About Mathematics Commentary
A Brief Content Overview
Many instructors will wish to interpret About Mathematics from their own point of view.
For them we offer this shortened summary of what we seek to accomplish in each chapter of the text.
The associated comments suggest aspects of the content that you should consider communicating.
Preface We recognize and seek to address the problem of communicating concepts we have mastered —
and thus appear simple to us — to students who lack our background.
Comment: Here we seek to position us with rather than against your students.
Introduction Mathematics, an essential component of the educated mind, has two important roles to play: as queen of the sciences
(that is as the logical structure that holds science together) and as handmaiden to those sciences
(carrying out the computations that apply them.) Unfortunately, a gulf exists between science and the humanities and
we seek to bridge that gap, just as humanities courses are offered to inform science students. And finally,
unlike most math texts, our listed goals do not include the usual new focus on content and technique.
Comment: Despite the inclusion of the Snow criticisms, this chapter is written from a humanities point of view.
Especially important is the idea that this course will be different as those goals point out.
Chapter 1 Getting StartedThe panels are introduced and some elementary calculation with calculators reviewed before turning
to a topic of 21st Century mathematical exploration: fractals.
Comment: This chapter is designed to capture the interest of students and to introduce them to some
significant ideas very different from those generally addressed in mathematics classes,
giving them a new fresh look at mathematics.
Chapter 2 Algorithms, Functions and Equations This chapter focuses on algorithms, providing insights into how arithmetic “works”
and how machines mirror mathematical processing including the solution of complex equations.
Comments: After Chapter 1, this may seem like a retreat to standard math content; however, whereas equations
and functions are familiar to students from secondary school mathematics, algorithms are less familiar to them.
Except for synthetic substitution, the examples are meant to be just that and, especially the square root algorithm,
not to be memorized.
Chapter 3 Dimensional AnalysisThis topic, normally shunned by mathematicians but often taught in physics classes, is simply the arithmetic of labels.
Comment: This approach provides a useful means to organize calculations with labels and to check work.
Chapter 4 Money MattersThis chapter introduces economic tools that can contribute to the lives of readers.
It also includes the lengthy derivation of the amortization formula.
Comments: Economic formulas can be complex and off-putting, but here and in future chapters the power of the
panels in which such formulas are embedded is evident. Going through the derivation of section 4.4 (the only one in the book)
is an illustration of “we need a reason to support line x implying line x+1.” It is important to point out that the difficulty
lay readers have with mathematical exposition is that the gap between those two lines increases with the sophistication
of the audience.
Chapter 5 The Science of SecrecyThis chapter provides an illustrated history of coding leading up to the contemporary RSA codes.
Comments:Only in the final sections does this chapter include any “usual” mathematics, but the logical structure
of the codes fully qualifies. The code-breaking examples require no prerequisite skills and thus provide excellent intellectual
challenges. The introduction to RSA coding is not designed to be useful for sending messages; rather, it is to show the extreme
difficulty it poses to code breaking.
Chapter 6 Calculus: The Smallest Pebble on the BeachHere are presented in simple form the ideas of differentiation and integration and the remarkable (and historically important)
association of the two. For comparison with the continuous mathematics underlying calculus a brief introduction to
discrete mathematics is included.
Comment: While this chapter may seem superficial to calculus teachers, it includes the important unifying ideas of the subject.
Chapter 7 Mathematical ModelsThis chapter is about simulations and applications using random numbers. They illustrate the remarkable convergence of random
number solutions and solutions based on mathematical derivations.
Comment: Here the panels play their most powerful role because they are driven by random numbers.
Chapter 8 ArithmeticThis chapter on the four basic arithmetic functions explores the algorithms that drive those functions and applies
them to other bases.
Comment:This chapter appears late in this text for a reason. Arithmetic is reconsidered here from this new and more
general perspective and not as a tool for the remainder of the text.
Chapter 9 Powers, Logarithms and Exponential ChangeBasing them on Newton’s Method for calculating square roots, computer techniques for calculating exponents and logarithms
are explained. Exponential and logistic growth are then considered.
Comment: If this content were taught with an expectation of student mastery, this would be far and away the most difficult
chapter in this text. Instead, it is designed only to illustrate how mathematics is applied to computing and to some real world
modeling related to exponential growth.
Chapter 10 Alternative GeometriesA consideration of Euclid’s fifth postulate leads to spherical and hyperbolic geometries.
Comments: This chapter also offers an additional perspective on how mathematics works. In this case long accepted beliefs
are reexamined and turn out to offer alternatives. Just as with Einstein’s relativity, the new conceptualization extends
our understanding but Newtonian concepts remain as useful as in the past: these new geometries do not replace Euclidean geometry.
About this Commentary
This commentary is in no way meant to serve as a single roadmap about how to teach with the text, About Mathematics; rather,
it offers classroom teaching suggestions that may selectively supplement the ideas of individual instructors.
Providing such a commentary is not common for a college text. They are generally designed for teachers who need step by step
guidance for teaching a subject with which they are uncomfortable. A different problem arises in teaching with this text, however,
and it is not with the content. Even chapters of this text that address mathematics unfamiliar to you — coding, for example —
will be well within your understanding. Rather, what may be new is the teaching style that best suits the approach
of About Mathematics.
Most of us who teach mathematics do so by straightforward lecturing. We seek to clarify the content of the text we are using
through classroom presentations that involve very little student participation. I have taught classes of 240 students by this
means. This text calls for a very different approach that will be more familiar to and thus more appropriate for humanities students:
settings that are often described as seminars. In these classrooms, teacher-student interaction is far more important.
It is also important that you understand the goal of this text. It is definitely NOT to give students additional mathematics tools.
The calculus chapter is not, for example, designed to give the students the power to solve advanced problems with this powerful tool.
Not this time: here the chapter is designed, as is the title of this text, to give students a sense of what calculus is ABOUT: that is,
what mathematicians do, how some of its ideas have been developed, and especially how mathematics can serve us. I am reminded in this
regard of the remark of a math major who served as my student grader. After we studied the calculus chapter he told me, “Despite the
two semesters I have studied calculus, only now do I have a sense of what this subject is about.” (There is that word again, and again
it breaks that rule about prepositions ending sentences.)
This commentary will then offer suggestions about how the topics of the text may be extended in such discussions. Yes, there will be
mathematics content aspects of the topics that you will wish to review with care to give your students better understanding of them,
but we hope that you will always keep in mind that you are not seeking additional skill at mathematical activities like proof and
additional mathematics content mastery. Instead you are giving your humanities students a sense of what mathematics is and an
appreciation for its contributions to our thinking, our society and them as individuals.
Important Invitation
This commentary is designed to be useful to anyone teaching with this text. Its content is not fixed and we invite you to share
classroom techniques that have served you in teaching this content. We promise that such suggestions will be carefully
reviewed and very likely incorporated here.
We also look forward to later editions of the text, About Mathematics, and we invite your suggestions — including criticisms —
that could affect text revisions. Such suggestions could include such things as: additional or substitute content, changed
approaches to concepts or additional exercises.
These communications may be directed to any of us. Our email addresses are in the text at the end of the Preface.