Basic Course Information
Schedule
Class Schedule: Wednesdays, 7:25
10:05 p.m., HH C-1
Instructor: B. Gold Office Location:
HH C-6 Office Telephone: 571-4451 E-mail Address:
bgold@monmouth.edu
Office Hours: Monday 1 2 p.m., Tuesday 5 6 p.m., Wednesday
noon 1 p.m., Thursday 4 5 p.m. or by appointment or chance.
Catalog Description: This course gives the middle-school teacher
a deeper understanding of the number systems (integers, rational numbers,
real numbers, complex numbers). Topics include number bases, divisibility,
factorization, Fundamental Theorem of Arithmetic, equivalence relations,
congruence, Chinese Remainder Theorem, decimal representation, axioms for
number systems, geometric representation of numbers.
Relationship to professional program: This course is an elective
course in the MAT-advanced program, recommended for all elementary school
teachers and middle- and high-school mathematics teachers.
Course Objectives: This course will
Introduce students to the properties of the assorted number systems
used in modern mathematics
Introduce concepts of number theory both to provide an introduction
to this college-level mathematical topic and give a deeper understanding
of elementary mathematics
Give examples of how to use these concepts to develop classroom materials
to enhance student learning
Introduce students to mathematical software available at the school
level for investigating number concepts
Involve students in problem solving activities via exploration and
experimentation to allow students to construct (and reconstruct) mathematics
understanding and knowledge
Encourage visual reasoning as well as symbolic deductive modes of
thought (by incorporating models, concrete materials, diagrams and sketches)
Introduce multiple strategies of approaching problems by discussing
and listening to how others think about a concept, problem, or idea
Involve students in small group work and cooperative learning
Help students become aware of their own mathematical thought processes
(and feelings about mathematics) and those of others
Introduce students to multiple methods of assessment in mathematics
Weekly written reports on mathematical investigations and problem
summaries
Written communication between instructors and individual students
Learner Outcomes: Students will develop the ability to:
Approach problems from multiple perspectives
Use mathematical language to correctly state mathematical definitions
and theorems
Begin developing mathematical proofs
Explain the relationship between the natural numbers, integers, rational
numbers, real numbers, and complex numbers
Demonstrate the properties of these number systems
Relate those properties to mathematics taught in elementary and middle
school
Methods of instruction: Each class will include answering questions from the previous class, lecture-discussion of the new mathematical topic of the day, one or more class activities to familiarize students with the material, a discussion of its connection to school mathematics, and some activities appropriate for use, at some grade level, with K-12 students. Some of the activities will involve use of appropriate computer software, both college level and K-12 level.
Required Texts: Frederick Stevenson, Exploring the Real Numbers, Prentice Hall, 2000; we will also be a field-test site for materials currently under development by Ira Papick at the University of Missouri under a grant from the National Science Foundation
Course Requirements:
You should read the section(s) for the week prior to class.
In-class discussion and problem-solving:
Roughly 30% of each class will be spent on class discussion of advanced
concepts and their relationship to the school classroom, and 40% on group
problem-solving activities. Because in-class activities are such
an important part of the course, if you need to miss a class, please contact
me (preferably prior to the class, but in any case prior to the next class)
for ways to make up this work.
Weekly homework assignments:
These assignments will reinforce the advanced mathematical concepts
learned in the class. Homework is due the week following discussion
of that section in class.
Reflective mathematics journal:
Weekly entries in these journals are to help students become aware
of their own mathematical thought processes (and feelings about mathematics).
I expect at least one entry per week, and will collect the journals one
week prior to midsemester and one week prior to the end of the semester.
You should reflect on what you found difficult and why, what you feel stretched
you, anything which seemed a waste of time and why, and anything else which
you believe will help you or me understand your mathematical growth during
the course.
In class examinations (2), Final Examination:
Examinations will cover correct use of mathematical terminology (especially
correct statement of definitions) and mathematical concepts studied.
Computer laboratory projects:
There will be 2 computer laboratory projects exploring computer software
which enhances students understanding of number concepts; these will be
done in class, with reflection on the uses, value, assessment mechanisms
and limitations of each after class
Curriculum project: 10%
Students will work in pairs beginning at mid-semester to develop week-long
curriculum projects which they will use with their classes and revise.
Methods of Evaluation and Grading Policy: 20% in-class work, 10% each in-class examinations (2), 20% final examination, 15% homework assignments, 5% journal, 10% computer projects, 10% curriculum projects.
Attendance Requirement: Attendance is very important, as
the ideas are not developed sufficiently fully in the text. Students missing
a class will be expected to consult with the professor about how to make
up the material.
Last date to Withdraw with automatic assignment of W grade:
Monday, March 31
Statement on Academic Honesty: You are welcome to consult
others, whether students in the class or tutors in the Mathematics Learning
Center. However, whenever you have had assistance with a problem,
you are to state that at the beginning of the solution to the problem.
Unless it becomes excessive, there will be no reduction in credit for getting
such assistance. This policy applies to both individual and group
work. (Of course groups only have to acknowledge help from outside the
group.)
Examination Rules: No student is permitted to have at his or
her desk any books or papers that are not given out or expressly permitted
by the instructor. Possession of such material will be regarded as
evidence of intent to use the information dishonestly. No communication
between students during the examination is permitted. If there are
questions, or if there is a need for additional material, the instructor
should be asked. If there is a need for calculations or notes, they may
be written on the pages of the exam.
The following pledge must be signed and submitted with the examination:
I, ____________________________, certify that I have read the above
rules for examinations, and that I have abided by them. By signing, I affirm
that I have neither given nor received aid during this examination, and
I understand that violation of this affirmation may result in suspension
or expulsion from Monmouth University.
Statement on Special Accomodations: Students with disabilities
who need special accommodations for this class are encouraged to meet with
me or the appropriate disability service provider on campus as soon as
possible. In order to receive accommodations, students must be registered
with the appropriate disability service provider on campus as set forth
in the student handbook and must follow the University procedure for self-disclosure,
which is stated in the University Guide to Services and Accomodations for
Students with Disabilities. Students will not be afforded any special
accommodations for academic work completed prior to the disclosure of the
disability, nor will they be afforded any special accommodations prior
to the completion of the documentation process with the appropriate disability
office.
Outline of Course Content and Schedule:
Weeks 1-3: The natural numbers: number bases, divisibility,
division algorithm, fundamental theorem of arithmetic (chapter 1)
Homework:
Week 1 (due week 2): Section 1.1 # 1abc, 3ace, , .
Week 2: Section 1.1 Induction; homework: 4ab, 5, 6, 8, 10, 14a, 15a
Week 3: Section 1.1 greatest common divisor, least common multiple;
homework: 17, 18
Also, prove or give a counter-example:
(a) If n = abc + 1, then gcd(n,a) = gcd(n,b) = gcd(n,c) = 1
(b) If d|a and d|b, then d2|ab.
(c) If a|c and b|c, then ab|c
Week 4: 1.2 # 1, 2bc, 3, 5, 6ac, 7bc, 8, 10ac, 11ab, 12b, 13
Also, prove
(a) If n is composite, then 2n 1 is also composite.
(b) If n is a square, each exponent in its prime-power decomposition
is even.
Week 5: The integers. Skim section 1.3 and 2.1. We'll do Computer lab 1. Homework is from attached sheet, problems 3, 4, 5, 7, 9, and 2.1 # 6, 12ab, 13ac, 14, 16
Week 6: Equivalence relations, congruence modulo n (supplement); also exam # 1, on material through week 5. Homework: Appendix D (supplement) # 1, 2, 4, 9
Week 7: 2.2: Congruence modulo n, Chinese Remainder Theorem. Problems 1, 5acd, 6acef, 7ac, 9
Week 8: 2.2: The integers: Fermats little theorem
Problems 12, 14, 15b, 16a, 17; 2.2 # 4
Also: 1. a. What is the remainder when 314162 is divided
by 163?
b. What is the remainder when 314162 is divided by 7?
c. What is the remainder when 314164 is divided by 165?
(CAREFUL: 165 is NOT prime!)
2. Suppose that
p
is an odd prime. Prove that 1p-1 + 2p-1
+ ... + (p-1)p-1 = -1 (mod p).
Week 9: 3.1: The rational numbers: building them from the integers, decimal representation. Homework: 1ab, 2bc, 3ab, 4bc, 6a
Week 10: 2.3 (pages ..) and 3.2 (pages ..) The rational numbers: representation in the plane. Handout, Moving Beyond the Integers. Homework: 2.3 # 29abc; 3.2 # 19, 20ab
Weeks 11-13: The real numbers: algebraic extensions of the rationals, decimals, Dedekind cuts, limits of sequences, axioms for the real numbers, existence of transcendental numbers. Computer lab 2. Semester project due week 13. Second exam week 12.
Week 14: The complex numbers: building them from the real
numbers, the complex plane, polar coordinate representation