What Do We Owe To Each Other?
A Blog Post for The Virtual Math Conference on Humanizing Mathematics
A recurring theme from the TV show The Good Place is the question “What do we owe to each other?” I have reflected on this question ever since it came up and have applied it to my life: What do I owe to my parents? My friends? My students? Strangers? Of course, there is no one right answer to this question, but I believe it is important to ponder this question to search for our own answers. Whatever our answers are, I believe they shape our values and our purpose in whatever we do, whether it is in our relationships or in our classroom. So when thinking of the question “What do I owe to my students?” I would like to share with you four ideas that came to mind.

I owe my students a better definition
Going through my bachelor’s program, I thought of being “good at math” as just being able to memorize and apply formulas, definitions, and theorems. Even in high school, I would get jealous of my friend who has a great memory; I considered her better at math simply because of her ability to memorize things really quickly. The issue that I had as a student was that I did not have a good sense of what being “good at math” actually meant. How can students say they are bad at math if they might have an inaccurate definition of what that phrase means?
I strongly believe in focusing on mindset before teaching content. On the first day of summer school this year, I told my students to think about someone they know personally who they believe is good at math. With that person in mind, I would have my students think about five qualities their person has that make them good at math and share their thoughts with a partner. Here are the top five my students came up with:
 Good problem solver
 Asks a lot of questions
 Practice
 Takes notes
 Organized
During this activity, I do not meddle unless there is a quality that I would like to address and perhaps diffuse as a top five quality, such as if someone thinks memorization or speed is a key characteristic to being good at math.
I tell my students to notice that none of these have to do with genetics; no one is born a “math person” and no one is born “not a math person.” I also note that all five of these qualities can be practiced. I ask them, “Can we focus on practicing these qualities during the time we have together?” and they all nod their heads. I write these five qualities on the board every day just to remind students to not give up, and that these qualities are achievable.

I owe my students representative role models
In 2000, Picker and Berry conducted a study in five countries and asked 1213 year old students to draw a mathematician. In this study, they noticed that most boys were drawing male mathematicians, and girls for the most part were drawing male mathematicians as well, as seen in the table below. (Some of the percentages do not add up to 100% because the researchers did not have the data to conclude the gender of the mathematician.)
(Source: Picker, S.H. & Berry, J.S. Educational Studies in Mathematics (2000) 43: 65. https://doi.org/10.1023/A:1017523230758)
I did this activity with my students by saying “Draw a mathematician. When I say the word “mathematician,” what comes to mind? What are they doing? Please take three minutes to draw your mathematician.” Though roughly 80% of my students are female, the majority still draw male mathematicians. The question is, “Can people see themselves as a mathematician?” Going back to response #1, I remind them that being good at math is not genetic, and we need to show that everyone has the capability to excel at math.
After this, I show some pictures included in this study. Here is one drawing by a male student from the UK:
(Source: Picker, S.H. & Berry, J.S. Educational Studies in Mathematics (2000) 43: 65. https://doi.org/10.1023/A:1017523230758)
(Starting from the top and going clockwise, the notes read as follows: Bored tired eyes, Dirty unwashed hair, An unshaved face, Pencils handy in case of math problem, Fat from doing nothing but math, Old math problems, Pants are too small, Hole in wrinkled pants, he’s too lazy to buy a new pair, An old stain, he’s too lazy to wash his shirt, Bad body posture, Wrinkles from thinking too hard)
Showing drawings like these one by one, I ask my students, “How does this person view a mathematician? Would they want to be a mathematician? Why do you think this person drew a mathematician like this?” Not only were the majority of students drawing male mathematicians, but some people saw mathematicians in a negative light as well.
One major reason for students drawing male mathematicians is that students are often not exposed to a diverse group of mathematicians. For example, after I completed my undergraduate degree, I could easily name 30 white male mathematicians but would have trouble naming three mathematicians that do not fit that category, which was embarrassing. We owe it to our students to make mathematicians who are women and people of color more visible. This is why I love what Chrissy Newell is doing. She has an online shop (https://shop.spreadshirt.com/womeninmathematics) along with the hashtag #MathGals. This is so important because wearing shirts like these will spark curiosity and hopefully start conversations. Women can, and always have been able, todo well in math. For quick access to mathematicians of color, Annie Perkins curated a great list found here: https://arbitrarilyclose.com/2016/08/21/themathematiciansprojectmathematiciansarenotjustwhitedudes/ (Thank you to Sam Shah to directing me to this website). We need to do our part in increasing visibility to show that mathematics is a human endeavor, no matter your background.
(For those who want to do this activity with your students, or to see more pictures from this study, I created a quick Google Slide here: https://docs.google.com/presentation/d/1yimmq1ptqytniGa0ZMuEkgD8Ei92qlSw5NObwFTO3Bs/edit?usp=sharing)

I owe to my students an open mind
I watched this video (https://www.youtube.com/watch?v=h00Ux1qx2zw) from Max RayRiek where he talks about the idea of “Listening to” versus “Listening for” and it has changed the way that I conduct class. The difference between these phrases, at least from my understanding, is that “Listening to” is really listening and honoring student thinking, whereas “Listening for” is looking for a student to simply match with the answer already in your head. I will share one story that really elicits this idea of “Listening to” versus “Listening for.”
Early on in the semester, I had my students look at patterns, so I put up a couple sequences on the board, asking them to find the 100th term. One was 3, 7, 11, 15, 19,…
If I had a “Listening for” mindset, knowing that we needed to cover the arithmetic sequence formula, I would be hoping for someone to come up with the thought process of, “Well…I see that we are jumping 4 each time, so we would need to jump 4, 99 times starting at 3 to get the answer” and then I would guide that thinking to the arithmetic sequence formula. A couple of students did share that exact reasoning, but one student raised his hand and said “I did it differently and got the same answer.” Intrigued, I told him to come up to the board to share. He said, “Well, I noticed that each of these numbers are one less than the multiples of 4, so I did 4 x 100, then subtracted 1 to get 399.”
When he said that, I was just blown away, and afterwards, it made me realize the strong connection to each arithmetic sequence formula and its relation to being a constant away from multiples (4n1 is just one less than the multiples of 4). When we ask for multiple strategies like this, it really humanizes math, and it shows that math is not just about memorization of formulas. Additionally, it invites our students in, honoring their thought process and show that math isn’t robotic, but rather, creative. Listening to students really opens up this idea of what math means to them, that their ideas matter. When we are listening for, it may feel like students are just guessing what the teacher wants them to say, and that, to me, is not what math is. When we listen to, we are building students up as their own critical thinkers, becoming their own mathematician, and as an added bonus, we teachers get to learn as well!

I owe my students a different lens
To quote Galileo Galilei, “To understand the Universe, you must understand the language in which it’s written, the language of Mathematics.” To this end, math is a practice, and as teachers, we should constantly look at our world with a math lens. We all have a different lens of how we view the world, but once we see the mathematics around us, we notice Earth is a beautiful and mathematical place. For students to practice this, I have them go on a walk and search for math around campus, taking pictures and including their noticings and wonderings on a shared Google Slide. When students reflected on this activity, they did not realize that math was everywhere; they had a hard time seeing math outside of the math classroom. For example, where do we see fractions outside the classroom? Can we readily see how we can use the four main operations outside the classroom? The stronger we see how natural and integrated math is to our lives, we invite our students to a more beautiful world.
Conclusion
Going back to the original question of “What do we owe to each other?”, for my students, I owe them a humanizing way to see and understand math. It is so easy to just tell students to memorize formulas and see math as black and white, but we dehumanize math if we approach it this way. I owe them the idea that they are whole and complete, and that their thoughts and opinions matter. I owe them the idea that mathematics is part of their identity, and mathematics opens up a lens to see and understand the world. I owe them the idea that mathematics and humans are more interconnected than they think. Math is a human endeavor and one that every student can pursue.
Thank you for reading.