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Mathematical Excursions

Richard Aufmann et al.
Houghton Mifflin
Publication Date: 
Number of Pages: 
[Reviewed by
Álvaro Lozano-Robledo
, on

Most instructors face an uphill battle when teaching mathematics to college students who need to fulfill a math requirement, outside of their major. What topics are suitable? What topics are fundamental? At what level should we teach these topics? These are hard and important questions and each department of mathematics should agree on the appropriate answers according to their student body.

Whatever the answers are, Mathematical Excursions attempts to provide a wide range of topics in mathematics to choose from (see the table of contents for a list) that might be taught in some sort of 100-level "Introduction to Mathematics" course. As the authors point out in the preface, "[the book] is similar to an English literature textbook, an Introduction to Philosophy textbook, or perhaps an Introduction to Psychology textbook". The authors also state as a goal the strengthening of the student's quantitative understanding of the world we live in. In that vein, they have included numerous examples and applications throughout the book.

The book is organized as most calculus textbooks (i.e. a brief introduction followed by the main definitions, followed by example — example — example), and each section ends with a large list of exercises. The pages are also sprinkled with "excursions" (extra topics, such as Conway's game of sprouts, or fuzzy sets) and "math matters" pieces and margin notes (explaining the relevance of a given topic in the real world, such as the invention of the Enigma machine). This is a very standard structure. A student who opens this book will be immediately reminded of his/her (often dull) high-school and calculus textbooks.

So, does the book achieve its goals? Yes and no. The book does cover a very wide and varied array of topics, and the authors do a good job at explaining the concepts throughout numerous examples.

The list of topics is varied, but the authors only seem to tap lightly into most of them, and many seem too elementary or even tedious.  Chapters 5, 6 and most of 8 (with titles "applications of equations and functions" and "geometry") merely cover high-school level topics, such as solving first and second degree equations; linear, quadratic, exponential and logarithmic functions, and area, perimeter of polygons and elementary trigonometry. The length and elementary nature of such chapters is in stark contrast with other topics. For example, chapter 2 includes an introduction to cardinality (7 pages) and Cantor's diagonalization argument (in one page), while chapter 5 explains how to solve first degree equations in one variable (in over ten pages!).

The biggest problem, however, is the tone. As I pointed out before, I would expect that most of the students enrolled in this class either dislike mathematics, or they fear mathematics, or, at best, the are indifferent about math. In order to reach out to these students, and to capture their imagination and attention, the lecturer needs to transmit the enthusiasm we share for math and explain to the students why mathematics is such an exciting and interesting subject. We need to convey the idea that math is very much alive. Ideally, I would like to have a textbook which already vibrates with such enthusiasm… but "Mathematical Excursions" does not vibrate.

Rating : 1 star (out of three)

Álvaro Lozano-Robledo is H. C. Wang Assistant Professor at Cornell University.

  • 1. Problem Solving
    1.1 Inductive and Deductive Reasoning
    1.2 Problem Solving with Patterns
    1.3 Problem-Solving Strategies
  • 2. Sets
    2.1 Basic Properties of Sets
    2.2 Complements, Subsets, and Venn Diagrams
    2.3 Set Operations
    2.4 Applications of Sets
    2.5 Infinite Sets
  • 3. Logic
    3.1 Logic Statements and Quantifiers
    3.2 Truth Tables, Equivalent Statements, and Tautologies
    3.3 The Conditional and the Biconditional
    3.4 The Conditional and Related Statements
    3.5 Arguments
    3.6 Euler Diagrams
  • 4. Numeration Systems and Number Theory
    4.1 Early Numeration Systems
    4.2 Place-Value Systems
    4.3 Different Base Systems
    4.4 Arithmetic in Different Bases
    4.5 Prime Numbers
    4.6 Topics from Number Theory
  • 5. Applications of Equations
    5.1 First-Degree Equations and Formulas
    5.2 Rate, Ratio, and Proportion
    5.3 Percent
    5.4 Second-Degree Equations
  • 6. Applications of Functions
    6.1 Rectangular Coordinates and Functions
    6.2 Properties of Linear Functions
    6.3 Finding Linear Models
    6.4 Quadratic Functions
    6.5 Exponential Functions
    6.6 Logarithmic Functions
  • 7. Mathematical Systems
    7.1 Modular Arithmetic
    7.2 Applications of Modular Arithmetic
    7.3 Introduction to Group Theory
  • 8. Geometry
    8.1 Basic Concepts of Euclidean Geometry
    8.2 Perimeter and Area of Plane Figures
    8.3 Properties of Triangles
    8.4 Volume and Surface Area
    8.5 Introduction to Trigonometry
    8.6 Non-Euclidean Geometry
    8.7 Fractals
  • 9. The Mathematics of Graphs
    9.1 Traveling Roads and Visiting Cities
    9.2 Efficient Routes
    9.3 Planarity and Euler's Formula
    9.4 Map Coloring and Graphs
  • 10. The Mathematics of Finance
    10.1 Simple Interest
    10.2 Compound Interest
    10.3 Credit Cards and Consumer Loans
    10.4 Stocks, Bonds, and Mutual Funds
    10.5 Home Ownership
  • 11. Combinatorics and Probability
    11.1 The Counting Principle
    11.2 Permutations and Combinations
    11.3 Probability and Odds
    11.4 Addition and Complement Rules
    11.5 Conditional Probability
    11.6 Expectation
  • 12. Statistics
    12.1 Measures of Central Tendency
    12.2 Measures of Dispersion
    12.3 Measures of Relative Position
    12.4 Normal Distributions
    12.5 Linear Regression and Correlation
  • 13. Apportionment and Voting
    13.1 Introduction to Apportionment
    13.2 Introduction to Voting
    13.3 Weighted Voting Systems
  • Appendix: The Metric System of Measurement
    Web Appendix: Algebra Review