# Teaching Opinions

I consider Federico Ardila's Axioms foundational:

Axiom 1. Mathematical potential is distributed equally among different groups, irrespective of geographic, demographic, and economic boundaries.

Axiom 2. Everyone can have joyful, meaningful, and empowering mathematical experiences.

Axiom 3. Mathematics is a powerful, malleable tool that can be shaped and used differently by various communities to serve their needs.

Axiom 4. Every student deserves to be treated with dignity and respect.

Before everything, grace. My course is just one part of the rich tapestry of students' lives.

Mathematics is for everyone and our mathematical spaces must reflect that. Broadly speaking, they do not, and it is on us to change both the formal rules and the culture of our mathematical communities to make room for everyone. The systemic barriers in mathematics require systemic solutions.

I am not actively political in the classroom, but I am openly anti-bigotry.

Every course is a dialogue; learning happens when students meaningfully engage with the material presented, and when they receive timely and appropriate feedback. Meaningful engagement requires that students understand the point of what they are doing, hence the effectiveness of a teaching strategy requires genuine buy-in from students. Because of this I am proactively open and honest with students about course design. When I set activities for students I also provide an explanation of their purpose and how the activity design aligns with that purpose.

For example I explain the difference between formative and summative assessment, pointing out which activities are formative and which are summative. I set weekly homework because learning mathematics requires sustained engagement and regular feedback. I tell my students that the purpose of the homework is not to assess them, it is to help them learn the material and so I can gauge what content the class is struggling with. I am open that the contribution of the homework to their final mark is there as an incentive for them to do the work.

Another reason to be open about course design is to help students engage in meta-cognition, something that all educators should explicitly discuss with their students. Many students locally optimise a poor set of study techniques. I cannot impose good practice upon students, and the first step is to convince them that it is worth their time to think critically about how they learn.

An aspect of mathematics is its use as a language, and the best way to learn a language is to communicate in it. Thus there need to be learning activities which give students the space to practice these skills. When appropriate I encourage my students to write in complete punctuated sentences. For tutorials I have students to work in small groups, with the tutor being there to facilitate and provide immediate feedback. When it comes to homework I encourage students to form study groups so they can discuss ideas together. Giving and receiving constructive criticism from their peers about their ideas hones their ability to state things precisely.

I believe that a focus on numerical marks can distract students from the more explicit and useful feedback they receive on assignments. Students are often focused on marks in a way that is detrimental to their learning. For this reason I think it is useful to release feedback before providing numerical marks.

I completed my undergraduate degree in mechanical engineering and worked for several years as an engineer before returning to further study. I have been a full-time student, a part-time student while working full-time, and a distance student. I find that letting students know that I have taken a non-traditional path in academia and that I have struggled with the mathematics they are now learning helps build rapport and make the material seem less daunting.

In the 2017 fall term and the 2018 winter term I lectured a 30 student college algebra course for mostly first year students at the University of Oregon. During the 2017 Fall term I attended a term long seminar about teaching this course and active teaching strategies more generally following Scientific Teaching. In the winter term I attempted to run a flipped class, in discussion with the class I returned to giving regular lectures after a month. However I learned a lot in the process. I do believe that properly supported and organised flipped classes can result in better learning outcome for all students, and I plan to return to this class structure in the future.