Tuesday, March 17, 2009
Making Pedagogical Choices in Algebra Class
From 2000 to 2003, I taught the same intermediate algebra course semester after semester to (mostly) high school sophomores whom I was trying to prepare to take and pass with at least a C the same course given by a community college mathematics department for dual-enrollment credit (this was at what is called a "middle college," located in Ann Arbor and serving a diverse population of students drawn from around eight counties in southeast Michigan).
I taught from a variety of materials during the nine semesters in which I taught this course, from very traditional to more contemporary and progressive textbooks, all with accompanying use of graphing calculators to varying degrees. Following the order of topics in the books always resulted in presenting quadratic equations, their graphs, and the relationships between their transformations and parameters before exploring the same issues with absolute value equations and their graphs in the Cartesian plane. And student understanding and mastery as evidenced by performance on assessments was often poor on the first topic and abysmal on the second. When we had to look at quadratic and absolute value inequalities and their respective graphs and transformations, things deteriorated further for many. (Of course, many kids "got it" all the time, based on their test results and occasional class participation, and overall my students did well both in my class and when they moved into the college course, but I'm speaking here of the ones who did not).
One semester, for reasons I don't recall, I reversed the order. Remarkably, or so it seemed to me at the time, many of the students I had at the time who had seemed indifferent and/or lost when we worked on linear equations gave evidence both in classroom discussions and on subsequent assessments of "getting" how the graphs of absolute value equations graphed and moved around as they played with the parameters (and vice versa). Later, when we looked at the same issues for quadratic equations, their understanding seemed to carry over. The overall success of the two units went up dramatically compared with past semesters. Had I inadvertently stumbled upon something of value, or was the result an utter fluke that could rarely, if ever, be replicated by other instructors or me?
Before considering that question, let me share my speculations on why things may have gone as they did. It struck me that for students who had some minimal understanding of the behavior of linear equations and their graphs, it might have been easier to move to a look at absolute value equations and their graphs because those graphs are comprised of two linear "legs" that meet at a vertex. A look at the interplay between the graphs and the algebraic expressions that produced them was easier to gain if one graphed by hand, because all that was needed once students understood the basic shape of these graphs was to find the vertex and one point on each leg. Naturally, with the use of graphing calculators (or computer software) it would be even easier to play with and think about the graphs, but for students who did not have access to these tools or who were expected to work with out them at the beginning of (or even entirely throughout) each unit, graphing a symmetric pair of line segments that meet at a common vertex is relatively easy by hand. It is also very easy to find the coordinate pairs needed to produce the graphs.
By contrast, calculating the y-values for quadratic equations can be more challenging for many students. And anticipating how the graphs will look seems to be complicated by how certain parts of the graph (e.g., for non-zero x-values between -1 and 1) will turn out because squaring numbers in that interval results in smaller absolute value outputs for simple quadratic expressions, a somewhat counter-intuitive concept for many students.
My sense was afterwards that students were able to deal with these absolute value equations and their graphs more easily when they saw them immediately after looking at linear expressions and graphs, and before they had been (possibly) confused by the quadratic ones. They were then more able to look at the transformations and subsequently apply what they learned to the quadratic situation.
Of course, it would be wildly irresponsible to claim that my experience with these students would obtain consistently or even in a majority of cases with other students and/or other instructors. While my analysis may be plausible, to know whether it's correct would require significant further research. To expect teachers, textbook authors, policy makers, and other stakeholder to consider seriously that this may be a more effective order to teach the topics in question, there would need to be reliable data that support the above. Indeed, for some, only controlled a double-blind experiment would suffice.
Unfortunately, it is not possible to conduct such an experiment. Teachers know the order in which they teach topics. Students know the order in which they are being taught them. If different classes in the same school are taught in different ways, it is extremely difficult to keep the differences between the approaches walled off from one another. Students have friends in other classes. These are only a few of the obstacles. Indeed, it would not be easy to conduct the simpler, non-blind experiment with controls. Because as a rule, parents are not happy about letting their children be subject to "educational experiments." And when it comes to investigating mathematics education, they are perhaps least inclined to do so, given the hostile propaganda against meaningful reform that has been spread by the American media, fed to them by conservative think-tanks, foundations, pundits, and propaganda groups like Mathematically Correct and NYC-HOLD. Testing even as simple a question as in which order is it more effective to teach two basic and related topics in elementary algebra would likely face real opposition, should it come to the attention of such groups. The seeds of suspicion have been sewn and in many places already have taken root.
Opposition aside, a sole practitioner would find it daunting to try to conduct research of this kind. Finding support for it within a public school setting would be far from trivial. Why, after all, should other teachers, let alone administrators, take the question seriously? And if they did, why should they take the risk of investigating it given some of the things I've raised above? Where would the funding come from, especially in these difficult economic times? Why not leave well enough alone?
Perhaps the most viable option for a single classroom teacher looking to investigate the sort of question I've raised here would be to connect with a university-based researcher who has or could obtain funding. With the funding and (relative) influence and authority of a professor, it might be possible to convince a district to allow such research, though many of the above-mentioned concerns and limitations would still obtain.
I'm not suggesting that research is impossible or that teachers shouldn't be reflecting on practice and using it to inform future teaching. But I do despair to some extent that in the recent and current educational climate, rhetoric about "data-based research" is probably the biggest obstacle to actually conducting meaningful research there is.