chapter 1

RUSLAN FLEK, YOEL RODRIGUEZ, FRANCISCO FERNANDEZ

CHAPTER 4.4 BRIDGING THEORY AND APPLICATIONS: IMPROVING STUDENTS' MATHEMATICAL PROFICIENCY FOR THE PHYSICAL SCIENCES WITH EMPHASIS ON PHYSICS The Birth of a Teaching-Research Question

The educational philosophy and the institutional structure of Eugenio Maria de Hostos Community College (HCC) is driven by its history, its student population and its overarching mission: Consistent with the mission of The City University of New York (CUNY) to provide access to higher education for all who seek it, Eugenio María de Hostos Community College was established in the South Bronx to meet the higher educational needs of people from this and similar communities who historically have been excluded from higher education. The mission of Eugenio María de Hostos Community College is to offer access to higher education leading to intellectual growth and socio-economic mobility through the development of linguistic, mathematical, technological, and critical thinking proficiencies needed for lifelong learning and for success in a variety of programs including careers, liberal arts, transfer, and those professional programs leading to licensure (Hostos, 2014) The college takes great pride in its historical role in cultivating students from diverse ethnic, racial, cultural and linguistic backgrounds, ― “An integral part of fulfilling its mission is to provide transitional language instruction for all English-as-aSecond-Language learners,” as well as a variety of different specialized “education offerings to foster a multicultural environment for all students.” (Hostos, 2014) To further advance the college’s goal and to foster a rich learning environment, equipping students with necessary tools and affording them accessibility to a wide spectrum of opportunities, HCC, together with some of its senior CUNY college partners, currently offers several joint-degree/dual-admission programs: A.A./B.A in Criminal Justice, jointly with John Jay College, A.S./B.S. in Forensic Science, also jointly with John Jay College, and several A.S./B.E. programs, jointly with City College (CCNY). The latter include degrees in Mechanical, Civil, Chemical, Electrical and Environmental Engineering. Additional joint A.S./B.S. programs, in Chemistry, Earth Science and Biology, are presently being developed in association with Lehman College with the assistance of an NSF grant, and will be available to HCC students soon (Project SEED, 2012). Unlike articulation agreements, the current HCC-CCNY engineering partnerships are jointly registered, dual admission programs designed to meet the licensure guidelines of the Accreditation Board for Engineering and Technology (ABET).

FLEK, RODRIGUEZ, FERNANDEZ

Students entering the program are granted dual admission to HCC and CCNY; furthermore, CCNY guarantees admission to HCC students who successfully complete the HCC A.S. degree. Students begin the program at HCC, and must fulfil the same curriculum requirements as the first two years of the corresponding licensure-qualifying CCNY engineering programs, earning an A.S. in Engineering in the process. Students continue their studies at CCNY, and, upon successful completion of the entire program, are awarded a B.E. degree. The collegial nature of the program facilitates the transition to the professional portion of the curriculum. Since its inception in 2007, the number of students enrolled in these programs has been growing significantly with every semester, successfully attracting more students to these disciplines. This, in turn, increases the demand for mathematics and science courses at HCC, as well as the need for an adequate amount of effective educational support services for STEM students. The workshops explored in this article were originally developed to meet this need. Based on the continuously increasing number of STEM students at HCC, and anticipating an even more rapid growth in the near future, the importance of productive academic support systems and/or interventions is self-evident. Counsellors, deans and faculty members in the related departments are actively invested in the college’s students’ success, and have been involved in several activities to enhance the educational experiences of HCC’s STEM students. The workshops discussed here were among the first of such support mechanisms, developed collaboratively by the Mathematics and the Natural Sciences departments. The teaching methodologies utilized and the point of view from which they were designed, discussed in detail later in the chapter, aim to address common difficulties that students, historically, encounter in chemistry and physics courses. In the authors’ opinions, all of the factors mentioned above, led to the development of very effective preparatory workshops relying heavily on the initial collaborative atmosphere surrounding their design as well as the highly collaborative and adaptive teaching approach employed on a daily basis during the actual instructional periods. THE HOSTOS INTERSESSION SCIENCE INSTITUTE: FOCUS ON PHYSICS

The Natural Science Department jointly with the Mathematics Department at Hostos Community College currently offers two semesters of Physics: Physics I and Physics II. Both courses are calculus-based with Calculus I being the pre-requisite for Physics I, and Physics I a pre-requisite for Physics II. As is commonly the case, HCC students find physics to be a very difficult and challenging subject. This assertion holds true for most engineering and science students across the nation (AACU 9). Physics is hard; no doubt about it! Physics as a discipline is one of the biggest obstacles in the progression and retention of STEM students. The most common difficulties found among HCC students taking physics courses, as is the situation nationwide, are inadequate problem-solving foundations, lack of abstract-analysis skills and a lack of sufficient ability to make connections with previous knowledge, specifically, with concepts in mathematics such as trigonometry, geometric and algebraic operations on vectors, and an adequate understanding of introductory calculus (Zumdahl A266). Some of these students have never taken physics, or any

BRIDGING THEORY AND APPLICATIONS: MATHEMATICS AND PHYSICS

science course, for that matter, during their previous educational endeavours. As a result, by the time they take the course, they often already have come to “dislike” physics, and feel frustrated and discouraged because they do not understand it. Furthermore, even when they understand the concepts, they find it very difficult to solve physics problems because of lack of visualization and critical thinking skills. These could be some of the reasons why many students do not consider science careers as professional options despite the great demand in the United States for potential scientists (AACU 9; Mervis, “NIH Told” 328; Mervis, “NIH Wants” 1119; Rochin and Mello 305), as well as mathematics and science middle school and high school teachers. To address this problem, among others, the Natural Sciences Department and the Mathematics Department, with the help of the Office of Academic Affairs at Hostos, created and began offering the Intersession Science Institute in the winter of 2010. This institute was tailored primarily for students enrolled in one of the four initial Hostos Engineering programs (Civil, Chemical, Electrical and Mechanical), but is open to all students who intend to take a Physics or Chemistry course for their respective degrees. Problem-Solving in Physics Problem-solving is a pervasive skill in all sciences, and, therefore, plays the central role in students’ successful learning and genuine understanding of physics. Not surprisingly, this presents the necessity of transforming students into proficient problem solvers as the primary challenge for the instructor, ― a theme common to almost all chapters of this book, and, certainly, not limited to mathematics and physics. This highly sought after student evolution is a common goal of all contributing authors in this publication, as well as most other well-informed and dedicated educators across many different disciplines. It has been the theme of numerous studies and publications, and yet still remains one of the most difficult and elusive objectives of education research. Mestre writes: Teaching students to become proficient problem solvers is [also] one of the most challenging tasks in science courses, especially in disciplines that are highly analytical, such as chemistry and physics. To understand why it is so difficult for high school and college students to develop problem-solving skills in the sciences, we need only examine the ingredients necessary to be proficient at solving problems in a discipline such as physics: (1) An understanding of physics principles and concepts; (2) Ability to recognize which principles and concepts apply to problems varying widely within the discipline; (3) Knowledge of procedures for applying the principles and concepts; (4) Knowledge of the mathematical forms, such as equations and formulas, for the principles and concepts, and (5) Proficiency in the mathematics necessary to execute solutions. (Mestre, 1994) While the first three ingredients above implicitly invoke mathematical concepts, ingredients (4) and (5) clearly affirm the pedagogical interdependence of mathematics and physics, and suggest a collaborative approach to the solution of the underlying research and practical goal of students’ acquisition and retention of knowledge. Profoundly guided by these ideologies, the authors of this chapter


adopted a highly interactive, reflective and collaborative configuration for the Science Institute. The structure of the institute heavily utilizes Collaborative Teaching (CT) and Cooperative Group Problem Solving (CGPS) methodologies: Cooperative Group Problem Solving (CGPS) is a mode of cognitive apprenticeship developed at the University of Minnesota. It is intended to draw students into engaging in expert practice by the use of problems and tasks chosen to illustrate the power of certain problem-solving methods, to give students practice in applying those methods in diverse settings, and to help them develop self-monitoring and correction skills and integrate the skills they need to advance toward expertise (McNeil, 2004). During the workshops, students are given problems to solve in small groups. In their work, they are expected to develop and follow a problem-solving strategy similar to the one explored and modelled during the instructor-guided interactive portion of the workshop session. The strategy consists of focusing and properly interpreting the presentation of the problem, describing the physics, planning a solution, executing the plan, and evaluating the solution. (Heller & Heller, 0000) An important component of this methodology, as it used at the Hostos Intersession Institute, is the incorporation of the collaborative teaching approach that allows the students to make the connections between the physics problem-solving objectives and the corresponding mathematical concepts needed to re-frame the problem and execute the solution in “real-time,” that is, at the pace that both are presented. See Figure 1 for a detailed structure of the Math & Physics workshop. We utilize a methodology that would lie somewhere between the cluster course model and the interdisciplinary model of teacher collaborations. These terms and their explanations are described by Austin and Baldwin (1991): While the term team teaching usually implies two or more faculty working together to teach one course, it also can refer to faculty efforts to coordinate separate courses in different disciplines. The faculty who teach such clusters of courses are engaged in a coordinative team teaching model (Rinn & Weir, 1984). The courses arranged in a cluster each relate to a broad topic and develop different aspects of the topic … The courses each have a departmental home, and each is taught by a different faculty member. The extent of faculty interaction in the cluster course model can vary from periodic meetings to inform each other what the other is discussing to more regular meetings to modify the courses so that common themes can be developed in a complementary way. They go on to say that with increased faculty interaction, as was the case in our workshop, the cluster course model approaches an interdisciplinary model: An interdisciplinary model of team teaching is planned and taught as if knowledge were one and the disciplines had not yet been invented (Rinn & Weir, 1984, p. 5). Typically, interdisciplinary courses draw on materials from various disciplines to explore and analyse a particular theme or issue. The emphasis, however, is not on the distinct disciplines, but on what light the disciplines, in concert, can shed on the topic or issue. Faculty teams usually


involve two to four faculty members, with different disciplinary homes, who meet frequently and teach together in the kind of integrated format described in the interactive team model (Rinn & Weir, 1984). The PI methodology actively involves students in their own learning during lecture and focuses their attention on underlying concepts (Mazur, 2011), and is found to decrease student attrition in introductory physics courses at both four-year and two-year institutions (Lasry 1066). During the Institute, some aspects of PI were used, by choosing a Peer Leader whose role is both that of a student and a session leader. In the past this student was selected prior to the beginning of the Institute based on the relative strength of his/her understanding of the topics to be covered. The intellectual, practical, theoretical and pedagogical connection between mathematics and physics was certainly appreciated by many mathematicians and scientists over the centuries. Our approach, and its underlying theoretical basis, in our opinion, is well-established but, unfortunately, not often practiced. Moore eloquently proposed such a program in his address to the American Mathematical Society in 1902. He referred to it as the Laboratory Method: This program of reform calls for the development of a thoroughgoing laboratory system of instruction in mathematics and physics, a principal purpose being as far as possible to develop on the part of every student the true spirit of research, and an appreciation, practical as well as theoretic, of the fundamental methods of science. In connection with what has already been said, the general suggestions I now add will, I hope, be found of use when one enters upon the questions of detail involved in the organization of the course. As the world of phenomena receives attention by the individual, the phenomena are described both graphically and in terms of number and measure; the number and measure relations of the phenomena enter fundamentally into the graphical depiction, and furthermore the graphical depiction of the phenomena serves powerfully to illuminate the relations of number and measure. This is the fundamental scientific point of view. Here under the term graphical depiction I include representation by models … Even under the present organization of the curriculum, the teachers will find that much improvement can be made by closer cooperation one with another. (Moore, 1903)


Figure 1. Math & Physics Workshop Target Topics Outline


THE HOSTOS INTERSESSION SCIENCE INSTITUTE: A QUICK LOOK AT CHEMISTRY

Figure 2. Math & Chemistry Workshop Target Topics Outline


Figure 3. Math for Chemistry Portion – Target Topics Outline

QUANTITATIVE ANALYSIS: ASSESSMENT STATISTICS

Since its creation, HCC has offered five intersession institutes: winter 2010, summer 2010, winter 2012, summer 2012 and winter 2013. Due to budgetary constraints, the institute was not offered in 2011. The data regarding student retention and course grades was collected, and is summarized below. It is important to note that some students from other CUNY colleges, such as BMCC and City College have also attended the workshops; however, they are excluded from the data analysis below. The data below includes only those students who were enrolled in a Physics I course during the semesters immediately following each of the workshops (spring 2010, fall 2010, spring 2012, fall 2012 and spring 2013). The total number of students enrolled in the Physics course during those five semesters was 107. Out of those 107 students, 29 students attended the institute before enrolling in the course. We have divided the students into two groups: The control group, consisting of those 78 students who took the class and did not take the workshop (No Workshop), and The treatment group, consisting of 29 students who took the class and did complete the workshop prior to enrolment (Workshop)


Workshop Participation and Retention Rates One of the most dramatic indicators is the Retention Rate. Figure 4. Retention Rates

Workshop Participation and Student Grades The next indicator is the difference in the grades earned by those students who did not withdraw from the class during the observed semesters (a total of 71 students). To this end, we used the following grade-point conversion scale (consistent with the HCC grading scheme):

Table 1. Letter Grade Conversion Table Letter Grade A AB+ B BC+ C D F

Points 4.0 3.7 3.3 3.0 2.7 2.3 2.0 1.0 0.0

For final reporting purposes we use the inverse conversion table on the right:

Point Range 3.85 – 4.00 3.50 – 3.84 3.15 – 3.49 2.85 – 3.14 2.50 – 2.84 2.15 – 2.49 1.50 – 2.14 0.50 – 1.49 0.00 – 0.49

Letter Grade A AB+ B BC+ C D F


Table 2. Grade Frequency Distributions for the Two Groups


Workshop Group Points Frequency 4.0 3.7 3.3 3.0 2.7 2.3 2.0 1.0 0.0

Mean = 2.92 (“B”) S.D. = 0.93

5 4 3 5 3 0 7 0 1


VS.

No Workshop Group Points Frequency 4.0 3.7 3.3 3.0 2.7 2.3 2.0 1.0 0.0

Mean = 1.91 (“C”) S.D. = 1.39

N = 28

2 5 3 3 3 5 9 2 11 N = 43

Figure 4. Comparison of Final Grades for the Workshop and the No Workshop Groups Final Grades of the Workshop Group 9

Final Grades of the No Workshop Group

Mean = 2.92

Mean = 1.91

12

8

10

7

Frequency

Frequency

6 5 4 3

8 6 4

2

2

1 0

0

1

2 3 Final Grade (Quality Points)

4

0

0

1

2 3 Final Grade (Quality Points)

4

It is clear from the tables and the associated histograms above that the Workshop group had a higher average, with a difference of 1.01 quality points, and the student results were more consistent for this group (demonstrated by a lower standard deviation). Next, by applying a two sample mean comparison test (including a 95% confidence interval for the mean difference) we obtain: Figure 6. Two – Sample t-Test for Difference in Means


With such a low p-value, we can certainly conclude that the average grade for the Workshop group is statistically significantly higher than the average grade for the No Workshop group, and, at a confidence level of 95%, we can reasonably expect the point difference to be approximately between 0.5 and 1.6. Next, we apply a Chi-Squared (𝜒 2 ) test for independence (see Figure 7 below), obtaining a p-value of 0.016: Figure 7. Pass Rates and Workshop Participation I: Chi-Square Hypothesis Test for Independence

From this assessment, we may conclude, at the significance level of 0.02, that there exists a relationship between taking the workshop and passing the class. In other words, we are at least 98% confident that those students who take the workshop are more likely to pass the course. Pass Rates and Workshop Participation II: Two Proportions Hypothesis Test Table 3. Passing Rates for the Two Groups

Pass Total Rate

Workshop 27 28 96.4%

No Workshop 32 43 74.4%

Total 59 71 + 22.0%

Figure 8. Bar Graph of Passing Rates and a Two Proportions Hypothesis Test


With such a low p-value, we can certainly conclude that the passing rate for the Workshop group is statistically significantly higher than that for the No Workshop group, and, at a confidence level of 95%, we can reasonably expect the passing rate difference to be approximately between 7.3% and 36.8%. ………….. ………….. ………….. ― Content Hidden ― ………….. ………….. …………..

LEARNING COMMUNITIES

The authors would strongly argue that the students attended the workshops for three main reasons, - to learn, to gain an academic edge in the science course they were taking during the semester following the workshop, and to engage with the instructors and other like-minded students, intentionally or subconsciously, beginning to form their own learning community. The latter observation was further evidenced by these students' mutual interactions, as well as their frequent, and welcomed, active pursuit of any necessary assistance and guidance with their respective mathematics or science course material from the workshop instructors throughout the following semester and beyond. The atmosphere of mutual respect and substantive discourse among all of the institute's participants, instructors and students, persisted well beyond its duration, establishing academically productive relationships that lasted all the way until the students' graduation or early transfer to a senior college. This, initially unanticipated, social consequence of the institute, is now properly recognized by the authors as one of its significant objectives. And it is worth noting that in the words of Church and Swain (2009): Co-teaching is often a natural extension of strong professional learning communities. When co-teaching evolves as a result of the ongoing interactions of a professional learning community, it is likely that team members will have worked together long enough to have a strong basis from which to develop well-functioning co-teaching arrangements. This type of collaboration moves the work of the professional learning community directly into the classroom and has the potential to exert a powerful impact on school culture and on student learning. ………….. ………….. ………….. ― Content Hidden ― …………..


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AFFILIATIONS

Ruslan Flek, Ph.D. Director of Quantitative Reasoning Interdisciplinary Science Department The New School University Yoel Rodriguez, Ph.D. Natural Sciences Department Eugenio Maria de Hostos Community College City University of New York Francisco Fernandez, Ph.D. Natural Sciences Department Eugenio Maria de Hostos Community College City University of New York