Discussions About Proofs Drive Student Learning
By Audrey Malagon
Instructors know that students need deep understanding in proof-based courses. The literature shows that active learning has serious potential to help students make these learning gains. Kate Melhuish, Paul Dawkins, Kristen Lew, and Robert Sigley see discussion as a key mechanism for inviting students into the generation and understanding of mathematical ideas. Their project, Orchestrating Discussions around Proof, focuses on providing tools for instructors in undergraduate Abstract Algebra courses to incorporate meaningful discussion based on models used in the K-12 setting. Here Dr. Melhuish walks us through the project and what they’re learning about using discussion in the classroom.
1) Why is student discussion important in mathematics classes?
A request as simple as, “turn to the person next to you and discuss what you think this symbol means” provides a space for students to not just take in a lecture but engage with the content personally. Through discussion, students can come to joint understandings, ask new questions of mathematics, author ideas into classroom content, and ultimately engage in more authentic activity.
2) Why did you choose to study Abstract Algebra courses for this project?
Abstract Algebra is a required and often pivotal course for mathematics majors and future high school teachers. For many, it is one of the first places they grapple with formal definitions and deductive proofs – a substantial shift in how we do mathematics. Here we can support students in a particularly challenging setting and learn about supporting productive discussion in proof-based courses more broadly.
3) Your project builds on research in K-12 around class discussions. What research or methods are also applicable to higher education?
We have found that teaching practices from K-12 classrooms are largely applicable to the college setting. The nature of the proof-based context requires attention to different types of teaching moves and instructional focus. For example, logical quantification becomes essential for engaging students in mathematical proving activity. Thus, asking questions about logical quantification (e.g., “Does this statement cover some or all objects?”) is an important driver in conversation. Additionally, the types of objects being discussed are often quite abstract and rely on layers of mathematical symbolization. If students are to have a productive conversation, it is important to develop a joint understanding of the symbols at play. Guiding questions might include, “What does this symbol represent?” or “What type of object is this?” Navigating between the formal mathematical system and informal systems became key to the implementation of nearly all the teaching practices we adapted from the K-12 settings.
We also found that the proof-based setting has potential to amplify status inequities between students. Students who more quickly adapt to the requirements of the formal proof may take a more substantial role in discussions. This led us to adapt existing mechanisms for balancing group work participation into the proof setting and to attend to ways students can show their competencies beyond just constructing proofs.
4) What can instructors who wish to incorporate more discussion in their classes take away from your project?
First, incorporating discussion does not mean that every aspect of instruction needs to change. There are small yet powerful ways of opening up a class, such as inviting discussions about the meanings of terms or symbols. If students have a chance to actively think and discuss questions about the symbols and meanings within proofs, they are in a position to better comprehend proofs.
Second, introducing good tasks and space for discussion does not guarantee that all or even most students will be actively engaging in authentic mathematical activity. When we used think-pair-share mechanisms, we found that students did not equally participate in discussions. In later stages, we developed specific roles and responsibilities to support students in more structured investigations and collaboration. For example, each member of a small group was given a card with a specific question about a complex proof. That student led that part of the group discussion. This provided students specific goals for digging into a proof and also set up specific roles for collaboration to increase the likelihood that everyone contributes meaningfully.
5) What impacts did you find student discussion had on comprehending, validating, and constructing proofs?
In proof-based classes, it is not uncommon for students to “not know where to start.” Developing a sense for what tools to use when is challenging. This is true not just of constructing proofs, but also validating and comprehending proofs. In our setting, we focused on three specific proofs — each of which were quite challenging in their own way and that prior research had shown present challenges for students. We found that students were able to make sense of the complex maps and elements in the First Isomorphism Theorem, recognize validity issues in common approaches to the structural property proofs, and develop intuition for the multiplicative structure of cosets, the key idea behind Lagrange’s Theorem.
Instructional mechanisms that supported student engagement in these activities included discussion comparing student proofs, comparing examples, anticipating proof structures, and recalling key terms and meanings. As students worked together, they engaged in rich interplays of making arguments (constructing), comprehending each other’s arguments, and then validating them to provide feedback. We see this as indicative of how they were getting access to authentic mathematical activity in their discussions.
6) What did you find most surprising in your project?
We were most surprised by the way our choice of analytic lens presented different answers as to whether students were engaged in authentic mathematical activity. We found that attending to if students were engaged in authentic activity without attention to which students were leading the discussions amounted to an overly positive interpretation. We were meeting our hypotheses about students doing all sorts of rich activity, but we were losing information about whether this was being accomplished by only some or all of the students. We kept having to change our analytic perspectives to get a more holistic (and realistic) view of what was going on in student discussion.
Learn more about NSF DUE (#) 1836559
Full Project Name: Orchestrating Discussions Around Proof
Abstract Link: https://nsf.gov/awardsearch/showAward?AWD_ID=1836559&HistoricalAwards=false
Project Contact: Dr. Kate Melhuish melhuish@txstate.edu
Access resources related to this project at https://rume.txst.edu/