Field Experiences and
Student Teaching:
In Math 460, Secondary Mathematics Methods, the students will participate
in field experiences such as attending the LATM state conference and the SEATM
local conference and reporting on their experiences, observing secondary
mathematics teachers, and evaluating their experiences.
In Education 485/488,
students microteach lessons in mathematics to their peers, then engage in an
extended (4-week) field experience in area middle/high schools during which they
are responsible for the entire class.
Student
Teaching[4]
Course Number
Course Title
Nature of Field Experiences
Education 486
Student Teaching
F0, F3
A full-time Director of Field Experiences in the College of Education
coordinates student teaching at Southeastern Louisiana University. Each student
teacher is placed in a school in 1 of 10 surrounding school systems (9
parish/county systems and 1 independent city system). School sites are selected
so that student teachers have experience working with students from diverse
cultures. Supervising teachers,
recommended by their principles, must have at least 3 years of teaching
experience, a masters degree which includes a course in supervising student
teachers, and exhibit personal and professional qualities of a role model for
preservice teachers.
Student teachers are placed at one location for the entire semester, and
are required to engage in a minimum of 270 hours of combined observation,
participation, and teaching. A minimum of 180 of these 270 hours is to be
teaching under the supervision of the assigned cooperating teacher, with a
substantial portion - a minimum of 3 consecutive weeks - in all-day teaching.
Hours are logged in four categories: (a) conference - meeting with the
cooperating teacher, administrator, or university supervisor to discuss lesson
plans, teaching performance, etc.; (b) observation - watching the cooperating
teacher or another teacher engaged in teaching; (c) participation - assisting
the cooperating teacher in duties other than teaching (e.g., duty, taking roll,
serving as teaching assistant, attending faculty meetings); and (d) teaching -
teaching whole classes from the student teacher’s lesson plans. Students
gradually assume responsibility for the class until they assume total
responsibility for the full day of class activities, then toward the end of the
semester, gradually relinquish teaching responsibilities back to the cooperating
teacher.
Student teachers are supervised and evaluated by their cooperating
teacher, an administrator at the school site, and a university supervisor . All completed evaluations are given to
the Director of Field Experiences. General responsibilities and characteristics
of those involved are listed below.
The Student
Teacher:
·
adheres to all school
policies and functions as an active participant in the school. This may involve
after-school activities, teacher-parent conferences, and other duties of a
teacher.
·
keeps a daily reflective
journal and log of hours.
·
completes a minimum of 8
observations of other teachers in the school or area
schools.
·
prepares daily lesson plans
and submits them to the cooperating teacher approximately 1 week prior to
teaching.
·
videotapes a lesson near
mid-semester and completes a written reflective critique of the lesson. This
written critique and videotape are reviewed by both the university supervisor
and cooperating teacher.
·
completes a self-evaluation
at midterm and the end of the semester.
The Cooperating
Teacher:
·
is the person primarily
involved with orienting the student teacher to the school and monitoring his/her
progress.
·
schedules the student
teacher’s assumption of teaching duties.
·
evaluates the student
teacher’s performance on a daily basis, conferences with the student teacher
regularly, and keeps a record of the conferences and content covered, and a
journal the student teacher’s progress.
·
completes two formal
evaluations of the student teacher (one at midterm, and one at the end of the
semester).
The University
Supervisor:
·
serves as liaison between
the university and the cooperating school.
·
observes each student
teacher a minimum of 6 times over the semester. At each observation, the student teacher
provides the written lesson plan for, and discusses with the university
supervisor the lesson that will be observed.
·
evaluates the student
teacher on 4 of these occasions.
The results of each evaluation is discussed with the student teacher
following observation, and copies of the evaluation are given to the student
teacher and the Director of Field Experiences.
·
monitors/checks student
teachers’ logs, lesson plans, and unit plans.
·
meets regularly over the
semester with a small group of student teachers in seminar
format.
The
School Administrator
·
regularly observes the
student teacher (a minimum of 4 times over the semester).
·
on two occasions, observes a
lesson taught by the student teacher and completes a written evaluation of the
lesson.
Deviation from Program
Standards
None
Location of Program in
Professional Education Unit
and Relationships with other
Unit Programs
Southeastern Louisiana University (SLU) is a comprehensive regional
university located in Hammond, Louisiana (halfway between Baton Rouge and New
Orleans). Due to its location and reputation as a student-centered university,
SLU has grown tremendously over the past two decades, from a total student
population of 7,706 in 1980, to its Fall 1999 enrollment of
15,334.
Academic programs at SLU are organized into 6 units: the Colleges of Arts
and Sciences, Basic Studies, Business and Technology, Education and Human
Development, Nursing and Health Sciences, and Graduate Studies. The Department of Mathematics is in the
College of Arts and Sciences
Administration of the Teacher Education Program is headed by
the Dean of the College of Education and Human Development, assisted by the
Council for Teacher Education and the College of Education and Human Development
Curriculum Committee. The Council for Teacher Education, considered the
institution-wide governing body for the Teacher Education Program, is composed
of representatives from each department in the College of Education and Human
Development as well as other departments offering teacher certification degree
programs. It is responsible for policy, philosophy, objectives, curricula,
student services, and the administration of the Teacher Education Program. The
College of Education and Human Development Curriculum Committee is composed of
the College Director of Performance Assessment, and department heads, faculty,
and students from each department in the college. It is the vehicle through
which curricular issues and problems are discussed, and changes
proposed.
Students pursuing the
Bachelor of Science in Mathematics Education degree must meet specific criteria
set by the Teacher Education Program for all students seeking degrees leading to
teacher certification. Students enter SLU in the College of Basic Studies, and
upon meeting the requirements to exit this college (normally during the
sophomore year), apply for admission into the Teacher Education Program. In
order to successfully matriculate (e.g., enroll in junior- and senior-level
Professional Education coursework, student teach, and complete the program),
they must meet several criteria, which include:
a)
achieving a minimum 2.5 GPA
on a 4-point scale,
b)
achieving a criterion score
on the General Knowledge and Communication Skills portions of the NTE/Praxis
Exam,
c)
achieving a grade of C or
better in all Professional Education courses (EDUC and
EPSY),
d)
achieving a grade of C or
better in all major coursework (KIN or HS),
e)
completing a speech and
language screening.
Faculty with primary responsibilities in
the Secondary Mathematics Education program
Name
Position
Course
Specialization
Tenure Status
Responsibilities
Elizabeth Gray
Associate Professor, Tenured
Math 460
Tena Golding
Associate Professor, Tenured
Math 460
Brian O’Callaghan
Associate Professor, Tenured
Math 311
Number of graduates
There were eight graduates
from the mathematics education program in 1998-99; there
were
11 graduates in the
mathematics education program in 1999-2000.
Narrative on Connections
The Mathematics Department of Southeastern Louisiana University is
committed to providing pre-service with opportunities to show how mathematics is
related to other disciplines and to discuss the connections of topics in diverse
areas of mathematics.
In the calculus sequence, an effort is made to work applied problems from
the social sciences, business and the natural sciences. A typical optimization problem relates
geometry, algebra, calculus techniques and real-life
situations.
In Math 309, the students use Geometer’s Sketchpad software to examine
relationships between quantities.
Students are always amazed at how much algebra is involved here. Easy access to the built-in calculator
to find lengths of segments and to compare ratios shows them the strong presence
of algebra in the geometry program.
Connections is a constant theme in Math 311 – the History of Mathematics
course. Students see the
development of different mathematical ideas from the perspective of other
disciplines. As they solve problems
using only the tools available at different periods of history, their
perspective on links increases.
In the Math 460 class, students are encouraged to sue a style of lesson
development that involves starting with a real life situation. The students collect data from an
experiment, examine the data for a pattern that they recognize, enter the data
into a calculator to examine its graph, determine a relation if it exists, and
write about their discoveries in light of the original
problem.
Students are invited to the Department of Mathematics Colloquia series in
which faculty from Southeastern and other Louisiana universities make
presentations on diverse topics. In
this setting, the students are encouraged to see how different branches of
mathematics are connected.
Narrative on
Reasoning
The Mathematics Department of Southeastern Louisiana University is
committed to having an educational program for mathematics education majors
which recognizes the important role of reasoning and proof in understanding
mathematics.
In the calculus sequence, Math 200, 201, and 312, students are given an
introduction to formal reasoning and proof through the presentation of lessons
with an analysis of “why” certain techniques work. The importance of reasoning is
emphasized in a less formal manner by requiring the development of logical
arguments in working problems posed in class.
In Math 223, the main goals are to teach students the logic of
mathematics and how to write proofs.
Discussion of the variety of methods for proving statements leads to a
better understanding of the importance of proof and of the acceptability of
different methods of thought processes.
In the geometry course, Math 309, students review the study of proofs and
use proofs in verifying geometrical statements. Additionally, students will work both
independently and in groups to give verbal presentations of their
proofs.
Students are required to complete at least two of the following courses:
Math 350 – Applied Differential Equations, Math 360 – Applied Linear Algebra,
and Math 370 – Introduction to Abstract Algebra. In each of these courses, the theory
behind the mathematics is explained by connecting the new information to the old
by the use of proofs. An emphasis
on concepts gives students a chance to explore their own reasoning and discuss
this with their fellow classmates.
In Math 460, the emphasis on problem solving naturally requires an
acquisition of reasoning capacity in the students. Students are required to be able to
explain their reasoning to others in the class, as they will be required to do
so in the school setting.
Additionally, mathematics education majors are encouraged to see and hear
about mathematics reasoning and proof outside of their classrooms through
discussions with faculty about their research interests and through attendance
at mathematics colloquia on the Southeastern campus.
Narrative of Problem
Solving
The Mathematics Department of Southeastern Louisiana University is
committed to providing an environment in all of its courses, which focuses on
the solving of problems as a means of learning mathematical concepts. Collaborative learning activities
encourage the sharing of ideas and strategies with fellow students as well as
the instructor. Through these
settings, students learn that there can be different approaches to the same
problem situation, and, as a result, they can incorporate the varying strategies
when useful to future problems, both within and outside of
mathematics.
In the calculus sequence, Math 200, 201, and 312, applied problems from
the fields of business, life sciences, and social sciences are used to build
problem-solving skills. Students
are given non-routine problems, which require investigative thought and
analysis. Oral presentations or
discussions of perspectives provide opportunities for monitoring and adjusting
strategies in problem solving.
Math 350, Applied Differential Equations, and Math 360, Applied Linear
Algebra, require problem-solving development and implementation in real-life
situations posed by the instructors.
Students are able to work independently and collaboratively on the
improvement of their analytical techniques.
In Math 460, the development of problem-solving techniques by the
pre-service teachers is intertwined with learning about lesson development. The student learns about posing problems
to elementary and secondary students and leading investigations as a facilitator
to their learning.
Students are also encouraged to speak with their professors regarding the
research questions that are being investigated by the departmental faculty. This gives them insight into how problem
solving is carried on by mathematics.
Narrative on
Communication
The Mathematics Department of Southeastern Louisiana University if
committed to providing diverse opportunities for pre-service mathematics
education majors to communicate with each other, with their professors and with
their future students. In all
courses beginning with Math 200 they are encouraged to solve problems in small
groups and to present solutions to the whole class.
In the discrete mathematics class, Math 223, students begin to structure
their thoughts into logical arguments that can be spoken or written. This process of communication is
continued in many courses, but in particular in Math 309 they are encouraged to
present logical arguments orally to the class. Once they are able to express their
arguments, the explanations can be written down.
In the history of mathematics class, Math 311, the students are required
to make oral presentations to the class.
The
mathematics department realized the need for a course in which the big ideas of
algebra, geometry, trigonometry and calculus are related to the secondary
curriculum. In Math 460, the
students are required to explain concepts to the whole class, make presentations
to the class on supplementary topics, and work with secondary school students
using technology. In this
environment, students are encouraged to use standard mathematical terminology
and conventions.
The students are also encouraged to attend the Department of Mathematics
Colloquia Series in which they can see and hear mathematicians speaking the
language of mathematics.
Department of
Mathematics
Course Outline for Math
200
Calculus I
·
Prerequisites: Math 165, OR a score of 27 or above on
the Mathematics section of the Enhanced ACT, OR permission of the Department
Head.
· Text: Larson, Hostetler, and Edwards, Calculus of a Single Variable, 6th Edition, Houghton Mifflin Company.
· Adopted: Fall 1998
· Course Overview: This is a beginning Calculus course for Mathematics, Mathematics Education, Physics, Chemistry, Computer Science, and Pre-Engineering majors. Topics include limits, derivative, rules of differentiation (sum, product, quotient, and chain rules), integration, Riemann sum, area, exponential and logarithmic functions, and inverse trigonometric functions. These topics can be found in Chapters 1– 5 of the text.
Course
Objectives:
At the end of the course, the student will be able to:
·
Determine one-sided and
two-sided limits from a graph.
·
Determine one-sided and
two-sided limits by completing a table of y-values.
·
Use the properties of limits
to calculate one-sided and two-sided limits algebraically.
·
Calculate the limit of a
linear function and use the delta-epsilon definition of limit to prove the
answer is correct.
·
Determine where a
polynomial, rational, or trigonometric function is
continuous.
·
Use the definition of
derivative to calculate derivatives of functions.
·
Use graphing calculator or
CAS to calculate the numerical value of a derivative at a
point.
·
Find equations of tangent
lines.
·
Use the basic rules of
differentiation (sum and difference rule, product rule, quotient rule, chain
rule) to compute derivatives of functions.
·
Find a derivative using
implicit differentiation.
·
Calculate critical numbers
of a function
·
Find extrema of a function
on a closed interval.
·
Determine the intervals
where a function is increasing, decreasing, concave up, and concave
down.
·
Use the First and Second
Derivative Tests to find local extrema.
·
Use limits, a graph, or a
table of y-values to find horizontal and vertical
asymptotes.
·
Sketch the graph of a
function given important features of the graph (e.g., intercepts, asymptotes,
intervals of increase/decrease or concavity, inflection
points).
·
Solve an optimization
problem (e.g., areas, volumes, profit, cost) using the First Derivative
Test.
·
Compute indefinite integrals
using basic formulas.
·
Compute the exact area under
the graph of a linear, quadratic, or cubic polynomial using the limit process
with Riemann sums, equal partitions, n rectangles, and right or left
endpoints of subintervals.
·
Use area formulas of common
geometric object to find definite integrals.
·
Use the Fundamental Theorem
of Calculus to evaluate definite integrals.
·
Use a graphing calculator or
CAS to compute a definite integral.
·
Compute an integral using
the method of substitution.
·
Use the properties of
logarithms to simplify logarithmic expressions and solve exponential
equations.
·
Use properties of
exponentials to solve logarithmic equations.
·
Find the equation of the
inverse of a function given the equation of the function.
·
Find the graph of the
inverse of a function given the graph of the function.
·
Solve a differential
equation using separation of variables.
Course
Outline:
The
following syllabus allows for 5 regular examinations and the final
examination.
·
Chapter P Preparations for
Calculus
·
1
Week
P.1 Graphs and
Models
P.2 Linear Models and Rates of
Change
P.3 Functions and Their
Graphs
P.4 Fitting Models to
Data
· This chapter is a concise review of what students should have learned in College Algebra and Trigonometry. Some instructors may choose not to include this chapter. Emphasis should be given on graphing elementary functions. A graphing calculator or a Computer Algebra System such as Derive, Mathematica, or Maple, may be used. It is also important that the student understand that a point (a, b) is on the graph of the function y = f(x) if and only if the point satisfies the equation; that is if and only if b = f(a).
·
Chapter 1 Limits and Their
Properties
·
2
Weeks
1.1 A Preview of
Calculus
1.2 Finding Limits Graphically
and Numerically
1.3 Evaluating Limits
Analytically
1.4 Continuity and One-Sided
Limits
1.5 Infinite
Limits
· This is the chapter where the use of a graphing calculator can greatly enhance student’s understanding of the material. This is also a good time to give students their first examination. You may choose to include questions from Chapter P in the examination. Emphasis should be placed on finding limits (if such a limit exists) using one of the three methods: graphically, numerically, or analytically.
·
Chapter 2
Differentiation
·
3
Weeks
2.1 The Derivative and the
Tangent Line Problem
2.2 Basic Differentiation Rules
and Rates of Change
2.3 The Product and Quotient
Rules and Higher-Order Derivatives
2.4 The Chain
Rule
2.5 Implicit
Differentiation
2.6 Related
Rates
·
We have three weeks to cover
this chapter, so two sections per week might be a good pace. You may even teach the first three
sections during the first week so as to give students enough time to assimilate
all materials. Exam 2 may be given during
the last week on this chapter.
Emphasis should be placed on sections 2.1, 2.2, 2.3, and 2.4. If you feel you need more time, then
teach only the first five sections.
·
Chapter 3 Application of
Differentiation
·
3
Weeks
3.1 Extrema on an
Interval
3.2 Rolle’s Theorem and the Mean
Value Theorem
3.3 Increasing and Decreasing
Functions and the First Derivative Test
3.4 Concavity and the Second
Derivative Test
3.5 Limits at
Infinity
3.6 A Summary of Curve
Sketching
3.7 Optimization
Problems
3.8 Newton’s
Method
3.9 Differentials
3.10 Business and Economics
Applications
·
Note: Sections 3.8, 3.9, and
3.10 are optional. Some instructors
may choose to do student projects using these topics. Emphasize the connection between the
graph of the function and the graphs of its first and higher order
derivatives. Exam 3 on the third week, or
a project due on the third week.
Emphasis should be placed on Sections 3.1, 3.3, 3.4, and 3.6. Students enjoy graphing the derivative
of a function whose graph is given, and graphing a function whose derivative is
given.
·
Chapter 4
Integration
·
3
Weeks
4.1 Antiderivatives and
Indefinite Integration
4.2 Area
4.3 Riemann Sums and Definite
Integrals
4.4 The Fundamental Theorem of
Calculus
4.5 Integration by
Substitution
4.6 Numerical
Integration
·
Section 4.6 is optional, but
again, can be a good source of student projects. Note that three weeks are given to each
of Chapters 2, 3, and 4. This
should give enough flexibility to the instructor to assign projects. Exam 4. Emphasis should be
placed on Sections 4.1– 4.5.
Students should be able to set up the integrals that would give the area
of a given region. After Section
4.1, the students should be taught to use a CAS or a graphing calculator to
evaluate an integral.
·
Chapter 5 Logarithmic,
Exponential, and Other Transcendental Functions
·
3
Weeks
5.1 The Natural Logarithmic
Function and Differentiation
5.2 The Natural Logarithmic
Function and Integration
5.3 Inverse
Functions
5.4 Exponential Functions:
Differentiations and Integration
5.5 Bases Other than e and Applications
5.6 Differential Equations:
Growth and Decay
5.7 Differential Equations:
Separation of Variables
5.8 Inverse Trigonometric
Functions and Differentiation
5.9 Inverse Trigonometric
Functions and Integration
5.10 Hyperbolic
Functions
·
Students at this point are
expected to see how integration and
differentiation are connected.
Instructors can go easy on the integration part, as this will be reviewed
again in Techniques of Integration in Math 201. Emphasis should be placed on Sections
5.1, 5.4, and 5.8. Students are expected to set up an integral, but a graphing
calculator or CAS may be used to evaluate the integral. Exam 5, and review for the
finals.
Department of
Mathematics
Course Outline for Math
201
Calculus II
·
Prerequisites: Math 200
· Text: Larson, Hostetler, and Edwards, Calculus of a Single Variable, 6th Edition, Houghton-Mifflin
· Adopted: Fall 1998
Course Overview: This is the second course in a standard three-semester calculus sequence. Topics to be covered include applications of the definite integral, techniques of integration, L’Hopital’s rule and improper integrals, sequences, series, conic sections, and the calculus of parametric and polar equations.
The following
is a list of chapters and sections to be covered in chapters 6 – 9 of the
text. Some sections are optional
and it is up to the individual instructor to decide whether or not to cover any
or all of these sections. Also,
sections in which a Computer Algebra System (CAS) such as Derive, Mathematica,
or Maple, would be useful are indicated.
Chapter 6: Applications of Integration
Time: 15 days (50 minute class
periods)
Exam 1 at the
end of the chapter
·
6.1 Area of
Region between Two Curves – Emphasis
should be placed on drawing the region and setting up the appropriate definite
integral(s). A CAS or graphing
calculator may be used to evaluate the definite
integral(s).
·
6.2 Volume: The Disk
Method –
Emphasis should be placed on the fact that when the disk method is used, the
region is sliced into rectangles which are perpendicular to the axis of
revolution. Also, emphasis should
be placed on drawing the appropriate region, appropriate definite integral to
calculate the volume of the solid.
A CAS or graphing calculator may be used to evaluate the definite
integral.
·
6.3 Volume:
The Shell Method – Emphasis
should be placed on the fact that when the shell method is used, the region is
sliced into rectangles which are parallel to the axis of revolution. Also, emphasis should be placed on
drawing the appropriate region, calculating the volume of a representative shell
and using this volume to set up an appropriate definite integral to calculate
the volume of the solid. A CAS or
graphing calculator may be used to evaluate the definite
integral.
·
6.4 Arc Length
and Surfaces of Revolution – Emphasis
should be placed on setting up an appropriate definite integral to calculate the
length of a given curve and an appropriate definite integral to calculate the
surface area of a surface of revolution.
A CAS or graphing calculator may be used to evaluate the definite
integral.
·
6.5
Work – Emphasis
should be placed on calculating the increment of force (usually weight), using
this result to calculate the increment of work, and using this result to set up
an appropriate definite integral to calculate total work. A CAS or graphing calculator may be used
to evaluate the definite integral.
·
6.6 Moments,
Centers of Mass, and Centroids –
Optional
·
6.7 Fluid
Pressure and Fluid Force – Optional
Chapter 7: Integration Techniques, L’Hopital’s Rule, and Improper Integrals
Time: 20 days (50 minute class
periods)
Exam 2 at the end of the chapter
·
7.1 Basic
Integration Rules – Emphasis should be placed
on rewriting the integrand so that one of the basic rules of integration from
Chapter 4 can be used.
·
7.2
Integration by Parts – Emphasis should be placed
on the connection between integration by parts and the product rule, and the
types of integrands for which this technique is suitable.
·
7.3
Trigonometric Integrals – Emphasis should be placed
on recognizing the form of the integrand and being able to apply the appropriate
technique to evaluate the integral.
·
7.4
Trigonometric Substitution – Emphasis should be placed
on recognizing when trigonometric substitution is necessary, making appropriate
substitutions, and for indefinite integrals rewriting the end result in terms of
the original variable (using appropriate right triangles and trigonometric
identities).
·
7.5 Partial
Fractions –
Emphasis should be placed on problems which have repeated linear factors. The case with repeated quadratic factors
can be considered, but we do not want to overwhelm students with tedious
algebraic manipulation.
·
7.6
Integration by Tables and other Integration Techniques –
Optional
·
7.7
Indeterminate Forms and L’Hopital’s Rule – Emphasize the difference
between determinate and indeterminate forms, when L’Hopital’s rule is
applicable, and techniques that can be used to transform indeterminate forms
into the forms 0/0 or ?????.
·
7.8 Improper
Integrals –
Emphasis should be placed on the difference between proper and improper
integrals, and how the evaluation of an improper integral is a limiting
process.
Chapter 8: Applications of Integration
Time: 25 days (50 minute class
periods)
Exam 3 after section 8.6
Exam 4 after section 8.10
·
8.1
Sequences –
Emphasis should be placed on the difference between a convergent and divergent
sequence, and how to determine, whether a given sequence converges, and if it
does how to determine its limit. A
CAS may be useful to generate examples and possibly evaluate various
limits.
·
8.2 Series and
Convergence
– Emphasis should be placed on the difference between a sequence and a series,
what it means for a series to converge in terms of taking the limit of the
sequence of partial sums, the divergence test for series, and how to sum
infinite geometric series. A CAS
may be useful to generate examples.
·
8.3 The
Integral Test and p-Series – Emphasis should be placed
when the integral test is valid and how it can be used to determine the
convergence/divergence of a p-Series for different values of
p.
·
8.4 Comparison
of Series –
Emphasis should be placed on how to use the various comparison tests, and how to
recognize an appropriate series to be used to compare with a given
series.
·
8.5
Alternating Series – Emphasis should be placed
on how to use the alternating series test and the difference between and
absolute and conditional convergence.
·
8.6 The Ratio
and Root Tests – Emphasis should be placed
on recognizing when to use the ration test, the root test, or some other method
previously learned to determine convergence.
·
8.7 Taylor
Polynomials and Approximations – Emphasis should be placed
on why Taylor/Maclaurin polynomials are the “best” polynomial approximations of
functions near some particular value of x.
Also, emphasis should be placed on finding explicit Taylor/Maclaurin
polynomials for various elementary functions. Determining the accuracy of
approximations can be treated lightly.
A CAS may be useful to generate examples.
·
8.8 Power
Series –
Emphasis should be placed on determining the power series representations of
elementary functions and the corresponding interval of convergence (including
endpoints). Also, finding a power
series representation of a function’s derivative and integral from the
function’s power series representation should be
discussed.
·
8.9
Representations of Functions by Power Series – Emphasis should be placed
on recognizing functions whose power series are geometric
series.
·
8.10 Taylor
and Maclaurin Series – Emphasis should be placed
on knowing the Taylor or Maclaurin series of elementary functions and the
intervals of convergence of these various functions. Using series to approximate definite
integrals can be treated lightly.
Chapter 9: Conics, Parametric Equations, and Polar Coordinates
Time: 10 days (50 minute class
periods)
Optional Exam 5 at the end of the chapter
·
9.1 Conics and
Calculus –
Emphasis should be placed on the equations and properties of the conic sections,
and solving application problems in which they appear.
·
9.2 Plane
Curves and Parametric Equations – Emphasis should be placed
on the drawing of plane curves defined parametrically, and finding parametric
equations for plane curves. A CAS
would be very useful.
·
9.3 Parametric
Equations and Calculus – Emphasis should be placed
on finding derivatives (slopes of tangent lines) of curves defined
parametrically, and the arc length of a curve defined
parametrically.
·
9.4 Polar
Coordinates and Polar Graphs – Emphasis should be placed
on converting polar coordinates to rectangular coordinates and vice-versa, polar
equations to rectangular equations and vice-versa, and the graphing of polar
equations. A CAS would be
useful.
·
9.5 Area and
Arc Length in Polar Coordinates – Emphasis should be placed
on finding the area of a region defined by one or more polar curve, and on
finding the arc length of a polar curve.
Department of
Mathematics
Course Outline for Math
223
Discrete
Mathematics
·
Prerequisites: Math 200
· Text: Susanna S. Epp, Discrete Mathematics with Applications, 2nd Edition, Brooks/Cole Publishing, 1995.
Course
objectives:
After completing this
course, students should be able to:
·
Construct truth tables for
compound statements involving negation, conjunction, and
disjunction
·
Use truth tables to
determine if two statements are logically equivalent
·
State and prove DeMorgan’s
Laws of Logic and use them to write negations of given compound
statements
·
Define tautological and
contradictory statements and give examples of each
·
Use identities (e.g.,
commutative, associative, distributive, double negative, and DeMorgan’s laws) to
prove that two statements are logically equivalent
·
Construct truth table for
statements involving the conditional, p à q, and the biconditional, p
ßà q.
·
Find the negation of a
conditional statement
·
Find the contrapositive,
converse, and inverse of a given conditional statement
·
Translate sentences
involving the phrases “only if”, “sufficient condition”, and “necessary
condition” into sentences involving the conditional.
·
Determine if an argument is
valid or invalid
·
Determine if a universal
statement, an existential statement, or a universal conditional statement, is
true or false using methods such as providing counterexamples or using the
method of exhaustion
·
Find the negations of
universal, existential, or universal conditional
statements
·
Give examples of finite,
infinite, and alternating sequences
·
Write a sum or product in
expanded form given an explicit formula in summation or product notation, and
vice-versa
·
Transform an expression
given in summation or product notation by changing the dummy
variable
·
Simplify expressions
involving factorials
·
Use mathematical induction
to prove a given statement
·
Find elements and subsets of
a given set
·
Prove whether a given set is
a subset of another given set
·
Construct unions,
intersections, differences, complements, and Cartesian products of given
sets
·
Prove whether two sets are
equal or not
·
Define the empty
set
·
Determine if a collection of
sets are pairwise disjoint
·
Find the power of a
set
·
Write proofs concerning
basic number theoretic concepts such as odd and even integers, rational numbers,
prime and composite numbers, divisibility, perfect squares, and floor and
ceiling
·
Write proofs by
contradiction
·
Determine the domain,
codomain, and range of a given function
·
Represent functions by arrow
diagrams or sets of ordered pairs
·
Determine if two given
functions are equal
·
Give examples of functions
that are not well-defined
·
Prove or disprove that a
given function is one-to-one or onto
·
Find the inverse of a
function, if it exists
·
Use the pigeonhole principle
in counting arguments
·
Compute the composition of
two functions
·
Determine whether a
composition of functions is one-to-one or onto
·
Define and give examples of
a relation
·
Find the inverse relation of
a given relation
·
Determine whether a relation
is reflexive, symmetric, transitive, or an equivalence
relation
Course
Outline:
3 Weeks Chapter 1 – The Logic and Compound Statements
Sections 1.1, 1.2, and 1.3
Chapter 2 – The Logic of Quantified Statements
(Optional)
Sections 2.1 and 2.2
Exam
1
4 Weeks
Chapter 3 – Elementary Number Theory and Methods of
Proof
Sections 3.1, 3.2, 3.3, 3.4, 3.5, 3.6, and 3.7
Exam
2
3 Weeks Chapter 4 – Sequences and Mathematical Induction (Optional)
Sections 4.1 and 4.2
Chapter 5 – Set
Theory
Sections 5.1, 5.2, and 5.3
Exam
3
4 Weeks Chapter 7 – Functions
Sections 7.1, 7.3, 7.4, 7.5
Chapter 10 – Relations
Sections 10.1, 10.2, and 10.3
Exam 4
1 Week
Final Exam Review
Department of
Mathematics
Course Outline for Math
309
College
Geometry
·
Prerequisites: Math 200
·
Text: Sibley, The Geometric Viewpoint, A Survey of
Geometries, Addison-Wesley, 1998.
Course
Objectives:
Introduction to Proof
·
Throughout the semester,
students will read, write, and critique proofs.
·
Students will work
independently and in groups to prove various lemmas, theorems, and
corollaries.
·
Students will give verbal
presentations of their proofs.
Introduction to Axiomatics
·
Students will be introduced
to the axioms of absolute geometry including the incidence axioms, betweenness
axioms, metric axioms, and congruence axioms. Students will see how from a few basic
axioms, the well-known theorems of geometry may be derived. Students will see that some results that
may seem to be “intuitive” can not be proven in absolute
geometry.
·
Students will regularly test
and apply the theorems, etc. in specific examples. In such computational settings, the
students will experience the interplay between geometry and
algebra.
·
Students will also see how
the axioms and resulting theorems can be used with compass and straightedge (or
equivalent software) to construct
particular figures.
·
Throughout the course,
students will have the opportunity to experiment and conjecture using the
computer program Geometer’s Sketchpad.
Euclidean parallelism
·
Students will explore
Euclidean geometry through a careful investigation of the Euclidean parallel
postulate.
·
In particular, they will
explore a litany of statements that are logically equivalent to the parallel
postulate (many of which are the “intuitive” results that could not be proven
earlier).
·
Students will also explore
parallel and central projection and their ratio preserving properties. This provides an opportunity for a
strong algebraic tie-in.
Non-Euclidean geometry
·
Students will explore
non-Euclidean geometries using well-known models. E.g., elliptic geometry will be modeled
on the sphere and hyperbolic geometry will be modeled using the
Beltrami-Poincare Half-Plane model.
·
Students will also
investigate a metric in the Beltrami-Poincare model that is based on the natural
logarithm, thus providing another tie-in to algebra.
Geometric transformations
·
Students will investigate
the basic isometries of the plane:
reflections, translations, and rotations.
·
Students will see the
transformations as functions mapping the plane itself.
·
We will see how we may
provide coordinate characterizations of the transformations – an excursion into
analytic geometry.
·
Finally, using the operation
of composition students they will see that isometries from a
group.
Independent Projects
·
Students will independently
research a topic in geometry.
·
The students will prepare a
written paper on their topic. The
paper will be graded on content as well as organization and
exposition.
·
The students will give a
presentation to the class on their topic.
This presentation may take the form of a lesson (with lesson plan) that
they might use in a middle or high school, a computer presentation, a web-site,
etc. Students are encouraged to be
imaginative.
Course
Outline:
Week Sections to be Covered
1 Sections 1.1 and 1.2
2
Sections 2.4, 2.5, 2.6, and
2.7
3
Sections 2.7 and
2.8
4
Review; Exam
#1
5
Sections 3.1, 3.3, 3.4, and
3.5
6
Sections 3.6, 3.7, and
3.8
7
Sections 4.1 and
4.2
8
Sections 4.3;
Review
9
Exam #2; Section
6.1
10
Sections 6.2 and
6.3
11
Sections 6.4 and
6.7
12
Sections 5.1, 5.2, 5.3, and
5.4
13
Review
14
Exam
#3
15
Projects
16
Projects
Department of
Mathematics
Course Outline for Math
311
History of
Mathematics
·
Prerequisites: Math 200
· Text: Howard Eves, An Introduction to the History of Mathematics, 6th Edition, Saunders College Publishing, Fort Worth, 1990.
· Adopted: Fall 2000
Course
objectives:
For Chapter 1, the student
should:
·
Be able to characterize a
numeration system.
·
Be familiar with types of
numeration systems other than positional systems.
· Be able to rename decimal numbers in bases other than ten and vice versa.
·
Know that properties of
numbers are independent of the numerals used to name them. For example, the property of being an
even or odd number.
For Chapter 2, the student should:
· Know some of the original sources of ancient Babylonian and Egyptian Mathematics
·
Know the motivation for the
origins of the mathematics of the ancient Babylonians and
Egyptians.
·
Compare and contrast the
development of mathematics in Egypt and Babylon.
For Chapter 3, the student should:
· Know the origin of demonstrative mathematics.
·
Know who the Pythagoreans
were and what the Pythagorean Philosophy was.
· Recognize the geometric character of early Greek algebra.
·
Be able to give geometric
demonstrations of such identities as
·
Be able to name and state
the three famous problems and know their significance to the development of
mathematics.
For Chapter 5, the student
should:
·
Be able to describe the
content of Euclid’s Elements.
·
Know and be able to use the
Euclidean algorithm.
·
Know and be able to apply
the Fundamental Theorem of Arithmetic.
For Chapter 6, the student
should:
·
Be knowledgeable of the
contributions of Archimedes.
·
Be knowledgeable of the
contributions of Appolonius.
·
Be knowledgeable of the
contributions of other Greeks after Euclid.
For Chapter 7, the student
should:
·
Know some of things the
Chinese before the Europeans
·
Be able to contrast between
Greek and Hindu mathematics.
·
Know the Arabian
contribution to the history of mathematics.
For Chapter 8, the student
should
·
Know about the contributions
of Fibonacci.
·
Know the beginnings of
algebraic symbolism.
·
Be familiar with early
arithmetics.
·
Know the Italian school
solutions for cubic and quartic equations.
For Chapter 9, the student
should:
·
Know some of the major
contributions to mathematics made in the 17th
century.
·
Have a general idea of how
the concept of logarithms developed.
·
Know about the many
contributions made by Galileo.
·
Know about the difficult
circumstances behind the contributions of Kepler.
·
Know about the contributions
of Desargues and Pascal.
For Chapter 10, the student
should:
·
Be able to distinguish
between analytic geometry and projective geometry.
·
Know about the contributions
of Descartes.
·
Know about the contributions
of Fermat.
·
Know about the contributions
of Huygens.
·
Know the origins of
probability theory
·
Know some of the significant
contributions from other 17th century
mathematicians.
For Chapter 11, the student
should:
·
Know the historical
development of the calculus.
·
Know the problems posed by
Zeno’s paradoxes.
·
Know how the method of
exhaustion works.
·
Know about the beginnings of
differention.
·
Know about the development
of calculus by Newton and Leibniz.
For Chapter 12, the student
should:
·
Be aware of the exploitation
of the calculus that occurred in the 18th
century.
·
Know about the contributions
of the Bernoulli family.
·
Know about the development
of series by Taylor and Maclaurin.
·
Know about the contributions
to the development of calculus by Euler.
·
Know about other significant
uses of calculus during the 18th century.
Course
Outline:
Week Chapter . Section
1
1.1 – 1.10
2
2.1 – 2.10
3
3.1 – 3.5
4 3.6 – 3.10, Test I
5
4.1 – 4.8
6
5.1 – 5.8
7 6.1 – 6.10
8
Test II, 7.1 – 7.7
9
7.8 – 7.14, 8.1 –
8.5
10
8.6 – 8.10, 9.1 – 9.5
11
9.6 – 9.9, Test III
12
10.1 – 10.8
13
11.1 – 11.10
14
12.1 – 12.6
15
12.7 –12.12,
Review
Department of
Mathematics
Course Outline for Math
312
Calculus
III
·
Prerequisites: Math 201
· Text: Larson, Hostetler, and Edwards, Multivariable Calculus, 6th Edition, Houghton-Mifflin, 1998
· Adopted: Fall 1998
Course
Outline:
Week 1
·
10.1 Vectors in the
Plane –
Emphasis should be placed on the computational aspects of vectors, e.g., sums,
scalar multiples, negatives, lengths, and unit vectors.
·
10.2 Space Coordinates and
Vectors in Space – This section should be
covered quickly.
·
10.3 The Dot Product of Two
Vectors –
Computing dot products, using the properties of the dot product, computing and
angle between vectors, and finding vector components should be emphasized. Direction cosines can be
omitted.
Week 2
·
10.4 The Cross Product of
Two Vectors in Space – Emphasis should be placed
on the computation of the cross product, the algebraic and geometric properties
of the cross product, and using the cross product to find a vector orthogonal to
two given vectors.
·
10.5 Lines and Planes in
Space –
Emphasis should be placed on finding equations of lines and planes in
space.
Week 3
·
10.6 Surfaces in
Space –
Emphasis should be placed on sketching quadric surfaces. A computer algebra system should be used
to generate the graphs.
·
10.7 Cylindrical and
Spherical Coordinates – Emphasis should be placed
on converting amongst rectangular cylindrical, and spherical
coordinates.
Week 4
·
Exam #1
Week 5
·
11.1 Vector-Valued
Functions –
Emphasis should be placed on basic properties of vector-valued functions, e.g.,
evaluating vector-valued functions at given points and finding domains. Graphing vector-valued functions can be
facilitated with a computer algebra system.
·
11.2 Differentiation and
Integration of Vector-Valued Functions – Emphasis should be placed
on the computations of differentiation and integration.
Week 6
·
11.3 Velocity and
Acceleration – Emphasis should be placed
on the computations of position, velocity, and acceleration given one of those
three vector-valued functions.
·
11.4 Tangent Vectors and
Normal Vectors – Emphasis should be placed
on finding unit tangent vectors and principal unit normal vectors. Tangential and normal components of
acceleration can be omitted.
Week 7
·
11.5 Arc Length and
Curvature –
Curvature may be omitted.
·
12.1 Introduction to
Functions of Several Variables – Emphasis should be placed
on sketching graphs of functions of several variables, including level
curves. A computer algebra system
will be helpful here.
·
12.2 Limits and
Continuity
– Emphasis should be placed on the definitions of limit and continuity and
computing limits utilizing different paths.
Week 8
·
12.3 Partial
Derivatives
– Emphasis should be placed on computing directional derivatives and gradients,
finding normal vectors to level curves, and using gradients to find directions
of maximum and minimum increase.
·
12.5 Chain Rules for
Functions of Several Variables – Emphasis should be placed
on computing derivatives using the chain rule, including implicit
differentiation.
·
Exam #2
Week 9
·
12.6 Directional Derivatives
and Gradients – Emphasis should be placed
on computing directional derivatives and gradients, finding normal vectors to
level curves, and using gradients to find directions of maximum and minimum
increase.
·
12.7 Tangent Planes and
Normal Lines – Emphasis should be placed
on finding equations of tangent planes and normal lines. The angle of inclination of a plane can
be omitted.
Week 10
·
12.8 Extrema of Functions of
Two Variables – Emphasis should be placed
on the definitions of critical points and relative extrema, the second partials
test, and finding relative and absolute extrema and saddle
points.
·
12.9 Applications of Extrema
of Functions of Two Variables – Standard word problems
should be used here. At least one
least squares problem should be done by hand; then subsequent least squares
problems could be computed with a calculator or computer.
Week 11
·
13.1 Interated Integrals and
Area in the Plane – Emphasis should be placed
on why a double integral gives an area and the computation of double
integrals. Special care should be
taken with determining the limits of integration.
·
13.2 Double Integrals and
Volume –
Emphasis should be placed on why double integrals give volume and the
computation of double integrals.
Special care should be taken with determining the limits of
integration.
·
Exam
#3
Week 12
·
13.3 Change of Variables:
Polar Coordinates – Emphasis should be placed
on describing regions with polar coordinates and converting rectangular
integrals to polar integrals.
Special care should be taken to emphasize the addition of the r dr d Q term.
Week 13
·
13.5 Surface
Area –
Emphasis should be placed on setting up the correct
integrals.
·
13.6 Triple Integrals and
Applications – Emphasis should be placed
on setting up the correct integrals.
Week 14
·
13.7 Triple Integrals in
Cylindrical and Spherical Coordinates – Emphasis should be placed
on volume problems. Attention
should be focused on setting up the correct integrals, particularly when
selecting which coordinate system to use.
·
14.1 Vector
Fields –
Emphasis should be placed on sketching vector fields and determining if a vector
field is conservative or not.
Week 15
·
14.2 Line
Integrals –
Emphasis should be placed on setting up correct integrals and computing line
integrals
·
Exam #4
Course Timeline (for TTH
Class):
Week Sections to Cover
1
10.1, 10.2,
10.3
2
10.4,
10.5
3
10.6,
10.7
4
Review, Exam
#1
5
11.1,
11.2
6
11.3,
11.4
7
11.5, 12.1,
12.2
8
12.3, 12.5, Exam
#2
9
12.6,
12.7
10
12.8,
12.9
11
13.1,
13.2
12
Exam #3,
13.3
13
13.5,
13.7
14
13.7,
14.1
15
14.2, Exam
#4
16
Final
Exam
Department of
Mathematics
Course Outline for Math
350
Applied
Differential Equations
·
Prerequisites: Math 201
· Text: Edwards and Penney, Elementary Differential Equations, 3rd Edition
· Course Objective: After the study of this course, you are expected to understand what a differential equation is, how to solve some basic differential equations analytically or numerically, how to use differential equations to model real world situations, use computer to solve them and interpret the results in real context.
Sections to
Cover:
·
Sections 1.1 –
1.8
·
Sections 2.1 –
2.8
·
Sections 3.1 –
3.3, 3.5 – 3.7
·
Sections 4.1 –
4.5
Department of
Mathematics
Course Outline for Math
360
Linear Algebra
I
COURSE CONTENT
Matrices and Systems of
Equations
1.
Systems of Linear
Equations
2.
Row Echelon
Form
3.
Matrix
Algebra
4.
Elementary
Matrices
5.
Partitioned
Matrices
In
this section, students are introduced to matrices using systems of linear
equations. Operations on matrices
such as addition, multiplication, scalar and vector multiplication are also
discussed. Topics also include
elementary matrices and operations on partitioned matrices. Matlab exercises will be assigned to
enhance students’ awareness.
Determinants
1.
The Determinant of a
Matrix
2.
Properties of
Determinants
Vector Spaces
1.
Definition and
Examples
2.
Subspaces
3.
Linear
Independence
4.
Basis and
Dimension
5.
Change of
Basis
6.
Row Space and Column
Space
In
this section, students are introduced to vector spaces and their subspaces. Topics also include linear independence,
basis and dimension of a vector space, linear combination, change of basis, and
the relationships between row spaces and column spaces. Problems involving proving will be
emphasized in this section. Also,
Matlab exercises will be assigned.
Linear Transformation
1.
Definition and
Examples
2.
Matrix Representations of
Linear Transformations
3.
Similarity
In
this section, students are introduced to linear transformations, and the matrix
representation of a linear transformation.
Also included in this section is similarity. Problems that require proving will be
emphasized.
Department of
Mathematics
Course Outline for Math
370
Introduction to
Abstract Algebra
·
Prerequisites: Math 201 and Math 223 or concurrent
enrollment
· Text: I.N. Herstein, Abstract Algebra, 3rd Edition, John Wiley & Sons, Inc.
· Adopted: Fall 2000
· Course Overview: An introduction to abstract algebra concentrating on elementary group theory. Students will be introduced to various groups, i.e., abelian, symmetric, permutations. Other concepts include properties of groups, mappings of groups and definition and examples of rings.
Course
objectives:
·
State the
definition of a group
·
Give examples
of finite, infinite, abelian, and cyclic groups
·
Given a set
with operation, determine whether it is a group
·
Prove that a
given subset of a group is a subgroup
·
Find the order
of a group or a subgroup of a group
·
State the
definition of an equivalence relation
·
Given a
relation on a set, determine whether it is an equivalence
relation
·
Find all left
and right cosets of a subgroup of a group
·
Define a
homomorphism
·
Construct
external direct product of groups
·
Prove that a
given subgroup of a group is normal
·
Find products
and powers of permutations
·
Decompose a
permutation into cycles
·
Find the
parity of a given permutation
·
Give the
definition and examples of rings
Course
Outline:
1.
Review
a)
Set
theory
b)
Mappings
(optional)
2.
Groups and
subgroupings
a)
Definition and
examples
b)
Properties
c)
Lagrange’s
Theorem
d)
Homomorphisms and normal
subgroups
e)
Factor
groups
f)
The homomorphisms theorems
(optional)
g)
Cauchy’s
theorem
h)
Direct
products
3.
Symmetric
groups
a)
Products and
powers
b)
Cycle
decomposition
c)
Parity
4. Definition and examples of
rings
Department of
Mathematics
Course Outline for Math
409
Linear Algebra
II
·
Prerequisites: Math 360
· Text: Leon, Linear Algebra with Applications, 5th Edition, Macmillan, NY, 1997
· Course Content: Systems of linear equations, matrices, and matrix algebra, determinants, vector spaces, subspaces, linear transformations, scalar products, eigenvalues and eigenvectors, diagonalization.
Sections to
Cover:
·
Sections 1.1 –
1.5
·
Sections 2.1 –
2.3
·
Sections 3.1 –
3.6
·
Sections 4.1 –
4.3
·
Sections 5.1 –
5.3, 5.5, 5.6
·
Sections 6.1,
6.3
COURSE
CONTENT
I.
A Review of
Linear Algebra I
A quick review
of Linear Algebra I that includes matrices and systems of equations, row echelon
form, elementary matrices, operations on partitioned matrices, determinants,
vector spaces, linear independence, basis and dimension, change of basis, and
linear transformation.
II.
Orthogonality
1. The Scalar
Product in Rn
2. Orthogonal
Subspaces
3. Inner Product
Spaces
4. Least Squares
Problems
5. Orthogonal
Sets
6. The
Gram-Schmidt Orthogonalization Process
7. Orthogonal
Polynomials
In this
section, students are introduced to scalar product of vectors. Topics to be discussed include
orthogonal subspaces and inner product spaces, and their applications to least
squares problems. Topics also
include orthogonal sets, the Gram-Schmidt orthogonalization process and
orthogonal polynomials.
III.
Eigenvalues
1. Eigenvalues
and Eigenvectors
2. Systems of
Linear Differential Equations
3. Diagnolization
4. Hermitian
Matrices
5. Quadratic
Forms
6. Positive
Definite Matrices
In this
section, students are introduced to the concepts of eigenvalues and eigenvectors
of a matrix. An application on
systems of linear differential equations will also be discussed. Further topics include diagonalization
of a matrix, Hermitian matrices, quadratic forms, and positive definite
matrices. Matlab exercises will be
used to enhance students’ learning.
IV.
Numerical
Linear Algebra
1. Floating-Point
Numbers
2. Gaussian
Elimination
3. Pivoting
Strategies
4. Matrix Norms
and Condition Numbers
5. Orthogonal
Transformations
6. The Singular
Value Decomposition
In this
section, students are introduced to
numerical linear algebra. Topics
include floating-point number, Gaussian elimination, matrix norms and condition
numbers, and orthogonal transformations.
We will also discuss the singular value decomposition and some of its
applications.
Department of
Mathematics
Course Outline for Math
410
Number
Theory
The main topics covered in this course are divisibility theory in the integers, primes and their distributions, the theory of congruences, Fermat’s Theorem, Number theoretic functions, Euler’s generalization of Fermat’s Theorem, Primitive roots and indices, perfect numbers, and the Fermat Conjecture.
Objectives
Divisibility Theory in the
Integers
We
will discuss the division algorithm, divisibility of one integer by another,
greatest common divisor of two integers, Euclidean division algorithm, and
Diophantine equations. The students
will be able to state and prove the division algorithm. Using the division algorithm, they will
establish certain properties of squares, cubes and fourth powers of
integers. For example, the will
prove that the square of any integer must be of the form 3k or 3k + 1 for some
integer k. The students will be
able to define the term ‘divisibility’, and will use the definition to prove
some properties of divisibility.
One such property is the transitive property of divisibility. The students will learn how to calculate
the greatest common divisor of two given integers using the definition. They will learn how to use the
definitions to prove the properties of greatest common divisors, including
Euclid’s Lemma. The students will
use the Euclid’s division algorithm to find the greatest common divisor of two
integers, and write the greatest common as a linear combination of the given two
integers. The students will solve
Diophantine equations such as 56x + 72y = 40.
Primes and Their
Distribution
We
will discuss the Fundamental Theorem of Arithmetic, Sieve of Eratosthenes, and
Goldbach Conjecture. The students
will compute the prime factorization of a given integer. They will use the Sieve of Eratosthenes
to find all the primes less than a given natural number. The students will learn to prove that
there is an infinite number of primes.
The students will also learn the Goldbach’s Conjecture and Dirichlet’s
Theorem.
The Theory of
Congruences
We
will discuss the basic properties of congruences, special divisibility tests,
and linear congruences. Using the
definition, the students will prove the basic property of congruences including
the transitive property. The
students will discover and prove special divisibility tests: For example, they
will prove that an integer is divisible by 3 if and only if the sum of its
digits is divisible by 3. The
students will solve the linear congruences, and also use congruences to solve
Diophantine equations.
Fermat’s
Theorem
We
will discuss Fermat’s Factorization Method, Fermat’s Little Theorem, and
Wilson’s Theorem. The students will
use the Fermat’s Factorization Method to factor a given integer. The students will be able to state and
prove Fermat’s Little Theorem. They
will use this theorem to derive certain congruences such as a7=a (mod 42) for all a. The students will also learn the
statement and the proof of Wilson’s Theorem.
Number Theoretic
Functions
The
students will learn about the number theoretic functions t
and
s. Given a positive integer n, the students will calculate t
(n) and s
(n). They will learn the meaning of a
multiplicative function, and prove that both t
and
s
are such
functions. The students will learn
about the Mobius m-function, and prove that it
is also a multiplicative function.
They will also learn and use the Mobius Inversion
Formula.
Euler’s Generalization to
Fermat’s Theorem
The
students will learn about the Euler phi-function f. Given a positive integer they will
calculate f (n). They will also prove that
f
is a
multiplicative function, and that f (n) is an even integer for integer n > 2. The students will state and prove
Euler’s Theorem. They will use this
theorem to reduce large powers of certain integers modulo a given prime, such as
reducing 2100000 modulo 77.
They will also learn about the connections between the Euler f-function, and the Mobius
m-function.
Primitive Roots and
Indices
Given an integer a, the students will be able to
calculate its order modulo another positive integer n.
They will also calculate primitive roots of prime numbers. They will also learn about the composite
numbers having primitive roots.
Perfect
Numbers
The students will learn about the Perfect Numbers, Mersenne Numbers and Mersenne Primes and Fermat Numbers. They will demonstrate examples of perfect numbers such as 6 and 28. They will also prove the fact that any even perfect number ends in digits 6 or 8.
Fermat’s Conjecture and
Pythagorean Triples
The
students will learn the statement of Fermat’s Conjecture (Fermat’s Last Theorem)
without proof. They will learn
about the general form of Pythagorean triples, and will generate several
examples using the general formula.
They will also study the Diophantine equations of the form x4 + y4 = z2 and x4 + y4 = z4.
Department of
Mathematics
Course Outline for Math
417
Mathematical
Statistics
Math 417 is a first course in the area of probability and statistics with a calculus prerequisite. The emphasis in Math 417 is on the foundations of statistics in probability, including both theory and applications. Math 417 is also provides an opportunity for students to see calculus in action and review many calculus topics, including sequences, series, integration, and differentiation. Consequently the students will be required to do proofs and computations involving calculus, as well as extensive modeling and problem solving.
Objectives
Counting/Combinatorics
Students will explore the
multiplication principle for counting, permutations, and combinations. Students will model real-life situations
and determine the number of outcomes to real-life experiments, which will
include cards, dice, lotteries, and coins.
Students will learn the binomial theorem and will find binomial and
multinomial coefficients. Pascal’s
triangle will also be discussed.
Probability
Students will learn the
definition of probability and the properties of probability as a set
function. Students will model
real-life situations (such as coins, dice, cards, and lotteries) and find
probabilities using the multiplication rule, the addition rule, conditional
probability, independent events, and Bayes’ Theorem. Students will understand the difference
between sampling with and without replacement. Students will understand the difference
between empirical probability and the theoretical probability determined from a
mathematical model.
Discrete
Distributions
Students will understand the
concept of random variable and the difference between an empirical probability
distribution and a theoretical probability distribution. Students will model real-situations
using discrete distributions, which will include the Bernoulli, binomial,
geometric, hypergeometric, negative binomial, Poisson, and uniform
distributions. Students will solve
probability problems using the discrete distributions. Students will study mathematical
expectation and use the concept to find mean, variance, and standard deviation
for discrete distributions.
Students will be able to prove the formulas for the mean, variance, and
standard deviation of the binomial, geometric, Poisson, and uniform
distributions. Students will find
moment generations functions for discrete distributions and use them to find the
mean and standard deviation.
Continuous
Distributions
Students will understand the
difference between continuous and discrete distributions and model real-life
situations using continuous distributions.
Students will explore the beta, chi-square, exponential, gamma, normal,
and uniform continuous distributions and solve probability problems using these
distributions. Students will be
able to prove the formulas for mean and standard deviation for the exponential
and normal distribution. Students
will study the gamma function and its properties. Students will find the mean and standard
deviation. Students will find
distributions of simple functions of random variables using the change of
variables technique with Jacobians.
Department of
Mathematics
Course Outline for Math
431
Numerical
Analysis
General Course
Objectives
After the study of this
course students should be able to know why numerical methods are necessary, how
numerical methods are developed, how to implement numerical methods on
computers, and how to evaluate numerical methods. Students are also expected to apply
various numerical computational methods to solve problems rising from several
branches of mathematics, physics and other applied
sciences.
Specific Course
Objectives
1.
Students will learn how to
solve algebraic equations by using numerical methods such as bisection methods,
Newton’s methods and fixed-point iteration methods.
2.
Students will explore
various methods for solving nonlinear systems of
equations.
3.
Students will learn various
techniques to approximate functions such as polynomial interpolation and least
square approximation.
4.
Students will explore how to
do data analysis and curve fitting.
5.
Students will be able to use
both direct and indirect methods for solving systems of linear
equations.
Computing project and
research project
1.
Students will be asked to
write a program to use and/or modify available computational method to solve a
realistic and challenging problem.
2.
The instructor will divide
the class into several groups by selecting the group leaders and letting
students form their groups. Each
group will be asked to write a research paper on a specific topic in numerical
analysis which is not covered in the class. The paper must address certain
issues. The students will present
their papers to the whole class.
Department of
Mathematics
Course Outline for Math
460
Secondary
Mathematics Methods
This course offers methods
and materials for teaching mathematics at the secondary level. It is designed to have students learn
how to have their pupils use problem solving, reasoning and communication in the
classroom. It is also designed to
show students how to make connections between all the big ideas of
mathematics.
Objectives
The students will understand the history of mathematics education and the development of the NCTM Standards for curriculum, teaching and assessment.
The
students will all receive starter packs from the NCTM and be encouraged to
become members of professional organizations.
The
students will read articles in professional journals and write abstracts to be
shared with the class.
The
students will attend the LATM state conference and the SEATM local conference
and report on their experiences.
The
students will use various manipulatives such as algebra tiles and geoboards and
geometry models to develop lessons in the development of number sense and in
various concepts in algebra, geometry and trigonometry.
The
students will work in cooperative groups and understand how to use cooperative
learning and peer tutoring in their classes.
The
students will do problem-solving daily and discuss how to use it in their
classes.
The
students will locate potential sites for good problems.
The
students will use writing to learn mathematics and they will keep portfolios of
their work.
The
students will discuss various alternative assessment techniques and derive
rubrics for assigning grades to students.
The
students will become proficient at using graphing calculators and various
computer programs including Geometer’s Sketchpad, Excel spreadsheets,
Mathematica, etc.
The
students will compile a list of good mathematical sites on the Internet. They will search these sites to find
answers to specific questions.
The
students will observe secondary mathematics teachers and evaluate their
teaching.
The
students will work with students in a local school system under the guidance of
the instructor.
The
students will prepare a lesson on an enrichment topic and teach it to the
class.