Math 3495

Spring 2005


I.                   Course:                     Math 3495   Advanced Perspective on School Mathematics I

                                                Department of Mathematics, College of Science and Mathematics

                                                Kennesaw State University


II.        Instructor:                  Dr. Mary Garner

                                                Office:  Science 522

                                                Phone:  770-423-6664


                                                Web Page:


                                                Office Hours:  2 PM – 3 PM Tues and Thurs

                                                                        10 AM – Noon Wed

                                                                        Other times by appointment

               Students are required to use WEBCT in this course,

            and through WEBCT, students can ask questions

            of the instructor or the class as a whole.


III.       Class Sessions:          Mondays and Wednesdays 5 PM – 6:15 PM. CL 1007           


IV.       Texts (required):        Mathematics for High School Teachers:  An Advanced Perspective  

By Zalman L. Usiskin, Anthony L. Peressini, Elena Marchisotto, and Dick Stanley.  Published by Prentice Hall.  2003.  Selections from this text will be used that specifically target the topics described below and where lower level high school and middle schools topics overlap.


You must access WEBCT to obtain the syllabus, assignments, announcements, important dates, and any other course materials. You also must access WEBCT to complete course requirements.


Also required is a TI-83 Graphing Calculator.


Geometer’s Sketchpad is recommended.




V.        Catalog Description: 3-0-3  Prerequisite:  Math 1190.

Students’ understanding of school mathematics will be deepened and broadened through the study of key topics in school mathematics, including algebra, linear functions, exponential functions, quadratic functions, number theory, discrete mathematics, and mathematical modeling. This course is designed so that students can revisit key ideas in school mathematics, bringing with them the skills and understandings of college course work in mathematics, deepening and broadening their understanding, and connecting more advanced ideas to the topics they will teach in middle school and high school.


VI.              Purpose/Rationale:


The purpose of this course is to prepare prospective 5-8 teachers and 7-12 teachers to become effective facilitators in the teaching of school mathematics.


According to a recent publication by the Mathematical Association of America (MAA)  “There is much evidence of a vicious circle in which too many future teachers enter college with serious holes in their understanding of school mathematics, have little college instruction focused on the mathematics they will teach, and so enter their classrooms inadequately prepared to teach mathematics.”  Furthermore, “Future teachers should learn mathematics in a coherent fashion that emphasizes the interconnections among theory, procedures, and applications.”  This course is designed so that students can revisit key ideas in school mathematics, bringing with them the skills and understandings of college course work in mathematics, deepening and broadening their understanding, and connecting more advanced ideas to the topics they will teach in middle school and high school.  Students may then leave the program with an integrated understanding of the mathematics learned in college and its relationship to the mathematics they will teach.


            Conceptual Framework Summary: 





The Professional Teacher Education Unit (PTEU) at Kennesaw State University is committed to developing expertise among candidates in initial and advanced programs as teachers and leaders who possess the capability, intent and expertise to facilitate high levels of learning in all of their students through effective, research-based practices in classroom instruction, and who enhance the structures that support all learning. To that end, the PTEU fosters the development of candidates as they progress through stages of growth from novice to proficient to expert and leader. Within the PTEU conceptual framework, expertise is viewed as a process of continued development, not an end-state. To be effective, teachers and educational leaders must embrace the notion that teaching and learning are entwined and that only through the implementation of validated practices can all students construct meaning and reach high levels of learning. In that way, candidates are facilitators of the teaching and learning process. Finally, the PTEU recognizes, values and demonstrates collaborative practices across the college and university and extends collaboration to the community-at-large. Through this collaboration with professionals in the university, the public and private schools, parents and other professional partners, the PTEU meets the ultimate goal of assisting Georgia schools in bringing all students to high levels of learning.


Knowledge Base: 

Teacher development is generally recognized as a continuum that includes four phases: preservice, induction, in-service, renewal (Odell, Huling, and Sweeny, 2000). Just as Sternberg (1996) believes that the concept of expertise is central to analyzing the teaching-learning process, the teacher education faculty at KSU believes that the concept of expertise is central to preparing effective classroom teachers and teacher leaders. Researchers describe how during the continuum phases teachers progress from being Novices learning to survive in classrooms toward becoming Experts who have achieved elegance in their teaching. We, like Sternberg (1998), believe that expertise is not an end-state but a process of continued development. 


Math 3495 develops a strong conceptual understanding of the algebraic concepts necessary for 5-8 and 7-12 teachers of mathematics, it also integrates various technologies that are used in the classrooms of today.  The instruction in MATH 3495 also models the “teacher as facilitator” as various instructional strategies are employed, such as cooperative groups and joint projects.  Students are actively engaged in all phases of their learning.


Use of Technology: 

The use of calculators and computers is an encouraged and accepted practice to enable students to discover mathematical relationships and approach real world applications.  Familiarizing the pre-service teacher with a variety of technological tools is an integral part of the math sequence for teachers.  Students in Math 3495 routinely use computer software and graphing calculators.


Diversity Statement

A variety of materials and instructional strategies will be employed to meet the needs of the different learning styles of diverse learners in class.  Candidates will gain knowledge as well as an understanding of differentiated strategies and curricula for providing effective instruction and assessment within multicultural classrooms.  One element of course work is raising candidate awareness of critical multicultural issues.  A second element is to cause candidates to explore how multiple attributes of multicultural populations influence decisions in employing specific methods and materials for every student.  Among these attributes are age, disability, ethnicity, family structure, gender, geographic region, giftedness, language, race, religion, sexual orientation, and socioeconomic status.  An emphasis on cognitive style differences provides a background for the consideration of cultural context. 

Kennesaw State University provides program accessibility and accommodations for persons defined as disabled under Section 504 of the Rehabilitation Act of 1973 or the Americans with Disabilities Act of 1990.  A number of services are available to support students with disabilities within their academic program.  In order to make arrangements for special services, students must visit the Office of Disabled Student Support Services (ext. 6443) and develop an individual assistance plan.  In some cases, certification of disability is required.


Please be aware there are other support/mentor groups on the campus of Kennesaw State University that address each of the multicultural variables outlined above. 

VII.           Goals and Objectives:


The KSU teacher preparation faculty is strongly committed to the concept of teacher preparation as a developmental and collaborative process. Research for the past 25 years has described this process in increasingly complex terms. Universities and schools must work together to successfully prepare teachers who are capable of developing successful learners in today’s schools and who choose to continue their professional development. 


Objectives for Math 3495:


The student will be able to:

·        Analyze problems from the middle school curriculum, identifying the skills required, exploring and analyzing alternate methods of solutions, connecting the problem to related problems and techniques, investigating the history of the problem and associated skills, and extending the problem to apply more complex modes of analysis. 

·        Analyze mathematical concepts that are used in middle school mathematics, exploring definitions, applications, and history of those concepts.

·        Understand rational and irrational numbers and their decimal representation.

·        Understand the ways that basic ideas of number theory and algebraic structures underlie rules for operations on basic symbolic expressions, equations and inequalities.

·        Use algebraic reasoning effectively for problem solving.

·        Understand the relationship between linear functions and arithmetic sequences, and between exponential functions and geometric sequences.

·        Understand and use linear, exponential, and quadratic functions, and understand the patterns they represent.

·        Use calculator and computer technology effectively and appropriately to study individual functions and classes of related functions.

·        Use basic principles of combinatorics in application to graph theory and finite functions on sets. 

·        Understand the basic principles of graph theory and its usefulness in applied problems.

·        Understand and use principles of recursion and iteration.

·        Read, discuss, and write about mathematics.


VIII.        Course Requirements/Assignments:


Class participation and assignments. The course is divided into 7 blocks (one to two weeks in length) that are spent analyzing a specific problem from the middle grades/high school curriculum.  In analyzing the problem, we’ll draw from our textbook, as well as Mathematics Teaching in the Middle School and Mathematics Teacher (both NCTM publications that are available very conveniently on-line to members of NCTM), and we’ll explore associated problems.


            Each of these blocks of time will be spent on two activities:


Problem analysis


Exploration of Connections


A “Problem Analysis” will include the following activities:

·        Solutions and Clarification: Discuss what assumptions are unstated or ambiguities must be resolved before solution of the problem. Describe different ways of solving or representing the problem, including ways involving diagrams or pictures, technology, algebra, guess and check, arithmetic approach. Always explain your reasoning and reflect on differences or similarities among approaches. Describe wrong approaches and why they are wrong.

·        Generalizations: Generalize the problem and its solution in several ways. Always explain your reasoning and reflect on differences or similarities between generalizations.

·        Functions/Variation: Formalize the mathematical notation, and analyze the problem and relationships between quantities in the problem using functions and variables.

·        Extension: Extend the problem and create a new problem so that a deeper understanding or more mathematics is required to solve the problem. Solve the problem. Describe how the problem is an extension of the original problem and discuss other possible extensions.

·        Summary and reflections: Discuss the significance of the problem and concepts required to solve the problem.


An “Exploration of Connections” will include the following:

·        Concept Analysis: Discuss what mathematical concepts must be understood to solve the problem. Choose a central concept and analyze that concept by describing and discussing definitions of the concept and applications of the concept in the curriculum. Also describe how that concept is extended or generalized in the curriculum.

·        Teaching: Articles from Mathematics Teacher and Mathematics Teaching in the Middle School are consulted to learn about teachers’ and researchers’ perspective on the concept or concepts.

·        Connections: Connect the problem to other problems. State the problems, their solutions, and specifically how the problems are the same, and how they are different.

·        History: Describe the mathematical history of the central concept of the problem.


Assignments associated with both of these activities will be due on Thursdays and will be announced at least a week in advance. All assignments will be graded for correctness and completeness; in other words, students won’t receive credit simply for submitting “something.” If you can’t do the assignment or feel unsure about its correctness, compare notes with colleagues and seek help. Students must use the opportunities provided in class on Tuesdays and through WEBCT to confer with colleagues about their solutions to the assignments and ask questions about the assignment. Late assignments will not be accepted. Please seek help with the assignments outside of class or through WEBCT.


Two tests and a final examinations. The final examination will be comprehensive, and the student will be allowed to consult notes only during the test (not text). There will be no reviews for any of the examinations. This is not a course in which students can simply review procedures before the test and then regurgitate those procedures on the examination. Students can best prepare for the exams by working on the assigned problems on a daily basis and participating in whole-class, small group, and WEBCT discussions. Dates for the exams will be announced in class and posted on WEBCT. Each exam will count 15 percent of the grade. Notes and text cannot be used except during the final examination. If a student missed a test, the final exam grade will be substituted for that test grade.


Problem analysis and exploration of connections.  Each student will choose a problem from the middle grades or high school mathematics curriculum and write both a problem analysis and an exploration of connections. The student will be responsible for finding the problem and getting approval from the instructor. Students are required to post their problems on WEBCT and ask for approaches/solutions from members of the class. Many students have access to middle school or high school students and would benefit from asking those students to solve the problem. See the calendar on WEBCT for due dates.


IX.   Evaluation and Grading:


There will be five grades: 

·        Participation and weekly assignments:          25%

·        Problem analysis

                        and exploration of connections: 40%

·        Test #1                                                       10%

·        Test #2                                                       10%

·        Final examination:                                        15%


Maximum participation points will be assigned to the student who attends class regularly, seeks instructor during office hours, and consistently participates in class, small-group and WEBCT discussions in a constructive manner.


Grading of assignments, quizzes, and exams will be based on both the correctness of the mathematical content and the quality of the associated write-up which may include proofs, explanations, justifications, and discussion of connections, history, generalizations, and extensions.


X.       Academic Honesty Statement:


Every KSU student is responsible for upholding the provisions of the Student Code of Conduct, as published in the Undergraduate and Graduate Catalogs.  Section II of the Student Code of Conduct addresses the University’s policy on academic honesty, including provisions regarding plagiarism and cheating, unauthorized access to University materials, misrepresentation/falsification of University records or academic work, malicious removal, retention, or destruction of library materials, malicious/intentional misuse of computer facilities and/or services, and misuse of student identification cards.  Incidents of alleged academic misconduct will be handled through the established procedures of the University Judiciary Program, which includes either an “informal” resolution by a faculty member, resulting in a grade adjustment, or a formal hearing procedure, which may subject a student to the Code of Conduct’s minimum one semester suspension requirement.


XI.              Class Attendance Policy:


Regular attendance is assumed and will be monitored.  Although it is impossible to reconstruct classroom lectures, discussions, and activities, in the event of unavoidable absence, the student will assume full responsibility for any material and/or announcement missed.  Daily assignments and quizzes that are missed cannot be made up.  The instructor strongly urges each student to form a study team with one or two other students.


Withdrawal Policy:

Students choosing to withdraw from this course without academic penalty must do so by March 4, 2005. Withdrawal forms can be obtained from the Office of the Registrar.  The completed form must be approved by the Registrar.  A student ceasing to attend class and completing course requirements will be assigned a failing grade if official withdrawal has not been completed.  There is a new University policy on the total number of withdrawals a student may have (see 2004-2005 Undergraduate Catalogue p. 42).


XII.           Course Outline:


Week 1 & 2: Problem analysis involving linear equation in one variable. Problem 1.

Chapter 4. Concepts include equations, solution sets, equivalent equations, links between graphs and equations, inequalities, systems of equations, calculator use, word problems involving linear equations.


Week 3 & 4: Problem analysis involving quadratic equation in one variable. Problem 2.

Chapter 4. Concepts include same as in Problem 1 but including the quadratic formula, zero product property, complex solutions, word problems involving quadratic equations.


Test  #1


Week 5:        Reading former problem analyses. Choose problem and start problem analysis.


Week 6 & 7: Problem analysis involving geometric patterns. Problem 3. Chapter 3. Concepts

include functions, linear functions, quadratic functions, exponential functions, recursive and closed forms of functions, discrete functions, sequences, method of finite differences, recursion, mathematical induction.


            Week 8 & 9:  Problem analysis involving data-fitting. Chapter 3. Concepts include functions,

regression, residuals, correlation, coefficient of determination.


Test #2


Week 10 & 11:  Analysis of word problem involving averages. Chapter 1 and Chapter 3.

Concepts include averages, rates, mixture word problems, algebraic and proportional reasoning.


Week 12 & 13: Problem analysis involving linear equation in two variables with only integer

solutions. Chapter 5. Concepts include divisibility, greatest common factor, prime numbers, fundamental theorem of arithmetic, division algorithm, Euclidean algorithm.


Week 14 & 15: Problem analysis involving fractions and decimals. Chapter 2.


            Final examination.


XIII.        References/Bibliography:


Fennema, E. & Romberg, T.A. Eds. (1999). Mathematics Classrooms that Promote

            Understanding. Mahwah, NJ: Lawrence Erlbaum Associates, Publishers.


       Mathematical Association of America & American Mathematical Society (2001).  The

                   Conference Board of the Mathematical Sciences: Mathematical Education of

                   Teachers Part I.  Washington, DC: MAA.


       National Council of Teachers of Mathematics (NCTM), Reston, VA.


                               Making Sense of Fractions, Ratios, and Proportions. 2002 Yearbook.

                                            Edited by Bonnie Litwiller and George Bright

                        Principles and Standards For School Mathematics Navigations Series:

                                     Navigating Through Algebra in Grades 6-8, 2001.                                       Principles and Standards For School Mathematics Navigations Series:

                                     Navigating Through Algebra in Grades 9-12, 2001.

                                    Principles And Standards For School Mathematics, 2000.

                                    The Ideas of Algebra, K-12. 1988 Yearbook. Edited by Coxford and Shulte.

                               Historical Topics For The Mathematics Classroom,

                                           edited by Baumgart, 1969.       


                               Mathematics Teacher

                               Mathematics Teaching in the Middle School



The syllabus is subject to change with notice.