IMPACT Live!

Thank you for sharing, Sidney.  Your experiences resonated with me, especially in taking the deep dive into and the research into what we term as “elementary mathematics” in grad school, which I also found to be sophisticated and very interesting! I had also found the core ideas of elementary mathematics carrying into the courses that we call “mathematically intensive”.

I loved re-reading the article by Ball and colleagues on content knowledge for teaching, it brought back memories, thank you for sharing!  This is one of the seminal works on characterizing the mathematical knowledge for teaching and characterizing the components that encompass it.  It had been some time ago and it was nice to re-read this resource. I then recalled reading other researchers who had expanding on this seminal work, such as Hill and colleagues (citation below) who delved deeper on the model of “egg” used to model mathematical knowledge for teaching.  At the time In reading these works, they also helped be become a better teacher, by helping me reflect on my own practice.

Source:

Hill, H. C., Ball, D. L., & Schilling, S. G. (2008). Unpacking Pedagogical Content Knowledge: Conceptualizing and Measuring Teachers’ Topic-Specific Knowledge of Students. Journal for Research in Mathematics Education, 39(4), 372-400.

I do NOT, at least now, teach the way I was taught. 

I had always been successful in math classes which were pretty exclusively taught in "traditional" ways. I was good at memorizing and applying formulas. I was able to extrapolate my learning into different and tangential areas of mathematics. 

However, after becoming a middle school math teacher and teaching the way I had been taught, I began getting peppered by my students about the why behind the formulas. I realized that I didn't actually have much genuine  understanding of the very ideas I was professing. 

I began working to wrap my mind around why all of the formulas we used genuinely make sense. The formula for the surface area of a pyramid

(SA =.5*slant height*perimeter of the base + area of the base) was the first one I went after. I became really obsessive about making sure I could help the students see why everything made sense.

This odyssey was given additional steam when, during my phd program, I was asked to teach math for elementary school teachers. I had to take a really deep dive into elementary school math. Elementary school math is far more interesting and sophisticated than we give it credit for. The deep dive into the "basics" has had a profound impact on how I teach college mathematics. My studies around fractions impacts the way I think about and teach rational functions. My work with division, partial quotients in particular, impact how I think about and teach division of polynomials and factoring higher degree polynomials. These are just a couple of examples, but ultimately studying elementary school math has significantly enhanced my teaching of college mathematics. 

From a more researchy perspective, my experience points towards what Ball calls "Mathematical Knowledge for Teaching." The knowledge to teach math is so much more robust than simply I know how to do some math. Do I know how to make the math accessible to students (especially students who have different backgrounds, cultures, experiences from me)? Do I know what students learned before they got to me and do I know the math they're taking after me so I can situate their learning in a broader context? 

Loewenberg Ball, D., Thames, M. H., & Phelps, G. (2008). Content Knowledge for Teaching: What Makes It Special? Journal of Teacher Education, 59(5), 389-407. https://doi.org/10.1177/0022487108324554

Thank you for sharing, Jillian.  My journey started somewhere similarly!  I did teach as I was taught, and in my early career I tried to emulate the teachers who I thought did an amazing job at teaching, which at the time meant to me to be the best "sage on a stage".

I think I became reasonably good as lecturer/"sage on a stage".  But about 10 years into teaching, I felt something was missing, and it really bothered me why all my students weren't successful!  I couldn't identify what core issue was and so I concluded there must be some gap in my teaching. That was beginning of my journey toward using research to informing my teaching.

In searching for that answer, the first resource I had picked up was "The Teaching Gap", and that set me on path of a complete revision of my teaching practice over the years. That research awakened me to the possibility of one, there are different ways to teach, as seen from examples of the three countries discussed, and two, that students can be active participants in their learning (and that students can be thinking about mathematics concepts differently - and correctly -  than me), and three, that teaching should not be a craft done in isolation.  This book completely changed my perspective, and thus began my journey toward becoming a better teacher using research to support student learning.

In addition to the "Teaching Gap" and "Knowing and Teaching Elementary Mathematics" (cited below) two of the seminal articles that personally helped revolutionize my support of student learning were Carlson and colleagues' article on co-variational reasoning and Thompson and Saldanha's article on multiplicative reasoning (citations below), as these were mathematical ideas I had not considered or thought of at the time.  However, by me being now able to formatively assess in real time how students are thinking about a mathematical concept that involve these mathematical ideas, I have been able to better support their ways of thinking in the process of active learning.  Currently my teaching involves hands on activities on using active learning strategies, and I also use projects as a key component in my courses.

Sources:

Carlson, M. P., Jacobs, S., Coe, E. E., Larsen, S., & Hsu, E. (2002). Applying covariational reasoning while modeling dynamic events: A framework and a study. Journal for Research in Mathematics Education, 33(5), 352-378. 

Ma, L. (1999). Knowing and Teaching Elementary Mathematics. Mahwah, New Jersey: Lawrence Erlbaum Associates.

Thompson, P. W., & Saldanha, L. (2003). Fractions and multiplicative reasoning. In J. Kilpatrick, G. Martin & D. Schifter (Eds.), Research companion to the Principles and Standards for School Mathematics (pp. 95-114). Reston, VA: National Council of Teachers of Mathematics.

When I first started teaching, I definitely DID teach the way I had been taught. I had always enjoyed the learning process and had been in classes with others who were much like me in terms of academic goals, so I expected all students to have this same perspective. WOW was I wrong. My first experience teaching was at the high school level and I quickly learned that most students did NOT share my perspective. It didn't take me long to learn that every student was different and I was going to need to adapt. I made minor changes here and there, but ultimately, my teaching methodologies remained largely the same. Several years later, I started diving deeper into pedagogical practices and learned that there was a WHOLE SLEW of methodologies that I could be employing that were VASTLY different than what I had been exposed to! Now, I use primarily active learning strategies, together with project-based-learning, to engage my students in the learning experience. It's the whole "sage on the stage" to "guide on the side" routine. I believe that my students and I am better for it!