MA 511 B, Analysis I, Fall 23
Welcome! If you need anything that you cannot find here, please email me.
"It is the pathologies that give rise to the need for rigor." - Stephen Abbott
"Analysis is the art of estimation." - Unknown
Embedded Files
This image is of the Weierstrass function, a continuous function that is nowhere differentiable! We will learn about this beast, and other curious creatures, this semester.
Logistics
Logistics
Syllabus: This webpage functions as our syllabus.
Meeting times and locations: Lectures are Tuesdays and Thursdays, 11am-12:15pm, in COM 215; the discussions are Thursdays, 3:35-4:25pm, in COM 217.
Instructor:
Margaret Beck (she/her); you're welcome to call me Margaret. (Prof Beck also works if you prefer.)
Office hours:
Mondays, 1030am-12pm (Starting Sept 11): drop in, CDS 537.
Thursdays, 130-230pm (Starting Sept 14): scheduled one-on-one 10-minute appointments, in CDS 537 or on Zoom. Use this link to schedule a time for an in-person or zoom meeting. If you would like to speak with me for longer than 10 minutes, you are welcome to schedule multiple, back-to-back, appointments. You can schedule appointments up to 14 days in advance.
Email: mabeck@bu.edu
Textbook: Understanding Analysis, Second Edition, by Stephen Abbott.
Course prerequisites: Multivariate Calculus, eg BU's MA 225 or MA 230.
Changes: I reserve the right to change any aspect of this course to better meet the needs of students. Should this become necessary, it will be clearly documented and students will be given sufficient notice so they may adapt accordingly.
Learning goals and objectives
Learning goals and objectives
I am confident that each student in this course will be able to achive the following by the end of the semester:
Knowledge: Students will obtain an understanding of analysis in the context of the real numbers, including basic properties of the real numbers; sequences and series of real numbers; basic topology of the real numbers; functional limits and continuity; derivatives; sequences and series of functions; and Riemann integreation.
Critical and Creative Thinking: Students will practice creative problem solving. Students will construct logically consistent and precisely communicated proofs within the context of analysis, as well as evaluate proofs to determine if they are successful.
Communication: Students will articulate their ideas clearly through written proofs, and possibly also through the presentation of mathematics at the board during an oral exam.
Collaboration: Students will work together through peer review of written mathematics and group problem solving. This includes providing feedback on the work of others critically yet respectfully, as well as receiving feedback from others with the understanding that the feedback is offered critically yet respectfully.
Evaluation: Students will identfy aspects of written mathematics that need improvement, such as gaps in a logical argument and imroper or unclear use of notation, and self-assess their own work in this regard.
Classroom environment, accessibility, and support
Classroom environment, accessibility, and support
I hope that all students in this course feel it is in an environment in which they can productively learn. To that end, I want to explicitly state that diversity of background (including, but not limited to: race, gender, ethnicity, sexual orientation, age, socioeconomic status, religion, ability) is an asset. Diversity of ideas makes our ability to do mathematics stronger. (See this article, which shows that "Being around people who are different from us makes us more creative, more diligent and harder-working.") Because this course will involve peer review and group work, it is extremely important that all members of our classroom community feel welcomed and respected. If there are any ways I can help facilitate this, I welcome that feedback. I hope that each student feels comfortable letting me know (in person or via email) if they feel that their learning is being adversely affected by any experiences, inside or outside of class. To that end, I would like to acknowledge and emphasize:
As Francis Su eloquently put it in his speech on The Lesson of Grace in Teaching, each of you is a valuable human, regardless of what your accomplishments may or may not be.
Frederico Ardila's Axioms: Axiom 1) Mathematical potential is distributed equally among different groups, irrespective of geographic, demographic, and economic boundaries. Axiom 2) Everyone can have joyful, meaningful, and empowering mathematical experiences. Axiom 3) Mathematics is a powerful, malleable tool that can be shaped and used differently by various communities to serve their needs. Axiom 4) Every student deserves to be treated with dignity and respect.
Labels like "good at math" are problematic. Each of us is capable of learning analysis in a deep and meaningful way. As explained in "The Secret to Raising Smart Kids" by Carol S. Dweck, "a focus on 'process' - not on intelligence or ability - is key to success."
Names and pronouns: You are welcome to tell me your preferred name and/or pronouns at any time.
Accessibility: BU's Disability and Access Services can provide services and support to ensure that students are able to access and participate in the opportunities available at Boston University. Please reach out to them if you need any additional support or accommodations. Please also feel free to reach out directly to me with requests for support and accommodation, regardless of whether or not you are in touch with the Disability and Access Services office.
Academic Support: ways to obtain support in your learning of analysis in the course include:
Office hours: please come to office hours! I welcome the opportunity to get to know my students outside of class. Please see the Logistics section above on this page for times/locations of office hours.
Tutoring Room in the Department of Mathematics and Statistics: The tutoring room is staffed by graduate students and provides drop-in help for students enrolled in any of our courses. Different tutors have different backgrounds and areas of expertise, so I recommend that you consult not only the schedule but also the list of tutoring expertise to find a time when a tutor with an appropriate background will be available.
Peer tutoring at the Educational Resource Center (ERC): The ERC Peer Tutoring program provides BU students an opportunity to meet with a fellow student and ask questions related to their course material. Peer Tutors are BU students just like you, and they are there to guide, encourage, and support you in your learning process. All tutors are undergraduates who excelled in their course and are recommended by BU faculty. These services are free to all BU students. If you are interested in this resource, we recommend that you arrange for such services early in the semester, as availability often becomes more limited as final exams approach.
Additional Support: additional types of support that you may find helpful can be found on the Academic Help and Wellness page of the Department of Mathematics and Statistics.
Academic integrity: I trust that you are all aware of BU's academic conduct code. I hope and expect that you will all uphold it. I would like to highlight two key expectations I have for each student. First, be honest with yourself and others about when you do not understand something. There is no shame in needing help with any aspect of this course. Second, do not present the work of anyone else (human or otherwise) as your own. It is natural in mathematics to collaborate with others, but you should acknowledge your collaborators, only write up ideas that you genuinely understand yourself, and those ideas should be written in your own words.
On the use of AI, like ChatGPT, and the use of other online resources like Chegg, MathOverflow, etc: Using these services to directly obtain answers to HW or HW* problems in this course will significantly compromise your learning. I obviously have no good way to prevent this, so I must trust you not to do this. However, there are productive ways to utilize online resources. Here are some examples:
Looking up definitions, theorems, etc, if for some reason you are not able to access our book in that moment, or if you would like a complementary persepctive on the topic.
Looking up LaTeX commands and debugging LaTeX code. You can even ask things like ChatGPT to generate LaTeX code for you. Just be aware that the code it generates might not be very good. But, nevertheless, it could be a good starting point for you.
Asking things like ChatGPT to prove results that we have already proven in class, or results for which a proof exists in the book, and then trying to determine yourself if the proof is correct. (It will likely be a different proof than the one you have already seen.) This is a great way to practice your ability to detect errors or logical inconsistencies in proofs. Some proofs I've tried in this way come out essentially perfect, but sometimes there are unexpected, and often interesting, errors.
On the use of cell phones, tablets, and laptops in class: Although there are many positive ways to use these devices (like note taking, typing up work in LaTeX, etc), there are also studies that show that using such devices for non-academic purposes leads to reduced long-term retention of in-class material, such as lower performance on exams. So I strongly encouage you to limit your device use to only academic purposes that are necessary for our class.
Assessment and grading
Assessment and grading
The grade requirements are summarized in the above table. To receive a grade of C, you need to complete all (except possibly 2) weekly HW assignments and regularly participate in class. To receive a grade higher than C, you'll need to earn grade bump tokens through achieving HW* problems and/or through taking a midterm and final oral exam. More details:
Assessment Items:
Weekly HW assignments: complete/not complete yet. Here complete means a reasonable attempt at each problem assigned, with 75% of the problems being essentially correct. All HW must be submitted via Blackboard. You do not have to type it in LaTeX, but if you choose to do so that's great. Feedback on HW assignments, and the label of complete/not complete yet, will be given by the course grader. A complete will be recorded in Blackboard as 1 point, and a not complete yet will be recorded in Blackboard as 0 points. Students who receive a not complete are able to resubmit the assignment once for a second attempt at obtaining a complete. Weekly HW will be due on Monday of most weeks. In general, I expect students to adhere to the weekly HW due dates. I understand that this will not always be possible, so if you need an extension please email me to let me know and to provide a brief explanation why.
HW* Problems: achieved/not achieved yet. Problems that can be selected for this purpose from each weekly HW assignment are indicated in the table below.
Assessment/Feedback: These will be assessed according to this rubric. You can achieve at most one HW* problem from any given HW assignment. For the overall problem to be achieved you must achieve each of the rows of the rubric table. I will adhere to a high standard when deterimining whether or not a student has achieved a HW* problem. I will provide you with honest feebdack about your submissions, with the goal being to help you ultimately achieve a successful proof, and with the understanding that you can revise your work if a given submission is not yet achieved.
Submission Instructions: You may submit as many revisions of a given HW* problem as you like, however you may only submit two HW* problems per week (regardless of whether they are new submissions or revisions). For any revision you submit, you must also include a paragraph explaining what changes you made and why. All HW* problems must be typeset in LaTeX and submitted via Blackboard. There will be one upload link for HW* problems on Blackboard most weeks, with a due date of Thursday. The due date for these submissions is strict; if you miss the deadline during one week, you must use the following week's submission link (and thus lose two of your HW* problem submissions). Blackboard will allow two "attempts" for each weekly submission, just in case you accidentally upload the wrong file or something the first time. Please note that only the most recent attempt will be graded. The submission for a given week should be a single pdf file containing all of your HW* problems for that week. In the pdf, please be sure to include your name, which HW* problems you are submitting, and if each problem is a first attempt or a revision. Please also include one copy of the rubric for each problem you are submitting. The HW* problems will be assessed by me. If the submission instructions are not followed I will return the submission ungraded and they can only be resubmitted using the following week's submission link (and thus you effectively lose two of your HW* problem submissions). You can expect to receive feedback on your HW* problems in Blackboard within 1-2 weeks from when they are submitted. Please remember:
Your submission must be a single pdf file that is typeset in LaTeX and submitted via Blackboard.
There will be one submission link per week (with some exceptions, like the first two weeks of the semester, and the weeks when the midterm oral exams will be held) with a strict deadline of Thursday night. If you miss the deadline you lose the opportunity to submit HW* problems that week and must wait until the following week.
The file must clearly indicate your name, which HW* problems you are submitting (maximum 2 per week), and whether the submissions are a first attempt or a resubmission. Please also include one copy of the rubric for each problem you are submitting.
For each resubmission you must include a paragraph explaining what changes you made to your previous attempt and why.
Midterm and final oral exam: These exams are optional, but they are required for any students wishing to be eligible for a grade of A- or A, or for any students who would like me to potentially write them a letter of recommendation at any point in the future. Each oral exam will be given 0, 1, or 2 points.
Midterm exam: You will be given a list of problems for the exam in advance, and you will be asked to answer two of the problems during your exam. When your exam begins, you will choose the first problem that I ask you, and then I will choose the second. To earn a 1 you must successfully answer both of these problems as stated on the problem list. Each problem will have follow-up questions, which you will not be given in advance. To earn a 2, you must successfully answer the follow-up questions for both problems. Thus, earning a 1 is relatively straightforward, but it will take a true understanding of the mateiral to earn a 2.
Final exam: For this exam you will not be given a list of problems, but you will be given a list of topics. You will be asked about two topics during your exam. When your exam begins, you will choose the first topic, and then I will choose the second. To earn a 1 you must successfully answer the initial questions I ask you about both topics, and to earn a 2 you must also succesfully answer the follow-up questions I ask you about both topics.
The midterm oral exams will be held during the weeks of Oct 30 and Nov 6, and the final oral exams will be held during the final exam period. The exams times for each student will be determined closer to the weeks the exams will take place, and I will make sure you have an exam time that fits reasonably into your schedule. All oral exams will be audio-recorded.
Regular participation in class: complete/not complete. If you are present and engaged during lecture and discussion that counts as complete. I understand that absences sometimes need to occur. Occasional absences will not prevent an assessment of complete for class participation. However, please email me each time you need to miss class and provide a reason. (I do not need a doctor's note or anything like that - just a brief email from you acknowledging your absence.) If I am concerned that an individual student will not receive a "complete" assessment for class participation I will let that student know and try to find ways to help facilitate their increased participation.
Grade bump tokens and determination of the final grade: Please note that
A grade bump toke can be obtained either (i) by achieving 2 HW* problems (a maximum of 3 grade bump tokens can be achieved in this way), or (ii) for each 2 points obtained on an oral exam.
You cannot obtain "half tokens". So 3 HW* problems just count for one token, as does 3 oral exam points.
I will not issue any grades of C+, C-, or D.
A failure to meet the requirements for a grade of C will result in a grade of F.
Therefore, to earn a grade of A at the end of the semester, in addition to completeing all (except possibly two) weekly HW assignments and regularly participating in class, a student would need to achieve 6 HW* problems and earn 2 points on each oral exam. To earn a grade of B+, in addition to completeing all (except possibly two) weekly HW assignments and regularly participating in class, a student would need to (i) achieve 6 HW* problems, (ii) achieve 4 HW* problems and 2 total oral exam points (eg 2 points on one exam, or 1 point on each exam), or (iii) achieve 2 HW* problems and 2 points on each oral exam.
The above assessment/grading scheme uses a combination of contract grading (the requirements for a C), mastery grading (the HW* problems), and more traditional grading (the oral exams).
Schedule and list of homework assignments
Schedule and list of homework assignments
Please note that the weekly topics are tentative, and they may be adjusted as the semester progresses. However, they and the HW assignments will be finalized at least one week in advance. I am including links to the tex files of the HW assignments; if you want to type up your solutions in LaTeX, you can upload the tex file for the HW assignment to overleaf and then type your solutions directly into the file. That way you will have the problems themselves included in your write-up, without having to type them out yourself. It will also give you a way of seeing more examples of how I type various math expressions in LaTeX.
Weekly Schedule and Weekly HW Assignments
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