My Math Life Story: A Story of Peaks and Pits

Part 1: Peak

 A brain like Fermi’s certainly wasn’t needed to estimate that, this first day of Plan II Math, in our classroom filled with forty Liberal Arts majors, the majority of hearts present beat more rapidly than usual.  When the bell rang in the dark UTC, there was no question for whom it was tolling:  freshman-semester Plan II majors about to face the inescapable--fearsome Plan II Math!  Most of us had dreaded this class; its presence on my own spring course schedule had cast an ominous shadow over my entire winter vacation.  My mind plagued me that whole break with images of math instruction befitting a medieval dungeon more than any classroom I’d heretofore experienced.  I worried that the humanity, creativity, and familiar ambiguity of words and ideas, which I’d come to value so much in my lit, philosophy, and history courses, were about to be replaced with the stern, black-and-white world of numbers, where formulas and rote memorization would rule.  Maybe these concerns were inevitable; after all, most of my past math experiences had been exactly that, all dry formulas and tedious memorization, my elementary school years spent solving thousands of timed Kumon problems, and my middle and high school years spent plugging numbers into formulas as practice for SAT and AP tests.

The big, fat, spiral notebook I opened as the instructor entered was a testament to exactly what I thought math was and what Plan II Math would be. I’d resigned myself to the onslaught of formulas that I thought would ensue, but which would never be explained, and to timed math tests that would be taken, and then, grades received, shoved into the same notebook, only serving to exacerbate the fear already so palpable in the classroom.  My consternation, however, was rapidly replaced with interest when my professor began to explain the purpose of the course and showed us the required texts.  Quickly, I realized that this class might require a reconsideration of my views regarding the study of math!  I’d expected to spend class time scrambling to write down every formula presented by the professor and every free moment after class memorizing and then applying these formulas. The professor, instead, engaged my classmates and me in a lively discussion about what math is, what numbers are, and what it means when we say we “solved” a problem, a discussion thread which was to dramatically embroider the entire semester. 

As the semester unfolded, my appreciation for mathematics continued to increase.  I’d never been in a math class where “solving a problem” first meant really thinking about it and considering all the novel ways that might exist to solve it.  Instead of just telling us to copy his particular methods for working out problems, this professor showed a genuine interest in the different ways each individual student in the class saw the math problems and how each arrived at solutions to them. Even when the solutions themselves were incorrect, the professor validated our contributions by allocating attention, not just to the answers to which we’d arrived, but also, to the thought processes we’d used.  Doing my homework each night, I began to feel a new sense of ownership for my solutions to math problems, and, when asked to explain how I arrived at my answers in class, the explanations actually came easily because the solutions were truly my own. 

Plan II Math helped make us, forty Liberal Arts majors, feel like a class of forty mathematicians!  This math class appealed to ur interests and strengths; for example, the professor connected math to the historical study of famous math equations’ origins, and he encouraged us to use the creative thinking we’d previously only employed in our humanities coursework.  A class that initially seemed so unfamiliar, frightening, and one-dimensional to me became a class I actually enjoyed, and its subject became an area in which I actually felt competent!  I learned a lot about math in my Plan II Math course, but perhaps the most important thing the course gave me was a new appreciation for what the essence of math really is and a surprising sense of accomplishment and interest in a field that previously had seemed so separate who I am.

Part 2: Pit

Thinking back to the peak and pit experiences in my “Math Life,” it’s strangely appropriate, that I entered both experiences with similar feelings of dread, and left both transformed, but in two very different ways.  During the two years after taking Plan II Math, I took biology, education, and literature classes, always aware that each class I completed and every semester that passed brought me that much closer to being forced to enroll in the requisite and infamous Plan II physics course. This was a science course by definition, but, by implementation, every bit a math course.  While Plan II Math had concerned me before I actually experienced it, this upcoming physics course was the stuff of nightmares, and I had yet to meet a senior who didn’t flinch when recounting his/her physics experiences to me and my classmates.  Still,  choosing to interpret my unfounded worry about and ultimate enjoyment of Plan II Math as a sign that perhaps my fears would prove unjustified yet again, I entered my physics class harboring some, albeit small, hope that everything would be ok.

That hope died around midnight after the first day of class.  Hunched over my physics homework, analyzing a page of math problems that remained as pristine as when originally handed to me for wont of my knowing what the heck to do with them, I finally accepted that my experience in this course would be quite different from that in Plan II Math.  The class session earlier that day had left me confused and anxious.  While the professor gesticulated wildly and rapidly transitioned from chalkboard to podium, lecturing in a manner more entertaining than informative, my pencil futilely hovered over my naked notebook page, waiting to transcribe anything that might resemble an explanation or formula.  My late night spent with physics homework sadly became a standing date; weeks and then months passed in the classroom without me actually being taught or effectively learning the course content.  Panic soon gave way to resignation for me and many of my classmates, all generally good students and diligent workers, but frustrated by trying to teach ourselves the course material, enduring the drudgery of yet another impossible weekly homework assignment, and suffering the horror of another stress-filled exam. 

By the end of the semester, I can’t say that my understanding of physics had grown significantly, but my feelings for the subject certainly had been colored by such a negative experience.  Employing what appeared to be a teaching philosophy at odds with everything my education courses have taught me makes someone a good teacher, the physics instructor laid all instructional responsibility upon his students. He, in effect, set us up for failure by designing a course that was inappropriate give the lack of necessary prior learning on the part of the students and by withholding clear explanations and instructional support throughout the course.  Rather than inspiring interest in the subject matter, the “fear of failure” and frustration that we continuously experienced during the semester translated for many students into general antipathy towards and a feeling of incompetence in physics, despite the high grades many ultimately received.  Before being given a scalpel, white coat, or degree, doctors must pledge to “do no harm.”  Thinking back to the harm done in my physics class, perhaps such a pledge might be warranted as well for others pursuing the title of “doctor,” including Ph.D.’s! 

Part 3: Turnabout

Some things are just difficult to teach.  For example, how do you explain to a preschooler the correct method for tying a shoe?  When the “bunny ear” trick fails, what keeps us from finding ourselves with a mangle of shoelaces and, ultimately, from making a trip to buy Velcro?  Sometimes it takes being asked to reconsider a concept, reevaluate a belief, or present something I think I know in new light to a new audience, to make me as a teacher realize that my own understanding of what I am teaching may require some reevaluation as well.  I know, because this happened to me last semester in my Life Skills placement, when, for the first time, I found myself initially but wholly at a loss when trying to define for a child with autism what numbers are and what it means when we say “add them.” 

The inherent difficulty in clearly defining and explaining math concepts and terms so that true comprehension and application on the part of the students is probably responsible for much of the “plug-in-and-go” approaches to math that you see in many schools today.  After all, when the formula or simple procedure for finding the correct answer is all that is given, you don’t really need to understand concepts to be successful.  Division of fractions becomes simply a matter of flipping one fraction and multiplying without regard to the underlying process.  

Last semester, I felt for the first time how scary it can be when I had no formula to guide my teaching.  As it became more and more apparent that my words were failing me, I did as I’d been taught in my special education training, and I considered what the concepts and processes I was presenting to my student might look like from his perspective.  In reflecting upon my own experience as a student in my Plan II Math class, I recalled that one way my professor had made his instruction more meaningful to me was by linking that instruction to things that interest me.  With this in mind, I implemented a math instruction approach where my words, or lack thereof, did not matter.  Using manipulatives and drawing on the child’s interest in technology and animated Pixar films, I was able to meaningfully demonstrate over time what the desired concepts and procedures looked like and actually teach him how to successfully add numbers in what was to him a meaningful and enjoyable way. 

I’m sure I’ll look back on the rudimentary instruction I gave that semester and think about all of the interventions and instructional techniques that I could have applied had I known them at the time.  Nevertheless, in considering why I found this event to be a turning point for me in my “math life,” I notice that it’s connected to my new-found awareness that effective math instruction is less about me and my understanding of math concepts and more about how well I am able to translate what I know into terms a diverse population of students can understand.  It is ironic that, in so many cases, math teachers understand perfectly everything they say when they teach a new math concept, while at the same time individual students too often understand none of it.  When this happens to me, I will remind myself not to blame my students or to resort to only formulaic approaches to teaching but, rather, to reconceptualize how I teach.  Finding myself without a formula was necessary before I could really formulate an understanding of the kind of math educator I hope to be.  When words fail me next time, I’ll take the opportunity to consider instruction from my individual student’s perspective.  I’ve learned that instruction which truly takes into consideration individual students’ needs, interests, learning style, developmental level, and degree of prior learning can be more effective and certainly more meaningful than somehow expecting students to change in order to receive inherently inappropriate instruction. 

Part 4: Influences

-Kumon, Or, The Hare Will Always Beat the Tortoise

When it came to math education, my elementary school seemed to subscribe to the same philosophy that many well-meaning mothers do when they tell their children to “pull the band-aid off quickly.”  After all, if pain must come, isn’t it better if it comes in a really fast burst rather than prolonging the agony?  While I’m not sure that there’s any real mitigation of pain with speed, I am sure that math at my school was over quickly.  Using the Kumon method, math “instruction” occupied approximately fifteen to thirty minutes of the school day during which time we sat in silence, furiously answering problem after problem with our trusty pencils before the timer rang.

Such was my introduction to the leveled world of math education, where students quickly and permanently got labeled as being “good at math” or “bad at math” in the classroom.  We all knew at what Kumon level the other students were; we marveled at the first graders who could already multiply and divide, and we felt deep sympathy for the other students who were still working on shapes and basic addition. 

Brought up in this system that reinforced speed over reasoning, it was easy to fall prey to the belief that a person’s skill at math could be largely appraised by the ticking of a clock.  Prizes were distributed to students who did particularly well in the program, correctly completing problem sets at the most advanced levels in the least amount of time.  The remediation for students who struggled was simple: more Kumon. 

There was a certain kind of payoff to being in this program.  The years of repetitive drill and practice meant that we became students who could work math problems very quickly, but our enjoyment of the subject was questionable.  Over time, however, just like aging track stars, our speed and agility faded, and we found ourselves in the unfamiliar ground of middle school and high school math, where thought and reasoning were more important to success than facility with math facts and where calculators supplanted stopwatches as necessary “tools of the trade.”

As I consider the impact upon my classmates and me of our school’s complete endorsement of the Kumon program, it was less the program itself, and more the message that the program espoused that had the longest-lasting and most damaging repercussions.  At a time when a supportive cognitively and affectively appropriate infrastructure would have helped us learn to appreciate the subject and feel confident in our skills in math, the judgmental nature of the program’s leveled framework-- and the message such levels gave us about our own abilities in mathematics-- reinforced at an early age the groupings and labeling that the education system sometimes appears to be all too willing to impose, and students seem all too quick to believe.

-AP Calculus, Or, There Was A Reason for Your Torture

If you had asked any one of my classmates what the bane of his/her seventeen-year-old existence was, no hesitation would have preceded their declaration of, “Calculus!”  Never in my thirteen years at the school had there ever been such a unifying force as our common dislike for first period’s AP Calculus class!  The fear of a pop quiz loomed over the door every day as we filed in, and the unrelenting glare of the overhead from the first day of school signaled that learning was definitely going to occur that day and every day, or else.  We progressed doggedly and systematically through the text book, and I came to see, that yes, it is possible to cover all content in a 400-page textbook when class always begins five minutes before the bell rings and ends five minutes into the next class.

Calculus ensured that even in the midst of applying to colleges and the creeping onset of the oh-so-contagious “Senioritis,” we still were kept aware that we were high-schoolers; as such, we had more important things to worry about, like the AP test in Calculus!  Without opening a practice guide or assigning practice AP problems, our teacher prepared us for the test, not by “teaching to the test,” but rather, by making sure that we understood the course content, and all of it, at that.  As spring and AP testing dates approached, I found myself questioning my readiness in certain content areas.  Had I learned sufficiently what the policies of the 16^th president of the United States were?  Did I remember the units for velocity or the formula for mass?  In the midst of scrounging about for missed information and of playing catch-up for what my other AP classes had let slide, I found myself >sigh of relief< at total peace with my understanding of Calculus. 

The week after the AP tests were over, my Calculus teacher was still hard at work preparing us for any Calculus II classes we might take in college.  In our final week of our final year in high school, it was Calculus that we were anxiously studying for a comprehensive test on the very last day of school.  The week following the AP test saw little change in my teacher’s passionate instructional practices, but it did see a huge change in our feelings towards her and her class.  Whereas before, it had been our intense hostility towards the class that united us, the feelings that we shared now were gratitude and appreciation for the teacher and the class that had really prepared us for our futures.

Part 5: Challenge

The single greatest challenge that I’ve faced in math is a challenge all too common among students, that is, successfully confronting the Correct Answer Syndrome (CAS).   Perhaps more so with math than with other areas of instruction, assessment in this field often becomes an absolute, daily occurrence, so that the need for accuracy and validation overrides the desire, or even ability, to truly learn and internalize the content being taught.  As I went through school, I spent hours each night on my math homework, solving the problems just as I had seen my teachers do, and panicking when I found myself confronted with a problem that I couldn’t solve.  In most math classes, what counts is the final product, the list of answers arrived at, but not the thinking process behind each answer.  Doing my math homework, I often remember feeling as though I were being set up for failure.  It wasn’t really until my Plan II Math course that I began to trust myself in math and to see that failing to arrive at the “correct answer” in math is really just another means for learning how to solve a problem.  Thanks in large part to my Plan II Math professor’s attitude towards –and even encouragement of— mistakes, especially when they stemmed from creative and well-reasoned thought, I was finally free to devote myself completely to the study of math and found that my critical thinking about math concepts actually improved, as did my enjoyment of the subject.  Most importantly I have overcome the challenge of CAS.

Part 6: Teaching

When I entered UT, I entered as a Plan II and Business major.  Devoting myself to my coursework, I kept telling myself that the nagging sensation that something was missing from my academic pursuits would disappear once I progressed further in my studies.  After going back home for winter break during my sophomore year, and after volunteering once more with my hometown’s Respite Care Program, in which I’d volunteered throughout high school to care for children with diverse disabilities, I finally realized that what was missing from my life in college was the fulfilling feeling that I was actually making a positive difference in peoples’ lives.  When I went back to college that spring, I dropped my Business major and added Special Education, a decision I’ve never regretted.  The opportunities to work with children with special needs and their families as they navigate through the education system has really reaffirmed for me my belief that each student has the right to learn, and that it is our responsibility as educators to ensure that this happens.  As I envision teaching mathematics to my students in the future, my greatest hope is that I will be able to instill in them confidence in their own abilities in math so that they never fear taking risks to try to solve problems.  Mathematics affords teachers and students alike the opportunity, not only to explore a field, but also, to really explore human thought and decision making through the development of critical thinking skills.  I’m dedicating myself to improving my students’ academic and social futures, and I can help ensure these by supplying a supportive and structured environment in which they can explore the field of math. My hope is that they will be able to look back at the time they spend with me as one of the peaks of their math lives!