Advanced Theory and Practice of Mathematics Teaching and Learning (EDUC 466) was a course I took in the Winter 2015 quarter. It was my second course with Professor Shawn Vecellio. This is one of a few required classes for the STEEM emphasis that focuses on how educational theories can be used in educational practices. I took the Advanced Theory and Practice of Science Teaching and Learning in the same quarter, which provided me with a rich exposure to the practices of mathematics and science teachers. It was incredibly valuable for me, working in curriculum development, to have insight into the experiences of classroom teachers who implement curriculum day-to-day.

My background is in mathematics; this was my undergraduate major at GWU. I did begin a teacher education program at Columbia University’s Teachers College for secondary mathematics education, so I had taken a handful of “teaching practice” courses, and I was a teaching assistant as a senior in college. However, I had never taught a full lesson in an authentic school setting before. There were lessons that I wrote and tested with coworkers, and I led tutoring sessions with students in my TA classes, and had presented lessons to fellow students, but this was the extent of my “teaching” experience. It turns out, I was the only student who had never been a teacher, which I felt put me in a different position as far as what I could both contribute to conversations (without experience to draw from) and what I could take away from the class (without a “practice” in which to implement what I learned).

I soon realized that this was not a constructive way to view myself in comparison to my peers. Instead, I realized that the course was an outstanding resource that could support my work – understanding the experiences of classroom teachers certainly can help me in developing curriculum that teachers use. In Advanced Mathematics, the practical readings and subsequent discussions gave me a window into what curriculum looks like when it is in use.

As a class, we delved into how teachers teach mathematics and how students learn mathematics. We discussed teaching for mathematical proficiency, the kinds of knowledge[1] (general pedagogy, content, and pedagogical content) that teachers need to have, selecting mathematical tasks, approaches for posing questions and problems, and strategies to motivate students. From my peers, I was able to get a good sense of how much work and how many decisions are made daily while teaching. Understanding the levels of demands of mathematical tasks (memorization, procedures without connections, procedures with connections, and doing mathematics)[2] was especially useful for me. I can compare these levels against problems in the curriculum I work on to assess the level of demand and get a sense of how much students would be expected to get out of it. Tasks at the highest level of demand require complex thinking and considerable exploration of “the nature of mathematics concepts, process, or relationships” (Smith & Stein, 1998, p. 348).

Applied learning is one of the Degree Qualifications from the Lumina Foundation; this project in particular met the Applied Learning goal of implementing a project in a setting outside of the class and articulating in writing what was learned from the experience. For the final project, we were given flexibility to choose how we wanted to synthesize the ideas covered throughout the quarter. I chose to see what an authentic teaching experience is like. One of the students in my class was an 8th grade teacher at the middle school I attended. Since I would like to shift my focus from college-level math curriculum to K-12 math curriculum, it was a good applied learning opportunity to write and teach a lesson for 8th graders. The focus of the lesson was on creating and solving systems of linear equations from real-world problems, so I concentrated on writing higher demand-level mathematical tasks: procedures with connections. I’ve always valued working together, so the teacher, Ms. Franklin, organized the students in pairs and they were able to complete the questions with one another. They were asked to show their linear models in more than one form (equations, graphing, or constructing a table, as well as describing it in words). I wrote two more challenging questions, with the assumption that students who worked through the first two problems could try the challenge problems.

Not everything went as expected, but in hindsight, I did not expect it to: teaching is not easy in my opinion, it is a set of very valuable, respectable skills that one develops throughout their career, it does not come naturally on one isolated day.

My artifact, a reflective essay on the experience and how it was a culmination of the topics covered in class, describes how students approached these tasks, highlights particular interactions and observations with specific students, and discusses my thoughts on teaching for a day and teaching in general. While I may not be a teacher myself, I have a great deal of admiration for teachers: I think they face and overcome great hurdles every day and care immensely not only for their students’ educations, but their health and well-being. I also believe that there is much for me to gain from teaching lessons like this on occasion, because good curriculum development is an iterative cycle—lessons are written, tested, and refined again and again to ensure they meet the intended learning goals and designed to support the learning needs of diverse groups of students.