Undergraduate Courses

course descriptions for Mathematics Lower & Upper Division, and PIC Classes

mathematics courses
General Course Outline

Course Description

(4) Lecture, three hours; discussion, one hour. Preparation: three years of high school mathematics. Requisite: successful completion of Mathematics Diagnostic Test at UCLA. Function concept. Linear and polynomial functions and their graphs, applications to optimization. Inverse, exponential, and logarithmic functions. Trigonometric functions. P/NP or letter grading.

 
Textbook(s)

D. Lippman & M. Rasmussen. Precalculus. An investigation of functions.
Available online at www.opentextbookstore.com/precalc

Outline update: P. Greene, 8/13

General Course Outline
Course Description

(4) Lecture, three hours; discussion, one hour. Preparation: three and one-half years of high school mathematics (including trigonometry). Requisite: successful completion of Mathematics Diagnostic Test (score of 35 or better) or course 1 at UCLA with a grade of C- or better. Not open for credit to students with credit in another calculus sequence. Modeling with functions, limits and derivatives, decisions and optimization in biology, derivative rules and tools. P/NP or letter grading.

Course Information:

The following schedule, with textbook sections and topics, is based on 24 lectures. The remaining classroom meetings are for leeway, reviews, and two midterm exams. These are scheduled by the individual instructor.

Math 3ABC is the “fast” calculus sequence at UCLA. It aims to provide students in three terms with the fundamental ideas and tools of calculus that will put them in a good position for understanding more technical work in their own areas. The course sequence covers basic topics in single-variable and multi-variable calculus. This includes some material on ordinary differential equations such as those governing population-growth models. The course also covers some material on calculus-based probability theory, including continuous probability distributions, the normal distribution, and the idea of hypothesis testing.

The course sequence 3ABC is suitable for students who want to be introduced to the powerful tools that the calculus provides without going through some of the more technical material required of the students in engineering and the physical sciences. While examples and illustrations are drawn from the life sciences when possible, the course sequence is also suitable for students in the social sciences and humanities who do not require a heavy mathematical background.

Students in 3ABC are expected to have a good background in precalculus mathematics, including polynomial functions, trigonometric functions, and exponential and logarithm functions. In order to enroll in 3A, students must either take and pass the Mathematics Diagnostic Test at the specified minimum performance level, or take and pass Math 1 at UCLA with a grade of C- or better.

Many of the students in Math 3ABC take Physics 6, either concurrently or later. The topics covered in 3ABC are selected so as to provide students with the prerequisite foundations for Physics 6.

Ample tutoring support is available for students in the course, including the walk-in tutoring service of the Student Mathematics Center at MS 3974.

Textbook(s)

S. J. Schreiber, Calculus for the Life Sciences, Wiley.

Outline update: P. Greene, 11/15

General Course Outline

Course Description

(4) Lecture, three hours; discussion, one hour. Requisite: course 3A with grade of C- or better. Not open for credit to students with credit for course 31B. Applications of differentiation, integration, differential equations, linear models in biology, phase lines and classifying equilibrium values, bifurcations. P/NP or letter grading.

Course Information:

The following schedule, with textbook sections and topics. The remaining classroom meetings are for leeway, reviews, and two midterm exams. These are scheduled by the individual instructor.

Math 3ABC is the “fast” calculus sequence at UCLA. It aims to provide students in three terms with the fundamental ideas and tools of calculus that will put them in a good position for understanding more technical work in their own areas. The course sequence covers basic topics in single-variable and multi-variable calculus. This includes some material on ordinary differential equations such as those governing population-growth models. The course also covers some material on calculus-based probability theory, including continuous probability distributions, the normal distribution, and the idea of hypothesis testing.

The course sequence 3ABC is suitable for students who want to be introduced to the powerful tools that the calculus provides without going through some of the more technical material required of the students in engineering and the physical sciences. While examples and illustrations are drawn from the life sciences when possible, the course sequence is also suitable for students in the social sciences and humanities who do not require a heavy mathematical background.

Students in 3ABC are expected to have a good background in precalculus mathematics, including polynomial functions, trigonometric functions, and exponential and logarithm functions. In order to enroll in 3A, students must either take and pass the Mathematics Diagnostic Test at the specified minimum performance level, or take and pass Math 1 at UCLA with a grade of C- or better.

Many of the students in Math 3ABC take Physics 6, either concurrently or later. The topics covered in 3ABC are selected so as to provide students with the prerequisite foundations for Physics 6.

Ample tutoring support is available for students in the course, including the walk-in tutoring service of the Student Mathematics Center at MS 3974.

Textbook(s)

S. J. Schreiber, Calculus for the Life Sciences, Wiley.

Outline update: P.Greene, 11/15

General Course Outline

Course Description

(4) Lecture, three hours; discussion, one hour. Requisite: course 3B with grade of C- or better. Multivariable modeling, matrices and vectors, eigenvalues and eigenvectors, linear and nonlinear systems of differential equations, probabilistic applications of integration. P/NP or letter grading.

Course Information:

The following schedule, with textbook sections and topics, is based on 25 lectures. The remaining classroom meetings are for leeway, reviews and two midterm exams. These are scheduled by the individual instructor.

Math 3ABC is the “fast” calculus sequence at UCLA. It aims to provide students in three terms with the fundamental ideas and tools of calculus that will put them in a good position for understanding more technical work in their own areas. The course sequence covers basic topics in single-variable and multi-variable calculus. This includes some material on ordinary differential equations such as those governing population-growth models. The course also covers some material on calculus-based probability theory, including continuous probability distributions, the normal distribution, and the idea of hypothesis testing.

The course sequence 3ABC is suitable for students who want to be introduced to the powerful tools that the calculus provides without going through some of the more technical material required of the students in engineering and the physical sciences. While examples and illustrations are drawn from the life sciences when possible, the course sequence is also suitable for students in the social sciences and humanities who do not require a heavy mathematical background.

Students in 3ABC are expected to have a good background in precalculus mathematics, including polynomial functions, trigonometric functions, and exponential and logarithm functions. In order to enroll in 3A, students must either take and pass the Mathematics Diagnostic Test at the specified minimum performance level, or take and pass Math 1 at UCLA with a grade of C- or better.

Many of the students in Math 3ABC take Physics 6, either concurrently or later. The topics covered in 3ABC are selected so as to provide students with the prerequisite foundations for Physics 6.

Ample tutoring support is available for students in the course, including the walk-in tutoring service of the Student Mathematics Center at MS 3974.

Textbook(s)

S. J. Schreiber, Calculus for the Life Sciences, Wiley.

Outline update: P. Greene, 3/16

General Course Outline

Course Description

Lecture, three hours; discussion, one hour. Requisites: courses 31A, 31B. Introductory number theory course for freshmen and sophomores. Topics include prime number theory and cryptographic applications, factorization theory (in integers and Gaussian integers), Pythagorean triples, Fermat descent (for sums of squares and Fermat quartic), Pell’s equation, and Diophantine approximation. P/NP or letter grading

 
Textbook(s)

J. Silverman, A Friendly Introduction to Number Theory (4th edition)

 
General Course Outline

Course Description

(4) Lecture, three hours; discussion, one hour. Preparation: at least three and one-half years of high school mathematics (including some coordinate geometry and trigonometry). Requisite: successful completion of Mathematics Diagnostic Test or course 1 with a grade of C- or better. Differential calculus and applications; introduction to integration. P/NP or letter grading.

Course Information:

The following schedule, with textbook sections and topics, is based on 26 lectures. The remaining classroom meetings are for leeway, reviews, and two midterm exams. These are scheduled by the individual instructor. Often there are reviews and midterm exams about the beginning of the 4th and 8th weeks of instruction, plus reviews for the final exam.

In certain cases (such as for coordinated classes), it may be possible to give midterm exams during additional class meetings scheduled in the evening. This has the advantage of saving class time. A decision on whether or not to do this must be made well in advance so that the extra exam sessions can be announced in the Schedule of Classes. Instructors wishing to consider this option should consult the mathematics undergraduate office for more information.

The goal of Math31AB is to provide a solid introduction to differential and integral calculus in one variable. The course is aimed at students in engineering, the physical sciences, mathematics, and economics. It is also recommended for students in the other social sciences and the life sciences who want a more thorough foundation in one-variable calculus than that provided by Math 3.

Students in 31AB are expected to have a strong background in precalculus mathematics, including polynomial functions, trigonometric functions, and exponential and logarithm functions. In order to enroll in 31A, students must either take and pass the Mathematics Diagnostic Test at the specified minimum performance level, or take and pass Math 1 at UCLA with a grade of C- or better.

Most students entering the 31-32-33 sequence at UCLA have taken a calculus course in high school and enter directly into Math 31B, for which there is no enforced prerequisite.

The course 31A covers the differential calculus and integration through the fundamental theorem of calculus. The first part of course 31B is concerned with integral calculus and its applications. The rest of the course is devoted to infinite sequences and series.

Single-variable calculus is traditionally treated at many universities as a three-quarter or two-semester course. Thus Math 31AB does not cover all of the topics included in the traditional single-variable course. The main topics that are omitted are parametric curves and polar coordinates, which are treated at the beginning of 32A.

Ample tutoring support is available for students in the course, including the walk-in tutoring service of the Student Mathematics Center.

Math 31A is not offered in the Spring Quarter. Students wishing to start calculus in the Spring may take 31A through University Extension in the Spring or in the Summer.

Please note: Students who are in the College of Letters and Science who will be enrolled at UCLA in Spring and wish to enroll in Extension simultaneously should meet with a College Counselor about whether they will be able to receive credit for the course because of concurrent enrollment restrictions: Concurrent Enrollment Information.

Textbook(s)

J. Rogawski, Calculus: Late Transcendentals Single Variable Calculus Fourth Edition, W.H. Freeman & CO

(a) Limits should be presented very informally with an emphasis on working with their properties: the “Limit Laws”.

(b) Section 6.2 should be restricted to the topic of average value.

Outline update: 3/15 R. Brown

General Course Outline

Course Description

Lecture, three hours; discussion, one hour; laboratory, one hour. Preparation: at least three and one-half years of high school mathematics (including some coordinate geometry and trigonometry). Requisite: successful completion of Mathematics Diagnostic Test or course 1 with grade of C- or better. Not open for credit to students with credit for course 31A. Intended for students who still need to review precalculus material (laboratory) while starting calculus. Differential calculus and applications; introduction to integration. P/NP or letter grading.

Course Information:

The following schedule, with textbook sections and topics, is based on 26 lectures. The remaining classroom meetings are for leeway, reviews, and two midterm exams. These are scheduled by the individual instructor. Often there are reviews and midterm exams about the beginning of the 4th and 8th weeks of instruction, plus reviews for the final exam.

In certain cases (such as for coordinated classes), it may be possible to give midterm exams during additional class meetings scheduled in the evening. This has the advantage of saving class time. A decision on whether or not to do this must be made well in advance so that the extra exam sessions can be announced in the Schedule of Classes. Instructors wishing to consider this option should consult the mathematics undergraduate office for more information.

The goal of Math31AB is to provide a solid introduction to differential and integral calculus in one variable. The course is aimed at students in engineering, the physical sciences, mathematics, and economics. It is also recommended for students in the other social sciences and the life sciences who want a more thorough foundation in one-variable calculus than that provided by Math 3.

Students in 31AB are expected to have a strong background in precalculus mathematics, including polynomial functions, trigonometric functions, and exponential and logarithm functions. In order to enroll in 31A, students must either take and pass the Mathematics Diagnostic Test at the specified minimum performance level, or take and pass Math 1 at UCLA with a grade of C- or better.

Most students entering the 31-32-33 sequence at UCLA have taken a calculus course in high school and enter directly into Math 31B, for which there is no enforced prerequisite.

The course 31A covers the differential calculus and integration through the fundamental theorem of calculus. The first part of course 31B is concerned with integral calculus and its applications. The rest of the course is devoted to infinite sequences and series.

Single-variable calculus is traditionally treated at many universities as a three-quarter or two-semester course. Thus Math 31AB does not cover all of the topics included in the traditional single-variable course. The main topics that are omitted are parametric curves and polar coordinates, which are treated at the beginning of 32A.

Ample tutoring support is available for students in the course, including the walk-in tutoring service of the Student Mathematics Center.

Math 31A is not offered in the Spring Quarter. Students wishing to start calculus in the Spring may take 31A through University Extension in the Spring or in the Summer.

Please note: Students who are in the College of Letters and Science who will be enrolled at UCLA in Spring and wish to enroll in Extension simultaneously should meet with a College Counselor about whether they will be able to receive credit for the course because of concurrent enrollment restrictions: Concurrent Enrollment Information.

Textbook(s)

J. Rogawski, Calculus: Late Transcendentals Single Variable Calculus Fourth Edition, W.H. Freeman & CO

ALEKS by McGraw-Hill Education, UCLA Calculus Preparation

(a) Limits should be presented very informally with an emphasis on working with their properties: the “Limit Laws”.

(b) Section 6.2 should be restricted to the topic of average value.

Outline update: 3/15 R. Brown

General Course Outline

Course Description

(4) Lecture, three hours; discussion, one hour. Requisite: course 31A with a grade of C- or better. Not open for credit to students with credit for course 3B. Transcendental functions; methods and applications of integration; sequences and series. P/NP or letter grading.

Course Information:

The following schedule, with textbook sections and topics, is based on 26 lectures. The remaining classroom meetings are for leeway, reviews, and two midterm exams. These are scheduled by the individual instructor. Often there are reviews and midterm exams about the beginning of the 4th and 8th weeks of instruction, plus reviews for the final exam.

In certain cases (such as for coordinated classes), it may be possible to give midterm exams during additional class meetings scheduled in the evening. This has the advantage of saving class time. A decision on whether or not to do this must be made well in advance so that the extra exam sessions can be announced in the Schedule of Classes. Instructors wishing to consider this option should consult the mathematics undergraduate office for more information

The goal of Math31AB is to provide a solid introduction to differential and integral calculus in one variable. The course is aimed at students in engineering, the physical sciences, mathematics, and economics. It is also recommended for students in the other social sciences and the life sciences who want a more thorough foundation in one-variable calculus than that provided by Math 3.

Students in 31AB are expected to have a strong background in precalculus mathematics, including polynomial functions, trigonometric functions, and exponential and logarithm functions. In order to enroll in 31A, students must either take and pass the Mathematics Diagnostic Test at the specified minimum performance level, or take and pass Math 1 at UCLA with a grade of C- or better.

Most students entering the 31-32-33 sequence at UCLA have taken a calculus course in high school and enter directly into Math 31B, for which there is no enforced prerequisite.

The course 31A covers the differential calculus and integration through the fundamental theorem of calculus. The first part of course 31B is concerned with integral calculus and its applications. The rest of the course is devoted to infinite sequences and series.

Single-variable calculus is traditionally treated at many universities as a three-quarter or two-semester course. Thus Math 31AB does not cover all of the topics included in the traditional single-variable course. The main topics that are omitted are parametric curves and polar coordinates, which are treated at the beginning of 32A.

Ample tutoring support is available for students in the course, including the walk-in tutoring service of the Student Mathematics Center.

Math 31A is not offered in the Spring Quarter. Students wishing to start calculus in the Spring may take 31A through University Extension in the Spring or in the Summer.

Please note: Students who are in the College of Letters and Science who will be enrolled at UCLA in Spring and wish to enroll in Extension simultaneously should meet with a College Counselor about whether they will be able to receive credit for the course because of concurrent enrollment restrictions: Concurrent Enrollment Information.

Textbook(s)

J. Rogawski, Calculus: Late Transcendentals Single Variable Calculus Fourth Edition, W.H. Freeman & CO

(a) The inverse trigonometric functions can be limited to the sine, cosine and tangent and the hyperbolic functions to the sine and cosine.

(b) The amount of time devoted to techniques of integration should be determined by the instructor

(c ) The topic of improper integrals is closely related to that of sequences and series, so it makes sense to postpone it until just before the chapter devoted to those subjects

(d) Although the formal definition of the limit is not included in Math 31A, the corresponding topic in the setting of infinite sequences is appropriate for 31B.

Outline update: 3/15 R. Brown

General Course Outline

Course Description

(4) Lecture, three hours; discussion, one hour. Enforced requisite: course 31A with grade of B or better. Honors course parallel to course 31B. P/NP or letter grading.

Course Information:

The following schedule, with textbook sections and topics, is based on 26 lectures. The remaining classroom meetings are for leeway, reviews, and two midterm exams. These are scheduled by the individual instructor. Often there are reviews and midterm exams about the beginning of the 4th and 8th weeks of instruction, plus reviews for the final exam.

In certain cases (such as for coordinated classes), it may be possible to give midterm exams during additional class meetings scheduled in the evening. This has the advantage of saving class time. A decision on whether or not to do this must be made well in advance so that the extra exam sessions can be announced in the Schedule of Classes. Instructors wishing to consider this option should consult the mathematics undergraduate office for more information

The goal of Math31AB is to provide a solid introduction to differential and integral calculus in one variable. The course is aimed at students in engineering, the physical sciences, mathematics, and economics. It is also recommended for students in the other social sciences and the life sciences who want a more thorough foundation in one-variable calculus than that provided by Math 3.

Students in 31AB are expected to have a strong background in precalculus mathematics, including polynomial functions, trigonometric functions, and exponential and logarithm functions. In order to enroll in 31A, students must either take and pass the Mathematics Diagnostic Test at the specified minimum performance level, or take and pass Math 1 at UCLA with a grade of C- or better.

The course 31A covers the differential calculus and integration through the fundamental theorem of calculus. The first part of course 31B is concerned with integral calculus and its applications. The rest of the course is devoted to infinite sequences and series.

Single-variable calculus is traditionally treated at many universities as a three-quarter or two-semester course. Thus Math 31AB does not cover all of the topics included in the traditional single-variable course. The main topics that are omitted are parametric curves and polar coordinates, which are treated at the beginning of 32A.

Ample tutoring support is available for students in the course, including the walk-in tutoring service of the Student Mathematics Center.

Math 31A is not offered in the Spring Quarter. Students wishing to start calculus in the Spring may take 31A through University Extension in the Spring or in the Summer.

Please note: Students who are in the College of Letters and Science who will be enrolled at UCLA in Spring and wish to enroll in Extension simultaneously should meet with a College Counselor about whether they will be able to receive credit for the course because of concurrent enrollment restrictions: Concurrent Enrollment Information.

Textbook(s)

J. Rogawski, Single Variable Calculus, (2nd Edition) , W.H. Freeman & CO

Click for information about an electronic version of the book

(A) Instructor should select from the material from Sections 8.3 – 8.5, since it is not possible to cover all integration techniques adequately in two lectures. In Section 8.3, you may require students to know how to integrate powers of (sin x)^m(cos x)^n and otherwise be able to evaluate trigonometric integrals given reduction formulas or a table of integrals. You may limit integration of rational functions to distinct linear factors or at most double linear factors, but require that students recognize the form of a partial fraction expansion in general (without having to find it).

(B) Students should learn to apply the Error Bound for Taylor polynomials.

� Although the :”epsilon-delta” definition of limits in section 2.8 is not covered in Math 31A, the “epsilon-N” definition of limits is appropriate for this course.

Outline update: 8/07, description updated 9/14

General Course Outline

Course Description

(4) Lecture, three hours; discussion, one hour. Enforced requisite for course 32AH: course 31A with grade of B or better. Honors sequence parallel to courses 32A. P/NP or letter grading.

Course Information:

The following schedule, with textbook sections and topics, is based on 26 lectures. The remaining classroom meetings are for leeway, reviews, and two midterm exams. These are scheduled by the individual instructor.

Math 32AB is a traditional multivariable calculus course sequence for mathematicians, engineers, and physical scientists.

The course 32A treats topics related to differential calculus in several variables, including curves in the plane, curves and surfaces in space, various coordinate systems, partial differentiation, tangent planes to surfaces, and directional derivatives. The course culminates with the solution of optimization problems by the method of Lagrange multipliers.

The course 32B treats topics related to integration in several variables, culminating in the theorems of Green, Gauss and Stokes. Each of these theorems asserts that an integral over some domain is equal to an integral over the boundary of the domain. In the case of Green’s theorem the domain is an area in the plane, in the case of Gauss’s theorem the domain is a volume in three-dimensional space, and in the case of Stokes’ theorem the domain is a surface in three-dimensional space. These theorems are generalizations of the fundamental theorem of calculus, which corresponds to the case where the domain is an interval on the real line. The theorems play an important role in electrostatics, fluid mechanics, and other areas in engineering and physics where conservative vector fields play a role.

Textbook(s)

G. Folland, Advanced Calculus, Pearson.

Outline update: O.Radko, 7/16

General Course Outline

Course Description

(4) Lecture, three hours; discussion, one hour. Requisite: course 31A with a grade of C- or better. Introduction to differential calculus of several variables, vector field theory. P/NP or letter grading.

Course Information:

The following schedule, with textbook sections and topics, is based on 26 lectures. The remaining classroom meetings are for leeway, reviews, and two midterm exams. These are scheduled by the individual instructor.

Math 32AB is a traditional multivariable calculus course sequence for mathematicians, engineers, and physical scientists.

The course 32A treats topics related to differential calculus in several variables, including curves in the plane, curves and surfaces in space, various coordinate systems, partial differentiation, tangent planes to surfaces, and directional derivatives. The course culminates with the solution of optimization problems by the method of Lagrange multipliers.

The course 32B treats topics related to integration in several variables, culminating in the theorems of Green, Gauss and Stokes. Each of these theorems asserts that an integral over some domain is equal to an integral over the boundary of the domain. In the case of Green’s theorem the domain is an area in the plane, in the case of Gauss’s theorem the domain is a volume in three-dimensional space, and in the case of Stokes’ theorem the domain is a surface in three-dimensional space. These theorems are generalizations of the fundamental theorem of calculus, which corresponds to the case where the domain is an interval on the real line. The theorems play an important role in electrostatics, fluid mechanics, and other areas in engineering and physics where conservative vector fields play a role.

Textbook(s)

J. Rogawski, Calculus: Late Transcendentals Multivariable, Fourth Edition, W. H. Freeman

1) Some problems may refer to polar coordinates. One only need inform the students that x = r cos q and y = r sin q. Polar coordinates are done in detail in 32B in order to help with areas, double integrals, etc.

2) The first two of Kepler’s Laws should be done if at all possible.

3) There are two lectures on limits and continuity, in order to introduce the concepts of open, closed sets, etc.

Outline update: R. Brown, 9/14

General Course Outline

Course Description

(4) Lecture, three hours; discussion, one hour. Requisite: courses 31B & 32A with a grade of C- or better. Introduction to integral calculus of several variables, line and surface integrals. P/NP or letter grading.

Course Information:

The following schedule, with textbook sections and topics, is based on 26 lectures. The remaining classroom meetings are for leeway, reviews, and two midterm exams. These are scheduled by the individual instructor.

Math 32AB is a traditional multivariable calculus course sequence for mathematicians, engineers, and physical scientists.

The course 32A treats topics related to differential calculus in several variables, including curves in the plane, curves and surfaces in space, various coordinate systems, partial differentiation, tangent planes to surfaces, and directional derivatives. The course culminates with the solution of optimization problems by the method of Lagrange multipliers.

The course 32B treats topics related to integration in several variables, culminating in the theorems of Green, Gauss and Stokes. Each of these theorems asserts that an integral over some domain is equal to an integral over the boundary of the domain. In the case of Green’s theorem the domain is an area in the plane, in the case of Gauss’s theorem the domain is a volume in three-dimensional space, and in the case of Stokes’ theorem the domain is a surface in three-dimensional space. These theorems are generalizations of the fundamental theorem of calculus, which corresponds to the case where the domain is an interval on the real line. The theorems play an important role in electrostatics, fluid mechanics, and other areas in engineering and physics where conservative vector fields play a role.

Textbook(s)

J. Rogawski, Calculus: Late Transcendentals Multivariable, Fourth Edition, W. H. Freeman

1) The section on polar coordinates should be used to emphasize areas inside polar curves, as a preview of polar double integrals and cylindrical coordinates, and not arcane polar coordinate curves.

2) The sections on Green’s Theorem, Stokes’ Theorem, and the Divergence Theorem are extremely important. Time must be left to cover these sections in detail.

Outline update: R. Brown, 8/12

General Course Outline

Course Description

(4) Lecture, three hours; discussion, one hour. Enforced requisite for 32BH: courses 31B and 32A, with grades of B or better. Honors sequence parallel to courses 32B. P/NP or letter grading.

Course Information:

The following schedule, with textbook sections and topics, is based on 26 lectures. The remaining classroom meetings are for leeway, reviews, and two midterm exams. These are scheduled by the individual instructor.

Math 32AB is a traditional multivariable calculus course sequence for mathematicians, engineers, and physical scientists.

The course 32A treats topics related to differential calculus in several variables, including curves in the plane, curves and surfaces in space, various coordinate systems, partial differentiation, tangent planes to surfaces, and directional derivatives. The course culminates with the solution of optimization problems by the method of Lagrange multipliers.

The course 32B treats topics related to integration in several variables, culminating in the theorems of Green, Gauss and Stokes. Each of these theorems asserts that an integral over some domain is equal to an integral over the boundary of the domain. In the case of Green’s theorem the domain is an area in the plane, in the case of Gauss’s theorem the domain is a volume in three-dimensional space, and in the case of Stokes’ theorem the domain is a surface in three-dimensional space. These theorems are generalizations of the fundamental theorem of calculus, which corresponds to the case where the domain is an interval on the real line. The theorems play an important role in electrostatics, fluid mechanics, and other areas in engineering and physics where conservative vector fields play a role.

Textbook(s)

G. Folland, Advanced Calculus, Pearson.

Outline update: O.Radko, 7/16

General Course Outline

Course Description

(4) Lecture, three hours; discussion, one hour. Requisite: course 3B, 31B or 32A with a grade of C- or better. Introduction to linear algebra: systems of linear equations, matrix algebra, linear independence, subspaces, bases and dimension, orthogonality, least-squares methods, determinants, eigenvalues and eigenvectors, matrix diagonalization, and symmetric matrices. P/NP or letter grading.

Course Information:

The following schedule, with textbook sections and topics, is based on 26 lectures. The remaining classroom meetings are for leeway, reviews, and two midterm exams. These are scheduled by the individual instructor.

The purpose of Math 33A is to provide mathematicians, engineers, physical scientists, and economists with an introduction to the basic ideas of linear algebra in n-dimensional Euclidean space. Abstract vector spaces are not covered; they are treated in Math 115A.

Students in the course should have covered the following topics in previous high school and college mathematics courses:

  • solving linear systems of equations,
  • matrices, matrix multiplication,
  • two-by-two and three-by-three determinants,
  • complex numbers,
  • complex polynomials, the fundamental theorem of algebra.

    This background material is reviewed in the course, though briefly.

Textbook(s)

O. Bretscher, Linear Algebra, 5th Ed., Prentice Hall.

Since the syllabus includes some important material for engineers at the end of the course (Chapter 8), the pacing of lectures is particularly important. Some time can be saved by synopsising the properties of determinants and leaving the details to the students. The students are already familiar with two-by-two and three-by-three determinants.

Most of the students are already familiar with matrix multiplication.

The ad hoc definition of “linear transformation” in Section 2.1 should be replaced by the correct definition, which can then be related to the definition given in the textbook.

Chapter 4 and Section 5.5 are generally not covered.

The QR decomposition in Section 5.2 is important for the engineers.

Most students will have seen the polar form of complex numbers given in Section 7.5 (in high school), but most students will not have seen the exponential form (Euler’s formula) in previous courses.

Positive-definite matrices (Section 8.2) and the singular-value decomposition (Section 8.3) are very important for the engineers.

Outline update: T. Gamelin, 3/04, C. Jung, 9/14

General Course Outline

Course Description

(4) Lecture, three hours; discussion, one hour. Enforced requisite: course 3B or 31B or 32A with grade of B or better. Introduction to linear algebra: systems of linear equations, matrix algebra, linear independence, subspaces, bases and dimension, orthogonality, least-squares methods, determinants, eigenvalues and eigenvectors, matrix diagonalization, and symmetric matrices. Honors course parallel to course 33A. P/NP or letter grading.

Course Information:

The following schedule, with textbook sections and topics, is based on 26 lectures. The remaining classroom meetings are for leeway, reviews, and two midterm exams. These are scheduled by the individual instructor.

The purpose of Math 33A is to provide mathematicians, engineers, physical scientists, and economists with an introduction to the basic ideas of linear algebra in n-dimensional Euclidean space. Abstract vector spaces are not covered; they are treated in Math 115A.

Students in the course should have covered the following topics in previous high school and college mathematics courses:

  • solving linear systems of equations,
  • matrices, matrix multiplication,
  • two-by-two and three-by-three determinants,
  • complex numbers,
  • complex polynomials, the fundamental theorem of algebra.

    This background material is reviewed in the course, though briefly.

    The topics in linear algebra that are covered in Math 33A include:

  • systems of linear equations, associated matrix equations,
  • row reduction of a matrix,
  • linear transformations,
  • invertible matrices,
  • subspaces, linear independence, bases, dimension,
  • row space, column space, rank-nullity theorem,
  • determinants,
  • orthogonality, orthonormal bases,
  • orthogonal matrices,
  • Gram-Schmidt process, QR factorization,
  • least-squares approximation, normal equations,
  • eigenvalues, eigenvectors, similarity, diagonalization,
  • applications to discrete dynamical systems,
  • diagonalization of symmetric matrices,
  • applications to quadratic forms, singular value decomposition.
Textbook(s)

O. Bretscher, Linear Algebra, 5th Ed., Prentice Hall. Check Schedule of classes for most current textbook.

Since the syllabus includes some important material for engineers at the end of the course (Chapter 8), the pacing of lectures is particularly important. Some time can be saved by synopsising the properties of determinants and leaving the details to the students. The students are already familiar with two-by-two and three-by-three determinants.

Most of the students are already familiar with matrix multiplication.

The ad hoc definition of “linear transformation” in Section 2.1 should be replaced by the correct definition, which can then be related to the definition given in the textbook.

Chapter 4 and Section 5.5 are generally not covered.

The QR decomposition in Section 5.2 is important for the engineers.

Most students will have seen the polar form of complex numbers given in Section 7.5 (in high school), but most students will not have seen the exponential form (Euler’s formula) in previous courses.

Positive-definite matrices (Section 8.2) and the singular-value decomposition (Section 8.3) are very important for the engineers.

Outline update: T. Gamelin, 3/04

General Course Outline

Course Description

(4) Lecture, three hours; discussion, one hour. Requisite: course 31B with a grade of C- or better. Highly recommended: course 33A. First-order, linear differential equations; second-order, linear differential equations with constant coefficients; power series solutions; linear systems. P/NP or letter grading.

Course Information:

In addition, two hour exams should be given. These exams are usually given in the fourth and eighth week; the exact time they are scheduled is up to the instructor. 24 of the 26 lectures are specified.

The course Math 33B has evolved over the years. At one time it was a course in infinite series, including power series solutions of differential equations. In the Fall of 1997 the infinite series course was renumbered as Math 31C, in hopes that students would take it earlier, but by the Fall of 1998 the course was back at the end of the calculus sequence with its original label Math 33B.

In 2004, the courses Math 33A and 33B were reorganized. The differential equations portion of Math 33A was moved to Math 33B, so that Math 33A is now a course in linear algebra and Math 33B is now a course in differential equations. The topics currently treated in Math 33B are as follows:

Introduction to first order differential equations

  • second order linear differential equations with constant coefficients
  • power series solutions of second order differential equations
  • linear systems of differential equations
Textbook(s)

Polking, Differential Equations, 2nd Ed., Prentice Hall.

Footnotes

1. On page 22 of the Polking text the author has a section on a ‘numerical solver”. He writes “We assume that each of our readers has access to a computer.” He also adds We assume that you have access to a solver [computer and software] that will draw direction fields, provide numerical solutions?, and plot solutions.” The author goes into detail on the vibrating spring example, pages 137-140. You might wish to put this off until 4.4 when he returns to the topic.

3. Math 33B does not have math 33A, linear algebra, as a prerequisite. This was a concession to the Chemistry Department. You will have to give a short, fast explanation of eigenvalues and eigenvectors.

4. All eiganvalue possibilities are discussed in this section.

Outline update: 9/14

General Course Outline

Course Description

(4) Lecture, three hours; discussion, one hour. Requisites: Math 31AB; 32AB; 33A; One of Statistics 10-15, Statistics 20; PIC 10A. This course gives an introduction to data-driven mathematical modeling and to combining data analysis with mechanistic modeling of phenomena from various applications. Topics include model formulation, data visualization, nondimensionalization and order-of-magnitude physics, introduction to discrete and continuous dynamical systems, and introduction to discrete and continuous stochastic models. The class will include examples drawn from many fields and practice problems from the Mathematical Contest in Modeling. P/NP or Letter grading.

Course Information:

Students will learn the basic principles of mathematical modeling and data visualization. The focus will be on mechanistic models, but in a data-driven and problem-driven way. They will get hands-on practice with problems from the Mathematical Contest in Modeling, including an in-depth exploration through a final project.

The grade will be determined based on homework, quizzes, a midterm, a final project (done in groups, with both written and oral components), and class participation.

 
Textbook(s)

Required:

(MS) “A Course in Mathematical Modeling”, by Douglas D. Mooney and Randall J. Swift
(Tufte) “The Visual Display of Quantitative Information” (2nd edition), by Edward R. Tufte

Important Supplementary Booklets:

(BFG) “Math Modeling & Getting Started”, by K. M. Bliss, K. R. Fowler, and B. J. Galluzzo (a free booklet from the Society for Industrial and Applied Mathematics)
(BGKL) “Math Modeling: Computing & Communicating”, by K. M. Bliss, B. J. Galluzzo, K. R. Kavanagh, & R. Levy (a free booklet from the Society for Industrial and Applied Mathematics)
Supplementary material through past Mathematical Contest in Modeling questions and handouts on specific topics.

 
General Course Outline

Course Description

(4) Lecture, three hours; discussion, one hour. Requisites: courses 31A, 31B. Not open for credit to students with credit for course 180 or 184. Discrete structures commonly used in computer science and mathematics, including sets and relations, permutations and combinations, graphs and trees, induction. P/NP or letter grading.

Course Information:

The following schedule, with textbook sections and topics, is based on 26 lectures. The remaining classroom meetings are for two midterm exams and review. These are scheduled by the individual instructor. Often there are midterm exams about the beginning of the fourth and eighth weeks of instruction.

Math 61 has two goals. One goal is the introduction of certain basic mathematical concepts, such as equivalence relations, graphs, and trees. The other goal is to introduce non-mathematicians to abstraction and rigor in mathematics. Finite graphs are well-suited to this purpose. Exercises asking for simple proofs are assigned where appropriate.

Roughly half of the students in Math 61 are engineering students in Computer Science. Of the remaining students, many are in business and economics. Relatively few (about one out of ten) have declared as Mathematics majors.

Math 61 is offered each quarter. Recent enrollment statistics are given in the following table.

Textbook(s)

R. Johnsonbaugh, Discrete Mathematics (8th Edition) , Prentice-Hall.

Outline update: I. Neeman 7/12

General Course Outline

Course Description

(4) Lecutre, 3 hours; Discussion, 1 hour. Requisites: courses 31A, 31B. Introduction to probability through applications and examples. Topics include laws of large numbers, statistics, chance trees, conditional probability, Bayes? rule, continuous and discrete random variables, jointly distributed random variables, multivariate normal and conditional distributions. In depth discussion of betting schemes in gambling, occurrence of rare events, coincidences and statistical predictions. P/NP or letter grading.

Course Information:

The course introduces a list of standard probabilistic problems and analyzes them in detail within the formalism of probability as a mathematical discipline. At the end of the course, the students will be able to demonstrate their understanding of the foundations and basic facts of probability as a mathematical discipline and apply them to resolve questions with probabilistic content.

 
Textbook(s)

Tijms, H. Understanding Probability, Chance Rules in Everyday Life, 3rd Edition. Cambridge University Press, 2012

 
General Course Outline

Course Description

(3) Seminar, two hours; fieldwork (classroom observation and participation), two hours. Introduce students to K-12 mathematics activity in the United States. Cultivate interest in teaching through exploration of the sequences of mathematical content and habits of mind taught in these grades. Analyze sequences of topics in the current California State Standards in Mathematics (CCSS-M), the mathematical structures that underlie these sequences and cognitive aspects of learning mathematics. Experience with professional mathematician?s habits of mind outlined in the California Standards for Mathematical Practice (including proof and mathematical modeling) and effective strategies for teaching mathematics to diverse student groups. Fieldwork in local mathematics classrooms arranged by Cal Teach program. P/NP (undergraduates) or S/U (graduates) grading.

 
Textbook(s)

National Research Counci How Students Learn: Mathematics in the Classroom. Washington, DC: The National Academies Press (https://doi.org/10.17226/11101), 2005.
Other reading materials to be provided

Online Resources:
National Governors Association & Council of Chief State School Officers Common Core State Standards for Mathematics (http://www.corestandards.org/Math/), 2010.

 
General Course Outline

Course Description

(3) (Formerly numbered Mathematics 71SL.) Seminar, two hours; fieldwork (classroom observation and participation), two hours. Facilitate development of professional mathematical and pedagogical understandings required to teach California?s K-5 mathematics curriculum. Exploration of K-5 mathematics, practice effective teaching strategies for all learners, and discuss current research and standards in math education. Fieldwork in local mathematics classrooms (observation and presenting lesson plan) arranged by Cal Teach program. P/NP (undergraduates) or S/U (graduates) grading.

 
Textbook(s)

Berlinghoff & Gouvea Math Through The Ages: A Gentle History for Teachers and Others. Oxton Publishers & MAA, 2015.
Other reading materials to be provided

 
General Course Outline

Course Description

(1) Tutorial, three hours. Limited to students in College Honors Program. Designed as adjunct to lower-division lecture course. Individual study with lecture course instructor to explore topics in greater depth through supplemental readings, papers, or other activities. May be repeated for maximum of 4 units. Individual honors contract required. Honors content noted on transcript. Letter grading.

General Course Outline

Course Description

(Formerly numbered 192.) Lecture, three hours. Requisite: course 31B with grade of C- or better. Problem-solving techniques and mathematical topics useful as preparation for Putnam Examination and similar competitions. Continued fractions, inequalities, modular arithmetic, closed form evaluation of sums and products, problems in geometry, rational functions and polynomials, other nonroutine problems. Participants expected to take Putnam Examination. P/NP grading.

Math 100 is a course in problem solving. The problems are more varied and unexpected than in a typical undergraduate mathematics course. Often an original or imaginative step is required. Some variations of topics from year to year are expected. Topics may include: explicit summations of series, spherical trigonometry, advanced Euclidean geometry, elementary number theory, combinatorial problems, inequalities, continued fractions. There is a lot of classroom discussion. Homework is assigned regularly and makes a large contribution to the course grade. One three-hour final is given.

Textbook(s)

Problem-Solving Through Problems by Loren C. Larson

Updated: 10/14 C. Manolescu

General Course Outline

Course Description

Lecture, three hours. Prerequisite: Math 100 or significant experience with mathematical competitions. Advanced problem solving techniques and mathematical topics useful as preparation for Putnam Competition. Problems in abstract algebra, linear algebra, number theory, combinatorics, probability, real and complex analysis, differential equations, Fourier analysis. Regular practice tests given, similar in difficulty to the Putnam Competition. Enrollment is by permission of the instructor, based on a selection test or past Putnam results. May be repeated for maximum of 12 units. P/NP or letter grading.

 
Textbook(s)

R. Gelca & T. Andreescu. Putnam and Beynd, Springer Verlag

Updated 10/14: C. Manolescu

General Course Outline

Course Description

(2) (Formerly Math 330.) Seminar, one hour; fieldwork (classroom observation and participation), two hours. Requisites: courses 31A, 31B, 32A, 33A, 33B. Course 103A is enforced requisite to 103B, which is enforced requisite to 103C. Observation, participation, or tutoring in mathematics classes at middle school and secondary levels. May be repeated for credit. P/NP (undergraduates) or S/U (graduates) grading.

General Information: The goal of this course is to expose prospective mathematics teachers to the field of secondary mathematics education. Among other things, students will observe classroom teachers, read mathematics education literature, do middle and high school level mathematics from an adult perspective, discuss mathematics education issues, and explore effective teaching strategies. Reflection and critical analysis, through written assignments and discussions, are key components of the course. Seminars for 103A and 103C meet seven times per quarter. Seminars for 103B meet six times per quarter and students attend the annual Curtis Center Conference. Active participation is expected.

 
Math 103A: General Course Outline
 
Assignments and Grading
  • Observations and Reflections: Observe at least 2 class periods between each meeting (a total of 12 class periods per quarter).  Six class periods will be in middle school classrooms and six in high school classrooms. Observe 12 different teachers. Following each observation, complete your assignment on the Online Information System (OIS), WeTeach website. (See Observation Protocol and Observation Reflection Guidelines).
  • Readings: Read the assigned articles for each session and write a reflection and critical analysis on each piece. (See Reading Reflection Guidelines.)
  • Problems of the Week (POW): Complete the POW assigned for each session. (See POW Guidelines.)
  • Attendance/Participation: Attend all scheduled classes and participate in discussions and critical analyses on observations, readings, mathematics problems, and other relevant education issues. Each student will facilitate one of the following discussions: observation reflection, reading reflection, or POW.
  • Mini-Portfolio: Compile a portfolio of personal highlights and reflections of the course. (See Mini-Portfolio Guidelines.)
  • Assignments will only be counted when turned in during class after discussions have taken place. Exception: all observation reflections are to be entered on the OIS WeTeach website the day before class. They will be graded on a scale of 1-3 (1—needs revision; 2—acceptable, meets requirement; 3—excellent, exceeds requirement). Work receiving a one (1) must be revised to receive credit. Math 103A is a two-unit pass/no pass course.
  • Grading: In order to pass the course, students must complete 80% of all assignments with a grade of at least 75%.
Summary of Course Requirements
Weekly Topics (Emphasis on the Teacher in the Classroom)

Session 1: General Overview

  • Mathematics Problem
  • Mathematics Autobiography
  • Introductions; Overview of Math 103A
  • Assignments
    • Write a one-page paper on the following: What are the characteristics of an effective teacher?
    • Problem of the Week
    • Observation Questions:
    • Look around the room. Describe the physical environment. Look at space, lighting, and safety. What might you change to make the environment more conducive to learning?
    • Describe the affective environment. Is this environment one that values every student? Would you be comfortable in this environment?
    • How and when does the teacher take roll? What other housekeeping items does the teacher need to take care of?
    • Are there any external interruptions (summons, PA system, bells)? How does the teacher react to these interruptions? How might you handle them?

Session 2: The Classroom Environment and Housekeeping

  • Mathematics Problem
  • POW
  • Observation Reflections
  • Reading Reflections
  • Assignments
    • Classroom Management That Works: Research-Based Strategies for Every Teacher by Robert J. Marzano, Jana S. Marzano and Debra J. Pickering (Chapter 1)
    • Problem of the Week
    • Observation Questions: The focus of the second set of observations is on classroom management. Include in your analysis whether you think the methods used for classroom management are effective. Why or why not? Think of the following questions as you observe.
    • How is the class managed? Who has control? What strategies does the teacher use to control the classroom? Which strategies are effective and which are not?
    • What type of classroom management do you think might be effective? What might you do to improve on the management of the particular class?

Session 3: Classroom Management

  • Mathematics Problem
  • POW
  • Observation Reflections
  • Reading Reflections
  • Assignments
    • Mathematics Framework for California Public Schools. Chapter 4.

      http://csmp.ucop.edu/downloads/cmp/MathematicsFramework2005.pdf

      and Kinds of Learning from http://www.indiana.edu/~idtheory/methods/m1d.html
    • Problem of the Week
    • Observation Questions: The focus for the third set of observations is methods of instruction.  Include in your analysis whether you think the instructors’ method of instruction is effective.  Why or why not? Think of these questions as you observe.
    • What types of instruction is being used? For example, lecture, demonstration, whole group instruction, or cooperative groups.
    • Who does most of the talking in the classroom?
    • Are all the students engaged in the learning?
    • Does the instruction include a focus on basic skills (especially procedural), conceptual understanding, and problem solving?
    • What is the level of cognitive learning? Memorization, understanding, and/or application?

Session 4: Methods of Instruction

  • Mathematics Problem
  • POW
  • Observation Reflections
  • Reading Reflections
  • Assignments
    • Koehler, Mary S. and Prior, Millie. (1993). “Classroom interactions: The heartbeat of the teaching/learning process.” In Research Ideas for the Classroom: Middle Grades Mathematics edited by Douglas T. Owens, pp. 280-298, NCTM: Reston VA.
    • Problem of the Week
    • Observation Questions: The focus for the fourth set of observations is teacher questioning and teacher wait time.
    • Does the teacher ask students questions during class? What types of questions are being asked? Does it focus on yes/no answers? Does it require students to think?
    • If none of the students answer the question, does the teacher answer the question? What might you do if none of the students answer a question that you ask?
    • If a teacher asks for any answer, and a student answers, does the teacher ask why the answer is correct?
    • Does the teacher give wait time? Is there more wait time given to some students over other students?

Session 5: Teacher Questioning; Wait Time

  • Mathematics Problem
  • POW
  • Observation Reflections
  • Reading Reflections
  • Assignments
  • Khisty, Lena L. and Viego, Gabriel. (1999) “Challenging conventional wisdom: A case study”. In Changing the Faces of Mathematics, Perspectives on Latinos edited by Luis Ortiz-Franco, Norma G. Hernandez, and Yolanda De La Cruz, pp. 71-80. NCTM: Reston VA.
  • Problem of the Week
  • Observation Questions: The focus for the fifth set of observations is teacher-student interaction.  Think of the following questions as you observe.
  • How does the teacher interact with the students?
  • Does the teacher interact with all of his/her students?
  • What is the nature of the interaction?
  • Which students does the teacher target?
  • Is there discourse between teacher and student? Between student and student? What is the nature of the discourse?

Session 6: (Teacher-Student Interaction)

  • Mathematics Problem
  • POW
  • Observation Reflections
  • Reading Reflections
  • Assignments
  • Brown, Catherine A. and Baird, Jayne. (1993). “Inside the teacher: knowledge, beliefs, and attitudes” In Research Ideas for the Classroom: High School Mathematics edited by Patricia S. Wilson, pp. 245-259, NCTM: Reston VA.
  • NOTE: Mini-portfolio due Session 7.
  • Observation Questions:
  • What evidence indicates teachers have the deep understanding of the mathematics they teach? What examples can you cite?
  • As part of teacher content knowledge, teachers need to understand their students’ thinking about mathematics. What examples can you cite?

Session 7: Teacher Content Knowledge and Final Reflection

  • Mathematics Problem
  • Observation Reflections
  • Reading Reflections
  • Mini-portfolio Reflections
  • My Comments
Observation Protocol
  • Observe at least 2 classroom periods between each UCLA class session for a minimum total of 12 sessions for the quarter.
  • No more than two students are to observe a specific classroom at the same time.
  • Consider carpooling to the school sites.
  • Check in at the main office each time you visit a school site and receive a visitor’s pass.
  • As representatives of UCLA and as prospective teachers, and under the guidance of the UCLA instructor, you must be professional at all times when dealing with school staff and secondary students. This includes being polite and courteous, being non-judgmental, and dressing appropriately.
  • All observation reflections are to be entered into the OIS.
  • Ask the teacher you observe to sign the Observation Record form.
  • You will be provided focus questions for each of the observations.  The reflections are to address these questions as part of the observation.
Observation Reflection Guidelines
  • Must be entered in the WeTeach website: https://tepd.ucop.edu/weteach.
  • Must use proper spelling, punctuation, and grammar.
  • Must address the focus question as part of the observation and be reflective.
  • Must include your description and your reflection/analysis.
Reading Reflection and Critical Analysis Expectations
  • Reflections and critically analyses are to be type written, approximately one to two pages in length, using 12-point type, single-spaced. The document can be sent electronically (Word Document) to the instructor prior to seminar.
  • Reflections are to reflect professional writing and academic language, including use of proper spelling, punctuation, and grammar.
  • Reflections and critical analyses are to address the following:
  • At least two ideas you gained from the reading.
  • At least one question that arose for you while reading this piece.
  • A general reflection and critical analysis on the reading as a whole (e.g., do you agree or disagree with the author? Why or why not?)
Problem of the Week Write-Up
  • Solve the problem using multiple methods.
  • Write a brief narrative on how you approached the problem and how you solved it describing your processes. Include any challenges that you faced and how you addressed them.
Mini-Portfolio Guidelines
  • Include your Mathematical Autobiography.
  • Include the assignment “What are the characteristics of an effective teacher?
  • Select one piece of writing from each of the following: Problem of the Day, Problem of the Week, Observation Reflection, and Reading Reflection/Critical Analysis to be included in the mini-portfolio.
  • Include your Observation Record Form.
  • Write a final reflection of the course. Include why you selected the pieces of work, what you learned and gained from the course and what questions you have remaining about teaching.

Comments

Outline update: 4/08

For more information, please contact
Student Services, ugrad@math.ucla.edu.
General Course Outline

Course Description

(2) (Formerly Math 330.) Seminar, one hour; fieldwork (classroom observation and participation), two hours. Requisites: courses 31A, 31B, 32A, 33A, 33B. Course 103A is enforced requisite to 103B, which is enforced requisite to 103C. Observation, participation, or tutoring in mathematics classes at middle school and secondary levels. May be repeated for credit. P/NP (undergraduates) or S/U (graduates) grading.

General Information: The goal of this course is to expose prospective mathematics teachers to the field of secondary mathematics education. Among other things, students will observe classroom teachers, read mathematics education literature, do middle and high school level mathematics from an adult perspective, discuss mathematics education issues, and explore effective teaching strategies. Reflection and critical analysis, through written assignments and discussions, are key components of the course. Seminars for 103A and 103C meet seven times per quarter. Seminars for 103B meet six times per quarter and students attend the annual Curtis Center Conference. Active participation is expected.

 
Math 103B: General Course Outline
 
Assignments and Grading
  • Observations and Reflections: Observe at least 2 class periods between each meeting (a total of 10 class periods). During this quarter, select two middle school or high school students per class and follow them the whole quarter for a total of 4 students.  Following each observation, complete your assignment on the Online Information System (OIS), WeTeach website. (See Observation Protocol and Observation Reflection Guidelines).
  • Readings: Read the assigned articles for each session and write a reflection and critical analysis on each piece. (See Reading Reflection Guidelines.)
  • Problems of the Week (POW): Complete the POW assigned for each session. (See POW Guidelines.)
  • Attendance/Participation: Attend all scheduled classes and participate in discussions and critical analyses on observations, readings, mathematics problems, and other relevant education issues. Each student will facilitate one of the following discussions: observation reflection, reading reflection, or POW.
  • Mini-Portfolio: Compile a portfolio of personal highlights and reflections of the course. (See Mini-Portfolio Guidelines.)
  • Assignments will only be counted when turned in during class after discussions have taken place. Exception: all observation reflections are to be entered on the OIS WeTeach website the day before class. They will be graded on a scale of 1-3 (1—needs revision; 2—acceptable, meets requirement; 3—excellent, exceeds requirement). Work receiving a one (1) must be revised to receive credit. Math 103B is a two-unit pass/no pass course.
  • Grading: In order to pass the course, students must complete 80% of all assignments with a grade of at least 75%.
Summary of Course Requirements
Weekly Topics (Emphasis on the Teacher in the Classroom)

Session 1: General Overview

  • Mathematics Problems

    Erickson, T. (1995) United We Solve. eeps media: Oakland, CA.
  • Introductions and Overview to Math 103B
  • Assignments
  • Lumsden, Linda (1994). “Student motivation to learn.” ERIC Digest, June 1994.

    https://scholarsbank.uoregon.edu/dspace/bitstream/1794/3313/1/digest092.pdf
  • Problem of the Week: The Month of January
  • Observation Questions: The focus of the first set of observations is student motivation. Think of these questions as you observe your two students in each class
  • Describe the students’ actions in the classroom.
  • Does the student appear motivated? Is the motivation intrinsic or extrinsic? What evidence supports your statements?
  • If the student is not motivated, what might you have done differently to motivate the student?
  • Suppose you were a student in this class. What could the teacher have done to increase your motivation?

Session 2: Student Motivation

  • Mathematics Problem

    Complete the Pre-Case Discussion Worksheet of What is Pi, Anyway?
  • POW
  • Observation Reflections
  • Reading Reflections
  • Assignments
    • Harvard Mathematics Case Development Project, 1994-97, Katherine K. Merseth, Project Investigator and Joan B. Karp, Project Manager. What is Pi, Anyway?
    • Problem of the Week
    • Observation Questions: The focus of the second set of observations is student understanding of mathematics. Think of these questions as you observe your two students in each class
    • Describe the students’ actions in the classroom.
    • Does the student understand the mathematics that is being taught?  What evidence gives you the indication that the student understands the mathematics?
    • What might you do differently to increase student understanding? It is not necessarily the case that “telling” the student will result in student understanding.

Session 3: Student Understanding

  • Mathematics Problem:
  • POW
  • Observation Reflections
  • Reading Reflections
  • Assignments
    • Thompson, D. R. and Rubenstein, R. N. “Learning mathematics vocabulary: Potential pitfalls and instructional strategies.” Mathematics Teacher (93) (October 2000): 568-574.
    • Problem of the Week
    • Observation Questions: The focus of the third set of observations is students’ mathematical literacy. Think of these questions as you observe your two students in each class
    • Describe the students’ actions in the classroom.
    • Does the student use the academic language of mathematics? Give evidence of mathematical language being used.
    • Is there opportunity for students to develop mathematical literacy?
    • How does the teacher assist students in developing mathematical literacy?
    • What might you do differently to increase mathematical literacy?

Session 4: Students? Mathematical Literacy

  • Mathematics Problem: Handshake problem
  • POW
  • Observation Reflections
  • Reading Reflections
  • Assignments
    • Lumsden, Linda (1997). “Expectations for students.” ERIC Digest, July 1997. https://scholarsbank.uoregon.edu/dspace/handle/1794/3338
    • Problem of the Week: Alternate Handshake Problem
    • Observation Questions: The focus of the fourth set of observations is student engagement and student expectations. Think of these questions as you observe your two students in each class
    • Describe the students’ actions in the classroom.
    • Are the students engaged in the mathematics? What evidence do you observe that indicates the students are engaged?
    • Is the student expected to achieve in this class?  Why or why not?
    • What might you do different to engage the students?
    • What strategies would you use to set high expectations of students? Going to students and informing them that you expect them to achieve doesn’t always work.

Session 5: Student Engagement/Student Expectations

  • Mathematics Problem
  • POW
  • Observation Reflections
  • Reading Reflections
  • Assignments
    • Problem of the Week: Hiker Problem
    • No reading assignment
    • Mini-portfolio due next week
    • Observation Questions: The focus of the fifth set of observations is student/student and student/teacher interaction. Think of these questions as you observe your two students in each class
    • Describe the students’ actions in the classroom.
    • Are the students interacting with the teacher? In what way?
    • Are the students interacting with other students? Is it about mathematics or something other than mathematics? What type of interaction is occurring among your two students and the other students?
    • What might you do different to ensure more interaction between teacher and student and among students?
    • What type of discourse and interaction do you expect in your future classroom?

Session 6: Student/Student and Student/Teacher Interaction and Final Reflection

  • Mathematics Problem
  • POW
  • Observation Reflections
  • Mini-Portfolio Due: discussion/reflection of portfolio 

Joint Math/Ed Breakfast and Mathematics for Teaching Conference: Winter Quarter

Observation Protocol
  • Observe at least 2 classroom periods between each UCLA class session for a minimum total of 10 sessions for the quarter.
  • No more than two UCLA students are to observe a specific classroom at the same time.
  • Consider carpooling to the school sites.
  • Check in at the main office each time you visit a school site and receive a visitor’s pass.
  • As representatives of UCLA and as prospective teachers, and under the guidance of the UCLA instructor, you must be professional at all times when dealing with school staff and secondary students. This includes being polite and courteous, being non-judgmental, and dressing appropriately.
  • All observation reflections are to be entered into the OIS.
  • Ask the teacher you observe to sign the Observation Record form.
  • You will be provided focus questions for each of the observations.  The reflections are to address these questions as part of the observation. 
Observation Reflection Guidelines
  • Must be entered in the WeTeach website: https://tepd.ucop.edu/weteach.
  • Must use proper spelling, punctuation, and grammar.
  • Must address the focus question as part of the observation and be reflective.
  • Must include your description and your reflection/analysis.  
Reading Reflection and Critical Analysis Expectations
  • Reflections and critically analyses are to be type written, approximately one to two pages in length, using 12-point type, single-spaced. The document can be sent electronically (Word Document) to the instructor prior to seminar.
  • Reflections are to reflect professional writing and academic language, including use of proper spelling, punctuation, and grammar.
  • Reflections and critical analyses are to address the following:
    • At least two ideas you gained from the reading.
    • At least one question that arose for you while reading this piece.
    • A general reflection and critical analysis on the reading as a whole (e.g., do you agree or disagree with the author? Why or why not?)
Problem of the Week Write-Up
  • Solve the problem using multiple methods.
  • Write a brief narrative on how you approached the problem and how you solved it describing your processes. Include any challenges that you faced and how you addressed them.
Mini-Portfolio Guidelines
  • Write a short paragraph on what you believe are the three most important characteristics of an effective mathematics teacher, building upon the assignment from last quarter but revising it based upon your experiences.
  • Write a one-page letter of recommendation of yourself in the third person.
  • Select one piece of writing from each of the following: Problem of the Day, Problem of the Week, Observation Reflection, and Reading Reflection/Critical Analysis to be included in the mini-portfolio.
  • Include your Observation Record Form.
  • Write a final reflection of the course. Include why you selected the pieces of work, what you learned and gained from the course and what questions you have remaining about teaching.

Comments

Outline update: 10/11

For more information, please contact
Student Services, ugrad@math.ucla.edu.
 
General Course Outline

Course Description

(2) (Formerly Math 330.) Seminar, one hour; fieldwork (classroom observation and participation), two hours. Requisites: courses 31A, 31B, 32A, 32B, 33A, 33B. Course 103A is an enforced requisite to 103B, which is enforced requisite to 103C. Observation, participation, or tutoring in mathematics classes at middle school and secondary levels. May be repeated for credit. P/NP (undergraduates) or S/U (graduates) grading.

General Information: The goal of this course is to expose prospective mathematics teachers to the field of secondary mathematics education. Among other things, students will observe classroom teachers, read mathematics education literature, do middle and high school level mathematics from an adult perspective, discuss mathematics education issues, and explore effective teaching strategies. Reflection and critical analysis, through written assignments and discussions, are key components of the course. Seminars for 103A and 103C meet seven times per quarter. Seminars for 103B meet six times per quarter and students attend the annual Curtis Center Conference. Active participation is expected.

 
Math 103C: General Course Outline
 
Assignments and Grading
  • Observations and Reflections: Observe at least 2 class periods between each meeting (a total of 12 class periods per quarter).  During this quarter, observe in multiple classrooms, both middle school and high school. Following each observation, complete your assignment on the Online Information System (OIS), WeTeach website. (See Observation Protocol and Observation Reflection Guidelines).
  • Readings: Read the assigned articles for each session and write a reflection and critical analysis on each piece. (See Reading Reflection Guidelines.)
  • Problems of the Week (POW): Complete the POW assigned for each session. (See POW Guidelines.)
  • Attendance/Participation: Attend all scheduled classes and participate in discussions and critical analyses on observations, readings, mathematics problems, and other relevant education issues. Each student will facilitate one of the following discussions: observation reflection, reading reflection, or POW.
  • Mini-Portfolio: Compile a portfolio of personal highlights and reflections of the course. (See Mini-Portfolio Guidelines.)
  • Assignments will only be counted when turned in during class after discussions have taken place. Exception: all observation reflections are to be entered on the OIS WeTeach website the day before class. They will be graded on a scale of 1-3 (1—needs revision; 2—acceptable, meets requirement; 3—excellent, exceeds requirement). Work receiving a one (1) must be revised to receive credit. Math 103C is a two-unit pass/no pass course.
  • Grading: In order to pass the course, students must complete 80% of all assignments with a grade of at least 75%.
Summary of Course Requirements
Weekly Topics (Emphasis on the Teacher in the Classroom)

Session 1: Overview of Assessment

  • Mathematics Problem:
  • Overview of Math 103C
  • Assignments
  • Assigned Reading: California Department of Education (2006). Mathematics framework for California public schools, kindergarten through grade twelve (220-227). Sacramento, CA: CDE Press.
  • Problem of the Week
  • Observation Questions: The focus of the first set of observations is to determine what students and teachers think about assessment in general. As the teacher and several students the following questions and record the responses.
  • Ask the teacher: what is assessment? What types of entry-level assessment did you administer at the beginning of this semester? What information do you use to determine your students’ prior knowledge? What types of assessment do you generally use in a mathematics classroom?
  • Ask three students per class: what is assessment? What types of assessment do you know of, both in class and out of class assessment?

    As you observe, do you see any assessment going on (of any kind)?

Session 2: What Is Assessment?

  • Mathematics Problem
  • POW
  • Observation Reflections
  • Reading Reflections
  • Assignments
    • Assigned Reading


      Mathematical Sciences Education Board, National Research Council (1993). Measuring What Counts. (1-13) Washington, DC: National Academic Press.
    • Problem of the Week: 12 dot problem
    • Observation Questions: The focus of the second set of observations is to determine what students and teachers think about classroom assessment.  Ask the teacher and several students the following questions and record the responses.
    • Ask the teacher: what is classroom assessment? How do you assess your students? How often do you assess your students?
    • Ask three students per class: what is classroom assessment? How does your teacher assess you? How often does your teacher assess you? Do the assessment results reflect what mathematics you know or learned?


      Also, as you observe, do you see any assessment going on (of any kind)?

Session 3: Formative Assessment

  • Mathematics Problem
  • POW
  • Observation Reflections
  • Reading Reflections
  • Assignments
    • Assigned Reading


      Mathematical Sciences Education Board, National Research Council (1993). Measuring What Counts. (67-90) Washington, DC: National Academic Press.
    • Problem of the Week (POW due May 7, 2007): Tic-Tac-X Continued
    • Observation Questions: The focus of the third set of observations is formative assessment. One type of formative assessment is the ongoing assessment that teachers need to make during their instruction.
    • Describe the mathematics lesson for the day.
    • Does the student understand the mathematics that is being taught?  What evidence gives you the indication that the student understands the mathematics?
    • What is the teacher doing to determine what the student is learning? What types of assessment is the teacher using?
    • What would you do to assess what students are learning that day?

Session 4: High Stakes Tests?for the Student

  • Mathematics Problem (CAHSEE problems)
  • POW
  • Observation Reflections
  • Reading Reflections
  • Assignments
    • Assigned Reading: CAHSEE Mathematics Study Guide. You will be assigned to one of five strands in groups of two or three. Select two released items to share with the whole group during the next class session.
    • Problem of the Week: Bag of Marbles
    • Observation Questions: The focus of the fourth set of observations is high stakes tests. Make sure that you are observing in classrooms where there are students in 10th grade or above.  Ask the following questions of the teacher and students (10th and 11th graders) and record their responses:
    • Ask the teacher: what do the students at this school think about the California High School Exit Exam (CAHSEE)? What do you do to prepare students for this exam?
    • Ask three students per class: Have you taken the CAHSEE? If yes, did you pass? Was it easy or hard? Did anyone help you prepare for the exam?

Session 5: High Stakes Tests?for the School

  • Mathematics Problem (CAHSEE and CST problems)
  • POW
  • Observation Reflections
  • Reading Reflections
  • Assignments
    • Assigned Reading: Articles focusing on standardized testing.
    • Problem of the Week
    • Observation Questions: The focus of the fifth set of observations is high stakes tests for the school. These observations can be in middle or high schools.  Ask the following questions of the teacher and students and record their responses:
    • Ask the teacher: what do the students at this school think about the standardized testing (particularly the California Standards Test [CST])? What do you think about the CST? What do you do to prepare students for the CST?
    • Ask three students per class: What do you think about the CST? If they are not sure what it is, it’s the testing they and the whole school take yearly that generally occurs in April or May (and not the CAHSEE).

Session 6: Summative Assessment

  • Mathematics Problem
  • POW
  • Observation Reflections
  • Reading Reflections
  • Assignments
    • Assigned Reading

      Silverman-Welty-Lyon (1992). Case study: Sarah Hanover. In Case Studies for Teacher Problem Solving, McGraw-Hill, Inc.
    • Assigned Reading: Go to http://csmp.ucop.edu/cmp/resources/CHASEE.html and scroll to the CST released questions. You are assigned to select either grades 6, 7, or geometry to read. Also look at the problems and select five different types of problems. Work out the problems. Look at the distractors and determine what students might do to obtain the distractors.
    • Problem of the Week (POW due May 28, 2007):
    • Observation Questions: The focus of the fourth set of observations is summative assessment—end of course exams and grading. Select two students per class and ask them the following:
    • What grade are you getting in this class? What grade do you think you should get?
    • What is used to determine your final grade?
    • How much does homework count toward the grade?

      As you observe the students you asked, do you think their behavior indicates that the grades they think they will or should get reflect what they know?

Session 7: Final Reflection

  • Mathematics Problem
  • POW
  • Observation Reflections
  • Reading Reflections
  • Mini-Portfolio Due: discussion/reflection of portfolio
  • Final Comments and Pizza
Observation Protocol
  • Observe at least 2 classroom periods between each UCLA class session for a minimum total of 12 sessions for the quarter.
  • No more than two UCLA students are to observe a specific classroom at the same time.
  • Consider carpooling to the school sites.
  • Check in at the main office each time you visit a school site and receive a visitor’s pass.
  • As representatives of UCLA and as prospective teachers, and under the guidance of the UCLA instructor, you must be professional at all times when dealing with school staff and secondary students. This includes being polite and courteous, being non-judgmental, and dressing appropriately.
  • All observation reflections are to be entered into the OIS.
  • Ask the teacher you observe to sign the Observation Record form.
  • You will be provided focus questions for each of the observations.  The reflections are to address these questions as part of the observation.
Observation Reflection Guidelines
  • Must be entered in the WeTeach website: https://tepd.ucop.edu/weteach. I do not need copies.
  • Must use proper spelling, punctuation, and grammar.
  • Must address the focus question as part of the observation and be reflective.
  • Must include your description and your reflection/analysis.
Reading Reflection and Critical Analysis Expectations
  • Reflections and critically analyses are to be type written, approximately one to two pages in length, using 12-point type, single-spaced. The document can be sent electronically (Word Document) to the instructor prior to seminar.
  • Reflections are to reflect professional writing and academic language, including use of proper spelling, punctuation, and grammar.
  • Reflections and critical analyses are to address the following:
    • At least two ideas you gained from the reading.
    • At least one question that arose for you while reading this piece.
    • A general reflection and critical analysis on the reading as a whole (e.g., do you agree or disagree with the author? Why or why not?)
Problem of the Week Write-Up
  • Solve the problem using multiple methods.
  • Write a brief narrative on how you approached the problem and how you solved it describing your processes. Include any challenges that you faced and how you addressed them.
Mini-Portfolio Guidelines
  • Select one piece of writing from each of the following: Problem of the Day, Problem of the Week, Observation Reflection, and Reading Reflection.
  • Write a final reflection of the course. Include why you selected the pieces of work, what you learned and gained from the course and what questions you have remaining about teaching.

Comments

Outline update: 10/11

For more information, please contact
Student Services, ugrad@math.ucla.edu.
 
General Course Outline

Course Description

(4) Lecture, four hours; fieldwork, 30 minutes. Requisites: courses 110A (or 117), 120A (or 123), and 131A, with grades of C- or better. Course 105A is requisite to 105B, which is requisite to 105C. Mathematical knowledge and research-based pedagogy needed for teaching key geometry topics in secondary school, including axiomatic systems, measure, and geometric transformations. Introduction to professional standards and current research for teaching secondary school mathematics. Letter grading.

Description

Math 105A is a team-taught course that aims to help you connect your undergraduate coursework to the secondary mathematics curriculum and to deepen your understanding of the mathematics you will teach. This course also aims to teach you new mathematics content using various research-based instructional strategies. It emphasizes problem solving and student presentation of solutions.

Math 105A also aims to teach you a variety of research based instructional strategies, skill with the technology and software used in schools, and skill with various models for secondary mathematics topics. This course includes readings of current math education research as well as state and national content standards for the teaching of secondary mathematics. It also requires observation in local secondary schools.

General Information

  • senior mathematics majors with demonstrated success in the above-mentioned upper- division mathematics coursework and demonstrated interest in mathematics teaching                                   
  • graduate students in the GSE&IS Teacher Education Program

Required Texts/Supplies:

Z. Usiskin, A. Perssini, E.A. Marchisotto, and D. Stanley, Mathematics for High School Teachers, An Advanced Perspective. (2003) Prentice Hall, Saddle River, NJ

The Mathematics Framework for California Public Schools  (available at http://www.cde.ca.gov/ci/ma/cf/documents/mathfrwkcomplete.pdf)



The National Council of Teachers of Mathematics Principles and Standards for School Mathematics (sign up for online access to this document at http://standardstrial.nctm.org/triallogin.asp)

 

J.D. Bransford, A.L. Brown, R.R. Cocking, Eds., How People Learn: Brain, Mind, Experience, and School, Expanded Edition. (2000) National Research Council, Washington, D.C.



J. Stigler, J. Hiebert, The Teaching Gap (1999) The Free Press, NY



TI 84 Plus graphing calculator (distributed by TI at a required training on October 28th)

Instructor Information:
Bruce Rothschild

Office: MS 6175

310) [82]5-3174

blr@math.ucla.edu   
Heather Dallas

Office: MS 2341

(310) [82]5-1702

dallas@math.ucla.edu   

Meeting Information:
Mondays, 4 – 8 PM, MS 6221. Usually there will be a 20 minute break for nourishment.

Course Grade Components:

Problems of the Week and Homework Exercises: 25%

Several homework exercises (mostly from the text) will be assigned each week, with solutions due the following week.  When a POW is assigned, a complete solution, including a thorough description of the solution process, and problem solving strategies used is due the following week.



Quizzes: 10%

A brief quiz covering straightforward mathematics material recently covered in the course will be given at the start of each class.

           

Reading Summaries: 10%                                                                            

Readings of math education research or professional standards will be assigned regularly, with brief summaries and reflections due via online submission.

Course and Lesson Design: 10%

Students will work in groups to write a course and unit plan for an Algebra or Pre-algebra course which is in accordance with the California Framework and the NCTM Standards and Principles.

Secondary Classroom Observations: 10%

Students will observe for 5 hours in an assigned secondary classroom. Observation notes will be taken. Students will write a Standards in Practice paper identifying the California Standards and NCTM Principles and Standards covered in the observed classes. 



Final: 25%

A final exam will be given in the first two quarters of the sequence and a final portfolio will be due in the third quarter of the sequence. Collection of the elements for the final portfolio will be incorporated throughout the three quarter 105 sequence, including work on a paper tracing the development of a mathematical idea through the secondary and undergraduate curriculum.



Participation: 10%

Attendance and promptness to class, active pursuit of problem solutions, presentation of problem solutions to fellow students, and engagement in and completion of the work of the model lessons plans will be assessed.

Please note the following policies:

No late assignments will be accepted.

A student who misses a final exam may receive an incomplete grade in the course providing the student (i) has completed all other grade components at a passing level, (ii) has an ironclad excuse (such as a medical emergency), and (iii), if possible, contacts one of the instructors on or before the day of the final exam to arrange a meeting.

For each of the above content pieces, the teaching, curriculum, and assessment of the content at the secondary level are introduced and analyzed in the context of current research and recommendations.

General Course Outline

Course Description

(4) Lecture, four hours; fieldwork, 30 minutes. Requisites: courses 105A, 110A (or 117), 120A (or 123), and 131A, with grades of C- or better. Mathematical knowledge and research-based pedagogy needed for teaching key polynomial, rational, and transcendental functions and related equations in secondary school; professional standards and current research for teaching secondary school mathematics. Letter grading.

Description

Math 105B is the second quarter in a team-taught course that aims to help you connect your undergraduate coursework to the secondary mathematics curriculum and to deepen your understanding of the mathematics you will teach. It also aims to teach you new mathematics content using various research-based instructional strategies and to emphasize problem solving and student presentation of solutions.

Math 105B aims to teach you a variety of research based instructional strategies, skill with the technology and software used in schools, and skill with various models for secondary mathematics topics. The course includes readings and discussion of current math education research and requires observation in local secondary schools.

General Information

  • senior mathematics majors with demonstrated success in the above-mentioned upper- division mathematics coursework and demonstrated interest in mathematics teaching                                   
  • graduate students in the GSE&IS Teacher Education Program

Required Texts/Supplies:
Z. Usiskin, A. Perssini, E.A. Marchisotto, and D. Stanley, Mathematics for High School Teachers, An Advanced Perspective. (2003) Prentice Hall, Saddle River, NJ

 

J.D. Bransford, A.L. Brown, R.R. Cocking, Eds., How People Learn: Brain, Mind, Experience, and School, Expanded Edition. (2000) National Research Council, Washington, D.C.



J. Stigler, J. Hiebert, The Teaching Gap (1999) The Free Press, NY



TI 84 Plus graphing calculator

Instructor Information:
Bruce Rothschild

Office: MS 6175

310) [82]5-3174

blr@math.ucla.edu   
Heather Dallas

Office: MS 2341

(310) [82]5-1702

dallas@math.ucla.edu   

Meeting Information:
Mondays, 4 – 8 PM, MS 6221. Usually there will be a 20 minute break for nourishment.

Course Grade Components:

Problems of the Week and Homework Exercises: 25%

Several homework exercises (mostly from the text) will be assigned each week, with solutions due the following week.  When a POW is assigned, a complete solution, including a thorough description of the solution process, and problem solving strategies used is due the following week.



Quizzes: 10%

A brief quiz covering straightforward mathematics material recently covered in the course will be given at the start of each class.

           

Reading Summaries: 10%                                                                            

Readings of math education research will be assigned regularly, with brief summaries and reflections due via online submission.

Course and Lesson Design: 10%

Students will work in groups to write two lesson plans employing methods taught in the course. After rounds of peer and instructor edits, groups will revise and submit final drafts. 

Secondary Classroom Observations: 10%

Students will observe for 5 hours in an assigned secondary classroom. Observation notes will be taken. Students will choose one student to focus on, ask the students to complete a written response problem and subsequently interview them.  Students will write a short paper analyzing the results of the interview.

Final: 25%

A final exam will be given in the first two quarters of the sequence and a final portfolio will be due in the third quarter of the sequence. Collection of the elements for the final portfolio will be incorporated throughout the three quarter 105 sequence, including work on a paper tracing the development of a mathematical idea through the secondary and undergraduate curriculum.  A number of the portfolio components will be due at the end of the second quarter.



Participation: 10%

Attendance and promptness to class, active pursuit of problem solutions, presentation of problem solutions to fellow students (at least twice in the quarter), and engagement in and completion of the work of the model lessons will be assessed.

Please note the following policies:

No late assignments will be accepted.

A student who misses a final exam may receive an incomplete grade in the course providing the student (i) has completed all other grade components at a passing level, (ii) has an ironclad excuse (such as a medical emergency), and (iii), if possible, contacts one of the instructors on or before the day of the final exam to arrange a meeting.

 
General Course Outline

Course Description

(4) Lecture, four hours; fieldwork, 30 minutes. Requisites: courses 105A, 105B, 110A (or 117), 120A (or 123), and 131A, with grades of C- or better. Mathematical knowledge and research-based pedagogy needed for teaching key analysis, probability, and statistics topics in secondary school; professional standards and current research for teaching secondary school mathematics. Letter grading.

Description

Math 105C is the third quarter in a team-taught course that aims to help you connect your undergraduate coursework to the secondary mathematics curriculum and to deepen your understanding of the mathematics you will teach. It also aims to teach you new mathematics content using various research-based instructional strategies and to emphasize problem solving and student presentation of solutions.

Math 105C aims to teach you a variety of research based instructional strategies, skill with the technology and software used in schools, and skill with various models for secondary mathematics topics. The course includes readings and discussion of current math education research and requires observation in local secondary schools.

In Math 105C, students will complete the following performance components:

  • presentation of model lessons both in class and in a secondary schoolroom
  • presentation of a paper which traces a mathematical topic through the secondary and undergraduate curricula

General Information

  • senior mathematics majors with demonstrated success in the above-mentioned upper- division mathematics coursework and demonstrated interest in mathematics teaching                     
  • graduate students in the GSE&IS Teacher Education Program

Required Texts/Supplies:

Z. Usiskin, A. Perssini, E.A. Marchisotto, and D. Stanley, Mathematics for High School Teachers, An Advanced Perspective. (2003) Prentice Hall, Saddle River, NJ

 

J. Stigler, J. Hiebert, The Learning Gap (1999) The Free Press, NY



TI 84 Plus graphing calculator

Instructor Information:
Bruce Rothschild

Office: MS 6175

310) [82]5-3174

blr@math.ucla.edu   
Heather Dallas

Office: MS 2341

(310) [82]5-1702

dallas@math.ucla.edu   

Meeting Information:

Mondays, 4:00 – 5:50 PM, MS 6221.

Tuesdays, 4:00 – 5:50, MS 6201

Course Grade Components:

Problems of the Week and Homework Exercises: 20%

Several homework exercises (mostly from the text) will be assigned each week, with solutions due the following week.  When a POW is assigned, a complete solution, including a thorough description of the solution process, and problem solving strategies used is due the following week.



Quizzes: 10%

A brief quiz covering straightforward mathematics material recently covered in the course will be given at the start of each class.

           

Reading Summaries: 10%                                                                            

Readings of math education research will be assigned regularly, with brief summaries and reflections due via online submission.

Final portfolio: 50%

A portfolio consisting of:

  • Two lesson plans (developed in Winter quarter and improved upon during spring quarter) along with analysis of video of one of these lessons in a secondary classroom (15%)
  • Paper tracing a mathematical topic through the secondary and undergraduate curricula (15%)
  • Exemplar Work including one POW, one Reading Summary and Reflection, the Winter quarter Student Interview Project, the Fall quarter Observation paper analyzing the CA and NCTM Standards addresses in secondary classrooms, and two class activities.  A short reflection on each piece of exemplar work will be included in this portion of the portfolio. (10%)
  • Presentation to the class of the analysis of the video of the lesson and subsequent improvements to the lesson (5%)
  • Presentation to the class of the paper tracing a mathematical topic through the secondary and undergraduate curricula (5%)

Participation: 10%

Attendance and promptness to class, active pursuit of problem solutions, presentation of problem solutions to fellow students (at least twice in the quarter), and engagement in and completion of the work of the model lessons will be assessed.

——————————————————————————————————————

Please note the following policies:

No late assignments will be accepted.

A student who misses their final presentations may receive an incomplete grade in the course providing the student (i) has completed all other grade components at a passing level, (ii) has an ironclad excuse (such as a medical emergency), and (iii), if possible, contacts one of the instructors on or before the day of the presentation to arrange a meeting.

 
General Course Outline

Course Description

Lecture, three hours; discussion, one hour. Requisites: courses 31A, 31B, 32A. Roots of modern mathematics in ancient Babylonia and Greece, including place value number systems and proof. Development of algebra through Middle Ages to Fermat and Abel, invention of analytic geometry and calculus. Selected topics. P/NP or letter grading.

General Information. Math 106 focuses on the development of mathematics and its role in society through the ages. The presentation of topics in the course varies according to the instructor. However, there are four major topic areas that form the core of the course.

1. The history of numeral systems through various early civilizations, and the development of place-value systems of numeration (the sexagesimal system of the Babylonians, and our own Hindu-Arabic system).

2. The origins and evolution of the axiomatic method and proof in mathematics, beginning with the Greeks (Thales, Eudoxus, Euclid), with major advances in the nineteenth century when calculus was placed on a rigorous footing through the efforts of Cauchy, Weierstrass, and others.

3. The evolution of symbolic algebra, which includes solution of equations and the work of Diophantus, Cardano, Viete, and Descartes (who gave us the unknown quantity “x”).

4. The development of the calculus, which demonstrated its power by explaining the motion of the planets.

Math 106 is particularly recommended for students who are planning to teach in middle school and high school, since many of the topics treated in the course are directly related to the mathematics taught in the schools.

Textbook(s)

Stillwell, J., Mathematics and its History, 3rd Ed., Springer.

General Course Outline

Course Description

(4) Lecture, three hours; discussion, one hour. Requisite: course 115A. Not open for credit to students with credit for course 117. Ring of integers, integral domains, fields, polynomial domains, unique factorization. Honors sequence parallel to courses 110A. P/NP or letter grading.

Course Information:

The following schedule anticipates 24 days of instruction, with 2 holidays and 4 days for exams and reviews. If there is extra time, one could do section 6.3 – the structure of R/I when I is prime or maximal and/or section 4.6 – irreducibility in R[x] or C[x].

Math 110ABC is the basic undergraduate course sequence in abstract algebra. Math 110A covers rings and fields, while Math 110B treats group theory.

An honors course sequence 110AH-110BH runs parallel to 110A-110B, however the order of topics is juxtaposed. Math 110AH is devoted to the study of group theory. Groups are a mathematical expression of symmetry and are vitally important in many areas of Mathematics, e.g. Number Theory, Topology and Geometry. Group theory plays an important role in Physics, especially in Quantum Theory. The course will cover the definition and properties of groups as well as the structure of finite groups. The honors sequence in Algebra is essential for those interested in pursuing pure mathematics at any higher level as well as being one of the most interesting and challenging mathematics courses at UCLA.

Math 110AH covers group theory in the Fall, while Math 110BH in the Winter covers rings and fields. Math 110BH is devoted to Ring Theory, especially commutative rings. Rings play a central role in many areas of mathematics, e.g. Algebra, Algebraic Geometry and Number Theory. The highlight of the course is the theory of modules over Principal Ideal Domains with applications to the theory of canonical forms in linear algebra and to the structure of finitely generated abelian groups.

Thus a student who has a difficult time surviving group theory in Math 110AH in the Fall can continue in Math 110B in the Winter and learn group theory really well. In the reverse direction, no student has ever taken 110A in the Fall and switched to 110BH in the Winter, though there always could be a first. The prerequisite for 110BH is 110AH or consent of instructor.

Students who take 110AH but not 110BH can take 110A or 117.

Math 110C, offered in the Spring, is designed for students completing either the 110A-110B or the 110AH-110BH sequence. Math 110C covers Galois theory. This is the theory initiated by Evariste Galois (killed in a duel at age 21), which laid an abstract foundation for proving the theorem of N. Abel (died of consumption at age 27) that the general quintic equation is not solvable by radicals.

Textbook(s)

R. Elman, Lectures on Abstract Algebra
Book is Subject to Change Without Notice

Outline update: Gieseker, D. 12/15

General Course Outline

Course Description

(4) Lecture, three hours; discussion, one hour. Requisite: course 115A. Not open for credit to students with credit for course 117. Ring of integers, integral domains, fields, polynomial domains, unique factorization. P/NP or letter grading.

Course Information:

The following schedule anticipates 24 days of instruction, with 2 holidays and 4 days for exams and reviews. If there is extra time, one could do section 6.3 – the structure of R/I when I is prime or maximal and/or section 4.6 – irreducibility in R[x] or C[x].

Math 110ABC is the basic undergraduate course sequence in abstract algebra. Math 110A covers rings and fields, while Math 110B treats group theory.

An honors course sequence 110AH-110BH runs parallel to 110A-110B, however the order of topics is juxtaposed. Math 110AH is devoted to the study of group theory. Groups are a mathematical expression of symmetry and are vitally important in many areas of Mathematics, e.g. Number Theory, Topology and Geometry. Group theory plays an important role in Physics, especially in Quantum Theory. The course will cover the definition and properties of groups as well as the structure of finite groups. The honors sequence in Algebra is essential for those interested in pursuing pure mathematics at any higher level as well as being one of the most interesting and challenging mathematics courses at UCLA.

Math 110AH covers group theory in the Fall, while Math 110BH in the Winter covers rings and fields. Math 110BH is devoted to Ring Theory, especially commutative rings. Rings play a central role in many areas of mathematics, e.g. Algebra, Algebraic Geometry and Number Theory. The highlight of the course is the theory of modules over Principal Ideal Domains with applications to the theory of canonical forms in linear algebra and to the structure of finitely generated abelian groups.

Thus a student who has a difficult time surviving group theory in Math 110AH in the Fall can continue in Math 110B in the Winter and learn group theory really well. In the reverse direction, no student has ever taken 110A in the Fall and switched to 110BH in the Winter, though there always could be a first. The prerequisite for 110BH is 110AH or consent of instructor.

Students who take 110AH but not 110BH can take 110A or 117.

Math 110C, offered in the Spring, is designed for students completing either the 110A-110B or the 110AH-110BH sequence. Math 110C covers Galois theory. This is the theory initiated by Evariste Galois (killed in a duel at age 21), which laid an abstract foundation for proving the theorem of N. Abel (died of consumption at age 27) that the general quintic equation is not solvable by radicals.

Textbook(s)

Hungerford, T.,Abstract Algebra, 3rd Ed., Brooks Col.

Outline update: 4/98

General Course Outline

Course Description

(4) Lecture, three hours; discussion, one hour. Requisite: course 110A or 117. Groups, structure of finite groups. P/NP or letter grading.

Course Information:

The course should cover essentially the material between pages 160 and 280 (excluding the section on the simplicity of the appropriate alternating groups; one can come back to this if there is enough time). If there is not enough time, the material at the beginning is more important than the material at the end.

Math 110ABC is the basic undergraduate course sequence in abstract algebra. Math 110A covers rings and fields, while Math 110B treats group theory.

An honors course sequence 110AH-110BH runs parallel to 110A-110B, however the order of topics is juxtaposed. Math 110AH is devoted to the study of group theory. Groups are a mathematical expression of symmetry and are vitally important in many areas of Mathematics, e.g. Number Theory, Topology and Geometry. Group theory plays an important role in Physics, especially in Quantum Theory. The course will cover the definition and properties of groups as well as the structure of finite groups. The honors sequence in Algebra is essential for those interested in pursuing pure mathematics at any higher level as well as being one of the most interesting and challenging mathematics courses at UCLA.

Math 110AH covers group theory in the Fall, while Math 110BH in the Winter covers rings and fields. Math 110BH is devoted to Ring Theory, especially commutative rings. Rings play a central role in many areas of mathematics, e.g. Algebra, Algebraic Geometry and Number Theory. The highlight of the course is the theory of modules over Principal Ideal Domains with applications to the theory of canonical forms in linear algebra and to the structure of finitely generated abelian groups.

Thus a student who has a difficult time surviving group theory in Math 110AH in the Fall can continue in Math 110B in the Winter and learn group theory really well. In the reverse direction, no student has ever taken 110A in the Fall and switched to 110BH in the Winter, though there always could be a first. The prerequisite for 110BH is 110AH or consent of instructor.

Students who take 110AH but not 110BH can take 110A or 117.

Math 110C, offered in the Spring, is designed for students completing either the 110A-110B or the 110AH-110BH sequence. Math 110C covers Galois theory. This is the theory initiated by Evariste Galois (killed in a duel at age 21), which laid an abstract foundation for proving the theorem of N. Abel (died of consumption at age 27) that the general quintic equation is not solvable by radicals.

Textbook(s)

Hungerford, T., Abstract Algebra, 3rd Ed., Brooks Col.

Outline update: 4/98

General Course Outline

Course Description

110B. Algebra (Honors). (4) Lecture, three hours; discussion, one hour. Requisite: course 110A or 117. Groups, structure of finite groups. P/NP or letter grading.

Course Information:

The course should cover essentially the material between pages 160 and 280 (excluding the section on the simplicity of the appropriate alternating groups; one can come back to this if there is enough time). If there is not enough time, the material at the beginning is more important than the material at the end.

Math 110ABC is the basic undergraduate course sequence in abstract algebra. Math 110A covers rings and fields, while Math 110B treats group theory.

An honors course sequence 110AH-110BH runs parallel to 110A-110B, however the order of topics is juxtaposed. Math 110AH is devoted to the study of group theory. Groups are a mathematical expression of symmetry and are vitally important in many areas of Mathematics, e.g. Number Theory, Topology and Geometry. Group theory plays an important role in Physics, especially in Quantum Theory. The course will cover the definition and properties of groups as well as the structure of finite groups. The honors sequence in Algebra is essential for those interested in pursuing pure mathematics at any higher level as well as being one of the most interesting and challenging mathematics courses at UCLA.

Math 110AH covers group theory in the Fall, while Math 110BH in the Winter covers rings and fields. Math 110BH is devoted to Ring Theory, especially commutative rings. Rings play a central role in many areas of mathematics, e.g. Algebra, Algebraic Geometry and Number Theory. The highlight of the course is the theory of modules over Principal Ideal Domains with applications to the theory of canonical forms in linear algebra and to the structure of finitely generated abelian groups.

Thus a student who has a difficult time surviving group theory in Math 110AH in the Fall can continue in Math 110B in the Winter and learn group theory really well. In the reverse direction, no student has ever taken 110A in the Fall and switched to 110BH in the Winter, though there always could be a first. The prerequisite for 110BH is 110AH or consent of instructor.

Students who take 110AH but not 110BH can take 110A or 117.

Math 110C, offered in the Spring, is designed for students completing either the 110A-110B or the 110AH-110BH sequence. Math 110C covers Galois theory. This is the theory initiated by Evariste Galois (killed in a duel at age 21), which laid an abstract foundation for proving the theorem of N. Abel (died of consumption at age 27) that the general quintic equation is not solvable by radicals.

Textbook(s)

Dummit and Foote, Abstract Algebra, 3rd Ed., Wiley & Sons.
Book is Subject to Change Without Notice

Outline update: 9/14

General Course Outline

Course Description

Lecture, three hours; discussion, one hour. Requisites: courses 110A, 110B. Field extensions, Galois theory, applications to geometric constructions, and solvability by radicals.

General Information. Math 110ABC is the basic undergraduate course sequence in abstract algebra. Math 110A covers rings and fields, while Math 110B treats group theory.

An honors course sequence 110AH-110BH runs parallel to 110A-110B, however the order of topics is juxtaposed. Math 110AH is devoted to the study of group theory. Groups are a mathematical expression of symmetry and are vitally important in many areas of Mathematics, e.g. Number Theory, Topology and Geometry. Group theory plays an important role in Physics, especially in Quantum Theory. The course will cover the definition and properties of groups as well as the structure of finite groups. The honors sequence in Algebra is essential for those interested in pursuing pure mathematics at any higher level as well as being one of the most interesting and challenging mathematics courses at UCLA.

Math 110AH covers group theory in the Fall, while Math 110BH in the Winter covers rings and fields. Math 110BH is devoted to Ring Theory, especially commutative rings. Rings play a central role in many areas of mathematics, e.g. Algebra, Algebraic Geometry and Number Theory. The highlight of the course is the theory of modules over Principal Ideal Domains with applications to the theory of canonical forms in linear algebra and to the structure of finitely generated abelian groups.

Thus a student who has a difficult time surviving group theory in Math 110AH in the Fall can continue in Math 110B in the Winter and learn group theory really well. In the reverse direction, no student has ever taken 110A in the Fall and switched to 110BH in the Winter, though there always could be a first. The prerequisite for 110BH is 110AH or consent of instructor.

Students who take 110AH but not 110BH can take 110A or 117.

Math 110C, offered in the Spring, is designed for students completing either the 110A-110B or the 110AH-110BH sequence. Math 110C covers Galois theory. This is the theory initiated by Evariste Galois (killed in a duel at age 21), which laid an abstract foundation for proving the theorem of N. Abel (died of consumption at age 27) that the general quintic equation is not solvable by radicals.

General Course Outline

Course Description

Lecture, three hours; discussion, one hour. Prerequisites: courses 110A or 117, and 115A. Divisibility, congruences, Diophantine analysis, selected topics in the theory of primes, algebraic number theory, Diophantine equations.

General Information. Number theory is among the oldest and broadest branches of mathematics. It has roots going back to ancient babylonic cuneiform tablets, and it is the subject of several books in Euclid’s Elements. Number theory has played an important role in the development of mathematics. Today number theory cuts across virtually every field of contemporary mathematics.

The most important mathematical event of the past decade has been the resolution of a famous problem that had been around since Fermat stated that a certain Diophantine equation (Fermat’s equation, x^n+y^n=z^n for n larger that 2) does not have any positive integer solutions. The assertion defied numerous proof attempts over a period of 400 years, until recently it was proved as a result of work of Andrew Wiles and other mathematicians, using many of the modern techniques of number theory that have been developed over the past 30 years.

Perhaps the most famous remaining open problem in mathematics is the Riemann hypothesis on the location of the zeros of a specific meromorphic function, the Riemann zeta function. The location of the zeros has consequences for the asymptotic distribution of prime numbers.

Prime numbers are of great concern in connection with mathematical cryptography, entering into the construction of public key encryption codes. This illustrates how number theory ties in with various areas, ranging in this case from complex analysis to areas of current business and governmental security interest.

Because number theory is so vast, there is no one course that could serve as a good introduction to the entire field. Several possibilities for class syllabi are given, each of which focuses on a different emphasis. It may be that the course instructor will follow yet a different path.

General Course Outline

Course Description

(Formerly numbered 114A). Lecture, three hours; discussion, one hour. Requisite: course 110A or 131A or Philosophy 135. Effectively calculable, Turing computable, and recursive functions; Church/Turing thesis. Normal form theorem; universal functions; unsolvability and undecidability results. Recursive and recursively enumerable sets; relative recursiveness, polynomial-time computability. Arithmetical hierarchy. P/NP or letter grading.

General Information. If a function can be precisely defined, does that mean we can write a computer program for it? Math 114C looks at Turing machines and other models for making the concept of effective computability into genuine mathematics. The unsolvability of the halting problem demonstrates the existence of purely theoretical barriers to computability. There are decidable sets, effectively enumerable sets, and others.

Computability theory originated in ground-breaking work by Alonzo Church, Stephen Kleene, Emil Post, Alan Turing, and others, beginning in 1936. The topic is relevant to pure mathematics, theoretical computer science, and the philosophy of mathematics.

General Course Outline

Course Description

Lecture, three hours; discussion, one hour. Requisite: course 110A or 131A or Philosophy 135. Introduction to mathematical logic, aiming primarily at completeness and incompleteness theorems of Godel. Propositional and predicate logic; syntax and semantics; formal deduction; completeness, compactness, and Lowenheim/Skolem theorems. Formal number theory: nonstandard models; Godel incompleteness theorem. P/NP or letter grading.

General Course Outline

Course Description

(Formerly numbered M112.) (Same as Philosophy M134.) Lecture, three hours; discussion, one hour. Prerequisite: course 110A or 131A or Philosophy 135. Axiomatic set theory as framework for mathematical concepts; relations and functions, numbers, cardinality, axiom of choice, transfinite numbers. P/NP or letter grading.

General Information Math 114S covers the basic facts about abstract sets, including the axiom of choice, transfinite recursion, and cardinal and ordinal arithmetic. It also makes a serious effort to explain how axiomatic set theory can be viewed as a “foundation of mathematics” — and, in particular, what this means.

Math 114S is especially useful for:

Undergraduate students who are preparing for graduate study in pure mathematics and graduate students in mathematics who have not had an opportunity to learn set theory in their undergraduate work. Real analysis, in particular, looks a lot more real if you know cardinal arithmetic and understand the meaning and uses of the axiom of choice.

Undergraduate students in mathematics or computer science who are preparing for graduate study in theoretical computer science, and CS graduate students who are veering towards theory and need to understand the mathematical justification of fixpoint theorems and the like.

Philosophy students with an interest in the philosophy of mathematics and a good mathematical background.

There is a strong tradition of research in logic — especially set theory — at UCLA, and both the Mathematics and Philosophy Departments offer a rich graduate program of study in the field.

Textbook(s)

Moschovakis, Y., Notes on Set Theory, 2nd Ed., Springer.

General Course Outline

Course Description

(5) Lecture, three hours; discussion, two hours. Requisite: course 33A. Techniques of proof, abstract vector spaces, linear transformations, and matrices; determinants; inner product spaces; eigenvector theory. P/NP or letter grading.

Math 115A is a core mathematics course required of all the various mathematics majors. The course material can be regarded as an elaboration of the linear algebra already covered in Math 33A. However, the level of abstraction and the emphasis on proof technique make this a difficult course for many students. Successful students emerge from the experience not only with a better understanding of linear algebra, but also with a higher level of mathematical maturity, better equipped to deal with abstract concepts.

The material covered in Math 115A includes linear independence, bases, orthogonality, the Gram-Schmidt process, linear transformations, eigenvalues and eigenvectors, and diagonalization of matrices. These topics are all covered in Math 33A though only in the context of Euclidean space. Topics in Math 115A that go beyond Math 33A include inner product spaces, adjoint transformations, and the spectral decomposition theorem for self-adjoint operators.

Three or four sections of Math 115A are offered each term. Also, an honors version Math 115AH runs parallel to Math 115A in some quarters. The content of Math 115AH is as follows:
Vector spaces, subspaces, basis and dimension, linear transformations and matrices, rank and nullity, change of basis and similarity of matrices, inner product spaces, orthogonality and, orthonormality, Gram-Schmidt process, adjoints of linear transformations and dual spaces, quadratic forms and symmetric matrices, orthogonal and unitary matrices, diagonalization of hermitian and symmetric matrices, eigenvectors and eigenvalues, and their computation, exponentiation of matrices and application to differential equations, least squares problems, trace, determinant, canonical forms. Systems of linear equations: solvability criteria, Gaussian elimination, row-reduced form, LU decomposition.

Textbook(s)

S. Friedberg, et al, Linear Algebra, 5th Ed., Pearson.

Outline Updated: June 2005

General Course Outline

Course Description

(5) Lecture, three hours; discussion, two hours. Requisite: course 33A with grade of B or better. Techniques of proof, abstract vector spaces, linear transformations, and matrices; determinants; inner product spaces; eigenvector theory. Honors course parallel to course 115A. P/NP or letter grading.

Math 115A is a core mathematics course required of all the various mathematics majors. The course material can be regarded as an elaboration of the linear algebra already covered in Math 33A. However, the level of abstraction and the emphasis on proof technique make this a difficult course for many students. Successful students emerge from the experience not only with a better understanding of linear algebra, but also with a higher level of mathematical maturity, better equipped to deal with abstract concepts.

The material covered in Math 115A includes linear independence, bases, orthogonality, the Gram-Schmidt process, linear transformations, eigenvalues and eigenvectors, and diagonalization of matrices. These topics are all covered in Math 33A though only in the context of Euclidean space. Topics in Math 115A that go beyond Math 33A include inner product spaces, adjoint transformations, and the spectral decomposition theorem for self-adjoint operators.

Three or four sections of Math 115A are offered each term. Also, an honors version Math 115AH runs parallel to Math 115A in some quarters. The content of Math 115AH is as follows:
Vector spaces, subspaces, basis and dimension, linear transformations and matrices, rank and nullity, change of basis and similarity of matrices, inner product spaces, orthogonality and, orthonormality, Gram-Schmidt process, adjoints of linear transformations and dual spaces, quadratic forms and symmetric matrices, orthogonal and unitary matrices, diagonalization of hermitian and symmetric matrices, eigenvectors and eigenvalues, and their computation, exponentiation of matrices and application to differential equations, least squares problems, trace, determinant, canonical forms. Systems of linear equations: solvability criteria, Gaussian elimination, row-reduced form, LU decomposition.

Textbook(s)

S. Friedberg, et al, Linear Algebra, 5th Ed., Pearson.
Book is Subject to Change Without Notice

Outline Updated: June 2005

General Course Outline

Course Description

(4) Lecture, three hours; discussion, one hour. Requisite: course 115A. Linear transformations, conjugate spaces, duality; theory of a single linear transformation, Jordan normal form; bilinear forms, quadratic forms; Euclidean and unitary spaces, symmetric skew and orthogonal linear transformations, polar decomposition. P/NP or letter grading.

 
Textbook(s)

S. Friedberg, et al, Linear Algebra, 5th Ed., Pearson.

 
General Course Outline

Course Description

(4) Lecture, three hours; discussion, one hour. Requisite: course 115A. Not open for credit to students with credit for Program in Computing 130. Introduction to mathematical cryptology using methods of number theory, algebra, probability. Topics include symmetric and public-key cryptosystems, one-way functions, signatures, key exchange, groups, primes, pseudoprimes, primality tests, quadratic reciprocity, factoring, rho method, RSA, discrete logs. P/NP or letter grading.

Course Information:

The course is planned for 28 lectures, 1 midterm exam, and 1 holiday.

Math 116 is the introduction to mathematical cryptology which uses methods of number theory, algebra, probability. Topics include: symmetric and public-key cryptosystems, one-way functions, signatures, key exchange, groups, primes, pseudoprimes, primality tests, quadratic reciprocity, factoring, rho method, RSA, and discrete logs.

Math 116 is not open for credit to students with credit for PIC 130.

Textbook(s)

Trappe, Intro to Cryptography with Coding Theory, Prentice Hall.

Outline update: D. Blasius, 2/02

NOTE: While this outline includes only one midterm, it is strongly recommended that the instructor considers giving two. It is difficult to schedule a second midterm late in the quarter if it was not announced at the beginning of the course.

General Course Outline

Course Description

(4) Lecture, three hours; discussion, one hour. Requisite: course 115A. Not open for credit to students with credit for course 110A. Integers, congruences; fields, applications of finite fields; polynomials; permutations, introduction to groups.

Course Information:

The following schedule is based on 26 lectures. The remaining three classroom meetings are for midterm exams and a review.

Math 117 is the “fast” course in abstract algebra, which focuses on topics that are of interest for applications. The topics covered include error correcting codes, fast polynomial multiplication, and the fast Fourier transform. The fast Fourier transform is absolutely critical for the efficient implementation of computer algorithms for signal processing and other engineering applications.

One section of Math 117 is offered each term. In the past several years the enrollments in the course have averaged about 35 students each term.

Textbook(s)

L. Childs, A Concrete Introduction to Higher Algebra, 3rd Ed., Springer-Verlag.

Note: The book contains a wealth of interesting topics (e.g. Sturm’s theorem, group theory), which can be substituted for material in the last five lectures at the instructor’s discretion.

Outline update: D. Gieseker, 1/97

General Course Outline

Course Description

(4) Lecture, three hours; discussion, one hour. Requisites: courses 42 and 115A. Introduction to computational methods for data problems with a focus on linear algebra and optimization. Matrix and tensor factorization, PageRank, assorted other topics in matrices, linear programming, unconstrained optimization, constrained optimization, integer optimization, dynamic programming, and stochastic optimization. P/NP or letter grading.

Course Information:

Students will learn key processes of optimization and linear algebra which underlies data science. These include linear programming, unconstrained optimization, constrained optimization, integer optimization, dynamic programming, stochastic optimization, integer optimization, dynamic programming, and stochastic optimization.

 
Textbook(s)

Required:
1. Elden, Lrs. Matrix Methods in Data Mining and Pattern Recognition. The Society for Industrial and Applied Mathematics, 2007.
2. Chong, E and S. Zak. An Introduction to Optimization, 4th edition. Wiley, 2013.

Supplemental:
3. Hillier, Frederick S. and Lieberman, Gerald J. Introduction to Operations Research, 9th edition. McGraw-Hill Higher Education, 2009.

 
General Course Outline

Course Description

(4) Lecture, three hours; discussion, one hour. Requisites: courses 32B, 33B, 115A, 131A. Course 120A is requisite to 120B. Curves in 3-space, Frenet formulas, surfaces in 3-space, normal curvature, Gaussian curvature, congruence of curves and surfaces, intrinsic geometry of surfaces, isometries, geodesics, Gauss/Bonnet theorem. P/NP or letter grading.

Differential geometry can be viewed as the study of space and curvature. The course depends heavily upon calculus, it uses the tools of linear algebra, and it develops geometric insight. As such it is a good course for students who want to strengthen their understanding of the core mathematics curriculum.

Differential geometry is a crucial tool in modern physics. The idea of curved space is at the foundation of Einstein’s theory of gravitation (general relativity). Several more recent developments in physics, as Yang-Mills theory and string theory, involve differential geometry.

The courses 120A and 120B deal with differential geometry in a special context, curves and surfaces in 3-space, which has a firm intuitive basis, and for which some remarkable and striking theorems are available.

The course begins with curves in the plane and in 3-space, which already have some interesting geometric features. Curvature and torsion measure how curves bend and twist. There are some beautiful theorems that if a curve in 3-space forms a closed loop, it has to bend at least a certain amount, and if it forms a knot, it has to bend at least a larger certain amount. Another beautiful theorem is the celebrated isoperimetric theorem, that among all closed curves of a fixed length, the circle encloses the largest area.

There are several notions of curvature for surfaces in 3-space. Mean curvature shows up in the problem of determining the surface of the smallest area with a fixed prescribed boundary. (The solution can be illustrated with soap bubbles.) Gaussian curvature shows up in the problem of determining which surfaces can be represented by a flat map.

Another problem treated in the course is how to determine the shortest route on a surface between two points. In the plane the shortest path is a straight line, and on a sphere the shortest path is an arc of a great circle.

The theorem of high-school geometry that the sum of the angles of a triangle is 180 degrees turns out to have a very beautiful generalization to a triangle on any surface (as a spherical triangle). The generalization is the Gauss-Bonnet theorem, which is one of the high-points of undergraduate mathematics. The theorem provides an identity with a sum of angles and a correction term that takes into account how curved the sides of the triangle are and how much the surface is curved inside the triangle. One of the remarkable features of the Gauss-Bonnet theorem is that it asserts the equality of two quantities, one of which comes from differential geometry and the other of which comes from topology.

Math 120AB is highly recommended for mathematics students who want to go on to graduate school.

Textbook(s)

Millman & Parker, Elements of Differential Geometry, Prentice Hall
Book is Subject to Change Without Notice

Outline update: P. Petersen, 9/14
(Requisites updated 5/98)

General Course Outline

Course Description

(4) Lecture, three hours; discussion, one hour. Requisites: courses 32B, 33B, 115A, 120A, 131A. Curves in 3-space, Frenet formulas, surfaces in 3-space, normal curvature, Gaussian curvature, congruence of curves and surfaces, intrinsic geometry of surfaces, isometries, geodesics, Gauss/Bonnet theorem. P/NP or letter grading.

Differential geometry can be viewed as the study of space and curvature. The course depends heavily upon calculus, it uses the tools of linear algebra, and it develops geometric insight. As such it is a good course for students who want to strengthen their understanding of the core mathematics curriculum.

Differential geometry is a crucial tool in modern physics. The idea of curved space is at the foundation of Einstein’s theory of gravitation (general relativity). Several more recent developments in physics, as Yang-Mills theory and string theory, involve differential geometry.

The courses 120A and 120B deal with differential geometry in a special context, curves and surfaces in 3-space, which has a firm intuitive basis, and for which some remarkable and striking theorems are available.

The course begins with curves in the plane and in 3-space, which already have some interesting geometric features. Curvature and torsion measure how curves bend and twist. There are some beautiful theorems that if a curve in 3-space forms a closed loop, it has to bend at least a certain amount, and if it forms a knot, it has to bend at least a larger certain amount. Another beautiful theorem is the celebrated isoperimetric theorem, that among all closed curves of a fixed length, the circle encloses the largest area.

There are several notions of curvature for surfaces in 3-space. Mean curvature shows up in the problem of determining the surface of the smallest area with a fixed prescribed boundary. (The solution can be illustrated with soap bubbles.) Gaussian curvature shows up in the problem of determining which surfaces can be represented by a flat map.

Another problem treated in the course is how to determine the shortest route on a surface between two points. In the plane the shortest path is a straight line, and on a sphere the shortest path is an arc of a great circle.

The theorem of high-school geometry that the sum of the angles of a triangle is 180 degrees turns out to have a very beautiful generalization to a triangle on any surface (as a spherical triangle). The generalization is the Gauss-Bonnet theorem, which is one of the high-points of undergraduate mathematics. The theorem provides an identity with a sum of angles and a correction term that takes into account how curved the sides of the triangle are and how much the surface is curved inside the triangle. One of the remarkable features of the Gauss-Bonnet theorem is that it asserts the equality of two quantities, one of which comes from differential geometry and the other of which comes from topology.

Math 120AB is highly recommended for mathematics students who want to go on to graduate school.

Textbook(s)

Millman & Parker, Elements of Differential Geometry, Prentice Hall
Book is Subject to Change Without Notice

Outline update: P. Petersen, 9/14
(Requisites updated 5/98)

General Course Outline

Course Description

(4) Requisite: course 131A. Metric and topological spaces, completeness, compactness, connectedness, functions, continuity, homeomorphisms, topological properties.

Course Information:

The following sample schedule, with textbook sections and topics, is based on 25 lectures. Assigned homework problems play an important role in the course, and there is usually a midterm exam.

Topology is the study of the properties of spaces (such as surfaces, or solids) that are invariant under homeomorphisms (such as stretchings). One striking theorem in topology is that any compact orientable two-dimensional surface is topologically a sphere with a certain number of handles attached. The number of handles completely characterizes the topological type of the surface. This leads to the adage that a topologist is a person who cannot tell the difference between a teacup and a doughnut. Topologically speaking, each is a sphere with one handle, and each can be continuously deformed to the other.

While topology is classified under geometry, the language of topology is fundamental to analysis. Many of the issues addressed by topology, such as compactness of spaces and continuity of functions, are treated in a simpler setting in the analysis courses 131AB.

One method for studying topological spaces is to assign algebraic objects, such as groups or vector spaces, to a topological space. One such object is the “fundamental group” of a topological space, which measures in some sense the number of holes in the space. Thus topology interacts also with algebra, leading to a branch of mathematics called “algebraic topology.”

Math 121 is a flexible course, and the selection of topics might be organized quite differently by different instructors. The subject matter for a standard syllabus breaks into three parts.

The first part treats metric spaces, which are closest to the intuition and to the development presented in 131AB. The fundamental concepts are completeness, compactness, continuity, and uniform continuity. The principal theorems are the Baire category theorem, the characterization of compact metric spaces, the theorem that continuous functions on a compact space are uniformly continuous, and the contraction mapping principle, which is perhaps the most important and useful tool in analysis.

The second part of the standard course covers point-set topology. Topological spaces are introduced, along with the separation axioms and various notions as compactness, local compactness, connectedness, and path connectedness. Product and quotient spaces are defined. The most important theorem in point-set topology is Tychonoff’s theorem that the product of a family of compact topological spaces is compact.

The third part of the standard course consists of an elementary introduction to algebraic topology. The fundamental group is introduced, and covering spaces are used to compute it for some special spaces. Some simple applications of the algebraic invariants are given.

Math 121 is offered once each year, usually in the Spring Quarter. Course enrollments run between 10 and 35.

Textbook(s)

T. Gamelin and R. Greene, Introduction to Topology, 2nd Ed., Dover.

Outline update: T. Gamelin, 5/96

NOTE: While this outline only suggests one midterm exam, it is strongly recommended that the instructor considers giving two. It is difficult to schedule a second midterm late in the quarter if it was not announced at the beginning of the course.

General Course Outline

Course Description

Lecture, three hours; discussion, one hour. Prerequisite: course 115A. Axioms and models, Euclidean geometry, Hilbert axioms, neutral (absolute) geometry, hyperbolic geometry, Poincare model, independence of parallel postulate.

Course Information:

The purpose of Math 123 is to study the classical geometries from an axiomatic perspective, with particular attention paid to Euclid’s parallel postulate and to geometric systems that violate it. These systems are called Non-Euclidean Geometries. Among them, the Hyperbolic Geometry is the most important today. Here is some background.

In his Elements, Euclid (~365BC-~300BC) built his geometry using five axioms. The first 4 are:

(1) Any two points can be joined by a (straight) line.

(2) Any segment can be extended continuously in a (straight) line.

(3) Given any point and distance, there is a circle centered at the point with radius equal to the distance.

(4) All right angles are equal to each other. These are easily understood as Euclid gave them.

The fifth was less obvious, but was found to be equivalent to (5) Given a line L and a point P not on the line, there exists one and only one line which passes through P and is parallel to (i.e. does not intersect) L. Axiom (5), in this version, is called the Parallel Postulate (and also Playfair’s Axiom).

From near the beginning, it seemed as if Euclid’s 5th axiom might be a consequence of the first 4, but no proof was ever found. Finally, in the nineteenth century Bolyai, Gauss and Lobachevsky independently put the question to rest by showing that a new geometry, Hyperbolic Geometry, satisfies the first 4 axioms but not the 5th. Thus, one of the goals of Math 123 is to study the concept of a “geometry” and to illustrate the implementation of this concept in examples.

The course can be useful for prospective secondary school teachers, in that it illustrates how a mathematical structure can be built upon an axiom system, and how the Euclidean geometry that is traditionally studied in the schools is only one of many possible “geometries”.

Math 123 is a flexible course, and it is taught quite differently by different instructors. For example, some instructors may approach the course primarily through the classical axiom systems, while others may take the Kleinian approach according to which geometries are classified by their symmetry groups.

General Course Outline

Course Description

(4) Lecture, three hours; discussion, one hour. Requisites: courses 32B, 33B. Recommended: course 115A. Rigorous introduction to foundations of real analysis; real numbers, point set topology in Euclidean space, functions, continuity.

Course Information:

The following schedule, with textbook sections and topics, is based on 26 lectures. The remaining three classroom meetings are for leeway, reviews, and midterm exams. These are scheduled by the individual instructor. Often there are midterm exams about the beginning of the fourth and eighth weeks of instruction, plus reviews for the final exam.

Math 131AB is the core undergraduate course sequence in mathematical analysis. The aim of the course is to cover the basics of calculus, rigorously. Along with Math 115A, this is the main course in which students learn to write logically clear and correct arguments.

There is an honors sequence Math 131AH-131BH running parallel to 131A-131B in fall and winter. 131AH: Rigorous treatment of the foundations of real analysis, including construction of the rationals and reals; metric space topology, including compactness and its consequences; numerical sequences and series; continuity, including connections with compactness; rigorous treatment of the main theorems of differential calculus. 131BH: The Riemann integral; sequences and series of functions; power series, and functions defined by them; differential calculus of several variables, including the implicit and inverse function theorems.

Math 131C is a special topics analysis course offered in the spring that is designed for students completing the honors sequence as well as the regular 131AB sequence. It traditionally covers Lebesgue measure and integration. Math 131A is offered each term, while 131B is offered only Winter and Spring.

Textbook(s)

K.A. Ross, Elementary Analysis: The Theory of Calculus, 2nd Ed.

Outline update: J. Ralston, 8/08(*1) Include Section 23, if time permits. The instructor can pick which convergence tests to cover in Sections 14 and 15.

General Course Outline

Course Description

(4) Lecture, three hours; discussion, one hour. Requisites for course 131AH: courses 32B and 33B, with grades of B or better. Recommended: course 115A. Honors sequence parallel to courses 131A. P/NP or letter grading. Rigorous introduction to foundations of real analysis; real numbers, point set topology in Euclidean space, functions, continuity.

Course Information:

The following schedule, with textbook sections and topics, is based on 26 lectures. The remaining three classroom meetings are for leeway, reviews, and midterm exams. These are scheduled by the individual instructor. Often there are midterm exams about the beginning of the fourth and eighth weeks of instruction, plus reviews for the final exam.

Math 131AB is the core undergraduate course sequence in mathematical analysis. The aim of the course is to cover the basics of calculus, rigorously. Along with Math 115A, this is the main course in which students learn to write logically clear and correct arguments.

There is an honors sequence Math 131AH-131BH running parallel to 131A-131B in fall and winter. 131AH: Rigorous treatment of the foundations of real analysis, including construction of the rationals and reals; metric space topology, including compactness and its consequences; numerical sequences and series; continuity, including connections with compactness; rigorous treatment of the main theorems of differential calculus. 131BH: The Riemann integral; sequences and series of functions; power series, and functions defined by them; differential calculus of several variables, including the implicit and inverse function theorems.

Math 131C is a special topics analysis course offered in the spring that is designed for students completing the honors sequence as well as the regular 131AB sequence. It traditionally covers Lebesgue measure and integration. Math 131A is offered each term, while 131B is offered only Winter and Spring.

Textbook(s)

Rudin, W., Principles of Mathematical Analysis, 3rd Ed, McGraw-Hill Higher Education
Copson, E. Metric Spaces, Cambridge University Press

Outline update:D. Gieseker, 9/14(*1) Include Section 23, if time permits. The instructor can pick which convergence tests to cover in Sections 14 and 15.

General Course Outline

Course Description

(4) Lecture, three hours; discussion, one hour. P/NP or letter grading. Requisites: courses 33B, 115A, 131A. Derivatives, Riemann integral, sequences and series of functions, power series, Fourier series.

Course Information:

The following schedule, with textbook sections and topics, is based on 26 lectures. The remaining classroom meetings are for leeway, reviews, and midterm exams. These are scheduled by the individual instructor. Often there are midterm exams about the beginning of fourth and eighth weeks of instruction, plus reviews for the final exam.

Math 131AB is the core undergraduate course sequence in mathematical analysis. The aim of the course is to cover the basics of calculus, rigorously. Along with Math 115A, this is the main course in which students learn to write logically clear and correct arguments.

There is an honors sequence Math 131AH-131BH running parallel to 131A-131B in fall and winter. 131AH: Rigorous treatment of the foundations of real analysis, including construction of the rationals and reals; metric space topology, including compactness and its consequences; numerical sequences and series; continuity, including connections with compactness; rigorous treatment of the main theorems of differential calculus. 131BH: The Riemann integral; sequences and series of functions; power series, and functions defined by them; differential calculus of several variables, including the implicit and inverse function theorems.

Math 131C is a special topics analysis course offered in the spring that is designed for students completing the honors sequence as well as the regular 131AB sequence. It traditionally covers Lebesgue measure and integration. Math 131A is offered each term, while 131B is offered only Winter and Spring.

Textbook(s)

Rudin, W., Principles of Mathematical Analysis, 3rd Ed
Copson, E. Metric Spaces, Cambridge University Press

Section 14.8 is the proof of the Weierstrass Approximation Theorem. This should probably be left for the Honors Section.
This is rather difficult, but it introduces summation by parts. Using summation by parts to prove Dirichlet’s Test (and hence the Alternating Series Test) is an alternative to Abel’s Theorem.
This is a lot, but Sections 17.1 is just a review of linear transformations and 17.2 and 17.3 contain only one theorem.

Outline update: D. Gieseker, 9/14

General Course Outline

Course Description

(4) Lecture, three hours; discussion, one hour. P/NP or letter grading. Requisites: courses 33B, 115A, 131A. Derivatives, Riemann integral, sequences and series of functions, power series, Fourier series.

Course Information:

The following schedule, with textbook sections and topics, is based on 26 lectures. The remaining classroom meetings are for leeway, reviews, and midterm exams. These are scheduled by the individual instructor. Often there are midterm exams about the beginning of fourth and eighth weeks of instruction, plus reviews for the final exam.

Math 131AB is the core undergraduate course sequence in mathematical analysis. The aim of the course is to cover the basics of calculus, rigorously. Along with Math 115A, this is the main course in which students learn to write logically clear and correct arguments.

There is an honors sequence Math 131AH-131BH running parallel to 131A-131B in fall and winter. 131AH: Rigorous treatment of the foundations of real analysis, including construction of the rationals and reals; metric space topology, including compactness and its consequences; numerical sequences and series; continuity, including connections with compactness; rigorous treatment of the main theorems of differential calculus. 131BH: The Riemann integral; sequences and series of functions; power series, and functions defined by them; differential calculus of several variables, including the implicit and inverse function theorems.

Math 131C is a special topics analysis course offered in the spring that is designed for students completing the honors sequence as well as the regular 131AB sequence. It traditionally covers Lebesgue measure and integration. Math 131A is offered each term, while 131B is offered only Winter and Spring.

Textbook(s)

Tao, T., Analysis II, 3rd Ed., Hindustan

Section 3.8 is the proof of the Weierstrass Approximation Theorem. This should probably be left for the Honors Section.
This is rather difficult, but it introduces summation by parts. Using summation by parts to prove Dirichlet’s Test (and hence the Alternating Series Test) is an alternative to Abel’s Theorem.
This is a lot, but Sections 6.1 is just a review of linear transformations and 6.2 and 6.3 contain only one theorem.

Outline update: J. Ralston, 9/19

General Course Outline

Course Description

 

(4) Lecture, three hours; discussion, one hour. Requisites: courses 131A, 131B or 131AH, 131BH. Covers multivariable calculus and applications to ordinary differential equations.

Math 131C studies primarily multivariable analysis: definition of differentiability in several variables, partial derivatives, chain rule, Taylor expansion in several variables, inverse and implicit function theorems, equality of mixed partials, multivariable integration, change of variables formula, differentiation under the integral sign, analysis on curves and surfaces. Further topics to be chosen, usually including basic applications to ordinary differential equations (existence and uniqueness theorems for solutions) and the Green, Gauss and Stoke theorems.

Textbook(s)

Conway, J., A First Course In Analysis, Cambridge University Press
Coddington, E., An Introduction to Ordinary Differential Equations, Dover Publications

General Course Outline

Course Description

(4) Lecture, three hours; discussion, one hour. Requisites: courses 32B, 33B. Introduction to basic formulas and calculation procedures of complex analysis of one variable relevant to applications. Topics include Cauchy/Riemann equations, Cauchy integral formula, power series expansion, contour integrals, residue calculus.

Course Information:

The following schedule, with textbook sections and topics, is based on 26 lectures. The remaining classroom meetings are for leeway, reviews, and a midterm exam. These are scheduled by the individual instructor. Often there are a review and a midterm exam about the end of the fifth week of instruction, plus a review for the final exam.

General Information. Complex analysis is one of the most beautiful areas of pure mathematics, at the same time it is an important and powerful tool in the physical sciences and engineering. The course Math 132 is aimed primarily at students in applied mathematics, engineering, and physics, and it is satisfies a major requirement for students in Electrical Engineering.

The topics covered in Math 132 include: analytic functions, Cauchy-Riemann equations, harmonic functions, branch points, branches of multiple-valued functions, Cauchy’s theorem, integral representation formulae, power series of analytic functions, zeros, isolated singularities, Laurent series, poles, residues, use of residue calculus to evaluate real integrals, use of argument principle to locate zeros, fractional linear transformations, and conformal mapping.

Students entering Math 132 are assumed to have some familiarity with complex numbers from high school, including the polar form of complex numbers. Students in Math 132 are also assumed to have a strong background in single and multivariable calculus, including infinite series, power series, radius of convergence (ratio and root tests), integration term by term of power series, parametrized curves, line integrals, and Green’s theorem. Some of this material is reviewed in Math 132, though at a fast pace.

Several sections of Math 132 are offered each term.

Textbook(s)

T. Gamelin, Complex Analysis, Springer/Verlag.

*The book is subject to change. Check with the UCLA Bookstore.

The students should be familiar with the elementary properties of complex numbers from high school. They have been introduced to the complex exponential function in Math 33B. They should be familiar with power series, including radius of convergence, the ratio and root tests, and integration term by term.

The idea of gluing sheets together at branch cuts to form a surface is important, but it can be omitted at this stage. At most it should be treated only at an intuitive level, to introduce the idea to the students and to arouse their interest.

The idea of conformality can be treated lightly if short on time. The results of the section on conformality are used primarily to see that fractional linear transformations map orthogonal circles to orthogonal circles.

With respect to uniform convergence, the only thing that is really needed is the Weierstrass M-test, together with the integration term by term of a uniformly convergent series of functions.

The material in Section VIII.1 on the argument principle is important to electrical engineers and should not be omitted. Rather omit Section VII.3 if short of time at the end of the course.

Outline update: T. Gamelin, 3/04

NOTE: While this outline includes only one midterm, it is strongly recommended that the instructor considers giving two. It is difficult to schedule a second midterm late in the quarter if it was not announced at the beginning of the course.

General Course Outline

Course Description

Lecture, three hours; discussion, one hour. Requisites: courses 32B, 33B, and 131A with grades of B or better. This course is specifically designed for students who have strong commitment to pursue graduate studies in mathematics. Introduction to complex analysis with more emphasis on proofs. Honors course parallel to course 132. P/NP or letter grading.

 
Textbook(s)

Complex Analysis by Stein and Shakarchi.

 
General Course Outline

Course Description

(4) Lecture, three hours; discussion, one hour. Requisites: courses 33A, 33B, 131A. Fourier series, Fourier transform in one and several variables, finite Fourier transform. Applications, in particular, to solving differential equations. Fourier inversion formula, Plancherel theorem, convergence of Fourier series, convolution. P/NP or letter grading.

Course Information:

This syllabus is based on a single midterm; instructors who wish to give a second midterm may adjust the syllabus appropriately, or give the second midterm in section. The lecturer may also wish to expand the applications components (lectures 11-12, 22-24, 26-28) or move them earlier in the course.

Math 133 is the introduction to Fourier series, the Fourier transform in one and several variables, finite Fourier transform, applications, in particular to solving differential equations. Fourier inversion formula, Plancherel’s theorem, convergence of Fourier series, convolution.

Textbook(s)

E. Stein and R. Shakarchi, Fourier Analysis: An Introduction (Princeton Lectures in Analysis, Volume 1), Princeton University Press.

 
General Course Outline

Course Description

(4) (Formerly numbered 135A.) Lecture, three hours; discussion, one hour. Requisites: course 33B. Dynamical systems analysis of nonlinear systems of differential equations. One- and two- dimensional flows. Fixed points, limit cycles, and stability analysis. Bifurcations and normal forms. Elementary geometrical and topological results. Applications to problems in biology, chemistry, physics, and other fields. P/NP or letter grading.

 
Textbook(s)

S. Strogatz, Nonlinear Dynamics and Chaos (2nd Ed.), Perseus Books Group.
J. Crawford, Introduction to Bifurcation Theory, Reviews of Modern Physics, vol. 63. (Recommended supplement).

For those instructors wishing to incorporate a final project, lectures 9 and 10 can be skipped and the last four lectures can be used for final project poster presentations.

If time is available for more lectures than those outlined, additional lectures could cover section 7.6 (on weakly nonlinear oscillations and perturbation theory) or selected sections from chapter 9 (on chaos and the Lorenz equations).

Outline update: C. Topaz, 4/04, updated, 3/05

NOTE: While this outline includes only one midterm, it is strongly recommended that the instructor considers giving two. It is difficult to schedule a second midterm late in the quarter if it was not announced at the beginning of the course.

General Course Outline

Course Description

(4) Lecture, three hours; discussion, one hour. Requisites: courses 33A, 33B. Selected topics in differential equations. Laplace transforms, existence and uniqueness theorems, Fourier series, separation of variable solutions to partial differential equations, Sturm-Liouville theory, calculus of variations, two point boundary value problems, Green’s functions. P/NP or letter grading.

General Information. Differential equations are of paramount importance in mathematics because they are equations whose solutions are functions – not numbers. Differential equations are thus widely used in mathematical models of systems where one wants to determine functional relationships. For example, the concentration of chemical reactants as a function of the time, the temperature on the surface of a heat shield as a function of position, or the size of a loan payment as a function of the interest rate. In fact, in nearly all of the courses in the physical sciences and engineering, and in many courses in the social sciences, differential equations play a fundamental role.

One of the goals of this course is to present solution techniques for differential equations that go beyond what is taught in 33B. In particular, the Laplace transform technique for solving linear differential equations is covered. This technique transforms the task of solving linear differential equations to one of solving algebraic problems. It is also a technique that can be used to solve differential equations containing generalized functions (e.g. discontinuous or Dirac delta functions). Other solution techniques include the method of Fourier series, the method of eigenfunction expansions and perturbation methods.

Another goal of this course is to introduce students to the theory of ordinary differential equations. A key part of this theory is the determination of the existence and uniqueness of solutions to differential equations. Just as it’s a fact that not all algebraic equations have solutions, it’s also a fact that not all differential equations have solutions. The theorems covered are especially useful, as they allow one to determine the existence and uniqueness of solutions without having to solve the differential equation.

Textbook(s)

G. Simmons, Differential Equations with Applications and Historical Notes, 3rd Ed., McGraw-Hill.

Footnotes

1. The book does not include a review of partial fractions. Most calculus textbooks provide a suitable discussion of the technique.

2. The book only states a limited form of the Heaviside expansion theorem in problem 5 of section 53. The more general statement can be found in standard texts devoted to Laplace transforms.

3. The book provides a limited description of the use of the unit-step function and unit impulse functions. A better treatment can be found in Redheffer’s book Differential Equations.

4. The proof of Theorem B is easier than Theorem A (the local existence theorem) since one doesn’t have to worry about the Picard iterates leaving the domain where f(x,y) is Lipschitz. Thus, discussing and proving Theorem B before Theorem A is recommended.

5. The book glosses over some of the mathematical details required by the convergence proofs so one must supplement the material in the text as needed.

Additional Notes

An energetic instructor may want to cover two point boundary value problems and Green’s functions in more depth instead of spending the last three lectures on the calculus of variations. Alternately, one could replace the lectures on the calculus of variations with lectures on regular perturbation theory. A reference for this latter topic is Bender and Orszag, Advanced Mathematical Methods for Scientists and Engineers, Chapter 7.

Outline update: C. Anderson, 5/05

NOTE: While this outline includes only one midterm, it is strongly recommended that the instructor considers giving two. It is difficult to schedule a second midterm late in the quarter if it was not announced at the beginning of the course.

General Course Outline

Course Description

Lecture, three hours; discussion,one hour. Prerequisites: courses 33A, 33B. Linear partial differential equations, boundary and initial value problems; wave equation, heat equation, and Laplace equation; separation of variables, eigenfunction expansions; selected topics, as method of characteristics for nonlinear equations.

General Information. Math 136 is offered once each year, in the Spring. Together with 135A in the Fall and 135B in the Winter, it is the third of a natural sequence of courses in differential equations. Note however that the courses 135AB are not required for 136.

Enrollments in Math 136 have oscillated between 30 and 100 over the past several years.

Textbook(s)

W.A. Strauss, Partial Differential Equations, 2nd Edition, John Wiley and Sons.
The course covers Chapters 1, 2, parts of 3, and most of 4-6.

 
General Course Outline

Course Description

Lecture, three hours; discussion, one hour. Prerequisites: courses 32B, 33B. Introduction to fundamental principles and spirit of applied mathematics. Emphasis on manner in which mathematical models are constructed for physical problems. Illustrations from many fields of endeavor, such as the physical sciences, biology, economics, and traffic dynamics.

General Information. One section of Math 142 is offered each term. For the past several years the enrollments in the course have run between 35 and 100 students each term.

Textbook(s)

Haberman, R., Mathematical Models, Society for Industrial and Applied Mathematics.

General Course Outline

Course Description

Lecture, three hours; discussion, one hour. Prerequisite: courses 32B, 33B. Integral equations, Green’s function, and calculus of variations. Selected applications from control theory, optics, dynamical systems, and other engineering problems.

General Information. The content of Math 146 varies depending on the instructor. The course is usually offered once each year, in Spring Quarter.

Textbook(s)

Troutman, J., Variational Calculus and Optimal Control: Optimization with Elementary Convexity, 2nd Ed., Springer.

General Course Outline

Course Description

Lecture, three hours; discussion, one hour. Prerequisites: course 115A, and Program in Computing 10A or equivalent knowledge of programming in either PASCAL or C language. Study of homogeneous coordinates, projective transformations, interpolating and approximating curves, representation of surfaces, and other mathematical topics useful for computer graphics.

General Information. The course Math 149 was originally designed by Kirby Baker, currently the Director of the Program in Computing. Usually taught by Baker, the course is offered once each year, usually in the Winter. Enrollments in the course have run between 20 and 50 students for the past several years.

In a recent Mathematics Department colloquium talk, Nathan Mhyrvald, who is one of the top executives at Microsoft, pointed to Math 149 as the most memorable undergraduate math course he had taken at UCLA.

To receive the instructor’s permission to enroll, students should have taken Math 115A with a grade of at least B-, and they should also have completed PIC 10A or the equivalent.

General Course Outline

Course Description

(4) Lecture, three hours; discussion, one hour. Requisites: courses 32B, 33B, 115A, Program in Computing 10A. Introduction to numerical methods with emphasis on algorithms, analysis of algorithms, and computer implementation issues. Solution of nonlinear equations. Numerical differentiation, integration, and interpolation. Direct methods for solving linear systems. Matlab programming. Letter grading.

Assignments Homework assignments in the course consist of both theoretical and computational work. The computational work is completed using Matlab.

General Information. Math 151AB is the main course sequence in numerical analysis, important for all of the applied mathematics majors. Mathematics majors who graduate and go into industry often find Math 151AB to be the most useful course for their work.

Math 151A is offered each term, and Math 151B is offered Winter and Spring.

Textbook(s)

R. Burden and J. Faires, Numerical Analysis, 10th Ed., Brooks/Cole.

Homework assignments in the course consist of both theoretical and computational work. The computational work is completed using Matlab.

* This topic is not in Burden and Faires. It can be found in Cheney-Kincaid, Numerical Mathematics and Computing, Brooks/Cole, section 4.2.

Topics in parenthesis are optional and can be included under the discretion of the instructor.

Outline update: J. Qin, 06/2015

NOTE: While this outline includes only one midterm, it is strongly recommended that the instructor considers giving two. It is difficult to schedule a second midterm late in the quarter if it was not announced at the beginning of the course.

General Course Outline

Course Description

(4) Lecture, three hours; discussion, one hour. Requisite: course 151A. Introduction to numerical methods with emphasis on algorithms, analysis of algorithms, and computer implementation. Numerical solution of ordinary differential equations. Iterative solution of linear systems. Computation of least squares approximations. Discrete Fourier approximation and the fast Fourier transform. Matlab programming. Letter grading.

Assignments Homework assignments in the course consist of both theoretical and computational work. The computational work is completed using matlab.

General Information. Math 151AB is the main course sequence in numerical analysis, important for all of the applied mathematics majors. Mathematics majors who graduate and go into industry often find Math 151AB to be the most useful course for their work.

Math 151A is offered each term, and Math 151B is offered Winter and Spring.

Textbook(s)

R. Burden and J. Faires, Numerical Analysis, 10th Ed., Brooks/Cole.

Homework assignments in the course consist of both theoretical and computational work. The computational work is completed using matlab.

AS: The topics of stiffness and of absolute stability are not well presented in Burden and Faires. Other textbooks should be consulted.

DLS: The matrix form of the discrete least squares problem is not presented in Burden and Faires. Other textbooks should be consulted.

Outline update: J. Qin, 06/2015

NOTE: While this outline includes only one midterm, it is strongly recommended that the instructor considers giving two. It is difficult to schedule a second midterm late in the quarter if it was not announced at the beginning of the course.

General Course Outline

Course Description

(4) Lecture, three hours; discussion, one hour. Requisites: courses 32B, 33B, 115A, Program in Computing 10A. Imaging geometry. Image transforms. Enhancement, restoration, and segmentation. Descriptors. Morphology. P/NP or letter grading.

Math 155 is an introductory course on mathematical models for image processing and analysis. The students will become familiar with basic concepts (such as image formation, image representation, image quantization, change of contrast, image enhancement, noise, blur, image degradation), as well as with mathematical models for edge and contour detection (such as the Canny edge detector), filtering, denoising, morphology, image transforms, image restoration, image segmentation, and applications. All theoretical concepts will be accompanied by computer exercises.

Textbook(s)

R. Gonzalez and R. Woods, Digital Image Processing, New edition, Prentice-Hall. Book is Subject to Change Without Notice.

Outline update: L. Vese, 2/03

General Course Outline

Course Description

Lecture, three hours; discussion, one hour. Requisite: course 115A, 164, 170E (or 170A or Statistics 100A) and Programming in Computing 10A of Computer Science 31. Strongly recommended requisite: Program in Computing 16A or Statistics 21. Introductory course on mathematical models for pattern recognition and machine learning. Topics include parametric and nonparametric probability distributions, curse of dimensionality, correlation analysis and dimensionality reduction, and concepts of decision theory. Advanced machine learning and pattern recognition problems, including data classification and clustering, regression, kernel methods, artificial neural networks, hidden Markov models, and Markov random fields. Projects in MATLAB to be part of final project presented in class. P/NP or letter grading.

 
Textbook(s)

?Pattern Recognition and Machine Learning?, by Christopher M. Bishop, Springer, 2006 (ISBN-13: 978-0387-31073-2), plus complementary sources where necessary (?n/a?).

 
General Course Outline

Course Description

Lecture, three hours; discussion, one hour. Prerequisites: courses 151A, Program in Computing 10C. Software structures, concepts, and conventions that support object-oriented programming. Identification of class structure, problem partitioning and abstraction. Design and implementation of computer applications requiring scientific computation, visualization, and GUI components. Inter-language interfacing. P/NP or letter grading.

General Information. The goal of Math 157 is to provide knowledge required for scientific/technical computing, which consists of combining mathematical models, algorithms, software and hardware units in such a way as to provide computational procedures that are useful for the solution of scientific/technical problems.

The course focuses on the techniques, both abstract and practical, that facilitate the process of “combining” constituent components to create complete applications. The primary theoretical emphasis is on object-oriented design principles and techniques. The primary practical emphasis is upon knowledge required to build applications for UNIX and PC platforms (organization of multi-component codes, inter-language calls, event-driven programming, and static and dynamic libraries).

Both C++ and Java are used in the course. Knowledge of C++ is assumed, but knowledge of Java is not assumed.

Students enrolling in the course should enjoy the intellectual challenge associated with programming.

This course was designed by Chris Anderson and offered for the first time Winter Quarter 1997. Enrollments have been about 30.

General Course Outline

Course Description

(4) Lecture, three hours; discussion, one hour. Requisite: course 115A, 131A. Not open for credit to students with credit for Electrical Engineering 136. Fundamentals of optimization. Linear programming: basic solutions, simplex method, duality theory. Unconstrained optimization, Newton’s method for minimization. Nonlinear programming, optimality conditions for constrained problems. Additional topics from linear and nonlinear programming. P/NP or letter grading.

Course Information:

The following schedule, with textbook sections and topics, is based on 27 lectures. The remaining classroom meetings are for leeway, reviews, and a midterm exam. These are scheduled by the individual instructor.

General Information. Math 164 provides an introduction to the theory and algorithms concerned with finding extrema (maxima and minima) of functions subject to constraints.

After a review of topics from multivariable calculus such as the gradient, Hessian, Jacobian, Taylor series, and linear algebra, the course offers the students a working knowledge of optimization theory and methods for linear and nonlinear programming, that is, how to find extrema of linear and nonlinear functions subject to various kinds of constraints.

There are ample opportunities for the students to improve their ability to read and write mathematical proofs as well as to solve applied and theoretical problems.

Textbook(s)

E. K.P. Chong and S. Zak, An Introduction to Optimization, 4th Edition, Wiley.

Outline update: W. Yin, 6/15

General Course Outline

Course Description

(4) Lecture, three hours; discussion, one hour. Requisite: course 115A. Quantitative modeling of strategic interaction. Topics include extensive and normal form games, background probability, lotteries, mixed strategies, pure and mixed Nash equilibria and refinements, bargaining; emphasis on economic examples. Optional topics include repeated games and evolutionary game theory. P/NP or letter grading.

 

Outline update: D. Blasius, 5/02

NOTE: While this outline includes only one midterm, it is strongly recommended that the instructor considers giving two. It is difficult to schedule a second midterm late in the quarter if it was not announced at the beginning of the course.

General Course Outline

Course Description

(4) Lecture, three hours; discussion, one hour. Requisites: courses 115A, 170A or Electrical and Computer Engineering 131A or Statistics 100A. Introduction to network science (including theory, computation, and applications), which can be used to study complex systems of interacting agents. Study of networks in technology, social, information, biological, and mathematics involving basic structural features of networks, generative models of networks, network summary statistics, centrality, random graphs, clustering, and dynamical processes on networks. Introduction to advance topics as time permits. P/NP or letter grading.

Course Information:

Students will develop a sound knowledge and appreciation of some of the tools, concepts, and computations used in the study of networks. The study of networks is predominantly a modern subject, so the students will also be expected to develop the ability to read and understand current research papers in the field. They will also have a chance to explore a topic in depth in a final project. Topics include basic structural features of networks, generative models of networks, centrality, random graphs, clustering, and dynamical processes on networks.

 
Textbook(s)

Mark E. J. Newmn, Networks: An Introduction, 2010 [primary text]
Mason A. Porter and James Gleeson, Dynamical Systems on Networks: A Tutorial, 2016
Supplementary material from survey, review, and tutorial articles.

 
General Course Outline

Course Description

(4) Lecture, three hours; discussion, one hour. Requisites: courses 32B, 33A, 131A. Not open to students with credit in course 170E, Electrical Engineering 131A or Statistics 100A. Rigorous presentation of probability theory based on real analysis. Probability space, probability and conditional probability, independence, Bayes? rule, discrete and continuous random variables and their distributions, expectation, moments and variance, conditional distribution and expectation, weak law of large numbers. P/NP or letter grading.

Course Information:

The course discusses the foundations of probability as a mathematical discipline rooted in undergraduate real analysis. At the end of the course, the students will have the tools and ability to formulate, analyze an answer questions in probability and prove the validity of their reasoning in full mathematical rigor.

 
Textbook(s)

Probability: An Introduction (2nd ed.). by Grimmett, G. R., & Welsh, D. J. (2014).Oxford: Oxford University Press.

Outline update: T. Austin, 01/20

 
General Course Outline

Course Description

(4) Lecture, three hours; discussion, one hour. Enforced requisite: courses 170A, 131A. Continuation of rigorous presentation of probability theory based on real analysis. Moments and generating functions; laws of large numbers, the central limit theorem, and convergence in distribution; branching processes; random walks; Poisson and other random processes in continuous time. Advance topics in probability theory. P/NP or letter grading.

Additional Information Advanced Topics:
Instructor selection of three more advanced topics. Possibilities include, but are not limited to:

(1) order statistics, extreme values, and Poisson processes;

(2) basics of entropy and information theory;

(3) theory of sampling and confidence in statistics;

(4) the probabilistic method in combinatorics, including lower bounds for Ramsey numbers;

(5) Borel-Cantelli lemmas, the strong law of large numbers, and Borel’s normal number theorem.

Instructor will provide notes or reference materials.

Textbook(s)

Probability: An Introduction (2nd ed.). Oxford: Oxford University Press. by Grimmett, G. R., & Welsh, D. J. (2014).

 
General Course Outline

Course Description

Lecture, three hours; discussion, one hour. Requisites: courses 31A, 31B. Not open to students with credit for course 170A, Electrical and Computer Engineering 131A, or Statistics 100A. Introduction to probability theory with emphasis on topics relevant to applications. Topics include discrete (binomial, Poisson, etc.) and continuous (exponential, gamma, chi-square, normal) distributions, bivariate distributions, distributions of functions of random variables (including moment generating functions and central limit theorem). P/NP or letter grading.

 
Textbook(s)

Hogg, Tanis, Zimmerman Probability and Statistical Inference (10th Edition)

Outline Updated 10/17

General Course Outline

Course Description

Lecture, four hours. Requisites: courses 31A, 31B, and 170E. The Math 170E and 170S two quarter probability and statistics sequence is aimed to equip Math-Econ and Financial Actuarial majors with essential skills in these areas. Math 170S is an introduction to statistics. Topics include sampling; estimation and the properties of estimators; construction of confidence intervals and hypotheses testing. It is designed to meet the Society of Actuaries’ VEE Requirements for Mathematical Statistics. Letter grading.

 
Textbook(s)

Hogg, Tanis, Zimmerman Probability and Statistical Inference (10th Edition)

Outline Updated 10/17

General Course Outline

Course Description

(Formerly numbered 151.)Lecture, three hours; discussion, one hour. Requisites: courses 33A, 170A (or Statistics 100A). Discrete Markov chains, continuous-time Markov chains, renewal theory. P/NP or letter grading.

Additional Information Probability and stochastic processes are used to create and analyze models in a broad range of fields, including statistics, economics, finance, engineering, biology and physics. Mathematics 170AB and 171 are designed to give a firm foundation in this area for students who will work and/or do graduate work in one of these fields. They also provide an excellent background for graduate work in probability and related areas of mathematics.

These courses are particularly well suited to students who plan to take the exams in actuarial science. The second exam in this series (number 110) is on probability and statistics. Mathematics 170AB covers roughly 2/3 of the material on that exam.

Course 170A is multiply listed with Statistics. Usually, two sections are offered each Fall Quarter, one by Mathematics and one by Statistics. Total enrollment in the two sections tends to be about 50.

The three courses are intended as a year-long sequence. However, it is possible, and not unusual, to take 171 without 170B. In fact, the enrollments in 171 are sometimes larger than in 170B (both are in the 10-20 range). Mathematics 170B is offered each Winter Quarter, and Mathematics 171 is offered each Spring.

General Course Outline

Course Description

(Formerly numbered 174.) Lecture, three hours; discussion, one hour. Enforced requisites: courses 33A, and 170E (or Math 170A or Statistics 100A). Not open for credit to students with credit for course 174A, Economics 141, or Statistics C183/C283. Mathematical modeling of financial securities in discrete and continuous time. Forwards, futures, hedging, swaps, uses and pricing (tree models and Black-Scholes) of European and American options, Greeks and numerical methods. P/NP or letter grading.

 
Textbook(s)

Hull, John C., Options, Futures and Other Derivatives, 10th Edition. Pearson 2018.

It is recommended to run course with one midterm in Week 6 and quizzes in discussion section in Weeks 2, 4, 8, and 10 whose total value is one midterm.

General Course Outline

Course Description

(4) Lecture, three hours; discussion, one hour. Requisite: course 32B. Types of interest, time value of money, annuities and similar contracts, loans, bonds, portfolios and general cash flows, rate of return, term structure of interest rates, duration, convexity and immunization, interest rate swaps, financial derivatives, forwards, futures, and options. Letter grading.

Course Information:

An introductory course on financial mathematics, Math 177 lays the foundation and prepares students for the series of courses required for the Financial Actuarial Mathematics major. By the end of this course, students should be familiar with numerous foundational concepts of financial mathematics, especially those from the theory of interest rates. Since one goal of the course is to help students prepare for the challenging Financial Mathematics (FM) exam) for the Society of Actuaries (SOA), two lectures before the midterms will be devoted to analysis of complex FM exam problems. While the basic ideas are mathematically elementary, their applications can be complex. The class is suitable for students who seek a career in financial engineering, the actuarial field, banking, etc., or are seeking to improve their financial literacy in a highly quantitative way.

 
Textbook(s)

Broverman, Samuel A. Mathematics of Investment and Credit. 7th ed., Actex Publications, 2017.

Bean, Michael A. (FSA, CERA FCIA, FCAS, PHD). Determinants of Interest Rates. Society of Actuaries, 2017. Education and Examination Committee of the Society of Actuaries – Financial Mathematics Study Note.
https://www.soa.org/Files/Edu/2017/fm-determinants-interest-rates.pdf

Alps, Robert (ASA, MAAA). Using Duration and Convexity to Approximate Change in Present Value. Society of Actuaries, 2017. Education and Examination Committee of the Society of Actuaries – Financial Mathematics Study Note.
https://www.soa.org/Files/Edu/2016/edu-2016-fm-24-17-using-duration-conv…

Beckley, Jeffrey (FSA, MAAA). Interest Rate Swaps. Society of Actuaries, 2017. Education and Examination Committee of the Society of Actuaries – Financial Mathematics Study Note.
Https://www.soa.org/Files/Edu/2017/fm-interest-rate-swaps.pdf

 
General Course Outline

Course Description

(4) Lecture, three hours; discussion, one hour. Requisite: course 32B, 175 or 177, 170A or 170E or Statistics 100A. An introductory course on to the mathematics associated with long term insurance coverages. Single and multiple life survival models, annuities, premium calculations and policy values, reserves, pension plans and retirement benefits. Letter grading.

Course Information:

A core sequence course for the Financial Actuarial Mathematics major, Mathematics 178A and the first half of Mathematics 178B cover the syllabus of the Society of Actuaries (SOA) Long-Term Actuarial Mathematics (LTAM) exam. By the end of this course, students will be able to value and set premiums for insurance instruments of numerous types using traditional actuarial models. They will also understand the typical models of life contingencies which are used in the calculations.

 
Textbook(s)

Dickson, David C.M., Hardy, Mary R. and Waters, Howard R, Actuarial Mathematics for Life Contingent Risks. 2nd ed., Cambridge University Press, 2013.

 
General Course Outline

Course Description

(4) Lecture, three hours; discussion, one hour. Requisite: 170S (or 170B or Statistics 100B), 178A. The second of the three quarter sequence 178ABC. Multiple state models, pensions, health insurances, profit testing. Topics in statistics used in actuarial work: methods of estimation and probability distributions. Letter grading.

Course Information:

Mathematics 178A and the first half of Mathematics 178B will almost completely cover the syllabus of the Long-Term Actuarial Mathematics exam by the Society of Actuaries. At the end of Mathematics 178A, students learned to value and set premiums for different types of insurances using traditional actuarial models. They were also exposed to typical models and calculations used in life contingencies. Mathematics 178B first extends this work to multistate models and then covers pensions, health insurances, and profit-testing. The last three weeks of the course will cover the probability distributions employed in most common actuarial theory and begins the study of the Short Term Actuarial Mathematics syllabus by the Society of Actuaries.

 
Textbook(s)

(DHW)
Dickson, David C.M., Hardy, Mary R. and Waters, Howard R., Actuarial Mathematics for Life Contingent Risks. 2nd ed., Cambridge University Press, 2013.

(Hardy)
Hardy, Mary R., Long-Term Actuarial Mathematics Study Note. Society of Actuaries, 2017. Education and Examination Committee of the Society of Actuaries – Long Term Actuarial Mathematics Supplementary Note.
https://www.soa.org/Files/Edu/2018/2018-ltam-supplementary-note.pdf

(KPW)
Klugman, Stuart A., Panjer, Harry H. and Willmot, Gordon E., Loss Models: From Data to Decisions. 3rd Edition, Wiley, 2012.

 
General Course Outline

Course Description

(4) Lecture, three hours; discussion, one hour. Requisite: 178B. This course is the third of the three quarter sequence 178ABC. 178C studies loss models associated with actuarial problems. It covers severity, frequency, and aggregate loss models, parameter estimation (frequentist, Bayesian), model selection and credibility. Letter grading.

Course Information:

The three quarter sequence 178ABC is the actuarial core of the FAM major. 178C covers topics associated with short term actuarial risk. With 178B, most of the topics 1-7 on the SOA STAM exam are covered.

 
Textbook(s)

S. Klugman, H. Panjer, G. Willmot, Loss Models: From Data to Decisions. 3rd Edition, Wiley, 2012.
Hardy, Mary R., Long-Term Actuarial Mathematics Study Note. Society of Actuaries, 2017.
Education and Examination Committee of the Society of Actuaries – Long Term Actuarial Mathematics Supplementary Note.
https://www.soa.org/Files/Edu/2018/2018-ltam-loss-models.pdf

 
General Course Outline

Course Description

(4) Lecture, three hours; discussion, one hour. Requisites: courses 174E. Continuation of Mathematics of Finance. In depth study of risk measures and the instruments of risk management in investment portfolios and corporate financial structure. Exotic and real options, value at risk, mean-variance analysis, portfolio optimization, risk analysis, capital asset pricing model, market efficiency and the Modigliani-Miller theory. P/NP or letter grading.

 
Textbook(s)

Hull, J. Optios., Futures and Other Derivatives, 10th edition. Pearson, 2018.

Berk, J. and P. DeMarz., Corporate Finance, 4th edition. Pearson, 2017.

White, Toby AMeasures of Investment Risk, Monte Carlo Simulation, and Empirical Evidence on the Efficient Markets Hypothesi Society of Actuaries, 2018. Education and Examination Committee of the Society of Actuaries ? Investment and Financial Markets Study Note.
https://www.soa.org/Files/Edu/2018/ifm-21-18-study-note.pdf

 
General Course Outline

Course Description

(4) Lecture, three hours; discussion, one hour. Requisites: courses 31A, 31B, and 61. Strongly recommended preparation: 115A. Graphs and trees. Planarity, graph colorings. Set systems. Ramsey theory. Random graphs. Linear Algebra methods. Ideal for students in computer science and engineering. P/NP or letter grading.

Course Information:

The following schedule, with textbook sections and topics, is based on 25 lectures. The remaining classroom meetings are for leeway, reviews, and midterm exams. These are scheduled by the individual instructor. Often there are reviews and midterm exams about the beginning of the fourth and eighth weeks of instruction, plus reviews for the final exam.

 
Textbook(s)

J. Matousek and J. Nesetril, Invitation to Discrete Mathematics, 2nd Ed., Oxford

Outline update: I. Pak, 12/15

General Course Outline

Course Description

(4) Lecture, three hours; discussion, one hour. Requisite: course 3C or 32A and 61. Lecture, three hours; discussion, one hour. Requisite: course 3C or 32A, and 61. Not open for credit to students with credit for Computer Science 180. Graphs, greedy algorithms, divide and conquer algorithms, dynamic programming, network flow. Emphasis on designing efficient algorithms useful in diverse areas such as bioinformatics and allocation of resources. P/NP or letter grading.

 
Textbook(s)

Kleinberg, Tardos: Algorithm Design, Addison Wesley

 
General Course Outline

Course Description

(Formerly numbered 180.) Lecture, three hours; discussion, one hour. Requisites: courses 31A, 31B, 61 and 115A. Permutations and combinations, counting principles, recurrence relations and generating functions. Application to asymptotic and probabilistic enumeration. Ideal for students in mathematics and physics. P/NP or letter grading.

 
Textbook(s)

M. Bona, Introduction to Enumerative Combinatorics , 2nd Ed., Chapman and Hall/CRC

Outline update: Pak, I., 12/15

General Course Outline

Course Description

(1) Tutorial, three hours. Limited to students in College Honors Program. Designed as adjunct to upper-division lecture course. Individual study with lecture course instructor to explore topics in greater depth through supplemental readings, papers, or other activities. May be repeated for maximum of 4 units. Individual honors contract required. Honors content noted on transcript. Letter grading.

General Course Outline

Course Description

(Formerly Math 197). Seminar, three hours. Math 191 is a variable topics research course in mathematics. Courses will cover material not covered in the regular mathematics upper division curriculum. Reading, discussion, and development of culminating project. May be repeated for credit with topic and/or instructor change. P/NP or letter grading.

General Course Outline

Course Description

(Formerly Math 190). Math Seminar, three hours. Participating seminar on advanced topics in mathematics. Content varies from year to year. May be repeated for credit by petition. P/NP or letter grading.

General Information Math 191H, the Honors Seminar: Mathematics, is offered once a year, quarter to be determined. The course is open to all Upper Division students who have done reasonably well in their other mathematics courses. Enrollment may be restricted by the instructor. Student participation is required, and the students are charged with presenting most of the material.

The instructor and the topic vary from year to year. The instructor for Spring 2004 is M. Takesaki. Topics treated in years past, and the instructors, are:

Spring 2005: The Banach-Tarski Paradox, G. Hjorth

Spring 2004: Introduction to Harmonic Analysis, M. Takesaki

Spring 2003: Introduction of Functional Analysis, M. Takesaki

Spring 2002: Matrix Groups, G. Hjorth

Spring 2001: Participating Seminar in Numerical Analysis, C. Anderson

Spring 2000: Introduction to Coding Theory and Information Theory, D. Blasius

Fall 1998: The Theory of Groups and Quantum Mechanics, M. Takesaki

Fall 1997: Basic Examples in Dynamical Systems, R. Perez-Marco

Fall 1996: Knots and their Invariants, S. Popa

Winter 1996: Control Theory: Pure and Applied, P. Petersen

Spring 1995: Rational Points on Elliptic Curves, R. Elman

Winter 1994: Fractal Geometry — Mathematical Foundations and Applications, L. Young

Winter 1993: Topics in Elementary Number Theory, M. Green

Winter 1992: Complex Dynamical Systems, T. Gamelin

Winter 1991: Continued Fractions, L. Carleson

Spring 1990: Mathematical Principles of Scientific Computing, B. Engquist

Winter 1990: An Introduction to Chaotic Dynamical Systems, D. Babbitt

General Course Outline

Course Description

Tutorial, to be arranged. Limited to juniors/seniors. Internship to be supervised by Center for Community Learning and Mathematics Department. Students meet on a regular basis with instructor, provide periodic reports of their experience, have assigned readings on mathematics education, and complete final paper. The final paper is a substantial part of course, and will require a significant investment of time during the quarter. May not be repeated and may not be applied toward major requirements. Individual contract with supervising faculty member required. P/NP grading.

General Course Outline

Course Description

(2 to 4 units). Tutorial, three hours per week per unit. Limited to juniors/seniors. At discretion of chair and subject to availability of staff, individual intensive study of topics suitable for undergraduate course credit but not specifically offered as separate courses. Scheduled meetings to be arranged between faculty member and student. Assigned reading and tangible evidence of mastery of subject matter required. May be repeated for maximum of 12 units, but no more than one 197 or 199 course may be applied toward upper division courses required for majors offered by Mathematics Department. Individual contract required. P/NP or letter grading.

General Information The Math 197 title has been used to cover coursework for a course that is listed in the catalog but not given in a particular year. However, Math 197 cannot be used to duplicate the coverage of a regularly offered course. University regulations specify that courses labeled 197 are open only to juniors and seniors with a 3.0 GPA in their major field. Math 197 is intended for students who have already taken a number of Math and PIC courses.

In order to enroll in a 197 course, the student’s petition must receive the approval of the sponsoring faculty member and of the Undergraduate Vice Chair. The petition should spell out student’s obligations are for successful completion of the course, including what will be covered in the course, how often the student will meet with the faculty sponsor, and what written material will be required.

While the 197 course is meant to be flexible, to cover students or groups of students with special interests and in special situations, there is a list of criteria that the Undergraduate Vice Chair considers before giving approval to a 197 petition. Some of these conditions have been mentioned above. Exceptions to these conditions are rare. The conditions are:

1. Math 197 is intended for students who have already taken a number of Math and PIC courses.

2. Math 197 cannot be used to duplicate the coverage of a regularly offered course.

3. The 197 course should be sponsored by a regular Mathematics Department faculty member.

4. Before agreeing to sponsor a 197 course, the faculty member should have some good grounds upon which to assess the student’s potential and level of ability, such as having had the student in another course.

5. There should be roughly 30 hours work for each unit credit.

6. The faculty sponsor and the student should meet on a regular basis, which should be specified in the petition. For four units credit, weekly meetings are appropriate, while for two units credit, biweekly meetings suffice.

7. There must be some written work, specified in the petition, that is submitted to the sponsoring faculty member and available to the Undergraduate Vice Chair upon the conclusion of the course.

8. Math 197 credit will not be given for work also turned in for another course.

9. Math 197 is not appropriate for field study credit, except in conjunction with a project for a Mathematics Department faculty member that also has a major component of reading on an advanced topic.

General Course Outline

Course Description

(2 or 4 units). Tutorial, three hours per week per unit. Limited to juniors/seniors. Supervised individual research under guidance of faculty mentor. Scheduled meetings to be arranged between faculty member and student. Culminating report required. May be repeated for maximum of 12 units, but no more than one 197 or 199 course may be applied toward upper division courses required for majors offered by Mathematics Department. Individual contract required. P/NP or letter grading.

General Information The research course, Math 199, provides an excellent opportunity for an advanced student to do research on a mathematical topic under the guidance of a faculty member. In a typical situation, the student finds an interesting topic in a mathematics course and wants to pursue the topic in the subsequent term. If the student has shown initiative and done well in the course, the faculty member may agree to direct the student’s further research through a 199 course. Occasionally a group of students will approach a professor to take a 199 course together, doing research on some aspect of a course they are currently taking from the professor.

University regulations specify that courses labeled 199 are open only to juniors and seniors with a 3.0 GPA in their major field. Math 199 is intended for students who have already taken a number of Math and PIC courses.

In order to enroll in a 199 course, the student’s petition must receive the approval of the sponsoring faculty member and of the Undergraduate Vice Chair. The petition should spell out student’s obligations are for successful completion of the course, including what will be covered in the course, how often the student will meet with the faculty sponsor, and what written material will be required.

While the 199 course is meant to be flexible, to cover students or groups of students with special interests and in special situations, there is a list of criteria that the Undergraduate Vice Chair considers before giving approval to a 199 petition. Some of these conditions have been mentioned above. Exceptions to these conditions are rare. The conditions are:

1.Math 199 is intended for students who have already taken a number of Math and PIC courses.

2.The 199 course should be sponsored by a regular Mathematics Department faculty member.

3.Before agreeing to sponsor a 199 course, the faculty member should have some good grounds upon which to assess the student’s potential and level of ability, such as having had the student in another course.

4.There should be roughly 30 hours work for each unit credit.

5.The faculty sponsor and the student should meet on a regular basis, which should be specified in the petition. For four units credit, weekly meetings are appropriate, while for two units credit, biweekly meetings suffice.

6.There must be some written work, specified in the petition, that is submitted to the sponsoring faculty member and available to the Undergraduate Vice Chair upon the conclusion of the course.

7.Math 199 credit will not be given for work also turned in for another course.

8.Math 199 credit will not be given for standard programming work alone. While a computer project can form part of the work, there should also be a major component of research on an advanced topic.

9.Math 199 is not appropriate for field study credit, except in conjunction with a project for a Mathematics Department faculty member that also has a major component of research on an advanced topic.

pic courses

General Course Outline

Catalog Description

(5 units). Lecture, three hours; discussion, two hours; laboratory, eight hours. Recommended requisite for students with no prior computing experience: course 1. Students with credit for course 3 will receive only two units of credit for this course. No prior programming experience assumed. Basic principles of programming, using C++; algorithmic, procedural problem solving; program design and development; basic data types, control structures and functions; functional arrays and pointers; introduction to classes for programmer-defined data types. P/NP or letter grading. Usually offered every quarter.

This class is intended for those of you who need to know how to write your own computer programs. This class will teach you how to design and develop computer programs using sound programming techniques. This class does not assume prior programming knowledge, but if you don’t have at least some familiarity with computers, consider taking PIC 1 first.

General Course Outline

Catalog Description

(5 units). Lecture, three hours; discussion, two hours; laboratory, eight hours. Enforced requisite: course 10A. Abstract data types and their implementation using C++ class mechanism; dynamic data structures, including linked lists, stacks, queues, trees, and hash tables; applications; object-oriented programming and software reuse; recursion; algorithms for sorting and searching. P/NP or letter grading

General Course Outline

Catalog Description

(5 units). Lecture, three hours; discussion, two hours; laboratory, eight hours. Enforced requisite: course 10B. More advanced algorithms and data structuring techniques; additional emphasis on algorithmic efficiency; advanced features of C++, such as inheritance and virtual functions; graph algorithms. P/NP or letter grading

General Course Outline

Catalog Description

(5 units). Lecture, three hours; discussion, two hours; laboratory, eight hours. Enforced requisite: course 10A. Introduction to symbolic computation using Lisp programming language. Basics: list structures, recursion, function abstraction. Advanced topics: knowledge representation, higher-order functions, problem-solving algorithms and heuristics. P/NP or letter grading.

This course is intended for those who are interested in cognitive science or any student who enjoys programming and mathematics. LISP is a functional programming language in which programs are written by a series of function calls instead of assignments. LISP is used heavily in Artificial Intelligence. Example programming assignments may include Eliza (a program which simulates a psychiatrist) and the heuristic based search of a graph (searching a graph for a “good” path instead of the “best” path).

General Course Outline

Catalog Description

(5 units). Lecture, three hours; discussion, two hours. Enforced requisite: course 10A, Computer Science 31, or equivalent, with grades of C- or better. Python programming and programming with Python packages. General Python programming constructs; standard data structures, flow control, exception handling, and input and output. Object oriented programming with Python. Application programming with commonly used Python modules such as PyQt or tkinter, NumPy, SciPy, and NLTK. P/NP or letter grading.

 
Textbook

A: The Python Tutorial (https://docs.python.org/2/tutorial/index.html)
B: RegexOne- Learning Regular Expressions (http://regexone.com/)
C: PyQt4 Tutorial (http://zetcode.com/gui/pyqt4/)

 
General Course Outline

Catalog Description

(5 units). (Formerly numbered 20.) Lecture, three hours; discussion, two hours; laboratory, eight hours. Enforced requisite: course 10A. Not open for credit to students with credit for course 3. Introduction to Java computer language. Class and interface hierarchies; graphics components and graphical user interfaces; streams; multithreading; event and exception handling. Issues in class design and design of interactive Web pages. P/NP or letter grading.

General Course Outline

Catalog Description

(5 units). Lecture, three hours; discussion, two hours; laboratory, eight hours. Enforced requisite: course 20A. Further aspects of use of classes, graphics components, exception handling, multithreading, and multimedia. Additional topics may include networking, servlets, database connectivity, and JavaBeans. P/NP or letter grading.

General Course Outline

Catalog Description

(5 units). Lecture, three hours; discussion, two hours; laboratory, five hours. Enforced requisite: course 20B. Overview of Enterprise Java APIs: remote method invocation, database access with SQL, servlets, and JSP. Issues in implementation of server-side Java applications. Use of Java in conjunction with XML. Individual or group projects and presentations. P/NP or letter grading.

General Course Outline

Catalog Description

(5 units). Lecture, three hours; discussion, two hours; laboratory, eight hours. Enforced requisite: course 10B. Not open for credit to students with credit for former Computer Science 30. Description of machine organization and operation. Representation of information, instruction sets and formats, addressing modes, memory organization and management, I/O processing and interrupts. P/NP or letter grading.

This course is intended for those who are interested in learning how the computer “really” works on the inside. The course will discuss how memory and the internal components of computers interact, as well as I/O processing and interrupt handling. The target processor for the assembly programming is the Intel family of processors using the Microsoft Assembler (MASM). Issues of target processor, memory model, data representation, and stack considerations are covered.

General Course Outline

Catalog Description

(5 units). (Formerly numbered 40.) Lecture, three hours; discussion, two hours; laboratory, eight hours. Enforced requisite: course 10A. Recommended: course 10B. Introduction to core technologies of Internet, with focus on client-side Web programming. Fundamental protocols, static Web pages, Perl language, Common Gateway Interface, XML. P/NP or letter grading.

General Course Outline

Catalog Description

(5 units). Lecture, three hours; discussion, two hours; laboratory, eight hours. Enforced requisite: course 40A. Study of advanced topics in Web programming, with focus on server-side technologies. P/NP or letter grading.

General Course Outline

Catalog Description

(4 units). Lecture, three hours; discussion, one hour, laboratory, five hours. Enforced requisite: course 10B, Mathematics 31A, 31B, 61. (Math 180, Formely 113 can be accepted in place of Math 61.) Review of basic data structures: arrays, stacks, queues, lists, trees. Advanced data structures: priority queues, heaps, balanced trees. Sorting, searching techniques. Corresponding algorithms.

General Course Outline

Catalog Description

(4 units). Lecture, one to three hours; discussion, zero to one hour. Enforced requisite: course 10A. Variable topics in programming not covered in regular program in computing courses. May be repeated for credit with topic change. P/NP or letter grading. (See schedule for any current listings.)

A section of PIC 97 is offered occasionally, for example, as an experimental version of a course before it becomes a regular course.

General Course Outline

Catalog Description

(5 units). Lecture, three hours; discussion, two hours; laboratory, eight hours. Requisite: course 10B or equivalent familiarity with programming in C or C++ language. Introduction to programming of parallel computers. Shared and distributed memory parallel architectures; currently available parallel machines; parallel algorithms and program development; estimation of algorithmic performance; distributed computing; selected advanced topics. P/NP or letter grading.

Students will have access to a large IBM SP-2 parallel computer for this course.

General Course Outline

Catalog Description

(5 units). Lecture, three hours; discussion, one hour; laboratory, three hours. Requisites: course 10B, Mathematics 115A. Design and analysis of cryptosystems for confidentiality and authentication. Classical cryptosystems and their security, modern private-key cryptosystems and applications, public-key cryptography and applications; generating prime numbers, factoring integers, discrete logarithms, digital signatures, perfect secrecy. P/NP or letter grading.

General Course Outline

Catalog Description

(Formerly numbered 197). Lecture, three hours; discussion, one hour. Variable topics in programming and mathematics of programming not covered in regular program in computing courses. May be repeated for credit with topic change. P/NP or letter grading. (See schedule for any current listings.)

Watch the Schedule of Classes each quarter for interesting offerings with this course number. Recent courses have included Modern Heuristics, Introduction to Scientific Research, and Introduction to UNIX System Administration.