Math 310 - Spring 2014 Proof Portfolio Instructions And Assessment Portfolio Description: Very few people are innately good writers, and that s even more true if we consider writing mathematical proofs. First you have to develop the skills necessary to come up with a proof, and then you have to figure out the best way to explain that proof to someone else using sentences. Most students in Math 310 will rarely (if ever) have been asked to write sentences communicating mathematical ideas, so we ve got a bit of a hill to climb. As a way to encourage you to work hard at this and consider feedback from me, you ll be building a portfolio of 8 well-written proofs this semester. Because you have more time to contemplate these problems and work on them, they ll generally require more work than homework problems. Problems will be assigned roughly weekly, but some weeks early in the semester may see two problems assigned. For some of the problems, you ll have a choice between two problems (2A or 2B, for instance); for other problems, you won t have an option. Your job will be to work on the problems and submit drafts to me for review. For all problems, you ll be typing your solutions using L A TEX, a (free) professional mathematics typesetting system. I ll give you feedback and record the fact that you submitted the draft, but the score I would have assigned to that draft won t count toward your course grade. You ll use the feedback to improve your proof and then turn in a revised version for further feedback before inclusion in the final portfolio. Around the middle of the semester, I ll ask you to answer some questions reflecting on your strengths and weaknesses in writing proofs at that stage, progress you feel you ve made so far, and plans for further improvement. The final portfolio, due at the end of the second to last week of the semester (i.e. the week before 15th week), will include your polished versions of all 8 proofs along with answers to more reflective questions. 1
Drafts and Draft Submission Drafts: For your portfolio, a draft of a proof is a complete, self-contained, well-written, and typeset solution. Scratch work and half-developed ideas are not drafts. If you feel you ve hit a roadblock on a problem, take a break and think about it more later. I ve had some of my best ideas walking around or standing in line to buy lunch. Consider coming to see me during office hours or by appointment, and bring your notes of what you ve tried and what you re thinking about. (Office hours are a great time to share your scratch work and half-developed ideas, but you must come prepared to discuss what you ve thought about and tried. I just don t know where to start is not the way to begin a discussion. Tell me what you ve tried and where you ve gotten stuck, for instance.) For the first three problems, if you ve tried all of this and are still stuck, meeting with me and talking about the problem by the deadline can count as a first draft. Turning in drafts: You ll submit drafts electronically by putting them a dropbox folder which I have set up for you. I ll email you a link to the folder as soon as you set up your dropbox account and email me to let me know that you ve got it set up, and you should join the shared folder and put your drafts in there. It will greatly help me if you name your files something meaningful, (for example, Problem1.Draft1.LastName.pdf) but I m flexible as to how you name them. You only need to put the PDF of your drafts in the folder as that s all I will look at, but if you want to put all of your files there, that s fine too. If you get stuck on compiling something, you can put your TEX files in here and I can help, but that can also be tricky as Dropbox doesn t know what to do if two versions are open at once. Draft Submission Details: Once you ve got your PDF prepared for submission, you will put it in a dropbox folder I ve created for you. Please name your files something that will help me identify them. Additionally, once you ve uploaded your files, please change the date on the README file I ve put in the folder for you so that I know to look at your new drafts. To be a bit more concrete, you ll do the following: 1. In the README file, you ll add a line to the top for the date and the problems you re submitting. It should look like this for the first week and be modified appropriately for other weeks: Due: 29 January == Problems: Problem1.Draft1 2. Change the name of the README file. In the first week it should be README Done 29Jan.txt and should be modified appropriately for other weeks. Questions regarding the draft should be put in the PDF file itself, and other questions should be emailed to the instructor at s-kshulti1@math.unl.edu. Returning drafts: I indent on grading drafts within 24 hours of the deadline. I ll grade your PDF by annotating it on my ipad and upload a PDF to your dropbox folder with a modified name for you to see as soon as I m done. Please check your dropbox folder before emailing me to ask if I ve graded your draft. To help you identify that I m done grading for you, I ll do the following: 1. I ll name the annotated version of your PDF to be Problem1 Draft1 Graded.pdf in the first week, and will name graded versions similarly for other weeks. 2. I ll edit the README file so that the line you added in the first week will read as follows Due: 29 January == Problems: Problem1.Draft1 == Graded and I will modify this appropriately for future weeks. 3. I ll change the name of the README file. When I m done grading (in any week) it will be README Grading Done.txt. 2
Proper Typesetting Each submission you make to me or a peer must be prepared using software capable of producing the proper mathematical symbols and equations. You must convert this to PDF format prior to submitting to me. You only need to put the PDF of your drafts in the folder as that s all I will look at, but if you want to put all of your files there, that s fine too. Handwritten work or typed work that does not use proper mathematical notation (for instance, using the calculator notation x^2 instead of x 2 ) will not receive credit. (For first or second drafts, no credit will be awarded for draft submission. For the final draft, you ll score a 0 on the problem on the 4 point rubric.) There is really only one full-capable system for typesetting mathematical notation. It s a free software package called L A TEX (pronounced LAH-tek or LAY-tek... which is correct is a frequent source of arguments). As the semester develops, you will be learning L A TEX in class, via tutorials (written or video), and via sample documents. There are three acceptable ways to run L A TEX for this course: 1. The free website http://www.sharelatex.com. You need to register, but then you can upload files (such as the templates I ll post on the course website) and download PDFs easily. An advantage to ShareL A TEX is that if you have a problem getting something to work properly, you can share the file with me on the website to expedite getting assistance in a way that should not break the file because it shouldn t allow the file to be changed in two different ways at the same time. 2. Free software available for Windows, Mac, and Linux. Instructions on how to download and install L A TEX can be found at http://danaernst.com/quick-latex-guide/. For those of you comfortable with installing software, this is probably the best approach. You can always upload something to ShareL A TEX if you get stuck. However, if you don t have your own computer or are having trouble getting the various components installed, ShareL A TEX is a great solution. (For those of you with a CS bent, I have heard that most text editors handle L A TEX code with syntax highlighting and such, so you can use whatever you want. Personally, I use TEXshop on my MacBook Pro, because it was the easiest way to install the files when I got started and haven t bothered trying anything else.) 3. Another free software program called LyX is available for Windows, Mac, and Linux from http: //www.lyx.org/. This provides a more point-and-click style editor for preparing your L A TEX documents. While it can be a good way to get started or figure out how to do something complicated, it is also far less efficient to have to use the mouse too much when preparing a document in L A TEX. However, I am not familiar with LyX (I was unaware of the existence LyX at the time when I was learning L A TEX), and so I cannot help you as readily if things go wrong and you re using it. As a result, I suggest this be the last resort method for preparing your documents. To give you time to get comfortable with L A TEX, you may prepare the first three drafts you submit (these are the three drafts due in the the first two weeks of drafts) using any software capable of producing proper mathematical notation. Microsoft Word has a (barely) functional equation editor, as does Google Docs. If you use one of these to prepare your first drafts, be sure to save to PDF before dropping them in your Dropbox folder! To incentivize using L A TEX as early as possible, students who submit PDFs generated from L A TEX for any of the first three drafts will receive a 0.5% bonus (per draft, up to 1.5% bonus) toward their proof portfolio score. (Thus, if you got a 80% per the rubric but turned in every draft using L A TEX, you would get a proof portfolio score of 81.5%.) Beginning with the fourth and fifth drafts (due 12 February 2014), all drafts must be done in L A TEX. 3
Rules and Expectations 1. This portfolio is to be done independently. The problems assigned will require you to develop a complete understanding of what we re learning, but they will not be so challenging that only a few students can do them. In developing your proofs, you may only discuss the problems with your Math 310 instructor, Kat Shultis. You may not discuss the problems with anyone else. For example, you may not discuss them with your parents, other students, other professors, or anyone in an online setting. (You re welcome to talk to your pet, if you have one, about the problem, as long as no one else is around and it s not a talking pet.) Action contrary to this policy will be considered academic misconduct and will be handled in accordance with UNL policy. You will face a minimum grade penalty of a full letter grade reduction in your course grade, but the penalty might be as severe as receiving an F in the course. (a) The one exception to this policy will be later in the semester when we exchange peer feedback on drafts. I will pair up students who have selected different problems (say 6A and 6B) to discuss the write-up of their proofs and specify what is and is not permitted during those discussions. 2. The only sources you may consult when preparing your proofs for the portfolio are the assigned textbook for the course, documents I post on the course web page, and your notes from class (of course if you are absent and get notes from a classmate this includes those notes as well). In particular, you are prohibited from looking online for sources unless I instruct you to do so in the problem. If you feel you must consult another source, please come speak with me about why you feel this is necessary and what you would like to consult. I will consider all requests, and if granted, you must properly reference the additional source in your proof. (A footnote may be the easiest way, but any clear reference will be fine.) Action contrary to this policy will be considered academic misconduct and will be handled in accordance with UNL policy. You will face a minimum grade penalty of a full letter grade reduction in your course grade, but the penalty might be as severe as receiving an F in the course. 3. You are expected to have two drafts, clearly labelled as Draft 1 or Draft 2, of each problem that will eventually be part of your portfolio reviewed before considering a proof to be in its final form. For six of your proofs, both reviews will be done by your instructor. For the remaining two, one review will be done by your instructor and the other will be done by a peer. The phase in which drafts can be submitted for peer review will begin during the week of 7 April 2014. Only drafts of problems 7 and 8 can be given to a peer for review, and I will arrange the review pairs prior to you exchanging drafts. Further instructions will be provided in March. I invite you to discuss your ideas with me in person at any stage prior to submitting your final proof portfolio. 4. Each draft you submit will be given a provisional rubric score. I will record this score (including the assessment given by a peer later in the semester), but it will not count toward your portfolio grade. The only scores that count toward your portfolio are those you receive for the final drafts of your proofs submitted during the last week of class. Failure to submit a final draft of a problem will result in a zero score for that problem, regardless of your draft scores. Please note that while I anticipate scores will increase from draft 1 to draft 2 to the final draft, it is possible that your final draft will receive a lower score if you make revisions that result in a proof of lower quality. 4
5. The deadlines for drafts will be 5pm on Wednesdays. On second to last page of the course policies handout, you ll see the suggestions I make regarding which drafts you turn in during a given week. I will ensure that you ve got at least one week to do any portfolio problem before the suggested first draft is due, but I may give you more time than that. As a result, it may be possible to turn in a different selection of problems in a given week than I ve suggested, and that is acceptable as long as you re turning in the right number of drafts each week. Any student who falls behind on drafts may submit them for feedback (but not draft credit) until 5pm on Wednesday 16 April 2014. 6. You may work ahead, but you may not submit more than three drafts in any one week. I ll keep a running list of drafts submitted, so if you turn in an extra draft some week, then you may turn in one fewer draft than required in any subsequent week. (You re welcome to ask me to verify that you re ahead on drafts at any point.) 7. I will not discuss portfolio problems in person or by email after Friday 18 April 2014. The final version of the proof portfolio must be submitted electronically by 11am on Wednesday 23 April 2013. The exact method of submission will be described later, but the basic idea is that you will upload a single file with all of your proofs and reflective statements to your dropbox folder. I d prefer that you merge everything into a single PDF, but a folder with PDF files will suffice. 8. My goal is to have drafts returned to you with feedback within two business days of submission. That means that I intend on having all drafts back to you by Fridays at 5pm. If I find that I have too much to grade in a given week, I ll ask for your input on what to prioritize. 5
Grading The proof portfolio is worth 20 percent of your course grade. That s more than your weekly quizzes. It s more than either of the two exams. The experience of writing and revising drafts will make you a better writer of proofs, and one of the most important outcomes of this course is that you develop that skill by the end of the semester. There will be four contributing factors in determining your grade on the proof portfolio: draft submissions, two sets of responses to reflective questions, and the overall assessment of your proofs. A rubric will be provided for each set of reflective questions at the time they are assigned. The emphasis will be on answering the questions thoughtfully and providing an appropriate assessment of your progress at writing proofs. Each of your proofs will be assessed against the 4 point grading rubric for homework and quizzes. The table below indicates how the final proof portfolio grade will be determined. Number of Drafts (as a percentage of 16) 8% Problem #1 10% Problem #2 10% Problem #3 10% Problem #4 10% Problem #5 10% Problem #6 10% Problem #7 10% Problem #8 10% Reflection #1 6% Reflection #2 6% Total 100% 6
Frequently Asked Questions Here are some questions I anticipate might come up. Please read them right away, as there is important information here that didn t fit anywhere else! Are there any other requirements I should be aware of for the proof portfolio? Your solutions/proofs must be properly typeset using complete sentences. You must adhere to the Elements of Style for Proof linked from the course web page. Proofread your drafts before submitting them. You must follow proper English mathematical grammar, use correct spelling, etc. Proofs that do not meet these expectations will receive few, if any, points. What if my proof is wrong? What if my proof is incomplete? That s the point of submitting the first two drafts. If you ve made a mistake or omitted something important, my feedback will attempt to steer you in the correct direction. If you are still stuck, you should come see me to discuss how to correct or complete your proof. The scores I give on drafts are just guidance on if you re on the right track. The only score that will count toward your course grade will be the one assigned to your final draft submitted during the last week of classes. May I collaborate? May I Google to find ideas or solutions? No. This is discussed extensively above. The only person you may discuss the problems with (in person, on the phone, online, via email, via telegram, via semaphore, etc.) is your instructor, Kat Shultis. You may use the assigned text and materials developed for this course and posted on the course web page. Anything else (from the Library, on the Internet, whatever, wherever) is off limits without prior permission. Can I get help from you? Absolutely! My office hours are published on the course web page, and you re always welcome to see me during office hours. You can also email me to make an appointment. (If you come up to me after class and ask about scheduling an appointment, I ll usually be trying to get out the door so the next professor can start class, so you ll need to email me a reminder of what we discussed regarding an appointment.) I prefer if you email at least 24 hours in advance of your proposed appointment, but I ll work with you as much as possible to find a convenient time. How should I prepare to visit you in office hours? Think about the problem. Write down very clearly what you re trying to prove. Make notes of things from the book or class that appear related. Write down what conclusions they allow you to draw that might start moving in the right direction, even if it feels like just a tiny baby step. Think of this a bit like brainstorming, where there s really nothing that can be declared wrong. You ll have lots of ideas that aren t ultimately helpful (I am working on a PhD in mathematics and have plenty of those ideas when I try to prove something.), but recording what you re thinking about will help organize the process. After you ve done all that, bring what you ve written down to my office. Don t be surprised if I sit at my desk and ask you to write on the board, explaining to me what your thought process has been so far. I ll help you get on the right track, and hopefully you ll then be able to write up a nice proof. If not, you can always come back again! (The idea for this proof portfolio project was given to me by Mitch Keller at Washington & Lee University. This file is a modified version of the project he assigned for this course while he was a postdoc at UNL in Spring 2013.) 7