We’ll use this page to keep track of what has happened each day in class. It won’t contain any of the nitty-gritty details, but will instead serve to summarize what has transpired each day.

## Week 1

• Wednesday, January 18: First day! After attempting to learn names, I discussed the first few homework assignments and summarized what content can be found on the course webpage. Next, we spent a few minutes on an activity that I called [Deep Learning and Productive Struggle](/teaching/mat320s17/DeepPractice.pdf). We wrapped up class by doing a quick overview of the syllabus.
• Friday, January 20: Great first day of student presentations! I was really pleased with how things went. After answering a few quick questions and making some announcements, groups of 2-3 spent a couple minutes discussing their homework. Next, we had LS volunteer to present a proof of Theorem 2.3. LS did a great job and her proof generated lots of good discussion. Afterwards, JN volunteered to present a proof of Theorem 2.4, which also provoked good discussion.

## Week 2

• Monday, January 23: Since classes starting after noon were canceled, we ended class 20 minutes early. With the time we had available, we revisited the definition of even and odd and took at look at the definition of "divides". In particular, we discussed some pitfalls related to the notation (see Question 2.11). In addition, we spent a little time discussing the difference between "proofs" and "counterexamples". Along the way, we tackled Problem 2.5, Problem 2.7, Theorem 2.14, and Problem 2.16. At some point, we still need to address Problems 2.8, 2.9, 2.12, and 2.13, as well as Corollary 2.15 and Theorem 2.17.
• Wednesday, January 25: I was really happy with how things went today. Lots of good discussion and comments. We jumped right in and had SC, DF, QC, SG, and JN1 present Problem 2.8, Problem 2.12, Problem 2.13, Corollary 2.15, and Theorem 2.17, respectively. JO, TC, and JR will tackle Theorem 2.18, Problem 2.19, and Theorem 2.20 next time.
• Friday, January 27: A productive day. We had JO, TC, JR, JQ, and RF presented Theorem 2.18, Problem 2.19, Theorem 2.20, Problem 2.21, and Theorem 2.22, respectively. Along the way, we discussed an alternate proof of Theorem 2.20. CP will tackle Theorem 2.23 at the beginning of class next time.

## Week 3

• Monday, January 30: We kicked off with CP presenting Theorem 2.25. Next, we split the class into several small groups. After several minutes, we had DD, FA, SR, and EV share out about Exercise 2.25. Next, SG, DF, LS presented Exercises 2.27, 2.28, and 2.31 respectively. We will get to Problem 2.32 and Theorem 2.34 next time.
• Wednesday, February 1: We cruised through Problem 2.32, Theorem 2.34, and Theorem 2.35, which were presented by CP, CK, and AV, respectively. Next, we collectively discussed Problem 2.36 and Exercises 2.38-2.41. This was followed by a presentation of Theorem 2.42 by JN2. I went to the board of presented a proof of Theorem 2.43 by proving the contrapositive. We spent the last 10 minutes of class discussing LaTeX. We'll get to Theorems 2.44 and 2.45 on Friday or Monday.
• Friday, February 3: While I was out of town, Dr. Falk covered for me. JO, SG, GC, JR/SR, LS, and AT presented Theorem 2.44, Theorem 2.45, Theorem 2.46, Corollary 2.47, Exercise 2.48, and Exercise 2.49, respectively. JR and SR presented two different approaches to Corollary 2.47.

## Week 4

• Monday, February 6: After checking in to see how Friday went while I was out of town, I summarized the three approaches we can take to proving a conditional statement (direct, contraposition, and contradiction). I also quickly discussed the big picture of Section 2.4. We kicked off with EV doing a quick presentation of Theorem 2.51. Next, FA and JN1 presented two alternate approaches to Theorem 2.55. We finished up with CD discussing Theorem 2.56. Unfortunately, we had to end class in the middle of a great discussion. AV will start us off with Exercise 2.57 next time.
• Wednesday, February 8: We started with a AV presenting Exercise 2.57. Next, we walked through Exercises 2.58, 2.60-2.61 together. This was followed by RF, GC, QC, and SC presenting Exercises 2.65(a), 2.65(b), 2.65(c), and 2.66, respectively.
• Friday, February 10: Busy day! After some discussion of Theorem 2.72, we split the class up into 9 small groups, where each group was tasked with discussing a chunk of the homework. We had CD, JR AT, TC, JQ, DD, and JH present Exercises 2.67, 2.68, 2.69(abc), 2.69(abc), 2.71, 2.73(abcd), and 2.73(efgh), respectively.

## Week 5

• Monday, February 13: After briefly discussing the upcoming exam, we divided the class up into 9 small groups, where each group was responsible for one of the homework questions. We had CK, DD, QC, JN1, CP, and JH Problem 2.79(a), (b), (c), (d), (f), and Theorem 2.80, respectively.
• Wednesday, February 15: The students took the in-class portion of Exam 1.
• Friday, February 17: After handing back the in-class portion of Exam 1, I lectured over the first bit of Chapter 3: Set Theory and Topology. In particular, we cover 3.1-3.12. We will pick up where we left off on Monday.

## Week 6

• Monday, February 20: I continued lecturing over Chapter 3: Set Theory and Topology. We covered 3.13-3.19.
• Wednesday, February 22: We had JO, DF, AV, QC, and EV present Theorem 3.20(a), Exercise 3.23(a), Exercise 3.23(b), Exercise 3.23(c), and Exercise 3.23(d), respectively. Along the way, I wrote out the proof of Theorem 3.21(a). We didn't get to Theorems 3.25-3.27, so we will have squeeze those in next time.
• Friday, February 24: As planned, I presented a proof of Theorem 3.25. Next, JR, LS, JN2, and TC presented Theorem 3.26, Theorem 3.27, Problem 3.28, and Problem 3.29, respectively.

## Week 7

• Monday, February 27: I had surgery on my back this morning and I will be out for 2 weeks. In my absence, Dr. Falk is covering this week. My understanding is that paradoxes, Russell's Paradox in particular, were discussed some more. With the time that remained, SR and CP presented 3.35 and 3.36. The plan for Wednesday will be to pick up where y'all left off.
• Wednesday, March 1: First, JN1, JN2, and SG presented 3.37, 3.38, and 3.39, respectively. Next, 3.41 and 3.42 were discussed as a class while Dr. Falk scribed. Class ended with RF presenting Theorem 3.44. Theorem 3.43 will likely get discussed next time.
• Friday, March 3: The word on the street is that, class started off with SG presenting Theorem 3.43. After an initial attempt, Dr. Falk pointed out some weaknesses and then SG was able to redo the whole thing and produce a valid proof. Next, DF, TC, and LS presented Exercise 4.2, Exercise 4.3, and Lemma 4.4. Like Theorem 3.43, Lemma 4.4 required two attempts, but in the end LS was able to nail it using the hint in the footnote.

## Week 8

• Monday, March 6: David Deville was there today to fill in for me. My understand is that RF, GC, AF, JN2, QC, and SG presented Exercise 4.6, Exercise 4.7, Exercise 4.9, Theorem 4.10, Problem 4.11, and Theorem 4.12, respectively.
• Wednesday, March 8: Today JH, JQ, and AV presented Theorems 4.14, 4.15, and 4.16, respectively. Problem 4.17 was discussed as a class.
• Friday, March 10: After checking in and summarizing a few things that were discussed while I was out for almost two weeks, we had CK and AT presented Lemma 4.19 and Theorem 4.20.

## Week 9

• Monday, March 20: We kicked off with me presenting a proof of Theorem 5.2 (Principle of Mathematical Induction), which was followed by proof of Theorem 5.4 and Theorem 5.5 by SC and FA, respectively.
• Wednesday, March 22: We started by dividing the class up into several small groups of size 2-3. Each group was charged with working on one of the induction problems from the homework. We didn't get to discuss them all, but we had JO, SR, LS, GC, and CD present Theorems 5.6, 5.8, 5.11, 5.14, and 5.21, respectively. I'm really impressed with how well things went.
• Friday, March 24: After a full class discussion of Theorem 5.25, we split the class up into 5 small groups, where each group worked on one of Theorems 5.27, 5.28, 5.29, 5.30, and 5.31. CD and AV carefully walked us through Theorems 5.28 and 5.31.

## Week 10

• Monday, March 27: We spent a few minutes discussing the upcoming exam and then split the class up into 12 small groups. We had QC, AV, JH, SC, AT, JN2, SG, DF, CP, CK, and JQ present solutions to Exercises 6.5, 6.6, 6.7, 6.8, 6.9, 6.10, 6.11, 6.12, 6.13, 6.20, and 6.22, respectively. We never got to Exercise 6.19.
• Wednesday, March 29: The students took the in-class portion of Exam 2.
• Friday, March 31: I lectured over the end of Section 6.1 and most of Section 6.2. In particular, we covered 6.23-6.38.

## Week 11

• Monday, April 3: I continued lecturing over Chapter 6. In particular, we covered 6.41-6.50.
• Wednesday, April 5: After splitting the class up into several small groups, we had JO, EV, FA, JR, and LS present Exercises 6.52, 6.54, 6.57, and 6.61, respectively. I presented a proof of Theorem 6.59 during the last few minutes of class.
• Friday, April 7: Today was a great day of student presentations! We had JQ, SG, GC, and TC present proofs for Theorems 6.51, 6.56, 6.58, and 6.60, respectively. We start tacking Chapter 7 next time.

## Week 12

• Monday, April 10: I was an action packed day. I think everyone that was present presented something. We had presentations from CD, SG, JR, SC, TC, AT, JN1, JQ, QC, AV, JH, LS, RF, JN2, DF, CP, FA, CK, and GC. We covered 7.3, 7.5-7.9, 7.13-7.15.
• Wednesday, April 12: Everyone was a bit worn out today. Instead of having students present, I slowly and methodically discussed all the parts of Exercise 7.16. The plan is to have people keep working on Daily Homework 28 for Friday.
• Friday, April 14: We had JN2, CP, LS, DF, SG, TC, and RF present 7.19(a), 7.19(b), 7.19(c), 7.19(d), 7.20, 7.21, and 7.22, respectively.

## Week 13

• Monday, April 17: Another productive day. We had JH, SR, RF, JR, JN1, and AV present 7.23(a), 7.23(b), 7.23(c), 7.23(d), 7.25, and 7.26, respectively. We'll tackle Theorem 7.27 next time.
• Wednesday, April 19: We didn't cover a lot of ground today, but we had productive discussions. JN2 and RF present Theorem 7.27 and Theorem 7.28(a), respectively.
• Friday, April 21: We finally finished tackling Chapter 7. We had SC, SG, and AT present Theorems 7.29, 7.30, and 7.31, respectively.

## Week 14

• Monday, April 24: Not a lot of energy in the room today. It's that time of the semester I guess. We spent the first few minutes discussing Wednesday's exam and then I briefly discussed the main points of cardinality. With the time we had left, we had CD, DF, RF, JR, and LS present various parts of Problem 8.2. We discussed Problem 8.2(a) as a class and we held off on part (f).
• Wednesday, April 26: The students took the in-class portion of Exam 3.
• Friday, April 28: After handing back the in-class portion of Exam 3, I lectured over the remaining bits of Section 8.1. In particular, we covered Theorems 8.3-8.7.

## Week 15

• Monday, May 1: We continued covering Chapter 8. In particular, I lectured over 8.8-8.14.
• Wednesday, May 3: More lecturing. We discussed 8.15-8.42, skipping some details along the way, especially towards the end.
• Friday, May 5: A hectic final day, but we got a lot done. We divided up into several small groups to discuss the day's problems. After a few minutes, we started discussing a few of the problems. In particular, we had JQ, JH, GC, and AV present Theorem 8.16, Exercise 8.29, Exercise 8.32, and Theorem 8.37, respectively. Thanks for a great semester!

# Dana C. Ernst

Mathematics & Teaching

Northern Arizona University
Flagstaff, AZ
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