Introduction
There are many challenges that confront us, math teachers. For instance, many students not only often resist learning challenging topics like mathematics but also do not realize how consequential poor mathematical preparation could be to their future selves. There are tremendous responsibilities that come with being an educator, and even though we are not capable of fully grasping and controlling the potential impact that educators may have on their students, a number of variables remain under our control. Among such variables, preparation ranks high in my book. As a starter, we need to have a firm grasp of the content that we are aiming to teach. In my view, the more we know, the better. Knowing mathematics in both depth and breadth provides us with a bird’s eye view of the materials, the inner connections between various subcomponents, and how to deliver relevant content to an audience at an appropriate level.
The case that I am attempting to make here, is for the importance and value of conceptual mathematics which is often viewed as a useless approach to learning since it is often unclear how to draw a straight line between conceptual understanding and the more algorithmic and ubiquitous approach of teaching mathematics. This begs the following question:
Is it important to learn mathematics at a conceptual level if teaching pre-higher ed is our main interest?
I suspect the answer to this question to be yes. However, I am unwilling to be presumptuous enough to impose this (informed) opinion of mine on you. As an educator that regards the classroom environment as a learning community, with great interest, I am turning to you to help me answer this thematic question.
Materials
Even though, I am not requiring you to purchase any particular textbook. Most of my lectures will be inspired by the following text: Understanding Real Analysis, Second Edition by Paul Zorn.
Office Hours
| Day | Time | Location |
|---|---|---|
| Monday | 2:30 PM – 3:30 PM | Lewis and Gaines Center for Inclusion and Equity |
| Thursday | 2:00 PM – 3:00 PM | DMF 453 |
Homework
| Platform | Gradescope |
| Entry Code | DK73VD |
Grading Scheme
| Category | Weight (%) | Details |
|---|---|---|
| Homework | 25% | – Weekly assignments on Gradescope. – Due dates aligned with topics covered in the schedule. |
| Midterm Exam | 25% | – One exam, administered during regular class time. – See the “Schedule of Topics” for the planned exam date. |
| Final Exam | 25% | – Comprehensive final exam during May 7 (Wed) – May 13 (Tue). – Date/time will follow the official final exam schedule or special arrangement for once-weekly classes. |
| Participation | 25% | – Attendance, engagement in discussions, in-class activities, and office hours. |
Schedule of Topics (Spring 2025)
This course meets on Mondays from 4:45 PM to 6:00 PM. Below is a weekly schedule reflecting holidays, breaks, and final exam dates from the official academic calendar. Note that each “Week” corresponds to a single meeting (unless a holiday disrupts the schedule). Adjustments may be necessary based on the class’s progress.
| Week / Date | Topics / Sections | Details / Remarks |
|---|---|---|
| Week 1 Jan 27 (Mon) | The Very Basics | First class meeting is Monday, Jan 27, 4:45–6:00. Course introduction, syllabus overview. Basic properties of numbers, notation. |
| (Note on Holidays) | Feb 17 (Mon) – Presidents’ Day: No classes. | |
| Week 2 Feb 3 (Mon) | Getting Started | Fundamental set concepts, subsets, set operations and notations. |
| Week 3 Feb 10 (Mon) | The Idea of a Function | Definition of functions, domain, codomain, range. Mapping diagrams and examples. |
| Week 4 Feb 24 (Mon) | Proofs and Proof-Writing | Structure of mathematical proofs, direct methods. Common pitfalls in logic. |
| (Feb 17 & Feb 19 Details) |
Feb 17 (Mon): No class (Presidents’ Day). | |
| Week 5 Mar 3 (Mon) | Types of Proof | Indirect proofs, contrapositive, contradiction. |
| Week 6 Mar 10 (Mon) | Midterm Exam | Take-home exam |
| Mar 10 – Mar 14 (Mon–Fri) | The official Spring Break begins Mar 10 (Mon). | |
| Week 7 Mar 17 (Mon) | Finite and Infinite Sets; Cardinality | Mathematical induction video lecture |
| Week 8 Mar 24 (Mon) | Bounds | Upper, lower bounds, max min, supremum, and infimum. |
| Week 9 Mar 31 (Mon) | Completeness | Completeness axiom, completeness of real numbers. Consequences of completeness. |
| Week 10 Apr 7 (Mon) | Sequences and Convergence, Working with Sequences | Definitions, convergence criteria. |
| Week 11 Apr 14 (Mon) | Subsequences, Cauchy Sequences | Subsequences and convergence properties. Cauchy criterion for convergence. |
| Apr 21 (Mon) |
Patriots’ Day – No classes. | |
| Week 12 Apr 28 (Mon) | Limits of Functions, Continuous Functions | ε-δ definitions of limits and continuity. Basic continuity theorems. |
| Week 13 May 5 (Mon) (Last Day of Instruction) | Final exam | Take-home |