Why are we doing any of the stuff below?

  1. With the change in major, we need an updated outcome list, which is very vague on purpose.
  2. For new teachers who want to take on this course, a topic list would help ensure a similar experience across several teachers. Having 461 teachers add topics to this list that they truly want would help the 301 teachers know how to focus their efforts. Generating a list of books that we've reviewed can help with this as well.
  3. Having some common exam questions can help us as teachers see where we are weak and where we can improve.

Topics

The objectives to the course are barebones. It would be nice to provide a more detailed list of specific mathematical topics/definitions/theorems that the students should have seen upon taking math 301.

The list below is fully editable. Just click "Edit" from the top menu, or bottom of this page.

At some point, I (Ben) will put the list into complete sentences. If there are specific definitions, theorems, examples/non examples, you want students to see/explain/proof/explore, please add them below.

  1. Set Theory
    • Know the definitions of and basic notation for inclusion, containment, intersections, unions, and complements.
    • Prove that a set is a subset of another.
    • Prove that two sets are equal.
    • This might be a good place to introduce the concept of infinite intersections, unions, nested subsets; in preparation for understanding limits.
    • Set notation. Example: $A = \{x \in \mathbb{R}| \; \lvert x+2 \rvert < 5\}$
  2. Logic
    • Accuracy and precision in thought and communication.
    • Construct truth tables and use the appropriate rules for $p\wedge q$, $p\vee q$, $p\implies q$, and $\sim p$. (Maybe truth tables are overkill? But students need to understand these logical operators and their negations.) Relationship between $\iff$ and if p then q.
    • For a conditional statement, give the contrapositive, converse, and inverse. In addition, be able to explain the logical equivalence between appropriate statements. (This is needed.)
  3. Quantifiers and Common Symbols.
    • $\forall$, and $\exists$, $\not\forall$, $\not\exists$---all four separately and in combination. (Order matters!)
    • $\in$, $\ni$, $\mathbb{Z}$, $\mathbb{Q}$, $\mathbb{R}$
  4. Direct and Indirect Proof Techniques
    • Mathematical Induction
    • Know what a counterexample is. Know how to construct a counterexample. Ditto for examples and nonexamples.
    • Recognize the difference between constructive proofs and nonconstructive proofs.
  5. Properties of Functions
    • Domain, codomain, image, preimage. Please do not use the word "range."
    • Injection, surjection, bijection (or equivalent vocabulary).
    • Larry: I refer to "Principles of Mathematical Analysis" by Walter Rudin (affectionately know as "Baby Blue Rudin" by many mathematician/analysts). See pp 24-25 for definitions and vocabulary related to functions in an analysis setting.
  6. Axioms of Number Systems (the Real numbers)
    • Use a list of axioms to prove other facts about a number system.
    • Since M301 is preparation for M461, perhaps we should focus on the Axioms of the Real numbers. (A response: Yes indeed, and part of the focus can involve contrasting the axioms for the reals with axioms for the rationals, the integers, and the natural or whole numbers.)
  7. Completeness Axiom (of the Real numbers)
    • Supremum and Infimum---The completeness of the reals is what underlies all of undergraduate calculus and analysis.
    • Emphasize concept of infinite versus finite, moving toward the concept of limit.
  8. Open Sets
    • (In what sense(s)? Sets without boundary points? Unions of open intervals?)
    • (Closed sets?)
  9. Limit Points
    • (Presumably, we mean limit points of sets. If so, should this come before or after limits of sequences?)
    • It seems safe to define a limit point of a set in terms of $\epsilon-$neighborhoods, allowing the study in open and closed sets, limit points, compact sets, etc. prior to the study of sequences.
  10. Sequences
    • Convergence, divergence, uniqueness of limits.
    • Every bounded monotone sequence converges.
  11. Limits of Functions
    • At some point, drawing a distinction between the discrete (sequences) and the continuous (functions) would be helpful.
  12. Other possibilities:
    • Absolute value as distance
  13. One way to describe real analysis that it is the study of infinite sets (of real numbers, real valued functions, ...) and their properties. Tightly coupled to that is the concept of limit. These two themes (the transition from finite to infinite, and limits) should be illustrated and emphasized at every possible opportunity throughout the courses.

Outcomes

Ben will send out an updated Outcomes list together with a course description.

  • One of the outcomes should be, "Students are well prepared for M461."
  • Outcome #4 in the current list refers to "$\epsilon-\delta$ techniques". Now that some M441 material is being removed, does "$\epsilon-\delta$ techniques" rise to the level of reality versus the current dream status?

Books

Ben and Paul will review the open source and free books, and provide brief reviews. Bonnie and Dave will review their list and then a suggested book list for the course will be the outcome.

Exams

Larry and Bonnie will create a few questions that should be administered to students in every section of Math 301. Dave will contribute.

LaTeX

Curtis will look at several universities to evaluate possible LaTeX resources we could use here on campus.

Name Change? Change to 201

Should we change the number to 201, instead of 301? Feel free to add any thoughts you have.

  • Ben: If the number is lower, students are more likely to take the class sooner. We want them in this course ASAP after having 113, so they can see what math is about before it gets too late to change their major.
  • Dave: Courses have numbers for reasons. A 200-level course should probably be intellectually less demanding in some sense than a 300-level course. Maybe a 201 should assume less about student preparation than a 301? Maybe a 201 should be a little gentler? Maybe this fits well with what we hope to accomplish here?
  • Larry: One of the effects of course numbers (intended or not) is that students interpret them as an indicator of the order in which they ought to take the course, supporting Ben's proposal. Personally, I am undecided. If that changes, I will weigh in (after I lose some weight).