COLLEGE OF EDUCATION DEPARTMENTAL COURSE SYLLABUS
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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; 6. An awareness of the relationship between number theory and the teaching of mathematics in the middle grades.
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 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)
9. Grading Criteria
Exams or Tests or Quizzes 50-55% of Grade Problem Sets and Journal 15-20% of Grade Historical Paper 10-15% of Grade External Project 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?
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.
3. 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 Matrix Course Objectives Evidence of Accomplished Achievement Practices 1.0 Knowledge of key concepts in number Exams #8 Knowledge of theory (e.g., primes, composites, Problem sets subject matter factors, multiples, greatest common factor, least common multiple, congruence)
2.0 Knowledge of key concepts and Exams #8 Knowledge of properties related to rational numbers Problem sets subject matter (e.g., terminating and repeating decimals).
3.0 The ability to solve problems by using Exams #8 Knowledge of number theory. Problem sets subject matter Journals Project
4.0 The ability to complete proofs related Exams #8 Knowledge of to basic number theory concepts. Problem sets subject matter
5.0 Knowledge of historical developments Historical paper #8 Knowledge of related to number and mathematical subject matter symbolism.
6.0 An awareness of the relationship Journals #8 Knowledge of between number theory and the Project subject matter teaching of mathematics in the middle #3 Continuous grades. improvement
Number Theory