By Katharine Ott, Department of Mathematics, Bates College
Real Analysis is a rite of passage for undergraduate math majors. It is one of my favorite courses to teach, but I recognize that the course is challenging for students, and, for many, downright intimidating. In Fall 2017 I was scheduled to teach Real Analysis for the third time in my career. Prior to the semester starting, I knew that I wanted to alter the grading scheme of the course to deemphasize exams in favor of effort. Ultimately, I wanted to promote a growth mindset and to help students identify their strengths and weaknesses independent of exam performance. During our annual summer visit, my good friend and graduate school classmate Matthew Pons described to me his new project with Allison Henrich, Emille Lawrence, and David Taylor called The Struggle is Real: Stories of Struggle and Resilience on the Path to Becoming a Mathematician. (For more information on their project, check out https://math.roanoke.edu/tsir/.) I loved their idea of gathering and sharing personalized stories around this topic and immediately thought of adapting the exercise for my students. Since I was teaching Real Analysis, I decided to include reflective homework problems and activities under the label #thestruggleisREAL. I was worried that the hashtag was too gimmicky, but decided that with the right sales pitch students would embrace the pun. In this post I describe how this welltrodden hashtag injected a great deal of reflection, and a bit of levity, into my students’ experience in Real Analysis.
I’ve spent a great deal of time privately considering my own struggles in mathematics. I’ve exchanged stories with classmates from graduate school, with my spouse and other confidantes, and oneonone with students during office hours. But as a student I never had these conversations as a part of a class, and as a teacher I have not offered a place for sharing such stories beyond oneoff conversations. Thus I decided, without figuring out any further details, to incorporate into the class writing prompts where students could share their own stories of struggle and resilience. The two things I did decide upon from the beginning were that the writing assignments would be regular throughout the semester and graded (but low stakes). The pertinent language on the syllabus read:
#thestruggleisREAL: Throughout the semester there will be homework problems and activities under this hashtag. At the end of the semester, students may elect to complete a reflective project elaborating on this work. For students who complete this option, the project will count towards 10% of the final grade, and the exams will each count 15%. [As opposed to the higher of two exam scores counting 25% and the lower exam score counting 15%].
I previously used different grading tracks in Calculus classes when including a Community Engagement component in the course, and I felt that this would be another appropriate time to give students agency in how their work was evaluated. It was important to me that they participate in reflection throughout the semester and that everyone was rewarded equally for this effort. At the end of the semester, students could then choose what type of work was the best measure of their performance in their class. As is to be expected there were questions regarding the mechanics of the grade distributions, but there were no complaints about separate grading tracks. Indeed, several students with high exam averages completed the final reflective project, and several with low exam scores opted out.
The Assignments
As in previous semesters, I assigned weekly homework in the class consisting mainly of proofs. This time around, I added to each assignment one problem labeled #thestruggleisREAL. In addition to the reflection that these problems prompted, I think that by putting narrative writing sidebyside with formal proof writing I was able to strengthen the case that mathematical writing is “writing”. In most cases, students received full credit for completing the problem. I relied on the fact that all Real Analysis students have taken a writing intensive firstyear seminar. This means that the quality of exposition was generally acceptable, and besides that I used the formal proofs in the homework to hold them accountable for writing mechanics.
Below is a selection of the writing prompts that I used during the semester.
First assignment: I teach at a small liberal arts college where course reputations carry a great deal of weight. With this in mind, the first reflection asked students to write down their preconceived notions of the class. Students did not turn in their assignments, but shared them in pairs during our second class meeting. I wrote my own reflection and shared it on the class webpage.
Graded assignments: The writing prompts asked students to reflect upon a particular homework problem or proof completed in class. I generally asked for a paragraph response and it counted as a regular homework problem. Some examples are:
 Which of the problems above was the most challenging? Approximately how much time did this homework take you to complete?
 What study strategy or food in Commons helped you most to get through this assignment? [Commons is the campuswide dining facility at Bates College. Many responses referenced the large wall of cereals available all day to students.]
 Can you locate a signpost in this proof, or something that you feel “unlocks” the proof? [This problem built upon a reading and discussion about signposts in mathematical proofs.]
Optional (fun) assignments:

 Write a tweet representative of your feelings about #thestruggleisREAL. It can be inspiring, funny, reflective, whatever you want! 140 characters or less. Please keep it clean and respectful.
 Create a meme under the theme #thestruggleisREAL. You can make it from scratch or use an online meme generator.
Final homework assignment: Should Real Analysis be a requirement for the math major? Why or why not?
Final project: Finally, as stated in the syllabus, students were given the option to complete a more formal writing piece reflecting on their semester in Real Analysis. I directed students who were interested in the project to first read an article from 2010 in Math Horizons, The View from Here: Confronting Analysis, by Tina Rapke. (If I had found the article sooner, I would have assigned it as reading in Week 1!) The written assignment required students to choose 35 of their own proofs from the semester as representative examples, and then to write 12 pages narrating their experience in Real Analysis using these proofs to illustrate their experiences. Roughly half of the class turned in a narrative and most exceeded the requested two pages in length. I was extremely pleased with the quality of reflection they demonstrated in the final projects. Yes, most students who did the final reflection did so in order to boost their course grade, but many of their writings achieved what I had hoped for at the beginning of the semester: they catalogued moments of growth, success, and failure while taking a look back on their experience in Real Analysis.
Overall Impressions
The most telling evaluation of #thestruggleisREAL is that word got back to me about the hashtag from colleagues in my department. Students were talking about the assignments outside of class, which gives me hope that they were sharing stories of their struggles with each other in addition to completing the writing. The class had a great sense of community throughout the semester and whether by chance or due to the invitation to voice their stories of struggle, they were very supportive of each other.
I use a wellknown text when teaching Real Analysis – Understanding Analysis by Stephen Abbott – so the Internet is always an issue when it comes to homework problems. This semester I noticed a decline in blatant misuse of online sources when grading homework. I would like to think that by being asked to document their struggle, and being incentivized to acknowledge the difficulty of the work, students were less inclined to simply copy an answer from an outside source.
I will certainly incorporate a regular reflective component into future Real Analysis classes, using #thestruggleisREAL as long as it has some relevance to students. The components that worked the best were the regularity of the reflective writing prompts, the periodic inclusion of “light” activities (such as the meme contest) to allow for creativity and humor, and the optional nature of the final project. In the future, I have three concrete ideas for improving upon #thestruggleisREAL:

 Participate in more of the reflective assignments myself, and invite other math faculty to make small contributions.
 Allow for more creativity in how students respond to reflective prompts
 Create a product from #thestruggleisREAL that can be used in future iterations of the class. One idea is to have students collaborate to compose a letter to future Real Analysis students.
I love this idea and find it very inspirational! Real analysis seems especially appropriate for what you did, but I may experiment with similar techniques in other math topics.
correct and as mathematicians we have to give our selftime to think because real analysis need someone who give him/her self to practise and to think in order to approach different problems