University of Connecticut College of Liberal Arts and Sciences
Department of Mathematics
Tom Roby's Math 2710 Home Page (Spring 2013)
Transition to Advanced Mathematics

Questions or Comments?

Class Information

COORDINATES: Classes meet Tuesdays and Thursdays 9:30–10:45 in MSB 319. The registrar calls this Section 003.

PREREQUISITES: MATH 1132 or 1152 (first year calculus).

TEXT: Gilbert & Vanstone: An Introduction to Mathematical Thinking Pearson Prentice Hall, 2005.

WEB RESOURCES: The homepage for this course is I may list some others below.

SOFTWARE: For generating data to understand properties of integers, or to solve equations, computer software can be a valuable tool. Doing some computations by hand is generally good for learning, but having software that can do bigger computations or check your work is very useful. One free source on the web is WolframAlpha. For a full-fledged programming environment, check out the free open-source computer algebra system called Sage. For something free and easy to use but with more limited capabilities, try GeoGebra.

GRADING: Your grade will be based on two midterm exams, one final exam, homework, & in class work (generally on Tuesdays).

The breakdown of points is:

HW InClassWork Midterms Final
25% 10% 20% each 25%

MIDTERM EXAMS: Will cover all the material up to that point in the term. They are currently scheduled for THURSDAY 28 FEBRUARY 2013 and THURSDAY 11 APRIL 2013 during our usual class meeting. Please let me know immediately if you have a conflict with those dates. There are no makeup exams.

HOMEWORK: Homework will be assigned most weeks, and should be attempted by the following TUESDAY, when I will be happy to answer questions or provide hints. It will generally be due Thursdays at the start of class. Since I may discuss the homework problems in class the day they are due, late assignments will be accepted only under the most extreme circumstances. (Please let me know as soon as possible if you find yourself with a situation that might qualify.) The lowest written homework score will be dropped in any event.

The assignments are listed below in the class schedule. Sometimes it's just problems from the text, sometimes there's a handout that includes other problems as well.

You may find some homework problems to be challenging, even frustrating, leading you to spend lots of time and effort working on them. This is a natural part of doing mathematics, faced by everyone from school children to top researchers. I encourage you to work with other people in person or electronically. It's OK to get significant help from any resource, but in the end, please write your own solution in your own words. Copying someone else's work without credit is plagiarism and will be dealt with according to university policy. It also is a poor learning strategy.

ACADEMIC INTEGRITY: Please make sure you are familiar with and abide by The Student Code governing Academic Integrity in Undergraduate Education and Research. For quizzes and exams you may not discuss the material with anyone other than the instructor or offical proctor, and no calculators, phones, slide rules or other devices designed to aid communication or computation may be used.

ACKNOWLEDGEMENTS: Many people have influenced my approach to this material, particularly Arnold Ross and others with whom I worked in the Ross Program at Ohio State, Hampshire College Summer Studies in Mathematics, and PROMYS Program at Boston University. I've also learned much from my UConn colleagues Keith Conrad, Johanna Franklin, Alvaro Lozano-Robledo, Steve Pon, and Reed Solomon.

CONTENT: The main learning objective of this math course is for students to become comfortable reading and understanding theoretical mathematics, and to gain facility in constructing their own mathematical arguments. It has many similarities with a writing course, and I will try to provide for some work to be rewritten after you have received feedback on it.

ACCESSIBILITY & DISABILITY ISSUES: Please contact me and UConn's Center for Students with Disabilities as soon as possible if you have any accessibility issues, have a (documented) disability and wish to discuss academic accommodations, or if you would need assistance in the event of an emergency.

LEARNING: The only way to learn mathematics is by doing it! Complete each assignment to the best of your ability, and get help when you are confused. Come to class prepared with questions. Don't hesitate to seek help from other students. Sometimes the point of view of someone who has just figured something out can be the most helpful.

CLASSTIME: We will sometimes spend classtime doing things in groups, presenting mathematics to one another, or having interactive discussions. There will not be time to "cover" all material in a lecture format so you will need to read and learn some topics on your own from handouts, web sources, or otherwise.

SCHEDULE: The following is a the start of a tentative schedule that I will update throughout the semester. If you have a religious observance that conflicts with your participation in the course, please meet with me within the first two weeks of the term to discuss any appropriate accommodations.

Date Section Topics HW, etc.
1/22T §1.1–4 Logic, sets, quantifiers
1/24R §1.5–6 Proofs & Counterexamples
1/29T §2.1 Divisibility & Properties of Z HW #2: Ch. 2 (p.50), #2–30even
1/31R §2.2–3 Division Algorithm & Euclid's Algorithm HW #1 due.
2/5T § 2.3 Solving Linear Diophantine Equations HW #3:
2/7R § 2.3 Consequences of Bezout HW #2 due.
2/12T Ch. 1 Going over sample proofs
2/14R § 2.5 Prime Numbers HW #3 due.
2/19T § 3.1–4; Modular arithmetic HW #4: Ch. 2: #70, 72, 93; Ch. 3: #2-10even
2/21R § 3.1–4; More mods & equivalence relations; HW #4 due.
2/26T § 3.5 Catchup & Review Day Do Practice Midterm 1 by today
THURSDAY 28 FEBRUARY 2013 MIDTERM EXAM 1 on Chapters 1–2
3/5T § 3.5–7 Equations in mods; Fermat's Little Thm HW #5: Ch. 3: #22–44even, 56, 68
3/7R § 3.7 Euler's Thm. and Phi-function HW #5 due
3/12T § 4.1–2 Mathematical Induction & Recursion HW #6: Ch. 3: 57,60,61,62,64,97,98,102,104
3/14R § 4.3 Binomial Theorem HW #6 due
3/26T § 5.1–2 Rational & Real Numbers HW #7: Ch. 4: #2–12even,28,36,58,65
3/28R § 5.3–4 Rational exponents & Decimal expansions HW #7 Due
4/2T § 6.1-3 Functions, Graphs, Composition HW #8: Ch. 5: #5,6,7,20,36,37; Ch. 6: #2-8even, 16, 22-34even
4/4R § 6.4–6 Inverse functions, Bijections, Cardinality
4/9T § Catchup & Review Day HW #8 due
4/16T § Hndout Ch. 13 Limits & Monotone Convergence. HW #9: Ch. 13: 8,10,16,22,25,27,30,32a,34,39
4/18R § Hndout Ch. 13 Decimal Expansions & Uncountability
4/23T § Hndout Ch. 14 Convergent Sequences; HW #9: Due
4/25R § Hndout Ch. 14 Convergent Series HW #10: Ch. 14: #8,9,11,13,18, 36, 47
4/30T § Hndout Ch. 14 Convergent Series Midterm 2 Rewrites due
5/2R § All Catchup & Review Day HW #10 due

Web Resources

Keith Conrad has an Expository paper website with many useful handouts on elementary number theory and abstract algebra. Some of these give alternate presentations to some of the topics of this course that some might find useful.

A discussion of Euclid's algorithm.
Here are some links to examples of proofs both well and poorly written:
  • Examples of bad proofs.
  • Eugenia Cheng's Guide to writing proofs
  • Dan Bach's Collection (some goofy, some hard to spot error)
  • Everything2's site

  • Anonymous Feedback

    Use this form to send me anonymous feedback or to answer the question: How can I improve your learning in this class?  I will respond to any constructive suggestions or comments in the space below the form. 

    Feedback & Responses

    1. I think that if you use the "manual clickers" more in class instead of just waiting and waiting for a response from us would be more effective.

      Thanks for mentioning this. I've tried to use the card clickers more during the past few lectures and hope this has helped.

    2. Grader is inconsistent in their grading.
    3. Time can be spent more wisely, sometimes goes off topic. I am starting to feel unprepared for the upcoming exam. Would like to see different proof techniques! More group work! Possibly breaking up into groups and working collaboratively on a proof with a couple other people and then comparing and critiquing them as a class. Grader seems to be a bit inconsistent, maybe you can review our work after it is graded. Thank you.

      Thanks for all this feedback! This is a particularly tricky course to teach, and I'm still feeling my way with a new book and a somewhat different approach than I've used in the past. Figuring out how to balance lecture, group work, etc., is part of that; maybe I can do more to poll the class about that in the next week or two.

      As far as the exam goes, it won't cover anything that you haven't had time to work through in class and HW. If you are feeling unprepared in other ways where I can help, please let me know. The nature of this course is that students need much individual feedback on their own work, and it's very hard to make that happen given other constraints on time and departmental resources.

      As far as consistency in grading goes, I'm always happy to take a look at anything you think wasn't graded properly. I would be love to hear more specifics, which I could then pass back to the grader while preserving anonymity (if that would help). My current sense is that the grader takes a good deal of time to write detailed comments about proofs, and unlike me, her handwriting is legible!

    4. I enjoy this class a lot. Most professors cover something briefly in class and expect you to know it in depth, but I feel like you actually care if we understand the material. I think the most helpful thing for me would be a practice exam. This would help me practice the material one last time before the test and give me more confidence going into it.

      Thanks! I hope some of the skills you learn and practice you get in grappling with mathematical reasoning from this class will serve you well in future math classes, where the pace will generally be much faster. I do plan to provide practice exams before each midterm and the final; thanks for the reminder.

    5. Please, no more group work than we currently have. The status quo is fine. Also: I am starting to think your purposeful mistakes aren't actually purposeful at all...

      Thanks for helping me figure out the right balance of lecture to group work. Today was pretty much all lecture, so I'm not sure what prompted the comment, maybe last week's looking student proofs to make improvements? As for my mistakes, I hope that catching them serves a purpose, even if it was perhaps not my intention at the time I made them...

    6. Can we go over #4 and #6 from the hw that is due on thursday? I don't know how to go about it, but I don't want to ask out loud in class.

      Thanks for mentioning that. I did go over them, though since we were rushed at the end, I also sent the email on Wednesday. I also got several email questions about these problems, which means perhaps there was no need to feel shy about asking?

    7. I am having difficulty solving problems involving sets and I feel >like we haven't spent sufficient time on them.

      Good point. I took some time today with the set proof from the HW. Did that help? Most proofs involving sets are long on technique and short on theory, which makes them less fun for class, but still important.

    8. I feel like this course is really similar to a W class in the way that we spend a lot of time trying to improve our writing. I feel like exams are not the best way to test our knowledge of the material. Maybe a packet of proofs to write or some sort of take home would be better.

      There's definitely truth in what you say, and I hope to write exams where time pressure won't be a major factor. On the other hand, most math courses for which 2710 is a prerequisite will have in-class exams, so this course should prepare you for those. I also think it's important to have samples of students work which were indisputibly created without any outside help in order to grade fairly. It's hard to guarantee that with a take-home exam.

    9. Can we do more examples in class that correspond to the homework? I feel like the homework asks us to do things that we have yet to learn and that aren't covered well in the text. Because the homework is actually graded (and graded strictly), I feel like we should learn more of how to do the problems it asks us, and those should be what we would see on the exams anyway, hopefully.

      We could, but I think it would be best if the students in the class took the initiative to ask about specific HW questions. As you progress to higher levels of math, your instructors will assume a increasingly greater ability for you to learn things on your own or from a text. So in this course we start to transition away from the earlier paradigm of "learning procedures and practicing them" to "grappling with material you don't understand, figuring some of it out, and getting help as needed from classmates, the web, and the instructor." Learning how to ask good questions, i.e., trying to pinpoint what might be confusing you, is an important first step in this process. I'm sympathetic to the challenge, but have confidence that you can achieve this greater independence (aka "mathematical maturity"), which will serve you well in math and your other learning endeavors.

    10. I really enjoy this class, and think it is taught well! Do you think you could send us an answer sheet to the practice midterm? I think it would help to be able to confirm we are doing it correctly!

      Thanks for the positive feedback. I've just posted solutions to the practice midterm. Please keep in mind that there are often multiple correct proofs or solutions, so don't take mine as gospel.

    11. Will there be rewrites on the exams? I am concerned because our homework is graded very strictly and exams are worth a large portion of our grade.

      Good question. It's a lot of extra work for the instructor, but there's definitely some pedagogical value. Generally I decide case-by-case, based on how the exam, went whether rewrites are worth the effort. An alternative if the grades seem too low because the exam was hard is to curve the scores in some way, or add a constant to everyone's scores. This is much less work (for everyone), but not as useful for learning.

    12. I am nervous about the final exam since there will not be any rewrites! In addition to no rewrites, there is a lot more material that it will cover than the midterms. I don't want it to have a huge negative effect on my grade, yikes!

      That's certainly understandable. I hope that all the work of writing and rewriting will pay off by the end of the term. When creating and grading the final I take into account the lack of rewrite. Even so sometimes I misjudge how hard a question will be and am forced to curve it when everyone does poorly. So as long as your performance on the final is not significantly lower than others, even if it seems low in an absolute sense, it shouldn't hurt your grade. Please study hard!

    13. I really appreciate your passion for teaching! It's encouraging to know you truly care about your students' success and developing mathematical maturity. Thank you!

      Thank you for such positive feedback! Are you sure you don't have anything to add? :-)


    Here are some other handouts:

    Back to my home page.

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