1.1: Subject Matter
“Is mathematics a discovery, or is mathematics an invention?” That is a question often asked of us as students of mathematics, especially those of us in education. It did not take long for me to decide that I do not believe specifically in either. For me, mathematics is a universal language that interprets discovery and is the foundation for invention. And with this definition in mind, I leave the door open for new “vocabulary” to be introduced to the language to further serve those purposes.
My interest and passion for the subject matter really emerged in junior high school, when some of the algebraic interpretations (abstractions) first begin to be taught more rigorously. I can recall as early as the seventh grade having fun while problem-solving and admiring my teacher’s technique while thinking to myself, “I could do that!” Often a step ahead of the work on the blackboard, I developed a keen understanding of the processes in mathematics, and I still value the process approach to mathematics over the product.
It is much more meaningful for students to understand how and why we approach a problem than it is to get the right answer. Of course, we want our students to eventually build bridges that do not crumble into the water, but we have computers that will generate mathematical solutions to problems that have already been solved, and our students know this fact well. Only a student who values the process will be able to write the program for that computer or possibly abstract the problem and eventually build a better bridge.
When I returned to school in 2001 ready to fulfill my teaching vision, I thought I would just pick up where I left off in 1987 with little consideration for my mathematical lapse. Wrong! After the second day of Calculus II, I immediately back-pedaled into Calculus I, where I was able to re-familiarize myself with many of the concepts and fundamentals. My brain worked in a way it had not worked in a long time, and it felt good.
So, over the recent years, my transcript and other test scores should demonstrate that I can “do” math. Yes, I know my subject matter, but again I defer to process. A more important subject matter evaluation may be, “Can I explain math?” From everyday problem-solving to helping my family and co-workers with mathematical work-related problems to tutoring students in subjects from Algebra to Pre-Calculus, I know that I can. For the first time, though, my efforts are being redirected to a classroom audience.
In Methods of Teaching Secondary Mathematics, we present daily micro-lessons to the class on various standards. This is very new to me, so the feedback I receive from my classmates and instructor are critical. Some things I inherently am doing well, and others I am being coached to improve. I believe my emphasis is in the right places. My organizational skills automatically direct my preparation, outlines and use of the board. But my fast rate of speech and enthusiasm sometimes turn an organized lesson into one that is difficult to follow. I also value the process of evaluating my classmates. It allows me to verbalize some of my ideas (and get immediate professional feedback), model the good practices of others, and learn from their mistakes.
As a high school educator, it is important to realize that you may be the only content expert in your students’ lives. Not many of them are receiving quality assistance outside the classroom, so it is important to take the time for supervised practice (homework) in class and be able to explain the subject matter using multiple representations.
I am also a huge proponent of reconstructing knowledge. I want to try to break my students of the need to memorize formulas. I know not all students can reconstruct a formula from a basic understanding of its purpose or origin, but I think it is important to lead them down that road. Reconstructed knowledge guarantees a deeper layer of understanding and longer retention. Eventually, the formula becomes inculcated through frequent use and is second nature. There are “big ideas” in mathematical subject areas, and if a student comes out of Algebra I or Geometry having memorized the Quadratic Formula or Pythagorean Theorem, but does not understand how to interpret (not calculate) the slope of a line or the area of a shape, that student is doomed to struggle through the remainder of his math courses.
So what does this mean for my future as a math teacher? My enthusiasm for the subject matter had better transform into my students’ enthusiasm. Math must be fun and meaningful to them and have some tangible relevance in their lives. So, first I have to delve into their world(s). The Psycho Math edition of the Wonton Scoop was admittedly more for my enjoyment than the staff’s, but it does demonstrate my ability to apply some math in a fun way to a seemingly unrelated environment. Instead of finding enthusiasm for math through other means, I was able to excite the staff about work through math. It was a fun contest in which a lot of people’s inner math geek emerged.
Learning materials become critical in this exchange of enthusiasm like this, so here is a sneak peek at what part of my Learning Materials expectation may eventually look like.
Early in the year, I would like the students to make a small personal collage without any instruction or rules. Throughout the year, I want to use them as demonstrations of statistical data, set theory, etc. We can group and/or overlap colors, ages, genders, geographic information, academics or other interests based upon the information contained in the collages. This will also be instrumental in learning names and acquiring some personal insight into what would be meaningful experiences for them. I also want the students to keep a math journal. There are two components (so far) to their weekly journal: 1) keep track of your own grade (understand and practice the math), and 2) thoroughly document at least one non-repeating, example of how you used math outside the curriculum. This expands my pool of insights and ideas to incorporate into examples, and encourages writing and verbalizing mathematical concepts.
Did you ever take a field trip in math class? We will. It is important to witness mathematics in action. Engineering, architecture/design, aviation, astronomy, museums (history, culture, art), scientific research and music are just a few fields that can offer practical applications and maybe even some career awareness to students. It will be important to find a field or guide who can demonstrate examples at a rigor that is appropriate to the current area of study. If the experience is over the students’ heads, it will become irrelevant and disengaging.
I was asked in my CoT program interview if I had envisioned my future classroom. Without hesitation I responded, because that vision has remained the same long before I ever decided to return to school to become a licensed teacher. One important component is that, as much as possible, the rings, sliced open cones, blocks, etc. that I described to the panel should all be constructed by the students as part of their instruction. I want my students to have as much ownership in the learning process and hands-on instruction as possible.
At the secondary level of study, the subject matter must become grounded in truth. It is important as a content expert to deconstruct any false information that may be getting in the way of meaningful understanding. Parents, elementary-level teachers, peers, siblings, web-based information and other well-intentioned resources may introduce tricks, short-cuts, or other advice that may or may not “work.” As the content expert, it is critical that these be recognized and discussed. If they work, the trick should be replaced with a sound explanation to offer deeper understanding. If they do not work, then an active deconstruction (a ritual burning of sorts) will take place in my class to demonstrate why it does not work and as aggressively, flamboyantly, interactively and memorably as possible to invalidate the absurdities.
The importance of subject matter should be evaluated in two ways. Can the instructor do the work? The level of understanding should be such that the instructor can present the work that is expected of the students in a clear manner. But, can the instructor explain the work? This implies not only a deeper understanding of the former, but suggests why it is relevant and why some approaches are better than others. Through reasoning, establishing rapport, communicating, using multiple representations and considering the students’ perspective, teachers who are problem-solvers rise to the top. Problem-solving is now recognized as a high standard of mathematics instruction, and it can only be explored when teachers step beyond doing the math, lead students down a curious path, and model an inquisitive approach in our lessons.
