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

• M361K: Introduction to Real Analysis
• M365C: Introduction to Analysis

• M361K: Introduction to Real Analysis: The term "analysis" refers to the part of mathematics that deals with infinite (or infinitesimal) processes (e.g., taking limits of functions, differentiation, integration, summing infinite series). Students are first exposed to such concepts in beginning calculus, where the emphasis is placed (and properly so) on using them to solve concrete problems (e.g., to find the tangent line to a curve, to find the area under the graph of a function). Real analysis revisits these topics but seeks to develop an understanding of the processes themselves. In particular, an attempt is made to give proofs for many of the theorems that are typically introduced in a nonrigorous fashion in calculus. Many teachers of M361K see as the course's main objective to give students a proof of the Fundamental Theorem of Calculus, which explains how the two basic processes of differentiation and integration are related to one another. Naturally, this proof requires that the stage be carefully set by discussing the many ideas that enter into the Fundamental Theorem, universally acknowledged as one of the crowning achievements of human thought.
• M365C: Introduction to Analysis: This course is a more advanced version of M361K. It includes the study of metric spaces and sequences and series of functions. M365C is intended to prepare students for graduate study in mathematics.

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

Both courses also address these items of communication:

• "5.21s translate mathematical statements among developmentally appropriate language, standard English, mathematical language, and symbolic mathematics;
• 5.24s use the language of mathematics as a precise means of expressing mathematical ideas."

How do these courses relate to the Texas Essential Knowledge and Skills for secondary mathematics?

These courses can help prepare the teacher to teach according to Section 111.34 (Geometry)(b)(3):

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

and Section 111.35 Precalculus (c)(4)(c)

• describe limits of sequences and apply their properties to ivnestigate convergent and divergent series.

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 both research mathematics and applied mathematics, and thus will help you teach from the perspective of a broad view of the field of mathematics.

• M408C/D (Calculus I and II) explains what calculus is and how to use it to solve problems. In these courses one learns about the deep principles underling calculus and why it works.

• M 325K (Discrete Mathematics) lays a foundation in areas such as logic, basic proof techniques, and mathematical induction that are built upon in these courses.

• M 360M (Mathematics as Problem Solving) helps develop problem solving and proof techniques that are useful in these courses.

• These courses give an 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 M361K. However, students who think they may want to go to graduate school, or who like a mathematical challenge, may opt to take M365C. This option should not be chosen unless you have had strong performance in both M341 and M325K.

You can only count one of these courses toward the BA-Teaching Option. If you are pursuing the BS-Teaching Option, you can take both 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 M325K and M341. Many 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.