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i CLEARING UP CALCULUS WITH ADAPTIVE LEARNING HABITS A Thesis by JENNIFER LYNN PATTERSON Submitted to the Honors College of Texas A&M UniversityCommerce in partial fulfillment of the requirements for the degree of BACHELOR OF SCIENCE December 2016 iii Copyright © 2016 JENNIFER LYNN PATTERSON iv ABSTRACT CLEARING UP CALCULUS WITH ADAPTIVE LEARNING HABITS Jennifer Lynn Patterson, BS Texas A&M UniversityCommerce, 2016 Advisor: Rebecca Dibbs, PhD One of the reasons for the exodus in STEM majors is the introductory calculus curriculum. Although there is evidence that curriculum like CLEAR calculus supports gains in students’ growth mindset, it is unclear how this curriculum promotes mindset changes. The purpose of this study was to investigate which features of CLEAR Calculus students notice and attribute to their success. After administering the Patterns of Adaptive Learning Scale to assess students’ initial mindset in one section of calculus, twelve students were selected for interviews. At the end of the semester, students were readministered the PALS. Students cited that the labs in the CLEAR Calculus curriculum challenge them in ways that facilitates deeper comprehensive learning than that of prior courses. In addition, students also noted that the many reflection components of the course were helpful in retention of the concepts. Successful students demonstrated an increase in adaptive learning habits throughout the semester. v ACKNOWLEDGEMENTS This thesis would not have been possible had it not been for the encouraging mentors and faculty members that pushed me to persevere through the difficult times, as well as celebrated with me through the fun times. To Dr. Raymond Green, for always having an open door and a listening ear for any concerns or questions. To Dr. Melanie Fields, for providing constant encouragement and support. To Mrs. Jennifer Hudson, for leading by example and being not only a mentor, but also a friend, and to Dr. RebeccaAnne Dibbs, for sharing your love of research with me. This experience has not only enhanced my college career, but it has also allowed me opportunities that I would have never otherwise encountered. vi TABLE OF CONTENTS LIST OF TABLES .................................................................................................................... vii LIST OF FIGURES ................................................................................................................... viii 1. INTRODUCTION ...................................................................................................... 1 2. LITERATURE REVIEW ........................................................................................... 2 Obstacles in Calculus .............................................................................................3 The Evolution of CLEAR Calculus: Calculus Reform ................................... 4 Formative Assessment ....................................................................................... 6 Research Stance ................................................................................................. 9 Theoretical Perspective ................................................................................... 12 3. METHODS ................................................................................................................. 15 Setting ................................................................................................................ 15 Data Sources ................................................................................................... 17 Data Handling .................................................................................................. 19 Instrumentation ................................................................................................ 20 Data Analysis .................................................................................................... 20 4. FINDINGS .................................................................................................................. 22 Challenge ........................................................................................................... 22 Instructor Caring .............................................................................................. 24 Reflection ........................................................................................................... 25 Adaptive Learning Habits .................................................................................. 27 vii 5. DISCUSSION ........................................................................................................... 29 REFERENCES .......................................................................................................................... 32 APPENDICES ........................................................................................................................... 35 Appendix A. Syllabus .............................................................................................................. 36 B. Labs .................................................................................................................... 39 C. Patterns of Adaptive Learning Scale survey ........................................................ 42 viii LIST OF TABLES TABLE 1. Interview participants....................................................................................................... 18 2. Standards of Evidence ..................................................................................................... 21 ix LIST OF FIGURES FIGURE 1. Typical CLEAR Calculus lab ............................................................................................ 6 2. Summary of mindset theory ............................................................................................ 15 3. Typical student postlab ................................................................................................... 17 10 INTRODUCTION One of the most pressing problems in higher education is that there are not enough science, technology, engineering or mathematics (STEM) majors in United States universities graduating to meet the demands of the industry. Students leaving STEM degrees after introductory calculus is mostly what causes this lack of STEM graduates (Bressoud, Rasmussen, Carlson, & Mesa, 2014). Because of the lack of efficiency of traditional calculus instruction, students leave their STEM majors to pursue fields of study that they perceive to require less effort in return for an acceptable amount of success (Ellis & Rasmussen, 2014; Oehrtman, 2009). The oldest, most common alternative to traditional calculus curriculum is the Treisman Model. This model suggests that students learn best when calculus is taught similar to a science class: with both a lecture and a lab (Treisman, 1992). However, more recent research suggests that these labs should cover a single theme (for example, limits) and mirror successful students’ actual learning processes (Oehrtman, 2009). It has been noticed that mindsets play a significant role in the overall success of calculus students. Carol Dweck (2006) defines mindset in two different ways: fixed mindset and growth mindset. Students classified under the fixed mindset, if not immediately successful in introductory calculus often leave the STEM field. However, growth mindset students can persist and succeed, even after failures as severe as failing a course (Dibbs, forthcoming). This study will identify students’ mindsets and monitor their success in a CLEAR Calculus classroom. A recent study found that all students who participated in CLEAR Calculus, a nontraditional calculus class utilizing the limit approximation calculus labs described in Oehrtman (2009), became more growth oriented by 1.6 standard deviations; this suggests that the curriculum itself is likely to be responsible for this positive mindset change (Dibbs, 2015). I 11 examined how this particular curriculum supports an increase in adaptive learning habits demonstrated by students of fixed and growth mindsets. This research will be guided by utilizing the research question: What are the features of CLEAR Calculus that promote positive changes in students’ mindsets? This topic is important because if it is known what causes students the most success, traditional calculus instruction methods can be altered to produce more effective calculus students. By understanding what makes CLEAR Calculus effective, it provides a strong argument for using this curriculum. Interested practitioners who are not willing to implement CLEAR Calculus can learn what components to add to their classes in an effort to keep students within the STEM field. In order to investigate this research question, I interviewed students enrolled in a CLEAR Calculus class in order to determine what features of the class promote positive mindset changes. A key component of the study was the participants’ mindsets. To measure this, a commercial survey instrument—the Patterns of Adaptive Learning Scale, called PALS—was used that identified and measured students’ mindsets. I argue that students noticed that the course contains components designed to challenge them at a rate above their average performance rate. In addition, students noticed that the instructor cares at a rate higher than their previous mathematics instructors, that reflection is a primary component of the course, and that they experienced an increase in adaptive learning habits. LITERATURE REVIEW There are several major topics related to the research question of this study: problems with traditional calculus instruction, the evolution of CLEAR Calculus, and formative 12 assessment. After discussing each of these topics in turn, the literature review concludes with my researcher stance. Obstacles in Calculus Of the 600,000 firstyear college students that take Calculus I, 250,000 of them fail in the first semester (Treisman, 1992). Fourteen years later, Ganter (2006) confirmed the calculus failure rate to be 4060%. Treisman (1992) speculated as to why there are such low success rates among introductory calculus courses. As a result of failing introductory calculus, students pursuing a STEM degree often leave the field citing the lectures were traditional and uninspiring, encouraging memorization instead of comprehensive application and understanding (Seymour, & Hewitt, 1997). However, an alternative instruction method has the potential for deeper comprehension and better appreciation of the subject (Treisman, 1992). By engaging the class in a labbased structure, encouraging group work and wholeclass discussions, students are more likely to retain information (Ellis & Rasmussen, 2014). Formative assessment can be utilized in the form of the CLEAR Calculus curriculum to ensure this retention among students (Oehrtman, 2009). Hutcheson, Pampka, and Williams (2011) found that students use the instruction experience of their first year of mathematics courses to decide to continue or discontinue their pursuit of a STEM degree. This is why a strong curriculum that fosters comprehension and understanding is vital. Many US universities utilize first year courses in STEM fields to discourage an overpopulation of students pursuing these degrees (Steen, 1988; Wake, 2011). It is important that students are successful in completing a STEM degree for many reasons. According to the US Presidents Council of Advisors on Science and Technology (PCAST, 2012), economic growth 13 can be traced to an increased number of students entering the STEM field. Without US STEM graduates, the job market—and by extension, the economy—will suffer greatly. The Evolution of CLEAR Calculus: Calculus Reform The major area of research in mathematics education in the 1990’s was the problem with introductory college calculus reform (Oehrtman, 2008). The most successful calculus reform to emerge from this reform movement was the Treisman Model of instruction (Ganter, 2006). The Treisman model emerged to address several problems in a traditional calculus course; the main problem being that students were more interested is passing calculus than they were in understanding the mathematics taught (Treisman, 1992). Although professors believed that the high failure rate was due to students’ low income, low motivation, poor academic preparation and lack of family support, further investigation revealed that that this was not the case (Treisman, 1992). Longitudinal studies at the University of CaliforniaBerkley revealed that students who were successful in calculus were so because they learned how to work and study in groups effectively, while students that failed were found to have worked alone, eventually becoming frustrated, and then quit trying (Treisman, 1992). The Treisman classroom model is designed to train all students to become effective group members in mathematics. The current iteration of the Treisman model used today is called CLEAR Calculus, which is a refinement of the original Treisman model based on the approximation framework (Oehrtman, 2008). According to Treisman, calculus instruction today does not teach problem solving in a way that promotes the development of generalized problem solving skills. Instead, departments are teaching students how to excel in routine computations rather than how to comprehend the subject matter (Treisman, 1992). The needs of the students 14 were not being met, and Treisman was determined to find a way to facilitate calculus comprehension rather than calculus regurgitation. However, even when students are taught with the Treisman model, many students have a great difficulty in articulating mathematical concepts (Oehrtman, 2008). Oehrtman believed mathematical careers should require much of students in the area of proof engagement and argumentation (Oehrtman, 2008). Students should be able to comprehensively defend mathematical theories by the end of their schooling. Traditional calculus instruction provides most students with a very superficial understanding of mathematical procedures, whereas it may equip only a few for further studies (Oehrtman, 2008). Upon entering Calculus I, students have no conceptual structure to comprehend the most basic components of calculus due to their lack of prior exposure to the content. Oehrtman (2008) suggests a solution that would aid students with their comprehension: engagement in multiple activities that reveal and encourage the abstraction of a common structure. By presenting all calculus concepts through a similar framework, students develop a more organized understanding of limits than they do in a traditional calculus course (Oehrtman, 2009). CLEAR Calculus achieves this through formative assessment (pre and post labs). In a typical lab, students are asked to approximate the water pressure of a dam (Figure 1). The first row would represent the prelab. Students are required to find the approximating by drawing a picture, graphing the data, finding the value algebraically, and again numerically (Oehrtman, 2008). 15 Figure 1. Typical CLEAR Calculus lab Formative Assessment During the CLEAR Calculus labs—such as the one described in the previous section—the instructor walks about the classroom conducting informal formative assessment, guiding the students through the labs; the pre and post lab assignments are considered written formative assessment. Although prior research does not indicate formative assessment is the cause of students’ mindset changes (Dibbs, 2015), research on noncognitive factors involved with students’ learning indicate that the use of formative assessment in the classrooms help students perceive their teacher as caring about them (Dibbs, 2014) and encourages them to take ownership of their own learning (Black & Wiliam, 2009). This makes formative assessment an important structural component of CLEAR Calculus. 16 Black & Wiliam (1998) define formative assessment as teaching to meet students’ academic needs (Black & Wiliam, 1998). After research and testing in the United Kingdom, Black & Wiliam (1998) have proven how formative assessment is essential in the raising of achievement standards in schools. Teachers play a large role in a students’ success. It is very important for teachers to be aware of their students’ strengths and weaknesses in learning, so that they can teach on the appropriate level (Black & Wiliam, 1998). The key component to formative assessment is the adaption of teaching to meet student needs (Black & Wiliam, 1998). By building on the successes of traditional instruction and reforming the less profitable practices, Black & Wiliam (1998) have created more efficient teaching methods through formative assessment in K6 British children. Formative assessment has been proven to assist low achievers more so than high achievers. This has aided in the reduction of the range of achievement with an overall higher standard. When teachers become aware of their pupils’ needs and begin to address those needs, a positive change has occurred in the classroom. Unfortunately, many teachers are out of touch with the amount of understanding the students are experiencing, therefore their teaching practices are inefficient (Black & Wiliam 1998; 2009). The way students’ perceive themselves is very important. If a student identifies his or herself as unable to learn, there begins to be a decrease in the seriousness they take in school (Black & Wiliam, 1998). This is more likely to happen to a student of the fixed mindset, because when confronted with a failure, students with fixed mindsets tend to retreat from the problem. Fixed mindset students with high confidence in the subject matter often have high persistence and seek challenges. However, if a fixed mindset student has low confidence in his or her ability to succeed, they may display a low persistence. Often, students that receive low marks will 17 continue the same behaviors, consistently receiving low scores. Many students are simply content to put forth the minimum effort that will be accepted. Students do so by avoiding difficult tasks – looking for clues and patterns to find the right answer without actually learning the material. Students of the fixed mindset may find Calculus I difficult if they do not initially understand it. The fixed mindset person believes that smartness cannot be obtained—it cannot be gained or lost; one either has the capacity (or potential?) to master a topic or not (Dibbs, 2012). Calculus I curriculum may be difficult to that of a fixed mindset because the student will not have entered the class with prior knowledge of the subject; he or she will be required to learn the new material. It is common for fixed mindset students—who have been successful in their school career—to drop Calculus I after the first test if they have been unsuccessful in the class. This is a pivotal point in the fixed mindset students’ academic career; many students change their major from that of a STEM field following the failure or drop of their introductory calculus course because they believe they are not mentally equipped to succeed. Growth mindset students often do poorly in the beginning of the semester, but what sets these students apart is the changes they make in their studying/notetaking after failure. The expectation is that these students will find more efficient ways to approach the class that will ensure learning and ultimate success. Black & Wiliam (1998) believe that the classroom is lacking a culture of success; simply assigning students grades is overemphasized, whereas advising students is underemphasized. When grades are assigned, this quantitative data can be used to accurately gauge the placement and success of a student in a class, but more is to be gained when academic advising from a teacher takes place (Black & Wiliam, 1998). Unfortunately, standardized testing in US high 18 schools often dominates teaching and assessment. Black & Wiliam (1998) states that testing does not provide teachers with an adequate model for formative assessment. Feedback improves learning when it shows the students their strengths and weaknesses without a number grade attached to it (Black & Wiliam, 1998). If pupils are to benefit from formative assessment, they should be trained in selfassessment so that they might comprehend what is needed to achieve success (Black & Wiliam, 1998). This can be facilitated by discussions in which students are to articulate their own understanding of the concept. This is beneficial to not only the student sharing, but also the students struggling with similar concepts. This kind of alteration of the original lesson plan is an investment of sorts. Teachers must invest time in their students to create a culture of questioning and deep thinking (Black & Wiliam, 1998). Black & Wiliam were quite thorough in their research; however their study focused on British children grades K6. Dibbs (2014) examined Black & Wiliam’s theoretical framework to see if it crossed age and cultural barriers. It did, but there were inconsistencies with postlab results. Though it appeared as if the inconsistencies were related to mindsets, it was found that changes in mindsets are more likely fostered by CLEAR calculus instruction rather than postlabs (Dibbs, 2015). This thesis project connected the previous work on mindsets and calculus by generating hypotheses about how CLEAR Calculus helps students make positive mindset changes. Research Stance When qualitative research is done, it is vital to disclose the researcher’s experience and expectations in the area of research so the reader understands where the writer stands on the topic. 19 I studied Calculus I, II, and III in a traditional classroom setting. In addition to attending class, I was also involved with the Texas A&M UniversityCommerce Calculus Bowl Team. This experience allowed me to have a larger exposure to calculus than the average calculus student. Regarding the pre and post labs, I have no personal experience. The only exposure I have had with formative assessment is casual conversation with classmates currently taking a nontraditional calculus course. In Calculus I, I struggled with the material. I did not immediately grasp the concepts presented; this was a novel and unpleasant experience. When I realized that I did not know the material as well as I would have liked, I spent hours studying each chapter in preparation for the assignments and tests. I received a B in the course, but I earned that B. I had to use the growth mindset to find more effective ways to study efficiently. My inner fixed mindset was saying, “If you can’t get it the first time, you’re not smart enough. Just give up.” Calculus II was a similar experience to Calculus I, except the material was more difficult, requiring more hours of studying for comprehension. In order to gain more experience and exposure to the material, I joined the Calculus Bowl team and Math Club to strengthen my calculus skills by surrounding myself with math intellectuals that shared similar misunderstandings of the material. This consisted of weekly evening practices that lasted for two hours. During these practices, we were presented with up to fifty calculus problems with content taken from Calculus I and Calculus II classes. These problems were to be solved in 60 seconds or less; precision and speed were a must. By participating in the Calculus Bowl, I increased my exposure to regular classroom material. During this study, I expected to encounter students with varying backgrounds in Calculus. College Algebra and/or PreCalculus classes are prerequisites to introductory Calculus 20 at this institution. Students were expected to be of the semifixed mindset immediately following Calculus prerequisite courses, because these courses require significantly less floundering to demand a change in mindset to earn a satisfactory grade. Upon entering introductory calculus, students were expected to have little to no understanding of limits, and struggle with complete comprehension of topics. In addition, it was expected that post College Algebra/PreCalculus students would struggle with trigonometric functions. I used the knowledge I gained from my own calculus experiences to provide me with an understanding of potential biases as I assessed students that participated in this study. Understanding my stance about the research allowed me to design a study that would provide a check on these potential biases. In the next section, I will explain the learning theory I used to structure student responses. Theoretical Perspective Before qualitative research is conducted, one must define the major constructs in the research question as well as define what learning means within the context of the research. After discussing which learning theory I used in this study, the remainder of this section explores the concept of mindsets in greater detail. Epistemology In order to frame participant’s responses, it was important to choose a learning theory. The theory chosen was Vygotsky on social constructivism. Vygotsky’s research, interpreted by Gredler and Shields (2008) found that all knowledge is constructed in a social setting. This theory claims two major components (Gredler & Shields, 2008). The first component is informal knowledge – knowledge gained implicitly by observation (i.e. monkey see: monkey do), also known as spontaneous knowledge (Gredler & Shields, 2008). One might not be able to articulate 21 why he or she knows a concept learned through spontaneous knowledge, however they are able to put it into practice. An example of this is the term “brother.” To a small child, he or she may be able to tell you if a person is their “brother” or not, but they would not be capable of defining what a “brother” is. The second component to Vygotsky’s theory is formal knowledge (Gredler & Shields, 2008). This is knowledge explicitly taught (i.e. calculus content). This component, also referred to as scientific knowledge, takes form as factual knowledge that one may be able to implement, but may not understand the reasoning behind it (Gredler & Shields, 2008). For example, Algebra I students are taught the mathematical mnemonic of FOIL. The students may be able to walk themselves through each step of the process, but they have no understanding of what they are doing, why it works, or whether it is relevant or not. Informal knowledge is not inferior to formal knowledge, but rather it is learned differently. Ideally, one must have both. This theory fits my study because I was looking for students’ spontaneous knowledge about their mindsets in the context of math classes with lots of scientific knowledge; Vygotsky captures both of these concepts. Theoretical Perspective Carol Dweck’s Mindsets (2006) showed that students’ learning patterns fall under two very different categories: the fixed mindset and the growth mindset. Dweck found in her research that a student’s mindset has a very large influence on the students’ success in a difficult class, such as Calculus I (Dweck, 2006). This section will explore the differences in two theories of intelligence: fixed and growth mindsets. The growth mindset primarily focuses on recovering from and learning from one’s failures for future improvement. However, the fixed mindset places much value on immediate 22 success; if a fixed mindset student believes he or she has a significant chance of failure, there will often not be an attempt of the task, assignment, class, etc. In her research, Dweck (2006) observed that the way a student perceives his or herself strongly affects the way they leads their life. Dweck (2006) discovered that the students that appear to be the most intelligent at the beginning of the semester might not always end the semester as the smartest. In a Calculus classroom, fixed mindset students may begin the semester as the smartest; however, when the coursework becomes difficult or if they find themselves failing – they give up. That’s when the growth mindset students seek ways to improve their methods, learn from their failures, and begin excelling in the class. The growth mindset students often end the semester with the label “smartest” because of their persistence (Dweck, 2006). Motivation is the overarching difference in these mindsets – what drives a student is how they’re going to react to failure. For instance, consider a class of 7th grade students is given a math exam where the overwhelming majority of the class fails. The fixed mindset students prepare for the next test by studying less – they could not do well on the first test, so they see no reason to try on the second. These students are under the fixed mindset belief that “smarts” are not to be gained; you either have the ability or you do not. However, the growth mindset students make a greater effort to do well on the next test by studying harder and more efficiently. When the second test is administered, the growth mindset students that spent extra time preparing for the second test excel well beyond those of the fixed mindset students that refused to study (Dweck, 2006). The fixed mindset student feels the constant need to be superior – which means failure is not an option. When they feel as if they are superior to others, they feel smart. Their view is that 23 “effort can reduce you” (Dweck, 2006) – why attempt something if failure is an option? They believe that only “dumb kids work hard in school. Is it better to fail on purpose than to try and show that you are dumb?” Fixed mindset students are prone to attempt only tasks that they are apt to succeed in; this allows for very high achievers. Many Honors College, Top 10% overachievers, and first semester STEM majors often have the fixed mindset. However, upon taking Calculus I, they do not experience immediate success. This unexpected difficulty (or failure) often discourages the fixed mindset students from continuing in their chosen STEM field (Dibbs, 2014; Dweck, 2006). In a Chemistry class, fixed mindset students were found to stay interested when they did well right away. However, if they were not successful from the start, they grew disinterested and distant (Dweck, 2006). These students were only engaged when they believed they would succeed. Dweck discovered that fixed mindset students have very low calibration, meaning they are often inaccurate when judging their own abilities (Dweck, 2006). Dweck defines the growth mindset as someone that can grow and change through application and experience. Students with the growth mindset view failure as a gift, an opportunity to get better. If not immediately successful, they don’t believe they are failing; they believe they are learning. Each student’s definition of failure plays a vital role in whether or not the student has a fixed or growth mindset. Dweck (2006) describes the growth mindset as unique students converting obstacles in life into future successes. Differing from fixed students, growth mindset students often have very high calibration; they are more realistic in selfassessment (Dweck, 2006). This is often helpful in the classroom as they assess the amount of learning that has taken place. 24 In summary, students’ mindsets play a significant role in the outcome of their overall classroom experience (Figure 2). Over time, students have been observed changing their mindset due to structure changes in the classroom (Dibbs, 2014). Formative assessment has been a crucial tool in facilitating a classroom environment conducive to positive mindset changes. Figure 2. Summary of Mindset theory METHODS The following describes the methods I used to find an answer to the research question “What are the features of CLEAR Calculus that promote positive changes in students’ mindsets?” I will identify the setting the study took place, as well as discuss the data sources, handling, and analysis. Setting This study took place at a midsized rural master’s comprehensive regional university in the South with approximately 11,000 students. There are 40.6% of the students at this university who are nonwhite, and 60.3% are female. The students who participated in this study were 25 enrolled in introductory calculus in the Fall 2015 semester. The students enrolled in introductory calculus are most often mathematics, physics, and engineering majors. Each week in a CLEAR Calculus course is comprised of the same schedule for each day: on Monday, students were given new material and a prelab to begin working on for the lab completed in class on Tuesday. The prelab, due before class on Tuesday graded on completion, required students to complete the Unknown Value of the approximation framework (Figure 1, found in the literature review) and to identify a quantity that could be used to approximate the unknown value. Lab days were comprised of group work in groups of 34 students. After the lab, students completed a postlab answering the following questions, graded on completion: 1. What did your group work on during the lab today? 2. How comfortable are you with the material in today’s lab? 3. What is the part of today’s lab you feel most comfortable with? 4. What part of the lab is most difficult so far? 26 A typical response on a postlab can be found in Figure 4. Figure 3. Typical student postlab The purpose of the postlabs is to encourage discussions in class on Wednesday, as well as to assist the instructor when prioritizing which material to review. The remainder of the week is spent on new material. For a more detailed layout of the course’s weektoweek activities, see Appendix A. The context of each approximation lab can be found in Appendix B. Data Sources I selected an introductory calculus course of primarily firstyear STEM majors to conduct my study. During the first three weeks of the semester, I obtained research consents 27 from the participants and administered a PALS study to assess initial student mindsets. Students also completed the PALS at the end of the semester. Upon reviewing the PALS results, I selected twelve students with trends in their mindset score to obtain a sample of maximum variation (Patton, 2002). The chosen students received a letter asking for their participation in a brief interview. I interviewed until data saturation was reached (Patton, 2002), and then selected the most interesting cases for analysis in my thesis. This particular class was comprised of mostly engineering students with few other majors such as computer science, mathematics, education, and physics. The participants’ classification ranged from freshman through graduate level students. Below is a table describing the participants, classification, major, mindset, and final grade in the course. Table 1 Interview participants Pseudonym Year Major Mindset Final Grade Kyler Freshman Engineering Strong Fixed A Taylor Sophomore Engineering Strong Fixed B Cameron Freshman Computer Science Strong Fixed A Michael Junior Engineering Strong Fixed B Lana Sophomore Mathematics Mixed Fixed A Chase Freshman Engineering Mixed Fixed B Sarah Sophomore Middle School Math Mixed Growth C Ricky Freshman Computer science Strong Growth A Jared Freshman Engineering Strong Growth B Lacey Freshman Physics/Math Strong Growth A Adam Graduate Physics Strong Growth A 28 Upon their consent, I conducted semistructured interviews to gather qualitative data regarding the effects of formative assessment in student learning and mindset. The following is an example of some sample interview questions used to conduct the first interviews: 1. What’s your major? 2. Why are you going into a mathbased field? 3. Tell me what you’ve been like in previous math classes 4. Tell me what this class is like. 5. If I had never seen one of Dr. Dibbs’ labs, tell me about them. What are they like? What do you do in them? 6. Why do you think you do labs? 7. Have labs changed you? How? 8. What do you do for calculus when you’re not in class? 9. How do you approximate a hole in the graph? 10. What’s a limit? 11. What will you remember about this class? 12. What should I have asked you but didn’t? 13. Do you think your experience in this class will make you approach your next class differently? The following are questions that were used to conduct the second interviews: 1. What have you learned in the class? 2. What are the features of the class that stood out to you? 3. Have you changed as a student as a result of this class? 4. A lot of people mentioned instructor caring? 5. What made you decide caring was a thing? 6. Would you recommend this class to another student? Data Handling Students were assigned ID numbers and pseudonyms at the beginning of the study to ensure anonymity. These pseudonyms are the way that the participants have been and will continue to be referred in the research. The key containing the students’ ID and pseudonyms is kept in Dr. Dibbs’ locked office to ensure security and confidentiality. Once ID numbers and pseudonyms were issued, students’ real names were not looked at again, and the key was discarded at the end of data collection. Upon being selected to participate in the interview process of this study, students were invited to participate via invitation letter. 29 Instrumentation The survey used to measure student mindset scores was the Patterns of Adaptive Learning Scale. This survey was developed for kindergarten through 5th grade students (Midgley, C., Maehr, M. L., Hruda, L. Z., Anderman, E., Anderman, L., Freeman, K. E., & Urdan, T., 2000). The PALS survey administered to students in this study was modified from the original survey to exclude the sections questioning students about their parents. The PALS test given to students had a Cronbach Alpha of .87, which is acceptable for an educational instrument (Gall, Gall & Borg, 1996). Appendix C contains a copy of the PALS survey used in this study. Data Analysis Data analysis for the thesis was conducted in two phases. First, I conducted a pilot study that allowed me to develop an analysis scheme for the main project. The data for the pilot was collected and analyzed in Spring 2015. The coding scheme that emerged from the pilot data is used appears in a standards of evidence table below (Table 2). This standards of evidence table was created from a topdown coding (Patton, 2002) of the interviews from the students participating in the pilot study. 30 Table 2 Standards of Evidence Code Definition Standards of Evidence Example Challenge is a feature Student’s perception of the class as having difficult problems as the core of the course AND seeing this as a positive thing Direct statements made by students that (challenging/difficult/real life) problems are important in the class Direct statements made by students indicating this is a good thing Tone of students indicated bullet point #2 “This class is challenging, but that is the point” Things that are challenging What parts of the class students find to take the most time outside of class/require the most work Direct statements/complaints about the things that are most difficult “Labs...they’ve been the most difficult challenges so far. Even more difficult than the test.” How labs changed me Student’s perceptions of the work habits and beliefs that participating in this curriculum changed for this semester &/or in the future Direct changes made by students about homework/study habits/beliefs that changed as the result of the class Direct statements about any beliefs that changed as a result of this class Evaluative statements about whether this belief will persist to the next class “I started studying more. Instead of studying the night before, I would study a few days before the test – it helps.” “Taught me to think through the problems more carefully...to figure out what’s wanted.” “I’ll think more in depth about...[future math classes].” Features of CLEAR calculus students notice Components of the labs and lectures students recognize as different from other math classes Direct statements made by students about the goals and effects of CLEAR Calculus “...so that I can understand the concepts better. So that I can visualize the concepts better.” “So that we can get almost real world experience so we can retain what we learn.” “To see how math is used in real world situations.” 31 In order to analyze the quantitative data, the following steps where taken. The PALS survey scores were entered into an excel spreadsheet. The numbers 15 were used to code the responses. When necessary, reverse coding was used so that 1 was always the most heavily fixed score. Participants that did not participate in both surveys (taken at the beginning and end of the semester) were eliminated. Because the n was small (n=12), a MannWhitney test was used instead of the ttest. Trustworthiness In an effort to protect against bias and ensure research was conducted rather than expectations being confirmed, it is important to have checks and balances in place for this thesis project. Dr. Dibbs and myself performed the data analysis independent of each other using an agreed upon chart. We reconciled our codes to verify consistency. FINDINGS There were four themes that emerged from the data that students noticed and valued as part of their introductory calculus experience: challenge, instructor caring, reflection, and adaptive learning habits. Students noticed that they were persistently challenged throughout this course and that the instructor demonstrated a heightened care for their success in the class. They also noted that the many reflection components of the course were very strong indicators of their overall success in the class. Students showed a very rapid increase in adaptive learning habits throughout the semester. Challenge A common term that is used during the participant interviews is “challenge.” Students identify this course as being “…just different,” in level of difficulty and structure. In his second interview, Taylor identified this course as being “not necessarily easy, just different from other 32 classes” in the level of difficulty and structure. One way that the CLEAR Calculus classroom varies from that of a traditional classroom is the weekly lab activity. The structure of the class is as follows: there are 3 instructorled lecture days, 1 day of lab group work, and 1 day dedicated to weekly review. Students work in groups to complete the weekly lab assignment on the second day of each week. These lab days are what students are referring to as being the most “challenging” portion of this course. Lana said in her first interview, “It was harder than I was expecting. It was definitely challenging. I liked and didn’t like the challenge because I’ve always been good at math, so struggling with it was difficult for me.” Lana had a fixed mindset and severely struggled in this class. There were times in the first 3 weeks of class that she refused to complete her work. She would cease to participate when she was not making progress in the lab. On more than one occasion, she abandoned her group mates to finish the lab without her. Midway through the course, Lana found that if she altered her habits outside of class, she was more successful and better understood the material. She states in her second interview: …a lot of the labs were really, really, really hard for me. Although it did help, I didn’t like them that much. But in the long run they really did help because it gives you more opportunities to see how the work can go in different ways from this class. Lana changed her success in the classroom by spending more time in the Math Lab on campus, in addition to attending office hours with her instructor. The extra time spent outside of class better prepared her to be more successful on lab day. Due to this change in her outside of class habits, Lana found herself to be a productive group mate. Several students mentioned their struggle during the labs has better prepared them to be successful on tests. Ricky mentioned in his second interview: The labs are interesting. They give me hard questions that are like—in other classes they give you super easy questions [before the test] but then on the test 33 they give really hard questions. Not here, she gives you really hard questions throughout the class so that the questions on the tests are easy. I like that. Kyler agreed with this thoughtprocess in his first interview by saying that he “believes that the interactive nature of the labs is more conducive to learning than the lecture time of the class.” In other words, students credit the difficulty of labs to their overall success in the class. Chase said in his second interview: After getting used to the level of difficulty of the labs, I got a lot better at the math. I just had to get used to how hard they were. They were really hard and I didn't like them. But they got easier as we went. I got better at them the farther we went in the semester. A trend immerged from students’ first and second interviews. During all first interviews, students seemed to fall under two categories: incredibly frustrated at the difficulty of lab work, or frustrated but understanding of the potential positive outcomes of the challenging work. In all postinterviews, students unanimously decided that labs were indeed at a heightened level of difficulty as previous math classes, but they understood and even advocated for the challenge. Each student was able to state why he or she felt labs were a vital component of their CLEAR Calculus class. Instructor Caring In this study, 5 students cited that they noticed the instructor cares more than previous instructors. Adam states in his second interview the ways in which he feels the instructor demonstrates care for the students: “She is caring…retention of who we are, characteristics about us—she remembers them, availability, she will talk you through a problem in a good way…an educational way.” It could be asked if the nature of this course and its features demonstrate increased interest in student success by the instructor than that of a traditionally taught calculus class, giving students a feeling that they are cared for. 34 In his second interview, when Taylor was asked about the ways in which the instructor demonstrates she cares for the students, he cited “she’s always available to help, she’s not distant…if you need anything she’ll put you on the right track to make sure you know how to do it.” He compares this instructor to previous instructors in her ability to follow through with her assistance. It is undetermined whether or not the instructor caring feature of this class is due to the CLEAR Calculus curriculum, or simply this particular instructor. Lana also agreed that the instructor demonstrated that she cared for the class. In her first interview, Lana said that the instructor was “open to answer any questions and explain. She would also check for understanding after she explained something.” It’s clear that the students felt that they were taken care of and had adequate amounts of help. When asked about why students may feel as if the instructor has a heightened sense of caring in this CLEAR Calculus class, the instructor of this class responded with: …This class has a lot more assignments than traditional calculus. So there is more paper shuffling. So I learn their names in the same number of repetitions as everyone else, but I hit the magic number soon, creating the impression of increased instructor caring. Also the increased numbers of assignments are perceived as hard, but not pointless busywork. A large number of assignments students thought were stupid probably wouldn’t help. It can be concluded that the setup of this class lends itself to demonstrate to students that the instructor is more intune with student performance than perhaps a traditional class that requires less paper traffic between student and instructor. The open door policy and availableness, however, is a more individual practice of any given professor. This particular instructor showed students that she cared by providing an increased amount of availability. Reflection In 12 interviews, 7 people attributed the reflection components of the class to be a primary reason for their success. The design of this class creates numerous opportunities for 35 students to reflect on their own work. After each lab, students complete a postlab where they list the components of the lab they felt they excelled in, as well as the parts they feel they struggled with the most. This formative assessment provides the instructor with information about how each student in the class is fairing within the unit. In his first interview, Cameron was very vocal about his thoughts on the reflection components of this class. He had even more to say during his second interview: This class has a lot of words. I have to write paragraphs...the reflection after we do the math, where we have to write about what we did. Used to be, I just had to do the problem and not know why it works. But now I have to do the problem and explain why it works. It’s difficult, but it helps. Cameron was emphatic about how the reflections he completed in this class enhanced his learning in an exponential way compared to previous class experiences. He attributed the reflection to his overall success in the class. The most prominent reflection components of this class include the postlab and test corrections. In a CLEAR Calculus class, students are permitted to correct their test to earn back up to 50% of the points they missed, but only if the corrections are wholly correct and accompanied by a sentence per question describing the error that the student originally made and what the student did to fix the error. In his second interview, Taylor speaks very candidly about how he feels about having a second try at test problems: I'm either on or I'm completely off. I'm either getting everything I should be getting or stuff I should be getting, I'm falling apart on the test. The testcorrections or whatever, I open them up and like—for related rates I opened the page and was like, "Gosh dangit, that was the freaking answer." I knew exactly what I did wrong. I think test corrections do a lot to help you understand how to do that so you don't miss it again. That's helped me with a lot of things like related rates and optimization. It's helped me understand what I did wrong. 36 Taylor continued to elaborate on how he puts forth extra effort to insure that his testcorrections are wholly correct before he submits them for grading. Because of the reflection component of test corrections, students are being cautious in their resubmissions, and it's having a very positive effect on their understanding. Sarah explained in her first interview that she is a poor test taker and enjoys test corrections because she strengthens her test taking ability as well as has an opportunity to see where she went wrong and how she needs to fix it. The instructor reviewed weekly postlab reflections after each lab day. Upon review, the instructor altered the lecture content for the remainder of the week to ensure that the topics mentioned in the “Understood least” section of the postlab would be covered in depth. This way, the content students struggle with the most was covered again; whereas class time was not spent on the content students struggled with the least. Students notice change in the weekly topics based on the items they write in their reflection pieces: In his first interview, Kyler noticed this and said that it has shown him that he can get help now. Students find that when they articulate their work, they are more effective in conceptualizing the content. Adaptive Learning Habits Students’ mindsets can be determined by administering a Patterns of Adaptive Learning Scale (PALS) survey. Mindsets were measured in the first 3 weeks and last 3 weeks of the semester in order to determine any change in mindset throughout the course by utilizing a Likert scale survey. There was no statistically significant change in median mindset (p=0.18). Although few students showed any increase or decrease in score by the end of the semester, students’ adaptive learning habits changed drastically, implying that students showed initiative in their own learning experience. 37 Students found themselves demonstrating traits that were uncharacteristic of their normal learning habits. For example, Taylor said in his second interview that he found he was “more successful when he asked more questions in class, worked problems repeatedly, sought help in the math lab, etc.” These habits are demonstrated in the growth mindset student, however the fixed mindset students were demonstrating these habits without a mindset score change. This implied that the course encouraged the development of adaptive learning traits. Lana was a very fixed mindset student. In her first interview, she stated that she had “never been pushed so hard in a math class before.” In high school, she was accustomed to little challenge and found great success in her math classes. However, in the first few weeks of this introductory calculus course she experienced a culture shock. She was no longer being successful with minimum effort. During a very trying semester, she found that in order to be successful in Calculus, she had to spend time preparing and studying outside of class. These newly adopted adaptive learning habits steered Lana into being a very successful student. She showed much improvement in her grade in the course, as well. Students stated in their interviews the many ways that they have shown growth or improvement in their adaptive learning habits. Chase said he “doesn’t wait until the last minute to start the homework anymore, and gets help whenever needed.” Ricky said that this class has taught him “better study habits.” Jayce said that because of the difficulty of the course, he has to “stay on top of his workload and his time management.” Adam, a nontraditional student, said, “It’s all about retention. There is absolutely no substitute for number crunching. I’m still trying to learn how to learn. I haven’t found a successful form of retention aside from working lots of problems.” These students have had to take ownership of their own learning in this class. Due to 38 the rigor of the course, many were challenged in a way they had not been before. This caused them to adapt to the difficulty by changing their learning habits. There was a positive shift of adaptive learning habits seen in students, however there was not a significant shift in the PALS scores. Students took the PALS survey at the beginning and end of the semester to measure any growth or decay in their mindset score. However, the only shift in the scores was within the margin of error. It can be concluded that the students demonstrated a shift in habits but not necessarily mindset. The findings were helpful in answering the research question: What are the features of CLEAR Calculus that promote positive changes in students’ mindsets? Although students did not show a significant numerical increase in mindset score, their actions and classroom behaviors were very clearly affected in this class. Students increased their adaptive learning habits dramatically, demonstrating a change in mindset. DISCUSSION It can be determined that students that are willing to apply themselves during the lab work, even though there is a larger degree of challenge than in comparison to previous math classes, are shown to experience more chronic success than those that are not challenged at a rate higher than their average performance rate. The students that had a high completion rate of the reflection components ended the semester with a more profound understanding of the mathematics. This is due to the fact that the students were required to articulate their understanding and comprehension of the concepts, resulting in a deeper understanding of the content. Students demonstrated a dramatic increase in adaptive learning habits, confirming the hypothesis that the CLEAR Calculus curriculum fosters a growth mindset in students. The 39 features of this class that resulted in a mindset increase were framed around the sense of challenge and rigor. Although some students did not respond positively to the increased challenge, the students that persevered through the difficult units saw great amounts of success in the class and an increase in adaptive learning habits. The increased adaptive learning habits should have been exhibited as an increase in the growth mindset score on the postPALS survey. However, this did not reflect on the postPALS survey administered in the last quarter of the semester. The lack of increase in mindset score was very surprising. It is believed that the increase in adaptive learning habits accounts for an increased mindset score, even though it was not reflected on the survey. Students demonstrated growth tendencies without actually experiencing a mindset change. The large amount of students that cited instructor caring to be a feature of the CLEAR Calculus class, a feature that is not found in other math classes, helped to discount the statistic that students often leave the STEM field due to the feeling that the instructor does not care and that the students feel they cannot get help (Ellis & Rasmussen, 2014). When asked, all students stated that they feel as if the instructor is very approachable, cares about the students’ success in the class, and is willing to help students. It could be speculated whether or not the feature of instructor caring is truly a feature of the CLEAR Calculus curriculum, or simply a feature of the particular instructor that taught this CLEAR Calculus class. These results suggest that the CLEAR Calculus curriculum creates an environment that fosters the growth mindset. Successful students in this class took ownership of their own learning in ways not found in a traditionally taught calculus class. Students were not merely given an answer, but rather they were required to search for and defend the answer they decided upon. The group work allowed students the opportunity to discuss and debate the inner workings of the 40 each calculus lab in ways that enhanced the learning experience for the students involved. It became clear that students that are given more opportunities to do exploratory learning through lab work and frequent reflections over such work are much more equipped to retain the calculus past a test, and quite possibly past a semester. A limitation of this study included the longterm retention of the material in students. If possible, it would have been rather interesting to quiz students based on knowledge and information they were taught in this untraditional Calculus I course a semester, or even a year after the class in order to determine the amount of information they truly retained. A student of a traditional calculus class could also be included in order to determine the percentage of retention of students in each class. It is vitally important for successful calculus programs to be found and promoted in order to strengthen the calculus foundation of STEM students in universities across the country. The ultimate goal is to retain students in the STEM field throughout the entirety of their degree. This starts with a thorough Calculus I foundation. By completing this thesis, I have learned plenty not only as a researcher, but also as an educator. The trends that I have observed will assist me in observing students and their motivation in the classroom. By knowing if a student is a fixed or growth minded student, I can do a more efficient job assisting them in areas that they may struggle. 41 References Black, P., & Wiliam, D. (1998). Inside the black box: Raising standards through classroom assessment. Phi Delta Kappa, October 1998, 139148. Black, P., & Wiliam, D. (2009). Developing the theory of formative assessment. Educational Assessment, Evaluation, and Accountability, 21(1), 531. Bressoud, D. M., Carlson, M. P., Mesa, V., & Rasmussen, C. (2013). The calculus student: insights from the Mathematical Association of America national study. International Journal of Mathematical Education in Science and Technology, 44(5), 685698. Dibbs, R. A. (2013). Students’ Perceived Utility of Precision Taught Calculus. The Qualitative Report, 18(51), 115. Dibbs, R. A., & Oehrtman, M. C. (2014) Formative assessment and students’ zone of proximal development in introductory calculus. Unpublished Doctoral Dissertation, University of Northern Colorado. Dibbs, R. A. (2015, January). How does this help me?" Modeling growth in introductory calculus by using participation in formative assessment. Paper presented at the MAAAMS Joint Meetings in San Antonio, TX. Dibbs, R. A. (forthcoming). Scarred by failure? Necessary and sufficient conditions for student success in a repeated course. Accepted pending revisions at Educational Studies in Mathematics. Dweck, C. (2006). Mindset: The new psychology of success. New York: Random House. Ellis, J., & Rasmussen, C. (2014). Student perceptions of pedagogy and associated persistence in calculus. ZDM, 46(4), 661673. 42 Gall, M. D., Borg, W. R., & Gall, J. P. (1996). Educational research: An introduction. Longman Publishing. Ganter, S. L. (2006). Calculus and introductory college mathematics: Current trends and future directions. MAA NOTES, 69, 46. Gredler, M. E., & Shields, C. C. (2008). Vygotsky's legacy: A foundation for research and practice. Guilford Press. Midgley, C., Maehr, M. L., Hruda, L. Z., Anderman, E., Anderman, L., Freeman, K. E., & Urdan, T. (2000). Manual for the patterns of adaptive learning scales. Ann Arbor, 1001, 481091259. Oehrtman, M. (2008). Layers of abstraction: Theory and design for the instruction of limit concepts. Making the connection: Research and practice in undergraduate mathematics, MAA Notes Volume, 73, 6580. Oehrtman, M. (2009). Collapsing dimensions, physical limitation, and other student metaphors for limit concepts. Journal for Research in Mathematics Education, 396426. Patton, M. (2002). Qualitative research and evaluation methods. Thousand Oaks, CA: Sage Publications. President’s Council of Advisors on Science and Technology (PCAST). (2012). Engage to excel: Producing one million additional college graduates with degrees in Science, Technology, Engineering, and Mathematics. Washington, DC: The White House. Seymour, E., & Hewitt, N. M. (1997). Talking about leaving: Why undergraduate leave the sciences. Boulder, CO: Westview Press. Steed, L. A. (Ed.). (1988). Calculus for a New Century: a Pump, not a Filter, Mathematical Association of America, MAA Notes Number 8. DC: Washington. 43 Treisman, U. (1992). Studying students studying calculus: A look at the lives of minority mathematics students in college. College Mathematics Journal, 362372. Wake. G. (2011). Introduction to the Special Issue: deepening engagement in mathematics in preuniversity education. Research in Mathematics Education 13(2). 109118. 44 APPENDICES 45 APPENDIX A: Syllabus Math 2143.002 Calculus I COURSE SYLLABUS: Fall 2015 COURSE INFORMATION Materials Textbook(s) Required: Calculus, 7th Edition, by James Stewart. Material covered during the session will be Sections 1.41.8, Chapters 2, 3, and 4, and 6.2, 6.3, and 6.4. We may occasionally cover enrichment activities not in the text. Optional: How to Ace Calculus/How to Ace the Rest of Calculus by Adams et al. Calculus II is split between the two books (Calculus I is entirely in the first book), but used copies can generally be found for under $5 on Amazon.com. Course Description: This course examines differential and integral calculus of functions of one variable, as follows. Topics include limits; continuity; derivatives; curve sketching; applications of the derivative; the definite integral; derivatives and integrals of trigonometric functions ; and use of computer technology. Prerequisite Two years of high school algebra and trigonometry or Math 142. Student Learning Outcomes 1. Students will demonstrate proficiency in the use of mathematics to structure their understanding of and investigate questions in the world around them. 2. Students will demonstrate proficiency in treating mathematical content at an appropriate level. 3. Students will demonstrate competence in the use of numerical, graphical, and algebraic representations. 4. Students will demonstrate the ability to interpret data, analyze graphical information, and communicate solutions in written and oral form. 5. Students will demonstrate proficiency in the use of mathematics to formulate and solve problems. 6. Students will demonstrate proficiency in using technology such as handheld calculators and computers to support their use of mathematics. Student Assessment Outcomes 1. Critical Thinking: The above learning objectives will be assessed for critical thinking in labs and other classroom activities. 2. Written, Oral, & Visual Communication: Students will be assessed on written, oral, and visual communication skills on their quizzes, exams, labs, and lab jigsaw activities. 3. Empirical and quantitative reasoning: All assessments in this course will contain a quantitative reasoning and empirical computation component. COURSE REQUIREMENTS Labs: On Tuesdays we will work in small groups on activities that develop the central concepts in the course. Attendance and participation is especially crucial on these days. You will turn in individual writeups of these labs activities. It is also important to ask questions of the other groups (who will generally work on related but slightly different problems than your own group) when they present as you will be responsible for all the problems on exams. Prelabs/Postlabs: The purpose of these assignments is to help me determine where the class is at and how much time we should spend on a particular topic. Prelabs are expected to be completed before class, and postlabs will be completed at the end of class on Tuesday. These assignments will be graded on completion. Attendance: There may be topics covered in class that are not in the text. You are responsible for all material covered. I don't take attendance, but there is a strong correlation between attendance and final grades. Missing class more than once or twice during the semester is likely to affect your grade, either directly or indirectly. If you do miss class, you should get notes and/or handouts from your classmates and see me during office hours. Homework: There will be suggested problems assigned for each section. The answers to most of these problems are in the text, so I will not collect them. However, you will see some of these problems (verbatim or with slight variations) on tests, so completing the problems is strongly encouraged! 46 The key to success in this course is regularly working with other students in the class, doing the homework early and asking questions when you have them!!! We will discuss homework problems in class, but there will often not be enough time to discuss all of them. Please come to office hours or visit the math tutoring lab if you have additional questions about the homework. Quizzes: There will be 11 take home quizzes based upon the suggested homework problems throughout the semester. Your best 10 scores will count for your final grade. Exams: We will have four inclass exams (roughly covering Chapters 7, 8, 9, and 10), and a comprehensive final exam. The final exam will be a joint final with the other section of Calc II. The data and time is TBD at the moment. Makeup exams are possible only if there is a documented emergency. Workload and Assistance: You should expect to spend 8 to 12 hours each week, outside of class, on the course material. This includes reading, homework, and studying for quizzes and exams. Some weeks (those in which an exam is scheduled, for instance) may require more of your time, other weeks may require less, but on average, budget 8 to 12 hours each week. I can’t stress enough that in order to be successful in this class you should spend much of this time working with other students in the class! Please ask questions and seek assistance as needed. You may email me at any time, and I encourage you to make use of my office hours Grading This class will be graded on a total points system. 1000 points are possible in the class. Assignments are weighted in the following manner: Assignments Total points possible Lab writeups 200 Reading sheets, 50 prelabs, postlabs Best 10 quizzes 200 Tests 100 each, 400 total Final 150 All point totals will be rounded to the nearest whole points before grades are assigned. Point ranges for final grades will be as follows: A: 900 – 1000 points B: 800  899 points C: 700  799 points D: 600  699 points F: 0 – 599 points TECHNOLOGY REQUIREMENTS Use of a graphing calculator having at least the capabilities of the TI83 will be helpful throughout the course. TI89 is highly recommended. A computer algebra system will be used for some problem exploration, enhanced conceptual understanding, and to engage students as active participants in the learning process. COMMUNICATION AND SUPPORT My primary form of communication with the class will be through Email and Announcements. Any changes to the syllabus or other important information critical to the class will be disseminated to students in this way via your official University Email address available to me through MyLeo and in Announcements. It will be your responsibility to check your University Email and Announcements. Students who Email me outside of regular office hours can expect a reply within 24 hours MF. Students who Email me during holidays or over the weekend should expect a reply by the end of the next regularly scheduled business day. myLeo Support Your myLeo email address is required to send and receive all student correspondence. Please email helpdesk@tamuc.edu or call us at 9034686000 with any questions about setting up your myLeo email account. You may also access information at https://leo.tamuc.edu. 47 COURSE AND UNIVERSITY PROCEDURES/POLICIES Academic Honesty Students who violate University rules on scholastic dishonesty are subject to disciplinary penalties, including (but not limited to) receiving a failing grade on the assignment, the possibility of failure in the course and dismissal from the University. Since dishonesty harms the individual, all students, and the integrity of the University, policies on scholastic dishonesty will be strictly enforced. In ALL instances, incidents of academic dishonesty will be reported to the Department Head. Please be aware that academic dishonesty includes (but is not limited to) cheating, plagiarism, and collusion. Cheating is defined as: Copying another's test of assignment Communication with another during an exam or assignment (i.e. written, oral or otherwise) Giving or seeking aid from another when not permitted by the instructor Possessing or using unauthorized materials during the test Buying, using, stealing, transporting, or soliciting a test, draft of a test, or answer key Plagiarism is defined as: Using someone else's work in your assignment without appropriate acknowledgement Making slight variations in the language and then failing to give credit to the source Collusion is defined as: • Collaborating with another, without authorization, when preparing an assignment If you have any questions regarding academic dishonesty, ask. Otherwise, I will assume that you have full knowledge of the academic dishonesty policy and agree to the conditions as set forth in this syllabus. Late Policy: Late work/Makeups will not be accepted without a documentable and valid excuse, because the lowest grade(s) in each category is dropped. Examples of documentable and valid excuses include: *car accident w/ police report *illness w/ doctor’s note (you or your child) *athletic or other mandatory extracurricular travel *field trip for another class *being detained upon entering the country by Homeland Security University Specific Procedures ADA Statement Students with Disabilities The Americans with Disabilities Act (ADA) is a federal antidiscrimination statute that provides comprehensive civil rights protection for persons with disabilities. Among other things, this legislation requires that all students with disabilities be guaranteed a learning environment that provides for reasonable accommodation of their disabilities. If you have a disability requiring an accommodation, please contact: Office of Student Disability Resources and Services Texas A&M UniversityCommerce Gee Library Room 132 Phone (903) 8865150 or (903) 8865835 StudentDisabilityServices@tamuc.edu Student Conduct All students enrolled at the University shall follow the tenets of common decency and acceptable behavior conducive to a positive learning environment. (See Code of Student Conduct from Student Guide Handbook). COURSE OUTLINE / CALENDAR 1). 1.4, 1.5, 1.6 2). 1.8, 2.1, 2.2 3). 2.3, 2.4, 2.5 4). Test I, 2.6 5). 2.7, 2.8, 2.9 6). 3.1, 3.2, 3.3 7). 3.4, Test II 8). 3.5, 3.6, 3.7, 3.8 9). 3.9, 4.1 10). 4.2, Test III 11). 4.3, 4.4 12). 4.5, 6.2 13). 6.3, 6.4, 1.7 14). Test IV 15). Review 16) FINAL WEEK 48 APPENDIX B: Labs The following labs are multiweek labs that deal with approximation. Each lab is accompanied by selected examples and starting problems for each major lab. Lab 1 – Rates and Amounts of Change Instructions: Work with your group on the problem assigned to you. We encourage you to collaborate both in and out of class, but you must write up your responses individually. Your work must be neat and include sufficient exposition to make the solution clear to another student who has not seen the assignment (for example, a sequence of equations without explanation will most likely receive zero credit). Pay particular attention to places where explanations using multiple representations are requested, and explicitly discuss the connections between your explanations using different representations. Type or write all of your work legibly on 8½"×11" paper with at least oneinch margins on all sides free of writing except your name, date, and assignment number, and staple all pages together. In the following, consider plotting height of water in a bottle vs. the volume of the water in the bottle. That is, height is on the vertical axis (dependent variable) and volume is on the horizontal axis (independent variable). Lab Preparation: The definition of an increasing function f is that 1 2 f (x ) f (x ) whenever 1 2 x x . The graph of height vs. volume must always be increasing (going up as we move from left to right) since more volume has to correspond to more height. 1) Explain the meaning of expressing the relationship between height and volume using the function notation h(V ) , and 2) include a description of the meaning of the equation h(3) 5 if volume is measured in cups and height is measured in inches. 3) Finally, rewrite the definition of an increasing function using h(V ) instead of f (x) and explain the meaning of this definition in terms of water in a bottle. 1. Steepness of the graph is related to the crosssectional area of the bottle. Explain why a steeper graph corresponds to a narrower bottle and a less steep graph corresponds to a wider bottle, as shown to the right. Make sure that you are talking meaningfully about the rate of change of height with respect to volume by breaking down your explanation in terms of amounts of change in height and amounts of change in volume. Lab 3: Locate the Hole Graph 1: The graph of 3 7 2 ( ) 1 x f x x has a hole. Your task is to determine the location of this hole using approximation techniques (no fancy limit computations allowed). Lab Preparation: Answer the following questions individually and bring your writeup to class. a. Draw a graph of f using an entire sheet of paper. Your graph should be drawn at a scale that gives a good sense of the x,ycoordinates of the hole. The x and y scales should be chosen so that your graph nearly extends between two diagonally opposites corners of the page. b. Identify what unknown numerical value you will need to approximate. Give it an appropriate shorthand name (that is, a variable name). c. Describe what you will use for approximations. Write a description of your answer using algebraic notation (for example, function notation, variables, formulas, etc.) Lab: Work with your group on the problem assigned to you. We encourage you to collaborate both in and out of class, but you must write up your responses individually. 1. Find an approximation to the height of the hole in your function (write out the approximation with several decimal places). Is this an underestimate or overestimate? Explain how you know. Find both an underestimate and an overestimate. Lab 5: The Derivative at a Point – Part 1 Lab Preparation: Answer the following questions individually and bring your writeup to class. A. Draw a fullpage picture of the physical context in three different configurations (specified in the context description) overlayed. This will help illustrate how the relevant quantities are changing in the region of interest. You will redraw and add to this picture during class. 49 B. What happens to the changes in the dependent quantity (volume, gravitational force, or mass of Iodine123) as the independent quantity (radius, separation distance, or time) is incremented by constant amounts? Is your rate of change constant, increasing or decreasing? C. Draw a fullpage graph showing the relationship between the two quantities involved in the instantaneous rate that you are asked to approximate. Add a point for each of the configurations you drew in your picture from A. Represent the changes in both quantities on your graphs as the length of short line segments. You will redraw and add to this graph during class. D. On your picture and your graph illustrate and label the changes in the relevant quantities to support your answer to B (using both Δnotation and numerical values). Lab 7: Linear Approximation The NASA Q36 Robotic Lunar Rover can travel up to 3 hours on a single charge and has a range of 1.6 miles. After leaving its base and traveling for t hours, the speed of the Q36 is given by the function 2 v(t) sin 9 t in miles per hour. One hour into an excursion, the Q36 will have traveled 0.19655 miles. Two hours into a trip, the Q36 will have traveled 0.72421 miles. Lab Preparation: Answer the following questions individually and bring your writeup to class. 1. Use your calculator to graph v(t) . Explain in words what the graph says about how the Q36 moves during a 3 hour trip starting with a full charge. 2. Using a full sheet of paper, carefully sketch a graph of the distance x(t) traveled by the Q36 measured in miles during this trip as a function of time in hours. Explain precisely why you drew the graph as you did. 3. Draw tangent lines to the graph of x(t) at times 0 t 0, 0 t 1, 0 t 2 , and 0 t 3. Label each tangent line with its equation. Use the variables x and t in these equations. Lab 8: Quadratic Approximation The NASA Q36 Robotic Lunar Rover can travel up to 3 hours on a single charge and has a range of 1.6 miles. After leaving its base and traveling for t hours, the speed of the Q36 is given by the function 2 v(t) sin 9 t in miles per hour. One hour into an excursion, the Q36 will have traveled 0.19655 miles. Two hours into a trip, the Q36 will have traveled 0.72421 miles. In Lab 7, you used tangent lines to approximate the distance x(t) traveled by the Q36 Rover. Lines with slope m through the point 0 0 (t , x ) can be written in pointslope form as 0 0 x x m(t t ) . You used the derivative v(t) x(t) to find the slope at 0 t . We could improve our approximations by using “best fit parabolas.” For the following problems, note that 2 0 0 0 x x a(t t ) b(t t ) is the equation of a parabola that passes through the point 0 0 (t , x ) . Changing the parameters a and b will change the shape of the parabola without changing the fact that it passes through that point. Lab Preparation: Answer the following questions individually and bring your writeup to class. a. Sketch a parabola on your large graph of x(t) that you think represents the best fit parabola at time 0 t 1. Then determine the equation of this parabola using the form 2 0 0 0 x x a(t t ) b(t t ) and the point 0 0 (t , x ) (1,0.19655) . To do this, find the first and second derivatives of the equation for this parabola, set x(1) v(1) and x(1) v(1) , then solve for a and b. b. Sketch a parabola on your large graph of x(t) that you think represents the best fit parabola at time 0 t 2 . Then determine the equation of this parabola using the form 2 0 0 0 x x a(t t ) b(t t ) and the point 0 0 (t , x ) (2,0.72421) . To do this, find the first and second derivatives of the equation for this parabola, set x(2) v(2) and x(2) v(2) , then solve for a and b. 50 Lab 12: Definite Integrals – Part 1 Work with your group on the context assigned to you. We encourage you to collaborate both in and out of class, but you must write up your responses individually. Context 1: Sam is tired of walking up two flights of stairs to get to calculus class every day, so he enlists Kelli to help him build a giant spring to lift him perfectly up to the second floor window. They order a twostory tall spring from Katelyn’s Giant Spring Limited Liability Co. When it arrives, it is packaged already compressed down 5 m shorter than its resting length. They figure they need to compress it another 5 m in order to climb on from ground level before launch. Tony walks by and points out this will take a lot of energy, saying: For a constant force* F to move an object a distance d requires an amount of energy** equal to E Fd . Hooke’s Law says that the force exerted by a spring displaced by a distance x from its resting length (compressed or stretched) is equal to F kx , where k is a constant that depends on the particular spring. *The standard unit of force is Newtons (N), where 1 N = 1 kg·m/s2 or the force required to accelerate a 1 kg mass at 1 m/s2. Increasing either the mass or the acceleration rate therefore requires a proportional increase in force. **The standard unit of energy is Joules (J), where 1 J = 1 N·m or the energy required to move an object with a constant force of 1 N a distance of 1 m. Increasing either the force or the distance requires a proportional increase in energy. Sam and Kelli’s spring has a spring constant of k 155N/m. Lab Preparation: 1. Draw and label a large picture of a spring initially compressed 5 m from its natural length then compressed to a displacement of 10 m. 2. Does it take less, the same, or more energy to compress the spring from 5 m to 7.5 m than it takes to compress the spring from 7.5 m to 10 m? Explain. 3. Explain why we cannot just multiply a force times a distance to compute the energy. 51 APPENDIX C: Patterns of Adaptive Learning Scale (PALS) Below is the PALS survey that was given to students in the first 3 weeks of the semester and again in the last 3 weeks of the semester. Part I: Information Gender: ☐ Male ☐ Female Class: ☐ Freshman ☐ Sophomore ☐ Junior ☐ Senior ☐ Graduate Ethnicity: ☐ Native American ☐ African American ☐ Latino/a ☐ Caucasian ☐ Asian ☐ Multiracial (Specify: _________________) ☐ Other: _____________ Native Language: ☐ English ☐ Spanish ☐ Other: _____________ ACT Math Score: ____________ Have you had a calculus class before this class? ☐ Yes ☐ No Here are some questions about you as a student in this class. Please circle the number that best describes what you think. 1. I would avoid participating in class if it meant that other students would think I know a lot. 2. It’s important to me that I don’t look stupid in class. 3. If other students found out I did well on a test, I would tell them it was just luck even if that wasn’t the case. 4. I would prefer to do class work that is familiar to me, rather than work I would have to learn how to do. 5. It’s important to me that other students in my class think I am good at my class work. 6. It’s important to me that I learn a lot of new concepts during this class. 7. I’m certain I can figure out how to do the most difficult class work. 1 2 3 4 5 Not true at all Somewhat true Very true 1 2 3 4 5 Not true at all Somewhat true Very true 1 2 3 4 5 Not true at all Somewhat true Very true 1 2 3 4 5 Not true at all Somewhat true Very true 1 2 3 4 5 Not true at all Somewhat true Very true 1 2 3 4 5 Not true at all Somewhat true Very true 1 2 3 4 5 Not true at all Somewhat true Very true I __________________________ (Print Name), having read this letter and having an opportunity to ask any questions, would like to participate in this research and my signature below indicates my informed consent to participate. A copy of this form will be given to you to retain for future reference. ________________________________ ___________________ Participant signature Date (month/day/year) 52 8. Some students fool around the night before a test. Then if they don’t do well, they can say that is the reason. How true of this of you? 9. Some students purposely get involved in lots of activities. Then if they don’t do well on their class work, they can say it is because they were involved with other things. How true is this of you? 10. When I’m working out a problem, my teacher tells me to keep thinking until I really understand. 11. Some students look for reasons to keep them from studying (not feeling well, having to help their parents, taking care of a brother or sister, etc.) Then if they don’t do well on their class work, they can say this is the reason. How true of this of you? 12. I wouldn’t volunteer to answer a question in class if I thought other students would think I was smart. 13. If I did well on a school assignment, I wouldn’t want other students to see my grade. 14. One of my goals in class is to learn as much as I can. 15. One of my goals is to show others that I’m good at my class work. 16. One of my goals is to keep others from thinking I’m not smart in class. 17. I like concepts that are familiar to me, rather than those I haven’t thought about before. 18. If I were good at my class work, I would try to do my work in a way that didn’t show it. 19. It’s important to me that I thoroughly understand my class work. 20. I sometimes copy answers from other students when I do my class work. 1 2 3 4 5 Not true at all Somewhat true Very true 1 2 3 4 5 Not true at all Somewhat true Very true 1 2 3 4 5 Not true at all Somewhat true Very true 1 2 3 4 5 Not true at all Somewhat true Very true 1 2 3 4 5 Not true at all Somewhat true Very true 1 2 3 4 5 Not true at all Somewhat true Very true 1 2 3 4 5 Not true at all Somewhat true Very true 1 2 3 4 5 Not true at all Somewhat true Very true 1 2 3 4 5 Not true at all Somewhat true Very true 1 2 3 4 5 Not true at all Somewhat true Very true 1 2 3 4 5 Not true at all Somewhat true Very true 1 2 3 4 5 Not true at all Somewhat true Very true 1 2 3 4 5 Not true at all Somewhat true Very true 53 21. I would choose class work I knew I could do, rather than work I haven’t done before. 22. One of my goals is to show others that class work is easy for me. 23. Some students let their friends keep them from paying attention in class or from doing their homework. Then if they don’t do well, they can say their friends kept them from working. How true is this of you? 24. Some students purposely don’t try hard in class. Then if they don’t do well, they can say it is because they didn’t try. How true is this of you? 25. One of my goals is to look smart in comparison to the other student sin my class. 26. One of my goals in class is to avoid looking smarter than other kids. 27. Some students put off doing their class work until the last minute. Then if they don’t do well on their work, they can say that is the reason. How true is this of you? 28. I can do almost all the work in class if I don’t give up. 29. One of my goals in class is to avoid looking like I have trouble doing the work. 30. Even if the work is hard, I can learn it. The following questions are about this class and about the work you do in class. Remember to say how you really feel. No one at school or home will see your answers. 31. In our class, trying hard is very important. 32. In our class, showing others that you are not bad at class work is really important. 33. In our class, how much you improve is really important. 1 2 3 4 5 Not true at all Somewhat true Very true 1 2 3 4 5 Not true at all Somewhat true Very true 1 2 3 4 5 Not true at all Somewhat true Very true 1 2 3 4 5 Not true at all Somewhat true Very true 1 2 3 4 5 Not true at all Somewhat true Very true 1 2 3 4 5 Not true at all Somewhat true Very true 1 2 3 4 5 Not true at all Somewhat true Very true 1 2 3 4 5 Not true at all Somewhat true Very true 1 2 3 4 5 Not true at all Somewhat true Very true 1 2 3 4 5 Not true at all Somewhat true Very true 1 2 3 4 5 Not true at all Somewhat true Very true 1 2 3 4 5 Not true at all Somewhat true Very true 54 34. In our class, getting good grades is the main goal. 35. In our class, really understanding the material is the main goal. 36. In our class, it’s important that you don’t make mistakes in front of everyone. 37. In our class, it’s important to understand the work, not just memorize it. 38. In our class, it’s important not to do worse than other students. 39. In our class, learning new ideas and concepts is very important 40. In our class, it’s OK to make mistakes as long as you are learning. 1 2 3 4 5 Not true at all Somewhat true Very true 1 2 3 4 5 Not true at all Somewhat true Very true 1 2 3 4 5 Not true at all Somewhat true Very true 1 2 3 4 5 Not true at all Somewhat true Very true 1 2 3 4 5 Not true at all Somewhat true Very true 1 2 3 4 5 Not true at all Somewhat true Very true 1 2 3 4 5 Not true at all Somewhat true Very true 1 2 3 4 5 Not true at all Somewhat true Very true
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Title  CLEARING UP CALCULUS WITH ADAPTIVE LEARNING HABITS 
Author  JENNIFER LYNN PATTERSON 
Directed by  RebeccaAnne Dibbs 
Department  Mathematics 
Date  20160603 
Subject  Calculus; 
Publisher of Collection  Texas A&M UniversityCommerce 
Type  Text 
Format  
Language  eng 
Rights  All rights to materials within this collection are held by respective holding institutions or individuals with the exception of public domain items. The materials contained within this collection are made available online for educational and/or personal research purposes only. 
Transcript  i CLEARING UP CALCULUS WITH ADAPTIVE LEARNING HABITS A Thesis by JENNIFER LYNN PATTERSON Submitted to the Honors College of Texas A&M UniversityCommerce in partial fulfillment of the requirements for the degree of BACHELOR OF SCIENCE December 2016 iii Copyright © 2016 JENNIFER LYNN PATTERSON iv ABSTRACT CLEARING UP CALCULUS WITH ADAPTIVE LEARNING HABITS Jennifer Lynn Patterson, BS Texas A&M UniversityCommerce, 2016 Advisor: Rebecca Dibbs, PhD One of the reasons for the exodus in STEM majors is the introductory calculus curriculum. Although there is evidence that curriculum like CLEAR calculus supports gains in students’ growth mindset, it is unclear how this curriculum promotes mindset changes. The purpose of this study was to investigate which features of CLEAR Calculus students notice and attribute to their success. After administering the Patterns of Adaptive Learning Scale to assess students’ initial mindset in one section of calculus, twelve students were selected for interviews. At the end of the semester, students were readministered the PALS. Students cited that the labs in the CLEAR Calculus curriculum challenge them in ways that facilitates deeper comprehensive learning than that of prior courses. In addition, students also noted that the many reflection components of the course were helpful in retention of the concepts. Successful students demonstrated an increase in adaptive learning habits throughout the semester. v ACKNOWLEDGEMENTS This thesis would not have been possible had it not been for the encouraging mentors and faculty members that pushed me to persevere through the difficult times, as well as celebrated with me through the fun times. To Dr. Raymond Green, for always having an open door and a listening ear for any concerns or questions. To Dr. Melanie Fields, for providing constant encouragement and support. To Mrs. Jennifer Hudson, for leading by example and being not only a mentor, but also a friend, and to Dr. RebeccaAnne Dibbs, for sharing your love of research with me. This experience has not only enhanced my college career, but it has also allowed me opportunities that I would have never otherwise encountered. vi TABLE OF CONTENTS LIST OF TABLES .................................................................................................................... vii LIST OF FIGURES ................................................................................................................... viii 1. INTRODUCTION ...................................................................................................... 1 2. LITERATURE REVIEW ........................................................................................... 2 Obstacles in Calculus .............................................................................................3 The Evolution of CLEAR Calculus: Calculus Reform ................................... 4 Formative Assessment ....................................................................................... 6 Research Stance ................................................................................................. 9 Theoretical Perspective ................................................................................... 12 3. METHODS ................................................................................................................. 15 Setting ................................................................................................................ 15 Data Sources ................................................................................................... 17 Data Handling .................................................................................................. 19 Instrumentation ................................................................................................ 20 Data Analysis .................................................................................................... 20 4. FINDINGS .................................................................................................................. 22 Challenge ........................................................................................................... 22 Instructor Caring .............................................................................................. 24 Reflection ........................................................................................................... 25 Adaptive Learning Habits .................................................................................. 27 vii 5. DISCUSSION ........................................................................................................... 29 REFERENCES .......................................................................................................................... 32 APPENDICES ........................................................................................................................... 35 Appendix A. Syllabus .............................................................................................................. 36 B. Labs .................................................................................................................... 39 C. Patterns of Adaptive Learning Scale survey ........................................................ 42 viii LIST OF TABLES TABLE 1. Interview participants....................................................................................................... 18 2. Standards of Evidence ..................................................................................................... 21 ix LIST OF FIGURES FIGURE 1. Typical CLEAR Calculus lab ............................................................................................ 6 2. Summary of mindset theory ............................................................................................ 15 3. Typical student postlab ................................................................................................... 17 10 INTRODUCTION One of the most pressing problems in higher education is that there are not enough science, technology, engineering or mathematics (STEM) majors in United States universities graduating to meet the demands of the industry. Students leaving STEM degrees after introductory calculus is mostly what causes this lack of STEM graduates (Bressoud, Rasmussen, Carlson, & Mesa, 2014). Because of the lack of efficiency of traditional calculus instruction, students leave their STEM majors to pursue fields of study that they perceive to require less effort in return for an acceptable amount of success (Ellis & Rasmussen, 2014; Oehrtman, 2009). The oldest, most common alternative to traditional calculus curriculum is the Treisman Model. This model suggests that students learn best when calculus is taught similar to a science class: with both a lecture and a lab (Treisman, 1992). However, more recent research suggests that these labs should cover a single theme (for example, limits) and mirror successful students’ actual learning processes (Oehrtman, 2009). It has been noticed that mindsets play a significant role in the overall success of calculus students. Carol Dweck (2006) defines mindset in two different ways: fixed mindset and growth mindset. Students classified under the fixed mindset, if not immediately successful in introductory calculus often leave the STEM field. However, growth mindset students can persist and succeed, even after failures as severe as failing a course (Dibbs, forthcoming). This study will identify students’ mindsets and monitor their success in a CLEAR Calculus classroom. A recent study found that all students who participated in CLEAR Calculus, a nontraditional calculus class utilizing the limit approximation calculus labs described in Oehrtman (2009), became more growth oriented by 1.6 standard deviations; this suggests that the curriculum itself is likely to be responsible for this positive mindset change (Dibbs, 2015). I 11 examined how this particular curriculum supports an increase in adaptive learning habits demonstrated by students of fixed and growth mindsets. This research will be guided by utilizing the research question: What are the features of CLEAR Calculus that promote positive changes in students’ mindsets? This topic is important because if it is known what causes students the most success, traditional calculus instruction methods can be altered to produce more effective calculus students. By understanding what makes CLEAR Calculus effective, it provides a strong argument for using this curriculum. Interested practitioners who are not willing to implement CLEAR Calculus can learn what components to add to their classes in an effort to keep students within the STEM field. In order to investigate this research question, I interviewed students enrolled in a CLEAR Calculus class in order to determine what features of the class promote positive mindset changes. A key component of the study was the participants’ mindsets. To measure this, a commercial survey instrument—the Patterns of Adaptive Learning Scale, called PALS—was used that identified and measured students’ mindsets. I argue that students noticed that the course contains components designed to challenge them at a rate above their average performance rate. In addition, students noticed that the instructor cares at a rate higher than their previous mathematics instructors, that reflection is a primary component of the course, and that they experienced an increase in adaptive learning habits. LITERATURE REVIEW There are several major topics related to the research question of this study: problems with traditional calculus instruction, the evolution of CLEAR Calculus, and formative 12 assessment. After discussing each of these topics in turn, the literature review concludes with my researcher stance. Obstacles in Calculus Of the 600,000 firstyear college students that take Calculus I, 250,000 of them fail in the first semester (Treisman, 1992). Fourteen years later, Ganter (2006) confirmed the calculus failure rate to be 4060%. Treisman (1992) speculated as to why there are such low success rates among introductory calculus courses. As a result of failing introductory calculus, students pursuing a STEM degree often leave the field citing the lectures were traditional and uninspiring, encouraging memorization instead of comprehensive application and understanding (Seymour, & Hewitt, 1997). However, an alternative instruction method has the potential for deeper comprehension and better appreciation of the subject (Treisman, 1992). By engaging the class in a labbased structure, encouraging group work and wholeclass discussions, students are more likely to retain information (Ellis & Rasmussen, 2014). Formative assessment can be utilized in the form of the CLEAR Calculus curriculum to ensure this retention among students (Oehrtman, 2009). Hutcheson, Pampka, and Williams (2011) found that students use the instruction experience of their first year of mathematics courses to decide to continue or discontinue their pursuit of a STEM degree. This is why a strong curriculum that fosters comprehension and understanding is vital. Many US universities utilize first year courses in STEM fields to discourage an overpopulation of students pursuing these degrees (Steen, 1988; Wake, 2011). It is important that students are successful in completing a STEM degree for many reasons. According to the US Presidents Council of Advisors on Science and Technology (PCAST, 2012), economic growth 13 can be traced to an increased number of students entering the STEM field. Without US STEM graduates, the job market—and by extension, the economy—will suffer greatly. The Evolution of CLEAR Calculus: Calculus Reform The major area of research in mathematics education in the 1990’s was the problem with introductory college calculus reform (Oehrtman, 2008). The most successful calculus reform to emerge from this reform movement was the Treisman Model of instruction (Ganter, 2006). The Treisman model emerged to address several problems in a traditional calculus course; the main problem being that students were more interested is passing calculus than they were in understanding the mathematics taught (Treisman, 1992). Although professors believed that the high failure rate was due to students’ low income, low motivation, poor academic preparation and lack of family support, further investigation revealed that that this was not the case (Treisman, 1992). Longitudinal studies at the University of CaliforniaBerkley revealed that students who were successful in calculus were so because they learned how to work and study in groups effectively, while students that failed were found to have worked alone, eventually becoming frustrated, and then quit trying (Treisman, 1992). The Treisman classroom model is designed to train all students to become effective group members in mathematics. The current iteration of the Treisman model used today is called CLEAR Calculus, which is a refinement of the original Treisman model based on the approximation framework (Oehrtman, 2008). According to Treisman, calculus instruction today does not teach problem solving in a way that promotes the development of generalized problem solving skills. Instead, departments are teaching students how to excel in routine computations rather than how to comprehend the subject matter (Treisman, 1992). The needs of the students 14 were not being met, and Treisman was determined to find a way to facilitate calculus comprehension rather than calculus regurgitation. However, even when students are taught with the Treisman model, many students have a great difficulty in articulating mathematical concepts (Oehrtman, 2008). Oehrtman believed mathematical careers should require much of students in the area of proof engagement and argumentation (Oehrtman, 2008). Students should be able to comprehensively defend mathematical theories by the end of their schooling. Traditional calculus instruction provides most students with a very superficial understanding of mathematical procedures, whereas it may equip only a few for further studies (Oehrtman, 2008). Upon entering Calculus I, students have no conceptual structure to comprehend the most basic components of calculus due to their lack of prior exposure to the content. Oehrtman (2008) suggests a solution that would aid students with their comprehension: engagement in multiple activities that reveal and encourage the abstraction of a common structure. By presenting all calculus concepts through a similar framework, students develop a more organized understanding of limits than they do in a traditional calculus course (Oehrtman, 2009). CLEAR Calculus achieves this through formative assessment (pre and post labs). In a typical lab, students are asked to approximate the water pressure of a dam (Figure 1). The first row would represent the prelab. Students are required to find the approximating by drawing a picture, graphing the data, finding the value algebraically, and again numerically (Oehrtman, 2008). 15 Figure 1. Typical CLEAR Calculus lab Formative Assessment During the CLEAR Calculus labs—such as the one described in the previous section—the instructor walks about the classroom conducting informal formative assessment, guiding the students through the labs; the pre and post lab assignments are considered written formative assessment. Although prior research does not indicate formative assessment is the cause of students’ mindset changes (Dibbs, 2015), research on noncognitive factors involved with students’ learning indicate that the use of formative assessment in the classrooms help students perceive their teacher as caring about them (Dibbs, 2014) and encourages them to take ownership of their own learning (Black & Wiliam, 2009). This makes formative assessment an important structural component of CLEAR Calculus. 16 Black & Wiliam (1998) define formative assessment as teaching to meet students’ academic needs (Black & Wiliam, 1998). After research and testing in the United Kingdom, Black & Wiliam (1998) have proven how formative assessment is essential in the raising of achievement standards in schools. Teachers play a large role in a students’ success. It is very important for teachers to be aware of their students’ strengths and weaknesses in learning, so that they can teach on the appropriate level (Black & Wiliam, 1998). The key component to formative assessment is the adaption of teaching to meet student needs (Black & Wiliam, 1998). By building on the successes of traditional instruction and reforming the less profitable practices, Black & Wiliam (1998) have created more efficient teaching methods through formative assessment in K6 British children. Formative assessment has been proven to assist low achievers more so than high achievers. This has aided in the reduction of the range of achievement with an overall higher standard. When teachers become aware of their pupils’ needs and begin to address those needs, a positive change has occurred in the classroom. Unfortunately, many teachers are out of touch with the amount of understanding the students are experiencing, therefore their teaching practices are inefficient (Black & Wiliam 1998; 2009). The way students’ perceive themselves is very important. If a student identifies his or herself as unable to learn, there begins to be a decrease in the seriousness they take in school (Black & Wiliam, 1998). This is more likely to happen to a student of the fixed mindset, because when confronted with a failure, students with fixed mindsets tend to retreat from the problem. Fixed mindset students with high confidence in the subject matter often have high persistence and seek challenges. However, if a fixed mindset student has low confidence in his or her ability to succeed, they may display a low persistence. Often, students that receive low marks will 17 continue the same behaviors, consistently receiving low scores. Many students are simply content to put forth the minimum effort that will be accepted. Students do so by avoiding difficult tasks – looking for clues and patterns to find the right answer without actually learning the material. Students of the fixed mindset may find Calculus I difficult if they do not initially understand it. The fixed mindset person believes that smartness cannot be obtained—it cannot be gained or lost; one either has the capacity (or potential?) to master a topic or not (Dibbs, 2012). Calculus I curriculum may be difficult to that of a fixed mindset because the student will not have entered the class with prior knowledge of the subject; he or she will be required to learn the new material. It is common for fixed mindset students—who have been successful in their school career—to drop Calculus I after the first test if they have been unsuccessful in the class. This is a pivotal point in the fixed mindset students’ academic career; many students change their major from that of a STEM field following the failure or drop of their introductory calculus course because they believe they are not mentally equipped to succeed. Growth mindset students often do poorly in the beginning of the semester, but what sets these students apart is the changes they make in their studying/notetaking after failure. The expectation is that these students will find more efficient ways to approach the class that will ensure learning and ultimate success. Black & Wiliam (1998) believe that the classroom is lacking a culture of success; simply assigning students grades is overemphasized, whereas advising students is underemphasized. When grades are assigned, this quantitative data can be used to accurately gauge the placement and success of a student in a class, but more is to be gained when academic advising from a teacher takes place (Black & Wiliam, 1998). Unfortunately, standardized testing in US high 18 schools often dominates teaching and assessment. Black & Wiliam (1998) states that testing does not provide teachers with an adequate model for formative assessment. Feedback improves learning when it shows the students their strengths and weaknesses without a number grade attached to it (Black & Wiliam, 1998). If pupils are to benefit from formative assessment, they should be trained in selfassessment so that they might comprehend what is needed to achieve success (Black & Wiliam, 1998). This can be facilitated by discussions in which students are to articulate their own understanding of the concept. This is beneficial to not only the student sharing, but also the students struggling with similar concepts. This kind of alteration of the original lesson plan is an investment of sorts. Teachers must invest time in their students to create a culture of questioning and deep thinking (Black & Wiliam, 1998). Black & Wiliam were quite thorough in their research; however their study focused on British children grades K6. Dibbs (2014) examined Black & Wiliam’s theoretical framework to see if it crossed age and cultural barriers. It did, but there were inconsistencies with postlab results. Though it appeared as if the inconsistencies were related to mindsets, it was found that changes in mindsets are more likely fostered by CLEAR calculus instruction rather than postlabs (Dibbs, 2015). This thesis project connected the previous work on mindsets and calculus by generating hypotheses about how CLEAR Calculus helps students make positive mindset changes. Research Stance When qualitative research is done, it is vital to disclose the researcher’s experience and expectations in the area of research so the reader understands where the writer stands on the topic. 19 I studied Calculus I, II, and III in a traditional classroom setting. In addition to attending class, I was also involved with the Texas A&M UniversityCommerce Calculus Bowl Team. This experience allowed me to have a larger exposure to calculus than the average calculus student. Regarding the pre and post labs, I have no personal experience. The only exposure I have had with formative assessment is casual conversation with classmates currently taking a nontraditional calculus course. In Calculus I, I struggled with the material. I did not immediately grasp the concepts presented; this was a novel and unpleasant experience. When I realized that I did not know the material as well as I would have liked, I spent hours studying each chapter in preparation for the assignments and tests. I received a B in the course, but I earned that B. I had to use the growth mindset to find more effective ways to study efficiently. My inner fixed mindset was saying, “If you can’t get it the first time, you’re not smart enough. Just give up.” Calculus II was a similar experience to Calculus I, except the material was more difficult, requiring more hours of studying for comprehension. In order to gain more experience and exposure to the material, I joined the Calculus Bowl team and Math Club to strengthen my calculus skills by surrounding myself with math intellectuals that shared similar misunderstandings of the material. This consisted of weekly evening practices that lasted for two hours. During these practices, we were presented with up to fifty calculus problems with content taken from Calculus I and Calculus II classes. These problems were to be solved in 60 seconds or less; precision and speed were a must. By participating in the Calculus Bowl, I increased my exposure to regular classroom material. During this study, I expected to encounter students with varying backgrounds in Calculus. College Algebra and/or PreCalculus classes are prerequisites to introductory Calculus 20 at this institution. Students were expected to be of the semifixed mindset immediately following Calculus prerequisite courses, because these courses require significantly less floundering to demand a change in mindset to earn a satisfactory grade. Upon entering introductory calculus, students were expected to have little to no understanding of limits, and struggle with complete comprehension of topics. In addition, it was expected that post College Algebra/PreCalculus students would struggle with trigonometric functions. I used the knowledge I gained from my own calculus experiences to provide me with an understanding of potential biases as I assessed students that participated in this study. Understanding my stance about the research allowed me to design a study that would provide a check on these potential biases. In the next section, I will explain the learning theory I used to structure student responses. Theoretical Perspective Before qualitative research is conducted, one must define the major constructs in the research question as well as define what learning means within the context of the research. After discussing which learning theory I used in this study, the remainder of this section explores the concept of mindsets in greater detail. Epistemology In order to frame participant’s responses, it was important to choose a learning theory. The theory chosen was Vygotsky on social constructivism. Vygotsky’s research, interpreted by Gredler and Shields (2008) found that all knowledge is constructed in a social setting. This theory claims two major components (Gredler & Shields, 2008). The first component is informal knowledge – knowledge gained implicitly by observation (i.e. monkey see: monkey do), also known as spontaneous knowledge (Gredler & Shields, 2008). One might not be able to articulate 21 why he or she knows a concept learned through spontaneous knowledge, however they are able to put it into practice. An example of this is the term “brother.” To a small child, he or she may be able to tell you if a person is their “brother” or not, but they would not be capable of defining what a “brother” is. The second component to Vygotsky’s theory is formal knowledge (Gredler & Shields, 2008). This is knowledge explicitly taught (i.e. calculus content). This component, also referred to as scientific knowledge, takes form as factual knowledge that one may be able to implement, but may not understand the reasoning behind it (Gredler & Shields, 2008). For example, Algebra I students are taught the mathematical mnemonic of FOIL. The students may be able to walk themselves through each step of the process, but they have no understanding of what they are doing, why it works, or whether it is relevant or not. Informal knowledge is not inferior to formal knowledge, but rather it is learned differently. Ideally, one must have both. This theory fits my study because I was looking for students’ spontaneous knowledge about their mindsets in the context of math classes with lots of scientific knowledge; Vygotsky captures both of these concepts. Theoretical Perspective Carol Dweck’s Mindsets (2006) showed that students’ learning patterns fall under two very different categories: the fixed mindset and the growth mindset. Dweck found in her research that a student’s mindset has a very large influence on the students’ success in a difficult class, such as Calculus I (Dweck, 2006). This section will explore the differences in two theories of intelligence: fixed and growth mindsets. The growth mindset primarily focuses on recovering from and learning from one’s failures for future improvement. However, the fixed mindset places much value on immediate 22 success; if a fixed mindset student believes he or she has a significant chance of failure, there will often not be an attempt of the task, assignment, class, etc. In her research, Dweck (2006) observed that the way a student perceives his or herself strongly affects the way they leads their life. Dweck (2006) discovered that the students that appear to be the most intelligent at the beginning of the semester might not always end the semester as the smartest. In a Calculus classroom, fixed mindset students may begin the semester as the smartest; however, when the coursework becomes difficult or if they find themselves failing – they give up. That’s when the growth mindset students seek ways to improve their methods, learn from their failures, and begin excelling in the class. The growth mindset students often end the semester with the label “smartest” because of their persistence (Dweck, 2006). Motivation is the overarching difference in these mindsets – what drives a student is how they’re going to react to failure. For instance, consider a class of 7th grade students is given a math exam where the overwhelming majority of the class fails. The fixed mindset students prepare for the next test by studying less – they could not do well on the first test, so they see no reason to try on the second. These students are under the fixed mindset belief that “smarts” are not to be gained; you either have the ability or you do not. However, the growth mindset students make a greater effort to do well on the next test by studying harder and more efficiently. When the second test is administered, the growth mindset students that spent extra time preparing for the second test excel well beyond those of the fixed mindset students that refused to study (Dweck, 2006). The fixed mindset student feels the constant need to be superior – which means failure is not an option. When they feel as if they are superior to others, they feel smart. Their view is that 23 “effort can reduce you” (Dweck, 2006) – why attempt something if failure is an option? They believe that only “dumb kids work hard in school. Is it better to fail on purpose than to try and show that you are dumb?” Fixed mindset students are prone to attempt only tasks that they are apt to succeed in; this allows for very high achievers. Many Honors College, Top 10% overachievers, and first semester STEM majors often have the fixed mindset. However, upon taking Calculus I, they do not experience immediate success. This unexpected difficulty (or failure) often discourages the fixed mindset students from continuing in their chosen STEM field (Dibbs, 2014; Dweck, 2006). In a Chemistry class, fixed mindset students were found to stay interested when they did well right away. However, if they were not successful from the start, they grew disinterested and distant (Dweck, 2006). These students were only engaged when they believed they would succeed. Dweck discovered that fixed mindset students have very low calibration, meaning they are often inaccurate when judging their own abilities (Dweck, 2006). Dweck defines the growth mindset as someone that can grow and change through application and experience. Students with the growth mindset view failure as a gift, an opportunity to get better. If not immediately successful, they don’t believe they are failing; they believe they are learning. Each student’s definition of failure plays a vital role in whether or not the student has a fixed or growth mindset. Dweck (2006) describes the growth mindset as unique students converting obstacles in life into future successes. Differing from fixed students, growth mindset students often have very high calibration; they are more realistic in selfassessment (Dweck, 2006). This is often helpful in the classroom as they assess the amount of learning that has taken place. 24 In summary, students’ mindsets play a significant role in the outcome of their overall classroom experience (Figure 2). Over time, students have been observed changing their mindset due to structure changes in the classroom (Dibbs, 2014). Formative assessment has been a crucial tool in facilitating a classroom environment conducive to positive mindset changes. Figure 2. Summary of Mindset theory METHODS The following describes the methods I used to find an answer to the research question “What are the features of CLEAR Calculus that promote positive changes in students’ mindsets?” I will identify the setting the study took place, as well as discuss the data sources, handling, and analysis. Setting This study took place at a midsized rural master’s comprehensive regional university in the South with approximately 11,000 students. There are 40.6% of the students at this university who are nonwhite, and 60.3% are female. The students who participated in this study were 25 enrolled in introductory calculus in the Fall 2015 semester. The students enrolled in introductory calculus are most often mathematics, physics, and engineering majors. Each week in a CLEAR Calculus course is comprised of the same schedule for each day: on Monday, students were given new material and a prelab to begin working on for the lab completed in class on Tuesday. The prelab, due before class on Tuesday graded on completion, required students to complete the Unknown Value of the approximation framework (Figure 1, found in the literature review) and to identify a quantity that could be used to approximate the unknown value. Lab days were comprised of group work in groups of 34 students. After the lab, students completed a postlab answering the following questions, graded on completion: 1. What did your group work on during the lab today? 2. How comfortable are you with the material in today’s lab? 3. What is the part of today’s lab you feel most comfortable with? 4. What part of the lab is most difficult so far? 26 A typical response on a postlab can be found in Figure 4. Figure 3. Typical student postlab The purpose of the postlabs is to encourage discussions in class on Wednesday, as well as to assist the instructor when prioritizing which material to review. The remainder of the week is spent on new material. For a more detailed layout of the course’s weektoweek activities, see Appendix A. The context of each approximation lab can be found in Appendix B. Data Sources I selected an introductory calculus course of primarily firstyear STEM majors to conduct my study. During the first three weeks of the semester, I obtained research consents 27 from the participants and administered a PALS study to assess initial student mindsets. Students also completed the PALS at the end of the semester. Upon reviewing the PALS results, I selected twelve students with trends in their mindset score to obtain a sample of maximum variation (Patton, 2002). The chosen students received a letter asking for their participation in a brief interview. I interviewed until data saturation was reached (Patton, 2002), and then selected the most interesting cases for analysis in my thesis. This particular class was comprised of mostly engineering students with few other majors such as computer science, mathematics, education, and physics. The participants’ classification ranged from freshman through graduate level students. Below is a table describing the participants, classification, major, mindset, and final grade in the course. Table 1 Interview participants Pseudonym Year Major Mindset Final Grade Kyler Freshman Engineering Strong Fixed A Taylor Sophomore Engineering Strong Fixed B Cameron Freshman Computer Science Strong Fixed A Michael Junior Engineering Strong Fixed B Lana Sophomore Mathematics Mixed Fixed A Chase Freshman Engineering Mixed Fixed B Sarah Sophomore Middle School Math Mixed Growth C Ricky Freshman Computer science Strong Growth A Jared Freshman Engineering Strong Growth B Lacey Freshman Physics/Math Strong Growth A Adam Graduate Physics Strong Growth A 28 Upon their consent, I conducted semistructured interviews to gather qualitative data regarding the effects of formative assessment in student learning and mindset. The following is an example of some sample interview questions used to conduct the first interviews: 1. What’s your major? 2. Why are you going into a mathbased field? 3. Tell me what you’ve been like in previous math classes 4. Tell me what this class is like. 5. If I had never seen one of Dr. Dibbs’ labs, tell me about them. What are they like? What do you do in them? 6. Why do you think you do labs? 7. Have labs changed you? How? 8. What do you do for calculus when you’re not in class? 9. How do you approximate a hole in the graph? 10. What’s a limit? 11. What will you remember about this class? 12. What should I have asked you but didn’t? 13. Do you think your experience in this class will make you approach your next class differently? The following are questions that were used to conduct the second interviews: 1. What have you learned in the class? 2. What are the features of the class that stood out to you? 3. Have you changed as a student as a result of this class? 4. A lot of people mentioned instructor caring? 5. What made you decide caring was a thing? 6. Would you recommend this class to another student? Data Handling Students were assigned ID numbers and pseudonyms at the beginning of the study to ensure anonymity. These pseudonyms are the way that the participants have been and will continue to be referred in the research. The key containing the students’ ID and pseudonyms is kept in Dr. Dibbs’ locked office to ensure security and confidentiality. Once ID numbers and pseudonyms were issued, students’ real names were not looked at again, and the key was discarded at the end of data collection. Upon being selected to participate in the interview process of this study, students were invited to participate via invitation letter. 29 Instrumentation The survey used to measure student mindset scores was the Patterns of Adaptive Learning Scale. This survey was developed for kindergarten through 5th grade students (Midgley, C., Maehr, M. L., Hruda, L. Z., Anderman, E., Anderman, L., Freeman, K. E., & Urdan, T., 2000). The PALS survey administered to students in this study was modified from the original survey to exclude the sections questioning students about their parents. The PALS test given to students had a Cronbach Alpha of .87, which is acceptable for an educational instrument (Gall, Gall & Borg, 1996). Appendix C contains a copy of the PALS survey used in this study. Data Analysis Data analysis for the thesis was conducted in two phases. First, I conducted a pilot study that allowed me to develop an analysis scheme for the main project. The data for the pilot was collected and analyzed in Spring 2015. The coding scheme that emerged from the pilot data is used appears in a standards of evidence table below (Table 2). This standards of evidence table was created from a topdown coding (Patton, 2002) of the interviews from the students participating in the pilot study. 30 Table 2 Standards of Evidence Code Definition Standards of Evidence Example Challenge is a feature Student’s perception of the class as having difficult problems as the core of the course AND seeing this as a positive thing Direct statements made by students that (challenging/difficult/real life) problems are important in the class Direct statements made by students indicating this is a good thing Tone of students indicated bullet point #2 “This class is challenging, but that is the point” Things that are challenging What parts of the class students find to take the most time outside of class/require the most work Direct statements/complaints about the things that are most difficult “Labs...they’ve been the most difficult challenges so far. Even more difficult than the test.” How labs changed me Student’s perceptions of the work habits and beliefs that participating in this curriculum changed for this semester &/or in the future Direct changes made by students about homework/study habits/beliefs that changed as the result of the class Direct statements about any beliefs that changed as a result of this class Evaluative statements about whether this belief will persist to the next class “I started studying more. Instead of studying the night before, I would study a few days before the test – it helps.” “Taught me to think through the problems more carefully...to figure out what’s wanted.” “I’ll think more in depth about...[future math classes].” Features of CLEAR calculus students notice Components of the labs and lectures students recognize as different from other math classes Direct statements made by students about the goals and effects of CLEAR Calculus “...so that I can understand the concepts better. So that I can visualize the concepts better.” “So that we can get almost real world experience so we can retain what we learn.” “To see how math is used in real world situations.” 31 In order to analyze the quantitative data, the following steps where taken. The PALS survey scores were entered into an excel spreadsheet. The numbers 15 were used to code the responses. When necessary, reverse coding was used so that 1 was always the most heavily fixed score. Participants that did not participate in both surveys (taken at the beginning and end of the semester) were eliminated. Because the n was small (n=12), a MannWhitney test was used instead of the ttest. Trustworthiness In an effort to protect against bias and ensure research was conducted rather than expectations being confirmed, it is important to have checks and balances in place for this thesis project. Dr. Dibbs and myself performed the data analysis independent of each other using an agreed upon chart. We reconciled our codes to verify consistency. FINDINGS There were four themes that emerged from the data that students noticed and valued as part of their introductory calculus experience: challenge, instructor caring, reflection, and adaptive learning habits. Students noticed that they were persistently challenged throughout this course and that the instructor demonstrated a heightened care for their success in the class. They also noted that the many reflection components of the course were very strong indicators of their overall success in the class. Students showed a very rapid increase in adaptive learning habits throughout the semester. Challenge A common term that is used during the participant interviews is “challenge.” Students identify this course as being “…just different,” in level of difficulty and structure. In his second interview, Taylor identified this course as being “not necessarily easy, just different from other 32 classes” in the level of difficulty and structure. One way that the CLEAR Calculus classroom varies from that of a traditional classroom is the weekly lab activity. The structure of the class is as follows: there are 3 instructorled lecture days, 1 day of lab group work, and 1 day dedicated to weekly review. Students work in groups to complete the weekly lab assignment on the second day of each week. These lab days are what students are referring to as being the most “challenging” portion of this course. Lana said in her first interview, “It was harder than I was expecting. It was definitely challenging. I liked and didn’t like the challenge because I’ve always been good at math, so struggling with it was difficult for me.” Lana had a fixed mindset and severely struggled in this class. There were times in the first 3 weeks of class that she refused to complete her work. She would cease to participate when she was not making progress in the lab. On more than one occasion, she abandoned her group mates to finish the lab without her. Midway through the course, Lana found that if she altered her habits outside of class, she was more successful and better understood the material. She states in her second interview: …a lot of the labs were really, really, really hard for me. Although it did help, I didn’t like them that much. But in the long run they really did help because it gives you more opportunities to see how the work can go in different ways from this class. Lana changed her success in the classroom by spending more time in the Math Lab on campus, in addition to attending office hours with her instructor. The extra time spent outside of class better prepared her to be more successful on lab day. Due to this change in her outside of class habits, Lana found herself to be a productive group mate. Several students mentioned their struggle during the labs has better prepared them to be successful on tests. Ricky mentioned in his second interview: The labs are interesting. They give me hard questions that are like—in other classes they give you super easy questions [before the test] but then on the test 33 they give really hard questions. Not here, she gives you really hard questions throughout the class so that the questions on the tests are easy. I like that. Kyler agreed with this thoughtprocess in his first interview by saying that he “believes that the interactive nature of the labs is more conducive to learning than the lecture time of the class.” In other words, students credit the difficulty of labs to their overall success in the class. Chase said in his second interview: After getting used to the level of difficulty of the labs, I got a lot better at the math. I just had to get used to how hard they were. They were really hard and I didn't like them. But they got easier as we went. I got better at them the farther we went in the semester. A trend immerged from students’ first and second interviews. During all first interviews, students seemed to fall under two categories: incredibly frustrated at the difficulty of lab work, or frustrated but understanding of the potential positive outcomes of the challenging work. In all postinterviews, students unanimously decided that labs were indeed at a heightened level of difficulty as previous math classes, but they understood and even advocated for the challenge. Each student was able to state why he or she felt labs were a vital component of their CLEAR Calculus class. Instructor Caring In this study, 5 students cited that they noticed the instructor cares more than previous instructors. Adam states in his second interview the ways in which he feels the instructor demonstrates care for the students: “She is caring…retention of who we are, characteristics about us—she remembers them, availability, she will talk you through a problem in a good way…an educational way.” It could be asked if the nature of this course and its features demonstrate increased interest in student success by the instructor than that of a traditionally taught calculus class, giving students a feeling that they are cared for. 34 In his second interview, when Taylor was asked about the ways in which the instructor demonstrates she cares for the students, he cited “she’s always available to help, she’s not distant…if you need anything she’ll put you on the right track to make sure you know how to do it.” He compares this instructor to previous instructors in her ability to follow through with her assistance. It is undetermined whether or not the instructor caring feature of this class is due to the CLEAR Calculus curriculum, or simply this particular instructor. Lana also agreed that the instructor demonstrated that she cared for the class. In her first interview, Lana said that the instructor was “open to answer any questions and explain. She would also check for understanding after she explained something.” It’s clear that the students felt that they were taken care of and had adequate amounts of help. When asked about why students may feel as if the instructor has a heightened sense of caring in this CLEAR Calculus class, the instructor of this class responded with: …This class has a lot more assignments than traditional calculus. So there is more paper shuffling. So I learn their names in the same number of repetitions as everyone else, but I hit the magic number soon, creating the impression of increased instructor caring. Also the increased numbers of assignments are perceived as hard, but not pointless busywork. A large number of assignments students thought were stupid probably wouldn’t help. It can be concluded that the setup of this class lends itself to demonstrate to students that the instructor is more intune with student performance than perhaps a traditional class that requires less paper traffic between student and instructor. The open door policy and availableness, however, is a more individual practice of any given professor. This particular instructor showed students that she cared by providing an increased amount of availability. Reflection In 12 interviews, 7 people attributed the reflection components of the class to be a primary reason for their success. The design of this class creates numerous opportunities for 35 students to reflect on their own work. After each lab, students complete a postlab where they list the components of the lab they felt they excelled in, as well as the parts they feel they struggled with the most. This formative assessment provides the instructor with information about how each student in the class is fairing within the unit. In his first interview, Cameron was very vocal about his thoughts on the reflection components of this class. He had even more to say during his second interview: This class has a lot of words. I have to write paragraphs...the reflection after we do the math, where we have to write about what we did. Used to be, I just had to do the problem and not know why it works. But now I have to do the problem and explain why it works. It’s difficult, but it helps. Cameron was emphatic about how the reflections he completed in this class enhanced his learning in an exponential way compared to previous class experiences. He attributed the reflection to his overall success in the class. The most prominent reflection components of this class include the postlab and test corrections. In a CLEAR Calculus class, students are permitted to correct their test to earn back up to 50% of the points they missed, but only if the corrections are wholly correct and accompanied by a sentence per question describing the error that the student originally made and what the student did to fix the error. In his second interview, Taylor speaks very candidly about how he feels about having a second try at test problems: I'm either on or I'm completely off. I'm either getting everything I should be getting or stuff I should be getting, I'm falling apart on the test. The testcorrections or whatever, I open them up and like—for related rates I opened the page and was like, "Gosh dangit, that was the freaking answer." I knew exactly what I did wrong. I think test corrections do a lot to help you understand how to do that so you don't miss it again. That's helped me with a lot of things like related rates and optimization. It's helped me understand what I did wrong. 36 Taylor continued to elaborate on how he puts forth extra effort to insure that his testcorrections are wholly correct before he submits them for grading. Because of the reflection component of test corrections, students are being cautious in their resubmissions, and it's having a very positive effect on their understanding. Sarah explained in her first interview that she is a poor test taker and enjoys test corrections because she strengthens her test taking ability as well as has an opportunity to see where she went wrong and how she needs to fix it. The instructor reviewed weekly postlab reflections after each lab day. Upon review, the instructor altered the lecture content for the remainder of the week to ensure that the topics mentioned in the “Understood least” section of the postlab would be covered in depth. This way, the content students struggle with the most was covered again; whereas class time was not spent on the content students struggled with the least. Students notice change in the weekly topics based on the items they write in their reflection pieces: In his first interview, Kyler noticed this and said that it has shown him that he can get help now. Students find that when they articulate their work, they are more effective in conceptualizing the content. Adaptive Learning Habits Students’ mindsets can be determined by administering a Patterns of Adaptive Learning Scale (PALS) survey. Mindsets were measured in the first 3 weeks and last 3 weeks of the semester in order to determine any change in mindset throughout the course by utilizing a Likert scale survey. There was no statistically significant change in median mindset (p=0.18). Although few students showed any increase or decrease in score by the end of the semester, students’ adaptive learning habits changed drastically, implying that students showed initiative in their own learning experience. 37 Students found themselves demonstrating traits that were uncharacteristic of their normal learning habits. For example, Taylor said in his second interview that he found he was “more successful when he asked more questions in class, worked problems repeatedly, sought help in the math lab, etc.” These habits are demonstrated in the growth mindset student, however the fixed mindset students were demonstrating these habits without a mindset score change. This implied that the course encouraged the development of adaptive learning traits. Lana was a very fixed mindset student. In her first interview, she stated that she had “never been pushed so hard in a math class before.” In high school, she was accustomed to little challenge and found great success in her math classes. However, in the first few weeks of this introductory calculus course she experienced a culture shock. She was no longer being successful with minimum effort. During a very trying semester, she found that in order to be successful in Calculus, she had to spend time preparing and studying outside of class. These newly adopted adaptive learning habits steered Lana into being a very successful student. She showed much improvement in her grade in the course, as well. Students stated in their interviews the many ways that they have shown growth or improvement in their adaptive learning habits. Chase said he “doesn’t wait until the last minute to start the homework anymore, and gets help whenever needed.” Ricky said that this class has taught him “better study habits.” Jayce said that because of the difficulty of the course, he has to “stay on top of his workload and his time management.” Adam, a nontraditional student, said, “It’s all about retention. There is absolutely no substitute for number crunching. I’m still trying to learn how to learn. I haven’t found a successful form of retention aside from working lots of problems.” These students have had to take ownership of their own learning in this class. Due to 38 the rigor of the course, many were challenged in a way they had not been before. This caused them to adapt to the difficulty by changing their learning habits. There was a positive shift of adaptive learning habits seen in students, however there was not a significant shift in the PALS scores. Students took the PALS survey at the beginning and end of the semester to measure any growth or decay in their mindset score. However, the only shift in the scores was within the margin of error. It can be concluded that the students demonstrated a shift in habits but not necessarily mindset. The findings were helpful in answering the research question: What are the features of CLEAR Calculus that promote positive changes in students’ mindsets? Although students did not show a significant numerical increase in mindset score, their actions and classroom behaviors were very clearly affected in this class. Students increased their adaptive learning habits dramatically, demonstrating a change in mindset. DISCUSSION It can be determined that students that are willing to apply themselves during the lab work, even though there is a larger degree of challenge than in comparison to previous math classes, are shown to experience more chronic success than those that are not challenged at a rate higher than their average performance rate. The students that had a high completion rate of the reflection components ended the semester with a more profound understanding of the mathematics. This is due to the fact that the students were required to articulate their understanding and comprehension of the concepts, resulting in a deeper understanding of the content. Students demonstrated a dramatic increase in adaptive learning habits, confirming the hypothesis that the CLEAR Calculus curriculum fosters a growth mindset in students. The 39 features of this class that resulted in a mindset increase were framed around the sense of challenge and rigor. Although some students did not respond positively to the increased challenge, the students that persevered through the difficult units saw great amounts of success in the class and an increase in adaptive learning habits. The increased adaptive learning habits should have been exhibited as an increase in the growth mindset score on the postPALS survey. However, this did not reflect on the postPALS survey administered in the last quarter of the semester. The lack of increase in mindset score was very surprising. It is believed that the increase in adaptive learning habits accounts for an increased mindset score, even though it was not reflected on the survey. Students demonstrated growth tendencies without actually experiencing a mindset change. The large amount of students that cited instructor caring to be a feature of the CLEAR Calculus class, a feature that is not found in other math classes, helped to discount the statistic that students often leave the STEM field due to the feeling that the instructor does not care and that the students feel they cannot get help (Ellis & Rasmussen, 2014). When asked, all students stated that they feel as if the instructor is very approachable, cares about the students’ success in the class, and is willing to help students. It could be speculated whether or not the feature of instructor caring is truly a feature of the CLEAR Calculus curriculum, or simply a feature of the particular instructor that taught this CLEAR Calculus class. These results suggest that the CLEAR Calculus curriculum creates an environment that fosters the growth mindset. Successful students in this class took ownership of their own learning in ways not found in a traditionally taught calculus class. Students were not merely given an answer, but rather they were required to search for and defend the answer they decided upon. The group work allowed students the opportunity to discuss and debate the inner workings of the 40 each calculus lab in ways that enhanced the learning experience for the students involved. It became clear that students that are given more opportunities to do exploratory learning through lab work and frequent reflections over such work are much more equipped to retain the calculus past a test, and quite possibly past a semester. A limitation of this study included the longterm retention of the material in students. If possible, it would have been rather interesting to quiz students based on knowledge and information they were taught in this untraditional Calculus I course a semester, or even a year after the class in order to determine the amount of information they truly retained. A student of a traditional calculus class could also be included in order to determine the percentage of retention of students in each class. It is vitally important for successful calculus programs to be found and promoted in order to strengthen the calculus foundation of STEM students in universities across the country. The ultimate goal is to retain students in the STEM field throughout the entirety of their degree. This starts with a thorough Calculus I foundation. By completing this thesis, I have learned plenty not only as a researcher, but also as an educator. The trends that I have observed will assist me in observing students and their motivation in the classroom. By knowing if a student is a fixed or growth minded student, I can do a more efficient job assisting them in areas that they may struggle. 41 References Black, P., & Wiliam, D. (1998). Inside the black box: Raising standards through classroom assessment. Phi Delta Kappa, October 1998, 139148. Black, P., & Wiliam, D. 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Necessary and sufficient conditions for student success in a repeated course. Accepted pending revisions at Educational Studies in Mathematics. Dweck, C. (2006). Mindset: The new psychology of success. New York: Random House. Ellis, J., & Rasmussen, C. (2014). Student perceptions of pedagogy and associated persistence in calculus. ZDM, 46(4), 661673. 42 Gall, M. D., Borg, W. R., & Gall, J. P. (1996). Educational research: An introduction. Longman Publishing. Ganter, S. L. (2006). Calculus and introductory college mathematics: Current trends and future directions. MAA NOTES, 69, 46. Gredler, M. E., & Shields, C. C. (2008). Vygotsky's legacy: A foundation for research and practice. Guilford Press. Midgley, C., Maehr, M. L., Hruda, L. Z., Anderman, E., Anderman, L., Freeman, K. E., & Urdan, T. (2000). Manual for the patterns of adaptive learning scales. Ann Arbor, 1001, 481091259. Oehrtman, M. (2008). Layers of abstraction: Theory and design for the instruction of limit concepts. Making the connection: Research and practice in undergraduate mathematics, MAA Notes Volume, 73, 6580. Oehrtman, M. (2009). Collapsing dimensions, physical limitation, and other student metaphors for limit concepts. Journal for Research in Mathematics Education, 396426. Patton, M. (2002). Qualitative research and evaluation methods. Thousand Oaks, CA: Sage Publications. President’s Council of Advisors on Science and Technology (PCAST). (2012). Engage to excel: Producing one million additional college graduates with degrees in Science, Technology, Engineering, and Mathematics. Washington, DC: The White House. Seymour, E., & Hewitt, N. M. (1997). Talking about leaving: Why undergraduate leave the sciences. Boulder, CO: Westview Press. Steed, L. A. (Ed.). (1988). Calculus for a New Century: a Pump, not a Filter, Mathematical Association of America, MAA Notes Number 8. DC: Washington. 43 Treisman, U. (1992). Studying students studying calculus: A look at the lives of minority mathematics students in college. College Mathematics Journal, 362372. Wake. G. (2011). Introduction to the Special Issue: deepening engagement in mathematics in preuniversity education. Research in Mathematics Education 13(2). 109118. 44 APPENDICES 45 APPENDIX A: Syllabus Math 2143.002 Calculus I COURSE SYLLABUS: Fall 2015 COURSE INFORMATION Materials Textbook(s) Required: Calculus, 7th Edition, by James Stewart. Material covered during the session will be Sections 1.41.8, Chapters 2, 3, and 4, and 6.2, 6.3, and 6.4. We may occasionally cover enrichment activities not in the text. Optional: How to Ace Calculus/How to Ace the Rest of Calculus by Adams et al. Calculus II is split between the two books (Calculus I is entirely in the first book), but used copies can generally be found for under $5 on Amazon.com. Course Description: This course examines differential and integral calculus of functions of one variable, as follows. Topics include limits; continuity; derivatives; curve sketching; applications of the derivative; the definite integral; derivatives and integrals of trigonometric functions ; and use of computer technology. Prerequisite Two years of high school algebra and trigonometry or Math 142. Student Learning Outcomes 1. Students will demonstrate proficiency in the use of mathematics to structure their understanding of and investigate questions in the world around them. 2. Students will demonstrate proficiency in treating mathematical content at an appropriate level. 3. Students will demonstrate competence in the use of numerical, graphical, and algebraic representations. 4. Students will demonstrate the ability to interpret data, analyze graphical information, and communicate solutions in written and oral form. 5. Students will demonstrate proficiency in the use of mathematics to formulate and solve problems. 6. Students will demonstrate proficiency in using technology such as handheld calculators and computers to support their use of mathematics. Student Assessment Outcomes 1. Critical Thinking: The above learning objectives will be assessed for critical thinking in labs and other classroom activities. 2. Written, Oral, & Visual Communication: Students will be assessed on written, oral, and visual communication skills on their quizzes, exams, labs, and lab jigsaw activities. 3. Empirical and quantitative reasoning: All assessments in this course will contain a quantitative reasoning and empirical computation component. COURSE REQUIREMENTS Labs: On Tuesdays we will work in small groups on activities that develop the central concepts in the course. Attendance and participation is especially crucial on these days. You will turn in individual writeups of these labs activities. It is also important to ask questions of the other groups (who will generally work on related but slightly different problems than your own group) when they present as you will be responsible for all the problems on exams. Prelabs/Postlabs: The purpose of these assignments is to help me determine where the class is at and how much time we should spend on a particular topic. Prelabs are expected to be completed before class, and postlabs will be completed at the end of class on Tuesday. These assignments will be graded on completion. Attendance: There may be topics covered in class that are not in the text. You are responsible for all material covered. I don't take attendance, but there is a strong correlation between attendance and final grades. Missing class more than once or twice during the semester is likely to affect your grade, either directly or indirectly. If you do miss class, you should get notes and/or handouts from your classmates and see me during office hours. Homework: There will be suggested problems assigned for each section. The answers to most of these problems are in the text, so I will not collect them. However, you will see some of these problems (verbatim or with slight variations) on tests, so completing the problems is strongly encouraged! 46 The key to success in this course is regularly working with other students in the class, doing the homework early and asking questions when you have them!!! We will discuss homework problems in class, but there will often not be enough time to discuss all of them. Please come to office hours or visit the math tutoring lab if you have additional questions about the homework. Quizzes: There will be 11 take home quizzes based upon the suggested homework problems throughout the semester. Your best 10 scores will count for your final grade. Exams: We will have four inclass exams (roughly covering Chapters 7, 8, 9, and 10), and a comprehensive final exam. The final exam will be a joint final with the other section of Calc II. The data and time is TBD at the moment. Makeup exams are possible only if there is a documented emergency. Workload and Assistance: You should expect to spend 8 to 12 hours each week, outside of class, on the course material. This includes reading, homework, and studying for quizzes and exams. Some weeks (those in which an exam is scheduled, for instance) may require more of your time, other weeks may require less, but on average, budget 8 to 12 hours each week. I can’t stress enough that in order to be successful in this class you should spend much of this time working with other students in the class! Please ask questions and seek assistance as needed. You may email me at any time, and I encourage you to make use of my office hours Grading This class will be graded on a total points system. 1000 points are possible in the class. Assignments are weighted in the following manner: Assignments Total points possible Lab writeups 200 Reading sheets, 50 prelabs, postlabs Best 10 quizzes 200 Tests 100 each, 400 total Final 150 All point totals will be rounded to the nearest whole points before grades are assigned. Point ranges for final grades will be as follows: A: 900 – 1000 points B: 800  899 points C: 700  799 points D: 600  699 points F: 0 – 599 points TECHNOLOGY REQUIREMENTS Use of a graphing calculator having at least the capabilities of the TI83 will be helpful throughout the course. TI89 is highly recommended. A computer algebra system will be used for some problem exploration, enhanced conceptual understanding, and to engage students as active participants in the learning process. COMMUNICATION AND SUPPORT My primary form of communication with the class will be through Email and Announcements. Any changes to the syllabus or other important information critical to the class will be disseminated to students in this way via your official University Email address available to me through MyLeo and in Announcements. It will be your responsibility to check your University Email and Announcements. Students who Email me outside of regular office hours can expect a reply within 24 hours MF. Students who Email me during holidays or over the weekend should expect a reply by the end of the next regularly scheduled business day. myLeo Support Your myLeo email address is required to send and receive all student correspondence. Please email helpdesk@tamuc.edu or call us at 9034686000 with any questions about setting up your myLeo email account. You may also access information at https://leo.tamuc.edu. 47 COURSE AND UNIVERSITY PROCEDURES/POLICIES Academic Honesty Students who violate University rules on scholastic dishonesty are subject to disciplinary penalties, including (but not limited to) receiving a failing grade on the assignment, the possibility of failure in the course and dismissal from the University. Since dishonesty harms the individual, all students, and the integrity of the University, policies on scholastic dishonesty will be strictly enforced. In ALL instances, incidents of academic dishonesty will be reported to the Department Head. Please be aware that academic dishonesty includes (but is not limited to) cheating, plagiarism, and collusion. Cheating is defined as: Copying another's test of assignment Communication with another during an exam or assignment (i.e. written, oral or otherwise) Giving or seeking aid from another when not permitted by the instructor Possessing or using unauthorized materials during the test Buying, using, stealing, transporting, or soliciting a test, draft of a test, or answer key Plagiarism is defined as: Using someone else's work in your assignment without appropriate acknowledgement Making slight variations in the language and then failing to give credit to the source Collusion is defined as: • Collaborating with another, without authorization, when preparing an assignment If you have any questions regarding academic dishonesty, ask. Otherwise, I will assume that you have full knowledge of the academic dishonesty policy and agree to the conditions as set forth in this syllabus. Late Policy: Late work/Makeups will not be accepted without a documentable and valid excuse, because the lowest grade(s) in each category is dropped. Examples of documentable and valid excuses include: *car accident w/ police report *illness w/ doctor’s note (you or your child) *athletic or other mandatory extracurricular travel *field trip for another class *being detained upon entering the country by Homeland Security University Specific Procedures ADA Statement Students with Disabilities The Americans with Disabilities Act (ADA) is a federal antidiscrimination statute that provides comprehensive civil rights protection for persons with disabilities. Among other things, this legislation requires that all students with disabilities be guaranteed a learning environment that provides for reasonable accommodation of their disabilities. If you have a disability requiring an accommodation, please contact: Office of Student Disability Resources and Services Texas A&M UniversityCommerce Gee Library Room 132 Phone (903) 8865150 or (903) 8865835 StudentDisabilityServices@tamuc.edu Student Conduct All students enrolled at the University shall follow the tenets of common decency and acceptable behavior conducive to a positive learning environment. (See Code of Student Conduct from Student Guide Handbook). COURSE OUTLINE / CALENDAR 1). 1.4, 1.5, 1.6 2). 1.8, 2.1, 2.2 3). 2.3, 2.4, 2.5 4). Test I, 2.6 5). 2.7, 2.8, 2.9 6). 3.1, 3.2, 3.3 7). 3.4, Test II 8). 3.5, 3.6, 3.7, 3.8 9). 3.9, 4.1 10). 4.2, Test III 11). 4.3, 4.4 12). 4.5, 6.2 13). 6.3, 6.4, 1.7 14). Test IV 15). Review 16) FINAL WEEK 48 APPENDIX B: Labs The following labs are multiweek labs that deal with approximation. Each lab is accompanied by selected examples and starting problems for each major lab. Lab 1 – Rates and Amounts of Change Instructions: Work with your group on the problem assigned to you. We encourage you to collaborate both in and out of class, but you must write up your responses individually. Your work must be neat and include sufficient exposition to make the solution clear to another student who has not seen the assignment (for example, a sequence of equations without explanation will most likely receive zero credit). Pay particular attention to places where explanations using multiple representations are requested, and explicitly discuss the connections between your explanations using different representations. Type or write all of your work legibly on 8½"×11" paper with at least oneinch margins on all sides free of writing except your name, date, and assignment number, and staple all pages together. In the following, consider plotting height of water in a bottle vs. the volume of the water in the bottle. That is, height is on the vertical axis (dependent variable) and volume is on the horizontal axis (independent variable). Lab Preparation: The definition of an increasing function f is that 1 2 f (x ) f (x ) whenever 1 2 x x . The graph of height vs. volume must always be increasing (going up as we move from left to right) since more volume has to correspond to more height. 1) Explain the meaning of expressing the relationship between height and volume using the function notation h(V ) , and 2) include a description of the meaning of the equation h(3) 5 if volume is measured in cups and height is measured in inches. 3) Finally, rewrite the definition of an increasing function using h(V ) instead of f (x) and explain the meaning of this definition in terms of water in a bottle. 1. Steepness of the graph is related to the crosssectional area of the bottle. Explain why a steeper graph corresponds to a narrower bottle and a less steep graph corresponds to a wider bottle, as shown to the right. Make sure that you are talking meaningfully about the rate of change of height with respect to volume by breaking down your explanation in terms of amounts of change in height and amounts of change in volume. Lab 3: Locate the Hole Graph 1: The graph of 3 7 2 ( ) 1 x f x x has a hole. Your task is to determine the location of this hole using approximation techniques (no fancy limit computations allowed). Lab Preparation: Answer the following questions individually and bring your writeup to class. a. Draw a graph of f using an entire sheet of paper. Your graph should be drawn at a scale that gives a good sense of the x,ycoordinates of the hole. The x and y scales should be chosen so that your graph nearly extends between two diagonally opposites corners of the page. b. Identify what unknown numerical value you will need to approximate. Give it an appropriate shorthand name (that is, a variable name). c. Describe what you will use for approximations. Write a description of your answer using algebraic notation (for example, function notation, variables, formulas, etc.) Lab: Work with your group on the problem assigned to you. We encourage you to collaborate both in and out of class, but you must write up your responses individually. 1. Find an approximation to the height of the hole in your function (write out the approximation with several decimal places). Is this an underestimate or overestimate? Explain how you know. Find both an underestimate and an overestimate. Lab 5: The Derivative at a Point – Part 1 Lab Preparation: Answer the following questions individually and bring your writeup to class. A. Draw a fullpage picture of the physical context in three different configurations (specified in the context description) overlayed. This will help illustrate how the relevant quantities are changing in the region of interest. You will redraw and add to this picture during class. 49 B. What happens to the changes in the dependent quantity (volume, gravitational force, or mass of Iodine123) as the independent quantity (radius, separation distance, or time) is incremented by constant amounts? Is your rate of change constant, increasing or decreasing? C. Draw a fullpage graph showing the relationship between the two quantities involved in the instantaneous rate that you are asked to approximate. Add a point for each of the configurations you drew in your picture from A. Represent the changes in both quantities on your graphs as the length of short line segments. You will redraw and add to this graph during class. D. On your picture and your graph illustrate and label the changes in the relevant quantities to support your answer to B (using both Δnotation and numerical values). Lab 7: Linear Approximation The NASA Q36 Robotic Lunar Rover can travel up to 3 hours on a single charge and has a range of 1.6 miles. After leaving its base and traveling for t hours, the speed of the Q36 is given by the function 2 v(t) sin 9 t in miles per hour. One hour into an excursion, the Q36 will have traveled 0.19655 miles. Two hours into a trip, the Q36 will have traveled 0.72421 miles. Lab Preparation: Answer the following questions individually and bring your writeup to class. 1. Use your calculator to graph v(t) . Explain in words what the graph says about how the Q36 moves during a 3 hour trip starting with a full charge. 2. Using a full sheet of paper, carefully sketch a graph of the distance x(t) traveled by the Q36 measured in miles during this trip as a function of time in hours. Explain precisely why you drew the graph as you did. 3. Draw tangent lines to the graph of x(t) at times 0 t 0, 0 t 1, 0 t 2 , and 0 t 3. Label each tangent line with its equation. Use the variables x and t in these equations. Lab 8: Quadratic Approximation The NASA Q36 Robotic Lunar Rover can travel up to 3 hours on a single charge and has a range of 1.6 miles. After leaving its base and traveling for t hours, the speed of the Q36 is given by the function 2 v(t) sin 9 t in miles per hour. One hour into an excursion, the Q36 will have traveled 0.19655 miles. Two hours into a trip, the Q36 will have traveled 0.72421 miles. In Lab 7, you used tangent lines to approximate the distance x(t) traveled by the Q36 Rover. Lines with slope m through the point 0 0 (t , x ) can be written in pointslope form as 0 0 x x m(t t ) . You used the derivative v(t) x(t) to find the slope at 0 t . We could improve our approximations by using “best fit parabolas.” For the following problems, note that 2 0 0 0 x x a(t t ) b(t t ) is the equation of a parabola that passes through the point 0 0 (t , x ) . Changing the parameters a and b will change the shape of the parabola without changing the fact that it passes through that point. Lab Preparation: Answer the following questions individually and bring your writeup to class. a. Sketch a parabola on your large graph of x(t) that you think represents the best fit parabola at time 0 t 1. Then determine the equation of this parabola using the form 2 0 0 0 x x a(t t ) b(t t ) and the point 0 0 (t , x ) (1,0.19655) . To do this, find the first and second derivatives of the equation for this parabola, set x(1) v(1) and x(1) v(1) , then solve for a and b. b. Sketch a parabola on your large graph of x(t) that you think represents the best fit parabola at time 0 t 2 . Then determine the equation of this parabola using the form 2 0 0 0 x x a(t t ) b(t t ) and the point 0 0 (t , x ) (2,0.72421) . To do this, find the first and second derivatives of the equation for this parabola, set x(2) v(2) and x(2) v(2) , then solve for a and b. 50 Lab 12: Definite Integrals – Part 1 Work with your group on the context assigned to you. We encourage you to collaborate both in and out of class, but you must write up your responses individually. Context 1: Sam is tired of walking up two flights of stairs to get to calculus class every day, so he enlists Kelli to help him build a giant spring to lift him perfectly up to the second floor window. They order a twostory tall spring from Katelyn’s Giant Spring Limited Liability Co. When it arrives, it is packaged already compressed down 5 m shorter than its resting length. They figure they need to compress it another 5 m in order to climb on from ground level before launch. Tony walks by and points out this will take a lot of energy, saying: For a constant force* F to move an object a distance d requires an amount of energy** equal to E Fd . Hooke’s Law says that the force exerted by a spring displaced by a distance x from its resting length (compressed or stretched) is equal to F kx , where k is a constant that depends on the particular spring. *The standard unit of force is Newtons (N), where 1 N = 1 kg·m/s2 or the force required to accelerate a 1 kg mass at 1 m/s2. Increasing either the mass or the acceleration rate therefore requires a proportional increase in force. **The standard unit of energy is Joules (J), where 1 J = 1 N·m or the energy required to move an object with a constant force of 1 N a distance of 1 m. Increasing either the force or the distance requires a proportional increase in energy. Sam and Kelli’s spring has a spring constant of k 155N/m. Lab Preparation: 1. Draw and label a large picture of a spring initially compressed 5 m from its natural length then compressed to a displacement of 10 m. 2. Does it take less, the same, or more energy to compress the spring from 5 m to 7.5 m than it takes to compress the spring from 7.5 m to 10 m? Explain. 3. Explain why we cannot just multiply a force times a distance to compute the energy. 51 APPENDIX C: Patterns of Adaptive Learning Scale (PALS) Below is the PALS survey that was given to students in the first 3 weeks of the semester and again in the last 3 weeks of the semester. Part I: Information Gender: ☐ Male ☐ Female Class: ☐ Freshman ☐ Sophomore ☐ Junior ☐ Senior ☐ Graduate Ethnicity: ☐ Native American ☐ African American ☐ Latino/a ☐ Caucasian ☐ Asian ☐ Multiracial (Specify: _________________) ☐ Other: _____________ Native Language: ☐ English ☐ Spanish ☐ Other: _____________ ACT Math Score: ____________ Have you had a calculus class before this class? ☐ Yes ☐ No Here are some questions about you as a student in this class. Please circle the number that best describes what you think. 1. I would avoid participating in class if it meant that other students would think I know a lot. 2. It’s important to me that I don’t look stupid in class. 3. If other students found out I did well on a test, I would tell them it was just luck even if that wasn’t the case. 4. I would prefer to do class work that is familiar to me, rather than work I would have to learn how to do. 5. It’s important to me that other students in my class think I am good at my class work. 6. It’s important to me that I learn a lot of new concepts during this class. 7. I’m certain I can figure out how to do the most difficult class work. 1 2 3 4 5 Not true at all Somewhat true Very true 1 2 3 4 5 Not true at all Somewhat true Very true 1 2 3 4 5 Not true at all Somewhat true Very true 1 2 3 4 5 Not true at all Somewhat true Very true 1 2 3 4 5 Not true at all Somewhat true Very true 1 2 3 4 5 Not true at all Somewhat true Very true 1 2 3 4 5 Not true at all Somewhat true Very true I __________________________ (Print Name), having read this letter and having an opportunity to ask any questions, would like to participate in this research and my signature below indicates my informed consent to participate. A copy of this form will be given to you to retain for future reference. ________________________________ ___________________ Participant signature Date (month/day/year) 52 8. Some students fool around the night before a test. Then if they don’t do well, they can say that is the reason. How true of this of you? 9. Some students purposely get involved in lots of activities. Then if they don’t do well on their class work, they can say it is because they were involved with other things. How true is this of you? 10. When I’m working out a problem, my teacher tells me to keep thinking until I really understand. 11. Some students look for reasons to keep them from studying (not feeling well, having to help their parents, taking care of a brother or sister, etc.) Then if they don’t do well on their class work, they can say this is the reason. How true of this of you? 12. I wouldn’t volunteer to answer a question in class if I thought other students would think I was smart. 13. If I did well on a school assignment, I wouldn’t want other students to see my grade. 14. One of my goals in class is to learn as much as I can. 15. One of my goals is to show others that I’m good at my class work. 16. One of my goals is to keep others from thinking I’m not smart in class. 17. I like concepts that are familiar to me, rather than those I haven’t thought about before. 18. If I were good at my class work, I would try to do my work in a way that didn’t show it. 19. It’s important to me that I thoroughly understand my class work. 20. I sometimes copy answers from other students when I do my class work. 1 2 3 4 5 Not true at all Somewhat true Very true 1 2 3 4 5 Not true at all Somewhat true Very true 1 2 3 4 5 Not true at all Somewhat true Very true 1 2 3 4 5 Not true at all Somewhat true Very true 1 2 3 4 5 Not true at all Somewhat true Very true 1 2 3 4 5 Not true at all Somewhat true Very true 1 2 3 4 5 Not true at all Somewhat true Very true 1 2 3 4 5 Not true at all Somewhat true Very true 1 2 3 4 5 Not true at all Somewhat true Very true 1 2 3 4 5 Not true at all Somewhat true Very true 1 2 3 4 5 Not true at all Somewhat true Very true 1 2 3 4 5 Not true at all Somewhat true Very true 1 2 3 4 5 Not true at all Somewhat true Very true 53 21. I would choose class work I knew I could do, rather than work I haven’t done before. 22. One of my goals is to show others that class work is easy for me. 23. Some students let their friends keep them from paying attention in class or from doing their homework. Then if they don’t do well, they can say their friends kept them from working. How true is this of you? 24. Some students purposely don’t try hard in class. Then if they don’t do well, they can say it is because they didn’t try. How true is this of you? 25. One of my goals is to look smart in comparison to the other student sin my class. 26. One of my goals in class is to avoid looking smarter than other kids. 27. Some students put off doing their class work until the last minute. Then if they don’t do well on their work, they can say that is the reason. How true is this of you? 28. I can do almost all the work in class if I don’t give up. 29. One of my goals in class is to avoid looking like I have trouble doing the work. 30. Even if the work is hard, I can learn it. The following questions are about this class and about the work you do in class. Remember to say how you really feel. No one at school or home will see your answers. 31. In our class, trying hard is very important. 32. In our class, showing others that you are not bad at class work is really important. 33. In our class, how much you improve is really important. 1 2 3 4 5 Not true at all Somewhat true Very true 1 2 3 4 5 Not true at all Somewhat true Very true 1 2 3 4 5 Not true at all Somewhat true Very true 1 2 3 4 5 Not true at all Somewhat true Very true 1 2 3 4 5 Not true at all Somewhat true Very true 1 2 3 4 5 Not true at all Somewhat true Very true 1 2 3 4 5 Not true at all Somewhat true Very true 1 2 3 4 5 Not true at all Somewhat true Very true 1 2 3 4 5 Not true at all Somewhat true Very true 1 2 3 4 5 Not true at all Somewhat true Very true 1 2 3 4 5 Not true at all Somewhat true Very true 1 2 3 4 5 Not true at all Somewhat true Very true 54 34. In our class, getting good grades is the main goal. 35. In our class, really understanding the material is the main goal. 36. In our class, it’s important that you don’t make mistakes in front of everyone. 37. In our class, it’s important to understand the work, not just memorize it. 38. In our class, it’s important not to do worse than other students. 39. In our class, learning new ideas and concepts is very important 40. In our class, it’s OK to make mistakes as long as you are learning. 1 2 3 4 5 Not true at all Somewhat true Very true 1 2 3 4 5 Not true at all Somewhat true Very true 1 2 3 4 5 Not true at all Somewhat true Very true 1 2 3 4 5 Not true at all Somewhat true Very true 1 2 3 4 5 Not true at all Somewhat true Very true 1 2 3 4 5 Not true at all Somewhat true Very true 1 2 3 4 5 Not true at all Somewhat true Very true 1 2 3 4 5 Not true at all Somewhat true Very true 



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