Spreadsheet Software as a Teaching Tool for Introductory Mathematics Courses Shuzo Takahashi Arizona Winter School 2002 March 10, 2002

Abstract I will talk about one way to use spreadsheets (Excel) in teaching mathematics, in particular, in teaching elementary probability theory to business majors. Other applications (i.e., calculus) will be briefly discussed.

Outline of Talk  Overview of the Business Mathematics course  The loan workout project and its solution using Excel  A fun approach to teaching expected value and a graphical illustration of the meaning of expected value using Excel.  Possible use of Excel in teaching calculus

The Business Mathematics Course at the University of Arizona This course was developed jointly by Professor Richard Thompson Department of Mathematics and Professor Chris Lamoureux Department of Finance

Business Mathematics Business Mathematics is the foundation for students’ future career in business, showing them how to apply the language of mathematics to business problems, and how to use computers to do mathematics.

Business Mathematics Business Mathematics centers around significant business problems and the tools, both mathematical and computer based, that are needed for their solutions.  Projects are the loan workout project, the stock option pricing project, etc.  Mathematical tools are elementary probability theory, calculus, etc.  Computer tools are Excel, PowerPoint, etc.

Business Mathematics  Using the new tools, students work in teams to complete projects and report their solutions both orally and in written form.  Student presentations will be given in the current business style, using PowerPoint and Excel.

Business Mathematics  Hyperlinks allow you to move easily between mathematical instruction and computer calculation, graphing, and simulations.  Computer animations clarify mathematical concepts.  Electronic searching can locate any given word or phrase.  You can customize your computer text by adding notes, extra pages, or additional files. All course material is in PowerPoint, Excel, Word, and Media files. Usefulness of Computer Text

Business Mathematics Course Web Site Business Mathematics

 You are loan officers of Acadia Bank.  Acadia Bank has a commercial loan with a full value of $4,000,000 outstanding to John Sanders’ entrepreneurial venture.  His business is not going well and he is not able to pay interest on his loan.  What are you going to do as loan officers? Loan Workout Project

Acadia has to decide whether to foreclose on the loan or to work out a new schedule of payments.  If Acadia forecloses on John’s loan, it will only recover a foreclosure value of $2,100,000.  If Acadia attempts a workout, and John's venture succeeds, then the bank will be repaid the full value of $4,000,000.  If Acadia attempts a workout, and John's venture fails, then the bank will only recover a default value of $250,000.

Decision How do you decide which one to choose?  Foreclosure  Workout

Expected Value E(W) = Full Value  Pr(S) + Default Value  Pr(F)  E(W) = Expected Value of Workout  Pr(S) = The probability of success  Pr(F) = The probability of failure

How to Calculate Pr(S) and Pr(F)  Historical data on past borrowers  Certain information about John Sanders  Current economic conditions  Conditional Probability and Bayes’ Theorem

A Decision Method  Compare Expected Value and Foreclosure Value.  If Expected Value < Foreclosure Value, then choose “Foreclosure.”  If Expected Value > Foreclosure Value, then choose “Workout.” Solution

Questions  What is the actual meaning of expected value?  Is our decision method sound? Issue  Students tend to focus on their individual case.  They are not able to recognize that the bank’s profit is based on many similar loans as a whole.

A Fun Approach to Teaching Expected Value One lecture I gave in class.

Let's play! Who wants to be a millionaire?

You have won $64,000 and are facing a tough $125,000 question. This means that you will receive $125,000 if you answer the question correctly. But you will receive only $32,000 if your answer is wrong. Also, you can walk away with $64,000 if you decide not to answer the question. What would be the best strategy to take? Game Situation

$125,000 Question Which city is the birthplace of the famous business math instructor Shuzo? A. WakayamaB. Iiyama C. KumayamaD. Matsuyama

 Suppose that you decide to answer the question. What is the expected value of the outcome?  Do you answer the question or do you walk away with $64,000? $125,000 Question Which city is the birthplace of the famous business math instructor Shuzo? A. WakayamaB. Iiyama C. KumayamaD. Matsuyama

Expected Value 125,000  1/4 + 32,000  3/4 = 55,250 Walk Away 64,000 $125,000 Question Which city is the birthplace of the famous business math instructor Shuzo? A. WakayamaB. Iiyama C. KumayamaD. Matsuyama

50:50 $125,000 Question Which city is the birthplace of the famous business math instructor Shuzo? A. WakayamaB. Iiyama C. KumayamaD. Matsuyama

B. Iiyama C. Kumayama Two Choices

B. Iiyama C. Kumayama Two Choices  Suppose that you decide to answer the question. What is the expected value of the outcome?  Do you answer the question or do you walk away with $64,000?

B. Iiyama C. Kumayama Two Choices Expected Value 125,000  1/2 + 32,000  1/2 = 78,500 Walk Away 64,000

Suppose all of you will have a turn playing the game. You got together and agreed that you will divide the total winnings equally. (We are assuming that each of you is facing exactly the same situation described above.) Now think about what the best strategy is. Should everyone walk away? In this case, everyone gets $64,000. Or should everyone answer the question? Game Simulation Does Teamwork Help?

Project Simulation Team Discussion Now think about how this can be applied to your project.

Some Uses of Excel in Teaching Calculus  Graphing  Midpoint Sums  Numerical Integration  Slopes and Derivatives Graphing Sums Slopes Integration