Earlier this week, I had the pleasure of hanging out at the the Mathematics Assistance Center (the MAC) at IUPUI. The MAC is a tutoring center available to any student taking a math class at IUPUI, but it’s quite unlike the dusty, desk-y sort of space one might imagine. Recently redone, the MAC is a large room that is open, well-lit, and full of movable tables and chairs. It’s noisy, the walls are all covered in whiteboard paint, and the space is partitioned by smaller movable white boards. Throughout my stay, there were perhaps fifty students to about ten tutors.
I found myself a table and sat down. Without delay, the person across the table introduced himself and asked what I was working on. I explained that I am a researcher trying to get a feel for the space. He immediately started raving about the MAC, noting that he liked to spend a lot of time there, found the tutors really helpful, and sometimes collaborated with other classmates.
While not a “creative learning space” in the same way, say, a makerspace is – students aren’t necessarily working on interest-driven projects – the MAC certainly is creative in its approach to how tutoring takes place and how math is learned in general. It is built to support participation and collaboration.
As an undergrad, I tutored at my school’s equivalent of the MAC. Although much smaller, I really enjoyed the experience. It was a place that engendered the questions about learning that pushed me towards the learning sciences, and the community it fostered was an important part of the larger community of our (small) department. I think about tutoring differently now, of course, but being back in such a space reminded me of my own experiences.
Projects, Interest, and Learning Community
As I foreshadowed, the MAC complicates the distinction between formal and informal learning spaces. It is formal in the sense that students work mostly on homework or studying for assessments, so their activities are dictated by the formal class activities. On the other hand, it is hardly a “formal” space in the way a lecture might be. Students aren’t always “on task” – and they’re not expected to be. Students eat lunch, chat about non-math ideas, and generally seem to be “hanging out.” I wouldn’t be surprised if many of them stayed at the MAC after they finished their math homework to continue working.
The social nature of the MAC is illustrated by a student who walked in and saw one of his friends. The former came up to the latter and they chatted for a bit before the new student went on his way (presumably to sit with others from his class). This small moment showcases both the social/”informal” nature of the space (catching up with friends) and the disciplinary/”formal” nature (but then moving on to work on assignments).
Although the activities were not strictly interest-driven, there were a wide variety of classes and students represented. On display were many approaches (sometimes learner-dictated) to tutoring. For some students, their problems were worked by tutors on the board. Then, the learner would copy down the answer and solution method. A student at my table displayed that even in this seemingly-cheap interaction, significant learning was taking place. The learner called the tutor back during the copying process to ask about a specific step, indicated that he was not blindly copying the solution but rather sought to understand it. Other students worked the problems with the tutor (in a call-and-response format; for example, the tutor would say “How do we do the limit of a square root” and the tutee might respond “multiply by the conjugate?”). Still others were at the board themselves, taking a greater ownership role over working the problem, while tutors looked over their shoulders or advised them further.
There was certainly a community among the tutors. They seemed to be friendly with one another and felt comfortable asking one another about problems if they didn’t know the answers. When no student had questions, the tutors varied in their activities. Sometimes, they asked each other about math content (e.g., “How do you prove to a student the derivative of log?”). Other times, they just played around (e.g., writing “sin(x)/n = 6” on the board). I cite these moments as evidence that there was a learning/teaching community being formed among the tutors.
The center is marked with different zones for different classes, so learners generally sit in the area designated for their class. This allows them to talk to other learners working on the same ideas and allows the staffing of tutors for only certain classes. For example, some courses that were required for exercise science majors were staffed by a tutor who was also an exercise science major. Because she had not taken all the math courses, she couldn’t reasonably be expected to tutor, say, calculus. But given the value of having tutors/mentors from one’s own background, the space was organized to allow for the hiring of a diverse array of tutors.
The space was deliberately designed to be open with movable chairs and tables. In the time I was there, however, no one took advantage of this mobility. Students settled in one place when they arrived and generally stayed there until they left.
One can still see the potential afforded by such flexibility. Tutors might reorganize students who they see are working on the same sorts of problems, or learners might push their chair over to work through just one problem before returning to their home base. I imagine the mobility of the chairs and tables is especially called upon before tests when large groups of classmates gather.
Another key feature of the space is its wall-to-wall whiteboards. Tutors seemed particularly comfortable using these to demonstrate solutions. By using whiteboards, they “publicized” ownership of the solution process. One could conjecture that such publicity could help learners feel more empowered to problem-solve.
I confess that in my observations, few learners went up to the board. However, I have anecdotal evidence to suggest that it is not an uncommon practice. In general, tutors I chatted with seemed to not have considered the possibility of the boards being used by learners. Baskets with markers and erasers are available at every table, suggesting that anybody can use these tools. Thus, access to the tools was not restricted even though in practice, their use was. In my own tutoring experience, many learners who feel comfortable in the tutoring space were more than happy to start working problems on the board on their own. It may also be that I was observing mostly individual students; when groups get together, they may choose to demonstrate more work on the board.
Values and the Culture
As I have tried to demonstrate, there certainly is a more social culture around the MAC. People joked and talked about non-math topics. This more playful (and presumably less intimidating) approach to math is exemplified in the (tutor-made) intro video posted on the MAC’s website:
Undoubtedly, the MAC is a welcoming, open sort of place that may help change students’ perceptions of what math is, how it is done, and how they might relate to it. Although the MAC’s values – of understanding over achievement – may greatly change learners’ relationships to math, in chatting with tutors I did not see tutors’ views of what tutoring is to have been complicated by this culture. There was still a general sense that you have to explain a problem in different ways until a student “gets it.” For some students, this might be the very value of the MAC. For others, I wonder if this approach might not fully exploit the affordances of a space that might easily lend itself to more constructivist ways of thinking about tutoring.
Constructionist Interpretations and Questions
In his 1980 work Mindstorms, Papert comments extensively on our cultural relationship to math. He couples our fear of math with a fear of learning more generally but points to the ways in which math has been disassociated from its roots and purposes. What might be going on at the MAC that Papert would laud or push back against? I have a few ideas for exploration:
- Body-Syntonic learning. When working with children learning math, Papert believed that relating concepts to one’s own body was a key part of the learning process (although not always explicitly necessary). For example, when trying to figure out how to draw a circle (with a turtle holding a pen in the language Logo), Papert encourage children to pretend to be the turtle (“play turtle”) and figure out how they walked in a circle. To what extent can this sort of body-syntonic learning be present in the MAC? Presumably, learners won’t want to pretend to be a parabola and throw their arms up like tangent lines (maybe some will; I don’t judge). But does the act of walking to the board and physically writing out the steps affect cognition? Surely it does in some way. A theoretical question is: how? A design question is: how might the MAC capitalize on the affordances for learning of students using the boards?
- Debugging. Center director Kevin Berkopes raised the point that the MAC might provide a setting in which mistakes are better valued. This seems relatively obvious for learners, but it also seems true for tutors. I extend this idea to probe the computational idea of “debugging” a solution. In my own math tutoring experience, students would come in all the time and say “I did this problem but WebWork says it’s wrong – what did I do wrong?” At the time, many tutors would look at the piece of paper and find the error and point it out. I, as a hard-and-fast rule, refused to debug (this didn’t earn me many fans) and insisted we work the problem from the top. Knowing what I know now, I think the optimal solution lives somewhere in between the two. Papert is really interested in the idea of debugging and I think the skill of debugging a math problem is one that is never explicitly taught but is a key part of the disciplinary practice. Can we conceptualize tutor-mistake-making as a kind of debugging process in which both tutors and tutees can engage?
- Objects-to-think-with. Papert outlined his love of gears (and in a previous post, I discussed my own love of toy airplanes) and how this helped him learn. How might objects-to-think-with be used in the MAC? Perhaps it is through the lens of shared majors and interests, as with the exercise science students discussed above. One could argue that a major is abstract (whereas the gears and airplanes are concrete). But Wilensky suggests that it is not objects or ideas that are concrete or abstract – rather, it is our relationship to them which can be “concretized.”
- Making in Social Contexts. I conjecture that the movement of problems from worksheets to board – where everyone can see them – changes the ownership of the problems and their solutions. This seems worth further study in the MAC. How is the movement of problems from private (worksheet) to public (board) spaces negotiated? What are its implications for learning? How does it support (or constraint) collaboration among multiple tutees in the space?
These are areas worth unpacking throughout our ongoing research in this space. All of these questions beg for further observation, but they will also require further thought about the underlying constructionist theory of how people learn.
(The author would like to thank the MAC director, tutors, and students for allowing him to observe and chat about the ideas discussed here.)