Thoughts on Business Calculus

The project of this text can be traced to a talk at a conference many years ago when the speaker said "Teaching business students math with graphing calculators is silly because no one in the business world does math with graphing calculators.  They use spreadsheets." 

Trying to implement that idea, we started with Networked Business Math by Richardson and Felkel from Appalachian State University.  When it was clear it would not be revised, we decided to write a book, but did a rethinking of Business Calculus. 

In rethinking a course on calculus for business students I looked at the Curriculum Foundations Project (CFP) reports of the Committee for Renewal  Across the First Two Years (CRAFTY) of the Mathematics Associations of America (MAA).  I also surveyed Business faculty at several institutions.  This leads to a number of principles.

• The primary computational tools for business students is a spreadsheet, so they should be taught math using that tool.
• Students should be taught practices of good spreadsheet usage.
  
• Using a spreadsheet enables more emphasis on numerical methods and conceptual approaches.
  
• Examples for business students should follow the terminology and usage from the business world.
• The selection and ordering of topics should be based on the needs of the business school.
• Problems should regularly start with raw data and include fitting the data to a model.
• The design of the course should be consistent with it be the idea that it is the last math course the students will take.
• The course should be taught assuming that the students are not familiar with Excel.
  

In contrast, most one semester calculus texts I have seen seem to have started as a three semester engineering calculus with trig functions removed.  They are designed to be technology agnostic, which means that little attention is paid to numerical methods.  They are often written for "Business and the Life Sciences".  They often are more dependent on symbolic manipulation techniques.

These principles behind the text have implications for the ordering and selection of topics in the course.

• Both derivatives and integrals are introduced through numerical methods. 
  
• The derivative is introduced with the informal definition of the slope of the line obtained by zooming.  It then moves to a balanced difference quotient.
• The integral is introduced as an approximation by Riemann sums.
• With a spreadsheet template for derivatives and integrals, students can be asked about most applications before they are given any symbol manipulation formulas.
• The numerical answers can be used to justify the symbol manipulation formulas.  (I find the argument that the computations work in lots of cases is more convincing to the students that the more rigorous mathematical proof.)
  
• Partial derivatives are introduced before integrals.
• The opinions of the faculty in partner disciplines should matter.  The consistent reply from business faculty is that partial derivatives and optimization of functions of several variable is more important than integration.
  
• Examples, definitions, and conventions used in business should be used whenever possible.
• Economics examples use the q-p axes with q as the independent variable.
• Marginal functions are not the same as derivatives.  A better description would be a difference quotient with denominator 1.
• For the most part, physics and geometry examples are not used.
  
• Students should be taught Excel and good Excel practices.
  
• The text does not assume any Excel skills.  They are taught through the text.
• However, the text is understood as a math text that uses Excel rather than a course in Excel.  The Excel skills needed are quite limited.
• Examples will have a single screen where inputs and outputs are gathered, with supporting work below.
• Unlike the student's previous experience, there is an emphasis on building templates that can be reused in the future and understood by the reader.
• Template production and report presentation are important.  Most math classes train students to do work that will never be looked at again.  In a last math course, more emphasis should be placed on re-useability and readability.
• There are about four templates that cover most of the material of the course,  There is an emphasis on solving classes of problems.  A well designed template for numeric differentiation or integration works for all reasonable functions.
• Students are encourages to build templates that can be reused and that have internal documentation.
• Text examples will tend to use long variable names and limit the number of new operations in a single cell.  Both practices improve readability.
• A standard comment in class has been that we use "Business rules".  If I, your boss, is confused, you are wrong.  It does not matter if you are correct if you cannot explain it to your boss.
  
• Problems routinely start with data points.  The student is then asked to find a best fitting formula for use in the problem.
  
• "Outside of math class" problems typically start with data.  We then need to decide on an appropriate model or symbolic formula, fit the model to the data, manipulate the symbolic formula for the model, find an answer, and interpret the answer in the original context.
• The students' experience is that problems are reduced to manipulate the formula.
• Reducing to symbol manipulation may make sense in a course designed to prepare students for more math classes.  It makes less sense in the student's last math class.
  
• Exercises start with nice numbers and then move to ugly numbers.
• Students often come to this course trained that problems should work out with nice numbers.  That is a product of teaching math in a manner that is technology agnostic.  We don't want the arithmetic to overwhelm the day's lesson.
• In building templates, we want to start with problems that can easily be done by hand.  We also want methods that work when the problem is not rigged.
  

I have been asked a number of times while I use Excel rather than other spreadsheet products.  The simple answer is that my business school want the course to use Excel.  From the viewpoint of mathematical instruction, the Solver tool in Excel is much more robust than the comparable tools in other spreadsheets.  In the text, Solver is used for optimization and finding critical points of functions of several variables.  However, most of the text can be done with any spreadsheet program.

This project started with an attempt to write a text.  It then turned into a project to put a textbook online.  It has become a project for an online text.  I am still thinking through how an online text differs from a text that has been put online.  This project will continue as I think continue to think about the difference.

Some features of an online text that have been implemented:

• An online text allows links rather than simple scrolling.
  
• An online text allows "hidden pages" for things like solutions.
• An online text allows links to videos for demonstrations and small lectures.
• An online text should be viewable on a variety of screen sizes.
• An online text can include files for students to use on their own machines.
  

Some other features are planned and are just beginning to be implemented:

• An online text should encourage active learning with checkpoint exercises.
• An online text should contain appropriate applets for experimentation by students and instructors.

How others can help:

• Please send me and typos, corrections, or critiques.
• This textbook was designed for a course at my school.  If it can be used as a base for the course designed for your school, let me know.  I would be happy to collaborate on the missing chapter that you need to make it work at your school.
  

Mike May - maymk@slu.edu