From my perspective, one of the shortcomings of even the best conversations about practice, however, is the lack of access to specific classroom data, particularly video with supporting documentation (teacher journals, samples of student work, assessment instruments, observer notes, etc.) that would allow other practitioners to engage in deep analysis of what goes on in mathematics teaching based on common access to the same materials. As a former literature doctoral student, I am reminded of the countless frustrating conversations I had with undergraduate students of mine (and occasional fellow graduate students) about the absurdity of discussing literature without making much or even any direct reference to things in the text(s) under examination. Hearing someone offer up "personal reactions" to Dylan Thomas' "And Death Shall Have No Dominion," or Marlowe's DR. FAUSTUS (e.g., "I could really relate to that Mephostophilis character. He reminded me of my best friend in high school") just doesn't quite measure up to the level of discourse one hopes for in serious literary analysis. I kept wanting students to show me where IN THE TEXT they were getting their ideas and HOW what they said was going on actually worked. Generally, I was very disappointed even when I gave meaty selections from the texts that we'd dealt with in class on exams and asked students to offer some sort of analysis that stayed primarily "on the page."

To expect deep conversations about teaching, one must have similarly detailed examples of practice about which to converse. There's really no substitute for specifics and no specifics better than seeing what teachers do, with supporting documentation to flesh out and give depth to one's analysis. But as of this writing, there seems to be no readily-accessible, free web resource that could be used by groups of teachers, parents, researchers, and other interested stake-holders to discuss actual mathematics practice grounded in the sort of data needed for "close textual analysis." (Of course, perhaps I'm in error: if so, I hope to be flooded with information about just such resources and will, you can be sure, flock to them post haste).

While waiting for Godot (or, in fact, trying to figure out how to create just such a resource), I will resort to the less-satisfactory standby of anecdotal evidence, offered strictly in writing and through the admittedly-flawed lens of my own memory for something that's now about five or six years in the past. Any inaccuracies in the recounting are, I hope, due merely to incipient senility rather than purposeful misrepresentation of fact.

A Fortuitous Decision

My focus is the decision I made in the winter of the 2002-2003 academic year to change the order in which I taught two topics in an intermediate algebra course I'd been teaching with various materials for 2 1/2 years at a public charter high school in Ann Arbor. The goal of the course was to prepare high school sophomores (for the most part) to take the same course subsequently with college instructors for dual-enrollment credit towards both high school graduation and credit at the community college that sponsored and hosted our school. My course was intended to help bridge the gap between taking first year algebra over the period of one year at the typical pace of high school and taking a second algebra course at a college pace in one semester.

I had been using various books for the course, including the same text mandated for the college class, an indifferent text to my mind written by past and current members of the math department, designed for use both in regular classes and in self-paced courses. As a result, it was similar to programmed instruction texts from the 1960s to some extent. I had also used COMAP's Mathematics: Modeling Our World, a more progressive text that was heavily grounded in applications and mathematical modeling. While each book worked well with a subset of my students (and, frankly, with my own development as a teacher), neither was without flaws and neither reached a core group of students I had each semester who were low achievers and/or what might politely be called slackers when it came to school in general and mathematics in particular.

Although I eventually wound up using a third text (ALGEBRA 1: A PROCESS APPROACH, from the University of Hawaii Curriculum Research and Development Group) to reach that group in a shortened spring semester, when other students had gone on to the college course, I'm not sure it or any book I've seen would have been right for all my students or been fair to all these groups of kids I wanted to reach in ONE semester.

An example of a difficulty none of the books seemed to address adequately was teaching absolute value inequalities and their graphs, as well as the broader question of helping students see the connections between linear and quadratic equations and their respective graphs. But merely solving absolute value inequalities was always a sticking point for all but my best students. For reasons I had not yet determined, this topic always came after we had covered quadradic equations and their graphs (though not immediately thereafter). And even for students who did reasonably well with the pure algebraic manipulation and solving required in the units on quadratics, understanding how changes in the coefficients impacted their graphs and vice versa proved more difficult. So I had two stubborn but seemingly unrelated problems to deal with in these units.

For reasons that have become lost in the mists of time (I wish I could say that I had some sort of brilliant insight following much mediation or had been led to an epiphany through conversations with colleagues, but if either was the case, I have no recollection to report), I decided to try teaching the absolute value inequality unit before we did quadratics. It may be that I just wanted to get it out of the way and move on, or that I wanted to weed out a few students who were annoying me. But most likely it was nothing quite so insidious or inspired.

To what I recall as my great surprise and greater pleasure, the students on the whole seemed to do much better on both units, including how they approached solving the absolute value inequalities. But the biggest payoff, based both on the sorts of things they said in class and how they performed on assessments was in their understanding of the graphs and their connections with their symbolic representations. In particular, ideas about horizontal shifts, which heretofore always seemed extremely elusive for the majority of my students, now seemed to make sense to a lot more of them. And when we revisited these and related issues in a subsequent unit on quadratic equations (and even later on quadratic inequalities), they were to a much greater degree than in previous semesters I'd taught able to grapple successfully with what was going on with the graphs as we fiddled with the parameters.

To what do I attribute the apparent changes? Naturally, I wish I could say it was a vast improvement in the quality of my instruction, and perhaps that was true in one particular way. Because as I taught the transformations of the graphs of the absolute value inequalities, it hit me for perhaps the first time (or at least for the first time so overwhelmingly clearly) that there were a host of parallels between these changes and those associated with quadratics. In fact, to my mind, it seemed that as we covered the topic in the unit on absolute value inequalities, it was somehow easier to communicate what was happening with the graphs as we changed the values of the coefficients.

What occurred to me, though I only had scattered informal feedback from some students to confirm this (and it was in retrospect, when we treated the quadratics unit later), is that absolute value graphs are extremely simple because they consist only of two straight lines. Students are comfortable with straight lines compared with curves, and of course a linear relationship is easier to plot and to draw the graph for with complete confidence compared with graphing any curve, which always seems imprecise by comparison when drawn by hand or even with tools to help aid in precision on graph paper. Students quite quickly "get" the symmetry in the absolute value graphs, and of course the fact that there are two predictable (and opposite-signed) slopes operating on the two "legs" of the graph makes hand-graphing a snap.

While it may seem like a minor point, contrast that with graphing parabolas. Although the symmetry is there, of course (in fact, ALL the same properties are there EXCEPT for the straightness), it's harder to graph individual points except for lattice points with complete confidence (it can be argued that the same is true with absolute value graphs, too, but consider how the issue of slope comes into play and it is likely that one will see how relatively easy these are compared with any graphs involving curves). Moreover, and this may be the crucial point, it's easier to picture mentally and to anticipate what's happening on a point-by-point basis with the absolute value graphs, which in their simplest forms (with "leading coefficient" of one) are, after all, pieces of y = x and y = -x, possible shifted vertically and/or horizontally.

And it's particularly those horizontal shifts that are difficult for students to grasp. The idea that adding a positive quantity to the independent variable "inside" the function results in a left shift and adding a negative quantity produces a right shift is counter-intuitive at first blush. Without having to deal with the complication of squaring, many students seemed more able to just think about the impact of this one change on the resulting y-values. Later, when the same issue arose with quadratics, most students seemed able to transfer what they'd already learned from their previous experiences with the absolute value graphs.

Research "Inside" Education?

Is the above "proof" that what I observed is definitely the case and that all teachers should follow suit? Of course not. It's simply food for thought and, I hope, someone's formal research. But this is one example of where classroom practice could inform research and that the subsequent research could later influence broad-scale practice. If someone decides to set up and conduct a sound comparative experiment, it is conceivable that they could do more deeply the kind of analysis that Deborah Ball and Francesca Forzani call research "inside" education rather than about it ("What Makes Education Research 'Educational'?", looking at the interplay between content, teacher, and student on this and related issues. Thus, I suggest that while what I have presented here is not a substitute for either formal research or the kind of deeper and broader data-base I mention in the beginning of this piece as a source for practitioners and others looking to improve classroom practice, what I've offered is a meaningful contribution to research and practice.

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