By Hortensia Soto

A couple weeks I ago I visited my best friend, Debbie, and as usual we sat around laughing and sharing about novel gadgets that simplify our lives, about the latest news regarding certain foods, about the new vitamins that we are taking, about new tricks that help us sleep, etc. Once I got home, I ordered the new gadgets, bought the vitamins, and tried the new sleeping tricks. My sister laughed when I said, “I learn so much from my conversations with Debbie.” That is when I knew what I would write about in this blog.

Learning happens in the conversations that we have, and our relationships determine the conversations. Moreover, if learning happens during our conversations, then teaching must also happen in these conversations. If my logic serves me well, then what I am saying is that relationships determine conversations, and these conversations lead to teaching and learning. Furthermore, the ability to have safe and meaningful conversations can forge healthy relationships, which can foster teaching and learning, which in turn leads to new conversations (see Fig. 1). Thus, what we say, how we say it, when we say it, and to whom we say it matters. This is also true in the mathematics classroom.

Conversations are pivotal in learning; thus, our mathematics courses must provide ample opportunities for conversations to occur between students and between instructor and students, inside and outside the classroom. Such course design must be intentional or as I like to say, “on purpose and with a purpose.” Here are a few teaching practices that I do on purpose.

1. Know students’ names at the end of the first class.

2. Provide opportunities for students to work with different students.

3. Ask students how they are doing, listen to their responses, and respond in a way that indicates that I heard them.

4. Be vulnerable.

For the past 26 years, I have memorized my students’ names by the end of the first class. I do this by requiring the students to work in groups of 4-6 students on a mathematical task that introduces overarching themes from the course. As the students converse and work together, I walk around to each table, make eye contact with each student, and say their name. I do this over and over until I have a name associated with a face and a location in the classroom. Before the students leave, I randomly call out all the students’ names and ask them to say my name. I also ask them to sit in the same place for at least three weeks. Knowing my students by name and taking the time to ensure that I pronounce their name correctly, allows me to ask questions such as, “Francisco, what might you add to Diana’s response?” Questions directed at specific students allow me to engage more students in classroom conversations, to make direct eye contact with a student (Francisco), and to acknowledge another student’s (Diana’s) response.

I acknowledge that such direct questioning might make some students nervous. Generally, this is not the case for me because I let students know that this is my questioning style, and I explain that it is on purpose to ensure that everyone contributes to classroom conversations. Every 3 weeks, I rotate members in groups so that students build different relationships and engage in different conversations with the purpose of creating different learning experiences. Explaining my rationale for such actions is important to me because it helps build trust, thereby strengthening our relationship.

The strong bond that we build as a class provides a safe space for students to share genuine concerns when I ask how they are doing. For example, recently an abstract algebra student bravely shared with the class how scared he was to tell his parents that he no longer wanted to pursue a physics degree. Several students encouraged him and told him that his parents would support his decision. He then asked me what I thought, and I replied, “Parents just want their kids to be happy. I bet that is what your parents want for you too.” A week later he walked into class and shared that he told his parents and “they were cool and understood.” The class clapped! These conversations are possible because we purposefully develop strong relationships in class to foster learning from one another.

Of course, students are not the only ones who need to be brave. As a teacher, I need to remember to be humble and vulnerable. I try to model this by letting students know that I make mistakes and to feel free to call me out on them. I also share stories about my own excitement about finally understanding a concept and my struggles in learning some concepts. This lets students know that learning takes time, that mistakes are learning opportunities, and that together we will learn from one another. Because of our relationships and through our conversations we learn mathematics and about life!

Hortensia Soto is a Professor of Mathematics at Colorado State University and the President-Elect of the MAA. Her research centers on the teaching and learning of undergraduate mathematics.