired him to study engineering and mathemafics. When studying for mathemafics. Student B mostly preferred being alone. Occasionally, he participates in-group study to reinforce his knowledge and understanding and to help those stmggling. The social atmosphere is a necessity in Student B's hfe. In high school, he was involved in many extracurricular acfivities for reasons of the social climate and the challenging tasks. He interacted mainly with math classmates and band members. In college, his social group consists of wide range of friends from a variety of majors. Among all his extracurricular acfivities both in high school and college, the marching band is his favorite.

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5.2 Conclusion Both students shared many common success factors. Both had high school teachers that were compafible ethnically, and according to Mayo, et al [1995], this relationship contributes to academic achievement. Both students chose their major before entering college and were strongly influenced by their HS teacher. This is consistent with Hinton's [1988] flnding. Although Terenzini and Pascarella [1997] considered informal contact with faculty a major contributor to academic success, neither student encountered much informal contact with the faculty. As a matter of fact. Student A is convinced that the faculty is impersonal and apathetic towards the students. Treisman [1992] emphasizes the importance of peer support as a key to academic success. Such support from his fellow students seemed to be more of a contributing factor in Student B's success than in the case of Student A. Both high school instmctors used cooperative leaming in a very limited way and it was not considered a factor with either student. Both students' personalities seemed to be the major success factor; however in very different ways. Student A exhibits an introverted personality, with his strong work ethic overcoming his mathematical inadequacies. Moreover, he was always very well guarded with respect to his social life for fear his social life might interfere with his academic life. This is an interesting observation in the light of the finding of Mayo et al [1995] that social life, especially among Hispanic students, can impede academic success.

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Student B on the other hand exhibits an extroverted personality, with his mathemafical prowess overcoming his lack of a strong work ethic. His social life was a significant part of his high school experience and by his own admission sometimes did impede his academic progress. Both students considered themselves to be highly self-motivated, howe\ er, it is clear that both were greatly influenced by their parents and by their high school mathematics instmctors. In the case of Student A, the high school instmctor seemed to provide somewhat more influence than did the parents, whereas with Student B the parental influence seemed to be the dominant factor. Both sets of parents were highly educated, with at least one holding a Masters degree. This is consistent with Anderson's [1990] finding that the level of parental education is directly related to student performance.

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REFERENCES Anderson, B. J. "Minorifies and mathemafics: The new frontier and challenge of the i^inefies," Journal of Negro Educafion. Vol. 59, pp. 260-272. (1990) Attinasi Jr., L. C. "Getting In: Mexican Americans' perceptions of university attendance and the implicafions for freshman year persistence," Journal of Higher Education. Vol. 60, pp. 247-277. (1989) Bennett, C. & A. M. Okinaka "Factors related persistence among Asian. Black, Hispanic & White undergraduates at a predominantly White Universit\: Comparison between first and fourth year cohorts," The Urban Review. \'ol. 22, pp. 33-60. (1989) Boli, John, Mary Lou Allen, and Adrienne Payne "High-Ability Women and Men in Undergraduate Mathematics and Chemistry Courses," American Educafional Research Journal. Vol. 22, No. 4, pp. 605-626. (1983) Carter, Carol J. "Strategies for Recmifing and Retaining Minorities and Women in Nontraditional Programs," New Direcfions for Higher Educafion, No. 57, pp. 75-81. (1987) Erekson, O. Homer "Joint Determination of College Student Achievement and Effort: Implications for College Teaching," Research in Higher Education. Vol. 33, No. 4, pp. 433-446. (1989) Fullilove, Robert E. "Mathematics Achievement Among African American Undergraduates at the University of California, Berkeley: An Evaluation of the Mathematics Workshop Program," Journal of Negro Education, 1990. Vol. 59, pp. 463^77. (1990) Gainen, J. "Barriers to success in quantitative gateway courses," New Directions for Teaching and Leaming, No. 61, Spring 1995. pp. 5-14. (1995) Galindo, R. and K. Escamilla "A biographical perspecti\ e on Chicano educational success," The Urban Review, Vol. 27, pp. 1-29. (1995) Hedges, Larry V. and Amy Nowell "Sex Differences in Mental Test Scores, Variability, and Numbers of High-Scoring Individuals," Science, 1995. V. 269, pp. 4 1 ^ 5 . (1995) Hilton, T. L. and V. E. Lee "Student interest and persistence in science: Changes in the educational pipelining in the last decade," Journal of Higher Education, Vol. 59, pp. 510-526. (1988) 60

Jay, G. M. and A. R. D'Augelli "Social support and adjustment to university life: A comparison of African American and White freshmen," Journal of Community Psychology, Vol. 19, pp. 95-108. (1991) Lopez, Edward M. "Challenges and Resources of Mexican American Students within the Family, Peer Group, and University: Age and Gender Patterns," Hispanic Journal of Behavioral Sciences, 1995, Vol. 17, pp. 499-508. (1995) Mayo, Judith R., Edward Murguia, and Raymond V. Padilla "Social Integration and Academic Performance Among Minority University Students," Journal of College Student Development, Vol. 36, pp. 542-552. (1995) Rendon, L. I. and E. M. Triana "Making Mathematics and Science Work for Hispanics," American Association for the Advancement of Science, 1989, pp. 1-25. (1989) Sax, Linda J. "Mathematical Self-Concept: How College Reinforces the Gender Gap," Research in Higher Education, Vol. 35, No. 2, pp. 141-166. (1992) Terenzini, P. T. and E. T. Pascarella "Voluntary freshman attrition and patterns of social and academic integration in a university: A test of a conceptual model," Research in Higher Education, Vol. 6, pp. 2 5 ^ 3 . (1997) Treisman, Uri "Studying Students Studying Calculus: A Look at the Li\ es of Minority Mathematics Students in College," The College Mathematics Journal, 1992, Vol. 23, pp. 362-372. (1992)

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APPENDIX A STUDENTS' INTERVIEW QUESTIONS

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1. Describe your high school preparafion in mathematics (e.g. what courses did you take, did you do well in those courses, etc.) [Have them list their mathemafics courses. If any seem to have been developmental in nature, invesfigate why they took the courses.] a. If you needed any help on the material, how did you get it? (i.e. tutorials, group leaming, or parental assistance) b. Did you study with others in high school? If yes, for what courses, how, and why? If no, why not? 2. Do you feel your high school preparation in mathematics was adequate? If yes, why? If not, why not? a. Did you have any ideas about what mathematics in college would be like? Who inspired those ideas? b. Would you describe the exams and homework assignments in high school as challenging? Why or why not? c. How do the exams and homework assignments in college differ from those you received in high school? 3. Would you describe your motivation level in studying mathemafics in high school as "high", "moderate", or "below average"? Why? a. Was your motivation level in high school influenced by your teachers? Why or why not? b. Did you feel pressure—in high school—(from society, former teachers, parents or siblings) to do well in mathematics? Why or why not? How did this "pressure" affect your motivation level if at all? 4. Describe your social group in high school, (i.e. what interests did you have in common?, are they now attending college?, how many of your close friends were in your mathematics courses?, etc.) a. If you studied with other people, how many of them were from your social group (i.e. friends)? Why were they the same? Or Why were they different? b. How important were your friends' opinions when it came to choosing a university to attend? [Why were they important? Why weren't they important?] 5. a. Are your parents' occupations related at all to your career choice/future occupafion goals? If yes, why? If no, why not? b. What are your parents wishes for your future career? [may want to pursue this with why?] How do they compare with your own career goals? 6. What role, if any, did your family play in supporting your decision to come to college? a. What family and peer relationships empower you in your academic pursuits? How? b. What family and peer relationships endanger or detract from your academic pursuits? How? Why?

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c. What kinds of responsibilifies (housework, child care, job, etc.) did you ha\ e in high school? Do you think that these responsibilities enhanced or detracted from your educational experiences? 7. Role models: If you had to name a particular teacher/person who influenced you to pursue mathematics at the college level, who would it be? Why? [e.g. Were they interested in mathematics and tell you about their own story of how they became interested in mathemafics] 8. Before coming to college (and now) how do/did you find out about financial aid, careers involving mathematics, the university system, etc. ? 9. Would you describe your motivafion level in studying mathemafics in college as "high", "moderate", or "below average"? Why? a. Would you describe yourself as intemally mofivated (motivated from within) or externally motivated (mofivated from without)? Why or why not? b. Is your motivation level in college influenced by your instmctors? Why or w hy not? c. At present, do you feel pressure (from society, former teachers, parents or siblings) to do well in mathematics? Why or why not? How does this "pressure" affect your motivation level if at all? 10. Describe your (current) social group at college? (i.e. what are their majors?, do you study together?, etc.) a. If you study with other people, how many of them are from your social group (i.e. friends)? Why are they the same? Or, why are they different? b. What kinds of information do you share (among you and your friends) about navigating the university (i.e. registration, financial aid, what professors to sign up for, what courses to take, emotional support, etc.)? 11. a. How many hours a week did you spend studying calculus? b. How many of these hours were spent studying with other people? c. How did you study when you studied? (read the book, do problems, etc.) What was your studying method? (working relatively easy problems progressing to harder problems, discussing the difficult problems with peers, etc.) d. When working in a group, who do you feel most comfortable with? (i.e. others of your ability level, others from your social background, others from high schools similar to your own, etc.) 12. Describe your experiences in your college calculus course, (i.e. did you like the professor, the textbook, the stmcture of the class, etc.) a. Did you feel like you were able to draw on any of your previous mathematical background? [Did your high school prepare you for college-level mathematics?] b. Did you feel like that you understood some, if any, of the concepts or applications of the course?

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APPENDIX B PARENTS' INTERVIEW QUESTIONS

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1. What factors contributed to your child's success in high school mathematics courses? (i.e. social group, amount of fime studying, parental involvement, etc.) a. Are there factors that had a negafive affect on your child? (i.e. too much time consumed in extracurricular acfivifies, other responsibilifies invading studying time) b. Did you encourage your child to take mathemafics all four years of high school or just the high school requirement (of three years)? Did you encourage your child to take any advance level mathemafics in high school? 2. Would you describe your child's mofivafion level in studying (mathematics) as "high", "moderate", or "below average"? Why? a. Is your child motivated from within or from without? Why? (Have you ever pressured your child to perform well academically? (i.e. punishment for poor performance, reward for good performance, etc.) b. How did you help your child excel in mathemafics? Did you provide your child w ith extra assistance? (provide tutors, talk with teachers, etc.) 3. Do you feel that your child's high school adequately prepares students for college? Did it adequately prepare your child, in particular in mathematics? Why or why not? a. How many and what type of mathematics courses did your child take? b. Are there other ways that have not been mentioned that you think that your high school could better prepare students for college? 4. In what ways do you promote pursuing a higher educafion among your child(ren)? a. Do you have definite ideas about the career path your child should follow? Why or why not? (Do you encourage them to do whatever they desire?) b. Are your ideas about your child's career path the same as your child's goals? c. Would you encourage a degree/career in the mathematical disciplines? Why or why not? 5. Describe your educational background? a. If you had to name factors that prevented or encouraged you to pursue your education what would they be? . Were you encouraged by teachers, friends, or relatives to succeed in school? Were there any factors/people that discouraged your pursuit of education? Did you actively try to circumvent those factors in your child's pursuit of education? b. What were your parents' views about education? 6. Did you like mathemafics when you studied it in high school (or college or both)? a. If not, what factors affected you in your interest in mathematics? b. If so, what was it that you liked about mathemafics?

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APPENDIX C TEACHERS' INTERVIEW QUESTIONS

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1. Describe Student X's performance in your mathemafics course(s)? a. If pooriy, did you encourage Student X to work harder or provide verbal encouragement? b. If good, did you encourage Student X to pursue more mathemafical courses and possibly a mathematical career? 2. Do you feel your mathemafics courses prepare students well for college calculus? Why or why not? a. Do you encourage academic compefifion in your class among your students or academic cooperation? b. Do your tests and homework prepare them for the tests and homework in college mathematics? 3. To the best of your knowledge, what role (if any) did the following have in contribufing to Student X's success in mathemafics: parents?; extra-curricular activities?; social group (friends/peers)?; apfitude for mathematics? a. Did any of these roles contribute to the student's lack of success? b. What was your role in contribufion to Student X's performance in mathematics? (a success or failure?) 4. Name (and rank) three factors that contributed to Student X's performance in your course(s)? (e.g. motivafion level, aptitude for mathematics, study habits, etc.) a. Name the factors that contribute to Student X's success? b. Name the factors that contribute to Student X's lack of success? 5. What qualities do you feel that students must possess in order to pursue a career in a mathematics-based discipline? How do you encourage/model these qualities? a. If a student lacks theses qualities, what advise do you give the student? b. Do you inform the students of the way you became interested in mathematics and what qualities you possess? 6. What suggestions do you usually give your students on how to study for mathematics courses? a. Do you advise them to organize into groups of race, sex, or level of understanding? b. Do you involve the students in group leaming or individual leaming or a combination?

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