AP Biology Quantitative Skills: A Guide for Teachers The College Board New York, NY AbouT The ColleGe boArd The College Board is a mission-driven not-for-profit organization that connects students to college success and opportunity. Founded in 1900, the College Board was created to expand access to higher education. Today, the membership association is made up of over 6,000 of the world’s leading educational institutions and is dedicated to promoting excellence and equity in education. Each year, the College Board helps more than seven million students prepare for a successful transition to college through programs and services in college readiness and college success — including the SAT® and the Advanced Placement Program®. The organization also serves the education community through research and advocacy on behalf of students, educators and schools. For further information, visit www.collegeboard.org. equiTy And ACCess PoliCy sTATemenT The College Board strongly encourages educators to make equitable access a guiding principle for their AP programs by giving all willing and academically prepared students the opportunity to participate in AP. We encourage the elimination of barriers that restrict access to AP for students from ethnic, racial and socioeconomic groups that have been traditionally underserved. Schools should make every effort to ensure their AP classes reflect the diversity of their student population. The College Board also believes that all students should have access to academically challenging course work before they enroll in AP classes, which can prepare them for AP success. It is only through a commitment to equitable preparation and access that true equity and excellence can be achieved. Pages 16, 17, 44, 45: Graphs from AP Biology Investigative Labs: An Inquiry-Based Approach. (New York: The College Board, 2012). Pages 18, 19, 20, 46, 47, 48, 77, 96, 97: Questions from AP Biology Practice Exam. (New York: The College Board, 2012). © 2012 The College Board. College Board, AP, Advanced Placement Program, SAT and the acorn logo are registered trademarks of the College Board. All other products and services may be trademarks of their respective owners. Visit the College Board on the Web: www.collegeboard.org. 1 2 3 4 5 Contents Acknowledgments ......................................................................................................................v introduction Problem Solving and Quantitative Skills in AP Biology .......................................................1 Science Practices and Quantitative Skills ................................................................................2 How to Use This Guide ..............................................................................................................2 Incorporating Quantitative Skills in the Classroom ..............................................................4 Chapter 1: Graphing Labs Using Graphing .................................................................................................................6 Components of Graphing and Preliminary Data Analysis ...................................................6 Elements of Effective Graphing .............................................................................................13 Using Graphing Programs .....................................................................................................14 Alignment to the Investigations ............................................................................................15 Alignment to the Exam ..........................................................................................................18 Resources ..................................................................................................................................21 Chapter 2: data Analysis Labs Using Data Analysis ......................................................................................................32 Components of Data Analysis ...............................................................................................32 Why Bother with Data Analysis? ..........................................................................................33 Using Data Analysis ................................................................................................................33 Examples of Data Analysis .....................................................................................................35 Alignment to the Investigations ............................................................................................43 Alignment to the Exam ..........................................................................................................46 Resources ..................................................................................................................................49 Chapter 3: hypothesis Testing Labs Using Hypothesis Testing .............................................................................................57 Introduction to Hypothesis Testing ......................................................................................57 Using Hypothesis Testing .......................................................................................................59 Examples of Hypothesis Testing ............................................................................................61 Applications of Chi-square Test Results ...............................................................................68 Alignment to the Investigations ............................................................................................75 Alignment to the Exam ..........................................................................................................77 Resources ..................................................................................................................................78 iii Contents Chapter 4: mathematical modeling Labs Using Mathematical Modeling . ...................................................................................84 Components of Mathematical Modeling .............................................................................85 Using Mathematical Modeling ..............................................................................................85 Examples of Mathematical Modeling ...................................................................................86 Alignment to the Investigations ............................................................................................94 Alignment to the Exam ..........................................................................................................96 Resources ..................................................................................................................................98 Appendix: AP biology equations and Formulas ....................................107 Contents iiivvv Acknowledgments The College Board would like to acknowledge the following individuals for their commitment and dedication toward the completion of this project: • Elizabeth Carzoli, Castle Park High School, Chula Vista, CA • John Jungck, University of Delaware, Newark, DE • Kim Foglia, Division Avenue Senior High School, Levittown, NY • Sharon Radford, The Paideia School, Atlanta, GA • Paul Rodriguez, Troy High School, Fullerton, CA • Jim Smanik, Sycamore High School, Cincinnati, OH • Gordon Uno, University of Oklahoma, Norman, OK • Brad Williamson, University of Kansas, Lawrence, KS • Julianne Zedalis, The Bishop’s School, La Jolla, CA v Acknowledgments Introduction Problem Solving and Quantitative Skills in AP Biology The BIO2010 report of the National Research Council (2003) describes the current trend in biology and biological research as follows: Biological concepts and models are becoming more quantitative, and biological research has become critically dependent on concepts and methods drawn from other scientific disciplines. The connections between the biological sciences and the physical sciences, mathematics, and computer science are rapidly becoming deeper and more extensive. Therefore, it is imperative that today’s students develop and apply quantitative skills as part of their exploration into biology. A good grasp of quantitative methodology and reasoning is particularly important in the laboratory experience. In fact, many statistical tests and other mathematical tools were developed originally to work out biological problems. America’s Lab Report: Investigations in High School Science (National Research Council, 2005) sums it up: Laboratory experiences provide opportunities for students to interact directly with the material world (or with data drawn from the material world), using the tools, data collection techniques, models, and theories of science. The new AP Biology curriculum framework; AP Biology lab manual, AP Biology Investigative Labs: An Inquiry-Based Approach; and AP Exam reflect this emphasis on inquiry and reasoning and the quantitative skills these processes entail. In their laboratory experience, students will now model the behavior of scientists. As they carry out independent investigations, students will observe, explore, and discover knowledge themselves as they engage in scientific problem solving. The focus of the labs has shifted from students achieving predetermined results to students making independent observations, posing their own questions, and determining how to investigate these questions. Students now must choose which variables they want to investigate (formerly, they often were told what they would investigate). They then conduct lab and field investigations and gather their own data. They must determine the appropriate way to record those data. (Formerly, tables and graphs were provided for students to fill out). They go on to analyze the results of their data gathering, test hypotheses, and use mathematical modeling. (Formerly, the necessary mathematical equations and other quantitative skills would be provided with step-by-step instructions on how to use them). Finally, students communicate the results and conclusions of their laboratory investigations the way scientists all over the world do—through peer review, mini-posters, and presentations, in addition to the more traditional lab notebooks, portfolios, reports, and papers. In the new laboratory environment, students are directed to resources, which they then must decide how to apply. The focus is on helping students develop the skills they need for using statistics and other techniques to inform their investigations. In so doing, they will develop the reasoning skills essential to good scientific practice. 1 Introduction Science PracticeS and Quantitative SkillS Practice can be defined as a way to coordinate knowledge and skills to accomplish a goal or task. The new AP Biology curriculum framework is based on seven science practices or necessary skills for the biology student of the 21st century. It is impossible to imagine engaging in any one of the science practices without a good grasp of quantitative skills. For example, the first science practice states, The student can use representations and models to communicate scientific phenomena and solve scientific problems. By creating and interpreting graphs, or visual representations, students can illustrate biological concepts and ideas, communicate information, make predictions, and describe systems to promote and document understanding. The second science practice says, The student can use mathematics appropriately. It expresses the fact that quantitative reasoning is crucial in solving problems, analyzing experimental data, describing natural phenomena, making predictions, and describing processes symbolically. The fifth science practice, The student can perform data analysis and evaluation of evidence, recognizes that quantitative analysis is involved in determining whether the data support a conclusion, drawing conclusions from a data set, assessing the validity of experimental evidence, identifying possible sources of error in an experimental procedure or data set, identifying outliers, and proposing explanations for them. The sixth science practice, The student can work with scientific explanations and theories, points to the importance of hypothesis testing and mathematical modeling in scientific inquiry. The other science practices similarly involve the use of the quantitative skills described in this guide. Perhaps the biggest challenge AP Biology instructors will face in implementing the new curriculum will be getting students to adopt the scientific inquiry method and its language. For example, students who may be used to “proving” that a hypothesis is true or false may struggle as they begin to work with the more scientifically accurate statements reject the null hypothesis or fail to reject the null hypothesis. In addition, students may or may not already have been introduced to the necessary quantitative skills in mathematics classes. And even students with the educational background to support the skills probably will not be familiar with their application to biological inquiry in the “real world.” How to uSe tHiS Guide This guide is an introduction to thinking about quantitative skills as they are used to help answer questions about biological phenomena. It explains how biology instructors can help students learn and apply four of the quantitative skills that students will rely on most heavily in their exploration of biology in the AP Biology curriculum framework: graphing, data analysis, hypothesis testing, and mathematical modeling. Chapter 1: Graphing introduces the graphical representation of data to allow students to discover patterns or relationships in that information and infer causal biological mechanisms. Chapter 2: Data Analysis helps students discover meaningful patterns in masses of data as they develop their skills in statistics. This quantitative skill helps students understand and deal with the variability and the sometimes ambiguous nature of data that are typical in scientific investigations. Chapter 3: Hypothesis Testing covers the application of statistical tests to help distinguish between working hypotheses. Introduction 2 Finally, Chapter 4: Mathematical Modeling helps students understand complex systems, explore various possibilities in those systems, develop conceptual frameworks, make accurate predictions, and generate explanations. When students learn to model, they improve their ability to solve problems, understand how they know what they know, and transfer that insight to other subject areas. Each of the four skills covered in this guide means something a little different depending on the field of study. This guide limits the discussion of data analysis primarily to descriptive statistics with a few specific examples of inferential statistics. Inferential statistics are tools that allow one to infer, or conclude, something about the true population—the population beyond the sample. The discussion of hypothesis testing is limited to a small sample of statistical tests and techniques that may be useful to the AP Biology student. Each chapter begins with a list of labs in AP Biology Investigative Labs: An Inquiry-Based Approach that use the skills covered in the text. An introductory explanation includes components of the quantitative skill, key terms and their definitions, a discussion of how the skill applies to biology, and information on how the skill can be practiced and used both in the classroom and in the wider world of biological inquiry. Next, examples illustrate the application of the quantitative skill in a lab setting. These examples reflect the type of analysis needed for most of the investigations in the lab manual and follow an imagined run-through to demonstrate the use of the quantitative skill in a lab setting. Preliminary data are provided for students to explore and analyze in order to discover patterns and make observations that can inform their hypotheses and experimental design. Students will then be ready to design and carry out an investigation using the same or similar techniques or procedures in conjunction with a question of their choice. Working through these examples before doing the actual lab allows students to get a feel for the process. Finally, each chapter ends with a list of resources that the instructor may tap into for classroom activities and further examples. Every laboratory in AP Biology Investigative Labs: An Inquiry-Based Approach encourages students to carry out independent investigations based on their own questions that arise after exploring a new technique or method in order to learn more about a biological concept. The resources in this guide will help students to develop good experimental designs to answer those questions, as well as to use quantitative methods to make sense of the results—a critical requirement for drawing valid conclusions. Of course, because students are asking their own questions, some of their investigations will require techniques and skills not covered here. In those cases, the teacher and student will need to build on the approaches presented here to explore further tests and quantitative methods. This is not meant to be a comprehensive guide to teaching quantitative skills. Instead, this document will provide enough background to allow you to seek out appropriate tools and methods suitable to your own classroom. One place to start seeking out those resources is the NIMBioS website: http://www.nimbios.org/. Furthermore, the information in this guide is not meant to replace the quantitative skills that have been emphasized in previous AP Biology labs, such as preparing buffers by use of logarithms, mathematically modeling the rate of enzymatic actions, and considering the magnification in microscopy. Rather, the information here is intended to help you in the transition to the new AP Biology teaching framework, which includes thinking conceptually about analyzing data, testing hypotheses, and building mathematical models to discriminate among possible causes. As such, the goal of this manual is 3 Introduction to guide you in planting the seeds for an intuitive understanding of statistics and mathematical modeling in your students. Be sure to explore the resources at the end of each of the skills chapters to further your own exploration into the new curriculum. Note that nearly all of the formulas required for the examples in this resource guide are on the AP Biology equations and formulas list, which was created to help teachers and students design their instruction and prepare for the exam (see the appendix). The intent of this guide is to provide a context for understanding how this kind of quantitative thinking informs biology. It is not intended that students should memorize a list of formulas. This list will be available to students when they sit for the exam. incorPoratinG Quantitative SkillS in tHe claSSroom Quantitative skills should be deeply embedded into all laboratory and content instruction as a reflection of how integral these skills are to the study of biological sciences. As such, student assessments that focus on quantitative skills alone are probably inappropriate. Instead, course and lab assessments should include evaluation of how well the appropriate quantitative skills have been applied to solving biological problems. Much of this assessment of quantitative skills should come in the form of feedback from peers and teachers to the student on the appropriate application of quantitative skills. It is essential that the classroom environment spark an ongoing, nonthreatening dialogue between students and teachers that explores the appropriate application of various skills. This interchange will vary by laboratory, by content, and by the skills that the student and teacher bring to the investigation. Think of a casual lab lunch meeting where members of the lab present their preliminary results and analysis to the entire group. Similarly, give students multiple opportunities to polish their research and analysis based on authentic peer review before making the final presentation of their results, as described at http://www.nabt.org/blog/2010/05/04/mini-posters-authentic- peer-review-in-the-classroom/. Students who are used to more formulaic science labs may question the need for the quantitative skills covered in this guide. It may be helpful to spend some time clarifying what happens when this type of reasoning is not applied or when it is distorted deliberately. Using a catchy or local news report to illustrate how and why quantitative intelligence is important to science and to the students’ own lives can provide a number of excellent avenues for discussion. Students will begin to understand why important health and governmental decisions that will affect them and their families should be based on a clear-minded assessment of the science behind the issues. Guide your students in acquiring the ability to detect when information is credible, incomplete, or suspect. Have the students read reports and ask themselves, What does this really mean? Does this represent a useful or meaningful result? How could I test this? Ask students to discuss the “scientific facts” in a news item and the ways in which quantitative analysis may have been used to arrive at the statements reported. Let the students debate their ability to even answer that question. Then remind them to reflect on this discussion as they prepare to undertake their own scientific experiments. You might want to have the students follow a story of interest to them. Many resources offer reports on, and analyses of, today’s scientific issues. A search of resources on the Internet can be instructive in its own right, as students will be faced with the task of assessing which sites come with their own agendas. Emphasize the need for analysis Introduction 4