COLLEGE OF EDUCATION
DEPARTMENTAL COURSE SYLLABUS
The College of Education CAREs
The College of Education is dedicated to the ideals of Collaboration, Academic Excellence, Research, and
Ethics/Diversity. These are key tenets in the Conceptual Framework of the College of Education. Competence
in these ideals will provide candidates in educator preparation programs with skills, knowledge, and
dispositions to be successful in the schools of today and tomorrow. For more information on the Conceptual
Framework, visit: www.coedu.usf.edu/main/qualityassurance/ncate_visit_info_materials.html
1.
Course Prefix and Number: MAE 6335
2.
Course Title
Number Theory for Middle Grades Teachers
3.
Regular Instructor(s)
Dr. Richard A. Austin
Dr. Helen Gerretson
Dr. Gladis Kersaint
Dr. Denisse R. Thompson
adjuncts credentialed by program faculty (Several faculty at Manatee Community
College have completed the Ph.D. in mathematics education at USF and will likely help
with the teaching of this course.)
4.
Course Prerequisites (if any)
Admission into the MAT in Middle Grades Mathematics or CI
5.
Course Description
This course examines in depth number theory concepts appropriate for middle grades
mathematics teachers, including historical connections. Topics studied include integers,
factors, primes, composites, congruence, figurate numbers, and rational numbers.
Teachers experience instructional approaches appropriate for use in middle grades
classrooms. This course is required in the MAT in Middle Grades Mathematics.
Prerequisite: Admission to the MAT program in middle grades mathematics or CI.
6.
Course Objectives
Upon completion of this course, students will demonstrate the following:
1.
Knowledge of key concepts in number theory (e.g., primes, composites, factors,
multiples, greatest common factor, least common multiple, congruence);
2.
Knowledge of key concepts and properties related to rational numbers (e.g.,
terminating and repeating decimals);
3.
The ability to solve problems by using number theory;
4.
The ability to complete proofs related to basic number theory concepts;
Number Theory
5.
Knowledge of historical developments related to number and mathematical
symbolism;
An awareness of the relationship between number theory and the teaching of
mathematics in the middle grades.
6.
7.
Content Outline
Week 1. Classifying numbers.
Even and odd numbers, including related proofs.
Figurate numbers, particularly square and triangular numbers
Week 2. Primes and composites, including sieve of Eratosthenes
Twin primes
Proofs about the infinitude of primes
Historical connections
Week 3. Unique factorization
Divisibility tests
Greatest common factor, least common multiple
Week 4. Euclidean algorithm and applications
Numeration systems, including the binary system
Sorting algorithms
Week 5. Games involving number theory
Applications in coding theory
Week 6. Number of factors of a given number
Perfect numbers and amicable numbers
Week 7. Modular arithmetic, including basic proofs
Applications to calendars
Week 8. MIDTERM
Week 9. Pythagorean triples
Generating primitive Pythagorean triples
Historical connections, Bablylonian tablets
Week 10. Squares on a geoboard
Week 11. Fibonnaci numbers
Week 12. Rational numbers and properties
Week 13. Terminating and repeating decimals
Week 14. Fermat primes
Constructability of regular polygons
Week 15. Connecting number theory concepts to the middle grades curriculum
8.
Evaluation of Student Outcomes

Exams or tests will evaluate students' content knowledge on the major content topics
in the course. Students will have to pass the final, comprehensive exam in order to
pass the course. (CF #2)

Problem sets will evaluate students' ability to explore open and extended problems.
(CF #2)
Number Theory
9.

Historical paper will give students an opportunity to explore the historical
background of a topic from number theory. (CF #2 and #4)

External project will have students engage in an number theory content project of the
instructor's design or of their own approved design. (CF #2, #4, and #6)

A journal will provide on-going evaluation of students' facility with the content of the
course and emphasize the importance of writing throughout the curriculum. (CF #4)
Grading Criteria
Exams or Tests or Quizzes
Problem Sets and Journal
Historical Paper
External Project
50-55% of Grade
15-20% of Grade
10-15% of Grade
15-20% of Grade
The university's plus/minus system of grading will be used.
USF Policy on Religious Observances
"No student shall be compelled to attend class or sit for an examination at a day or time
prohibited by his or her religious belief. In accordance with the University policy on
observance of religious holy days, students are expected to notify their instructors if they
intend to be absent for a class or announced examination prior to the scheduled meeting."
10.
Required Textbooks
Sample text: The following units from the Connected Mathematics Project are possible
texts.
Prime Time
Bits and Pieces I
Bits and Pieces II
Comparing and Scaling
Accentuate the Negative
(Glenda Lappan, James T. Fey, William M. Fitzgerald, Susan N. Friel, and Elizabeth
Difanis Phillips, Menlo Park, CA: Dale Seymour Publications, 1998.)
A packet of readings related to activities on the number theory concepts in the course and
to historical topics from the course will also be used. Sample readings might include the
following:
Lamon, Susan J. "Presenting and Representing: From Fractions to Rational Numbers." In
Albert A. Cuoco and Frances R. Curcio, The Role of Representation in School
Mathematics, pp. 146-165. Reston, VA: National Council of Teachers of Mathematics,
2001.
Number Theory
Sgroi, Laura. "An Exploration of the Russian Peasant Method of Multiplication." In
Lorna J. Morrow and Margaret J. Kenney, The Teaching and Learning of Algorithms in
School Mathematics, pp. 81-85. Reston, VA: National Council of Teachers of
Mathematics, 1998.
Note: The actual text and readings will be selected at the time the course is offered in
order to permit the most current materials to be used. The above samples reflect the
nature of the materials intended for use to help teachers address the content of this course.
11(a) ADA Statement: Students with disabilities are responsible for registering with the
Office of Student Disabilities Services in order to receive special accommodations and
services. Please notify the instructor during the first week of classes if a reasonable
accommodation for a disability is needed for this course. A letter from the USF
Disability Services Office must accompany this request.
11(b). USF Policy on Religious Observances:
Students who anticipate the necessity of being absent from class due to the observation of
a major religious observance must provide notice of the date(s) to the instructor, in
writing, by the second class meeting.
Number Theory
COLLEGE OF EDUCATION
DEPARTMENTAL COURSE SYLLABUS
Graduate Level Course
ATTACHMENT I
Please respond to each of the following questions and complete the attached Matrix:
1.
Rationale for Setting Goals and Objectives: What sources of information (e.g.,
research, best practices) support the formulation and selection of course goals and
objectives?
The aim of the course is to provide middle grades mathematics teachers with a solid
background related to the number content they would be expected to teach in the middle
grades. The course will not only focus on providing a solid foundation in number theory
but will approach that content from pedagogical perspectives that are appropriate for use
in the middle grades classroom. In this way, teachers experience learning mathematics
through the types of approaches they are expected to use when they teach.
2.
What aspects of the COE conceptual framework is/are specifically addressed in this
course?


3.
USF prepares professionals who know the content they teach. USF education
candidates demonstrate an understanding of his/her subject field, its linkage to other
disciplines, and applications to real world, integrated settings.
USF professionals are reflective and analytical problem-solvers. USF education
candidates engage in continuous professional improvement for self and school
through a commitment to life-long learning.
List the specific competencies addressed from the relevant national guidelines.
National Council of Teachers of Mathematics
1.5.1 "apply concepts of number, number theory, and number systems;"
1.5.2 "apply numerical computation and estimation techniques and extend them
to algebraic expressions;"
1.5.11 "use mathematical modeling to solve real-world problems;"
1.6
"Programs prepare prospective teachers who have a knowledge of
historical development in mathematics that includes the contributions of
underrepresented groups and diverse cultures."
4.
Are there field-based experiences in this course? If so, please briefly indicate nature
and duration.
No.
Number Theory
5.
a.
Is technology used in this course?
Students will be expected to have access to a graphing calculator throughout the duration
of the course. Graphing calculators will be used to explore number theory concepts.
Although scientific calculators could be used, graphing calculators have built-in functions
that are particularly helpful to this course and students will need a graphing calculator for
other courses.
b.
Are students required to access and demonstrate use of technology in
instruction or record keeping in this course?
No.
6.
How are issues of diversity addressed in this course? Indicate which aspect of the
course (e.g., instructional strategies and/or experiences) provides the candidate the
opportunity to acquire and/or apply knowledge, skills and/or dispositions necessary
to help all students learn. ("All students" includes students with various learning
styles, students with exceptionalities and different ethnic, racial, gender, language,
religious, socioeconomic, and regional/geographic origins and achievement levels.)
A variety of instructional methods are used in the course to accommodate a learning
environment with diverse kinds of students.
Students will explore historical connections to number theory, including the contributions
made by diverse peoples and cultures. Students will be expected to complete a historical
paper related to some number theory topic.
7.
(For initial certification programs)
a.
List the specific competences addressed from the Florida Adopted Subject
Matter Content Standards or the Florida Adopted Subject Area Competencies.
Knowledge of mathematics as problem solving (1-4)
Knowledge of mathematics as communication (1)
Knowledge of mathematics as reasoning (1, 3)
Knowledge of numbers and number relationships (1, 3, 4)
Knowledge of number systems and number theory (2, 3)
b.
Describe any component of the course designed to prepare teacher
candidates to help PK-12 students achieve the Sunshine State Standards.
The Sunshine State Standards are drawn from the national content recommendations
from the National Council of Teachers of Mathematics. This course will address aspects
from Strand A on Numbers and Operations.
Number Theory
Course Objectives
Matrix
Evidence of
Achievement
Accomplished
Practices
1.0 Knowledge of key concepts in number
theory (e.g., primes, composites,
factors, multiples, greatest common
factor, least common multiple,
congruence)
Exams
Problem sets
#8 Knowledge of
subject matter
2.0 Knowledge of key concepts and
properties related to rational numbers
(e.g., terminating and repeating
decimals).
Exams
Problem sets
#8 Knowledge of
subject matter
3.0 The ability to solve problems by using
number theory.
Exams
Problem sets
Journals
Project
#8 Knowledge of
subject matter
4.0 The ability to complete proofs related
to basic number theory concepts.
Exams
Problem sets
#8 Knowledge of
subject matter
5.0 Knowledge of historical developments
related to number and mathematical
symbolism.
Historical paper
#8 Knowledge of
subject matter
6.0 An awareness of the relationship
between number theory and the
teaching of mathematics in the middle
grades.
Journals
Project
#8 Knowledge of
subject matter
#3 Continuous
improvement
Number Theory