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  1. 1. Draft Report (2nd Version) CRAFTY Workshop for Business Faculty October 28-29, 2000 Tucson, Arizona Goals Mathematics is an integral component of the business school curriculum. Mathematics Departments can help prepare business students by stressing conceptual understanding of quantitative reasoning and enhancing critical thinking skills. Business students must be able not only to apply appropriate abstract models to specific problems but also to become familiar and comfortable with the language of and the application of mathematical reasoning. Business students need to understand that many quantitative problems are more likely to deal with ambiguities than with certainty. The use of business-appropriate technologyi is clearly important and supports the primary objective of developing quantitative literacy. Even though many business students may have an inherent fear of mathematics, we hope that reforms in the business math curriculum can help students appreciate the relevance of mathematics. In the spirit that less is more, coverage is less critical than comprehension and application. As a broad set of broad guidelines, Math faculty should try to keep the following objectives in mind:  Courses should stress problem solving, with the incumbent recognition of ambiguities.  Courses should stress conceptual understanding (motivating the math with the “whys” – not just the “hows.”)  Courses should stress critical thinking.  An important student outcome is their abilities to develop appropriate models to solve defined problems.  Courses should use industry standard technology (spreadsheets).  An important student outcome is their abilities to become conversant with mathematics as a language. Business faculty would like its students to be comfortable taking a problem and casting it in mathematical terms.  The Math faculty should recognize that business students have an inherent distaste for math. (This does not imply anything about their abilities.) As such, the courses should strive to lessen math phobia, and enable students to be more comfortable with math. Content In general, business faculty are less concerned with specific course content than with adherence to the notion of developing quantitative literacy and analytical ability in our students. Upon completion of a business math curriculum the students should be comfortable with using mathematics as a tool to communicate analytical concepts. A measure of the success of the business math curriculum is the level of comfort of students when exposed to a new formula in a business class.
  2. 2. - page 2 - When in doubt, math faculty should cover less material–and treat the material covered with respect–imbuing in the students a sense of the importance of mathematics as a necessary part in the development of successful business people. Business decisions are most commonly made under conditions of uncertainty and risk. Inferences must be drawn from data and information that is incomplete, inconclusive, and most likely imprecise. Wherever possible, math courses should attempt to illustrate this ambiguity and provide guidance in dealing with such uncertainty and variation. Sensitivity analysis can be used to demonstrate how changes in assumptions, variation in data, and the influence of contextual variables will affect outcomes. This approach will facilitate a more in-depth understanding of the interrelationships and interdependencies embedded in real world situations. In order to achieve the desired outcomes, the business faculty recommends that the curriculum include:  Realistic business problems (take a business person to lunch).ii  Spreadsheet solutions.  Real data sets.  Problem motivated modeling.  Development of students’ abilities to express things analytically using algebraic articulation and symbolic notation.  Sensitivity analysis. Knowledge of the concepts and applications of algebra at the college level should be considered a serious prerequisite to study in business. If this prerequisite is not validated by a placement test upon entering college, algebra should be taken before the business math curriculum. It is imperative that students be able to:  Solve simultaneous equations.  Understand intuitively the concept of a function and functional relationships.  Understand the use of common functions in modeling business concepts.  Graph and understand graphs.  Use abstraction to build simple models. Calculus should emphasize the basic concepts and how they apply to business problems. Give more attention to numerical methods and less to techniques of symbolic differentiation and integration.iii  Introduction to rates of change  Dynamic nature of real world systems  Understanding of constrained optimization model  Understand areas under a curve (Example: learning curve) Business students need to acquire a practical and conceptual understanding of statistical concepts including measures of central tendency and dispersion, probabilities, probability distributions and hypothesis testing. This understanding should enable students to examine,
  3. 3. - page 3 - summarize, analyze, graph and interpret real data sets used in business. Among the specific topics to be addressed are:  Know how to describe real data (central tendency and dispersion).  Be able to present and explain data.  Dealing with uncertainty and risk.  Conditional probabilities (Example: Decision tree analysis).  Discrete probability distributions.  Continuous probability distributions. In virtually all schools, there is a requirement for an additional statistics course, beyond the introductory course in basic probability. In most business schools business faculty teaches this course. Regardless of who teaches this statistics course, it should also be integrated seamlessly into the business math curriculum. How do we get there? There are a few key elements that are critical to the accomplishment of the goals detailed above. Central to this effort is the cooperation and communication between the business and the mathematics faculties. The willingness to change will be built upon personal relationships. Collaborative identification of clear objectives, and creation of the corresponding pedagogy, will build the foundation for successful implementation. The probability of lasting change will depend on the monitoring and assessment of relevant performance measures. Specific suggestions for moving towards the goal of improved mathematical competence of undergraduate business students include:  Communication and collaboration between business and mathematics faculty is critical.  Staffing of business math courses should be treated equally with other math courses.  The faculties from math and business should develop a seamless education process. Students and faculty should understand that the math curriculum in integrated with the business curriculum.  Change perceptions: The purpose of business math is to add value - not to weed out students.  Math faculties have the right to ask business for examples to motivate the mathematics including background on topics.  An active dialogue between MAA and AACSB should be established.  Establish dialog between math and business at regional and national business conferences.  To avoid disasters and disappointment, it should be recognized that every school is unique. No one recipe will work.
  4. 4. - page 4 - Instructional techniques Educational reform in many business schools has changed the role of the instructor. Useful pedagogical techniques include questions to make students think about a problem or a concept, asking students to discuss questions amongst themselves, and inviting a student to share their proposed solution to a problem with the class and asking the rest of the class to critique it. Students feel as part of the process in active and cooperative learning exercises, while the traditional passive learning characterized by an instructor lecturing may fail to achieve a similar buy-in from students. A very effective active learning method is real-time problem solving or model building using student input in class. Starting with a blank spreadsheet on the screen and filling it out using student suggestions and instructions gives students a sense of control over the lecture and draws them in. In contrast, displaying completed sheets on a screen may result in information overload and rejection. Another potentially useful method for drawing students into the lecture is to start the lecture with a real-world (or realistic) business problem. If students are convinced that the problem is worthy of their attention, and that they do not know how to solve it, they are much more likely to pay attention and to retain what they learn. The same effect cannot be achieved by merely mentioning some applications after completing the coverage of a topic. It is important to get the buy-in at the beginning. Many business courses attempt to develop and reinforce skills that are valued highly by employers such as group work and communication skills. We believe that these skills can and should be emphasized in Math courses as well. Students could work on group projects or assignments, and be asked to prepare written reports, or even oral presentations, to communicate the results of their analysis. The ability to work effectively in groups and the ability to explain quantitative concepts and results in plain English are skills that are highly valued in Business schools, and emphasis of these skills in first-year courses would achieve a more seamless transition to Business school for students. We believe that spreadsheets should be used in the delivery of quantitative material to the fullest extent. Spreadsheets are very useful in charting data, conducting exploratory data analysis, developing models, demonstrating impact of changes on the inputs on the outputs, carrying out parametric analysis, as well as more sophisticated applications such as optimization and simulation. Their potential should be fully exploited in the classroom, and the classroom use should be complemented with hands-on experience for the students in computer labs. The labs can be self-directed and rely on tutorials available on the web or prepared for the class, or they can take the form of instructional labs or workshops supported by teaching assistants, depending on the needs and resources of the institution. Students could repeat the in-class exercises or work on assignments and projects in the labs. Some specific recommendations for pedagogy include the recognition that:  Active methods are good. We seem to rely too much on passive methods.  Working in teams will help the learning process for business students.  Business students should be required to communicate concepts with oral and written reports.  Business students should be held to professional standardsiv for the work they submit.  Emphasize problem solving to build the mechanics and improve them.
  5. 5. - page 5 - Technology Technology has several roles to play in the business math curriculum. First, it provides tools that students will encounter in the work place. Second, it enhances the effectiveness and efficiency of the learning process. Third, it can help to deepen and maintain student interest.v Goals for the curriculum vis-à-vis technology include:  Students should become comfortable with the use of technology as an analytical tool.  Integrated spreadsheet experiences starting with math and continued in subsequent course.  Using technology to answer “what-if” questions.  Encouraging students to experiment and try alternative approaches to a problem.  The use of real world data.  Enhancing students’ ability to design an experiment and find data, e.g., from the web.  Enhancing students’ abilities to explain calculations and prepare a clear effective presentation.  Enhance students’ abilities to display results graphically. Research Business faculty on pedagogical issues conducts substantial research. Periodic reviews of business education journals by CRAFTY would keep MAA current on business pedagogical issues and provide CRAFTY with mathematics initiatives and excellent business examples. The business education journals include: Financial Practices and Education, Journal of Financial Education, Journal of Economic Education, Accounting Educators Journal, Issues in Accounting Education, Journal of Management Education, Journal of Marketing Education, INFORMS Transactions on Education, and Journal of Education for Business. Research on business education is taking on an increasing importance as AACSB assessment expectations rise. In reaction, pedagogy is rising in importance and new pedagogical journals are under consideration; for instance, the Decision Sciences Institute is planning an education journal for Information Sciences. The business environment is rich with a variety of activities. There are a number of business pedagogical seminars sponsored by foundations such as the Lilly Foundation. Book publishers often conduct seminars on learning theory and the learning impact of their products, and professional associations such as the American Marketing Association, Academy of Management, Decision Sciences Institute, Financial Management Association, American Accounting Association, and the American Economic Association routinely sponsor teaching and learning sessions. AACSB constantly monitors changes in the business environment and the corresponding changes in business education. Through networking with the AACSB staff, CRAFTY can stay apprised of the latest expectations in business content and technology. Then, real-time changes in the content-pedagogy-technology triad can be promulgated to departments of mathematics. CRAFTY reviews of the two-year mathematics curriculum thereby become semiannual or annual rather than decennial reviews. An added benefit of this reduced review period is that mathematics faculty becomes increasingly knowledgeable business school partners.
  6. 6. i This specification is meant to focus math faculty on the tools of business – discussed in more detail below – as distinct from software whose primary applications are in science. i ii This pithy phrase is meant to highlight the notion that we do not expect the math faculty to develop these problems on their own. We envision a partnership. Business faculty must add value to the courses. i iii Calculus should emphasize the basic concepts, with minimum attention paid to details or computational techniques. Basic concepts can, of course, be organized under differential and integral calculus. The exact content of any course might differ among universities determined by local needs and demands. Rates of change should normally be included among the basic concepts. So also should changes in rates of change, the sign of the second derivative. What is a slippery slope? Will population, stock prices, profits, costs, debt, the current account deficit (surplus), inflation rates, the government budget surplus (deficit), pollution counts, P/E ratios, aids cases, oil prices, the professor’s “uhs” per minute, or anything else rise (fall) indefinitely, or is there an end in sight. Optimization and constrained optimization models should also be included. Firms seek to maximize profits, revenues, shareholder value, market share or output, or to minimize costs, time to market, errors in quality control, etc. Non-profit organizations seek to minimize costs constrained by their limited budgets. No organization can escape the constraints of their budgets or production functions. Special topics of interest to business students may arise. For example, the chain rule may seem needlessly complex, but could be applied in quantitative analysis of multi-stage cause & effect processes such as value chains, supply chains or even the chain of events in the life of a students that begins with the first Business Mathematics course and ends with the BBA. Integral calculus may be less important quantitatively than it is qualitatively. However, areas under curves, representing as they do the cumulative values of the curve, are of great importance in many problems. New learning appears on a learning curve (height) but cumulated wisdom is underneath. Deficits appear regularly, but debt accumulates as the sum of all deficits. New AIDS cases are added to the total burden of our health industry. Population growth (aging) rates accumulate as population or as threats to the solvency of Social Security. Benefits accumulate under the demand curve; costs accumulate under the supply curve. Hence, benefit-cost analysis becomes feasible. Real world systems, normally expressed in static terms, can be understood dynamically using differential equations. If students are expected to know enough algebra to solve a system of simultaneous linear equations, why not substitute in a few rates of change for the parameters? i iv Business students are anticipating entry into the business world. Faculty should expect that the materials that students are asked to prepare are analogous to materials that they will hand to their bosses in the business world. This expectation, coupled with the enhanced focus on fewer – but deeper – problems v Technology has revolutionized the way in which business is undertaken. No longer are hours spent determining a loan payment, the value of a bond, the effect of a change in sales on net income or the price of an option. Now with a financial calculator or a spreadsheet, these tasks are preformed with the touch of a few buttons. Business education has strived to keep up with these changes. Debits and credits, and journal entries no longer dominate the teaching of accounting classes. The algebraic manipulation necessary to compute present and future value no longer dominates finance courses.
  7. 7. Today, business students are introduced to these concepts, but the tireless hours devoted to their calculation are left to the spreadsheet and the financial calculator. The business executive of the twenty-first century will certainly have a personal computer on her desk, and a financial calculator in her purse. It is important that the businessperson of today have the conceptual basis of their area to use these tools. The business leader of tomorrow and therefore the business student of today need to understand the conceptual basis of algebra, calculus and statistics. Thanks to technology, they no longer will need to waste time on the repetitive mundane calculations necessary to get an answer, but they must be able to interpret and use this information. The accountant has changed from a bean counter that reports what has already occurred into a business planner who assists charting the future course of the company. The finance executive has to be able to quickly evaluate several competing alternatives with risk and expected returns. For these and other business executives to be successful in the highly competitive environment of today and tomorrow, they need to be able to use technology to give them the data they need, and an understanding of the algebra, calculus and statistics underlying this data, and knowledge on how sensitive the results are to changes in the input data. As educators in business and mathematics, our role is to provide students with this understanding.