You may choose one of the following courses to satisfy this requirement:

• M328K: Introduction to Number Theory, or
• M343K: Introduction to Algebraic Structures, or
• M373K: Algebraic Structures I

• M328K: Introduction to Number Theory: Number Theory is the study of various topics involving integers. Examples of topics that might be studied in this course are: Why can every integer be factored uniquely into primes? How frequently do primes occur as we look at large and larger numbers? Which equations with integer coefficients have integer solutions?
• M343K: Introduction to Algebraic Structures: "Algebraic structures" refers to the study in an abstract setting of certain properties (e.g., the commutative, distributive, and associative properties) that are familiar from numbers. The two main structures that might be studied in this course are called rings and groups. A ring is a system that has both an addition and a multiplication that interact in certain ways. The set of all integers and the set of all rational numbers are two examples of rings. A group is a system that has only one operation (for example, addition, or multiplication, or composition) that satisfies certain properties. The integers, if we ignore multiplication and just look at addition, provide an example of a group. So does the set of all rotations about the origin in the plane, with composition of functions as the operation. Some instructors include the study of both groups and rings in M343K; some instructors specialize to just one of these topics.
• M373K: Algebraic Structures I: This course is a more advanced version of M343K. It includes the study of both rings and groups. M373K is intended to prepare students for graduate study in mathematics

 All three courses are proof courses, and therefore address the Logical Reasoning portion of SBEC Standard V: Mathematical Processes:

They also address the following item in the Communication portion of Standard V:

All three courses also address the first part of item 6.7s of SBEC Standard VI. Mathematical Perspectives:

• "The beginning teacher of mathematics is able to … analyze the structure of mathematical systems and use the structural properties of mathematical systems … "

 M328K also addresses SBEC Standard I. Number Concepts, especially item 1.28s:

• "The beginning teacher of mathematics is able to … investigate and apply fundamental number theory concepts and principles (e.g., divisibility, Euclidean algorithm, congruence classes, modular arithmetic, the fundamental theorem of arithmetic) in a variety of situations."

M 343K and M 373K may address item 1.23s of Standard I:

• "The beginning teacher of mathematics is able to … demonstrate an understanding of the real and complex number systems as algebraic fields."

These courses can help prepare the teacher to teach according to Basic Understanding (2) of the TEKS for Algebra II:

• "(2) Algebraic thinking and symbolic reasoning. Symbolic reasoning plays a critical role in algebra; symbols provide powerful ways to represent mathematical situations and to express generalizations. Students study algebraic concepts and the relationships among them to better understand the structure of algebra."

In addition, since they are proof courses, they help prepare the teacher to teach according to item b(3) of the Geometry TEKS:

• The student understands the importance of logical reasoning, justification, and proof in mathematics.

These courses can help prepare you to teach according to the Represent and Analyze portion of the 9-12 Algebra standard:

• "Represent and analyze mathematical situations and structures using algebraic symbols"

Most importantly, these courses can help you develop your skills in reasoning and proof so that you can teach according to the 9-12 Standard for Reasoning and Proof.

This course will give you an introduction to one of the main areas of research mathematics, and thus will help you teach from the perspective of a broad view of the field of mathematics

• M 325K (Discrete Mathematics) lays a foundation in areas such as logic, basic proof techniques, and mathematical induction that are built upon in these course.
• M 326K (Foundations of Number Systems) studies the algebraic properties of numbers, which are then studied more deeply and more abstractly in these courses.
• M 341 (Linear Algebra and Matrix Theory) provides an introductory experience with proofs which is built on in these three courses. It also develops the basic theory of matrices, which is then used to provide examples in M 343K and M 373K.
• M 360M (Mathematics as Problem Solving) helps develop problem solving and proof techniques that are useful in these three courses.
• These courses give in introduction to the methods of mathematical research. This helps give you background to understand the difference between research methods in math and the various sciences in CH 368/PHY 341 (Research Methods)

Most students in these degree plans take either M 328K or M 343K. However, students who think they may want to go to graduate school, or who like a mathematical challenge, may opt to take M 373K. This option should not be chosen unless you have had strong performance in both M 341 and M 325K.

The choice between M 328K and M 343K is up to you. Sometimes one of these courses is taught by a discovery method called the Moore Method. These sections are especially recommended for future teachers.

You can only count one of these courses toward the BA-Teaching Option. If you are pursuing the BS-Teaching Option, you can take two of these courses and count the second one toward the five hours of math electives.

• First, be sure to get the most out of the courses M 325K and M 341 that help prepare you for the algebra courses.
• Remember that most students find these courses challenging, so be prepared for a challenge. Don't expect the ideas to come automatically; you will need to work at them. Be sure to devote adequate time to studying. This usually means more than just doing homework and studying for exams.
• Review the SBEC standards quoted above to help keep your focus on the importance of developing logical reasoning skills, and not just narrowly learning the topics in the course.