Ingredients of mathematical modeling

Applied mathematics refers to the portion of mathematics most often useful in representing the physical world. Many scientists spend most of their time working on the theories of physics, chemistry, geology, astronomy, or biology; most of their time is spent with mathematics as well. Statistical analysis is a portion of that analysis, but there is much more as well. Another name for this sort of activity is mathematical modeling.

Numbers

The most useful idea in mathematical modeling is also the oldest and most basic. It is the idea of using numbers to represent the world. There are many different sorts of numbers, and it may be useful to review them before proceeding to more complicated operations.

Positive integers These represent increasing quantities of essentially identical objects. The basic idea is simple yet abstract, that one object can be essentially the same as another, while not literally being identical. Even two symbols, X, X, are different if only because they are in different places, but everyone seems to understand what it means to say that they are the same, and there are two of them.

Zero This is a number representing the absence of some quantity. It makes it possible to state that something is not present.

Negative integers These numbers seem to have been conceived in India during our Medieval period, along with the Arabic numerals (brought by Arabs from India to Europe).

Course goals

Science starts with curiosity about the world. It starts with questions about why the sky is blue, why dogs wag their tails, whether electric power lines cause cancer, and whether God plays dice with the universe. But while curiosity is necessary for science, it is not enough. Science also depends upon logical thought, carefully planned experiments, mathematical and computer models, specialized instruments, and many other elements. In this course, you will learn some of the research methods that turn curiosity into science. In particular, you will learn to

• Create your own experiments to answer scientific questions.

• Design experiments to reduce systematic and random errors and use statistics to interpret the results.

• Use probes and computers to gather and analyze data.

• Treat human subjects in an ethical fashion.

• Apply safe laboratory procedures.

• Find and read articles in the scientific literature.

• Create mathematical models of scientific phenomena.

• Apply scientific arguments in matters of social importance.

• Write scientific papers.

• Review scientific papers.

• Give oral presentations of scientific work.

You will not just be learning about these skills. You will be acquiring and applying them by carrying out scientific inquiries.

Kinds of questions

Testable questions Every scientific inquiry answers questions, and most scientific inquiries begin with specific questions.

The following quotation is from Galileo's final book, Dialogues on Two New Sciences [213] (179) translated by Henry Crew and Alfonso de Salvo (Macmillan, New York, 1914). Salvatore, a character who stands in for Galileo himself, has just been challenged to justify the claim that falling objects accelerate uniformly.

These grading rubrics provide an attempt to formalize the way that scientists evaluate scientific work. Most of the rubric is subtractive. This means that the rubric describes various sorts of mistakes, and the numbers of points that can be deducted for them. The total number of points that can be deducted from each category adds to much more than 100 because scientific work can be rendered invalid by poor performance in any of these areas. Unsafe practices or plagiarism lead immediately to a failing grade.

At the very end of the rubric, there is room for additions of points. Adding points accounts for exceptional or innovative work. This part of the rubric captures the idea that the best scientific work has an element of creativity and effort that cannot be captured by the simple avoidance of doing anything wrong.

To use the rubric, the instructor will decide on the baseline value from which points will be subtracted for errors and omissions. This value may be less than 100. Getting the maximum score of 100 may require the addition of points from the final section of the rubric.

Please note that no rubric or checklist can fully capture the full range of strengths and difficulties that describe independent inquiries. This rubric provides a guide to help you identify common misconceptions and errors, but cannot fully cover all cases.

This book accompanies a one-semester undergraduate introduction to scientific research. The course was first developed at The University of Texas at Austin for students preparing to become science and mathematics teachers, and has since grown to include a broad range of undergraduates who want an introduction to research. The heart of the course is a set of scientific inquiries that each student develops independently. In years of teaching the course, the instructors have heard many questions that students naturally ask as they gather data, develop models, and interpret them. This book contains answers to those most common questions.

Because the focus is on supporting student inquiries, the text is relatively brief, and focuses on concepts such as the meaning of standard error, p-values, and deterministic modeling. If a single statistical test, such as χ2, is adequate to deal with most student experiments, the text does not introduce alternatives, such as ANOVA, even if they are standard for professional researchers to know.

The mathematical level of the book is intermediate, and in some places presumes knowledge of calculus. It could probably be used with students who don't know calculus, skipping these sections without great loss.

There is an instructor's manual that describes daily activities for a 14-week class that meets two hours per week in a classroom and two hours per week in a lab. It is available at www.cambridge.org/Marder.

Introduction

If a scientist carries out a major research project, but no one knows about it, or no one can understand it, the research is of little use. So a large part of science is communication. The communication happens informally between colleagues in the hallway, at conferences through presentations, and most durably with global distribution of information through articles and books. Scientists have developed many conventional ways to explain their results, including equations, figures, and specialized vocabulary. Once you have completed a research project, you will need to practice communicating the results, to round out the set of scientific skills you have acquired. Each of the sections of this chapter touches just on a few essentials. You can find much more complete discussion of all the topics mentioned here in Valiela (2001).

Writing a proposal

Scientists spend a large fraction of their time writing proposals, whether they want to or not. Proposals are necessary to apply for grant funding, or for permission to conduct research. The research proposals needed for funding from federal agencies are usually a minimum of 15 pages in length, not including budgets, bibliography, and supporting documents. In this class you will not have to write proposals of that length. However, your instructors may ask you to write proposals a page or two long as you are beginning your inquiries.

Spreadsheet programs

Spreadsheets record and manipulate data. The purpose of this appendix is to emphasize the features of spreadsheets that make them useful for manipulating scientific data, and to perform computations based on scientific theories.

Excel for Windows

By far the most common spreadsheet is Excel, which is developed and marketed by the Microsoft Corporation as part of Microsoft Office. There are many different versions of Excel in circulation, but all of them have all the functions described here. Excel is available for all the different Windows operating systems Microsoft has written for IBM-compatible personal computers. The major releases of Excel in current use are Excel 2003, and Excel 2007. The latest release, Excel 2007, involves a completely new way of accessing menu items, but all functions of Excel 2003 are still available. Note that the native file format of Excel 2007 is a new format (.xlsx) that can cause difficulties for other programs. Although this format is becoming universal, you may still sometimes find it useful to read and write files in the old Excel 2003 format (Compatibility Mode).

Excel for Macintosh

Excel 2008 is available for computers running Apple's OSX operating system. The menu items are arranged somewhat differently from the Windows releases, but all the same functions are available in a nearly identical way. This version of Excel can read the .xlsx files of the recent Excel releases for Windows.

Introduction

One plus one makes two, doesn't it? Wrong! One plus one can make a lot more than two and that is what synergy is all about. Combine your talent for science with that of other people.

As a Bachelor, Master's, or PhD student you will primarily be examined on the basis of what you have done (in an excellent way, e.g. your exam) or contributed (something novel and creative). Your results always build on top of what other talented people have contributed at an earlier stage. How can you make the most of other people's findings? As a postdoc or a professor you may be supervising students working on Bachelor, Master's, and PhD thesis projects, and you may also be collaborating with colleagues on a national or international scale. How can you make the most of your interactions with other people? The factors I see as being most essential are:

• Read. Scientists write a lot. You may easily suffer from data overload. So what should you read, and how?

• Listen. Scientists present their work at all kinds of meetings. Good listeners learn fast. You will experience ultimate moments of “eureka!” when a solution for your problem appears, or moments of “how curious” if a new view manifests itself. Can you learn to listen better?

• […]

Introduction

A top talent has smart ideas on how to crack scientific problems. But top talents have more than that. Einstein (apparently) said something like: “I have no special talent, I am only passionately curious”, which proves that other talents than just your intellect (or IQ) matter, too, if not more. Another putative quote from Einstein supports this: “Imagination is more important than knowledge.” In the three previous chapters, I have outlined the essentials for a career in research. Great, but how do you put theory into practice, in your day-to-day life? Here is the strategy:

• Dream. What do you want to achieve? Imagine, like Einstein did. Dream your greatest future. At this point, don't limit yourself by “yes, but…”.

• Count. Learn from the past. Count your blessings. What was positive, what was negative?

• Believe. President Obama of the USA caught the American people's imagination in 2008 with his slogan “Yes, we can.” Do you think you can? Which beliefs, thoughts, and emotions play a role? Are they helpful or limiting?

• Act. Activate yourself. How can you make and execute plans?

• Care. Don't forget to look after yourself. Body and mind go hand in hand. Make and execute a plan to keep yourself in good mental and physical condition.

• Succeed. Which concrete actions are you really (yes really!) going to take now (right now!) to make the most of your talent?

Introduction

Are you a student? Then stay on board and read on: this chapter is highly relevant for you too. Why? Do you have to be a supervisor, professor, institute director, or dean, before you can start inspiring and changing other people? No, wake up! Inspiration can go in all directions, top down and bottom up. You can inspire a fellow Bachelor, Master's, or PhD student, or your team mates. You can bring change to them, to your supervising professor, perhaps even to your university.

This chapter is also relevant to you if you are not a student but an employee. Developing the other people around you, that's what this chapter is about. What's good for them is good for you.

Of course, changing someone else without his or her commitment is a hopeless, impossible mission. The “other” person should first become aware of the possibility or need to change, then want to change, be able to change, and act to change. You can support him or her by creating the right conditions, the breeding ground, and the treatment which will help them develop their own talent.

Who is this book for?

This book has been written for people serious about science: students, postdocs, professors, trainers, and support and other staff. So what's in it for you? I'll outline this briefly in the next three paragraphs. You could read only the paragraph relevant to your involvement in science but why not read the other paragraphs as well.

• You're a student. As a Bachelor, Master's, or PhD student you take classes in science subjects. You are (for now) at the bottom of your career ladder, but it's good to know what's going on higher up the ladder, so that you can better understand, appreciate, and communicate with your teachers and supervisors in that special type of organization called a university. It may also help you in deciphering whether a career in science in a university or in a company would be attractive if it's not your vocation. Some of you will already be involved in research projects and indeed feel you want to become a scientist at a university. Many of you will look for jobs outside the university and perform tasks in which you will nonetheless benefit greatly from having developed your talent for science as much as possible. So if you are a student, starting off on the road to becoming a scientist or something else, you will likely benefit both personally as well as professionally from reading this book.

• […]

Introduction

For the sake of clarity, let me check that you haven't misunderstood me. In many places this book has emphasized the “extra-scientific” skills and the importance of “playing the game” in the business of science, where how many scientific papers you have written, the impact factors of the journals, and your academy memberships do count. There is nothing wrong with this, unless it becomes your one-dimensional view of what science is all about. Doing your work with the dominating purpose of having papers in high-impact journals such as Nature and Science, perhaps more papers than your colleague next door, is empty and has no intrinsic value. It would be trivializing science. So what really counts? Your idealism, your curiosity, your intellectual endeavor. To potentially push forward the frontiers of knowledge or to use this knowledge to the benefit of humankind. Yes, do keep doing this.

So now this book can really draw to a close?

Actually, no, the most critical part is still to come. You used the four web figures to visualize your strengths and weaknesses. Now it is time to set your ambitions for preferred scores in, say, one year from now, and to define the appropriate actions to get there: dream, count, believe, act, and then succeed.

Introduction

Maybe you're following a scientific course at a university as a Bachelor, Master's, or PhD student. Or you're already working as a postdoc or professor at a university, or in industry, or in the service sector. At each stage of your career other people are appealing to your unique knowledge and talent for science. This book cannot tell you how talented you are in math, biology, physics, chemistry, behavioral sciences, or any other subject. So what can it do? It emphasizes the essentials you need to add to your talent. It's all about doing the right things right. You do the right things if you combine a basic talent with a strong passion for the chosen subject. You do things right if you have acquired and improved the essential skills such as prioritizing, giving presentations, and writing. So the emphasis in this chapter is on you. The factors I see as being most essential are:

• Passion. Is science your ultimate job vocation or do you want to use your science training in other ways? Does scientific thinking energize you? Do you say YES to science?

• Prioritize. Can you do more than one or two things at the same time at a top level? Do you know what to do if you run out of time? Do you know when to say NO?

• […]