Some other books Springer Brooks-Cole Self-Published
Building an Applied Mathematics and Statistics Major at a Liberal Arts College Macalester College St. Paul, Minnesota, USA Oct. 11, 2008
Discipline 1. the practice of training people to obey rules or a code of behavior, using punishment to correct disobedience : a lack of proper parental and school discipline. the controlled behavior resulting from such training : he was able to maintain discipline among his men. activity or experience that provides mental or physical training : the tariqa offered spiritual discipline Kung fu is a discipline open to old and young. a system of rules of conduct : he doesn t have to submit to normal disciplines. 2. a branch of knowledge, typically one studied in higher education : sociology is a fairly new discipline. origin Middle English (in the sense [mortification by scourging oneself] ): via Old French from Latin disciplina instruction, knowledge, from discipulus (see disciple ). from New Oxford American Dictionary
2008-10-11 Discipline Discipline 1. the practice of training people to obey rules or a code of behavior, using punishment to correct disobedience : a lack of proper parental and school discipline. the controlled behavior resulting from such training : he was able to maintain discipline among his men. activity or experience that provides mental or physical training : the tariqa offered spiritual discipline Kung fu is a discipline open to old and young. a system of rules of conduct : he doesn t have to submit to normal disciplines. 2. a branch of knowledge, typically one studied in higher education : sociology is a fairly new discipline. origin Middle English (in the sense [mortification by scourging oneself] ): via Old French from Latin disciplina instruction, knowledge, from discipulus (see disciple ). from New Oxford American Dictionary Mathematics: Rigor/proof/correctness, Aesthetics. Much mathematical work involves conjecture, but the goal is to resolve the issue definitively. Unsolved problems are cherished only insofar as they lead to the development of new mathematics. Think of Fermat s Last Theorem. Applied mathematics: Utility.
Interdisciplinary: Of or relating to more than one branch of knowledge. But what is a branch of knowledge? Is branch a useful metaphor isn t knowledge more of a network? [See the Wikipedia entry for more nuance.] What we typically mean is spans more than one department. Departments are divisions drawn up during the first decades of the 20th century. They are much like the political boundaries drawn at the end of the First World War: sometimes they make sense, sometimes they separated groups that want to be together. They aren t fluid enough to respond to changes (in demographics for geography, in the organization of knowledge for science). Example: Information sciences not in physics but in EE. Fluids in ME.
A Short History of Macalester s Math Curriculum Trends: At founding, in 1880s, all students were required to take 3 years of mathematics, whether they were in the Ancient Classical or Modern Classical track. Algebra, Trig, Conic Sections, Analytic Geometry, Surveying, Field Work, Differential Calculus, Integral Calculus, Analytical Mechanics. 1910-1919: Majors introduced. Gradual addition of more advanced topics and dropping of topics we now consider high-school math (e.g., trigonometry). Emphasis on practical subjects (surveying, accountancy, investment) dropped by 1950s. Mathematical Statistics introduced around 1950, Introductory Statistics introduced in 1960s. Computation introduced in early late 1970s. New topics (statistics, computation) were not linked to old ones.
Is our Department Interdisciplinary? No, because it is a department, but Yes in other senses... A Pragmatic Definition of an Academic Discipline Publish in the same journals. Go to the same conferences. Can reasonably teach the same courses. Would trust your colleagues to hire someone new while you are away on sabbatical. Math, Applied Math, Statistics, and CS are different disciplines by each of these standards.
Why Majors are Important The idea of a major has been around since about 1900. Advantages: Can define a core of related subjects. Allows courses to build on earlier studies and skills. Can require co-curricular activities (e.g., internships) and additional course work in other departments. Can create community among students: role models, peer study resources. Give introductory courses a target. When there are only introductory courses, what is the criteria for selecting subjects? How to you avoid drift from relevance to teach-ability and test-ability? Advanced courses let faculty connect students & teaching to their research. Give the program a set at the table when hiring and for tenure/promotion.
2008-10-11 Why Majors are Important Why Majors are Important The idea of a major has been around since about 1900. Advantages: Can define a core of related subjects. Allows courses to build on earlier studies and skills. Can require co-curricular activities (e.g., internships) and additional course work in other departments. Can create community among students: role models, peer study resources. Give introductory courses a target. When there are only introductory courses, what is the criteria for selecting subjects? How to you avoid drift from relevance to teach-ability and test-ability? Advanced courses let faculty connect students & teaching to their research. Give the program a set at the table when hiring and for tenure/promotion. They are a tremendous curricular success. Ubiquitous at colleges and universities. Possibly only the course is a more central organizational structure. But majors (and courses) are not without costs. They encourage isolation.
Why not a Flexible Math Major Math departments are not a natural setting for applied work. Culture emphasizes proof and aesthetics, not utility. Algebra and Analysis are the main structure, not computation & statistics. Few mathematicians have worked in science or applied areas. Many have no interest. Applicants for math jobs are predominantly pure mathematicians for whom education is a relatively attractive field. Hostility of mathematicians to statistics. At many schools, applied math is housed in physics.
What is Applied Math? Conventional answers: Applied Differential Equations? Too narrow. Anything that can connect to the world, science or technology? Too broad. Number theory is not applied math even if much of computer commerce is based on it. Anything at all? Common claim: Mathematics is the Queen of the Sciences : there would be no science without math. But this doesn t mean that the math that is being taught supports science.
What is Applied Math? Two closely interconnected ideas: Modeling. The construction of representations for a purpose. Specifically, models where the representation is rooted in the topics that are historically mathematical. (A non-mathematical model: He is a maverick. ) Extraction of information from the models. Sometimes called solving. These two ideas show up in diverse courses in the math and allied curriculum: Calculus & DE, Linear Algebra, Discrete Math, Analysis, Modeling Computation: programming but also computer graphics, AI,... Statistics (and econometrics, psychological research methods, etc.)
Fluxions for Describing Motion F = ma describes a relationship between force, mass, and acceleration. Galileo emphasized the idea of momemtum: mass speed. Newton developed a definition of acceration related to the change in momentum. In order to do calculations on this, he developed a mathematical representation which he called the theory of fluxions. We call these derivatives. Velocity is the first derivative of position, acceleration is the derivative of velocity. Newton developed calculational techniques to solve relationships like a = const. This became calculus. Newton and Leibnitz were informal about the meaning of small. But their calculus worked. 150 years later, mathematicians developed a framework for arguing rigorously about small. Newton s and Leibnitz s calculus is clearly applied math. But is the mathematics of small?
Mathematics and Computation Close historical tie between the two. Calculus comes from the same root as calculate : stone. This comes from the small stones used in counting tables and in abaci. The logarithm was developed c. 1600 as a computational tool. Napier s sticks, slide rules. Newton and Leibnitz both deeply concerned with computations. Modern era: von Neuman, Turing, RSA. Yet, many mathematicians are weak in working with computers. Consider top 10 US News Liberal Arts colleges: Amherst, Williams, Swarthmore, Wellesley, Middlebury, Bowdoin, Pomona,Carleton,Davidson, Haverford CS required: Davidson (1 course), CS counts toward major: Amherst None requires statistics!
2008-10-11 Mathematics and Computation Mathematics and Computation Close historical tie between the two. Calculus comes from the same root as calculate : stone. This comes from the small stones used in counting tables and in abaci. The logarithm was developed c. 1600 as a computational tool. Napier s sticks, slide rules. Newton and Leibnitz both deeply concerned with computations. Modern era: von Neuman, Turing, RSA. Yet, many mathematicians are weak in working with computers. Consider top 10 US News Liberal Arts colleges: Amherst, Williams, Swarthmore, Wellesley, Middlebury, Bowdoin, Pomona,Carleton,Davidson, Haverford CS required: Davidson (1 course), CS counts toward major: Amherst None requires statistics! Think of the stony build-up on your teeth: calculus.
Newton and Computation Newton s method of divided differences for evaluating polynomials is the methodological foundation for Charles Babbage s difference engine, proposed in 1822.
2008-10-11 Newton and Computation Newton and Computation Newton s method of divided differences for evaluating polynomials is the methodological foundation for Charles Babbage s difference engine, proposed in 1822. It s tempting to think of the link between computation and Newton in terms of Newton s method for finding zeros. This is a mainstay of scientific computation. But it s not clear that the method Newton used is the same method used today under the name Newton-Raphson.
Leibniz and Computation. For it is unworthy of excellent men to lose hours like slaves in the labour of calculation which would safely be relegated to anyone else if machines were used. Gottfried Wilhelm Von Leibniz - 1685 Leibniz built a mechanical calculator that could add, subtract, multiply and divide.
Analog Computation Was Important Historically Examples: I The slide rule (derived from Napier s bones, c. 1610) I Kelvin s Fourier analyzer (c. 1880) I Hartree s D.E. solver (c. 1930)
Structure of the Major Course Work: Mathematics (5 courses) Calc III Linear Algebra 3 of D.E.*, Math Modeling*, Continuous Applied Math*, Discrete Applied Math*, Probability*, Math Stats* Statistics (3 courses) Intro. to Statistical Modeling Multivariate Statistics At least one more upper-level course, e.g., Math Stats*, Econometrics*, Computation (3 courses) Introductory programming* Scientific computation AI*, databases*, (* = elective choice)
Structure of the Major Other requirements: Integrative Experience An internship or summer research project approved by the department; A minor or major in another department tied to applied mathematics or statistics (e.g., physics, economics, psychology, sociology, chemistry, geology, geography, environmental studies) approved on a case-by-case basis by the department. A preceptorship in two of the courses included in the applied mathematics requirement A Capstone research project. This focusses on communication: a public (30-min) presentation to a lay audience (first-year students), a technical report evaluated by faculty.
How We Built The Major The goal was to integrate three disciplines : applied mathematics (which draws heavily on some aspects of mathematics), statistics, and computation. We NEVER made a long-term plan to establish this program. Instead, over the period of a decade, slow, steady pressure was made in hiring and the program. 2002/3 hire a PhD statistician. Until then, statistics was taught by non-statisticians. 2003/4 establish statistics minor. Becomes 2nd most popular in the college and double enrollments in mid-level statistics courses. 2004/5 hire a computer scientist expert in databases. (Also bioinformatics.) 2004/5/6 hire a second statistician, but an applied statistician 2006/7 hire an applied mathematician. Each of the hires was difficult. There was pressure always to hire
Accidents that Helped First statistician left for Haverford. This was for family reasons, but we knew we would face difficulties hiring a replacement. Motivated an increase in the visibility of statistics within the department. A theoretical mathematician hire Andrew Beveridge had deep interests in some areas of applied math and industrial experience.
The situation now Faculty: Two statisticians (Addona and open position) & Kaplan Three computer scientists (Fox, Shoop, Sen) & Kaplan One applied mathematician (Topaz) plus support from Wagon (modeling, algorithms), Beveridge & Kaplan. This is the first year for the major. We ll see what happens.