By Richard Delaware

In fall 2014, I taught an introductory real analysis course (Math 402) at the University of Missouri--Kansas City. One of my students, Amber, struggled with the material, as many students do. In analysis, students need to prove or provide a reasoned argument for statements that it seems just a moment ago in their mathematical lives they were allowed to take for granted.

As a young mother of three children at the time, she had parental responsibilities that she was navigating while she was in my class.  Since she was often unable to attend office hours, she would email me with questions over the material.  Looking back on our correspondence now, I see that the exchanges provide a glimpse not only into Amber’s growth in understanding the material but also in her figuring out how to study mathematics over the course of a semester, while revealing her persistence and humor as healthy survival tactics for both the course and her outside life.

Amber’s words are used with her permission. I’ve selected which emails to include here, lightly edited them, and left out her descriptions of the mathematics so that we can more plainly see her progression. Each block of text is from a different email.

The semester started out with her unable to get a foothold; everything seemed elusive or slippery:

I didn't get too far with this one. I get confused right here. What am I missing? Please help.

Issues are abundant with the homework. I wondered if you could give me a bit of guidance to help me get rolling. I’m going to try a little bit more.

But within a week she understood the need to nail down definitions:

Quick question regarding sequences.  I want to make sure I am not missing the true concept here.

And she started really investing the time:

I've been pondering on this too long and think I may be getting myself more confused at this point.  Any little hint appreciated.

Her candid self-assessment and self-deprecating humor now began to sneak into her signatures: she signed the previous email “Your (slightly frustrated) 402 Student, Amber.”

Next, she read again my lecture notes and tried her own hand at the tools I used in class:

I have been reviewing proofs in hopes of finding techniques to help with homework.  It seems this should be easy and I am somehow missing something.

Thank you for your help and hints but I am still having a bit of trouble. I tried to draw out a picture to show where 1/m might live and have a drawing similar to class notes. Is this correct?

A month into the course, she began to seek help from her classmates as well:

I need to know what our boundaries are on discussing homework content on our Facebook group page. If we say, well I used this type of technique or I tried to show N did this or ... whatever. What is okay and what is not?

Difficulties remained, but her personal strength helped her push through them:

I wake up every morning saying I'm going to prove something and seem to be meeting roadblocks.

It's your daily email from Amber (ha, ha, sigh).  If you have a quick moment to clarify.  Just wanted to see if I’m doing this correctly and thinking along the right lines.

I have been trying to not be totally mad at myself for not getting back to one of the problems on the exam but that second page kind of threw me for a loop and then I kind of forgot on top of running out of time. I know it probably doesn't really matter now but I did work on this proof at home prior to the exam and did come up with a decent proof (I thought). Your ‘Gonna Tough It Out’ 402 Student, Amber.

Three months into the course, she was combining ideas, and understood the value of additional practice and careful review of her own work:

Sorry to bother you on [Thanksgiving] break but I have a quick thing here and wanted to make sure I’m not thinking incorrectly. This is how I am relating the definitions in my head. I always ask myself ‘what about x?’ I want to make sure I’m understanding correctly here.

I am working on a few practice problems.

I have a question on Homework 4 and I swear I didn't wait until the last minute to do this. I just found my proof to be shady after writing out the final copy. Any quick help would be appreciated.

I loved her insightful use of the word “shady.” But, as with most of our students, her classwork could not be disentangled from life outside the classroom. Later that day:

 (I would pop by for office hours but am going to leave for campus late today so I can pick up my kids from school. Don't want them walking home in this cold today.)

A couple of weeks later saw some exhaustion and her harnessing more grit and humor to continue:

I am hesitant to ask questions on [take-home] exams, but I need help. I have gotten 0 out of 5 done so far and am feeling hopeless. I don’t know… I’m getting pooped out. My mind feels like it needs a vacation. Your Hopelessly Stumped Student, Amber.

Thank you for all your help and positive words of encouragement. I've decided that you aren't really wearing a Dr. Evil costume sitting in your recliner eating popcorn doing the Dr. Evil laugh. [A character in the Austin Powers film series.] Also, I loved the grand finale proofs on the last [lecture] day. :)

I didn't really want to ask for any more help Dr. D., but it is apparent that I need it. I have to go to the hospital this morning for what I'm hoping is a quick test and then plan to come home and get my exam finished up. I'm considering bringing it after I pick up the kids so I will have until 3 pm to work on the exam but that will not put me at the college until right around 3:50 p.m., if I'm lucky. If by chance I get done early I will try to bring it earlier.

[Later that day:]

My issues involve my ability to take two defs I have written out and finagle them into what I want. I feel that since this is a definitions proof I need to let epsilon > 0 but I don’t really know where to go from there. I know I need to pull my known stuff down to give me what I want but am having trouble. Sigh.   Your “Trying Real Hard” Student, Amber.

Near the end of the course she started to gain some confidence:

:) So I can kinda prove stuff. :) The kids got a huge kick out of coming down to the college with me yesterday.

To which I replied:

Amber:

In spite of some deficits we can continue to work on, you can certainly "kinda prove stuff." I think of you as one of my success stories, because what we work on sinks in and you can think for yourself. That means you can and want to get better. The only way anyone succeeds in mathematics in the long run is to have that attitude and persistence.

[Later that day:]

Aww, and thank you so much. Those words mean more to me than you know. I work so hard and this means more to me than anything in the world right now, other than my family. I know I may not have the GPA the Math Dept. desires but to me this is big, this is a real big achievement for me. I have to do this and I'm almost there! Anyways, need to study [for the Final Exam], not get all mushy. Your Humble Student, Amber.

Epilogue

In May 2015, Amber received her Bachelor of Science degree in Mathematics and Statistics from UMKC.

In 2019, she received her Master of Arts in Curriculum and Instruction, Middle School Mathematics from UMKC.

On July 20, 2020, Amber wrote me one more time:

Hi Dr. Delaware.

Feels like old times emailing you. You are still the best mathematics teacher in the world (to me), and I hope that you have been doing well and surviving the madness going on in our world right now.

I have been out of Grad school since December and my brain feels like it is starting to deteriorate, very bored. I have learned a lot more about the concepts of mathematics in my educational journey and realize how much I didn't understand in my mathematics education due to lack of good conceptual knowledge. I was taught processes and not concepts growing up. Now I want to retake my math courses but can't ... but I can study on my own.

I am needing to know what book we used for the sets and proof course I took with you. I feel like that is where I got lost with some things and feel that I could better understand now that I have picked up some conceptual knowledge. If you get the chance, could you please let me know.

I truly appreciate your time and help. I also hope you are doing well.

Thanks so much.

Your old favorite student, Amber.

It's rare that we as teachers have a record of our everyday interactions with students, and it may be hard to remember how unique each one is and how unique their life circumstances are. We should take a moment during every class to celebrate their strengths, often hidden to them, their inspiring candidness and quirkiness, and that most, with our guidance, can cultivate reserves of persistence and humor to carry with them beyond the content of our classes. The habits of mind and of living that we nurture are retained and contribute to the content of their character, just as that character enriches us and our profession of mathematics, something I learned from my “(slightly frustrated), Gonna Tough It Out, Hopelessly Stumped, Trying Real Hard, Humble” student, Amber.

Richard Delaware is a Teaching Professor of Mathematics in his fortieth year at the University of Missouri – Kansas City. In addition to teaching real analysis and other courses, he created and has taught for over twenty-five years a writing-intensive history of mathematics course. See Convergence.