Biocalculus Calculus For Life Sciences Stewart Pdf Download
Betty Neyhart
Hosts of reports over the past few decades have pointed out the need for quantitative skills and conceptual mathematical foundations for undergraduates studying life science (American Association for the Advancement of Medicine, 2009; American Association for the Advancement of Science [AAAS], 2011; National Research Council [NRC], 2003; Steen, 2005). With the continuing growth of computational and data science approaches across the life sciences, these reports broadly agree that 21st-century biologists will be well-served through enhanced comprehension of the core quantitative concepts used throughout biology. Calculus provides one of the most fundamental mathematical frameworks that underlie science and is universally included as a core course for science, technology, engineering, and mathematics (STEM) students around the world. Calculus is a major component of quantitative training for biology undergraduates (Bressoud et al., 2013, 2015).
For decades calculus has been a required quantitative course for biology undergraduates, and biology students make up nearly 30% of all students taking Calculus 1 across all types of U.S. undergraduate institutions (Bressoud et al., 2013, 2015). The standard mechanism for teaching calculus in the United States has been through formal course sequences designed for a broad collection of science and engineering students. Historically, some institutions have either included life science students in these courses or have developed specialized courses for these students separate from and with somewhat different topic coverage than the standard science and engineering courses. Such specialized courses have sometimes been broadly inclusive of social science students as well, but some have focused explicitly on life science students, because they often make up a significant fraction of all STEM students at an institution. Over recent decades several biocalculus texts were developed that focus on standard calculus topics (Neuhauser, 2011; Adler, 2012; Schreiber et al., 2014) or take a somewhat broader perspective of quantitative topics to include linear algebra, probability, and discrete-time modeling (Bodine et al., 2014; Stewart and Day, 2015).
Our purpose is to explore the development and initial validity assessment of the BioCalculus Assessment (BCA), which aims to evaluate, in a comparative approach, undergraduate student understanding of calculus concepts embedded in the context of life science examples. Our objective is to develop a tool that can be effective in comparing alternative formats for student comprehension of concepts from calculus, particularly the alternative courses available to life science students at many U.S. institutions. Thus, the BCA has been developed explicitly to provide a means to assess the impact on calculus concept comprehension of different modalities of calculus instruction arising from different emphases and inclusion of concrete biological contexts. Options for students in this study include a standard science and engineering calculus sequence as well as a sequence designed specifically for life science students that emphasizes biology applications to enhance comprehension of calculus concepts. Three calculus concepts formed the basis of the BCA: rates of change, modeling, and interpretation of data and graphs.
It is our expectation that instruments such as the BCA and SRBCI can be applied to develop guidance regarding the impact of inclusion of life science disciplinary examples in calculus and statistical reasoning courses. Given the importance of quantitative methods across the life sciences, biology faculty may use the results of more expansive applications of the BCA to encourage their faculty colleagues who teach calculus to do so in a manner that is most effective for their students. This should also contribute to broader educational research questions regarding the impact of learning methods on student conceptual comprehension (Koedinger et al., 2013).
We conducted two focus groups with students to ensure that the students interpreted test items as intended, ensure that the language and notation used on the test were familiar to students, and obtain feedback from students about question wording and distractor choices. Criteria for student participation in the focus groups included undergraduates who had declared a biological science major and who had either taken 1) the AP Calculus exam but who had not taken calculus at the university, 2) two semesters of calculus at the university level, or 3) two semesters of Mathematics for the Life Sciences (a calculus course for life science majors that teaches calculus concepts in biological context). These criteria ensured that the focus group students would have the relevant educational background in mathematics to understand the concepts represented in the assessment. Email invitations to participate were sent to 463 prospective students, and a total of 19 students participated in one of two focus groups in Spring 2016 (10 and 9 students, respectively). All students had declared a biological sciences major, except one student who was from an environmental and soil science major. Nine of the students met the criteria of having AP Calculus exam credit, eight had taken two semesters of calculus at university, and two had taken two semesters of Mathematics for the Life Sciences. Ten of the focus group participants were female.
Calculus has historically been a major component of quantitative training for biology undergraduates. Because the majority of undergraduate life science curricula require calculus in some form, there continues to be a need for the BCA to assess student comprehension of calculus with different teaching methods and different levels of biological applications. The BCA measures a subcomponent of the broader range of quantitative skills to which life science students are exposed. Tools to assess conceptual understanding of calculus in a cross-disciplinary way are needed to assess changes in student understanding, examine potential advantages of pedagogical interventions, and explicitly evaluate whether placing quantitative concepts in this discipline-specific domain enhances student comprehension of calculus concepts. The availability of the BCA provides opportunities for faculty and other researchers to participate in the ongoing national experiment in life science quantitative education through which institutions offer different routes for calculus training for life science students, which may be broadly useful, particularly as new fields such as data science emerge and are connected to life science programs.
We created the BCA to fill a measurement gap for assessing learning gains of students with life science backgrounds who may learn calculus within an interdisciplinary quantitative biology course or within a traditional university calculus course (often geared toward mathematics and engineering majors). Our assessment of the validity of the BCA indicates the instrument is a valid diagnostic tool to assess calculus comprehension in undergraduate biology majors who learn calculus within a quantitative course designed specifically for life science students, but also is appropriate to compare scores for students from traditionally taught calculus courses. Together, results from our assessment of the content validity, response processes, and internal structure of the instrument provide accumulated support for the validity of the BCA (American Educational Research Association et al., 2014).
Comparing the difficulty scores for the items indicates that students across the set of courses for this study can understand rates of change from simple data in a chart or a graph (e.g., items 1 and 2) and the implications of exponential rates of growth from simple population data to estimate population sizes at various times (e.g., item 3). At the other end of the difficulty scale, the representation of functions using log-log graphs is not readily understood (e.g., item 19). This topic of nonlinear scaling, though arising in many areas of biology, is not generally emphasized in standard calculus courses (e.g., C1 and C2). Even though log-log plots are emphasized in BioCalc, the results indicate that these students generally also did not obtain conceptual understanding of nonlinear scaling. Similarly, integration of trigonometric functions scored at high difficulty for all courses (e.g., item 15). The implication of these results is that an emphasis on nonlinear functions in calculus courses for life science students should be encouraged. Conceptually, item 14 required responses about the assumptions in a simple population growth model and was of medium difficulty across all courses, while other items dealt with particular numeric or symbolic answers. So emphasis on determining basic underlying model assumptions might be appropriately increased to enhance student conceptual foundations.
BIOCALCULUS: CALCULUS, PROBABILITY, AND STATISTICS FOR THE LIFE SCIENCES shows you how calculus relates to biology, illustrating the topics of calculus with [real-life?] examples drawn from many areas of biology, including genetics, biomechanics, medicine, pharmacology, physiology, ecology, epidemiology, and evolution, to name a few. The text provides you with a sound knowledge of mathematics, an understanding of the importance of mathematical arguments, and a clear understanding of how these mathematical concepts and techniques are central in the life sciences.
Designed for students specializing in the life sciences. Topics include: limits, growth rate and the derivative; elementary functions, optimization and approximation methods, and their applications, integration, and differential equations; mathematical models of biological processes and their implementation and analysis using software. Students with credit for either MATH 150, 151 or 157 may not take MATH 154 for further credit. Quantitative.
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