Normalcy Versus Eccentricity: Case Study in Math Applications

The story is told of a man driving on the I-5 freeway, who received a cell phone call from his panicked wife. The woman warned her husband, "I am scared; I heard a news report that there is a crazy driver driving the wrong way on the freeway, right in your vicinity!" "It's not just one," replied the man, "there are hundreds of them!" In my math classes, it is oftentimes an interesting experience to pose unexpected questions to my students and make note of their replies. My math students are accustomed to classes commencing with a warm-up quiz via Kahoot. In lieu of a traditional multiple-choice practice quiz, I recently preceded a pre-calculus lesson on conic sections with a single-question philosophical question distinguishing a "normal" versus an "eccentric" person. This would lay the groundwork for the formulae on ellipses, especially that of eccentricity of an ellipse. I challenged students to select and justify which scenario is the more reasonable: "The distinction (between a normal and eccentric person) is mathematically describable", or "I'm normsl and everyone else is crazy; the Voices told me so." Students seemed divided in opinion, with perhaps a slight preference of the latter scenario (eccentricity being arbitrary). One student pointed out that since every individual is unique, complete normalcy is unattainable. Another student suggested a mathematical quantification of normalcy is possible from the standpoint of allowing deviation from the norm within certain limits. This was highly reminiscent of the normal distribution function, despite the fact that such a function is beyond the scope of this course. Why do I discuss this particular application concept here? We math teachers often contend with students who opt to avoid or minimize work done in math classes. Justifications for such inaction are disturbing. "I hate math." "I can't do math." "The subject is boring." "The subject is dry." Then, of course, we have the evolved versions of the "homework dogs" that back in the 1980s would pop up in front of unsuspecting students, snatch and eat their homework, then vanish into the parallel universes from which they had come. In this generation, we have instead computer bugs that delete files from students' computers, render computers totally inoperable for weeks at a time, mysteriously delete submitted assignments from drop boxes and delete all evidence of tests taken and messages sent to teachers about missing items. I have heard all of this, and more. Give me any excuse for not working; I have probably heard it, or some slight variation, at some point in time. In addition to establishing and maintaining standards of thoroughness in math classes, it is highly advisable to convey to students the applications of the content material to their everyday lives. This is not always easy. Many math lessons are invariably "dry," as they often involve proofs, mechanics, simplifying expressions, and so on. However, whenever possible, if one can effectively relate mathematical concepts to real life scenarios, there is the greater likelihood of better engaging the students. Humor can be very helpful. Oh- by the way, I am not eccentric in the slightest degree. I am perfectly normal. You are all crazy. The Voices told me so.

Mathematics Education and Virtual Schools

In an earlier post, I had given a somewhat cynical depiction of mathemtics education in the traditional public school setting. With 29 nations (including such countries as Vietnam and Poland) having surpassed the US in math based on tests administered worldwide, there would seem to be cause for concern and considerable room for improvement. These national rankings highlight educational stagnation in the US since 2003. I thought it appropriate, given my nearly 1 1/2 years at California Virtual Academy, to consider the value of virtual schools such as this one in improving US students' performance in mathematics. It should be acknowledged at the outset that there is likely no "magical" solution, unless someone invents a Star Trek device such as the "Teacher" (episode: Spock's Brain) to rapidly save course content directly into the human brain. Stagnation in US performance has not been owing to a lack of innovative educational theory. "New math" curricula attempting to replace "old-fashioned" textbooks and guide students to "discover" mathematical concepts and formulae in lieu of memorization have unsuccessfully been implemented in prior decades, including as recently as the 1990s. Most of these, upon their failure, were followed by a return to traditional instruction, with textbooks, notes, old-fashioned homework/practice problems. We have all seen the results of "No Child Left Behind", attempting to require schools to improve performance based on uniform "standards", guaged by students' performance on high-stakes standardized testing, from 2004 through the present. A key concept incorporated into teacher trainings over the past decade is that teachers must now relinquish their traditional role as "Sage on the Stage" in lieu of their new role as "Guide on the Side", the latter role being deemed more effective in this day and age. Our being outperformed by countries, including many that cling to old-fashioned techniques, would seem to throw this concept into question. Also, it is noteworthy that in most public school classrooms there seems to be little evidence of this new concept's effective implementation. Traditional textbooks, practice work, tests, and so on are still followed, with relaxation of behavioral expectations being the primary evidence of "facilitating" student learning. Since it is acknowledged that students will not shut their mouths for more than a few minutes, teachers endeavor to complete required instruction within that student-imposed time constraint, and then turn them loose for "cooperative learning" (usually a lot of talking accompanied by sporadic work on practice problems, etc.). Many teachers "choose their battles", allow such items as headphones, food/drink, etc. in class provided students do their work and are not overtly disruptive or disrespectful. The question to consider, then, is whether virtual schools represent a possible solution, or at least a positive factor in improving US students' performance in mathematics overall. I do not pretend to know the answer to this. I can point out some major advantages to students pursuing this option. The setup here seems to genuinely embody the "Guide on the Side" teacher model, rather than in most brick-and-mortar schools characterized by lip service to this notion in the absence of its effective implementation. Class attendance is often optional. Students who would normally disrupt physical classrooms usually do not attend virtual class sessions; this, they are not in the classroom to cause trouble or detract from the education of the motivated, well-behaved students. Those who in rare cases attend and then cause trouble can be private-messaged, silenced, and even discreetly ejected from the room with a push of a button. Indeed, some students who once were at-risk and disrupted traditional classrooms are thriving within the virtual school environment and are on track to themselves become strong candidates for college admission. Students aspiring to attain their mathematical potential enjoy the benefit of small live class sessions, ample opportunity to ask questions, even (if desired) questions beyond the scope of the course. It is noteworthy that in some local school systems (at least in California) students are being assigned individual laptops in lieu of multiple textbooks, must carry these laptops to and from school daily, and even submit assignments online into "dropboxes", much like assignments are submitted in virtual schools. Thus, the virtual school model is interestingly being adopted at least to some degree within brick-and-mortar schools.