Alarming! the Chasm Separating Education of Applications of Finite Math from It's Necessities

By Ramune B. Adams and William J. Adams

William J. Adams, Professor of Mathematics at Pace University,
is a recipient of Paces Outstanding Teacher Award. He was
Chairman of the Pace N.Y. Mathematics Department from 1976
through 1991.
Professor Adams is author or co-author of over twenty books
on mathematics, its applications, and history, including Elements
of Linear Programming (1969), Calculus for Business and Social
Science (1975), Fundamentals of Mathematics for Business, Social
and Life Sciences (1979), Elements of Complex Analysis (1987),
Get a Grip on Your Math (1996), Slippery Math in Public Affairs:
Price Tag and Defense (2002) ; Think First, Apply MATH, Think
Further: Food for Thought (2005), The Life and Times of the
Central Limit Theorem Second Edition(2009), and Alarming! The
Chasm Separating Basic Statistics Education from its Necessities
(2013). His concern with the slippery side of math and what math
can do for us and its limitations is a prominent feature of his writings
on applications.
Concerning higher education in general, he is the author of The
Nitty-Gritty in the Life of a University (2007).

• Language: English
• Publisher: Xlibris US
• Release date: Jul 15, 2013
• ISBN: 9781479799947

Ramune B. Adams

William J. Adams, Professor of Mathematics at Pace University, is a recipient of Pace’s Outstanding Teacher Award. He was Chairman of the Pace N.Y. Mathematics Department from 1976 through 1991. Professor Adams is author or co-author of over twenty books on mathematics, its applications, and history, including Elements of Linear Programming (1969), Calculus for Business and Social Science (1975), Fundamentals of Mathematics for Business, Social and Life Sciences (1979), Elements of Complex Analysis (1987), Get a Grip on Your Math (1996), Slippery Math in Public Affairs: Price Tag and Defense (2002) and Think First, Apply MATH, Think Further: Food for Thought (2005), The Life and Times of the Central Limit Theorem Second Edition(2009). His concern with the slippery side of math and what math can do for us and its limitations is a prominent feature of his writings on applications. Concerning higher education in general, he is the author of The Nifty-Gritty in the Life of a University (2007).

Book preview

ALARMING!

THE CHASM SEPARATING

EDUCATION OF APPLICATIONS

OF FINITE MATH FROM ITS NECESSITIES

Beware the Assumptions,

It’s All in the Assumptions

WILLIAM J. ADAMS

Mathematics Department, Pace University

with illustrations by

Ramunė B. Adams

Copyright © 2013 by William J. Adams.

Library of Congress Control Number: 2013903350

ISBN: Hardcover 978-1-4797-9993-0

Softcover 978-1-4797-9992-3

Ebook 978-1-4797-9994-7

All rights reserved. No part of this book may be reproduced or transmitted in any form or by any means, electronic or mechanical, including photocopying, recording, or by any information storage and retrieval system, without permission in writing from the copyright owner.

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116073

Contents

PREFACE

PART 1

Is Math Modeling a Necessity?

1

Is It Just a Word Problem, or Is There More to It Than That?

2

Which, if Either, Is the Right Linear Program Model?

3

Are Leontief Input - Output Models Realistic?

4

Equally-Likely Outcome Probability Models Über Alles,

No Ifs-ands-or Buts?

5

An Approach to Probability Modeling which Puts the Equally-Likely Outcome Model in Proper Perspective

6

Which Is the Right Probability Model?

7

The Other Side of the Coin: Equally-Likely Outcome Models, Conditionally

PART 2

If the Hypothesis of a Theorem in an Application Setting

is NOT Satisfied, What Then?

8

Is Random Sampling in Practice as Simple as It Sounds in Theory?

PART 3

What’s the Price Tag for Unrealistic Assumptions and Models?

9

Consider:

To Aleksa, Gaja, Sabrina, and Veronika

Acknowledgment: I am indebted to my daughter Ramunė for preparing the illustrations.

PREFACE

As an introduction to applied mathematics the sample of recently published finite math books that I gave thought to have two major shortcomings that make them unsatisfactory for this purpose.

1. Mathematical modeling which distinguishes a conclusion’s validity from its realism is not taken up.

2. The issue, if the hypothesis of a theorem in an application setting is not satisfied, what then?, is not addressed.

Truth in book title description requires that the afore sample of finite math books that I examined be called finite math with illustrations rather than applications since the difference is significant.

The purpose of this book is to address the afore dimensions of applied finite mathematics by providing illustrations and food-for-thought questions with answers/discussion to assist colleagues and students who share my concerns to become better acquainted with these dimensions.

As a point of information, for discussion of these dimensions in a finite math book I note W.J. Adams, Finite Mathematics, Models, and Structure, Revised Edition (Xlibris, 2009).

W.J.A.

Brief Contents

Preface

Contents

Part 1: Is Math Modeling a Necessity?

1 Is It Just a Word Problem or Is There More to It Than That?

2 Which, if Either, Is the Right Linear Program Model?

3 Are Leontief Input-Output Models Realistic?

4 Equally-Likely Outcome Probability Models Über Alles, No Ifs-ands-or-Buts?

5 An Approach to Probability Modeling which Puts the Equally-Likely Outcome Model in Proper Perspective

6 Which Is the Right Probability Model?

7 The Other Side of the Coin: Equally-Likely Outcome Probability Models, Conditionally

Part 2: If the Hypothesis of a Theorem in an Application Setting

is NOT Satisfied, What Then?

8 Is Random Sampling in Practice as Simple as It Sounds in Theory?

Part 3: What’s the Price Tag for Unrealistic Math Assumptions and Models?

9 Consider:

PART 1

Is Math Modeling a Necessity?

PREFACE

YES, absolutely. Consider the difference between It’s a Word Problem View and It’s a Math Model View.

Word Problem View

1. No distinction is made between the mathematical validity and real-world realism of the solution obtained.

2. The underlying statements that lead to valid conclusion are viewed as (real-world) facts, not as assumptions whose real-world realism may be open to question.

3. Since a valid conclusion viewed as based on facts is a fact, there is no question about implementing it.

Math Model View

1. Mathematical modeling rests on a distinction made between the mathematical validity of a conclusion and its real-world realism.

2. What is viewed as facts from the word problem view is viewed as assumptions whose realism may be open to question.

If the assumptions are unrealistic, the realism of a valid conclusion is open to question.

3. If the assumptions are realistic, this provides a GO-AHEAD to implementing a valid conclusion.

If some of the assumptions are not realistic, this provides a RED-ALERT that the realism of any valid conclusion is open to question and that its implementation might prove to be disastrous.

Shades of Validity: Be on Guard

I employ the term valid in this book in a very specific manner––roughly, that the so labeled statement follows as an inescapable consequence of a number of statements taken as a starting point (called a hypothesis), and individually termed assumptions, postulates or axioms. It is this sense of the word valid that is to be understood throughout the book.

As is the case with many words, valid is used in a number of ways in our everyday language. We might hear, for example, Jim had valid reasons for not following instructions; we must validate our assumptions. Uses for the word valid in our language include sound, cogent, convincing, telling and, unfortunately for discussions of applications of mathematics, true/realistic.

In reading any exposition some degree of misunderstanding is to be expected because of the many ways that words are used in our language. Beware.

1

Is It Just a Word Problem, or Is There More to It Than That?

1.1 PREFACE

In the 1960’s mathematics and mathematics education organizations endorsed mathematical modeling as part of their New Math proposals. The acceptance and implementation of this part of their propsals was very limited. The focus of mathematics education continued to involve translating a problem and its background into math form, solving it, and noting how to implement the solution. My book Elements of Linear Programming¹ is a book of this kind. A review² noted that The authors present an elementary account of linear programming and two-person zero-sum games with the primary intent being to teach the non-mathematics major how to set up and solve linear programs… . this reviewer feels that the authors have selected an interesting topic and presented it on a level which non-mathematics majors should be able to understand.

It may not seem surprising that what was standard mathematics education half a century ago will not suffice for the needs of the 21st century. To obtain a concrete sense in considering applications of finite mathematics I invite you to consider the case of the Last National Bank.

1.2 THE LAST NATIONAL BANK

The Last National Bank has assets in the form of loans and securities that, it is assumed, bring returns of 10 and 8 percent, respectively, in a certain time period. The bank has a total of $60 million to allocate between loans and securities. Two major guidelines are imposed by the bank on its lending activity: (1) a securities balance equal to or greater than 25 percent of total assets must be maintained, and (2) at least $15 million must be available for loans.

The bank is interested in determining, with respect to these conditions, how funds should be allocated between loans and securities so that investment income