Draft Report (2nd Version)
CRAFTY Workshop for Business Faculty
October 28-29, 2000 Tucson, Arizona
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
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
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.
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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
Realistic business problems (take a business person to lunch).ii
Real data sets.
Problem motivated modeling.
Development of students’ abilities to express things analytically using algebraic
articulation and symbolic notation.
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
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,
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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
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
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
To avoid disasters and disappointment, it should be recognized that every school is
unique. No one recipe will work.
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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
Business students should be held to professional standardsiv for the work they submit.
Emphasize problem solving to build the mechanics and improve them.
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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
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
Enhance students’ abilities to display results graphically.
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.
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.
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
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
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
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?
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
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.
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
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.