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.

Personal Finance- National & Global Finance

Just as there are indicators to gauge a business's finances, so are there indicators to measure the financial health at the national and even global level. The U.S. government- particularly the Federal Reserve, White House, and Congress- monitor these indicators and act (hopefully) to minimize the damage of economic crises such as the recent "Great Recession" (or Not-So-Great Depression), as well as to promote the long-term well-being of the U.S. economy. Oftentimes these indicators lead to disagreement amongst government officials and members of the Federal Reserve, as the course of action in each such case may be debatable and involve weighing one risk versus another. In response to a weak economy, the Federal Reserve will increase the supply of money by buying government securities, lowering the interest rate, and lowering the reserve requirement. This tends to increase economic activity, including investment, but runs the risk of undermining the economy by causing inflation. In response to a superheated economy, often characterized by excessive inflation, the Federal Reserve can contract the economy through reducing the supply of money by selling government securities, raising the interest rate, and raising the reserve requirement. As happened in 2001, this has the danger of causing a recession. Determining the ideal course of action is a non-trivial problem, involving navigating between the dangers of each possible scenario. The U.S. is one major participant in this global economy. Given the interconnectedness characterizing the current global economy, economic crises in one country can impact large portions of the world. The health of the global economy is thus of importance to everyone. Free trade is regarded as beneficial to everyone, whereas tariffs are seen as ultimately detrimental to all countries levying them. The reason is that while a country may levy tariffs and other protective measures to protect domestic companies from international competition, other countries will ultimately retaliate and impose protective trade measures of their own; the result (akin to the Prisoner's Dilemma in game theory) is that all players in the national economy lose in the end. The relative strength of one country's currency relative to another can be seen in shifts in the exchange rate between countries. If country A's currency gains strength while currency B's currency weakens buy comparison, $1 in country (using $ in the generic sense) will buy more $s in country B. This also provides an exchange rate risk in investing international; a country A resident who invests $1000 in country B and then tries to liquidate his investment will recover an amount less than his original investment. In addition to exchange rate risk, an international investor will want to invest in a country with a low inflation rate, high interest rate (thus, high real interest rate, defined as nominal interest rate - inflation rate), stable government (i.e. no regular unrest in the streets) which respects the market economy and will not arbitrarily seize assets held by others.

Personal Finance- Business Finance

The "finance" side of business involves a great amount of mathematical content. The financial health or situation of a business is evaluated mathematically. Crucial to this analysis is the basic equation Assets = Liabilities + Equity Put quite simply, this equation declares that the total value of assets (cash, equipment, inventory, real estate, etc.) of a business are the sum of the value owned by the business owners plus the amount borrowed from others, usually at a significant rate of interest. In any given period, profit = revenues (usually from sales) - expenses. We can then examine the financial well-being of the business based on any combination of ratios. Common ratios to consider are Return on Assets or Return/Assets, and Return on Equity or Return / Equity, both ratios usually measured over a period of 1 year. These ratios provide a benchmark measurement for investors, whose valuation of any particular company is largely based on these ratios. Financial analysts regard the value of a company, as with any investment, as equal to the expected present value of all future cash flows. Such cash flows can come from regular dividend payments and/or increase in market value (i.e. Stock price) in the company. The ratios which investors like to see will naturally vary by industry, as they are likely to demand lower return on equity of more conservative companies as compared to riskier, high-growth companies. Debt ratio = liabilities/assets is a measure of what proportion of a company's assets is financed through debt; the higher this ratio (and especially if this ratio is more than 1), the more leveraged the company is to be, and thus the riskier an investment the company becomes.

Personal Finance- Insurance

Throughout one's life, unexpected events inevitably occur. Accidents happen, things break down spontaneously, illnesses occur, and eventually we face death. Ultimately, though caution may reduce the frequency of such events and increase lifespan, the risk of such events will always exist. Insurance - car insurance, medical insurance, property insurance, life insurance- help by compensating us financially for unexpected major events in return for regular premium payments made to the insurance companies. (Car insurance is legally required, at least in California). Possession of insurance helps to provide a degree of security, by guarding us against the risk of severe financial losses resulting from unexpected, often catastrophic, events. We pay insurance companies to take the risk for us; the payments- or "premiums"- are the price the insurance companies demand of us as compensation for this risk. True, given that the insurance companies must make a profit (if only to pay the salaries and operating expenses within the business), the premium must represent an amount greater than the expected value of the company's payout to you. However, given that we are risk averse- particularly if we have family members who may suffer if we incur financial losses- most of us are wise to have insurance, particularly medical insurance, life insurance, property insurance, and the like. Having insurance also allows us to more accurately budget based on our expenses. Our insurance premiums are fairly predictable; we can surely add up the premiums for each of our insurances and determine the total cost of "insurance" for the year. This value is subject to far less variation compared to owning no insurance (except car insurance, as the law requires that), and then having to prepare for each of the possible contingency scenarios of facing unexpected huge medical expenses; having one's house damaged or destroyed in a fire, earthquake, flood, etc.; or laying the groundwork for your family members to continue on in the event of your disability or passage. Thus, it is accurate to say that the primary advantages to purchasing insurance are the security of not needing to fear financial harm resulting from major losses combined with allowing for more accurately budgeting (for a household) based on anticipated expenses, with far less variation than would be the case if deciding to decline insurance and simply "take one's chances."

Personal Finance- The Business World

Owning and/or managing a business can be highly profitable, but requires accepting the risk of failure with the accompanying heavy financial losses owing to any of a number of factors. A successful business follows such practices (not all-inclusive) as a clearly defined set of short- and long-term goals, effective marketing to attract the business's target market, responsible financial decision-making, differentiating one's product from that offered by one's competitors. Given the potential of accidents and/or lawsuits owing to numerous unanticipated incidents, purchasing of liability insurance is essential. The smallest businesses are typically sole proprietorships, owned by a single person and essentially inseparable from the owner. Sole owners can enjoy huge profits, but also assume full personal and financial liability for the company. In lieu of being the sole owner, small firms can be owned by 2 or more partners, sharing the profits but also the liability. Large companies, "corporations", are usually owned primarily by a large number of investors who own equity in the company in the form of shares of stock, each representing a tiny piece of the company. Corporations have an existence separate from that of the owners; thus, although the owners share the profits or losses sustained by the company as a whole, personal liability attached to the company does not apply to the owners directly. Thus, fraud within a company does not implicate each of the individual owners of that company.

Personal Finance- Careers

The very nature of the "career" has changed over the past generation. Loyalty of employers to employees, and vice versa, declined during the 1980s and 1990s. With the exception of government positions, teaching, military careers, or the like, it became increasingly common for people to switch companies a few times or more during the course of their careers. Unfortunately, during the boom of the late 1990s, new college graduates, even with math or math-related degrees from highly rated universities were informed they were at a disadvantage when applying for the entry to junior level positions they were targeting owing to their lack of "experience." These companies were not interested in investing any significant amount of training into these highly intelligent young people who had followed society's "rules" and focused on their well-rounded studies while in college. Ironically, the young people who ultimately had the advantage were the less studious C-level college students, who barely passed their humanities classes, but stayed awake extremely late in their dorm rooms, engaging in such activities as computer games, experimenting with various cutting-edge computer technology and software, or other similar endeavors. Meanwhile, the very process of applying for employment changed rapidly. Up through the early 1990s, a motivated qualified job-seeker (especially with a strong college degree) who scanned through the Sunday classified section of the newspaper, cutting out all relevant job postings, and applied by sending an updated resume with appropriate cover letter to every job for which he was a good match could reasonably expect to be called in for interviews and secure employment within weeks to perhaps a few months. Each such job posting attracted perhaps a few dozen applicants, if that. By the late 1990s, online job postings replaced newspapers as the more effective method of accessing job openings. A single search engine had the capability of immediately accessing many individual newspapers or job listings, combined with the convenience of allowing keyword searches. This initially seemed to simplify the job search process. However, from 1999 on, as the internet increasingly became the primary method of applying for employment, employers became less receptive to being contacted by job seekers through any other means. "No phone calls" appeared frequently on job postings. Even at job fairs, recruiters could increasingly be heard to advise most job seekers to "apply online" and provide them with the company's career website. The "Great Recession" or "Not So Great Depression" (however you prefer to term it) struck in full force from 2007 through 2009, severely affecting the careers, job prospects, and financial well-being of a large proportion of the American population. An overly simplistic analysis would attribute the resulting and subsequent long-term unemployment and underemployment- 2 to 5 years in many cases- of even highly educated Americans to this period of extreme economic weakness. However, a more thorough analysis- as pointed out by spokespeople in the National Employment Council- would consider the developments of the prior 15 years, particularly the factors enumerated above.