For many of us, especially those over forty, our elementary, junior high and high school teachers mostly served as the sole arbiter of truth in our learning, especially so in our math classes. The extent to which we bought into the validity of their occupying that role, or not, was likely a strong determinant of our ability to succeed while “doing school.” Impacting whether that turned out well for us, or not, in many ways, were other factors such as our own personal characteristics (personality, preferences, tendencies, attitudes, moods, learning styles, beliefs, etcetera); as well as the impact on our characteristics by our: teachers, school environment, home environment, socioeconomic status, and etcetera. In spite of our academic record, as reflected in our grade point averages, or SAT / ACT / other standardized test scores, it is arguable whether we truly developed a deep understanding of the various subjects to which we were exposed, at least that is the inference one might draw from much of the academic research of educational reform; again, especially so in mathematics. While I succeeded in high school, college, and graduate school, as measured by academic performance, and in my 25-year career in high-tech, as measured by income growth, I wonder whether I truly possessed a deep understanding of the various academic subjects I studied, and if not, was that really problematic as I progressed through life? Hence, by extension, is it really a potential problem for my future students?

Daniel Chazan (2000), in *Beyond Formulas in Mathematics and Teaching: Dynamics of the High School Algebra Classroom* would likely believe it is a problem, since the instruction I received through my elementary, secondary, college, and graduate school (MBA), predominantly consisted of direct instruction which could have limited my ability to experience true understanding. Chazan advocates the approach proffered by Lampert (1990) where “[teacher’s relinquish their] role as sole arbiter of mathematical truth and insist that students help decide what is true.” Chazan writes that Lampert went on to suggest that “Rather than have students memorize procedures for finding solutions, she asks students to conjecture and invent as they solve problems.” In this way, students join with the teacher as arbiters of mathematical truth with the teacher facilitating the group process. Chazan further explains Lampert’s role expansion by highlighting the pedagogical inversion Lampert introduces where “student exploration of a [mathematical] problem [is placed] prior to classroom discussion during which students share the results of their exploration” through which “the class as a whole (the teacher included) develops a shared set of mathematical understandings.” Hence, this method definitely distinguishes itself from earlier direct instruction methods allowing students to view mathematics as more than boring, rote recitation and application, but as a “living and growing field in which developments occur when people create solutions to [real] problems.”

Upon further reflection, as I write this posting, I must agree with the literature that direct instruction, alone, is ripe for students to miss out on true, meaningful student learning. While some teachers may do well with direct instruction by weaving in ample real-world applications to which students may relate, hence develop deep understanding, their numbers seem to be few and far between. Additionally, their pedagogy seems to resonate more with students of privilege who understand how to learn effectively, and seek deep understanding. However, for those absent such privilege, the likelihood of establishing meaningful connections with the mathematical material is minimal. My personal experience serves as a testament; the means by which I learned mathematics was insufficient for a time in my life, so much so that I required remedial instruction, which emphasized students’ wrestling with topics before any formal instruction, so deeper understanding could take root. Once I experienced this instructional style, my ability to comprehend and connect mathematical concepts grew immensely. The learning experience, absent the content, was powerful as well. So much so, that I am an advocate of instruction akin to Lampert, Chazan, and others, with the wish to light a fire within all of my current, and future students, so they may strive to develop deep understanding of the mathematics present in their current and future courses, and more importantly, to carry that understanding forward with them in life, expanding their horizons as wide as they so wish.

Having been the kid in class who always seemed to have the bizarre, circuitous route to the right solution to any math problem, I pride myself on working patiently with kids who take a different road. The beaten path is not the road to innovation, and there are myriad ways to “skin a cat”.

My mentor spent minimal time with direct instruction, preferring to let students lead the way solve problems. His approach was to act as scribe and guide, letting the student explain his steps from his seat as the teacher copied them onto the board. When a blunder occurred, he would invite other students to spot the error.

I’ve tried to follow this example. Another approach I find interesting is to work a homework problem on the board, and then ask: “who got the same answer another way?”.

It’s very enlightening to investigate “alternative” algorithms for basic elementary arithmetic – Austrian Subtraction or Lattice Multiplication, for instance. Setting aside your calculator and playing with a slide rule or an abacus can be similarly instructive.

When it comes to mathematical truth, questions involving the number zero have a rather nasty tendency to get deep fast, I find, and can be a real challenge. “Why can’t I divide by zero?” or “How can ANY number raised to the power of zero possibly be equal to ONE!” (This last one tormented me as a 7th grader). These are hard questions because they threaten to expose mathematics as a humble human invention, rather than the deep cosmic truth we prefer to present it as.

The problem with being in the truth business is that it might eventually force you to be honest.

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