Adventures in Corequisite STEM Calculus: Working with an Embedded Tutor

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Two women smiling with T-shirts "It's a good day to do math"

Welcome back, my fellow adventurers! Last post, I introduced some of the ways my Embedded Tutor, Allison Stern, works with students during our Building Thinking Classroom activities. This time, you get to hear from her directly. 

I met Allison years ago when she was a math student at San Diego Mesa College. She never took any classes with me and didn’t work as my embedded tutor then, but she was always around the math department so we became friends. She has since transferred to and finished a degree in math at UC Berkeley before heading back to San Diego. We’re fortunate that our tutoring program allows tutors who are not currently students – this gives us experienced tutors for all classes including statistics and higher level math courses like differential equations. I was lucky enough to get matched with Alli in Spring 2023 and she worked as an embedded tutor in my partially online Calculus class. Since then, we’ve worked together in our supported Calculus class the last two semesters. 

Important Context: Embedded tutors at San Diego Mesa College attend the classes that they are assigned to and then hold one hour-long session per week, typically after a class meeting. Our tutors also do not have to be current students, so we have several tutors who have returned to work on campus after finishing their bachelor degrees.

Kelly: I mentioned that you have successfully transferred from Mesa and completed a degree in Mathematics. Is there anything else about your lived experiences that you believe helps you support students?

Allison: Math did not start to ‘click’ for me until my senior year of high school. I spent years with private tutors, hearing the same steps over and over but not understanding the ‘why’ behind what I was supposed to be doing. In 4th grade I remember forming a strong “I am not a math person” belief once memorization and timed multiplication facts came into play. Because I spent so many years feeling quite frankly bullied by math and math classrooms, I know math is not always an enjoyable activity and the math classroom is not always a place where students feel safe to explore or share ideas. I believe that my not so pleasant history with math as a learner plays a huge role in how I support students in and outside of the classroom.

I am living proof that our relationship to math is not set in stone. We can change it when we understand how we learn, and what we need to succeed. I try to understand how students process material and how they  relate to the subject. If something is not clicking for a student, I verbalize that I am trying to think of ways to explain the idea differently, so that they see that the learning process is shared between us. I try my best to be vulnerable and truly listen to their experiences.

Kelly: You were my embedded tutor in my partially online Calc 1 class, prior to AB1705. Have you noticed a difference in the students these last two semesters compared to that first semester?

Allison: I see a larger gap in some algebra skills, but many of the mistakes are the same as I saw with students who had more math preparation. Calculus students often make more mistakes with algebra than they make with calculus. For example, a student can understand the product rule for taking the derivative and find the derivative of y=sin(x)ln(x). Yet, they may incorrectly see the function  y=sin(3x) as y=(sin)(3x) and try to use the product rule again instead of the chain rule. They have a shaky understanding that “sin” does not mean anything without an argument. Of course, it also doesn’t help that we often just say ‘sin’ when talking about ‘six(x)’! 

an embedded tutor at a whiteboard while 8 students look on

Kelly: From my perspective, Allison works more closely with groups when they are working on whiteboards, while I take a bigger picture view, trying to look for quick nudges, whether a hint or extension would be useful, and if I need to pause the work to discuss something as a class. 

What do you do to support students during our board work? How do you choose a group to work with?

Allison: Lots of ideas about this one!

First off, if a group asks for my help, I will help them out as long as I am not working with another group. In times when a few groups ask at once, usually by the time I get around to them all, they have already solved the problem together or asked other students.

When a group doesn’t ask for my help but I join them, there are a few things I might be doing. If I notice an error, I like to ask the group to tell me about their ideas or their process. Sometimes, as they are explaining, someone in the group will notice the error, and they take it from there. Other times, no one in the group sees the mistake, so I will ask more pointed questions to prompt them to notice it. And other times, it’s just a really small thing and I simply point it out to the group.

Something I do while Kelly is starting class is make sure there are plenty of markers at every whiteboard. I do this so that a single student cannot  hog the marker and so that I can shift who is leading the group when I notice only one person has been writing on the board. I might go over to a group, ask them a question or make a note on their board, and then hand my marker to someone else in the group to encourage them to write on the board. This helps get more group members involved.

I also like to use board work as a time to check in with students. How did they feel about the lecture or the discovery-based activity? Do they have any questions they wanted to ask but didn’t during the lecture? This way board work is a space for group learning, as well as independent reflection. I take away observations and questions and raise them with Kelly if they feel like a good topic for a whole class discussion, or I may just keep them in mind for the group tutoring session. 

Kelly: While I try to minimize the time spent lecturing, there are times it has to happen – usually to solidify understanding about what was discovered in the day’s activity or to demystify new mathematical language. 

What do you see your role as being during a lecture? What are some of the moves you make as an embedded tutor to support students during these more teacher-centered moments?

Allison: During lecture I try to hold both the perspective of a student and of a teacher. I try to recall common misconceptions and clarifying questions students have asked in the past about the topic, then I raise them during Kelly’s lecture. Because this is a supported class where students may have different levels of math preparation, I also look for places where little algebra steps are glossed over or where students may not follow the jumps, and I might ask Kelly to explain how she did certain steps or why she did them. Also, I sit towards the back of the room, so I can see all of the students. Students will sometimes signal to me if they have a one-on-one type of question and I will go to them and answer it. I also check-in with students near me to get an idea of whether the material is landing for them. In general, I am trying to take a birds eye view, thinking of the big things Kelly is sharing with students and where students may need support with previous topics embedded within the lecture material.

Kelly: Allison holds sessions twice a week for our students, Monday and Wednesday after our class meetings in our STEM Center. Our class ends at 11am and her sessions run from 11:15am-12:15pm.

students smiling while sitting around a table studying

What does a typical embedded session look like?

Allison: I like to think there is a certain structure, but in reality it depends on what we are covering, what students need, as well as how big the group is. But I always start by bringing out the snacks!

I usually begin with some sort of check in to see what students need from me, what topics they want to cover, as well as how they are feeling about what we covered that day. If we are in a topic where we are learning a lot of new vocabulary, I ask students what words they heard in class and write them up on the board. We then work together to connect them to calculus concepts. I like this  little brain dump as it is a chance for students to tell me what they know and how they are understanding ideas. Even if it was not a vocabulary heavy day, I usually will do some sort of mini review at the board asking students the main ideas so I know where they are at. This is usually a place where students will also say they want to talk more about a specific word or idea that was shared.

For a handful of sessions when we are first learning differentiation and integration, I will write up common derivatives or integrals that we expect students to know , such as d/dx(x^n)= or the integral of sin(x) =, but leave them unanswered. I then have students grab markers and fill in ones that they know. This is a good way for me and for students to check in on what they know and what they need more time on. It is also nice to have the derivatives/integrals to refer back to during the session, and it’s something students made rather than me!

We also spend a lot of time practicing problems. We will usually do one together, with students sharing ideas about how to move through the problem and me annotating their ideas up on the board. This is a place where students will often ask why another student did what they did or how they knew to do it. Often the best explanations come from students who are just learning the material. Then, I will write a new problem up and have students try it on their own, and I check in as needed while they are working. When students finish, I will ask for different students to explain what they did and see if they all got the same answer. If they say different answers, we will usually spend some time understanding the different ideas they tried before moving on.

I think it is important to share that my tutoring sessions were quite different when I worked with instructors who do not do much collaborative work during class time. For such classes, my sessions were focused more on group work. It was all about helping students build a community together since there had been fewer opportunities for that to happen in the classroom. We often spent many sessions breaking down fears around making mistakes when working in groups. By the end of the semester, students were usually self-sufficient and were leading the tutoring session as a group with me sitting back and monitoring. Because of the way that Kelly runs her class, students are already comfortable working together, and this allows them to use the time in session to really  assess where they are individually with understanding the material.

Kelly: How do you encourage students to attend your embedded tutoring sessions?

Allison: At the beginning of the semester I make sure to make many announcements about my session times and write them up on the whiteboards, as well as my general tutoring hours in our STEM center. I try to be personable in class and develop relationships with students so they hopefully feel safe and see that I am a potential resource for them. During board work or lecture when students are asking me questions, I will remind them that we may have more time to dive deeper on that topic in session if they are able to attend. I have found that when I connect one on one with students, get to know what they need in class, and let them know how we can work on such topics in session, they more often than not will come to sessions. All of that is with the caveat that they need to be free during my session time. I have a handful of students who want to come to sessions but have other classes during that time, or even work and are not able to attend.

Kelly: Allison has become an integral part of my class – not just all the amazing things she does during class, but as a sounding board for ideas. I get her opinion on ideas for ways to introduce topics or problem sets when we check in before class each morning. Additionally, she lets me know what the students are struggling with either in class and in her sessions. I will never forget that first semester together – I had students working on something and she came up to me to let me know that the students in the back of the room were lost. (This was before I randomized groups.) I would have never realized that because the front half of the room was doing well, and I wasn’t moving around the room like I do now after that feedback.

What other advice do you have for tutors and faculty working to support students in these calculus classes?

Allison: Be aware of the underlying knowledge we use in calculus that often goes unmentioned during practice problems, and pull back the curtain on such topics. For example, if you know you are going to start chain rule, it would likely benefit students to do a few problems with composite functions first. Another big one is notation! Being so immersed in math language and notation, we (tutors and instructors) know when a certain notation means something different in different contexts. For example, y(x) can mean multiplication of y and x or it can mean a function value depending on the context. Another example is (a, b), which can refer to an ordered pair or an interval. Our students may need support in unscrambling ambiguous notation. Mostly, when you notice a mistake or error, instead of just correcting it, ask students to explain their thoughts back to you so you can hear how they are seeing and understanding the material.

Want to discuss calculus with corequisite support? Join Kelly during community hours on Thursdays from 12-1pm in a Coreq Calc Discussions Zoom Room every week until May 23rd.

Kelly Spoon

Kelly Spoon is a Professor of Mathematics at San Diego Mesa College in the San Diego Community College District. She has been at the forefront of her institution’s response to AB705 and AB1705, playing a key role in coordinating Mesa’s corequisite classes for Intermediate Algebra, Statistics, and, most recently, Calculus.

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