San José State University
Department of Mathematics
Math 10
Mathematics for General Education
Prepared by Tim Hsu, Spring 2009
This course outline is much more subjective and opinionated than usual, and many instructors
will disagree with some part of what is said here, which is OK (see Course Objectives below). If
you have further questions, please email hsu@math.sjsu.edu.
Catalog Description: Topics from: methods of proof, problem solving, trigonometry,
probability, statistics, applications to scheduling and apportionment, population studies,
consumer math, theory of games, polyhedra, networks, graph theory, linear programming. (3
units) (Note: As you’ll see, the catalog description is badly outdated and should be ignored. It’s
just not worth filling out the paperwork to change, as no one actually uses the description to
decide to take the course.)
Prerequisite: Satisfaction of the ELM requirement.
Course Objectives
In Math 10, students need to learn new and interesting math in a manner that satisfies the
General Education Area B4 requirement. Exactly how this is achieved is up to you. (See
below for several approaches that have worked recently.)
The GE Area B4 requirements are described as follows: The major goal is to enable the student
to use numerical and graphical data in personal and professional judgments and in coping with
public issues. More specifically, a mathematical concepts course should prepare the student to
use mathematical methods to solve quantitative problems, including those presented in verbal
form; demonstrate the ability to use mathematics to solve real life problems; and arrive at
conclusions based on numerical and graphical data.
Also: Issues of diversity shall be incorporated in an appropriate manner. There is also a
minimum writing requirement of 1500 words “in a language and style appropriate to the
discipline”; in practice, the 1500 word requirement is met either by “word equivalents” (i.e.,
math problems) in homework and exams, or more literally by requiring projects, papers, etc.
Students in Math 10
Probably the most important thing to understand about Math 10 is the student population.
Students typically take Math 10 for precisely one reason: because they need to take some kind of
math class at some point to graduate. Therefore, students usually major in subjects that do not
fulfill the math GE requirement in some other fashion, like English, graphic design, history,
dance, recreation and leisure, foreign languages, and so on. Note that science, engineering,
business, and psychology majors all have classes as part of their majors that fulfill the math GE
requirement, so you are unlikely to encounter any of them.
My personal experience is that Math 10 students are bright and able to understand substantial
mathematical ideas, but can have severe technical limitations; for example, I had a student who
gave great explanations of the Cantor diagonalization argument and how the 4th dimension would
allow you to walk through walls, but could not reliably compute powers of fractions (e.g.,
(2/3)^9). One positive surprise to me was that very few of them fear math or have “math
anxiety”, especially if algebra is avoided; in general, they seem to be more “math-indifferent”,
and many of them do come in thinking that math is boring. My standard of success has always
been that students should leave knowing more math than when they came in, and liking math
more than when they came in. (Though of course, if you can only get one of those to happen,
“knowing more” comes first.)
What should you cover?
A few self-contradictory tips. (For every Math 10 guideline, there is an equal and opposite reguideline!)
 Cover topics that you find interesting. If you think something is boring, there is little
hope that your students will find it interesting.
 Avoid subjects that require a lot of algebra, multiple layers of definitions, or are
otherwise technical. If you cover something that has some technical difficulty to it (e.g.,
cardinality, probability, counting), be prepared to spend a lot of time on it.
 Be practical: Math 10 students tend to prefer topics that seem relevant to their daily lives.
For example, basic practical probability works well with almost every approach,
especially when combined with gambling. (As of 2009, poker is very popular with
students.)
 Be aware of what practical means to students: Sometimes they don’t immediately see the
relevance of certain “practical” topics, like compound interest, and on the other hand, I
have had students list topology and infinity as real-life subjects. It may be that some
students don’t have the life experience needed to relate to things like loans.
 Don’t worry too much about covering too little material. There is no Math 11 that you’re
preparing them for. If you’re doing something interesting, but they don’t get it, just take
a few more classes to cover it and cut some stuff at the end.
 Avoid too much repetition of high school. In my experience, if students encounter
something they saw in high school, like percentages, they can start to tune out/veg out,
even if they didn’t understand it the first time.
Sample approaches, textbooks, and coverage guidelines
Approaches to Math 10 are generally some linear combination of super-practical topics (e.g.,
compound interest), “cocktail party” versions of college math topics (e.g., how can there be
different kinds of infinity?), and cultural topics (e.g., the Babylonian number system). Some
sample approaches:
Real-world mathematics
 Basic idea: Teach students math they will use in real life, like basic financial math (a.k.a.
consumer math) and statistics.
 Textbook: Using and Understanding Mathematics (4th ed.), Bennett and Briggs
 Typical material covered (Ng, Fall 2008):
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Sec 2A
Sec 2B
Sec 3A
Sec 3B
Sec 3C
Sec 3D
Sec 3E
Sec 4B
Sec 4C
Sec 4D
Sec 5C
Sec 6A
Sec 6B
Sec 6C
Sec 7A
Sec 7B
Sec 7E
Sec 10A
Sec 10B
The problem-solving power of units
Standardized units: More problem-solving power
Uses and abuses of percentages
Putting numbers in perspective
Dealing with uncertainty
Index numbers: The CPI and beyond
How numbers deceive: Polygraphs, mammograms, and more
The power of compounding
Savings plans and investments
Loan payments, credit cards, and mortgages
Statistical tables and graphs
Characterizing data
Measures of variation
The normal distribution
Fundamentals of probability
Combining probabilities
Counting and probability
Fundamentals of geometry
Problem solving with geometry
Mathematics as a liberal art
 Basic idea: Teach serious topics in college math (countable and uncountable sets,
numbers mod n, topology, probability) in a nontechnical manner, emphasizing essaystyle explanations and creative thinking instead of computations. Class should be as
algebra and computation-free as possible.
 Greensheet description: “What is mathematics? If this course succeeds, you'll have a
different answer to that question at the end of this course than you had at the beginning.
Along the way, you'll encounter some of the great ideas of mathematics, unlock your
mathematical creativity and imagination, and learn deep principles that will help you not
only in math, but also in many other aspects of life.” (I know, shameless, but I meant
every word.)
 Textbook: The Heart of Mathematics: An invitation to effective thinking, Burger and
Starbird
 Typical material covered (Hsu, Spring 2004): Puzzles/introduction (Ch. 1), recreational
number theory (2.1-2.4, 2.6-2.7), countable and uncountable numbers (3.1-3.3), topics
from non-high school geometry (4.1, 4.3, 4.5, 4.7), probability and statistics (7.1-7.3,
7.5), topology (5.1-5.3). One could also substitute voting (8.4), fair division (8.5), etc.,
for some of these topics. Take about twice as much time on everything as you’d think,
especially countability.
 Problems on HW and exams are either very short computations, or more commonly, brief
(one-paragraph) explanations. (The latter are dangerously close to proofs, but avoid the
p-word like the plague.)
 Personal note: In my opinion, this approach is the most fun way to teach Math 10. It is
also, however, the most draining teaching experience I have ever had, and requires loads
of preparation. If you’re interested in trying this out, please feel free to contact me at
hsu@math.sjsu.edu.
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Mathematics and culture
 Basic idea: Combining both of the above approaches (real-life math and math as a liberal
art), plus some Math 8 (Algebra) and Math 19 (Precalculus).
 Typical textbooks: The Mathematical Palette (3rd ed.), Staszkow and Bradshaw; Thinking
Mathematically (4th ed.), Blitzer. Both books cover roughly the same material.
 Typical material covered (Maddux, Fall 2008, Blitzer):
o Review of the number system (4.1, 4.2, 4.4, 5.1-5.6)
o Solving equations and inequalities (6.1-6.3, 6.5-6.6)
o Factoring
o Graphs, functions, linear systems (7.1-7.5)
o Consumer and financial mathematics (8.1-8.4)
o Geometry, triangle trig (10.6, Law of sines/cosines)
o Counting and probability (11.1-11.5, 11.8)
o Statistics (12.1-12.4)
 A slightly more liberal arts-inclined list of topics (Staszkow and Bradshaw):
o Number systems through history (Ch. 1)
o Logic and truth tables (Ch. 2)
o Sets and counting (Ch. 3)
o Probability (Ch. 4)
o Statistics (Ch. 5)
o Geometry and Art (Ch. 7)
o Modern mathematics: Voting, apportionment, linear programming (Ch. 10)
Note that this second topic list is not taken from an actual syllabus, but is still about the
right amount of material for a semester.
Tim Hsu
Mathematics Department, SJSU
March 23, 2009
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